MODELING OF GAS SORPTION AND DIFFUSION BEHAVIOR …

The Pennsylvania State University

The Graduate School

MODELING OF GAS SORPTION AND DIFFUSION BEHAVIOR AND

IMPLICATIONS ON COALBED METHANE PRODUCTION

A Dissertation in

Energy and Mineral Engineering

by

Yun Yang

copy 2020 Yun Yang

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2020

ii

The dissertation of Yun Yang was reviewed and approved by the following

Shimin Liu

Associate Professor of Energy and Mineral Engineering

Dissertation Advisor

Chair of Committee

Derek Elsworth

Professor of Energy and Mineral Engineering

Sekhar Bhattacharyya


Chair of Mining Engineering Program

Chris Marone

Professor of Geosciences

Mort Webester

Professor of Energy Engineering

Associate Department Head for Graduate Education

iii

ABSTRACT

Exploration of coalbed methane (CBM) in North America started from the 1970s

as the oil crisis shifted the interest to potential natural gas resources in coalbeds Unlike

conventional natural gas reservoirs coal acts as both source and reservoir for hydrocarbon

where 90-98 of gas in the coal seam is adsorbed at its internal surface of coal matrices

Previous studies have demonstrated that pore structure is a key factor determining gas

storage and transport behaviors of CBM reservoirs This study established an analytical

relationship between pore structure and gas sorption and diffusion characteristics of coal

My holistic study can be broadly divided into two parts including theoretical modeling

(Chapter 2) and experimental study (Chapter 3) Theoretical models have been proposed

to quantify gas storage capacity and diffusion coefficient of coal by directly using pore

structure parameters as physical inputs The proposed models are calibrated and validated

by laboratory data and the results are presented in Chapter 4 The theoretical analysis and

experimental work conducted in these three Chapters are further coupled into gas

production simulator to define the unique production profile for mature CBM wells in San

Juan basin (Chapter 5) The knowledge of pore structure alteration and its influence in

gas-solid interactions of coal is employed to examine the applicability of a waterless

fracturing technique cryogenic fracturing in CBM reservoirs (Chapter 6)

A pore structure-gas sorption model has been proposed in Chapter 2 This model

is validated against experimental data measured by sorption apparatus depicted in Chapter

3 and the validation results are presented in Chapter 4 Here presents an abstract of the

iv

findings of my thesis on the relationship between pore structure and gas sorption behavior

Gas adsorption volume has long been recognized as an important parameter for CBM

formation assessment as it determines the overall gas production potential of CBM

reservoirs As the standard industry practice Langmuir volume (VL) is used to describe the

upper limit of gas adsorption capacity Another important parameter Langmuir pressure

(PL) is typically overlooked because it does not directly relate to the resource estimation

However PL defines the slope of the adsorption isotherm and the ability of a well to attain

the critical desorption pressure in a significant reservoir volume which is critical for

planning the initial water depletion rate for a given CBM well Qualitatively both VL and

PL are related to the fractal pore structure of coal but the intrinsic relationships among

fractal pore structure VL and PL are not well studied and quantified due to the complex

pore structure of coal In this thesis a series of experiments were conducted to measure the

fractal dimensions of various coals and their relationship to methane adsorption capacities

The thermodynamic model of the gas adsorption on heterogonous surfaces was revisited

and the theoretical models that correlate the fractal dimensions with the Langmuir

constants were proposed Applying the fractal theory adsorption capacity ( 119881119871 ) is

proportional to a power function of specific surface area and fractal dimension and the

slope of the regression line contains information on the molecular size of the adsorbed gas

We also found that 119875119871 is linearly correlated with sorption capacity which is defined as a

power function of total adsorption capacity (119881119871) and a heterogeneity factor (ν) This implies

that PL is not independent of VL instead a positive correlation between 119881119871 and 119875119871 has been

noted elsewhere (eg Pashin (2010)) In the Black Warrior Basin Langmuir volume is

v

inversely related to coal rank (Kim 1977 Pashin 2010) and Langmuir pressure is

positively related to coal rank It was also found that 119875119871 is negatively correlated with

adsorption capacity and fractal dimension A complex surface corresponds to a more

energetic system which results in an increase in the number of available adsorption sites

and adsorption potential which raises the value of 119881119871 and reduces the value of 119875119871

A pore structure-gas diffusion model is developed in Chapter 2 This model is

validated against experimental data measured by sorption apparatus depicted in Chapter


findings of the research on the relationship between pore structure and gas diffusion

behavior Diffusion coefficient is one of the key parameters determining the coalbed

methane (CBM) reservoir economic viability for exploitation Diffusion coefficient of coal

matrix controls the long-term late production performance for CBM wells as it determines

the gas transport effectiveness from matrix to fracturecleat system Pore structure directly

relates to the gas adsorption and diffusion behaviors where micropore provides the most

abundant adsorption sites and meso- and macro-pore serve as gas diffusive pathway for

gas transport Gas diffusion in coal matrix is usually affected by both Knudsen diffusion

and bulk diffusion A theoretical pore-structure-based model was proposed to estimate the

pressure-dependent diffusion coefficient for fractal porous coals The proposed model

dynamically integrates Knudsen and bulk diffusion influxes to define the overall gas

transport process Uniquely the tortuosity factor derived from the fractal pore model

allowed to quantitatively take the pore morphological complexity to define the diffusion

for different coals Both experimental and modeled results suggested that Knudsen

vi

diffusion dominated the gas influx at low pressure range (lt 25 MPa) and bulk diffusion

dominated at high pressure range (gt6 MPa) For intermediate pressure ranges (25 to 6

MPa) combined diffusion should be considered as a weighted sum of Knudsen and bulk

diffusion and the weighing factors directly depended on the Knudsen number The

proposed model was validated against experimental data where the developed automated

computer program based on the Unipore model can automatically and time-effectively

estimate the diffusion coefficients with regressing to the pressure-time experimental data

This theoretical model is the first-of-its-kind to link the realistic complex pore structure

into diffusion coefficient based on the fractal theory The experimental results and

proposed model can be coupled into the commercially available simulator to predict the

long-term CBM well production profiles

Chapter 5 presents a field case study to model long-term production behavior for

mature CBM wells CBM wells in the fairway of the San Juan basin are in the mature stage

of pressure depletion experiencing very low reservoir pressure These mature wells that

have been successfully producing for more than 20 years exhibit long-term hyperbolic

decline behavior with elongated production tails Permeability growth during primary

production is a well-known characteristic of fairway reservoirs and was historically

interpreted to be the dominant factor causing the production tail Several experimental

works observed that the diffusion coefficient of the San Juan coal sample also varied with

pressure However the pressure-dependent nature of gas diffusion in the coal matrix was

neglected in most simulation works of CBM production This may not significantly mis-

predict the early and medium stage of production behavior when permeability is still the

vii

primary controlling parameter of gas flow Prediction errors are elevated considerably for

these late-stage fairway wells when diffusion mass flux takes the predominant role of the

overall flowability A novel approach to implicitly incorporate the evolution of gas

diffusion during pressure depletion in the flow modeling of fairway reservoirs was

proposed in this Chapter where the derived diffusion-based matrix permeability model

converts gas diffusivity into Darcys form of matrix permeability This modeling of matrix

flow enables the direct use of lab measurements of diffusivity as input to the reservoir

simulator The calculated diffusion-based permeability also increases with pressure

decrease The matrix and cleat permeability growths are then coupled into the numerical

simulator to history-match the field production of multiple CBM wells in the fairway

region The established numerical model provides satisfactory matches to field data and

accurately predicts the elongated production tail in the late decline stage Sensitivity

analyses were conducted to examine the significance of accurate modeling of gas diffusion

flow in CBM production throughout the life span of the fairway wells The results show

that the assumption on constant matrix flowability leads to substantial errors in the

prediction of both peak gas production rate and long-term declining behavior Accurate

modeling of gas diffusive in the matrix is essential in production projection for the mature

fairway CBM wells The integration of gas diffusivity growth into production simulation

improves the prediction of gas production rates and the estimation of ultimate recovery for

the late-stage fairway reservoirs

Chapter 6 investigates the applicability of cryogenic fracturing in exploiting CBM

plays using the theoretical and experimental analyses conducted in Chapter 2 and Chapter

viii

3 Cryogenic fracturing using liquid nitrogen is a waterless and environmentally-friendly

formation stimulation method to effectively create a complex fracture network and

dilatated nano- and micropores within coal matrix that greatly enhances gas transport in

coal matrix as well as cleats However the development of cryogenic fracturing is still at

its infancy Before large-scale field implementation a comprehensive understanding of the

fracture and pore alteration will be essential and required For pore-scale investigation this

chapter focuses on the induced pore structural alterations due to cryogenic treatment and

their effects on gas sorption and diffusion behaviors The changes in the pore structure of

coal induced by cyclic nitrogen injections were studied by physical adsorption at low

temperatures A micromechanical model was proposed to simulate the microscopic process

and predict the degree of deterioration due to low temperature treatments As a common

characteristic of modeled results and experimental results the total volume of mesopore

and macropore increased with cryogenic treatment but the growth rate of pore volume

became much smaller as freezing-thawing were repeated Pores in different sizes

experienced different degrees of deterioration In the range of micropores no significant

alterations of pore volume occurred with the repetition of freezing and thawing In the

range of mesopores pore volume increased with the repetition of freezing and thawing In

the range of macropores pore volume increased after the first cycle of freezing and thawing

but decreased after three cycles of freezing and thawing Because of pore structural

alterations cryogenic treatment enhanced gas transport process as the diffusion coefficients

of the freeze-thawed coal samples were increased by 1876 and 3018 in the adsorption

and desorption process For the studied Illinois coal sample repetitive applications of

ix

cryogenic treatment reduced macropore volume and increase mesopore volume For the

tested sample the diffusion coefficient of the coal sample that underwent three cycles of

freezing-thawing was lower than that of the coal sample that underwent a single cycle of

freezing and thawing The outcome of this study provides a scientific justification for post-

cryogenic fracturing effect on diffusion improvement and gas production enhancement

especially for high ldquosorption timerdquo CBM reservoirs

For fracture-scale investigation Chapter 6 develops a non-destructive geophysical

technique using seismic measurements to probe fluid flow through coal and ascertain the

effectiveness of cryogenic fracturing A theoretical model was established to determine

fracture stiffness of coal inverted from wave velocities which serves as the nexus that

correlates hydraulic with seismic properties of fractures In response to thermal shock and

frost forces visible cracks were observed on coal surfaces that deteriorated the mechanical

properties of the coal bulk As a result the wave velocity of the frozen coal specimens

exhibited a general decreasing trend with freezing time under both dry and saturated

conditions For the gas-filled specimen both normal and shear fracture stiffness

monotonically decreased with freezing time as more cracks were created to the coal bulk

For the water-filled specimen the formation of ice provoked by cryogenic treatment leads

to the grouting of the coal bulk Accordingly fracture stiffness of the wet coal initially

increased with freezing time and then decreased for longer freezing time Coalbed with

higher water saturation is preferred in the application of cryogenic fracturing because fluid-

filled cracks can endure larger cryogenic forces before complete failures and the contained

water aggravates breaking coal as ice pressure builds up from volumetric expansion of

x

water-ice phase transition and adds additional splitting forces on the pre-existing or

induced fracturescleats This study also confirms that the stiffness ratio is sensitive to fluid

content The measured stiffness ratio varied between 07 and 09 for the dry coal and it

was less than 03 for the saturated coal The outcome of this study provides a basis for a

realistic estimation of stiffness ratio for coal for future discrete fracture network modeling

xi

TABLE OF CONTENT LIST OF FIGURES xiv

LIST OF TABLES xx

ACKNOWLEDGEMENTS xxii

Chapter 1 INTRODUCTION 1

11 Background 1

12 Problem Statement 3 13 Organization of Thesis 7

Chapter 2 THEORETICAL MODEL 9

21 Gas Sorption Modeling in CBM 9 211 Literature Review 9 212 Fractal Analysis 12

213 Pore Structure-Gas Sorption Model 13 22 Gas Diffusion Modeling in CBM 22

221 Literature Review 22 222 Diffusion Model (Unipore Model) 28 223 Pore Structure-Gas Diffusion Model 33

23 Summary 41

Chapter 3 EXPERIMENTAL WORK 45

31 Coal sample procurement and preparation 45 32 Low-Pressure Sorption Experiments 47

33 High-Pressure Sorption Experiment 48 331 Void Volume 49 332 AdDesorption Isotherms 51

333 Diffusion Coefficient 53 34 Summary 54

Chapter 4 RESULTS AND DISCUSSION 56

41 Coal Rank and Characteristics 56 42 Pore Structure Information 57

421 Morphological Characteristics 57 422 Pore size distribution (PSD) 58

423 Fractal Dimension 60 43 Adsorption Isotherms 64

xii

44 Pressure-Dependent Diffusion Coefficient 67 45 Validation of Pore Structure-Gas Sorption Model 70 46 Validation of Pore Structure-Gas Diffusion Model 78 47 Summary 87

Chapter 5 FIELD APPLICATION TO CBM WELLS IN SAN JUAN BASIN 90

51 Overview of CBM Production 90 52 Reservoir Simulation in CBM 92

521 Numerical Models in CMG-GEM 92 522 Effect of Dynamic Diffusion Coefficient on CBM Production 94

53 Modeling of Diffusion-Based Matrix Permeability 97 54 Formation Evaluation 101 55 Field Validation (Mature Fairway Wells) 103

551 Location of Studied Wells 105 552 Evaluation of Reservoir Properties 107

553 Reservoir Model in CMG-GEM 114 554 Field Data Validation 116 555 Sensitivity Analysis 121

56 Summary 127

Chapter 6 PIONEERING APPLICATION TO CRYOGENIC FRACTURING 129

61 Introduction 129 62 Mechanism of Cryogenic Fracturing 130

63 Research Background 132 631 Cleat-Scale 132

632 Pore-Scale 133 64 Experimental and Analytical Study on Pore Structural Evolution 134

641 Coal Information 136

642 Experimental Procedures 137 643 Micromechanical Analysis 142

65 Freeze-thawing Damage to Nanoporous Network of Coal Matrix 146

651 Gas Kinetics 146 652 Pore Structure Characteristics 155

653 Application of Micromechanical Model 169 66 Experimental and Analytical Study on Fracture Structural Evolution 174

661 Background of Ultrasonic Testing 174 662 Coal Specimen Procurement 176 663 Experimental Procedures 177

664 Seismic Theory of Wave Propagation Through Cracked Media 179 67 Freeze-thawing Damage to Cleat System of Coal 193

671 Surface Cracks 194 672 Wave Velocities 197

xiii

673 Fracture Stiffness 201 68 Summary 214

Chapter 7 CONCLUSIONS 219

71 Overview of Completed Tasks 219 72 Summary and Conclusions 220

APPENDIX A USER INTERFACE IN MATLAB FOR THE ESTIMATION OF

DIFFUSION COEFFICIENT 231

APPENDIX B MATLAB PROGRAM TO DERIVE FRACTURE DENSITY 238

REFERENCE 241

xiv

LIST OF FIGURES

Figure 1-1 Illustration of multi-scale and multi-mechanism gas flow in a CBM

reservoir CBM production data Source DringInfoinc 3

Figure 1-2 Workflow of the theoretical and experimental study 8

Figure 2-1 Graphical illustration of fractal dimension (Df) (a) For a smooth

surface Df = 2 (b) For irregular surfaces 2 lt Df lt 3 13

Figure 2-2 Conceptual model of collisonal frequency at smooth and rough

surfaces 16

Figure 2-3 Diffusion regimes in coal matrix can be categorized as Knudesn

diffusion viscous diffusion and bulk diffusion controlled by Knudsen number

24

Figure 2-4 User interface of unipore model based effective diffusion coefficient

estimation in MATLAB GUI 31

Figure 2-5 Flowchart of the automated computer program for effective diffusion

coefficient estimation in MATLAB GUI 32

Figure 2-6 Fractal pore model 35

Figure 2-7 Diagnostic plot of reciprocal diffusion coefficient (119863119901 minus 1) vs 119875 to

determine the dominant diffusion regime Plot (b) is updated from plot (a) by

considering the weighing factor of individual diffusion mechanisms and

Knudsen diffusion coefficient for porous media 41

Figure 3-1 Location and geologic information of Luling Sijaizhuang and Xiuwu

coalmine The Luling coal mine is located in the outburst-prone zone as

separated by the F32 fault 46

Figure 3-2 (a) Experimental adsorption setup (reference cell and sample cell) (b)

Data acquisition system (c) Schematic diagram of an experimental adsorption

setup 49

Figure 4-1 N2 adsorption-desorption results from four coal samples from Northeast

China 58

Figure 4-2 The pores size distribution of the selected coal samples calculated from

the desorption branch of nitrogen isotherm by the BJH model 60

xv

Figure 4-3 Fractal analysis of N2 desorption isotherms 62

Figure 4-4 Results of methane adsorption tests and the corresponding Langmuir

isotherm curves 65

Figure 4-5 Sample experimental results of CH4 sorption rate at 055 MPa for

Xiuwu-21 and Luling-10 68

Figure 4-6 Variation of the experimentally measured methane diffusion

coefficients with pressure 70

Figure 4-7 Relationships of fractal dimension (D1 D2) and Langmuirrsquos parameters

(VL PL) 72

Figure 4-8 Relationship of Langmuir pressure (PL) to sorption capacity (X1=VLν) 76

Figure 4-9 Relationships of Langmuirrsquos volume (VL) to monolayer coverage

estimated by gas molecules with unit diameter (X2=σDf2) 76

Figure 4-10 Relationship of Langmuir pressure (PL) to sorption capacity evaluated

from monolayer coverage (X3 = (SσDf2 + B)ν) 77

Figure 4-11 Variation of bulk diffusion coefficient (DB) and Knudsen diffusion

coefficient (DKpm) at different pressure stages for Sijiazhuang-15 80

Figure 4-12 Diagnostic plot of reciprocal diffusion coefficient (DP-1) vs P to

specify pressure interval of pure Knudsen flow (P lt P) and critical Knudsen

number (Kn= Kn (P)) 81

Figure 4-13 A plot of wk as a piecewise function of Kn The horizontal tails at the

low and high interval of Kn correspond to pure bulk and Knudsen diffusion

respectively 83

Figure 4-14 Comparison between experimental and theoretical calculated

diffusion coefficient for methane diffusion in Xiuwu-21 Wheeler (1955) is

described by Eq (4-2) and this work is given by Eq (2-41) 85


diffusion coefficients of the studied four coal samples at same ambient

pressure 85

Figure 5-1 (a) Structure contour map of San Juan Fruitland Formation (b)

Application of Arps decline curve analysis to gas production profile of San

Juan wells The deviation is tied to the elongated production tail 92

xvi

Figure 5-2 Modelling of gas transport in the coal matrix 98

Figure 5-3 Workflow of simulating CBM production performance coupled with

pressure-dependent matrix and cleat permeability curves 104

Figure 5-4 Blue dots correspond to the production wells investigated in this work

The yellow circle marked offset wells with well-logging information available

105

Figure 5-5 The production profile of the studied fairway well with the exponential

decline curve extrapolation for the long-term forecast 106

Figure 5-6 Example of using a gamma-ray log and bulk density log to identify coal

layers and determine the net thickness of the pay zone for reservoir evaluation

The well-logging information is accessed from the DrillingInfo database

(DrillingInfo 2020) 108

Figure 5-7 Fairway coalbed pressure-dependent permeability multiplier curve

Po=1542 psi 113

Figure 5-8 Evolution of diffusion coefficient and corresponding equivalent matrix

permeability with pressure for San Juan coal Data on the diffusion coefficient

is provided by Wang and Liu (2016) 114

Figure 5-9 Rectangular numerical CBM model with a vertical production well

located in the center of the reservoir 116

Figure 5-10 Relative permeability curves for cleats used to history-match field

production data 119

Figure 5-11 Matrix permeability growth during pressure depletion employed in the

matching process 119

Figure 5-12 History-matching of the field gas production data of two fairway

wells (a) Well A and (b)Well B (shown in Figure 5-4) by the numerical

simulation constructed in CMG 121

Figure 5-13 Effect of cleat and matrix permeability growth on gas production The

solid grey lines correspond to comparison simulation runs with constant

matrixcleat permeability evaluated at initial condition The grey dashed lines

correspond to comparison simulations runs with constant matrixcleat

permeability estimated at average reservoir pressure of the first 4000 days 125

Figure 6-1 Mechanism of cryogenic fracturing Damage mechanism A derives

from the volume expansion of LN2 Damage mechanism B is the thermal

xvii

contraction applied by sharp heat shock Damage mechanism C is stimulated

by the frost-heaving pressure 132

Figure 6-2 The experimental system (a) is a freeze-thawing system where the

coal sample is first water saturated in the glassware beaker and then subject to

cyclic liquid nitrogen injection In between the successive injections the

sample is thawed at room temperature The freeze-thawed coal samples and

the raw sample are sent to the subsequent measurements ((b) and (c)) (b) is

the experimental setup for measuring the gas sorption kinetics This part of the

experiment is to evaluate the change in gas sorption and diffusion behavior of

coal after cryogenic treatment (c) is the low-pressure adsorption system for

the determination of surface area and porosimetry of pore structure of the coal

sample This step is to evaluate the pore-scale damage caused by the cryogenic

treatment to the coal sample 140

Figure 6-3 The process diagram of freeze-thawing treatment (a) freezing

operation (b) thawing operation 141

Figure 6-4 Hori and Morihirorsquos model of fractured micropore (Hori and Morihiro

1998) The nanopore system of coal is modeled as a micro cracked solid The

pair of concentrated forces normally acting on the crack center represents the

crack opening forces produced by the freezing action of pore water 143

Figure 6-5 Results of methane addesorption tests and the corresponding Langmuir

isotherm curves for raw 1F-T and 3F-T coal 149

Figure 6-6 The role of PL acting on the adsorption and desorption process 150

Figure 6-7 Measured CH4 diffusion coefficients for raw coal 1F-T coal and 3F-

T coal at different pressure stages 151

Figure 6-8 Effect of surface heterogeneity on surface diffusion (a) and (c) describe

surface diffusion along a rough surface (b) describes surface diffusion along

a flat surface Less energy is required to initiate surface diffusion along a flat

surface than a rough surface 154

Figure 6-9 The role of multilayer of adsorption on surface diffusion In desorption

the already built-up multiple layers of adsorbed molecules smoothened the

rough pore surface Greater surface diffusion happens in the desorption process

than the adsorption process 154

Figure 6-10 Low-pressure N2 adsorption isotherm at 77K for the raw 1F-T and

3F-T coal samples 156

xviii

Figure 6-11 The adsorption isotherm of nitrogen at 77K on raw coal sample fitted

by the BET equation and GAB equation The solid curves are theoretical and

the points are experimental The grey area Aad is the area under the fitted

adsorption isothermal curve by the GAB equation 160

Figure 6-12 The desorption isotherm of nitrogen at 77K on raw coal sample fitted

by the GAB equation (n=0) and the modifed GAB equation (n=1 2) The

grey region is the area under the desorption isothermal curve fitted by the

quadratic GAB equation 163

Figure 6-13 PSD calculated from N2 sorption isotherm using the BJH model for

the raw 1F-T and 3F-T coal samples 165

Figure 6-14 CO2 sorption isotherms at 273 K of the raw 1F-T and 3F-T coal

samples 166

Figure 6-15 Micropore size distribution curves of CO2 adsorption for the raw 1F-

T and 3F-T coal samples 167

Figure 6-16 Proportional variation of pore sizes for different F-T cycles 169

Figure 6-17 Various estimates of pore volume expansion (Upper case and Lower

case) due to cyclic liquid nitrogen injections according to the micromechanical

model (solid line) The grey area is the range of estiamtes specified by the two

extreme cases The computed results are compared with the measured pore

volume expansion determined from experimental data listed in Table 6-4

(scatter)Vpi is the intial pore volume or the pore volume of the raw coal sample

Vpf is the pore volume after freezing and thawing corresponding to the pore

volume of 1F-T sample and 3F-T sample 173

Figure 6-18 An intact coal specimen (M-2) before freezing 177

Figure 6-19 Experimental equipment and procedure 179

Figure 6-20 The fracture model random distribution of elliptical cracks in an

otherwise in-contact region 180

Figure 6-21 The workflow of seismic interpretations of fracture stiffness for coal

specimens subject to cryogenic treatments 194

Figure 6-22 Evolution of surface cracks in a complete freezing-thawing cycle for

(a) dry coal specimen (b) wet coal specimen Major cracks are marked with

red lines in the images of surface cracks taken at room temperature ie pre-

existing surface cracks and surface cracks after completely thawing 196

xix

Figure 6-23 Recorded waveforms of compressional waves at different freezing

times for (a) 1 dry coal specimen and (b) 2 saturated coal specimen 198

Figure 6-24 Variation of seismic velocity with freezing time for (a) dry coal

specimen (b) wet coal specimen 200

Figure 6-25 Under dry condition (M-1) the variation of normal and tangential

fracture stiffness and tangentialnormal stiffness ratio with freezing time 204

Figure 6-26 Under wet condition (M-2) variation of normal and tangential fracture

stiffness and tangentialnormal stiffness ratio with freezing time 209

Figure 6-27 Effect of the presence of water and ice on fracture stiffness A saw-

tooth surface represents the natural roughness of rock fractures 211

xx

LIST OF TABLES

Table 2-1 Sorption kinetic experiments of methane performed in the various

literature HVB and LVB are high and low volatile bituminous coals Sub is

sub-bituminous coals Diffusion coefficient is derived from unipore model 27

Table 3-1 Proximate analyses and vitrinite reflectance of the coal samples used in

this study 46

Table 3-2 Void volume for each sample estimated with multiple injections of

Helium 51

Table 4-1 Mean pore diameter specific surface area and pore volume of the coal

samples analyzed during this study 59

Table 4-2 Fractal dimensions of the four coal samples 62

Table 4-3 The fractal dimension mean free path and tortuosity factor based on the

fractal pore model and estimated at the specified pressure stage (ie 055 138

248 414 607 and 807 MPa) 63

Table 4-4 Langmuir parameters for methane adsorption isotherms 66

Table 4-5 Parameters used in the analysis of pore characteristics and its effect on

CH4 adsorption on coal samples 74

Table 4-6 Theoretically calculated bulk diffusion coefficient (DB) and Knudsen

diffusion coefficent of porous media (DKpm) 79

Table 5-1 Investigated logs for coalbed methane formation evaluation 102

Table 5-2 Coal characteristics interpreted from well-logging information in four

offset wells 109

Table 5-3 Input parameters for Liu and Harpalani model on the permeability

growth 113

Table 5-4 Coal seam properties used to history-match field data of two fairway

wells 118

Table 6-1 Langmuirrsquos parameters for raw 1F-T and 3F-T coal The bracket

indicates the percentage increase in PL of 1F-T and 3F-T coal with respect to

PL of raw coal An increase in PL is preferred in gas production as it promotes

the gas desorption process 149

xxi

Table 6-2 Measured diffusion coefficients of raw coal 1F-T coal and 3F-T coal

(Draw D1F-T D3F-T) in the adsorption process and desorption process and the

corresponding increase in the diffusion coefficient due to freeze-thawing

cycles (ΔD1F-T ΔD3F-T) 152

Table 6-3 BET surface area parameters of GAB adsorption model and quadratic

GAB desorption model of nitrogen experimental sorption data with their

corresponding correlation coefficients (R2) the areas under the best adsorption

and desorption fitting curves (Aad Ade) and the respective hysteresis index of

raw coal 1F-T coal and 3F-T coal samples 157

Table 6-4 Peak pore diameter mean pore diameter total pore volume with its

distribution in different pore sizes after the different number of freeze-thawing

cycles 168

Table 6-5 Coal properties used in the proposed deterioration analysis 171

Table 6-6 Physical properties of two coal specimens used in this study 177

Table 6-7 Crack density (119873 ) and average half-length (119886 ) aperture (119887 ) and

ellipticity (119890) of cracks determined from the automated computer program 202

Table 6-8 Thermophysical parameters used in modeling heat transfer in the

freezing immersion test The heat capacity (Cp) and heat conductivity (119896119888) of

the saturated coal specimen (M-2) were measured at room temperature of 25

following the laser flash method (ASTM E1461-01) 208

xxii

ACKNOWLEDGEMENTS

I would like to express my gratitude to my primary supervisor Dr Shimin Liu who

guided me throughout this entire PhD study for three and half years His patience

enthusiasm and immense knowledge make me passionate about my research and my PhD

life an enjoyable journey I could not have a better advisor and mentor

I would also like to thank my doctoral committee members Dr Derek Elsworth

Dr Sekhar Bhattacharyya and Dr Chris Marone who have provided their valuable

suggestions and insights on this research and taught me a great deal about scientific

research I also wish to acknowledge the help provided by Dr Luis Ayala and Dr Hamid

Emami as my master advisor Their advice and assistance taught me the way to conduct

professional research

I am also grateful for my colleagues Ang Liu Guijie Sang Qiming Huang Long

Fan Xiaowei Hou who were good colleagues and provided me kind help in the laboratory

work A special thank also goes to my best friends in the US and China Yuzhe Cai and

Peiwen Yang for their support and time spending with me during my graduate study

I would also like to thank my parents in China Chunhe Yang and Jun Yang They

always listened to my words and helped me get through all the hard times I encountered

during my life in the US Thanks for their unconditional love I also want to thank my

boyfriend Haoming Ma as a perfect companion of my life

Chapter 1

INTRODUCTION

11 Background

Exploration of coalbed methane (CBM) in North America started with the early

activities conducted by US Bureau of Mines experiments in Alabama and Pennsylvania

Then it came to prominence in the 1980s as the oil crisis shifted the interest to potential

natural gas resources in coalbeds CBM classified by energy industry is an unconventional

resource and an important natural gas source According to Energy Information

Administration (EIA) the proven coalbed methane reserves of the US was 118 trillion

cubic feet (TCF) in 2017 The CBM production in 2017 was 098 TCF that accounted for

30 of total natural gas production in the US (EIA 2018) CBM is considered as an

environmentally friendly fuel because its combustion emits no ash no toxins and less

greenhouse gas emission compared to oil coal or even wood (Al-Jubori et al 2009) The

extraction of CBM from coal seam also prevents underground coal-mine gas outbursts and

benefits safe mining operations For these advantages CBM is expected to be an essential

sector in the future energy portfolio

Coalbed incorporate unique gas transport and storage mechanism that differs from

conventional reservoirs Coal acts as both source and reservoir for the gas where 90-98

of methane is adsorbed in a liquid-like dense phase at the internal surface of coal matrix by

2

physical adsorption with the remaining small amount of gas compressed in open void

spaces in the natural fracture network by pressure mechanism (Gray 1987 Harpalani and

Chen 1997a Levine 1996) The sorbed gas content of coal depends on mineral content

total organic content coal rank moisture content petrology gas composition as well as

reservoir conditions (Busch and Gensterblum 2011 Yee et al 1993) Migration of

methane in a CBM reservoir starts from desorption from the internal coal surface followed

by the diffusion in coal matrix which is subject to the diffusion coefficient and gas

concentration gradient After diffusing through the matrix the gas reaches the naturally

occurring fractures (cleats) and evolves to Darcy flow controlled by the permeability of

coal and pressure gradient (Figure 1-1) The rate of viscous Darcian flow through the cleat

network depends on the distribution of cleat presented in coalbed The rate of gas diffusion

depends on the pore properties of the coal matrix Production of gas from a CBM reservoir

is intuitively affected by both diffusion coefficient and permeability of coal (King 1985

Kumar 2007) At the late stage of a CBM production well (ie mature wells) coal

permeability might not be the bottle-neck for the overall gas production as commonly

believed and instead diffusion process dominates overall well production performance

since the matrix to cleat influx is limited (Wang and Liu 2016)

3

Figure 1-1 Illustration of multi-scale and multi-mechanism gas flow in a CBM reservoir

CBM production data Source DringInfoinc

12 Problem Statement

Coal is a complex polymeric material with a convoluted pore structure (Clarkson and

Bustin 1999a) Coal exhibits a broad pore size distribution ranging from micropores (lt 2

nm) to mesopores (2-50 nm) and macropores (gt50 nm) according to the International

Union of Pure and Applied Chemistry (IUPAC) classification (Schuumlth et al 2002) As

0 5 10 15 20 25 30

0

50

100

150

Pro

duct

ion r

ate

(M

cfd

ay)

time (yrs)

Desorption from

internal pore surface

Diffusion in coal matrix

Butt cleat

Face cleat

Darcyrsquos

flow

Log (nm) 012gt3

Dominated by

Darcyrsquos flow Dominated by

Diffusion + Desorption

Short-term Long-term

Well information

Pennsylvanian FormationCentral Appalachian Basin

Total producing life 28 yrs

4

micropores provide the greatest internal surface area the proportion of microporosity is a

dominant factor of gas storage in coal The distribution of mesopores and macropores

provide free gas storage and transport pathway for gas molecules that dominates gas

diffusion rate in coal Pore structure has an immerse effect on gas storage and transport

behavior in coal matrix (Smith and Williams 1984)

Extensive research have been performed on understanding the effect of pore

structure on gas sorption and diffusion behavior of coal Pore structure of coal is known to

be complex in occurrence that does not converge to a traditional Euclidean geometry The

application of fractal theory provides an intuitive description of heterogeneous structure of

coal (Pfeifer and Avnir 1983) Coal with a convoluted pore structure typically have high

adsorption energy a great number of adsorption sties as well as elevated gas storage

capacity On the other hand coal with a homogenous structure is favorable for gas

desorption and diffusion Fractal analysis serves as a powerful tool of characterizing the

complexity of pore structure of coal The effect of fractal dimension on gas adsorption

capacity has been studied in several works (Cai et al 2013 Li et al 2015 Liu and Nie

2016 Wang et al 2018a Wang et al 2016 Yao et al 2008) However their works were

limited to qualitative analysis derived from experimental measurements A quantitative

modeling of gas sorption capacities by using pore structure information as direct inputs is

still lacking in the literature For CBM production diffusion coefficient is another

important parameter as it directly related to the matrix permeability and is a required input

in most reservoir simulators such as CMG-GEM ARI-COMET IHS-FASTCBM

However as coal exhibits ultralow matrix permeability direct permeability measurements

5

on coal matrix is subject to great uncertainties As an alternative diffusion coefficient

measured by particle method varies with pressure but no unified trend persists (Charriegravere

et al 2010 Mavor et al 1990a Nandi and Walker 1975 Pillalamarry et al 2011 Wang

and Liu 2016) Theoretical understanding on the change of diffusion coefficient of coal

during pressure depletion is still obscure in the previous studies

A mechanistic based understanding on the correlation between pore structure and

gas transport mechanism of coal is highly desireable to be established This is because pore

structural parameters including pore size pore shape and pore volume is closely related to

coal rank and coal composition (eg fixed carbon moisture mineral constituent vitrinite

inertinite and others) that control gas diffusion characteristics of coal A dual porosity

model (Warren and Root 1963) that depicts coal as large fractures (secondary-porosity

system) and much smaller pores (primary-porosity system) is commonly applied to

describe the physical structure of coal for gas transport simplification which is widely

adopted in commercial CBM simulators such as CMG-GEM IHS-FASTCBM Diffusion

coefficient or sorption time is a required input in all these numerical simulations Therefore

it is critical to couple gas diffusion into CBM simulation that requires a comprehensive

understanding on the pressure-dependent diffusion behavior Nevertheless the application

of dual-porosity model to simulate CBM production always treats the high-storage matrix

as a source feeding gas to cleats with a constant diffusion coefficient which violates its

pressure-dependent nature As discussed the traditional modeling approach may not

significantly mis-predict the early and medium stage of production behavior since the

permeability is still the dominant controlling parameter However the prediction error will

6

be substantially elevated for mature CBM wells which diffusion mass flux dominates total

gas production It is crucial to accurately model gas diffusion in coal matrix and properly

weigh the contribution of diffusional flux from matrix to cleats and Darcian flux through

cleats to the overall gas production

Even with the improved understanding of gas sorption and diffusion on coal the

CBM development is still challenging due to the low permeability high fracture density

high formation compressibility CBM reservoir stimulation is commonly required for the

coal formations The conventional hydraulic fracturing can effectively increase the

stimulated reservoir volume (SRV) through fracture generation however it has no

influence on the diffusion enhancement for low diffusion coals Therefore the exotic

formation stimulation should be pursued and investigated for simultaneously increasing

SRV as well as the micropore dilation for the diffusion enhancement Cryogenic fracturing

is one of candidates for this purpose and its effectiveness should be investigated for future

application

The objective of this Dissertation was to predict gas storage and transport properties

of coalbed based on pore structure information The study aimed at an improved

understanding on the change of gas diffusion coefficient or matrix permeability of coal

during CBM production that is critical for accurate analysis of production data and

forecasting of well performance

7

13 Organization of Thesis

The present study can be separated into four tasks theoretical models experimental

work field application and fundamental research on cryogenic fracturing Figure 1-2

outlines the workflow of the theoretical (Chapter 2) and experimental studies (Chapter

3) Two sets of theoretical models were developed for both gas sorption and diffusion

characteristics and their relationship with pore structure of coal (Chapter 2)

Correspondingly sorption experiments were conducted at high-pressure for obtaining

sorption isotherms and diffusion coefficient and at low-pressure for characterizing

nanoporous network of coal (Chapter 3) Then theoretical models were validated against

laboratory data (Chapter 4) The theoretical and analytical methodology developed in

Chapter 2 and Chapter 3 on the quantification of gas diffusion in coal matrix was applied

to history-match field production for mature CBM wells in San Juan Basin (Chapter 5)

Chapter 6 presents another application of theoretical and analytical methodology

developed in Chapter 2 and Chapter 3 which is the development of cryogenic fracturing

in CBM exploration This research is conducted at multi-scale ranging from micropores to

large apertures of coal utilizing the experimental setup depicted in Chapter 3 and the

theoretical analysis in Chapter 2 to evaluate the effectiveness of this waterless fracturing

technique on the enhancement of gas production Chapter 7 presents the conclusion based

on the results of experimental and analytical work

8

Figure 1-2 Workflow of the theoretical and experimental study

Validation of Theory2

Understanding gas production mechanism

regarding to pore structure of coal

Theory Experiment

Pore structure-Gas

kinetic ModelGas Kinetic Pore Structure

Theory 1 Theory 2High P Sorption

Experiment (CH4)Low P Sorption

Experiment

Adsorption

Capacity

Adsorption

Rate

Transport

RateHeterogeneity

Pore structure-

Sorption Model

Pore structure-

Diffusion Model


9

Chapter 2

THEORETICAL MODEL

21 Gas Sorption Modeling in CBM

Modeling of gas adsorption behavior is critical for resource assessment as well as

production forecasting of coal reservoirs As coal incorporates a nanoporous network

sorption characteristics including adsorption capacity and adsorption pressure are closely

related to pore structure attributes However the mechanism of how these microscale

characteristics of coal affect gas adsorption behavior is still poorly understood This section

develops a pore structure-gas sorption model that can predict gas sorption isotherms based

on pore structure information This model provides a direct evaluation method to link the

micro-pore structure with the sorption behavior of coal

211 Literature Review

Extensive research (Budaeva and Zoltoev 2010 Cai et al 2013 Li et al 2015

Wang et al 2018a Wang et al 2016) have been performed on the fundamental

relationship between methane adsorption and pore structure in coals where a dual porosity

model describes the complex structure of coal (Warren and Root 1963) Typically macro-

(gt50 nm) and mesopores (2-50 nm) most likely provide transport pathways and

micropores (lt 2 nm) give the greatest internal surface area and hence gas storage capacity

(Ceglarska-Stefańska and Zarębska 2002 George and Barakat 2001 Harpalani and Chen

1997 Laubach et al 1998) Coal pores distributed in a three-dimensional (3D) space are

10

hard to model accurately using traditional Euclidean geometric methods and do not

converge to Euclidean geometry (Mandelbrot 1983 Wang et al 2016) The concept of

fractal geometry raised by Mandelbrot (1983) proves to be a powerful analytical tool that

provides an intuitive description of the pore structure of coal by characterizing the pore

size distribution over a range of pore sizes with a single number (ie fractal dimension

119863119891) Different values of 119863119891 were found to be between 2 and 3 for different sized pores

which is frequently applied to quantify the heterogeneity of pore surface and volume for

coals (Pfeifer and Avnir 1983) A value of fractal dimension close to 2 corresponds to a

more homogenous pore structure Otherwise the pore structure becomes more complex as

119863119891 approaches 3 Among different methods of quantifying fractal dimension low-pressure

N2 adsorptiondesorption is the most time- and cost-effective technique where fractal

Brunauer-Emmett-Teller (BET) model and fractal FrenkelndashHalseyndashHill (FHH) models

have been effectively applied to evaluate irregularity of pore structure (Avnir and Jaroniec

1989 Brunauer et al 1938a Cai et al 2011) In the fractal analysis two distinct values

of fractal dimensions (1198631 and 1198632) can be derived from low- and high-pressure intervals of

N2 sorption data The two fractal dimensions reflect different aspects of pore structure

heterogeneity interpreted as the pore surface (1198631) and the pore structure fractal dimension

(1198632) (Pyun and Rhee 2004) Higher value of 1198631 characterizes more irregular surfaces

giving more adsorption sites Higher value of 1198632 corresponds to higher heterogeneity of

the pore structure and higher liquidgas surface tension that diminishes methane adsorption

capacity (Yao et al 2008) It has been shown that sorption mechanisms may change at

different pressure stages that causes the fractal dimension of pore surface (1198631) differs from

11

fractal of pore volume (1198632) (Li et al 2015) Clearly fractal dimensions are closely tied to

adsorption behavior of the coal

The sorption isotherm is commonly used to describe gas sorption capacity Different

adsorption models are developed to mathematically model the gas sorption isotherms of

coals including Langmuir BET Barrett-Joyner-Halenda (BJH) density functional theory

(DFT) model etc (Zhang and Liu 2017) Among all these models the Langmuir model

is the most straightforward and widely accepted model Langmuirrsquos constants 119875119871 and 119881119871

define the shape of sorption isotherm where 119881119871 describes the ultimate gas storage capacity

and 119875119871 changes the slope of the sorption isotherm Some works (Cai et al 2013 Li et al

2015 Liu and Nie 2016 Wang et al 2018a Wang et al 2016 Yao et al 2008) have

attempted to correlate fractal dimension with Langmuirrsquos parameters but only based on

experimental results with limited theoretical analysis Among these reported studies the

empirical correlations were not universally consistent for different sets of coal samples

Specifically Yao et al (Yao et al 2008) found significant binomial correlations between

119881119871 and fractal dimensions (1198631 and 1198632 ) Liu and Nie (Liu and Nie 2016) claimed 119881119871

increased linearly with fractal dimensions but Li et al (Li et al 2015) observed that 119881119871

was affected negatively by 1198632 and correlated positively with 1198631 Some qualitative

interpretations were made on these relationships as a high value of 1198631 means irregular

surfaces of coals which provides abundant adsorption sites for gas molecules resulting in

high adsorption capacity but the physical mechanism of 1198632 acting on 119881119871 was not well

analyzed Besides 119875119871 was observed to be strongly related to 1198632 in Liu and Nie (Liu and

Nie 2016) and was weakly correlated with 1198632 by Fu et al (Fu et al 2017) These

12

inconsistent empirical correlations imply that the mechanism of fractal dimensions acting

on gas sorption behavior is still not clearly understood

212 Fractal Analysis

The fractal dimension (119863119891) of surfaces characterizes surface irregularity and it has a

value between 2 and 3 (Pfeifer and Avnir 1983) A rougher surface incorporates a value

of 119863119891 approaching 3 as illustrated in Figure 2-1 For coal the fractal surface is analyzed

using a fractal BET model and a fractal FHH model (Avnir and Jaroniec 1989 Brunauer

et al 1938a Cai et al 2011)

In this current study the FHH model was used to determine surface fractal dimension

from 1198732 sorption isotherm data The fractal dimension is determined by

ln (V

V0) = 119860 ln (ln (

P0119875)) + 119864 ( 2-1 )

where 1198811198810 is the relative adsorption at the equilibrium pressure 119875 1198810 is a monolayer

adsorption volume 1198750 is gas saturation pressure 119864 is the y-intercept in the log-log plot

and 119860 is the power-law exponent used to determine the fractal dimension of the coal

surface (119863119891) (Qi et al 2002) Two distinct formulas were proposed to correlate 119860 to 119863119891 by

(Liu and Nie 2016)

119863119891 = 119860 + 3 ( 2-2 )

and

119863119891 = 3119860 + 3 ( 2-3 )

13

Eq (2-2) was used to determine 119863 from the slope 119860 as Eq (2-3) would consistently

yield an unreasonably high value of fractal dimension (Yao et al 2008) Typically two

linear parts were observed in the log-log plot of ln(119881

1198810) vs ln (ln (

P0

P)) corresponding to

high- and low-pressure adsorption The fractal dimension (119863 ) of the coal surface is

obtained from the slope of the straight line (119860)

Figure 2-1 Graphical illustration of fractal dimension (Df) (a) For a smooth surface Df =

2 (b) For irregular surfaces 2 lt Df lt 3

213 Pore Structure-Gas Sorption Model

Langmuir Isotherm on Heterogenous Surfaces

A type I isotherm describes the sorption behavior of microporous solids where

monolayer adsorption forms on the external surface of the adsorbent (Gregg et al 1967)

Coal is typically treated as a microporous medium and behaves like a type I isotherm

without exhibiting significant hysteresis in pure component sorption The most widely

applied adsorption model for a type I isotherm is the Langmuir isotherm Numerous studies

(Bell and Rakop 1986b Clarkson et al 1997 Mavor et al 1990a Ruppel et al 1974) on

methane adsorption on coal have shown that Langmuir isotherm accurately fits over the

range of temperatures and pressures applied The surface of the adsorbent is assumed to

119863 = 2

(a)

2 119863 3

(b)

14

be energetically homogenous and only a single layer of adsorbate is considered to form

(Langmuir 1918) In this study the Langmuir isotherm equation is used to model the coal

adsorption isotherm from high-pressure gas sorption data of dry coals The classic form of

this equation is expressed as

119881 =

119875

119875 + 119875119871119881119871

( 2-4 )

where 119881119871 and 119875119871 are two regressed parameters to fit experimental adsorption data in the

plots of 119875119881 vs 119875

Langmuir constants (119881119871 and 119875119871) are important parameters that greatly impact the field

development of coal reservoir Langmuir volume (119881119871) is a direct indicator of the CBM gas

storage capacity Langmuir pressure (119875119871) is closely related to the affinity of a gas on the

solid surface and the energy stored in the coal formation 119881119871 is proportional to total number

of available sites for adsorption and is further affected by surface complexity total

adsorption volume and coal composition (Cai et al 2013) The relationship between 119881119871

and pore structure was analyzed where specific surface area (SSA) is comprised of the

mesopore and micropore SSA estimated using BET and Dubinin-Radushkevich (DR)

models respectively (Clarkson and Bustin 1999a Zhao et al 2016) 119875119871 is an important

parameter in CBM production Mavor et al (1990a) shows that 119875119871 along with gas content

data helps determine critical desorption pressure This pressure is an important parameter

that affects the pressure decline performance of CBM reservoirs as discussed in Okuszko

et al (2007) However how pore structure relates to 119875119871 is still poorly understood and no

quantitative relationship was reported to link the 119875119871with the pore structure

15

Crickmore and Wojciechowski (1977) implied that for a system with high enough

number of types of adsorption sites the total rate of the adsorption process is approximated

as

119877119905 =1198891205791119889119905

= 119896119886 119875(1 minus 1205791)119908+1 minus 119896119889 1205791

119898+1 ( 2-5 )

where 1205791 is surface coverage 119908 and 119898 are the coefficients of variation of the rate

constants of adsorption and desorption and 119896119886 and 119896119889 are the adsorption and desorption

constants respectively which are averaged over the heterogeneous surfaces Commonly

the spread of these two distributions are similar or are even treated as equivalent (ie 119908 =

119898) Then the expression of total rate can be simplified to the following equation by

replacing coefficient w by coefficient m

119877119905 =119889120579119905119889119905

= 119896119886 120583(1 minus 1205791)119898+1 minus 119896119889 1205791

119898+1 ( 2-6 )

where 120583 is the number of moles of molecules striking a smooth surface per unit area per

second and can be determined from molecular dynamics as

120583 =119875

(2120587119872119877119879)12 ( 2-7 )

where P is the pressure of the gas in free phase M is the molecular weight R is universal

gas constant T is temperature

For a rough surface the number of collisions would be expected because of multi-

reflection as illustrated in Figure 2-2 A surface heterogeneity factor (120584) (Jaroniec 1983) is

introduced to characterize the roughness of coal surfaces with a value ranging from 0 to 1

A ν of 1 corresponds to a perfect smooth surface For a first-order of approximation the

16

striking frequency is assumed to increase exponentially with surface heterogeneity which

is expressed as 1205831120584

Figure 2-2 Conceptual model of collisonal frequency at smooth and rough surfaces

At equilibrium surface coverage (1205791) is determined by

1205791 =

(119896119886 prime

119896119889 )120584

119875

1 + (119896119886 prime

119896119889 )120584

119875

( 2-8 )

where 120584 = 1(119898 + 1) and 119896119886 prime= 119896119886 (2120587119872119877119879)

minus12120584

Compared with Langmuirrsquos equation the expression of Langmuirrsquos coefficient (119886)

for a heterogenous surface is (Avnir and Jaroniec 1989)

119886 =1

119875119871= (

119896119886 prime

119896119889 )

120584

( 2-9 )

The value of 120584 ranges from 0 to 1 When 120584 = 1 Eq (2-8) reduces to Langmuirrsquos

model equation which agrees with the assumption made in the development of Langmuirrsquos

equation (Langmuir 1918) 120584 may be determined from surface roughness or fractal

dimension (119863119891) with the value ranging between 2 and 3 (Avnir and Jaroniec 1989) High

17

120584 (relatively small 119863119891) values indicate a smooth pore surface and a low 120584 value represents

an irregular surface Based on this interpretation and assuming a linear correspondence 120584

can be made a function of 119863119891 as

120584 = 1 minus (119863119891 minus 2

2) ( 2-10 )

Two interpretations of 120584 are given as measures of surface complexity and variation

of the reaction rate constants In most cases the latter one may not be directly identical to

the former one A coefficient (119862) may be necessary to describe the dependence of the

spread of reaction rate constants on surface roughness Langmuirrsquos coefficient is then given

by

119886 = (119896119886 prime

119896119889 )

119862120584

( 2-11 )

If a two-dimensional potential box is used to describe an adsorption site then the

adsorption rate constant (119896119886 prime) is proportional to the rate of molecules impinging on the site

(Hiemenz and Hiemenz 1986)

119896119886 prime = 1198921198730(2120587119872119877119879)minus12119862120584 ( 2-12 )

where 1198730 is the total available sites for adsorption evaluated by Langmuirrsquos volume (119881119871)

and 119892 is the fraction of the molecules that condenses and is held by surface forces

Desorption rate constant (119896119889 ) is composed of a frequency factor (119891) and a Bolzmann

factor (119896119861)

119896119889 = 119891119890minus119876119896119861119879 ( 2-13 )

18

where 119891 is the frequency with which the adsorbed molecules vibrate against the adsorbent

and 119876 is the activation energy of desorption which is approximated by adsorption heat

The ratio of 119896119886 prime and 119896119889 is directly related to the Langmuir coefficient 119886 as

119886 = (119896119886 prime

119896119889 )

119862120584

=1

radic2120587119872119877119879(119892

119891119881119871119890

119876119896119861119879)119862120584

( 2-14 )

where 1198730 is replaced by 119881119871

Both 119891 and 119892 depend on the affinity of the adsorbate to gas molecules For many

systems it is expected that these two constants would be equal resulting in the modified

form of Langmuirrsquos constant

119886 =1

radic2120587119872119877119879(119881119871119890

119876119896119861119879)119862120584

( 2-15 )

As explained in Crosdale et al (1998) methane adsorption onto the pore surfaces of

coal is dominated by physical adsorption indicated by the reversibility of the equilibrium

between free and adsorbed phase the relatively rapid sorption rate when pressure or

temperature are the varied and low heat of adsorption For a physisorption dominated

system only physical structural heterogeneity is considered neglecting the effect of

surface geochemical properties and functional groups on adsorption energy As a result

adsorption heat released at a smooth surface is constant for different coal species denoted

as 119876119904119905 In the aspects of physical structural heterogeneity coal surface with a low value of

120584 corresponds to a more heterogeneous structure with a substantial amount of adsorption

energy which may be approximated as proportional to the inverse of heterogeneity factor

19

(1120584) Based on this 119876 is related to the heat of adsorption measured at a perfect smooth

surface (119876119904119905) as

119876 = 119870119876119904119905119862120584

( 2-16 )

where 119870 is a constant that evaluates how severe 119876 changes in response to surface

complexity (120584) and 119876119904119905 may be approximated as the latent heat of vaporization

However an accurate evaluation of the activation energy of adsorption is related to

an energy distribution function (119891(휀) ) As explained by Jaroniec (1983) an explicit

solution of 119891(휀) on microporous media is hard to obtain and for the purpose of a first order

approximation the activation energy of adsorption may be treated as a constant for given

gas species and for the temperature at surfaces with similar properties

Then the Langmuir constant (119886) can be expressed as a function of the heterogeneity

factor (120584) Langmuirrsquos volume (119881119871) and temperature (119879) as

119886 =1

119875119871= (119881119871)

119862120584119865(119879) ( 2-17 )

119865(119879) =1

radic2120587119872119877119879119890minus119870119876119904119905(119896119861119879) ( 2-18 )

where 119865(119879) is a temperature-dependent function and becomes a constant under isothermal

condition

The Langmuirrsquos volume (119881119871) is a measure of ultimate adsorption capacity which is

affected by specific surface area pore size distribution and fractal dimension (Zhao et al

2016) Research has been performed (Avnir et al 1983 Fripiat et al 1986 Pfeifer and

Avnir 1983) to quantify the sorption capacity of a heterogenous surface where the number

20

of gas molecules held by the adsorbent has a power-law dependence on surface area and

the exponent describes the irregularity of the surface ie fractal dimension The adsorption

capacity in multilayer adsorption is hard to accurately derive and instead the power-law

relationship is commonly used to correlate the monolayer coverage with the surface area

and fractal dimension This simplification agrees to the assumption made in the

development of Langmuirrsquos isotherm and can be accurately applied in methane adsorption

isotherm In this work for a two-dimensional surface a fundamental straight line between

log(119881119871) and log(120590) is used to describe the power-law relationship as

119881119871 = 119878(120590)1198631198912 + 119861 ( 2-19 )

where 120590 is the specific surface area determined from the monolayer volume of the adsorbed

gas by the BET model 119878 and 119861 are the slope and intercept in the plot of 119881119871 vs (120590)1198631198912

119878 contains all the information of the effect of gas molecular size dependence on

adsorption capacity and thus the fractal dimension is an intensive property (Pfeifer and

Avnir 1983) 119861 is a correction factor to consider the variation of gas molecular size among

different gas species It should be noted that in classical fractal theory the number of

adsorbed molecules is related primarily to the surface area of the gas molecules where the

specific surface area of adsorbent measured by the BET model is inversely proportional to

the cross sectional area of different molecules (Pfeifer and Avnir 1983)

To separate the effect of temperature from pore structure on Langmuir pressure (119875119871)

Eq (2-17) may be rearranged as

ln(119875119871) = minus119862 ln(119881119871120584) + ln(119865(119879)) ( 2-20 )

21

The term ln(119881119871120584) is a lump sum of surface roughness and sorption capacity

interpreted as a measure of characteristic sorption capacity For 120584 = 1 log 119875119871 is linearly

related to log 119881119871 corresponding to an energetically homogeneous and smooth surface

which agrees with the assumption made in the Langmuir equation For a complex

surfacelog(119875119871) would change linearly in response to log(119881119871120584) In the above equation 119875119871

is correlated with sorption capacity and fractal dimension as a representation of surface

roughness The sorption capacity may be approximated by surface area and fractal

dimension with Eq (19) The expression 119875119871 could be further expanded as

ln(119875119871) = 119862 ln((119878(120590)1198631198912 + 119861)120584) + 119865(119879) ( 2-21 )

The pore structure-gas sorption model given in Eqs (2-19 2-20 2-21) were applied

to quantitatively investigate the relationship of Langmuirrsquos constants and pore

characteristics The value of 119863119891 and 120590 were measured directly through low-pressure N2

adsorption experiments The Langmuirrsquos constants were determined by high pressure

methane adsorption data 119881119871 and 119875119871 are important parameters in CBM production As

mentioned before 119881119871 indicates the maximum adsorption capacity of coalbed 119875119871 describes

the changing slope of the isotherm across a broad range of pressures and addresses gas

mobility 119875119871 determines the desorption rate and the higher the PL value is the easier the

CBM well arrives the critical desorption pressure Besides it has been shown that 119875119871 is

inversely related to coal rank (Pashin 2010) Typically a Langmuir isotherm with a larger

value of PL maintains slope at higher pressure which corresponds to a higher initial gas

production under the same pressure drawdown which is preferred for CBM wells

22

22 Gas Diffusion Modeling in CBM

This section develops a pore structure-gas diffusion model that correlates gas

diffusion coefficient with pore sturctural characteristics of coal The proposed model

provides an intuitive and mechanism-based approach to define the gas diffusion behavior

in coal and it can serve as a bridge from pore-scale structure of mass transport for the CBM

gas production prediction


Diffusion is the process that matter (gases liquids and solids) tends to migrate and

eliminate the spatial difference in composition in such a way to approach a uniform

equilibrium state with maximum entropy (Fick 1855 Philibert 2005 Sherwood 1969)

The study of diffusion in nanoporous solids came to prominence as such materials have

sufficient surface area required for high capacity and activity with extensive application in

the petroleum and chemical process industries (Kaumlrger et al 2012) For transport through

the pores with size comparable to diffusing gas molecules diffusion effects or may even

dominate the overall transport rate (Kaumlrger et al 2010) A comprehensive understanding

of the complex diffusional behavior lies the foundation for the technological development

of porous materials in adsorption and catalytic processes (Kainourgiakis et al 2002) As a

natural polymer-like porous material coal behaves like man-made nanoporous materials

for its exceptional sorption capacity contributed by nano- to micron-scale pores (Gray

1987 Harpalani and Chen 1997 Levine 1996) Dual porosity model proposed by Warren

and Root (1963) well represents the broad size distribution of coal pores where macro-

23

(gt50 nm) and mesopores (2-50 nm) most likely provide transport pathway and micropores

(lt 2 nm) provide the greatest internal surface area and gas storage capacity (Ceglarska-

Stefańska and Zarębska 2002 George and Barakat 2001 Harpalani and Chen 1997

Laubach et al 1998) The International Union of Pure and Applied Chemistry (IUPAC)

(Schuumlth et al 2002) classification of pores is closely related to the different types of forces

controlling the overall adsorption behavior in the different sized pore spaces Surface force

dominates the adsorption mechanism in micropores and even at the center of the pore the

adsorbed molecules cannot break from the force field of the pore surfaces For larger pores

capillary force becomes important (Kaumlrger et al 2012) Different diffusion mechanisms

occur in different sized pores governing the overall gas mass influx through coal matrix

(Clarkson et al 2010 Harpalani and Chen 1997 Liu and Harpalani 2013b Wang and

Liu 2016) Gas transport within coal can occur via diffusion through either pore volumes

or along pore surface or combined these two At temperatures significantly higher than the

normal boiling point of sorbate diffusion happens mainly in pore volumes where the

diffusional activation energy is negligible compared with the heat of adsorption

(Dvoyashkin et al 2007 Evans III et al 1961b Kaumlrger et al 2012 Valiullin et al 2004)

Two forms of diffusion modes are generally considered in diffusion in pore volume

which are bulk and Knudsen diffusions (Mason and Malinauskas 1983 Welty et al 2014

Zheng et al 2012) As shown in Figure 2-3 the relative importance of the two diffusion

modes depends on Knudsen number (Kn) which is the ratio of the mean free path (λ) to

pore diameter (119889) for porous rocks (Knudsen 1909 Steckelmacher 1986) Two extreme

scenarios are given in the discussion of the prevalence of the two diffusion mechanisms

24

(Evans III et al 1961b Kaumlrger et al 2012) For nanopores with 119889 ≪ 120582 the frequency of

molecule-wall collisions far exceeds the intermolecular collisions resulting in the

dominance of Knudsen diffusion In the reverse case (ie 119889 ge 120582) the contribution from

molecule-wall collisions fades relative to the intermolecular collisions and the diffusivity

approaches the molecular diffusivity As a rule of thumb molecular diffusion prevails

when the pore diameter is greater than ten times the mean free path Knudsen diffusion

may be assumed when the mean free path is greater than ten times the pore diameter (Nie

et al 2000 Yang 2013) In the intermediate regime both the Knudsen and molecular

diffusivities contribute to the effective diffusivity

Figure 2-3 Diffusion regimes in coal matrix can be categorized as Knudesn diffusion

viscous diffusion and bulk diffusion controlled by Knudsen number

Most real cases of diffusion in CBM are intermediate between these two limiting

cases (Shi and Durucan 2003b) The mean free path of gas molecules is a function of

pressure (Bird 1983) and as a result a transition of flow regime from Knudsen diffusion

to molecular diffusion will occur as pressure evolves Diffusion coefficient (119863) governs

the rate of diffusion and in CBM it can be determined from desorption time (Lama and

Bodziony 1998 Wei et al 2006) A significant amount work (Bhowmik and Dutta 2013

25

Busch et al 2004b Charriegravere et al 2010 Clarkson and Bustin 1999b Cui et al 2004

Kelemen and Kwiatek 2009 Kumar 2007 Marecka and Mianowski 1998 Mavor et al

1990a Nandi and Walker 1975 Naveen et al 2017 Pillalamarry et al 2011 Pone et al

2009 Salmachi and Haghighi 2012 Smith and Williams 1984 Wang and Liu 2016 Zhao

et al 2014) has reported the diffusion coefficient (119863) of methane in coal at different

pressures as summarized in Table 2-1 and the measured diffusion coefficient of methane

ranges from 10minus11 to 10minus15 1198982119904 Many parameters influence the gas diffusion

characteristics of coal and they include moisture content (Pan et al 2010) coal types

(Crosdale et al 1998 Karacan 2003) coal rank (Keshavarz et al 2017) sample size

(Busch et al 2004a Han et al 2013) and others In this study we are particularly

interested in the influence of pressure as it determines the mean free path and the dominant

diffusion regime

Due to the complex pore morphology of coal D is closely related to the coal pore

structure (Cui et al 2009) To our best knowledge limited efforts have been devoted to

study the quantitative inter-relationship bween pore structure and gas diffusivity in coal

Yao et al (2009) observed a strong negative correlation between the permeability and

heterogeneity quantitatively defined by fractal dimension for high-rank coals whereas a

slightly negative relationship was found for low-rank coals However the work does not

provide detailed quantitative analyses to define the fundamental mechanism for the

experimental observations A study conducted by Li et al (2016) found that coals with

higher fractal dimensions have smaller gas permeability because of complex pore shape

for tectonically deformed coals During a tectonic event such as deformation open pores

26

or semi-open pores may develop into ink-bottle-shaped pores or narrow slit pores These

pore morphological modificaitons result in a loss of pore inter-connectivity and a more

heterogenous pore structure (ie high fractal dimension) Although a lot of inroads were

achieved to uncover the relationship between the micropore structure and gas diffusivity

the quantitative linkage between them is lacking

27

Table 2-1 Sorption kinetic experiments of methane performed in the various literature

HVB and LVB are high and low volatile bituminous coals Sub is sub-bituminous coals

Diffusion coefficient is derived from unipore model

List of Works Year Location Rank Avg Particle size

119898119898

Pressure

MPa

Range of

119863 1198982119904 Nandi and Walker

(1975) 1975 US coals

Anthracite to

HVB 0315 119898119898

114minus 252

10minus13

minus 10minus14

Smith and

Williams (1984) 1984

Fruitland San

Juan Basin Sub 19119898119898 57

10minus13

minus 10minus14

Mavor et al

(1990a) 1990

Fruitland San

Juan Basin Sub to LVB 025119898119898 01 minus 136 10minus13

Marecka and

Mianowski (1998) 1998 Unknown

Semi-

anthracite 125 062 02 0032119898119898 0-01

10minus10

minus 10minus15

Clarkson and

Bustin (1999b) 1999

Lower

Cretaceous

Gates

Formation

Canada

Bituminous 021119898119898 09 minus 11 10minus11

minus 10minus13

Busch et al

(2004b) 2004

Silesian Basin

of Poland HVB 3119898119898 338 10minus11

Cui et al (2004)

(further reworked

by (Pillalamarry et

al 2011) )

2004 Unknown HVB 025119898119898 054minus 782

10minus13

minus 10minus14

Kumar (2007) 2007 Illinois Basin Bituminous 0125119898119898 030minus 476

10minus13

minus 10minus15

Pone et al (2009) 2009 Western

Kentucky

Coalfield

Bituminous 025119898119898 31 10minus11

Charriegravere et al

(2010) 2010

Lorraine

Basin France HVB 048119898119898 01 minus 53 10minus13

Pillalamarry et al

(2011) 2011 Illinois Basin Bituminous 0143119898119898 0 minus 7

10minus13

minus 10minus14

Salmachi and

Haghighi (2012) 2012

Australian

coal seam HVB 0294119898119898

0014minus 4678

10minus12

Bhowmik and

Dutta (2013) 2013

Raniganj

Coalfield

Jharia

Coalfield

Gondwana

Basin of India

Sub to HVB 01245119898119898 036minus 461

10minus12

minus 10minus13

Zhao et al (2014) 2014 Shanxi China Bituminous 0225119898119898 105minus 456

10minus11

minus 10minus12

Wang and Liu

(2016) 2016

San Juan

Basin and

Pittsburgh


minus 10minus14

Naveen et al

(2017) 2017

Jharia

Coalfield

Gondwana

Basin of India

HVB 023119898119898 0 minus 7 10minus13

28

222 Diffusion Model (Unipore Model)

Fickrsquos second law of diffusion for spherically symmetric flow (Fick 1855) is

widely applied to describe gas diffusion process across pore space where a diffusion

coefficient (119863 ) governs the rate of diffusion Mathematically the diffusion can be

described as

119863

1199032120597

120597119903(1199032

120597119862

120597119903) =

120597119862

120597119905

( 2-22 )

where 119903 is the radius of the pore 119862 is the adsorbate concentration and 119905 is the diffusion

time

lsquoUniporersquo and lsquobidisperse porersquo models are two widely adapted solutions to the

above partial differential equation (PDE) to quantify the diffusive flow (Nandi and Walker

1975 Shi and Durucan 2003b) As the name suggests the unipore model assumes a

unimodal pore size distribution while the bidisperse model considers a bimodal pore size

distribution The bidisperse model can provide a better modeling result to the entire

sorption rate curve than the unipore model for most of the coals (Smith and Williams

1984) Different from unipore model the bidisperse model separates the macropore

diffusivity from the micropore diffusivity and a ratio of microporemacropore relative

contribution to overall gas mass transfer has been included in the model The bidisperse

model is a more robust model than the unipore model because it reflects the heterogeneous

nature of the coal pore structure Nevertheless the bidisperse model requires to regress

multiple modeling parameters (ie micropore diffusivity macropore diffusivity and

volume ratio of micropore to macropore) to the experimental data and it may potentially

29

encounter non-uniqueness solution sets (Clarkson and Bustin 1999b) Besides the

bidisperse model assumes the independent process of rapid macropore diffusion and slow

micropore diffusion which cannot be always true (Wang et al 2017) The unipore model

is simple and has been successfully used to coal kinetic analysis of CH4 sorption in several

previous studies as summarized in Table 2-1 In this study the unipore model was selected

to analyze the sorption data with two reasons (1) unipore model gives reasonable accuracy

over the whole range of coal desorption and (2) unipore model is the model adapted by

commercial production simulators (Pillalamarry et al 2011) In unipore model (Crank

1975) constant gas surface concentration is assumed at the external surface and the

corresponding boundary condition can be expressed as

119862(119903 119905 gt 0) = 1198620 ( 2-23 )

where 1198620 is the concentration at the external surface of the pore In the sorption

experiment this is known to be valid since the coal particles will have a constant pressure

at the surface of the particle throughout the experimental procedure

With assumption on uniform pore size distribution the unipore model is given by

119872119905119872infin

= 1 minus6

1205872sum

1

1198992119890119909119901(minus119863119890119899

21205872119905)

infin

119899=1

( 2-24 )

119863119890 = 1198631199031198902 ( 2-25 )

where 119903119890 is the effective diffusive path 119872119905

119872infin is the sorption fraction and 119863119890 is apparent

diffusivity

30

In order to automatically and time-effectively analyze the sorpiton-diffuiosn data

we develop a matlab-based computer program (Figure 2-4) in this study based on a least-

squares criterion to regress the experimental gas sorption kinetic data and determine the

corresponding diffusion coefficient An automated computer code was programmed to

estimate the apparent diffusivity and the program is listed in the Appendix A The apparent

diffusivity (1198631199031198902) was adjusted using the Golden Section Search algorithm (Press et al

1992) until the global minimum of the objective function was reached The least-squares

function (119878) was chosen to be the objective function and described as

119878 =sum((119872119905119872infin)119890119909119901

minus (119872119905119872infin)119898119900119889119890119897

)

2

( 2-26 )

where (119872119905

119872infin)119890119909119901

and (119872119905

119872infin)119898119900119889119890119897

are experimentally measured and analytically determined

sorption fraction

In this computer program the primary input is the experimental sorption rate data

stored inrdquo diffusiontxtrdquo composed of two columns of experimental data The fist column

of entry is the sorption time and the second column is the corresponding sorption fraction

((119872119905

119872infin)119890119909119901)obtained from high-pressure sorption experiment Then the user specifies a

search window of the apparent diffusion coefficient as upper (119863ℎ119894119892ℎ) and lower (119863119897119900119908)

limits for the targeted value 119863ℎ119894119892ℎ and 119863119897119900119908 should be a reasonable range of typical values

of diffusion coefficient Based on the reported data as shown in Table 2-1 we recommend

setting 119863ℎ119894119892ℎ and 119863119897119900119908 to be 1e-3 and 1e-8 1s The last required input is the number of

terms in the infinite summation term (n119898119886119909) of the unipore model (Eq (2-24)) to fit the

31

experimental data A good entry of 119899119898119886119909 is 50 to truncate the infinite summation term and

the rest terms with large 119899 are negligible Following the Golden Section Search Algorithm

the diffusion coefficient is determined at the best fit that minimizes the difference between

experimental and analytical sorption rate data modeled by unipore model The flowchart

(Figure 2-5) shows the algorithm of the automated computer program

Figure 2-4 User interface of unipore model based effective diffusion coefficient estimation

in MATLAB GUI

32


coefficient estimation in MATLAB GUI

33

223 Pore Structure-Gas Diffusion Model

As discussed gas diffusion in coalbed during reservoir depletion typically are

intermediate between these two limiting cases (Shi and Durucan 2003b) The mean free

path of gas molecules is a function of pressure (Bird 1983) and as a result a transition of

flow regime from Knudsen diffusion to molecular diffusion will occur as pressure evolves

Knudsen diffusion (Kaumlrger et al 2010 Kaumlrger et al 2012) is the dominant

diffusion regime when the mean free path is about or even greater than the equivalent

effective pore diameter at which the pore wall-molecular collisions outnumber molecular-

molecular collisions For the gas transport in coal Knudsen diffusion dominates the overall

mass transport in small pores or under low pressure A critical point about Knudsen

diffusion is that when a molecule hits and exchanges energy with the pore wall the velocity

of molecule leaving the surface is independent of the velocity of molecule hitting the

surface and the reflecting direction is arbitrary As a result Knudsen diffusivity (Dk) is

only a function of pore size and mean molecular velocity and can be expressed as

(Knudsen 1909)

119863119870 =1

3119889119888 ( 2-27 )

where 119889 is the pore diameter and 119888 is the average molecular velocity determined from gas

kinetic theory assuming a Maxwell-Boltzmann distribution of velocity and it is given by

119888 = radic8119877119879120587119872 ( 2-28 )

where 119877 is the universal gas constant 119879 is the ttemperature and 119872 is the gas molar mass

34

The Knudsen diffusivity (119863119896) for porous media have been proposed and applied to

numerous pervious works (Javadpour et al 2007 Kaumlrger et al 2012) where the porous

media is assumed to consist of open pores (ie porosity) of the mean pore diameter and

have a degree of interconnection resulting in a tortuous diffusive path longer than an end

to end distance (ie tortuosity)

The Knudsen diffusion coefficient in porous and rough media is derived as

119863119870119901119898 =

120601

120591119863119870

( 2-29 )

where 120601 is the porosity and 120591 is the tortuosity factor

Eq (2-29) relates the diffusivity in a porous medium to the diffusivity in a straight

cylindrical pore with a diameter equal to the mean pore diameter under comparable

physical condition by a simple tortuosity parameter (120591) 120591 considers the combined effects

of increased diffusive path length the effect of connectivity and variation of pore diameter

However the definition of the tortuosity factor is not universally accepted (Wheatcraft and

Tyler 1988) Instead of using simple bodies from Euclidean geometry Coppens (1999)

successfully applied fractal geometry to describe the convoluted pore structure of

amorphous porous coal and conducted quantitate study of the effect of the fractal surface

on diffusion In this current study we would use the fractal pore model proposed by

Wheatcraft and Tyler (1988) to determine the tortuosity of the diffusive path of the pore

within coal matrix A schematic of the fractal pore model is shown in Figure 2-6

35

Figure 2-6 Fractal pore model

The key concept behind this model is that the tortuosity is induced by the surface

roughness This model provides a practical and explicit approach to quantify tortuosity by

relating it to the surface fractal dimension as developed below This model depicted in

Figure 2-6 considers a line having a true length 119865 and fractal dimension 119863119891 which is an

intensive property and independent of the size of the measuring yardstick molecules (휀)

The expression of 119865 is given by (Avnir et al 1984)

119865 = 119873휀119863119891 = 119888119900119899119904119905119886119899119905 ( 2-30 )

where 119873 is the number of yardsticks required to pave completely the line and varies with

휀

The number of yardsticks (119873 ) multiplied by the size of a yardstick (휀 ) is an

approximate or measured length (119871(휀)) of the line and can be expressed as

119871(휀) = 119873휀 ( 2-31 )

Combining Eqs (2-30) and (2-31) the measured length (119871(휀)) is related to the

fractal dimension as

119871

119903

36

119871(휀) = 119865휀1minus119863119891 ( 2-32 )

The characteristic length (119871119904) is defined as the length of the line segment holding a

constant 119863119891 If 휀 = 119871119904 then 119873 = 1 and the expression of 119865 can be written as

119865 = 119871119904119863119891 ( 2-33 )

Then 119871119904 is determined as

119871(휀) = 119871119904119863119891휀1minus119863119891 ( 2-34 )

At 119863119891 = 1 119871119904 is the end-to-end distance ( 119903) For practical application the axial

length of the pore segment ( 119871) was approximated by 119871(휀) (Welty et al 2014)

The tortuosity factor (120591) the ratio of the measured length to the end-to-end distance

is then determined to be

120591 = 119871

119903=119871119904119863119891휀1minus119863119891

119871119904= (

휀

119871119904)1minus119863119891

( 2-35 )

where 119863119891 is the fractal dimension of a line with a value between 1 and 2

The fractal dimension derived from the Nitrogen sorption data is the surface fractal

dimension with a value ranging from 2 to 3 (Avnir and Jaroniec 1989) Taking this into

account the expression of 120591 can be updated to

120591 = (휀

119871119904)2minus119863119891

( 2-36 )

Eq (2-34) provides an intuitive estimation of the tortuosity factor through the

correlation with surface fractal dimension Combing Eqs (2-27) (2-29) and (2-34) the

Knudsen diffusion coefficient of porous media (119863119870119901119898) is then found as

37

119863119870119901119898 =1

3120601 (119871119904휀)2minus119863119891

119863119870 =2radic21206011198891198770511987905

31205870511987205(119871119904휀)2minus119863119891

( 2-37 )

where 119863119870 is the Knudsen diffusion coefficient in a smooth cylindrical pore (Coppens and

Froment 1995)

Eq (2-37) has the same formula as the fractal pore model proposed in Coppens

(1999) except that porosity was introduced to consider mass transport exclusively in pore

space not through the solid matrix 119871119904 is the outer cutoff of the fractal scaling regime ie

the size of the largest fjords (Coppens 1999) In this current study as the structural

parameters were obtained from low pressure nitrogen sorption data 119871119904 was treated as the

largest cutoff of the pore size (ie maximum pore diameter) in the pore size distribution

(PSD) The other parameter 휀 is the molecular diameter of adsorbed molecules At

reservoir condition methane diffusion in free phase and pore volume dominates the overall

mass transport process (Dvoyashkin et al 2007 Evans III et al 1961b Kaumlrger et al 2012

Valiullin et al 2004) and as a result 휀 was estimated to be the mean free path of transport

gas molecules as the distance between successive collisions and the effective diffusive

diameter of the gas molecules The mean free path (120582) for real gas given in Chapman et al

(1990) is determined as

120582 =

5

8

120583

119875radic119877119879120587

2119872

( 2-38 )

where 120583 is the viscosity of the transport molecules 119875 is the pressure The factor 58

considers the Maxwell-Boltzmann distribution of molecular velocity and correct the

problem that exponent of temperature has a fixed value of 12 (Bird 1983)

38

Bulk diffusion is the dominant diffusion regime when the mean free path is far less

than the pore diameter which is usually found in large pores or for high pressure gas

transport Gas-gas collisions outnumber gas-pore wall collision The present work focuses

on gas self-diffusion in coal as only one species of gas is involved Considering Meyerrsquos

theory (Meyer 1899) the bulk or self-diffusion coefficient (119863119861) was derived neglecting

the difference in size and weight of the diffusing molecules as (Jeans 1921 Welty et al

2014)

119863119861 =1

3120582119888

( 2-39 )

When gas transport includes both aforementioned diffusion modes the relative

contribution on the overall gas influx should be quantified For free gas phase the

combined transport diffusivity (119863119901) including the transfer of momentum between diffusing

molecules and between molecules and the pore wall is given as (Scott and Dullien 1962)

1

119863119901=1

119863119870+1

119863119861 ( 2-40 )

Eq (2-40) stated that the resistance to transport the diffusing species the is a sum

of resistance generated by wall collisions and by intermolecular collisions (Mistler et al

1970 Pollard and Present 1948) One main implicit assumption behind this reciprocal

addictive relationship is that Knudsen diffusion and bulk diffusion acts independently on

the overall diffusion process In reality the probabilities between gas molecules colliding

with each other and colliding with pore wall should be considered (Evans III et al 1961a

Wu et al 2014) Then a weighing factor (119908119870) was introduced to consider the relative

39

importance of the two diffusion mechanisms as referred to Wang et al (2018b) Wu et al

(2014)

1

119863119901= 119908119870

1

119863119870119901119898+ (1 minus 119908119870)

1

119863119861 ( 2-41 )

The relative contribution of individual diffusion regime is dependent on the

Knudsen number (Kn) which is the ratio of pore diameter to mean free path It is critical

to identify the lower and upper limits of Kn where pure Knudsen and bulk diffusion can be

reasonably assumed Commonly when Kn is smaller than 01 the diffusion regime can be

considered as pure bulk diffusion (Nie et al 2000) Then 119908119870 is written in a piecewise

function 119891(119870119899) and takes the form as

119908119870 = 119891(119870119899) =

1(119870119899 gt 119870119899lowast) pureKnudsendiffusion(01)(01 119870119899 119870119899lowast) transitionflow0(119870119899 01) purebulkdiffusion

( 2-42 )

where 119870119899lowast is the critical Knudsen number of pure Knudsen diffusion

To estimate the contribution of each mechanism one should examine the manner

in which 119863119901minus1 varies with pressure From general kinetic theory (Meyer 1899) the bulk

diffusion coefficient is inversely proportional to pressure whereas the Knudsen diffusion

coefficient is independent of pressure A diagnostic plot of 119863119901minus1 obtained at a single

temperature vs various pressures (Figure 2-7(a)) is useful to identify the diffusion

mechanism as suggested by Evans III et al (1961a) A horizontal line corresponds to pure

Knudsen flow a straight line with a positive slope passing the origin represents pure bulk

flow and a straight line with an appreciable intercept depicts a combine mechanism as

illustrated in Figure 2-7(a) These interpretations are based on Eq (2-41) rather than Eq

40

(2-40) In fact the diagnostic plot simplifies the real case as it does not consider the

dependence of 119863119870119901119898 and 119908119870 at various pressures The weighing factor is subject to Kn

and pressure and a straight line will not persist for a combined diffusion Besides the

combined diffusion should be a weighted sum of pure bulk and Knudsen diffusion The

line of combined diffusion will lie between rather than above the pure bulk and Knudsen

diffusion On the other hand Knudsen diffusion in porous media also depends on the

tortuosity factor which varies with pressure As a result a horizontal line will not present

for pure Knudsen diffusion It should be noted that 119863119870119901119898 is not that sensitive to the change

in pressure as 119863119861 and a relative flat line may still occur at low pressure corresponding to

pure Knudsen flow But it needs to be further justified through our experimental data as

the flat region is important to specify the critical Knudsen number (119870119899lowast) for pure Knudsen

diffusion Considering the effect of weighing factor and tortuosity factor on the overall

diffusion process the diagnostic plot is updated from Figure 2-7(a) to Figure 2-7(b)

41

Figure 2-7 Diagnostic plot of reciprocal diffusion coefficient (119863119901minus1) vs 119875 to determine the

dominant diffusion regime Plot (b) is updated from plot (a) by considering the weighing

factor of individual diffusion mechanisms and Knudsen diffusion coefficient for porous

media

23 Summary

This chapter presents the theoretical modeling of gas storage and transport in

nanoporous coal matrix based on pore structure information The concept of fractal

geometry is used to characterize the heterogeneity of pore structure of coal by pore fractal

dimension The methane sorption behavior of coal is modeled by classical Langmuir

isotherm Gas diffusion in coal is characterized by Fickrsquos second law By assuming a

unimodal pore size distribution unipore model can be derived and applied to determine

diffusion coefficient from sorption rate measurements This work establishes two

theoretical models to study the intrinsic relationship between pore structure and gas

sorption and diffusion in coal as pore structure-gas sorption model and pore structure-gas

diffusion model Based on the modeling major contributions are summarized as follows

Pressure

minus

Pure Knudsen Diffusion


Pressure

minus

(a)(b)

Considering

tortuosity factor

Considering weighing factor

42

Gas Sorption Behavior

bull The pore structure-gas sorption model relates Langmuir parameters to pore structure

characteristics including fractal dimension and specific surface area Langmuir

constants can be estimated by only pore structure information

bull Adsorption capacity (VL) is proportional to a power function of specific surface area

and fractal dimension and the slope contains the information of on the molecular size

of the sorbing gas molecules 119875119871 also exhibits a power law dependence of 119881119871 and the

exponent is a normalized parameter of fractal dimension

bull Coal with a more heterogeneous pore structure and a more significant proportion of

microporosity have greater surface area for gas molecules to adsorb and higher

adsorption capacity So a larger fractal dimension typically corresponds to higher

sorption capacity

bull 119875119871 is negatively correlated with adsorption capacity and fractal dimension A complex

surface corresponds to a more energetic system resulting in multilayer adsorption and

an increase total available adsorption sites which raises the value of 119881119871and reduces the

value of 119875119871

bull Pore structure-gas sorption model provides an effective approach that correlates the

pore structure with the gas sorption behavior This guides the gas drainage in outburst-

prone coal mines and gas production planning in CBM reservoirs

Gas Diffusion Behavior

bull A theoretical pore-structure-based model is proposed to estimate the pressure-

dependent diffusion coefficient for fractal coals The proposed model takes the pore

43

structure parameters including porosity pore size distribution and fractal dimension

as inputs to predict the pressure-dependent gas diffusion behavior Knudsen and bulk

diffusion influxes are properly integrated to define the overall gas transport process

dynamically

bull A fractal pore model is applied to define tortuosity of diffusive path and quantitatively

correlates pore morphological complexity with diffusion coefficient of coal

bull The contribution of Knudsen diffusion and bulk diffusion to total mass transport

depends on Knudsen number a ratio of mean free path to pore size At low pressures

gas molecules collide with pore wall more frequently than intermolecular collisions

and Knudsen diffusion dominates overall gas transport At high pressures

intermolecular collisions are more significant than collisions with pore wall and bulk

diffusion dominates overall gas transport So the diffusion coefficient of coal varies

with pressure

bull The overall diffusion coefficient is modeled as a weighted sum of the Knudsen and

bulk diffusion coefficient Errors elevate towards low-pressure stages as Knudsen

diffusion becomes significant and pore structure becomes an increasingly important

role in the gas diffusion process This is when the exact characterization of the pore

structure is critical for predicting gas flow in a porous network and the proposed fractal

averaging method may not be applicable

bull This theoretical model is the first-of-its-kind to link the realistic complex pore structure

into the diffusion coefficient based on the fractal theory The proposed model can be

44

coupled into the commercially available simulator to predict the long-term CBM well

production profiles

45

Chapter 3

EXPERIMENTAL WORK

In this Chapter low-pressure N2 gas adsorption and desorption data were analyzed

through fractal analysis to characterize the pore structure of coal High-pressure methane

sorption expereiments were conducted to characterize gas sorption beahvior of coal

Specifically Langmuir isotherm was applied to model ad-de-sorption isotherms and

unipore model was employed to fit experimental sorption kinetic data and determine

diffusion coefficients The two sets of data from low-pressure and high-pressure sorption

experiments were then interrelated with theoretical model developed in Chapter 2 which

demonstrates the validity of the pore-structure based models

31 Coal sample procurement and preparation

Fresh coal blocks were collected from four different locations at three different coal

mines in China as shown in Figure 3-1 ie Luling mine in Hebei province (No 9 and No

10 coal seam) Xiuwu mine in Henan province (No 21 coal seam) and Sijiazhuang mine

in Shanxi province (No 15 coal seam) The coal samples were then pulverized to powders

for subsequent experimental tests including proximate analysis (10 g of the sample 70-

200 mesh) methane adsorption testing (40g 40-60 mesh) and N2 adsorption-desorption

test (1 g 60-80 mesh) According to the standard ISO 172462010 (Coal Proximate

analysis) (Thommes et al 2011) a 5E-MAG6600 proximate analyzer was used to

46

determine the proximate contents of the four different coal samples Table 3-1 summarizes

the experimental results from the proximate analysis

Figure 3-1 Location and geologic information of Luling Sijaizhuang and Xiuwu coalmine

The Luling coal mine is located in the outburst-prone zone as separated by the F32 fault

Table 3-1 Proximate analyses and vitrinite reflectance of the coal samples used in this

study

Nos Coal sample

Moisture

content

()

Ash

content

()

Volatile

matter

()

Fixed

carbon

()

Ro max

()

1 Xiuwu-21 149 2911 1037 6303 402

2 Luling-9 125 754 3217 6104 089

3 Luling-10 137 1027 3817 5119 083

4 Sijiangzhuang-15 203 3542 1223 549 311

47

32 Low-Pressure Sorption Experiments

Nitrogen adsorptiondesorption experiment was conducted using the ASAP 2020

apparatus at Material Research Institute Penn State University following the ISO 15901-

32007 (Pore size distribution and porosity of solid materials by mercury porosimetry and

gas adsorption Part 3 Analysis of micropores by gas adsorption) (ISO 2016) Each coal

sample was initially loaded into a sample tube which was required to remove moisture and

degas the sample prior to pore structure analysis (Busch et al 2006 Bustin and Clarkson

1998) Liquid N2 at 77 K was added to the sample following programmed pressure

increments within a wide range of relative pressure of N2 from 0009 to 0994 After each

dose of N2 the equilibrium pressure was recorded to determine the quantity of adsorbed

gas The Brunauer-Emmett-Teller (BET) model and density functional theory (DFT)

model were used to analyze the adsorption data and determine surface area and pore size

distribution (PSD) as discussed in the previous study (Gregg et al 1967)

Fractal analysis using FrenkelndashHalseyndashHill (FHH) models have been effectively

applied to evaluate irregularity of pore structure using low-pressure adsorption data (Avnir

and Jaroniec 1989 Brunauer et al 1938a Cai et al 2011) For N2 sorption isotherms the

sorption mechanism at low pressure is the Van der Waals force formed between gas

molecules and coal surfaces which mainly occurs in micropores At high pressure

capillary condensation in mesopores and macropores becomes the dominant sorption

mechanism In fractal analysis two distinct values of fractal dimensions (1198631 and 1198632) can

be derived from low- and high-pressure intervals of N2 sorption data The two fractal

48

dimensions reflect different aspects of pore structure heterogeneity interpreted as the pore

surface (1198631) and the pore structure fractal dimension (1198632) (Pyun and Rhee 2004) Higher

value of 1198631 characterizes more irregular surfaces giving more adsorption sites Higher

value of 1198632 corresponds to higher heterogeneity of the pore structure and higher liquidgas

surface tension that diminishes methane adsorption capacity (Yao et al 2008)

33 High-Pressure Sorption Experiment

Volumetric sorption experimental setup was employed to measure the sorption

isotherms Many previous studies have used volumetric methods to measure sorption

isotherms (Fitzgerald et al 2005 Ozdemir et al 2003) Figure 3-2 shows the experimental

apparatus with four sets of reference and sample cells maintained at a constant temperature

water bath (T = 54567K) The data acquisition system allows connecting eight pressure

transducers and measuring adsorption isotherms of four different coal samples

simultaneously

49

Figure 3-2 (a) Experimental adsorption setup (reference cell and sample cell) (b) Data

acquisition system (c) Schematic diagram of an experimental adsorption setup

331 Void Volume

The four coal samples are loaded into the sample cells and placed under vacuum

before gas is introduced to the sample cell The volumetric method involves three steps of

measurement including the determination of cell volumes sample volumes and the

amount of adsorbed gas (Ozdemir et al 2003) In the first two steps Helium is used as a

non-adsorbing inert gas with a small kinetic diameter that can access to micro-pores of the

coal samples easily (Busch and Gensterblum 2011) For the determination of empty cell

volumes a certain amount of Helium is introduced into the reference cell and injection

pressure is recorded as 119875119903 Then the reference cell is connected to the sample cell and the

Sample Cell

Reference

Cell

Pressure Transducer

1

23

4

Water Bath

(Constant T)

Data Acquisition

System

Connect to Data Acquisition System(a) (b)

(c)

Gas supply system Analysis system Data acquisition system

Reference cell

ValvePressure

transducer

Water bath

Sample cell

Pressuretransducer

50

pressure is equilibrated at 119875119904 The ratio of the volume of the sample cell (119881119904) to the reference

cell (119881119903) is then determined using ideal gas law A steel cylinder of known volume is then

placed in the sample cell to solve for the absolute values of cell volumes The applied gas

law can be written as

119875119881 = 119885119899119877119879 ( 3-1 )

where 119875 is the reading of the pressure transducer and 119881 is the participating volume or the

void volume of the system

In the above equation gas compressibility factor (119885) is dependent on gas species

temperature and pressure as estimated by the equation of state (119864119874119878) In our case we used

the Peng-Robinson EOS (Peng and Robinson 1976) which is a cubic equation of state

(119885)119875119903 and (119885)119875119904 are compressibility factors at injection pressure and equilibrium pressure

respectively The same notation is applied in the rest of this paper In the determination of

sample volume coal samples were put in the sample cells and the same experimental

procedures were applied to determine the sample volume (119881119904119886119898) Void volume (119881119907119900119894119889) as

the available space for free gas is determined by deducting the sample volume from total

cell volume which greatly affects the accuracy with which estimate the methane adsorption

capacity can be estimated in the next step Multiple cumulative injections of Helium into

the sample cell are recommended to reduce the experimental error and consider the matrix

shrinkage of coals (Table 3-2) With multiple injections of Helium 119881119904119886119898 is evaluated as an

average value from individual injections and the matrix to solve for 119881119904119886119898 is given by

119860119881 = 119861 ( 3-2 )

51

119860 =

[ 119875119904 minus

(119885)119875119904(119885)119875119903

119875119903 119875119904

119875119903119894

(119885)119875119903119894minus

119875119904119894

(119885)119875119904119894

119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894]

( 3-3 )

119861 = [

0119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894] 119881119904119886119898 ( 3-4 )

119881 = [119881119903119881119904] ( 3-5 )

Here 119894 is the index indicating the number of injections For the first injection (i = 1) 119875119904119894minus1

is set to be zero

Table 3-2 Void volume for each sample estimated with multiple injections of Helium

Coal Sample Xiuwu-21 Luling-9 Luling-10 Sijiazhuang-15 Injection times Void Volume V

void (cm

3)

1 27582 31818 26631 27611 2 27665 31788 26660 27666 3 27689 31782 26648 27688

Average 27645 31796 26647 27655

332 AdDesorption Isotherms

After determination of void volume adsorptive gases like methane nitrogen or

carbon dioxide were injected and the amount adsorbed at a given pressure was determined

using the basic calculations described above The experimental procedures were repeated

as the previous two steps Injection pressure was recorded as 119875119903 With the sample cell

connected pressures in the reference cell and the sample cell equilibrated and this pressure

52

was recorded as 119875119904 These values were used to construct adsorption isotherms The Gibbs

adsorption at a given pressure was calculated assuming constant void space The applied

molar balance to determine the amount adsorbed ( 119899119886119889119904119894 ) at the 119894119905ℎ injection is given by

119899119886119889119904119894 = 119899119900

119894 minus 119899119906119899119886119889119904119894 ( 3-6 )

The original amount of gas in the system prior to opening the connection valve is a

summation of the injection amount of gas from the pump section into the cell section and

the amount of free gas presenting in the cell section prior the injection given as

119899119900119894 =

119875119904119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119894minus1119877119879+

119875119903119894119881119903

(119885)119875119903119894119877119879 ( 3-7 )

The amount of free gas in the system at equilibrium pressure is determined by

119899119906119899119886119889119904119894 =

119875119904119894(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894119877119879 ( 3-8 )

The cumulative amount of adsorption (119899119886119889119904119894 ) is used to construct the adsorption

isotherm and measure the adsorption characteristics for individual coal samples

119899119886119889119904119894 = 119899119886119889119904

119894 + 119899119886119889119904119894minus1 ( 3-9 )

For the 1st injection no gas is adsorbed on the coal sample and 119899119886119889119904119894minus1 = 0 In

desorption experiment each time a known amount of gas is released from the cell section

into the vent to reduce the pressure in bulk and same preliminary experimental procedures

and calculations are conducted to determine the amount of gas desorbed from the coal

sample

53

333 Diffusion Coefficient

The sorption capacity and diffusion coefficient were measured simutaneously using

high-pressure sorption experimental setup depicted in Figure 3-2 The particle method was

adopted to quantify the diffusive flow for coal powder samples Numerous studies have

used this technique to characterize the gas diffusion behavior of coal (Pillalamarry et al

2011 Wang and Liu 2016) This method requires pulverizing the coal to powders and

ensures transport of gas is purely driven by diffusion However grinding the coal increases

the surface area for gas adsorption The change is considered to be minimal as the increase

for 40 minus 100 mesh coal size ranges from 01 to 03 (Jones et al 1988 Pillalamarry et

al 2011) and it still meets the purpose of this experiment to reduce the diffusion time and

ensure diffusion-driven in nature

In the adsorption experiment the pressure in the cell section was continuously

monitoring through the data acquisition system (DAS) After each dose of methane the

pressure in the reference cell was higher than in the sample cell When they were

connected a step increase in pressure occurred following by a gradual decrease in pressure

until equilibrium was reached The decrease in pressure was generated by the adsorption

of methane occurring at the pore surface of coal matrix and was measured very precisely

Constant pressure boundary condition was controlled by isolating the cell section from the

gas supply system This ensures a direct application of the diffusion models and the

simplest solution of diffusion coefficient (119863) is given when the constant concentration is

maintained at the external surface (Pan et al 2010) The real-time pressure data were used

54

to calculate the sorption fraction versus time data which is a required input of the unipore

model

At the ith pressure stage the sorption fraction (119872119905

119872infin) was gradually increasing with

time corresponding to a gradual decrease in pressure The sorption rate data was calculated

from the pressure-time data (119875119904119894(119905)) injection pressure (119875119903

119894) equilibrium pressure in the

previous pressure stage (119875119904119890119894minus1 ) and saturated or maximum amount of adsorbed gas

molecules in the current pressure stage (119899119904119886119905119894 )

119872119905119872infin

=1

119899119904119886119905119894 119877119879

(119875119904119890119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119890119894minus1+119875119903119894119881119903

(119885)119875119903119894minus119875119904119894(119905)(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894) ( 3-10 )

where 119872119905 is the adsorbed amount of the diffusing gas in time t and 119872infin is the adsorbed

amount in infinite time 119899119904119886119905119894 is a maximum adsorbed amount at the 119894119905ℎ pressure stage and

directly obtainable from the adsorption isotherm as the step change in cumulative

adsorption amount of the two neighboring equilibrium points

The experimentally measured value of 119872119905

119872infin was then fitted by the analytical solution

of unipore model (Mavor et al 1990a) to determine the diffusion coefficient of the coal

samples at the best match A computer program given in Appendix A can automatically

calculate diffusion coefficient from the experimental sorption rate data with least error

34 Summary

This chapter presents the experimental method and procedures to obtain gas

sorption kinetics and pore structural characteristics of coal Major achievements

accomplished in the experimental work can be summarized as follows

55

bull A high-pressure sorption experimental apparatus based on the volumetric method is

designed and constructed to measure the sorption kinetics of multiple coal samples (up

to four samples) at the same time

bull Addesorption isotherms are determined when the gas pressure in the sorption system

reaches an equilibrium condition Diffusion coefficients of coal are derived from the

sorption rate measurements when experimental systemrsquos pressure approaches to

equilibrium Specifically the analytical solution of the unipore model is utilized to

obtain the diffusion coefficient at the best fit to sorption kinetic data at each pressure

stage Therefore this high-pressure sorption experiment is able to predict the change

of diffusion coefficient or equivalent matrix permeability of coal during pressure

depletion The experimental measurements can be coupled into the commercially

available simulator to predict the long-term CBM well production profiles

bull Low-pressure sorption experiment using different gases such as N2 and CO2 is

employed to study the pore structure of coal a time- and cost-effective technique to

characterize pores with diameter 100119899119898 The fractal geometry is used to quantify

the complexity of pore structure of coal from the low-pressure adsorption data Fractal

analysis proves to be an effective approach to characterize the heterogenous structure

of coal matrix It allows quantifying and predicting the adsorption behavior of coal with

pore structural parameters

56

Chapter 4

RESULTS AND DISCUSSION

41 Coal Rank and Characteristics

The mean maximum vitrinite reflectance for samples tested are 402 (1)089

(2) 083 (3) and 311 (4) indicating they are anthracite (1 4) and high volatile

A bituminous coals (2 3) Coal rank has an important effect on the pore structures The

previous study showed that there is a ldquohookrdquo shape relationship between coal rank and

porosity and adsorption capacity is correlated positively with the coal rank (Dutta et al

2011) Based on the results of isotherm testing it is easy to obtain a positive correlation

between 119881119871 and 119877119900119898119886119909 The volatile matter content (ranging from 1037 to 3542 ) is also

a measure of coal rank The lower the volatile matter content the higher the coal rank In

addition moisture content is expected to affect adsorption capacity and the flow properties

(Joubert et al 1973 1974 Scott 2002) For samples studied they are 149 (1) 125

(2) 137 (3) and 203 (4) respectively These values are low and they may

suggest that moisture content have minimal impact of on adsorption capacity and volatile

matter content has a greater impact than moisture content on adsorption capacity Besides

higher ash content may decrease the adsorption capacity The Luling-9 sample has the

lowest ash content (754 ) while the Sijiazhuang-15 sample has the highest ash content

(3542 )

57

42 Pore Structure Information

421 Morphological Characteristics

The morphological parameters of pores including mean pore diameter specific

surface area and fractal dimensions were obtained from the low pressure N2 sorption

experiment (77 K and lt122 kPa) Figure 4-1 shows N2 adsorption-desorption isotherms of

the four coal samples that have type II isotherms with obvious hysteresis loops It is

worthwhile to demonstrate that micropores can fill with gas at low relative pressures where

the adsorption isotherm has a steep slope This mechanism may be attributed to the

presence of a hysteresis loop higher pressure where condensation builds at the walls of

pores and reduces the effective diameter of pore throat and impeding the desorption

process At lower pressure the overlapping of adsorption and desorption isotherms would

be expected as the capillary effect occurs beyond critical pressure illustrated by Kelvinrsquos

equation Following the De Boer (1958) scheme to classify the shape of hysteresis loop N2

adsorption-desorption isotherm (Everett and Stone 1958 Sing 1985) the coal samples

could be categorized into Type H3 (formerly known as Type B) For Type H3 samples

adsorption and desorption branches are parallel at low to medium pressure with negligible

hysteresis and an obvious yield point at medium relative pressure Hysteresis becomes

evident near saturation pressure which may be attributed to the difference in evaporation

and condensation rate at the walls of plate-like particles and slit-shaped pores Slit-shaped

pores are favorable for gas transport for their high connectivity (Fu et al 2017) If sharp

jumps are observed in the desorption isotherms (Luling-9 and Sijiazhaung-15) ink-bottled

58

shape pores may be present In this situation gas suddenly breaks through the pore throat

as indicated in Figure 4-1 These kinds of pores are a favor in CBM accumulation over gas

transport (Fu et al 2017)

Figure 4-1 N2 adsorption-desorption results from four coal samples from Northeast China

422 Pore size distribution (PSD)

In this study we used the classical pore size model developed by Barret Joyner and

Halenda (BJH) in 1951 (Barrett et al 1951) to obtain the pore size distribution of the coal

samples This model is adjusted for multi-layer adsorption and based on the Kelvin

equation The ready accessibility in commercial software makes the BJH model be

extensively applied to determine the PSD of microporous material (Groen and Peacuterez-

59

Ramırez 2004) The desorption branch of the hysteresis loop considers the evaporation of

condensed liquid (Gregg et al 1967) and thus the shape of desorption branch was directly

dependent on the PSD of adsorbent (Oulton 1948) The bimodal nature of PSDs is apparent

from the two peaks observed in most samples The pore volume was primarily contributed

by adsorption pores for all coal samples (ie pore diameter lt 100 nm) According to the

IUPAC classification the pore volumes of different sized pores (micro- meso- and macro-

pores) were listed in Table 4-1 Meanwhile it also reports the average pore diameter (119889)

and lower and upper cutoff of pore diameter (119889119898119894119899 119889119898119886119909 respectively) for the studied four

coal samples Figure 4-2 presents the PSDs of the four coal samples obtained from the BJH

desorption branch The average pore diameter (PD) varies between 761 to 2604 nm the

BJH pore volume (PV) varies from 000033 to 001569 cm3g The BET surface area of the

four coal samples ranges from 081 to 511 m2g The BET specific surface area (BET σ)

was estimated to be the monolayer capacity with the low-pressure sorption data up to

031198751198750 in the isotherms (Figure 4-1) and this capacity is provided by micropores

Table 4-1 Mean pore diameter specific surface area and pore volume of the coal samples

analyzed during this study

Coal

sample

Mean PD

(nm)

Pore Volume (cm3100 g) 119889119898119894119899

(nm)

119889119898119886119909

(nm)

BET σ

Vtotal Vmicro Vmeso Vmacro (m2g)

Xiuwu-21 761 1178 00247 0703 0451 1741 83759 485

Luling-9 1249 0395 000330 0172 0220 1880 115440 081

Luling-10 1505 0393 000372 0149 0240 1870 112430 089

Sijiangzhu

ang-15 46 2772 00537 0456 2262 1565 132447 511

60

Figure 4-2 The pores size distribution of the selected coal samples calculated from the

desorption branch of nitrogen isotherm by the BJH model

423 Fractal Dimension

The log-log plots of ln(119881

1198810) against ln (ln (

P0

P)) (Figure 4-3) were reconstructed

from the low-pressure N2 desorption data where two linear segments were observed with

the breakpoint around ldquo ln(ln(P0P)) = minus05 rdquo which corresponds to pores with a

diameter of about 5nm The behavior of two distinct linear intervals were interpreted as a

Luling-10

( )10 50 100 500 1000

00000

00005

00010

00015

00020

00025

00030

00035

00040

00045

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

10 50 100 500 1000

0000

0001

0002

0003

0004

0005

0006

0007

0008

0009

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Luling-9

( )10 50 100 500 1000

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

0020d

Vd

log

(W

) P

ore

Vo

lum

e (

cm

3g

)

Pore Width

Xiuwu-21

( )

10 50 100 500 1000

000

002

004

006

008

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Sijiazhuang-15

( )

micropores mesopores macropores micropores mesopores macropores


61

result of different mechanisms for low-pressure and high-pressure N2 sorption The




mechanism In the calculation individual values of fractal dimension were obtained for

different intervals of pressure to reflect different aspects of pore characteristics Two fractal

dimensions ( 1198631 and 1198632 ) were derived by curve-fitting the two linear segments

corresponding to multi and monolayer coverage in micropores and capillary condensation

in mesopores and macropores Besides an average fractal dimension (119863119891) was obtained

from linear regression of the entire pressure interval to evaluate the overall heterogeneity

of pore structure and applied to determine the heterogeneity factor (ν) as a measure of the

spread of reaction rate coefficients in all scales The results were listed in Table 4-2 1198631

and 1198632 are frequently referred to the pore surface and the pore structure fractal dimension

respectively (Pyun and Rhee 2004) Both 1198631 and 1198632 are values between 2 and 3 A smaller

value of 1198631 represents a smoother surface and as the value of 1198632 is lower pore size

distribution becomes narrower The pore surface fractal dimension of the 4 coal samples

varies from 213 to 257 along with pore structure fracture fractal dimension ranging from

232 to 269 Based on the interpretations Luling-10 provides the roughest pore surfaces

and Xiuwu-21 has the most heterogenous pore structure The influence of pore surface and

structure on methane adsorption behavior will be discussed further

62

Figure 4-3 Fractal analysis of N2 desorption isotherms

Table 4-2 Fractal dimensions of the four coal samples

Fractal analysis was also applied to determine tortuosity of gas diffusive path

which is a critical parameter to estimate gas transport rate in nanoporous network of coal

through pore structure-gas diffusion model The average fractal dimension ( 119863119891 )

characterizing the overall heterogeneity of the pore structure provides a quantitative

description of the tortuous diffusive path in the complex pore structure through the fractal

Coal sample A1 D1=A1+3 R2 A2 D2=A2+3 R2 A D=A+3 R2

Xiuwu-21 -0868 2132 0981 -0313 2687 0983 -0772 2229 0967

Luling-9 -0445 2555 0980 -0439 2561 0998 -0505 2495 0989

Luling-10 -0426 2574 0971 -0468 2532 0997 -0504 2496 0975

Sijiangzhuang

-15-0452 2547 0972 -0677 2324 0983 -0425 2575 0932

63

pore model developed in section 223 Based on fractal pore model (Eq (2-27)) the

tortuosity factor (τ) derived from the fractal pore model depends on the fractal dimension

and a normalized parameter (ie 120582119889119898119886119909 ) Apparently mean free path (λ) varies with

pressure In this study the diffusion coefficients were measured at six different pressures

which are 055 138 248 414 607 and 807 MPa Along with the pore structural

parameters the pressures were used to calculate the mean free path and corresponding

tortuosity factors The results were listed in Table 4-4 The average fractal dimension of

the four coal samples ranges from 2229 to 2496 From fractal results Luling-10 provides

the most complex pore structure with the Df of 2496 Combing with the pore structural

information from PSD we could see that Sijiazhuang-15 provides the most tortuous

diffusive path with a highest value of τ for all pressures As a result the diffusion time in

Sijaizhuang-15 is expected to be longest and this was confirmed by our experimental

results

Table 4-3 The fractal dimension mean free path and tortuosity factor based on the fractal

pore model and estimated at the specified pressure stage (ie 055 138 248 414 607

and 807 MPa)

Coal sample A 119863119891 = 119860 + 3 R2P (MPa) 055 138 248 414 607 807

Mean free path λ (nm) 6595 2660 1503 0924 0656 0516

Xiuwu-21 -0772 2229 0967

Tortuosity factor τ

1787 2199 2506 2800 3029 3199

Luling-9 -0505 2495 0989 4128 6472 8587 10924 12948 14576

Luling-10 -0504 2496 09754078 6395 8486 10798 12800 14409

Sijiangzhuang-

15-0537 2463 0932

5606 9444 13111 17336 21114 24223

64

43 Adsorption Isotherms

The methane adsorption measurements were conducted to further investigate the

effect of the fractal characteristics of coal surfaces on methane adsorption Figure 4-4

shows the experimental results of the high-pressure CH4 isothermal experiments At low

pressures adsorption of methane showed an almost linear increase with increasing

pressure The shape of the adsorption isotherm indicates that the adsorption rate of methane

adsorption decrease as pressure increases The adsorption isotherms become flat as

adsorption capacity is approached Langmuirrsquos parameters (119881119871 119875119871) were obtained by linear

fitting the curve of 119875119881 vs 119875 where 119875 and 119881 are the equilibrium pressure and the

corresponding adsorption volume The results are listed in Table 4-4 and the degree of fit

(1198772 gt 098) illustrates that Langmuir model described the adsorption behavior of the four

coal samples well indicating that monolayer coverage of coal surfaces corresponding to

the Type-I isotherm of physical adsorption

65

Figure 4-4 Results of methane adsorption tests and the corresponding Langmuir isotherm

curves

Ideally sorption in nature should be reversible where there is no adsorption-

desorption hysteresis However except for the methane isotherm of sample Sijiazhuang-

15 desorption isotherms generally lie above the excess sorption isotherms at high pressure

which is consistent with the experimental results from the low-pressure N2 sorption

experiment (Figure 4-1) and other works on methane adsorption (Bell and Rakop 1986a

Harpalani et al 2006) The deviation of desorption isotherm from adsorption isotherm

indicates that the sorbentsorbate system is in a metastable state where the activation

66

energy of desorption exceeds the heat of adsorption and the additional energy comes from

the activation energy of adsorption (Bell and Rakop 1986a) For a reversible adsorption

process the acitivation energy of desorption should equal to the heat of adsorption marked

as the thermodynamic equilibrium value (Busch et al 2003) For a non-reversible

adsorptoin process with hysteris effect the heat of adsorption with an additional activation

energy of adsorption are composed of the activation energy of desorption The small

amount of additional activation energy of adsorption explains the phenomena that the

desorption branch lies above the adsorption isotherm Thus gas is not readily desorbed to

the thermodynamic equilibrium value which is the equivalent desorption amount with the

same pressure drop found in the adsorption branch Other factors such as sample properties

(coal rank moisture) and experimental variables (coal particle size maximum equilibrium

pressure) may also affect the extent of the hysteresis effect in which the underlying

physical mechanisms are not well understood (Fu et al 2017) The irreversibility of

adsorption isotherm could be further quantified by hysteresis index and derived from

adsorption isotherms (Zhang and Liu 2017)

Table 4-4 Langmuir parameters for methane adsorption isotherms

Coal sample VL (m3 ∙ t-1) PL MPa R2

Xiuwu-21 2736 069 0984 1

Luling-9 1674 134 0987 2

Luling-10 1388 123 0986 8

Sijiangzhuang-15 3332 090 0980 1

67

44 Pressure-Dependent Diffusion Coefficient

Following the procedure depicted in the particle method (Pillalamarry et al 2011)

high-pressure methane adsorption rate data were collected at six different pressure steps

from initial pressure at 055 MPa up to the final pressure at 807 MPa With eight

transducers connecting to the data acquisition system twenty-four sorption rate

measurements were performed in this study For each pressure the apparent diffusion

coefficient is assumed to be constant As a result the estimated diffusion coefficient is an

average of the intrinsic diffusivity at a specific pressure interval The stepwise adsorption

pressure-time data were modeled by the unipore model described in Section 222 (Eq (2-

24)) and the pressure-dependence apparent diffusivity (1198631199031198902) was estimated by pressure

and time regression using our proposed automate Matlab program Figure 4-5 shows two

of the twenty-four rate measurements with modeled results based on the unipore model

These measurements were for Xiuwu-21 and Luling-10 at 055 MPa It can be seen that

the unipore model can accurately predict the trend of the sorption rate data with less than

1 percent error Due to the assumption on uniform pore size distribution the unipore

model was found to be more applicable at high pressure steps (Clarkson and Bustin 1999b

Mavor et al 1990a Smith and Williams 1984) The lowest pressure stage in this study

was 055 MPa and the unipore model gave convincible accuracy to model the sorption rate

data (Figure 4-6) Thus for higher pressure stage the unipore model should still retain its

legitimacy in this application In this work other measurements exhibited the same or even

68

higher accuracy when applying the unipore mode although they had different length of

adsorption equilibrium time

Figure 4-5 Sample experimental results of CH4 sorption rate at 055 MPa for Xiuwu-21

and Luling-10

Figure 4-6 shows the results of the estimated diffusion coefficients at different

pressures for the four tested coal samples where the effective diffusive path was estimated

to be the radius of the particle (Mavor et al 1990a) The diffusion coefficient values

exhibited an overall negative trend when the gas pressure was above 248 MPa The

decreasing trend is consistent with the theoretical bulk diffusion coefficient in open space

(Eq (2-39)) which is dependent on the mean free path of the gas molecule and gas

pressure The diffusion coefficient became relatively small at pressures higher than 6 MPa

when the coal matrix had high methane concentration and a low concentration gradient

The initial slight increasing trend were observed in the diffusion curves when the pressure

was below 248 MPa The same experimental trend was reported in Wang and Liu (2016)

0 20000 40000 60000 80000 100000

00

02

04

06

08

10

Adsorp

tion F

raction

Adsorption time (seconds)

Exp

Unipore Model

0 20000 40000 60000 80000 10000003

04

05

06

07

08

09

10

11

Adsorp

tion F

raction


Exp

Unipore Model

Xiuwu-21 Luling-10

69

and they explained that as the exerted gas pressure on the coal samples may open the

previously closed pores and more gas pathways were created to enhance the diffusion flow

Besides the relative contribution of Knudsen and bulk diffusions to the gas transport

process changes at various gas pressures Knudsen diffusion loses its importance in the

overall diffusion process as gas pressure increases and molecular-molecular collisions are

more frequent At the same time bulk diffusion becomes important at higher pressure and

typically it has faster diffusion rate than the Knudsen diffusion which explains diffusion

coefficient increase with pressure increase when pressure is less than 248 MPa The

underlying fundamental mechanism will be further discussed in the next subsequent

section The values of diffusivity range from 105 times 10minus13 to 977 times 10minus121198982119904 At all

pressure steps Xiuwu-21 had the highest diffusivity and two Luling coals have low

diffusivity because both Luling coals have high Df as reported in Table 4-4

70

Figure 4-6 Variation of the experimentally measured methane diffusion coefficients with

pressure

45 Validation of Pore Structure-Gas Sorption Model

Based on the fractal analysis 1198631 and 1198632 were determined using low-pressure 1198732

sorption data which illustrates various adsorption mechanisms at different pressure stages

associated with distinct pore surface and structure characteristics Therefore fractal

dimensions are closely tied to the adsorption behavior of the coal samples Figure 4-7

showed the correlations among fractal dimensions and Langmuirrsquos parameters From

Figure 4-7 (a) and (b) weak negative correlations were observed among Langmuirrsquos

volume and the fractal dimensions (11986311198632) which agrees with the results in Yao et al

times 10minus12

0 2 4 6 8

0

2

4

6

8

10

Measure

d D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

71

(2008) for coals with a low degree of heterogeneity but not exactly consistent with Li et al

(2015) where 1198631 positively correlates with adsorption capacity Based on the available

data 1198631and1198632 potentially have different influences on the sorption mechanism since the

dominant adsorption force may change at different pressure stages A high value of 1198631

signifies irregular surfaces of micropores of coals which provides abundant adsorption

sites for gas molecules A high value of 1198632 represents heterogenous structures in the larger

pores resulting in more capillary condensation and reduced CH4 adsorption capacity Thus

coal with high adsorption capacity typically has a large value of 1198631 and a small value of

1198632 In this study the coal samples have a fractal dimension less than 25 and the correlation

is very weak between 119881119871 and 1198631 which is found by Yao et al (2008) This may due to the

fact that the influence of 1198631 on adsorption capacity was not significant compared with the

effect of pore structures and coal compositions which leads to poor negative trend between

1198631 and 119881119871 as seen in Figure 4-7 (a) In Figure 4-7 (c) and (d) 119875119871 increases with the increase

in 1198631 and weakly correlated to 1198632 The correlation between fractal dimensions and

Langmuirrsquos parameters should be conspicuous which has led to inconsistent empirical

observations in the literature such as 119875119871 is strongly related to 1198632 in a negative way reported

by Liu and Nie (2016) and it has an extremely weak correlation with 1198632 found by this study

and Fu et al (2017) These poor regressions in Figure 4-7 imply that a simple one to one

correspondence of fractal dimension and Langmuirrsquos parameters is not sufficient to

comprehensively interpret the underlying mechanism Theoretical development of these

correlations is necessary to form an in-depth understanding of how pore structural

characteristics affect methane sorption

72

Figure 4-7 Relationships of fractal dimension (D1 D2) and Langmuirrsquos parameters (VL

PL)

Langmuirrsquos parameters are important in CBM exploration where 119881119871 determines the

maximum gas sorption capacity and 119875119871 defines the slope of the isotherm at any given

pressure As mentioned the experimental results did not provide good empirical

correlations between fractal dimensions and Langmuir variables In this section a

comprehensive analysis of pore characteristics and their effect on adsorption behavior was

determined using Eqs (2-19) (2-20) and (2-21) It is worthwhile to mention that 1198631 which

is derived from low-pressure 1198732 adsorption data is related to the fractal properties of pores

where adsorption takes place (ie micropores) whereas 1198632 obtained at a higher pressure

more closely reflects the surface properties of larger pores (ie mesopores and

macropores) Micropores provide abundant sites for adsorption because the specific

Rsup2 = 0138

0

10

20

30

40

15 17 19 21 23 25 27

VL m

3 to

n

D1

Rsup2 = 01642

0

10

20

30

40

50

15 17 19 21 23 25 27

VL m

3 to

n

D2

Rsup2 = 06301

0

04

08

12

16

15 17 19 21 23 25 27

PL M

Pa

D1

Rsup2 = 00137

0

04

08

12

16

15 17 19 21 23 25 27P

L M

Pa

D2

(a) (b)

(c) (d)

73

surface area of these pores is inversely related to pore size The adsorption capacity of coal

is dominated by micropores with greater adsorption energy and surface area than meso-

and macropores of similar composition (Clarkson and Bustin 1996) Thus 1198631 reflecting

the morphology of micropores influences the adsorption capacity and Langmuir volume

(119881119871 ) 119863119891 is specifically designated by 1198631 and the pore structure-adsorption capacity

relationship is expressed as

119881119871 = 119878(120590)11986312 + 119861 ( 4-1 )

On the other hand the heterogeneity factor (ν) developed as the spreading coefficient

of the distribution of the adsorption-desorption rate in the determination of 119875119871 which can

be interpreted as a combined contribution from micropores mesopores and macropores

Roughness of pores at all scales affects the values of ν and 119875119871 which can be estimated from

the lsquolsquomeanrdquo fractal dimension (Df) instead of distinct values related to the irregularity pore

surfaces (1198631 1198632) In Figure 4-3 119863119891 is determined by linear fitting the entire pressure

interval of 1198732 adsorption data in the log-log plot and the linear regression coefficient is

convincible (R2 gt 090) Therefore the ldquomeanrdquo fractal dimension is an effective way to

quantify the roughness of pores at all scales

Table 4-5 summarizes the parameters in the theoretical model and the meaning of

these parameters will be discussed Three variables (11988311198832 1198833) are defined and used to

plot the relationship between Langmuir variables and pore characteristics Two equivalent

parameters (1198831 and 1198833) represent the characteristic sorption capacity of a coal sample with

74

the heterogeneous surfaces where in the determination of 1198833 the sorption capacity is

approximated by a function of the fractal dimensions given by Eq 2-20

Table 4-5 Parameters used in the analysis of pore characteristics and its effect on CH4

adsorption on coal samples

Figure 4-8 demonstrates the application of the relationship (Eq 2-19) to determine

Langmuir pressure (119875119871) where the x-variable (1198831) is a measure of adsorption capacity on

a heterogenous surface 119875119871 is negatively correlated to 1198831 (R2 gt 09) A large value of

sorption capacity typically corresponds with an energetic adsorption system with high

interaction energy which increases the adsorption reaction rate and reduces the value of

119875119871 For the special case where 120584 = 1 only a monolayer of adsorbed gas molecules is

developed at the energetically homogeneous surface of coal and 119875119871 is then correlated to

119881119871 with slope equal to unity in the logarithmic plot This implies that coal with complex

structure would have both higher adsorption capacity and adsorption potential As a result

119875119871 decreases as 1198831(119881119871ν) increases Taking a closer look at 1198831 methane adsorption capacity

(119881119871) is a variable that depends on the number of available adsorption sites and the roughness

of the pore surface

Coal sample Df ν X1 = VLν X2 = σ

D12 X3 = (Sσ11986312 + 119861)ν

Xiuwu-21 223 089 1874 581 293

Luling-9 250 075 833 077 205

Luing-10 250 075 723 087 206

Sijiangzhuang-15 257 071 1217 818 250

75

As derived in section 213 Eq 4-1 describes the dependence of Langmuirrsquos volume

on fractal dimension In Figure 4-9 a linear relationship exists between the adsorption

capacity of coal samples and defined x-variable (1198832 ) which exhibits a power-law

dependence on monolayer surface coverage and the exponent is the fractal dimension The

two fitting parameters of 119878 and 119861 are determined to be 24119898 and 1331198983119892 respectively

The sorption capacity of coal would increase in response to an increase in specific surface

area or fractal dimensions A large value of fractal dimension typically represents a surface

with irregular curvature and thus has the ability to hold more gas molecules In this study

119881119871 is predicted by the linear correlation with a convincible coefficient of determination

(R2gt095) which updates the expression of 119875119871 in Eq 2-19 to Eq 2-21 119875119871 then can be

evaluated by fractal dimensions and specific surface area of the coal samples

With sorption capacity replaced by pore structural parameters (Eq 4-1) 119875119871 is only a

function of pore characteristics (ie specific surface area and fractal dimension) as

described by Eq 2-21 and shown in Figure 4-10 The same as previous observation 119875119871

exhibits a linear correlation with defined pore characteristic variable (1198833) A large value of

1198833 typically corresponds to a more heterogeneous coal sample which reduces the

adsorption desorption rate and lower the value of 119875119871 Physically this is an important

finding that the complex pore structure will have lower critical desorption pressure and

thus the CBM well will need to have a significant pressure depletion before the gas can be

desorbed and produced Even through the CBM formation with complex pore structure

can ultimately hold higher gas content these adsorbed gas will be expected to be hard to

produce due to the lower critical desorption pressure Therefore the CBM formation

76

assessment needs be to conjunctionally evaluate the Langmuir volume and pressure In

other words the high gas content CBM formation may not be always preferable for the gas

production due to the lower Langmuir pressure

Figure 4-8 Relationship of Langmuir pressure (PL) to sorption capacity (X1=VLν)

Figure 4-9 Relationships of Langmuirrsquos volume (VL) to monolayer coverage estimated by

gas molecules with unit diameter (X2=σDf2)

y = -06973x + 16643

Rsup2 = 09324

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

ln(X1)ln(1198831)

ln(119875119871)

ln 119875119871 = minus07ln (119881119871ν) + 17

1198772 = 093

y = 24372x + 133

Rsup2 = 09804

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

VL

m3

ton

X2 106 m2ton

VL m3tminus1

119883210 (m2 tminus1)

119881119871 = 24 1205901198632 + 133

1198772 = 098119881119871 = 24120590

1198631198912 + 133

1198772 = 098

77

Figure 4-10 Relationship of Langmuir pressure (PL) to sorption capacity evaluated from

monolayer coverage (X3 = (SσDf2 + B)ν)

The proposed pore structure-gas sorption model has been successfully applied to

correlate the fractal dimensions with the Langmuir variables Specifically gas adsorption

behavior was measured from high-pressure methane adsorption experiment and the

heterogeneity of pore structure of coal was evaluated from low-pressure N2 gas

adsorptiondesorption analysis Based on the FHH method two fractal dimensions 1198631 and

1198632referred as pore surface and structure fractal dimension were obtained for low- and

high- pressure intervals which reflects the fractal geometry of adsorption pores (ie

micropores) and seepage pores (ie mesopores and macropores) An average fractal

dimension (119863119891) is obtained from a regression analysis of the entire pressure interval as an

evaluation of the overall heterogeneity of pores at all scales Fractal dimensions alone

however appear not to be strongly correlated to the CH4 adsorption behaviors of coals

Instead this work found that adsorption capacity (119881119871) exhibits a power-law dependence on

y = -0723x + 17268

Rsup2 = 09834

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

X3

ln(119875119871)

ln 1198833

ln 119875119871 = minus07 ln 24 1205901198632 + 133

120584

+17

1198772 = 098

119891

78

specific surface area and fractal dimension where the slope contains the information of on

the molecular size of the sorbing gas molecules

Based on pore structure-gas sorption model 119875119871 is linearly correlated with

characteristic sorption capacity defined as a power function of total adsorption capacity (119881119871)

and heterogeneity factor (ν) in logarithmic scale This implies that PL is not independent of

VL Indeed these parameters are correlated through the fractal pore structures Fractal

geometry proves to be an effective approach to evaluate surface heterogeneity and it allows

to quantify and predict the adsorption behavior of coal with pore structural parameters We

also found that 119875119871 is negatively correlated with adsorption capacity and fractal dimension

A complex surface corresponds to a more energetic system resulting in multilayer

adsorption and an increase total available adsorption sites which raises the value of 119881119871 and

reduces the value of 119875119871

46 Validation of Pore Structure-Gas Diffusion Model

As the diffusion process controls the gas influx from matrix towards the

cleatfracture system it dominates the long-term well performance of CBM after the

fracture storage is depleted (Wang and Liu 2016) The estimation of diffusion coefficient

based on pore structure is critical to determine the production potential of a given coal

formation Apparently diffusion process is slower for coal pore in a smaller size or having

a more complex structure As mentioned above the diffusive gas influx is controlled by

combined Knudsen and bulk diffusions The theoretical values of the diffusivity under

79

these two diffusion modes was calculated based Eq (2-37) and Eq (2-39) and the results

are listed in Table 4-6 It should be noted that the expression of 119863119861 given in Eq (2-37) is

derived for open space and independent of the solid structure For porous media a

multiplication of porosity is added to the expression of 119863119861 that considers volume not

occupied by the solid matrix (Maxwell 1881 Rayleigh 1892 Weissberg 1963)

Table 4-6 Theoretically calculated bulk diffusion coefficient (DB) and Knudsen diffusion

coefficent of porous media (DKpm)

The overall diffusion coefficient (119863119901 ) was then defined as a weighted sum of

Knudsen diffusion and bulk diffusion given in Eq (2-41) To estimate the weighing factor

(119908119870) of each mechanism it is critical to determine the critical Knudsen number (119870119899lowast) and

for 119870119899 gt 119870119899lowast a pure Knudsen diffusion can be assumed Examination of the manner in

which 119863119901 varies with pressure using the diagnostic plot (Figure 2-7(b)) is intuitively

helpful to identify the pressure interval for pure Knudsen flow One challenging aspect of

applying the diagnostic plot is the uncertainty about the sensitivity of 119863119870119901119898 to the change

in pressure If 119863119870119901119898 is not very sensitive to pressure a small variation in pressure will not

have an apparent change of 119863119901 at low pressure stages and under pure Knudsen diffusion

Then a relative flat line can be found in a plot of 119863119901minus1 vs P at low pressure It corresponds

Pressure [MPa] 055 138 248 414 607 807

Theoretical Diffusion

Coefficient

[times10101198982119904]

DB DKpm DB DKpm DB DKpm DB DKpm DB DKpm DB DKpm

Xiuwu-21 10477 6760 4227 5494 2388 4822 1469 4315 1042 3990 820 3777

Luling-9 4187 1922 1689 1226 954 924 587 726 416 613 328 544

Luling-10 3847 2154 1552 1373 877 1035 539 813 383 686 301 610

Sijiazhuang-15 26248 5102 10589 3029 5982 2181 3679 1650 2611 1355 2056 1181

80

to a pressure interval of pure Knudsen flow and the contribution from bulk diffusion is

ignored as the intermolecular collision strongly correlated with pressure Figure 4-11

shows the change in 119863119861 and 119863119870119901119898 with pressure for Sijiazhuang-15 sample Figure 4-12

demonstrates the application of using diagnostic plot to identify diffusion mechanism

Figure 4-11 Variation of bulk diffusion coefficient (DB) and Knudsen diffusion coefficient

(DKpm) at different pressure stages for Sijiazhuang-15

0 2 4 6 8

0

5

10

15

20

25

30

DB

DKpm

Diffu

sio

n C

oeff

icie

nt

(m2s

)

Pressure (MPa)

times 10minus9

81

Figure 4-12 Diagnostic plot of reciprocal diffusion coefficient (DP-1) vs P to specify

pressure interval of pure Knudsen flow (P lt P) and critical Knudsen number (Kn= Kn

(P))

In Figure 4-11 bulk diffusion was subject to much greater variation than Knudsen

diffusion over the pressure range of interest Consequently a relatively flat line was found

at low pressure interval (119875 119875lowast) in the diagnostic plot (Figure 4-12) for a pure Knudsen

diffusion Effective diffusion coefficient (119863119901minus1) is then equivalent to 119863119870119901119898 and weighing

factor (119908119870 ) equals to one The critical Knudsen number (119870119899lowast ) is determined at the

inflection point where 119875 = 119875lowast As pressure increases pore wall effect diminishes as mean

free path of gas molecules shortens and bulk diffusion becomes important Then at about

25 MPa 119863119901minus1 was subject to a greater variation in terms of pressure variation since 119863119861 is

directly proportional to mean free path and inversely proportional to the pressure The

dividing pressure between pure Knudsen diffusion and combined diffusion for tested coal

Horizontal

pure Knudsen

diffusion

combined

diffusion

pure bulk diffusion

119875lowast

Non-linear Linear

times 1012

0 2 4 6 8 10

0

2

4

6

8

10

Re

cip

rocal D

iffu

sio

n C

oeff

icie

nt

(sm

2)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

82

samples were all determined to be 25 MPa ie 119875lowast = 25MPa For even higher pressure

the effect of pore wall-molecular collisions can be neglected and 119863119901minus1 was estimated by

119863119861minus1 As a result a linear trend was noted at pressure greater than 6 MPa when bulk

diffusion dominates the overall diffusion and 119908119870 equals to zero Using Figure 4-12 we

would be able to identify the dominant diffusion mechanism at different pressure stages

and evaluate the relative contribution of each mechanism or 119908119870 as dictated by Eq (2-42)

119908119870 equals to one for pure Knudsen diffusion and zero for pure bulk diffusion In the

transition regime no theoretical development has been made on the prediction of diffusion

coefficient in coal matrix

For catalysis Wheeler (1955) proposed an empirical combination of Knudsen and

bulk diffusion coefficient to determine the effective diffusion coefficient of combined

diffusion as

119863119901 = 119863119861(1 minus eminus1119870119899) ( 4-2 )

In Eq (4-2) 119863119901 approaches to 119863119861 as 119870119899 approaches to zero and mean free path is

far less than the pore diameter 119863119901 approaches to 119863119870 as 119870119899 approaches infinity since

119890minus1119870119899 asymp 1 minus 1119870119899 Correspondingly the weighing factor of Knudsen diffusion (119908119870)

grows towards higher 119870119899 However some built-in limitations are also observed for this

theoretical formula First it fails to consider the change in the effective diffusive path at

different pressures as 119863119870119901119898 rather than 119863119870 should be involved to describe the diffusion

rate under Knudsen regime Besides it underestimates 119908119870 as Eq (4-2) implicitly states that

pure Knudsen diffusion only occurs for flow with infinite value of 119870119899 In fact Knudsen

83

flow dominates the overall diffusion once 119870119899lowast is reached as illustrated in Figure 4-12

Instead 119908119870 is assumed to have a linear dependence on 119870119899 in the transition pressure range

and for a combined diffusion This assumption would be further justified by comparing

with the experimental data Figure 4-13 is a plot of 119908119870 vs 119870119899 applied to quantify the

relative contribution of each diffusion mechanism

Figure 4-13 A plot of wk as a piecewise function of Kn The horizontal tails at the low and

high interval of Kn correspond to pure bulk and Knudsen diffusion respectively

Once the 119908119870 is given the overall diffusion coefficient can be theoretically

determined by Eq (2-41) Experimentally measured diffusion coefficients for methane are

presented in Figure 4-6 The results were then compared with theoretical values predicted

00 01 02 03 04 0500

02

04

06

08

10

Wk

Kn

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

pure bulk

combined

pure Knudsen

84

by the relationships proposed by Wheeler (1955) and this study as given in Eq (4-2) and

Eq (2-41) respectively Figure 4-14 indicates that the theory of 119908119870 developed in this study

provided better fit to the experimental measured diffusion coefficient than the one proposed

by Wheeler (1955) The improvement in the prediction of diffusivity was more obvious

towards low pressure and Knudsen diffusion becomes predominant This is because our

method allows for the expected changes in the effective diffusion path Nevertheless great

discrepancy was still found at low pressure stages compared with the experimental

diffusion coefficient The source of error originates from the accuracy in the estimation of

pore structural parameters which is critical in Knudsen diffusion when pore morphology

is important Besides the scale of measured diffusion coefficient is three order of

magnitudes smaller than the predicted one This is caused by the presence of surface

diffusion Movement of gas molecules along the pore wall surface contributes significantly

to the gas transport of adsorbed species in micropores where gas molecules cannot escape

from the potential field of pore surface (Do 1998 Dutta 2009) The relative contribution

of surface diffusion and diffusion in pore volume is related to the volume ratio of gas in

adsorbed phase and free phase (Kaumlrger et al 2012) The primary purpose of this work is

to predict diffusion behavior of coal based on pore structure Surface diffusion as an

activated diffusion is mainly a function of adsorbate properties rather than adsorbent

properties To eliminate the effect of the variation in surface diffusion we conducted the

analysis under the same ambient pressure In Figure 4-15 the experimental measured

diffusion coefficients are plotted against the theoretical values determined by Eq (2-41)

for the four coal samples at each pressure stages

85

0 2 4 6 8 10

0

2

4

6

8

10

Experimental Diffusion Coefficient

Experim

enta

l D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

0

2

4

6

8

This Work

Wheeler (1955)

Theore

tical D

iffu

sio

n C

oeffic

ient (m

2s

)

Figure 4-14 Comparison between experimental and theoretical calculated diffusion

coefficient for methane diffusion in Xiuwu-21 Wheeler (1955) is described by Eq (4-2)

and this work is given by Eq (2-41)


coefficients of the studied four coal samples at same ambient pressure

0 2 4 6 80

2

4

6

8

10

Exp

erim

enta

l D

iffu

sio

n C

oe

ffic

ien

t (m

2s

)

Theoretical Diffusion Coefficient (m2s)

055 MPa

138 MPa

248 MPa

414 MPa

607 MPa

807 MPa

1198772 = 0782

1198772 = 09801198772 = 0992

1198772 = 0963

1198772 = 0926

1198772 = 0997

times10minus12

times10minus9

86

The experimental diffusion coefficients were measured at six pressure stages

ranging from 055 MPa to 807 MPa Therefore six isobaric lines are presented in Figure

4-15 and each line is composed of 4 points corresponding to the four studied coal samples

The theoretical diffusion coefficient derived from Eq (2-41) is a function of pore structural

parameters Overall it provides good fits to the experimental diffusion coefficients Due to

the presence of surface diffusion the scale of the theoretical values does not agree with it

of the experimental values But the linear relationships in Figure 4-15 inherently illustrates

that pore structure has negligible effect on the transport of gas molecules along the pore

surface Otherwise the contribution from surface diffusion should vary for different coal

samples and the four points will not stay in the same line

There is a compelling mechanism that determines the steepness of the linear

relationships Generally surface diffusion becomes predominant as surface coverage

increases and multilayer of adsorption builds up at higher pressure stages The slope is

reduced towards high pressures due to an increase in the contribution from surface

diffusion On the contrary as the pore surface is smoothed and the effective diffusive path

is shortened with a reduction in the induced tortuosity This leads to a faster diffusion

process with greater mass transport occurring in pore volume and the lines are expected to

be steeper as pressure increases Under these mechanisms the lines are steeper at lower

pressure stages (119875 4MPa) in Figure 4-15 For higher pressures reverse trend can be

found as the lines tend to be horizontal as pressure increases

87

47 Summary

This chapter investigates the validity of theoretical models developed in Chapter 2

using the laboratory measurements from high-pressure and low-pressure sorption

experimental setup presented in Chapter 3 This work aims at investigating the effect of

pore structure on methane adsorption and diffusion behavior for coal Major findings of

this chapter can be summarized as follows

bull Langmuir isotherm provides adequate fit to experimental measured sorption isotherms

of all the bituminous coal samples involved in this study Based on the FHH method

two fractal dimensions 1198631 and 1198632 referred as pore surface and structure fractal

dimension are obtained within low- and high- pressure intervals which reflects the

fractal geometry of adsorption pores (ie micropores) and seepage pores (ie

mesopores and macropores) However fractal dimensions alone appear not to be

strongly correlated to the CH4 adsorption behaviors of coal

bull The pore structure-gas sorption model developed in Chapter 2 well predicts Langmuir

constants including gas sorption capacity and gas adsorption pressure based on pore

structure information which is very easy to obtain Langmuir volume appears to have

a linear correspondence with a lump of specific surface area and fractal dimension in a

log-log plot Langmuir pressure is also linearly correlated with a lump of Langmuir

volume and fractal dimension in a log-log plot The correlation is valid for a set of coal

with similar rank and composition

88

bull The application of the unipore model provides satisfactory accuracy to fit lab-measured

sorption kinetics and derive diffusion coefficients of coal at different gas pressures A

computer program in Appendix A is constructed to automatically and time-effectively

estimate the diffusion coefficients with regressing to experimental sorption rate data

bull The pore structure-gas diffusion model developed in Chapter 2 is applied to model the

pressure-dependent diffusion behavior for fractal coals where diffusion coefficients

are measured from the high-pressure experimental setup constructed in Chapter 3 The

proposed model takes the pore structure parameters including porosity pore size

distribution and fractal dimension as inputs and it provides accurate modeling of the

variation of diffusion coefficients at different pressures and for different coals

bull Based on fractal pore model the determined tortuosity factors range from 1787 to

24223 for the tested pressure interval between 055MPa and 807 MPa The results

suggest that the increase in pressure and pore structural heterogeneity resulted in a

longer effective diffusion path and a higher value of tortuosity factor affecting the

Knudsen diffusion influx in porous media The pore structural parameters lose their

significance in controlling the overall mass transport process as bulk diffusion

dominates

bull Both experimental and modeled results suggest that Knudsen diffusion dominate the

gas influx at low pressure range (lt 25 MPa) and bulk diffusion dominated at high

pressure range (gt6 MPa) For intermediate pressure ranges (25 to 6 MPa) combined

diffusion should be considered as a weighted sum of Knudsen and bulk diffusion and

the weighing factor directly depends on Knudsen number The overall diffusion

89

coefficient was then evaluated as a weighted sum of Knudsen and bulk diffusion

coefficient At individual pressure stages from 055MPa and 807 MPa it provided

good fits to the experimentally measured overall diffusion coefficient which varied

from 105 times 10minus13 to 977 times 10minus121198982119904

90

Chapter 5

FIELD APPLICATION TO CBM WELLS IN SAN JUAN BASIN

51 Overview of CBM Production

San Juan Fruitland formation (see Figure 5-1(a)) is the worlds leading producer of

CBM that surpasses lots of conventional reservoirs in production and reserve values and

numerous wells in this region are at their late-stage being successfully produced for more

than 30 years (Ayers Jr 2003 Cullicott 2002) Figure 5-1(b) presents the typical

production profile of CBM wells in the San Juan region The production characteristics of

San Juan wells are the elongated production tails that deviate from the prediction of Arps

decline curve A brief overview of the CBM production profile is given later followed by

an analysis of the occurrence of the production tail As Fruitland coal reservoirs are initially

water-saturated water drive is responsible for early gas production in the de-watering stage

controlled by cleat flow capacity Short-term production is governed by cleatfracture

permeability whereas long-term production is related to gas diffusion in matrices dictating

gas supply to cleats and wellbore The production performance and reservoir characteristics

of Fruitland coalbed depend on interactions among hydrodynamic and geologic factors

Thus different producing areas have distinct coalbed-reservoir characteristics As marked

in the grey shade in Figure 5-1 the optimal producing area in San Juan Basin is commonly

referred to as the fairway which has an NW-SE oriented trend passing through the border

of New Mexico and Colorado Fairway wells have the most extended production history

and remarkably high rates of production in the San Juan Basin (Moore et al 2011)

91

However production now becomes challenging for these fairway wells maintaining at

extremely low reservoir pressures (lt100 psi for some mature wells ) for years or even

decades (Wang and Liu 2016) Correspondingly an elongated production tail in concave-

up shape typically presents in the production history that deviates from the exponential

declining trend given by Arps curve indicated in Figure 5-1(b) It was historically believed

to be caused by the growth of cleat permeability with reservoir depletion (Clarkson et al

2010 Palmer and Mansoori 1998 Palmer et al 2007) A contradicting mechanism against

the increase of permeability would be a failure of coal induced by a lowering of pressure

Coal failure exerts a potent effect on the mature fairway coalbed for its friable

characteristic and direct evidence is the increased production of coal fines during the

depletion of fairway wells (Okotie et al 2011) Permeability increase in cleats may

become marginal for those old fairway wells and an alternative mechanism needs to be

investigated for the elongated production tail As discussed gas diffusivity acting on the

coal matrix varies with reservoir pressure and it dominates gas production of coal

reservoirs in the mature stage of pressure depletion Since matrix conductivity dictates the

amount of adsorbed gas diffused out and supplied to cleats its increase with pressure

decline observed in San Juan coal (Smith and Williams 1984 Wang and Liu 2016) is

another important factor contributing to the hyperbolic or concave-up production curves in

the decline stage

92

Figure 5-1 (a) Structure contour map of San Juan Fruitland Formation (b) Application of

Arps decline curve analysis to gas production profile of San Juan wells The deviation is

tied to the elongated production tail

52 Reservoir Simulation in CBM

521 Numerical Models in CMG-GEM

Coal is heterogeneous comprising of micropores (matrix) and macropores (cleats)

Cleats is a distinct network of natural fractures and can be subdivided into face and butt

cleats Typically cleats are saturated with water in the virgin coalbeds of the US and no

methane is adsorbed to the surface of cleats (Pillalamarry et al 2011) It is not possible to

explicitly model individual fractures since the specific geometry and other characteristics

of the fracture network are generally not available To circumvent this challenge a dual-

93

porosity model (Warren and Root 1963) was proposed to describe the physical coal

structure for gas transport simplification This model does not require the knowledge of the

actual geometric and hydrological properties of cleat systems Instead it requires average

properties such as effective cleat spacing (Zimmerman et al 1993) Based on this model

gas transport can be categorized into three stages as desorption from coal surface diffusion

through the matrix and from the matrix to cleat network and Darcys flow through cleat

system and stimulated fractures towards wellbore (King 1985 King et al 1986) The rate

of viscous Darcian flow depends on the pressure gradient and permeability of coal In

contrast gas diffusion is concentration-driven and the diffusion coefficient quantitatively

governs its rate However the application of Warren and Root model (cubic geometric

model) to CBM reservoirs depicts matrix as a high-storage low-permeability and primary-

porosity system and cleats as a low-storage high permeability and secondary-porosity

system (Thararoop et al 2012) Based on this concept matrix flow within the primary-

porosity system is ignored and gas flow can only occur between matrix and cleats and

through cleats (Remner et al 1986) In fact the assumption that the desorbed gas from the

coal matrix can directly flow into the cleat system has been shown to frequently engender

erroneous prediction of CBM performance where gas breakthrough time was

underestimated and gas production was overestimated (Reeves and Pekot 2001)

Especially for those mature CBM fields at low reservoir pressure gas diffusion through

coal matrix cannot be ignored and it can be the determining parameter for the overall gas

output from the wellbore For mature wells gas deliverability of cleats can be orders of

magnitude higher than it of the matrix due to sorption-induced matrix shrinkage (Clarkson

94

et al 2010 Liu and Harpalani 2013b) Thus coal permeability may not be as the limiting

parameter for gas flow and production and the ability of gas to desorb and transport into

cleatfracture system takes the determining role to define the late stage production decline

behavior of CBM wells A better representation of CBM reservoirs as a dual-porosity dual-

permeability systems has been implemented in the latest modeling works (Reeves and

Pekot 2001 Thararoop et al 2012) with the implication that matrix provides alternate

channels for gas flow on top of fluid displacement through cleats Their study showed a

promising agreement between simulated results and the field productions with

consideration of diffusive flux from the matrix to the cleatfracture system

522 Effect of Dynamic Diffusion Coefficient on CBM Production

Gas in coal primarily resides in the adsorbed phase on the surface of micropores

where sorption kinetics and diffusion process control gas transport from matrices towards

cleats Diffusion rate is typically characterized by sorption time By definition sorption

time is a function of the diffusion coefficient and cleat spacing (Sawyer et al 1987) is

commonly used to quantify gas matrix flow in commercial CBM simulators The past

simulation results proved that CBM reservoirs with a shorter sorption time (faster

desorptiondiffusion process) would have a higher peak gas production rate as well as

higher cumulative gas production at the early production stage (Remner et al 1986

Ziarani et al 2011) The underlying mechanism of this phenomenon is that desorbed gas

would accumulate in the low-pressure region around the wellbore until critical gas

saturation was reached The formulation of the gas bank would inhibit the relative

95

permeability of water At the same time increase the mobility of gas such that a higher

diffusion rate or smaller sorption time with a stronger gas bank is expected to have a higher

gas production rate at the de-watering stage These results demonstrated that the diffusional

flow of gas in the coal matrix has a significant influence on gas production behavior within

the CBM well throughout its life cycle Diffusion coefficient (119863) as discussed describes

the significance of the diffusion process and varies with pore structure and pressure of

matrix Albeit the sorption time or diffusion coefficient can be a dominant factor

controlling the gas production of a CBM well most reservoir models are comparable to

Warren and Root (1963) model These models always assume that total flux is transported

through cleats and the high-storage matrix only acts as a source feeding gas to cleats with

a constant sorption time It is apparent that this traditional modeling approach violates the

nature of gas diffusion in the coal matrix where the diffusion coefficient is a pressure-

dependent variable rather than a constant during gas depletion as discussed in Chapter 2

and Chapter 4 As expected the traditional modeling approach may not significantly

mispredict the early and medium stage of production behavior since the permeability is

still the dominant controlling parameter However the prediction error will be substantially

elevated for mature CBM wells which the diffusion mass flux will take the dominant role

of the overall flowability This prediction error will result in an underestimation of gas

production in late stage for mature wells

This study intends to investigate the impact of the dynamic diffusion coefficient on

CBM production throughout the life span of fairway wells The numerical method was

adopted to simulate the gas extraction process as the complexity of sorption and diffusion

96

processes make it is impossible to solve the analytical solutions explicitly (Cullicott 2002)

Currently cleat permeability is still the single most important input parameter in

commercial CBM simulators including the CMG-GEM and IHS-CBM simulator to

control the gas transport in coal seam (CMG‐GEM 2015 Mora et al 2007) Numerous

studies (Clarkson et al 2010 Liu and Harpalani 2013a 2013b Shi and Durucan 2003a

Shi and Durucan 2005) reported the cleat permeability growth during depletion in San

Juan Basin that has been elaborately implemented in current CBM simulators Regarding

the mass transfer through the coal matrices we want to point out that these simulators

always assume a constant diffusion coefficientsorption throughout the simulation time

span This assumption contradicts both the experimental observations in literatures (Mavor

et al 1990a Wang and Liu 2016) and this work in Chapter 4 and theoretical studies in

Chapter 2 on gas diffusion in the nanopore system of coal where the diffusion coefficient

was found to be highly pressure- and time-dependent There are minimal studies on the

dynamic diffusion coefficient of coal and how it affects CBM production at different stages

of depletion This current study provides a novel approach to couple the dynamic diffusion

coefficient into current CBM simulators The objective is to implicitly involve the

progressive diffusion in the flow modeling to enable the direct use of lab measurements on

the pressure-dependent diffusion coefficient in the numerical modeling of CBM and

improve the well performance forecasting For this purpose numerically simulated cases

are critically examined to match the field data of multiple CBM wells in the San Juan

fairway region The integration of pressure-dependent diffusion coefficient into coal

reservoir simulation would unlock the recovery of a larger fraction of gas in place in the

97

fairway region which also improves the evaluation of the applicability of enhanced

recovery in San Juan Basin

53 Modeling of Diffusion-Based Matrix Permeability

Gas transport in coal can occur via diffusion and Darcys flows Mass transfer

through viscous Darcian flow in cleats is driven by the pressure gradient and controlled by

permeability In contrast mass transfer through gas diffusion is governed by the

concentration gradient and regulated by the diffusion coefficient Both flow mechanisms

can be modeled by the diffusion-type equation as gas pressure and concentration are

intercorrelated by real gas law We note that current reservoir simulators such as CMG-

GEM simulator still treat permeability as the critical parameter dictating gas transport in

coal As gas diffusion in the coal matrix controls the gas supply from matrices to cleats it

is crucial to accurately weigh the contribution of diffusion and Darcys flow to the overall

gas production This can be simply achieved by converting the diffusion coefficient into a

form of Darcy permeability based on mass conservation law and without a significant

modification of current commercial simulators Here we would introduce the modeling of

the gas diffusion process in the coal matrix with Ficks law and Darcys law and obtain an

equivalent matrix permeability in the form of gas properties and diffusion coefficient As

shown in Figure 5-2 gas transport in the coal matrix starts with desorption from gas in the

adsorbed phase at the internal pore surface to gas in the free phase Then these gas

molecules are transported in pore volume via diffusion (King 1985 King et al 1986)

98

Figure 5-2 Modelling of gas transport in the coal matrix

Assuming that pores in the microporous coal matrix have a spherical shape the

principle of mass conservation can be applied as

119902120588|119903+119889119903 minus 119902120588|119903 = 4120587119903

2119889119903120601120597120588

120597119905+ 41205871199032119889119903(1 minus 120601)

120597119902119886119889119904120597119905

( 5-1 )

where 119905 is time 119903 is the distance from the center of a spherical cell 119902 is the volumetric

flow rate of gas in free phase 120588 is the density of gas in free phase 119875 is pressure and 119902119886119889119904

is the density of gas in the adsorbed phase per unit volume of coal

Eq (5-1) can be simplified into

120597(119902120588)

120597119903= 41205871199032120601

120597120588

120597119905+ 41205871199032(1 minus 120601)

120597119902119886119889119904120597119905

( 5-2 )

To derive the equivalent matrix permeability (119896119898) for diffusion in nanopores we

first assume Darcys flow prevails in gas transport through coal matrix and 119902 is given by

(Dake 1983 Whitaker 1986)

99

119902 =

41205871199032119896119898120583

120597119875

120597119903

( 5-3 )

where 119896119898 is matrix permeability

Substituting Eq (5-3) into Eq (5-2) reduces the latter into

1

1199032120597

120597119903(1199032119896119898120583

120588120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-4 )

Diffusion is the dominant gas flow regime in the ultra-fine pores of the coal matrix

and rate of diffusion through a unit area of a section under a concentration gradient of 120597119862

120597119903

is given by (Crank 1975)

119869 = 119863

120597120588

120597119903

( 5-5 )

where 119869 is diffusion flux defined to be the rate of transfer of gas molecules per unit area 119863

is the diffusion coefficient and 120588 is gas concentration or gas density

The corresponding 119902 of diffusion flux in Eq (5-4) can be found as

119902 =

119860

120588119869

( 5-6 )

where 119860 is the sectional area available for diffusing molecules passing through and 119860 =

41205871199032120601

By applying Ficks law for spherical flow it is possible to substitute for 119902 in Eq (5-

2) with Eq (5-3) as

1

1199032120597

120597119903(1199032119863120601

120597120588

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-7 )

The isothermal gas compressibility factor (119888119892) is defined as

100

119888119892 = minus

1

119881

120597119881

120597119875=1

120588

120597120588

120597119875

( 5-8 )

Substituting the 119888119892 into Eq (5-3) gives

1

1199032120597

120597119903(1199032119863120601119888119892120588

120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-9 )

Eq (5-9) has a similar form to Eq (5-4) except for the prevailing flow regime that

results in different derivations of gas transport rate Comparing these two equations 119896119898

can be directly related to 119863 by

119896119898 = 120601119888119892120583119863 ( 5-10 )

With Eq (5-10) the equivalent matrix permeability can be determined as a function

of gas properties ( 119888119892and120583 ) porosity (120601 ) and diffusion coefficient (D) The same

relationship was also presented in Cui et al (2009) The pressure-dependent diffusion

coefficients can be obtained from high-pressure sorption experiment in Chapter 3 In

general permeability is a function of rock properties and independent of fluid properties

Here 119896119898 also depends on gas properties and reservoir conditions which reflects the nature

of gas diffusion driven by collisions between gas molecules or between gas molecules and

pore walls The derived 119896119898 will be used to simulate the gas diffusion process in numerical

models of this study This is because in current numerical simulators while the modeling

of gas diffusion is always programmed based on constant diffusion coefficient the

modeling of Darcys flow has the capacity of coupling the geomechanical effect on gas

flow and considering the dependence of permeability on stress Therefore the conversion

of 119863 into 119896119898 is the most effective and practical pathway to implement variation of

101

diffusion coefficient in gas production with minimum modifications to current numerical

simulators Using this proposed 119896119898 can offer a unique opportunity to couple the pressure-

dependent diffusion dynamics into the flow modeling under the real geomechanical

boundaries

54 Formation Evaluation

The application of wireline logs offers a timely-efficient and cost-effective method

of estimating reservoir properties when compared with core analysis Usually the location

of the coal layer can be accurately resolved with relatively basic logs (Scholes and

Johnston 1993) As shown in Table 5-1 gamma-ray log bulk density log and resistivity

log all have drastic and responses to coal and in turn utilized to specify coal depth and

thickness (Mavor et al 1990b) Gamma-ray logging measures the natural radiation of rock

and is traditionally used to identify shale with high gamma-ray counts Pure coal has a low

gamma-ray response of less than 70 API units for lack of naturally radioactive elements

unless some impurities such as clay exist (Mullen 1989) Bulk density log evaluates

formation porosity as rocks with low density are rich in porosity Coal can be very easily

identified from the density log as the adjacent shale formation typically has a density of

265 gcm3 and coal has an average density of 15 gcm3 For most coalbeds in the San Juan

Basin their density is less than 175 gcm3 (Close et al 1990 Saulsberry et al 1996) It

should be emphasized that the apparent porosity read from the density log is different from

actual coal porosity The nanopores in coal are too small to be detected with conventional

density log devices

102

Nevertheless the bulk density log is still useful in pinpointing coal zones A logging

suite consisting of a gamma-ray and a density log is sufficient for coal identification and

basic description Sometimes a resistivity log is also applied to identify coal formation

Pure coal reads high in resistivity log for its low conductivity However some thin layers

cannot be detected by resistivity log with standard vertical resolution This study chooses

to use open source well logs accessed from DrillingInfo database (DrillingInfo 2020) and

focuses the discussion on the interpretation of high-resolution bulk density log and gamma-

ray log with a resolution down to 1 ft referring to Schlumbergerrsquos handbook on locating

coal layers and determining the net thickness of the formation pay zone Although other

tools or sources such as drill stem testing may provide additional quantitative analyses for

well configuration the investigation on the coalbed in San Juan basin is quite mature and

such information can be easily referred to previous studies (Ayers Jr 2003 Ayers and

Zellers 1991 Clarkson et al 2011 Liu and Harpalani 2013a)

Table 5-1 Investigated logs for coalbed methane formation evaluation

Log type Log response to coal Purpose

Gamma-ray log reads low radioactivity (lt 70

API)

coal depth and thickness

Density log reads low density (lt175

gcc) and high porosity

coal depth thickness and

gas content

Resistivity log reads high resistivity coal depth thickness

Production log Reads bottom hole

temperature

formation temperature

Mud log Reads mud density formation pressure

minimal logging suite for coalbed methane production decisions

103

55 Field Validation (Mature Fairway Wells)

In this study we applied a novel approach to couple the equivalent diffusion-based

matrix permeability model into numerical simulation of CBM production as illustrated in

Figure 5-3 This approach aims to quantify the competitive flow between Darcian and

diffusive fluxes at different pressure stages The proposed model was validated in an effort

to history-match coalbed methane production data of two high productive fairway wells

As shown in Figure 5-4 Fruitland Total Petroleum System (TPS) is outlined by the black

line and sweet spot of the fairway region is denoted by the green line Figure 5-3 outlines

the workflow of implementing the lab-measured diffusivity and sorption strain curves into

the numerical simulation of CBM production where diffusivity is related to matrix

permeability through the proposed equivalent diffusion-based matrix permeability

modeling (Eq (5-10)) and sorption strain dictates the variation of sorption strain via the

analytical modeling of cleat permeability increase during depletion (Liu and Harpalani

2013b) This proposed method allows us to use the pressure-dependent diffusivity to

implicitly compute and forecast production behavior and define long-term production

behavior for mature CBM wells

104

Figure 5-3 Workflow of simulating CBM production performance coupled with pressure-

dependent matrix and cleat permeability curves

105

Figure 5-4 Blue dots correspond to the production wells investigated in this work The

yellow circle marked offset wells with well-logging information available

551 Location of Studied Wells

The targeted wells in this study are in the New Mexico portion of the fairway

indicated in Figure 5-4 Coal reservoirs in the fairway typically are well-cleated with high

permeability thick coal deposit and high gas content relative to other producing regions

of San Juan basin (Moore et al 2011) Figure 5-5 presents a typical production profile for

the studied wells The production performances of these wells are characterized by high

peak production rates high cumulative recoveries and rapid de-watering process

Currently they are at their mature stage of pressure depletion as being continuously

produced for more than 20 years For these depleted wells their declining production

106

curves show a significant discrepancy from the forecasting of Arps curve (Arps 1945)

Arps decline exponent extrapolated from the semi-production plot (Figure 5-5) evolves

over time where the early declining behaviors collapse to exponential decline curves and

tend to be more hyperbolic later throughout well life (Rushing et al 2008) Many

researchers believed that the permeability growth of fairway coalbeds (Clarkson and

McGovern 2003 Gierhart et al 2007 Shi and Durucan 2010) led to the deviation from

the long-term exponential decline behavior But as matrix shrinkage opens up cleats

Darcys flow in cleat network no longer restricts long-term gas production and instead

matrix flow by diffusion becomes the limiting factor In this work we intend to investigate

the pressure-dependent diffusive flux as an alternate mechanism responsible for the late-

stage concave up production behavior or the so-called elongated production tail marked

in Figure 5-1

Figure 5-5 The production profile of the studied fairway well with the exponential decline

curve extrapolation for the long-term forecast

107

552 Evaluation of Reservoir Properties

The first step of history matching is the collection of reservoir description data that

includes gas in place and rock and fluid properties affecting fluid flow As the vast majority

of the gas is adsorbed at the coal matrix surface an estimate of gas in place depends on the

drainage area coal thickness coal density and gas content The location and net thickness

of coal layers can be readily accessed from the evaluation of well logs as discussed in

Section 55 Since no logging data is available for the producing wells we used nearby

offset wells marked in Figure 5-4 as a surrogate for the formation evaluation Since no

logging data is publicly available for the targeted producing wells we used neighboring

well-logging information as a surrogate for the formation evaluation Figure 5-6 shows an

example of a coal analysis presentation for one offset well located in the Colorado portion

of the fairway marked in Figure 5-4 (DrillingInfo 2020) Coal intervals are identified by

densities of less than 175 gcc and low gamma-ray responses (APIlt70) The implemented

coal interval from a logging suite of high-resolution gamma-ray log and density log is from

3147 ft to 3244 ft with a net coal thickness of 40 ft

Table 5-3 lists the reservoir parameters determined from the integration of high-

resolution gamma-ray log and density log and well log header Based on the interpretation

of wireline logs the investigated wells are located in the regionally overpressured area

characterized by pressure gradients of 045 to 049 psift with reservoir pressure exceeding

1500 psi which is consistent with previously reported ranges (Ayers Jr 2003)

108

Figure 5-6 Example of using a gamma-ray log and bulk density log to identify coal layers

and determine the net thickness of the pay zone for reservoir evaluation The well-logging

information is accessed from the DrillingInfo database (DrillingInfo 2020)

109

Table 5-2 Coal characteristics interpreted from well-logging information in four offset

wells

Well Index Depth Net

Thickness Log date Density

Pressure

gradient

Reservoir

Pressure

(ft) (ft) (ft) (gcc) (psift) (psi)

1 3205 40 1181988 140 0478 1552

2 3440 26 1211995 157 0432 1508

3 3414 72 5291994 150 0458 1562

4 3495 34 12311993 155 0442 1527

Apart from the estimate of gas storage reservoir properties that are components of

Darcys and Ficks laws need to be evaluated appropriately The absolute and relative

permeability of cleats controls Darcy flow and these rock properties serve as calibration

parameters over the course of history matching This is because they are the least well-

defined reservoir properties in the literature and these simulated permeability values

should fall into the reported ranges documented in Ayers work (Ayers Jr 2003) for the

San Juan fairway region By incorporating the matrix strain model into the analytical

permeability model the growth of absolute permeability during pressure depletion is

predicted by Liu and Harpalani model (Liu and Harpalani 2013b)

119896119891

119896119891119900= (

120601119891

120601119891119900)

3

= [1 +119862119898120601119891119900

(119875 minus 119875119900) +1

120601119891119900(119870

119872minus 1) 휀]

3

( 5-11 )

and 119862119898 is defined as

119862119898 =

1

119872minus (

119870

119872+ 119891 minus 1) 119888119903

( 5-12 )

where 119896119891

119896119891119900 is the ratio of cleat permeability at initial reservoir pressure to it at current

pressure of 119875 120601119891

120601119891119900is the corresponding cleat porosity ratio119870 and 119872 are the bulk modulus

110

and constrained axial modulus 휀 is the sorption-induced matrix strain 119891 is a constant

between 0 and 1

Based on surface energy theory the sorption-induced volumetric strain 휀 can be

quantified by the Langmuir-type model (Liu and Harpalani 2013a) as

휀 =

3119881119871120588119904119877119879

119864119860119881119900int

1

119875119871 + 119875119889119875

119875

1198751

( 5-13 )

where 119881119871 and 119875119871 are Langmuir constants 120588119904 is the density of solid matrix 119864119860 is the

modulus of solid expansion associated with desorption or adsorption 119881119900 is gas molar

volume 119875120576 is the pressure when strain equals to half of 휀119871 and 1198751 and 1198752 defines the

pressure interval for evaluating the change in sorption strain

The setting of required input parameters for the prediction of permeability was

referred to Liu and Harpalanis work (Liu and Harpalani 2013b) and Table 5-4 lists the

values of these parameters for matching the field data Figure 5-7 indicates that 119896 increased

by a factor of 14 relative to 119896119900 at initial reservoir pressure (119875119900) and this increase is a typical

value estimated by previous researchers (Shi and Durucan 2010) for the San Juan fairway

area The well log derived value of 119875119900 for the two producing wells was 1542 psi averaged

from the formation pressures of the four offset wells given in Table 5-3 prior to production

On the other hand the ability of gas transport in the coal matrix controlling the amount of

gas fed into cleats was quantified by the diffusion coefficient measured from the sorption

kinetic experiment in Chapter 3 In general the diffusion coefficient of the San Juan coal

sample was negatively correlated with pressure as reported in our previous laboratory

work (Wang and Liu 2016) The measured diffusion coefficient would then be converted

111

into equivalent matrix permeability using Eq (5-10) which requires a reasonable estimate

of matrix porosity (120601119898)

120601119898 =

119881119901

119881119901 + 119881119892119903119886119894119899

( 5-14 )

where 119881119901 is pore volume available for gas transport in matrix and 119881119892119903119886119894119899 is the solid grain

volume of the coal matrix

The grain volume of the coal matrix was estimated from the sorption kinetic

experiment when helium was injected as a non-adsorbing gas prior to adsorption for the

determination of total void volume in the experimental system The grain density was

measured to be 133 gcc and 119881119901 was the inverse of density with a value of 0016 ccg The

total pore volume of the coal matrix was determined from the low-pressure nitrogen

sorption experiment The measured 119881119901 for San Juan coal was 000483 ccg Input these

measured volume values into Eq (5-14) yielded a matrix porosity of 002 This value would

be used as a starting point to calculate the equivalent matrix permeability with Eq (5-10)

and model its variation during reservoir depletion

Figure 5-8 plots the change of matrix flowability characterized by both diffusion

coefficient and equivalent matrix permeability at different pressure stages Together with

the cleat permeability growth model Figure 5-7 summarizes matrix and cleat permeability

multiplier curves with the pressure decline The multiplier was defined as the ratio of

permeability at current pressure to its initial value at virgin reservoir pressure As pressure

decreased matrix experienced a much greater increase in its equivalent permeability than

cleat since coal matrix shrinkage may significantly open up micropore and increase gas

112

mobility through the coal matrix (Cui et al 2004) Owing to compaction gas production

results in an increase in effective stress or even a failure of coal and in turn it leads to a

decrease in coal flowability Simultaneously the enhancement of permeability occurs due

to the matrix-shrinkage effect For coalbed wells in the fairway matrix shrinkage

dominates the mechanical compaction of coal leading to the positive trend of permeability

during depletion These two distinct phenomena are also expected to take place in the coal

matrix but at the pore scale The increase in effective stress during pressure depletion

causes pores to contract and inhibits the ability for gas molecules to flow through At the

same time the extraction of adsorbed gas molecules gives more free pore space for gas

transport related to matrix shrinkage effect Besides the diffusing species itself exhibits a

pressure-dependent nature where the diffusion rate increases as intermolecular collisions

and molecule-pore wall collisions become more frequent at lower gas pressures The

measured diffusion coefficient of San Juan coal shows an overall increasing trend with a

reduction in gas pressure (Figure 5-8) This positive trend implies that the effect of

mechanical compression of pores on gas flowability is canceled by matrix-shrinkage and

the pressure-dependent diffusive properties of gas molecules As with the cleat

permeability the equivalent matrix permeability was also observed to increase during

reservoir depletion (Figure 5-7) but to a higher degree This is contributed mainly by the

fact that diffusive flow occurring at a much smaller scale than Darcian flow is driven by

molecular collisions and therefore strongly depends upon gas pressure The observed

growth in matrix permeability is a potent indication that accurate modeling of the ability

113

of gas transport in coal matrix is critical for mature well gas production prediction in late

production stage

Table 5-3 Input parameters for Liu and Harpalani model on the permeability growth

s VL P

L E EEA c

r f T (gcc) (scft) (psi) (psi)

(psi-1

) (F) 14 674 292 290E+05 03 5 201E-06 07 107

Figure 5-7 Fairway coalbed pressure-dependent permeability multiplier curve Po=1542

psi

greater growth in matrix flowability

114


permeability with pressure for San Juan coal Data on the diffusion coefficient is provided

by Wang and Liu (2016)

553 Reservoir Model in CMG-GEM

Numerical simulation was applied to match field data of two mature fairway wells

and to examine the significance of the equivalent matrix permeability modeling in CBM

production The use of a reservoir simulator is the practical method to circumvent the

complexity of solving the partial differential equation concerning gas desorption and

diffusion in coal (Paul and Young 1990) Only limited analytical solutions existed for this

type of gas transport and they were often derived for the equilibrium sorption process with

instantaneous gas desorption (Clarkson et al 2012a Clarkson et al 2008) which differed

115

from the interest of this study A three-dimensional two-phase (gas-water) finite-

difference model was built with Computer Modeling Groups GEM (Generalized Equation-

of-State Model) simulator (CMG‐GEM 2015) As noted by Rushing et al (2008) GEM

can simulate every storage and flow phenomena characteristics of coalbed methane

reservoirs Specifically this reservoir simulator can couple geomechanical responses and

sorption induced swelling in cleat and matrix into the modeling of gas and water production

process A simulator built-in dual permeability model was applied to simulate Darcys flow

in the cleats and Ficks mass transfer in the matrix where two rock types were specified

separately for matrix and cleat systems The uniqueness of this simulation work was that

the stress-dependent and sorption-controlled permeabilities were modelled both for cleat

and matrix through the permeability analytical model (ie Liu and Harpalani model) and

the equivalent matrix permeability modeling whereas previous simulation studies focused

on the permeability growth only for cleats By converting the diffusion coefficient into

matrix permeability the effect of matrix flowability increase during reservoir depletion can

be easily incorporated into the current simulator and the required input for modeling this

phenomenon is a table of permeability multiplier with pressure As shown in Figure 5-7

cleat and matrix undergo a different degree of growth in permeability with continuous

pressure depletion separate tables would be applied to characterize the variation of

permeability in these two rock constituents

All simulations were constructed for a single-well on a spacing of 320 acres per

well which is a typical value of well spacing for San Juan wells drilled before 1999 (US

Department of the Interior 1999) Cartesian grids were employed since the face and butt

116

cleats are approximately orthogonal to each other The grid dimension was designed with

23 grids in both the x-direction and y-direction and utilized 9 layers for modeling of the

multi-layers of the coal seam A vertical production well was located in the center of the

reservoir As shown in Figure 5-9 the individual grid size was finer around the wellbore

It increased geometrically towards the edge of the reservoir to accurately capture

substantial changes in pressure and saturation adjacent to the well

Figure 5-9 Rectangular numerical CBM model with a vertical production well located in

the center of the reservoir

554 Field Data Validation

Coal properties listed in Table 5-4 were reservoir parameters used to match the field

data of the two fairway wells depicted in Figure 5-4 The reservoir model was set to be

fully water-saturated at the initial condition which is a typical characteristic in fairway

coalbeds (Ayers et al 1990) Overburden pressure of 1542 psi determined at an average

117

depth of 3460 ft and the pressure gradient of 0441 psift was considered as the initial

reservoir pressure Porosity cleat and matrix permeability relative permeability were the

key calibrating parameters in the history-matching process Estimates of these parameters

were derived during the matching process of the simulated production data with the field

production data accessed from the DrillingInfo database (Cui et al 2004) The resulting

relative permeability curves are presented in Figure 5-10 and the derived values for both

matrix and cleat porosity are summarized for the two wells in Table 5-4 For gas transport

properties cleat and matrix permeability evaluated at the initial reservoir condition would

be adjusted to achieve an agreement between simulated and recorded rates and their values

are summarized in Table 5-4 The horizontal permeability of cleats parallel to the bedding

plane was 100 times greater than the vertical permeability (Gash et al 1993) The cleat

permeability curve utilized in the previous history-matching work (Liu and Harpalani

2013b) (see Figure 5-7) was assumed to be the true characteristic of fairway reservoirs and

kept as an invariant in the matching process We want to point out that this simulation study

incorporates a lab-measured diffusivity curve plotted in Figure 5-8 and the corresponding

matrix permeability curve into a numerical model to forecast CBM production This is the

first of its kind for taking the dynamic diffusivity into the flow modeling for the gas

production simulation

Figure 5-11 presents the resulting growing trend of matrix permeability with

pressure decrease where the equivalent matrix permeability modeling was employed to

determine matrix permeability by substituting history-matched matrix porosity and lab-

measured diffusivity data into Eq (5-10) Other reservoir parameters such as net thickness

118

and fracture spacing were also adjusted slightly and their values derived at the matching

case were consistent with the range of reported reservoir properties in the San Juan fairway

region (Ayers Jr 2003)

Table 5-4 Coal seam properties used to history-match field data of two fairway wells

Input Parameters Values for Well A Well B

Drainage Area (acre) 320 320

Depth (ft) 3460 3460

Thickness (ft) 54 74

Fracture Spacing (ft) 008 006

Initial Reservoir Pressure (psi) 1542 1542

Reservoir Temperature (F) 120 120

Gas Content (scfton) 585 585

Langmuir Sorption Capacity (scfton) 695 695

Langmuir Pressure (psi) 292 292

Initial Water Saturation in Cleat 1 1

Initial Water Saturation in Matrix 0 0

Methane Composition 100 100

Fracture Porosity 010 008

Matrix Porosity 45 40

Pore Compressibility (1psi) 370E-4 620E-4

Horizontal Fracture Permeability (mD) 35 30

Vertical Fracture Permeability (mD) 035 03

Diffusion Coefficient (m2s) 138E-12 423E-13

Equivalent Matrix Permeability (mD) 930E-11 550E-11

Sorption Time (days) 415 762

Bottom-hole Pressure (psi) 600 (up to 710 days) 100 100 (beyond 710 days)

Skin Factor -2 -2

Key history-matching parameters set at initial reservoir condition

119

Figure 5-10 Relative permeability curves for cleats used to history-match field production

data

0 400 800 1200 1600

0

20

40

60

80

100

Matr

ix P

erm

ea

bili

ty M

ultip

lier

Pressure (psi)

Well A

Well B


matching process

The history matching results for the two fairway wells are shown in Figure 5-12

where the simulated gas production rate was compared against field data It is noted that

120

monthly data of the gas production rate is generally available for an entire well life In

contrast monthly data on water production is of poor quality especially for early time

Therefore the gas rate was used as a reliable source of field data in the history-matching

process Simulations were performed for 4000 days of production since the sorption

kinetics had a negligible effect on depleted coal reservoirs with a small concentration

gradient between matrix and cleats (Ziarani et al 2011) For Well B a sharp increase in

gas production occurred at around 710 days in the field production history which was

believed to be arisen by varying bottom hole conditions This is a common field practice

in operating CBM wells as documented in Young et al (1991) As indicated by Figure 5-

12 the modeled gas production rates well agree with field data for both Well A and Well

B for the entire 4000 days period There is less 10 error and the error was very likely

brought by an inexact determination of bottom hole condition But key characteristics in

the de-watering stage including peak gas rate and the corresponding peak production time

rate were accurately forecasted by the numerical model This indicated that initial gas and

water storage and their relative permeability curves were well approximated In the decline

stage the established numerical model was able to predict the concave up behavior of the

gas production curve This implied that permeability increased as the reservoir was

depleted The match to late time production data illustrated that the sorption kinetics were

accurately implemented in the numerical model where the amount of desorbed gas

diffused out to cleats was adequately evaluated In other words the equivalent matrix

permeability modeling can accurately dictate matrix flow during production through this

dual permeability modeling approach

121

Figure 5-12 History-matching of the field gas production data of two fairway wells (a)

Well A and (b)Well B (shown in Figure 5-4) by the numerical simulation constructed in

CMG

555 Sensitivity Analysis

As seen from Table 5-4 it can be observed that the permeability of cleats is much

greater than the equivalent matrix permeability converted from the diffusion coefficient

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For this reason matrix flow is historically neglected in the reservoir simulation assuming

that desorption and diffusion processes occur rapidly enough to ignore the sorption kinetics

process in the modeling of gas transport If reservoir simulation only considers the cleat

permeability growth mechanism and neglects the simultaneous change of matrix

flowability it generally yields an ultra-small initial porosity (lt005) at the best match

lower than the acceptable range of 005 to 05 for fairway wells (Palmer et al 2007)

This small porosity match suggests that there may exist an alternate mechanism on the

hyperbolic decline behavior In this work the observed pressure-dependent diffusion

coefficient was implemented in the reservoir simulation through the equivalent matrix

permeability modeling as a secondary mechanism on the conductivity increase during

pressure depletion As summarized in Table 5-4 the resulting initial cleat porosity had

values of 01 and 008 for the two target wells and these values were within the

acceptable range of 005 to 05 (Palmer et al 2007) The traditional purely cleat-flow

control production model must lower the porosity to compensate for the excessive outflow

due to the matrix gas influx This may lead to the erroneous analysis of the late gas

production behavior due to the lack of variation of matrix-to-cleat flows

Nevertheless one may still question whether an accurate characterization of matrix

flow is imperative to the simulation of CBM production This work would conduct

sensitivity analysis separately for the matrix permeability curve and the cleat permeability

curve and examine their effect on gas production for highly productive fairway wells with

mature depletion The impact of matrix permeability curves on gas production was

examined by conducting comparison simulation cases where either matrix permeability or

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cleat permeability was set as a constant and the rest of reservoir parameters were kept as

the same as the matching cases listed in Table 5-4 The intent was to isolate the smoothing

of the decline curve that arose by matrix permeability increase from cleat permeability

increase Figure 5-13 shows the simulated production curves with constant cleatmatrix

permeability and their comparison against field data A total number of 8 additional runs

were conducted to investigate the potential errors associated with the inaccurate modeling

of cleat or matrix flow Figure 5-13 (a) and (c) correspond to the simulation runs with

growing matrix permeability predicted by Figure 5-11 and constant cleat permeability for

Well A and Well B Figure 5-13 (b) and (d) show the simulation results of keeping matrix

permeability as an invariant whereas incorporating cleat permeability growth presented in

Figure 5-7 into the numerical models

Each scenario contained two cases of constant permeability that is one evaluated at

the initial condition and the other one valued at average reservoir pressure over the length

of simulation time As shown in Figure 5-13 (a) and (c) the simulated production curves

associated with constant kf evaluated at average pressure were almost not distinguishable

from the matched cases with dynamic fracture permeability and still provided satisfactory

matches to field data This implied that the average permeability over the entire production

history could practically provide reasonable gas production profiles which is the reason

why the constant permeability is commonly used for CBM simulation and the predict

production was found acceptable Besides even for the case with a constant and

underestimated cleat permeability evaluated at initial pressure it only triggered an

erroneous prediction of gas production in the de-watering stage and the discrepancy

124

diminished in the decline stage for highly permeable formations with promising production

potentials in San Juan basin

Early gas production was driven by the displacement of water that heavily

depended on cleat permeability Following the de-water stage pressure depletion was the

dominant production mechanism that relied on the gas desorptiondiffusion process to

supply flow in cleats and to the wellbore As a result cleat permeability had a limited effect

on gas declining behavior whereas accurate predictions of matrix flowability were

essential to long-term production prediction This was confirmed by simulation results

presented in Figure 5-13 (b) and (d) with constant matrix permeability and growing cleat

permeability assumed in the production process Although the stress-dependent and

sorption-controlled cleat permeability were precisely modeled they in general did not

provide good fits to field data except for the initial inclining rate period As explained

earlier the primary production mechanism in the decline stage would be gas

desorptiondiffusion as the majority of gas was stored in the matrix Due to this

phenomenon it could be expected that an increase in cleat permeability would have a

minimal effect on slowing down the depletion rate of gas production Instead the growth

of the matrix diffusion coefficient induced by evacuation of pore space and potential

change of pore shape was the key gas transport characteristic for production at the decline

stage

125

Figure 5-13 Effect of cleat and matrix permeability growth on gas production The solid

grey lines correspond to comparison simulation runs with constant matrixcleat

permeability evaluated at initial condition The grey dashed lines correspond to comparison

simulations runs with constant matrixcleat permeability estimated at average reservoir

pressure of the first 4000 days

It should also be noted that simulations with the same values of cleat permeability

and different matrix permeability would predict the peak production very differently This

was because matrix permeability would determine the amount of gas diffused to cleats

under a certain pressure drop Higher matrix permeability would allow a fast pressure

transient process and impose a steeper concentration gradient between the free space and

surface of the coal matrix Accordingly more gas would desorb and flow into cleats as

126

fracture water was running out The difference in simulated production curves became

smaller for longer production time and even disappeared when equilibrium sorption

condition was achieved and no more gas could be desorbed

When comparing the simulation results of cases with constant fracture permeability

and those with constant matrix permeability (eg Figure 5-13 (a) and (b)) accurate

modeling of matrix permeability growth is essential to the prediction of gas production in

decline stage for CBM wells in well-cleated fairway area For such wells gas can easily

transport through the cleat system but the gas desorptiondiffusion process controls its

supply Production projection for coal reservoirs with high cleat permeability is subject to

significant discrepancy without cognitive modeling of gas transport in the matrix

This modeling study demonstrates that the gas diffusion is a critical gas transport

process to control the overall gas production behavior both in the early time for determining

the peak production and the late time for the sustainable stable production tails The gas

diffusion mass transport has been theoretically and experimentally studied but

unfortunately it has been used neither for practical gas production forecasting nor for

reservoir sweet spot identification The reason why the dynamic diffusivity has been

historically ignored is due to no model framework has been set for diffusion-based matrix

flow in a commercial simulator This work fills this gap by using the equivalent matrix

permeability as a surrogate for the diffusion coefficient This method implicitly takes the

pressure-dependent gas parameters into the equivalent matrix permeability However we

want to point out that further studies will be required to establish an explicit multichemical

model and simulator which can directly account for multi-mechanism flow

127

56 Summary

This chapter investigated the impact of the pressure-dependent diffusion coefficient

on CBM production An equivalent matrix permeability modeling was proposed to convert

the measured diffusion coefficient into a form of Darcys permeability through the material

balance equation The equivalent matrix permeability and the dynamical cleat permeability

were integrated into reservoir simulation constructed in CMG-GEM simulator History-

match to field data were made for two mature San Juan fairway wells to validate the

proposed equivalent matrix modeling in gas production forecasting Based on this work

the following conclusions can be drawn

1) Gas flow in the matrix is driven by the concentration gradient whereas in the

fracture is driven by the pressure gradient The diffusion coefficient can be

converted to equivalent permeability as gas pressure and concentration are

interrelated by real gas law

2) The diffusion coefficient is pressure-dependent in nature and in general it

increases with pressure decreases since desorption gives more pore space for gas

transport Therefore matrix permeability converted from the diffusion coefficient

increases during reservoir depletion

3) The simulation study shows that accurate modeling of matrix flow is essential to

predict CBM production For fairway wells the growth of cleat permeability during

reservoir depletion only provides good matches to field production in the early de-

watering stage whereas the increase in matrix permeability is the key to predict the

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hyperbolic decline behavior in the long-term decline stage Even with the cleat

permeability increase the conventional constant matrix permeability simulation

cannot accurately predict the concave-up decline behavior presented in the field gas

production curves

4) This study suggests that better modeling of gas transport in the matrix during

reservoir depletion will have a significant impact on the ability to predict gas flow

during the primary and enhanced recovery production process especially for coal

reservoirs with high permeability This work provides a preliminary method of

coupling pressure-dependent diffusion coefficient into commercial CBM reservoir

simulators

5) The equivalent matrix permeability is a variable approach to implicitly take the

pressure-dependent parameters such as compressibility and viscosity into gas

production prediction This modeling results demonstrate that the diffusivity has

not only an impact on the late stable production behavior for mature wells but also

has a considerable effect on the peak production for the well In conclusion the

pressure-dependent gas diffusion coefficient should be considered for gas

production prediction without which both peak production and elongated

production tail cannot be modeled

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Chapter 6

PIONEERING APPLICATION TO CRYOGENIC FRACTURING

61 Introduction

As coal is highly compressive coal permeability depends on burial depth (Enever

et al 1999 Somerton et al 1975) In general coal permeability decreases with burial

depth that limits CBM production (Liu and Harpalani 2013b) The application of hydraulic

fracturing greatly enhances the permeability of the virgin coalbed However it comes with

the environmental concerns arising from heavy water usage and intractable formation

damage (King et al 2012) The other issues related to hydraulic fracturing is that it

exhibits poor performance on water-sensitive formations This is because capillary and

swelling forces leads to the water blocking around the induced fractures and restrict the

flow of hydrocarbon

Fracturing using cryogenic fluid is a remedy to this issue and the field study in

CBM and shale reservoirs proved its feasibility as a stimulation method (Grundmann et al

1998 McDaniel et al 1997) But this stimulation method is still at its scientific

investigation stage for combining factors such as low energy capacity or viscosity of

cryogenic fluids and the cost and difficulty in handling such fluids as well as the safety

concerns for the gas fracturing Theoretically the contact of the extremely cold fluid with

the warm reservoir rocks generates a severe thermal shock and opens up self-propping

fractures (Grundmann et al 1998) As the fluid heat up to reservoir temperature its volume

expansion in the liquid-gas phase transition immensely boosts the flow rate and gives the

130

potential of adequate transportation of light proppants The balance between expenditure

on the cryogenic fracturing itself and the resultant gas production is the key to promote the

industrial scale and commercial application of this waterless stimulation technique As

most gas is stored as the adsorbed phase in coal the reduction in the reservoir pressure

causes the incremental desorption determined by the sorption isotherm Both cleat and

matrix permeability are important factor controlling production performance of CBM

wells Specifically gas deliverability of coal matrix dominates long-term CBM production

as sufficient cleat openings are induced by the matrix shrinkage whereas cleat permeability

dominates short-term production (Clarkson et al 2010 Liu and Harpalani 2013b Wang

and Liu 2016) Therefore the evaluation of the effectiveness of cryogenic fracturing

should conduct at a broad scale from visible cracks to micropores

The goal of this study is to investigate the critical theoretical background of

cryogenic fracturing We give an outline of the interaction forces between reservoir rock

and cold injected fluid where heat transfer and frost-shattering effect are two critical

fracturing mechanisms However the development of cryogenic fracturing is still at its

infancy and the best approach for fracturing is not yet available As coal incorporates a

dual-porosity structure this work will present a comprehensive analysis of accessing the

effectiveness of cryogenic fracturing on coal at pore-scale and fracture-scale

62 Mechanism of Cryogenic Fracturing

Figure 6-1 presents a graphical illustration of various fracturing mechanisms

associated with cryogenic fluid injections at macro- and microscale When liquid nitrogen

131

(LN2) is introduced into the reservoir a severe thermal shock is generated by the rapid heat

transfer from reservoir rock to the cool injected fluid with a normal boiling point of

minus196 (McDaniel et al 1997) The surface of the rock matrix in contact with the

cryogenic fluid shrinks and it pulls inward upon the surrounding warm rock This

contraction induces tensile stress around the cooled rock ie thermoelastic stress and

eventually causes the rock fracture surface to fail and induce microcracks within the rock

matrix (Clifford et al 1991 Detienne et al 1998 Perkins and Gonzalez 1985)

Meanwhile the volume expansion ratio of LN2 upon vaporization is 1 694 (Linstrom and

Mallard 2001) The vaporized gas within a confined space imposes a high localized

pressure and serves as a penetration fluid for the fracture propagation (Perkins and

Gonzalez 1984)

An alternative fracturing mechanism is frost shattering by freezing of formation

water in fractures and pore spaces (French 2017) At micro-scale or pore-scale not all the

pore space in coal is accessible to water due to capillary effect (Dabbous et al 1976) For

water-wet pores water can intrude into pore space even at low pressure and frost shattering

becomes prominent A ~9 volumetric expansion is related to the water-ice phase

transition which produces high stress within the confined space and disrupts the rock

(Chen et al 2004) The presence of dissolved chemicals in micropores reduces the freezing

point of pore water which may be lower than 0 The hydraulic pressure associated with

the movement of the unfrozen water due to capillary and adsorptive suction causes

additional damage to the reservoir rock (Everett 1961) Numerous literature indicates that

132

volumetric expansion of freezing water and water migration are the leading causes of frost

shattering (Fukuda 1974 Matsuoka 1990)

Figure 6-1 Mechanism of cryogenic fracturing Damage mechanism A derives from the

volume expansion of LN2 Damage mechanism B is the thermal contraction applied by

sharp heat shock Damage mechanism C is stimulated by the frost-heaving pressure

63 Research Background

631 Cleat-Scale

To study the initiation and growth of fracture previous laboratory works (Cha et

al 2017 Cha et al 2014 Qin et al 2018a YuShu Wu 2013) focused on the rock thermal

133

fracturing mechanism of cryogenic fracturing Fractures were generated in the rock sample

in response to the thermal shock The Leidenfrost effect might restrict the heat transfer

process but efficient insulation and delivery of the cryogenic fluid would substantially

eliminate this effect Other experimental works studied the frost shattering mechanism of

cryogenic fracturing (Cai et al 2014a Cai et al 2014b Qin et al 2017a Qin et al 2018b

Qin et al 2016 Qin et al 2018c Qin et al 2017b Zhai et al 2016) The moisture content

intensified the frost action and aggravated the breakdown of coal For moderately saturated

coal samples moisture present in the open space promoted the damage process of

cryogenic fracturing where the degree of damage depended on water content

632 Pore-Scale

The pore structural evolution is a merit of cryogenic fracturing that alters the

sorption and diffusion behaviors of the coal matrix Previous study (Cai et al 2014a Cai

et al 2014b Qin et al 2018c Xu et al 2017 Zhai et al 2016 Zhai et al 2017) showed

that cryogenic fracturing enhanced the microporosity along with a variation in the pore size

distribution (PSD) based on nuclear magnetic resonance (NMR) method Based on the

NMR results inconsistent observations were reported on micropore damage stimulated by

cryogenic fracturing Cai et al (2016) indicated that the cooling effect increased the

micropore volume whereas Zhai et al (2016) Zhai et al (2017) found that cryogenic

treatment reduced the proportion of micropores The micropore deterioration measured by

NMR was subject to great uncertainty as this testing method is not suitable for very fine

pores (AlGhamdi et al 2013 Strange et al 1996)

134

To date the induced deterioration on pore structure was not fully understood

especially for micropores The investigation of induced pore structural variation requires

an alternative characterization method that can obtain insight into the microstructure of

coal Among various characterization methods (eg small-angle scattering SEM TEM

and mercury porosimetry) physical adsorption is the most employed technique for

characterization of porous solids (Gregg et al 1967 Lowell and Shields 1991 Okolo et

al 2015) yielding information about pore size distribution and surface characteristics of

the materials In this study the porous texture analysis of coal samples was carried out by

N2 adsorption at 77 K and CO2 adsorption at 273 K for the assessment of the pore structure

(Lozano-Castelloacute et al 2004 Solano et al 1998) In contrast to the well-accepted N2 at

77 K the higher adsorption temperature of CO2 yields larger kinetic energy of the

adsorptive molecules allowing to enter into the narrow pores (Garrido et al 1987 Lozano-

Castelloacute et al 2004) Owing to the inhomogeneities and polydispersity of the microporous

structure of coal CO2 adsorption serves as a complement to N2 adsorption that provides

micropore volume and its distribution of coal samples with narrow micropores (Clarkson

et al 2012b Dubinin and Plavnik 1968 Dubinin et al 1964 Garrido et al 1987)

64 Experimental and Analytical Study on Pore Structural Evolution

This section presents an experimental study on pore structural evolution stimulated

by cryogenic fracturing through gas adsorption measurements at low and high pressures

A micromechanical model is then developed based on stress analysis to determine the

induced pore structural deterioration by cyclic cryogenic fluid injections Although

135

cryogenic treatment has been shown to cause the degradation of mechanical properties of

coal its effect on small pores in terms of size shape and alignment has not been

investigated In this study a pulverized coal sample was processed and used with cryogenic

treatments The reason for using coal particles was to eliminate the pre-existing fracturing

network to exclude the pressure-driven Darcy and viscous flow and to secure the

dominance of diffusion flow in the gas transport of coal (Pillalamarry et al 2011) After

freezing and thawing subsequent experiments were conducted to analyze the deterioration

of pore structure Specifically the low-pressure physical adsorption analysis studied the

pore characteristics of raw and freeze-thawed coal samples The high-pressure sorption

experiment measured the sorption and diffusion behavior of the raw and LN2 treated coal

samples The experimental results were then presented with an emphasis on the change in

pore structural characteristics after cryogenic treatment and their corresponding alterations

on gas flow in the matrix Early research conducted by McDaniel et al (1997)

demonstrated that repeated contact with LN2 causes coal samples to break into smaller

units continuously Additionally numerous studies in other fields (Ding et al 2015

Kueltzo et al 2008 Stauffer and Peppast 1992 Watase and Nishinari 1988) demonstrate

that cyclic freeze-thaw treatment results in additional damage to the structure of polymers

and their porous nature is akin to the reservoir rock used in the present study Instead of a

single freezing treatment of LN2 the effectiveness of cyclic cryogenic fracturing was

studied

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641 Coal Information

Fresh coal blocks were acquired from Herrin coal seam in the Illinois Basin

Specifically the coal found in the middle and upper lower of the strata has the potential for

gas production (Treworgy et al 2000) The commercial CBM production is still at an early

stage in the Illinois Basin Fall-off tests (Tedesco 2003) indicate that the permeability of

the higher gas content area ranges from micro darcy to less than 10 millidarcys and thus

commercial CBM production needs to be aided by some stimulation methods such as

hydraulic fracturing As the dewatering of CBM wells generates large volumes of

formation water the wastewater discharge requirements impose significant burdens on the

economic viability of CBM in the Illinois basin (EPA 2013) Illinois State Geological

Survey (ISGS) (Morse and Demir 2007) reported the production history of several CBM

wells drilled in Herrin coal seam where gas pressure was maintained in a small but steady

value whereas water was produced in a high volume The steady flow of water

demonstrates that Herrin coal seam has good permeability and the bottleneck of the current

CBM production is the extraction and delivery of the sorbed gas It is quite challenging to

increase the gas desorption kinetics and gas diffusion because it requires the micropore

dilation which cannot be achieved through traditional reservoir stimulation Instead

cryogenic fracturing has potential to inflate the micropores which will increase the

diffusivity of coal as illustrated in Figure 6-1

The freshly collected coal sample was pulverized to 60-80 mesh Although

pulverizing the coal may modify the pore structure this modification is negligible for coal

137

particles down to a size of 0074 mm (Jin et al 2016) Besides the increase in surface

area for adsorption is only about 01 to 03 area for coal particles between 40 to 100

mesh (Jones et al 1988 Pillalamarry et al 2011) The crushed Herrin coal sample was

then examined by the proximate analysis following ASTM D3302-07a (Standard Test

Method for Total Moisture in Coal 2017) The Herrin coal is a high-volatile bituminous

coal with a moisture content of 362 ash content of 858 volatile matter of 3703

and the fixed carbon content is 2077 The pulverized coal samples were processed with

cyclic freeze-thawing treatments to study the effect of cryogenic fracturing on pore

structure

642 Experimental Procedures

A comprehensive experimental system (Figure 6-2) is designed to investigate the

effectiveness of cyclic cryogenic fracturing in terms of the deterioration of pore structure

and the change in gas sorption kinetics The experimental platform consists of three main

parts as freeze-thawing (F-T) system gas addesorption isotherm and kinetic

measurements pore structural characterization The F-T system is composed of a vacuum

insulated thermal bottle with double-wall stainless steel interior and exterior for freezing

and a glassware beaker for thawing The double-layer insulator provides enough

temperature retention time for freezing and strength for the endurance of the F-T forces

The gas addesorption isotherm and kinetic measurements were obtained using a high-

pressure sorption experimental apparatus presented in Chpater 3 This apparatus allows

measuring gas sorption up to 3000 psi which can simulate gas sorption addesorption

138

behavior of coal at both saturated and undersaturated conditions Besides the data

acquisition system employed in this experimental sorption system continuously delivers

the pressure readings to user-interface with a rate of up to 1000 data points per second

This allows for accurate measurements of gas sorption kinetics and diffusion coefficient

In the determination of pore characteristics physical sorption of N2 at 77 K and CO2 at 273

K were conducted with an ASAP 2020 physisorption analyzer (Micromeritics USA)

following the testing procedure documented in the ISO (2016)

The prepared coal sample was evenly divided into two groups One is the reference

group as the raw coal sample and the other is the experimental group that would undergo

a series of freeze-thawing cycles In order to include the water-ice expansion force in the

freezing process the experimental group was first saturated with water by fully immersing

the sample in the distilled water Once an apparent boundary forms between the clear water

and coal particles the water-saturated sample was made by filtering out from the

suspension and air-drying and then subject to F-T cycles Figure 6-3 displays the

experimental images captured at different times during the freezing and thawing

operations The coal sample was frozen in the thermal bottle filled with LN2 for 60 mins

(see Figure 6-3(a)) where the fluid level of LN2 kept almost the same for the entire one-

hour freezing This was desired since heat transfer mostly occurred between LN2 and the

coal sample rather than the atmosphere otherwise LN2 would vanish soon to cool the

surrounding air The frost started to form around 10 mins indicating the production of the

frost-shattering forces Followed by the freezing operation the coal sample was thawed at

room temperature of 25 The thawing operation lasted about 240 mins until a thermal

139

equilibrium was reached as shown in Figure 6-3(b) For multiple F-T cycles the same

freeze-thawing procedures would be repeated and a portion of the coal sample was

retrieved after one and three cycles (1F-T and 3F-T coal)

The freeze-thawed and raw coal sample were dried in the vacuum drying oven at

minus01 MPa and 60 degC for subsequent measurements on pore structure and gas sorption

behavior The coal samples subject to the different number of F-T cycles were used to study

the effectiveness of cyclic cryogenic treatments on the pore structural deterioration and

modification of gas sorption kinetics

140

Figure 6-2 The experimental system (a) is a freeze-thawing system where the coal sample

is first water saturated in the glassware beaker and then subject to cyclic liquid nitrogen

injection In between the successive injections the sample is thawed at room temperature

The freeze-thawed coal samples and the raw sample are sent to the subsequent

measurements ((b) and (c)) (b) is the experimental setup for measuring the gas sorption

kinetics This part of the experiment is to evaluate the change in gas sorption and diffusion

behavior of coal after cryogenic treatment (c) is the low-pressure adsorption system for

the determination of surface area and porosimetry of pore structure of the coal sample This

step is to evaluate the pore-scale damage caused by the cryogenic treatment to the coal

sample

141

Figure 6-3 The process diagram of freeze-thawing treatment (a) freezing operation (b)

thawing operation

0 minDumping

Freeze

1 min 10 min

30 min 20 min

Freeze Freeze

Freeze

FreezeFreeze

40 min

Freeze

50 min

Freeze

Freeze

Finish Freeze-Start Thaw

(a)

60 min

1 minThaw at room temperature

Thaw

10 min 20 min

40 min 30 min

Thaw

Thaw

ThawThaw

50 min

Thaw

60 min

Thaw

240 min

Finish 1 F-T cycle

Thaw

(b)

142

643 Micromechanical Analysis

The effects of freeze-thaw on the pore structure of coal have been extensively

studied in laboratories as presented in this work and various studies (Cai et al 2014a Xu

et al 2017 Zhai et al 2016) However a mechanistic model of the involved multi-physics

is sparely discussed in the literature A rational evaluation of pore structural deterioration

is essential in predicting the induced change in gas sorption and transport properties in

CBM reservoirs by cyclic liquid nitrogen injections Hori and Morihiro (1998) proposed a

micromechanical model to study the mechanical degradation of concrete at very low

temperatures and their analysis was employed by this work to estimate the damage degree

of the nanopore system of coal in response to the repetition of freezing and thawing In

their model a nanopore with a radius of ao is modeled as a microcrack with half crack

length of ao ao becomes an after nth cycle of freezing and thawing ie an = an(ao) Figure

6-4 is a graphical illustration of a deteriorating nanopore of coal where the fractured pore

is represented by a growing microcrack The growth of cracks can be solved with fracture

mechanics For simplicity we neglect the interaction among different pores and the

solution is obtained by treating each pore as an isolated crack in an infinite medium The

extremely low-temperature environment created by liquid nitrogen gives rise to a rapid

cooling rate and yields a sudden thermal shock to the coal matrix Water contained in the

nanopores expands as the temperature of the coal matrix is lowered to sufficiently cold

temperature This volume expansion induces local tensile stress and causes damage to the

143

pores which are depicted in Figure 6-4 as a pair of concentrated forces acting on the crack

center

Figure 6-4 Hori and Morihirorsquos model of fractured micropore (Hori and Morihiro 1998)

The nanopore system of coal is modeled as a micro cracked solid The pair of concentrated

forces normally acting on the crack center represents the crack opening forces produced by

the freezing action of pore water

We first develop a mechanistic model for determining the deterioration degree due

to the freezing of water and then couple it with heat conduction analysis Under the

application of a pair of concentrated forces the crack opening displacement ([119906(119909)]) is

given by (Sneddon 1946)

[119906(119909)] =

4(1 minus 1205842)

120587119864119875119908 (ln |

119886

119909| + radic120587(1 minus (119909119886)2))

( 6-1 )

where 120584 and 119864 are the elastic moduli of the coal matrix 119875119908 is the magnitude of crack

opening forces ie the frost pressure induced by the freezing of water 119886(1198860) is the half

crack length of a crack with an initial crack length of 1198860 before 119899th freeze-thawing cycles

ie 119886(1198860) = 119886119899minus1(1198860)

The crack opening displacement ([119906(119886)] ) of a single microcrack with half crack

length of 119886 can be found as

144

[119906(119886)] = int [119906(119909)]

119886

minus119886

119889119909 =2radic120587(1 minus 1205842)

119864119875119908119886

( 6-2 )

The overall crack strain ( 휀119888 ) for a collection of cracks in different sizes is

determined by (Hori and Morihiro 1998 Nemat-Nasser and Hori 2013)

휀119888 = int

[119906(119886)]

119886119889120588(1198860)

120588(119886119898119886119909)

120588(119886119898119894119899)

=2radic120587(1 minus 1205842)

119864int 119875119908119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-3 )

where 120588(1198860) is the crack density function In this work it is set as porosity and can be

extrapolated from pore size distribution measured from low pressure gas sorption

The deterioration degree is characterized by the magnitude of 휀119888 which is

dependent upon the evaluation of 119875119908 119875119908 increases as pore water are being frozen and some

portion of it remains after thawing The residual strain due to the generation of residual

stress characterizes the constant expansion of pore volume after freezing and thawing and

its magnitude corresponds to the deterioration degree of pore structure This residual stress

is crack opening forces acting at the crack center as shown in Figure 16 and its magnitude

is 119875119908 Hori and Morihiro (1998) showed that 119875119908 is proportional to the maximum pressure

for the freezing of water (119875119888)

Thus

119875119908 = 119860(119879 119886)120573119898119875119888 ( 6-4 )

where 119860 is the frozen water content in a micropore with a radius of 119886 at temperature 119879 120573119898

is the fraction of stress retained after completely thawing of the coal matrix and the removal

of 119875119888 The magnitude of 120573119898 depends on the material heterogeneity that different parts

undergo different deformations (Beer et al 2014)

145

Although the deterioration only proceeds when the water content exceeds 90

(Rostasy et al 1979) we assume 100 saturation for simplicity For this reason the

maximum pressure due to the freezing of pore water (119875119888 ) can be approximated by the

strength of a nanopore with a radius of 119886 Nielsen (1998) showed that for a porous material

the pore strength exhibited an inverse relationship with the pore size which took a form of

119875119888 = 119870119888radic1119886 ( 6-5 )

where 119870119888 is the fracture toughness of the material or the coal matrix

With Eq (6-3) ndash Eq (6-5) the internal pressure of nanopore as well as the crack

strain induced by the freezing of water (119875119908) can be determined

휀119888 = 2radic120587119860(119879 119886)120573119898

(1 minus 1205842)119870119888119864

int radic1119886119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-6 )

The deterioration analysis will be coupled with the heat conduction analysis As

with the crack strain only a portion of the thermal strain remains after thawing The

residual thermal strain is proportional to the temperature gradient and 120573119898 as

휀119905 = 120573119898120572119871 119879 ( 6-7 )

where 120572119871 is the linear coefficient of thermal expansion Due to a drop in temperature 휀119905 is

a negative value

The overall nanopore dilation (휀) due to the repetition of freezing and thawing is a

sum of thermal strain and crack strain in response to the freezing of pore water and it

reflects the deterioration degree and the effectiveness of cyclic liquid nitrogen injections

휀 = 휀119905 + 휀119888 ( 6-8 )

146

Practically volumetric strain (휀119907) may be more useful For spherical pores 휀119907can

be approximated as 43120587휀3 The magnitude of 휀 characterizes the deterioration degree of

pore structure induced by cyclic liquid nitrogen injections

65 Freeze-thawing Damage to Nanoporous Network of Coal Matrix

651 Gas Kinetics

With the high-pressure sorption experimental setup the addesorption isotherm was

constructed at the equilibrium condition when the pressure reading was stabilized At each

pressure stage the diffusion coefficient was evaluated from the equilibrating process of

pressure Langmuirrsquos equation and Fickrsquos law were applied to model the gas sorption and

diffusion behavior of the raw 1F-T 3F-T coal samples

Figure 6-5 is the adsorption and desorption isothermal analyses of raw 1F-T and

3F-T coal samples The hysteresis loop was more apparent in the raw sample than those

freeze-thawed samples suggesting the pore connectivity improved after freeze-thaw cycles

The adsorption capacity increased after the cyclic cryogenic operations After the first

freeze-thawing cycle further cycles did not impose additional changes to the sorption

behavior that could be seen from the overlapping of addesorption isotherms of 1F-T and

3F-T samples The fitted Langmuir curves are also shown in Figure 6-5 and the numerical

values of Langmuir parameters (ie 119881119871 and 119875119871) are summarized in Table 1 119881119871 is the total

adsorption sites depending on the accessible surface area and the heterogeneity of the pore

structure (Avnir and Jaroniec 1989) 119875119871 defines the curvature of the isotherm reflecting

147

the overall energy level of the adsorption system The results presented in Table 6-1

demonstrates that the cyclic cryogenic operation alternates both the ultimate adsorption

capacity and the adsorption potential The Langmuir volume was increased by 1515 and

Langmuir pressure experienced an increase of 2315 In the freeze-thawing treatment

the increase in 119881119871 implied an increase in the total available adsorption sites which could

be caused by the increase in accessible surface area as well as the heterogeneity of pore

system The associated forces in cryogenic treatment may cause some larger pores to

collapse into smaller pores creating more surface area Besides these forces may enhance

the overall pore accessibility by turning the isolated pores into accessible pores A rougher

surface may occur after the freeze-thawing treatment and the pore surface can adsorb more

gas molecules which is also a potential mechanism for the increase in 119881119871

In terms of 119875119871 its change reflects a change in adsorption potential Figure 6-6

demonstrates the role of 119875119871 acting on the adsorption and desorption processes When

subject to the same change in pressure ( 119875119886119889119904 or 119875119889119890119904) the adsorbent with an isotherm of

greater 119875119871 holds less gas in the adsorption process or smaller 119881119886119889119904 while it produces more

gas in the desorption process or larger 119881119889119890119904 The isotherm approaches a linear relationship

with a larger value of 119875119871 The ideal isotherm for CBM production is a linear isotherm

following Henryrsquos law that incorporates the fastest desorption rate For CBM production

an isotherm with a larger value of 119875119871 is preferred Table 6-1 shows that 119875119871 increases when

subject to more freeze-thawing cycles implying an increase in gas desorption rate with the

same pressure drop 119875119871 is defined to be a ratio of desorption rate constant to adsorption rate

constant dependent on the energy level of the system As defined in Langmuir (1918)

148

adsorption rate constant has a unit of 1MPa and desorption rate constant is dimensionless

Stronger adsorption force as well as higher adoption potential occurs at a rough pore

surface than a smooth pore surface So surface complexity directly affects the energy level

of adsorption field and the value of 119875119871 where the isotherm of a coal sample with a

convoluted pore structure typically incorporates a small 119875119871 The increase in 119875119871 induced by

freeze-thawing treatment was interpreted as a result of pore structural evolution When

imposing a low-temperature environment to the coal sample a drastic temperature gradient

was created between the warm sample and the surrounding and pore water was evolved

into ice There were two forces acting on the pore wall which were the thermoelastic forces

associated with the stimulated thermal shock and the expansion forces of pore water

associated with the phase transition into ice Pore shape and size would be affected once

these two forces exceeded the strength of coal pore Besides these two forces may

potentially eliminate surface irregularity Apparently the cryogenic treatment

homogenizes the convoluted structure of coal which explains the increase in 119875119871

149

0 2 4 6 8 10

0

5

10

15

Ad

so

rption

Cap

acity (

mlg

)

Equilibrium Pressure (MPa)

CH4 ad-desorption excess data of raw coal

Langmuir Isotherm for CH4 adsorption

CH4 ad-desorption excess data of 1F-T coal





isotherm curves for raw 1F-T and 3F-T coal

Table 6-1 Langmuirrsquos parameters for raw 1F-T and 3F-T coal The bracket indicates the

percentage increase in PL of 1F-T and 3F-T coal with respect to PL of raw coal An increase

in PL is preferred in gas production as it promotes the gas desorption process

Coal

Sample

119881119871 ml g

119875119871 MPa

R2

Raw 1446 091 0998 5

1F-T 1643 099 (79) 0998 5

3F-T 1665 112 (232) 0997 9

150

Figure 6-6 The role of PL acting on the adsorption and desorption process

Once the gas is desorbed from the surface of the coal matrix it is the gas diffusion

process that diffuses out the desorbed gas The gas diffusion coefficient was obtained from

the measurement of sorption kinetics where unipore model (Fick 1855 Nandi and Walker

1975 Shi and Durucan 2003b) was applied Figure 6-7 presents the results of the measured

diffusion coefficient of raw 1F-T and 3F-T coal samples at different pressure stages At

all pressure stages the freeze-thawed coal (1F-T and 3F-T coal) had higher diffusion

coefficients than the raw coal in both the adsorption and desorption process The measured

diffusion coefficients are listed in Table 6-2 Relative to the diffusivity of raw coal the

151

diffusion coefficients of 1F-T coal and 3F-T coal were improved on average by 1876

and 939 respectively in the adsorption process and by 3018 and 1496 respectively

in the desorption process This indicates that cryogenic treatment enhances the gas

diffusion in the coal matrix Overall the increase in the diffusion coefficients was more

apparent at lower pressure stages as indicated in Table 6-2 After the first cryogenic

treatment more cycles of freeze-thawing operation exerted a negative impact on the gas

diffusion rate as the 3F-T coal consistently had lower diffusion coefficients than the 1F-T

coal Cyclic cryogenic fracturing appears not to benefit the diffusion process in the coal

matrix compared with a single injection of LN2

Figure 6-7 Measured CH4 diffusion coefficients for raw coal 1F-T coal and 3F-T coal at

different pressure stages

0 2 4 6 8 10

2

4

6

8

ad-desorption diffusivity of raw coal

ad-desorption diffusivity of 1F-T coal


Diffu

sio

n C

oeff

icie

nt

(1e-1

3 m

2s

)


Improve by

1876

Improve by

3018

152

Table 6-2 Measured diffusion coefficients of raw coal 1F-T coal and 3F-T coal (Draw

D1F-T D3F-T) in the adsorption process and desorption process and the corresponding

increase in the diffusion coefficient due to freeze-thawing cycles (ΔD1F-T ΔD3F-T)

P DRaw D1FminusT D1FminusT D3FminusT D3FminusT

[MPa] [1e-13

m2s]

[1e-13

m2s] [1e-13

m2s]

Adsorption 049 157 186 1832 174 1056 103 189 240 2659 219 1550 209 269 326 2111 296 986 352 316 374 1859 344 895 559 377 462 2251 408 816 842 535 564 544 553 333

Desorption 052 189 258 3680 218 1562 106 243 321 3226 290 1919 205 310 414 3363 353 1386 338 357 475 3313 433 2114 535 563 648 1511 591 501

For all coal samples the diffusion coefficient showed an increasing trend with

pressure Gas diffusion in coal matrix can occur in either pore volume andor along pore

surface Fick and Knudsen diffusion are generally considered in diffusion in pore volume

or gas phase (Mason and Malinauskas 1983 Welty et al 2014 Zheng et al 2012)

whereas surface diffusion is considered in adsorbed phase behaving like a liquid (Collins

1991) It is well known that a major fraction of porosity of coal resides in micropores (less

than 2 nm in diameter) and indeed in ultra-micropores (less than 08 nm in diameter)

(Walker 1981) Considering micropore filling mechanism the gas molecules within

micropores cannot escape from the force field of the surface and the movement of

adsorbed molecules along the pore surface contributes significantly to the entire mass

transport (Krishna and Wesselingh 1997) Surface diffusion then became the dominant

153

diffusion mechanism in the overall gas transport in coal matrix and the diffusion coefficient

increases with surface coverage and gas pressure (Okazaki et al 1981 Ross and Good

1956 Sladek et al 1974 Tamon et al 1981) This transport requires the gas molecules to

surmount a substantial energy barrier that is diffusional activation energy and therefore

is an activated process (Gilliland et al 1974 Sladek et al 1974) Figure 6-8 demonstrates

the effect of surface heterogeneity on gas transport along the pore surface The higher the

extent of surface heterogeneity of coal the more energy is needed to initiate the movement

of the adsorbed molecules and the lower is the surface diffusivity at a given coverage

(Kapoor and Yang 1989) In response to the cryogenic environment coal matrix surfaces

could be modified and the surfaces became smooth Figure 6-8(a) and (b) illustrate the

potential modification trend of surface morphology occurred between the raw and 1F-T

coal sample The pore wall surface was modified toward the smoother direction and the

transport of gas molecules became relatively easier after the first freeze-thawing cycle

This explains why 1F-T coal sample had higher diffusion coefficients than the raw sample

In the subsequent freeze-thawing cycles coal matrix continued to have thermal shock and

water phase change forces which may increase the surface roughness because of the

inhomogeneous nature of the coal structure as illustrated from Figure 6-8(b) to (c)

Consequently surface diffusion capacity was suppressed as the surface became more

complex which illustrates the reduction in the diffusion coefficient of the 3F-T coal

sample For the same reason the diffusion coefficient measured from the desorption rate

was consistently higher than from the adsorption rate as the already built-up of multilayer

of adsorbed molecules in the desorption process smoothened the heterogeneous pore

154

surface of the coal sample as shown in Figure 6-9 Clearly the effect of surface

heterogenicity was hidden by the formulation of layers of adsorbed molecules and it

became negligible at the saturated condition or high-pressure stage So the improvement

of the diffusion coefficient was more apparent at lower pressure stages as shown in Figure

6-7


surface diffusion along a rough surface (b) describes surface diffusion along a flat surface

Less energy is required to initiate surface diffusion along a flat surface than a rough surface

Figure 6-9 The role of multilayer of adsorption on surface diffusion In desorption the

already built-up multiple layers of adsorbed molecules smoothened the rough pore surface

Greater surface diffusion happens in the desorption process than the adsorption process

By examining gas sorption and diffusion behaviors of freeze-thawed and raw coals

a single freeze-thawing treatment appears to be more effective than multiple freeze-

thawing treatments in terms of diffusion coefficient enhancement Besides the sorption rate

(a) rough surface (b) flat surface (c) rough surfacesurface diffusion

gas molecules

surface diffusion in adsorption

rough pore surface multilayer of adsorbed molecules smoothened out rough pore surface

surface diffusion in desorption

155

testing direct measurements of pore structural characteristics would provide an intrinsic

view on the change of coal matrix in micro-scale induced by cryogenic fracturing

652 Pore Structure Characteristics

The nitrogen adsorption isotherms of the raw 1F-T and 3F-T coal samples are

shown in Figure 6-10 The two freeze-thawed coal samples had greater adsorption amount

than the raw coal sample The sorption amounts were almost the same for 1F-T and 3F-T

treated coal samples The adsorption branch of the studied three coal samples were all in

sigmoid shape and categorized as Type II isotherm where the adsorption curve increases

asymptotically at the saturation pressure at 119875119875119900 asymp 1 At low relative pressure due to the

presence of micropores and fine mesopores within the samples micropore filling

mechanism is responsible for the plateau of the adsorbed amount At high relative pressure

capillary condensation occurring in the large mesopores and macropores leads to the rapid

rise in adsorption volume at the saturation pressure The amount of gas adsorbed at

different pressure stages correlates with multi-scale pore characteristics The enlargement

of the accessible surface area and the expansion of the pore volume are the two dominant

mechanisms that increase the adsorption capacity The change in surface area was

examined through the widely accepted BrunauerndashEmmettndashTeller (BET) method (Brunauer

et al 1938b) Empirical and theoretical work (Brunauer and Emmett 1937 Brunauer et

al 1938b Emmett and Brunauer 1937) indicated that the turning point from monolayer

adsorption to multilayer adsorption appeared at the beginning of the middle the nearly

linear portion of the isotherm at which the BET monolayer capacity (119899119898) was directly

156

related to the specific surface area (119886119861119864119879) The determined 119886119861119864119879 of the studied coal sample

was increased by 475 after the 1st F-T cycle and 505 after the 3rd F-T cycle which is

summarized in Table 6-3 Great stress can be induced by the cryogenic treatment because

of water-to-ice phase volumetric expansion coupled with the thermal shock across the coal

samples As this value exceeded the tensile strength of some pore walls large pores would

collapse into smaller pores and isolated pores would be connected which explains the

enlargement of accessible surface area for adsorption

Figure 6-10 Low-pressure N2 adsorption isotherm at 77K for the raw 1F-T and 3F-T coal

samples

00 02 04 06 08 10

000

005

010

015

020 Raw Coal

1F-T Coal

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Type B hysteresis loop

slit shaped pores

157

Table 6-3 BET surface area parameters of GAB adsorption model and quadratic GAB

desorption model of nitrogen experimental sorption data with their corresponding

correlation coefficients (R2) the areas under the best adsorption and desorption fitting

curves (Aad Ade) and the respective hysteresis index of raw coal 1F-T coal and 3F-T coal

samples

For all coal samples the desorption isotherms lagged the adsorption isotherms

suggesting the occurrence of irreversible adsorption process as shown in Figure 6-10 The

steep increase of the adsorption branch at saturation pressure associated with the steep

decrease of the desorption branch at intermediate pressures implied that the analyzed coal

samples had Type B hysteresis loops according to De Boer (1958) classification The lower

closure point of hysteresis loop for nitrogen adsorption at 77K typically occurs at 1198751198750 =

042 (Sing 1985) as a property of adsorbate and is independent of the nature of adsorbent

The studied three coal samples all exhibited well-defined hysteresis loops at the same

relative pressure of 047 which fell in the multilayercapillary condensation range rather

than the normal monolayer range Thus the occurrence of adsorption hysteresis is

predominantly associated with capillary condensation One critical aspect of this

adsorption mechanism in large assemblies of pores is all pores always have direct access

to vapor (Gregg et al 1967) The profile of adsorption branch primarily depends on the

density function of all pore radius or simply pore size whereas the shape of desorption

158

branch depends on both pore size and connectivity as not all pores are in contact with vapor

(Mason 1982) The desorption process starts with a stage that the pore space is full of

capillary condensed liquid As the relative pressure progressively reduces the outer surface

of pores in contact with vapor may be empty The partially emptied pores may not have

sufficient connectivity with the pores that have fully vacated to provide the general access

of the cavities to the vapor If the relative pressure is further dropped below the

characteristic percolation threshold a continuous group of pores is open to the surface that

causes the percolation effect and produces a steep ldquokneerdquo in the desorption isotherm as

presented in Figure 6-10 The connectivity of pore network is greatly affected by the pore

throat size where the steep slope of desorption branch is typically associated with the ink-

bottle-type pore (Ball and Evans 1989 Cole and Saam 1974 De Boer 1958 Evans 1990

Neimark et al 2000 Ravikovitch et al 1995 Thommes et al 2006 Vishnyakov and

Neimark 2003) Therefore the quantification of the hysteresis effect is important to

evaluate the overall pore connectivity which explains the variation in methane diffusion

coefficient given in Figure 6-7

Hysteresis index (HI) is a common parameter defined to quantify the extent of

hysteresis Several expressions of HI have been proposed based on the difference between

adsorption and desorption isotherms which can be evaluated through various aspects

including Freundlich exponent (Baskaran and Kennedy 1999 Ding et al 2002 Ding and

Rice 2011 Hong et al 2009) equilibrium concentration (Bhandari and Xu 2001 Ma et

al 1993 Ran et al 2004) slope of the isothermal curves (Braida et al 2003 Wu and

Sun 2010) and area under the isotherms (Wang et al 2014 Zhang and Liu 2017 Zhu

159

and Selim 2000) Referring to Wang et al (2014) this study utilized the area ratio to

evaluate the degree of hysteresis over the entire pressure range and developed a new

expression of HI specifically for nitrogen sorption isotherms The hysteresis index (HI)

determined from the areas under the isothermal curves is expressed as (Zhu and Selim

2000)

119867119868 =

119860119889119890 minus 119860119886119889119860119886119889

( 6-9 )

where 119860119886119889 and 119860119889119890 are the areas under the adsorption and desorption isothermal curves

respectively

The determination of these areas (ie 119860119889119890 119860119886119889) requires an accurate analytical

model to fit the nitrogen experimental sorption isotherm The two-parameter BET model

(Brunauer et al 1938b) has been extensively applied to model Type II isotherms however

it fails to predict the sorption behavior for relative pressures higher than 050 (Pickett

1945) (see Figure 6-11) The discrepancy of BET model in the multilayer region sources

from the assumption that infinite liquid layers are adsorbed at saturation pressure where

liquid and adsorbed layers are indistinguishable (Brunauer et al 1969) In fact only

several layers of adsorbed molecules can build up at saturation pressure limited by the

available capillary spaces (Pickett 1945) The three-parameter Guggenheim-Anderson-

DeBoer equation (GAB model) (Anderson 1946 Boer 1953 Pickett 1945) was then

modified from the BET equation that includes a third parameter 119896 to separate the heat of

adsorption in excess of the first layer from the heat of liquification As shown in Figure 6-

160

11 the GAB equation is successful in modeling the experimental adsorption data over a

whole range of vapor pressures which is written as

119907

119907119898=

119888119896119909

(1 minus 119896119909)(1 + (119888 minus 1)119896119909)

( 6-10 )

where 119909 is the relative pressure 1198751198750 119907 is the total adsorbed gas volume at a given relative

pressure of 119909 119907119898 is the monolayer adsorbed gas volume 119888 is the characteristic energy

constant of the BET equation and 119896 is the characteristic constant of the GAB equation

00 02 04 06 08 10

000

004

008

012

016

Experimental Adsorption Isotherm

BET

GAB

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Aad

Figure 6-11 The adsorption isotherm of nitrogen at 77K on raw coal sample fitted by the

BET equation and GAB equation The solid curves are theoretical and the points are

experimental The grey area Aad is the area under the fitted adsorption isothermal curve by

the GAB equation

Table 6-4 presents the GAB fitting parameters of nitrogen adsorption data for raw

1F-T and 3F-T coal samples with their respective determination coefficients (1198772) greater

161

than 099 The gray region corresponds to the area under the adsorption isothermal curve

(119860119886119889) which is determined as

119860119886119889 = int 1199071

0119889119909 =

119907119898

119896(119888minus1)(119897119899(1 minus 119896)minus119888119897119899(1 minus 119896) minus 119897119899(119888119896 minus 119896 + 1)) ( 6-11 )

However the GAB model fails to predict the desorption isotherm with a strong

hysteresis loop The constant 119888 in GAB equation characterizes chemical potential

difference between the first layer and superior layers (Timmermann et al 2001) where

the state of adsorbate molecules in the second or higher layers is identical to each other but

different from the liquid state While general accessibility to vapor phase is always

provided in the adsorption process not all pores are in contact with the bulk phase in the

desorption process over the entire pressure range especially for those occurring on the

porous adsorbent The postulation on equivalent adsorption potential of higher layers or

the constant value of 119888 is not valid for the desorption isotherm In order to remove this

rigidity 119888 was expressed as a polynomial function of relative humidity to model the water

desorption isotherm in the previous study (Blahovec and Yanniotis 2008)

In this study we adopt this concept to model the nitrogen desorption isotherm where

119888 depends on the relative pressure 119909 The formula of 119888 is given by

119888 = 119888119900

1

1 + 1198861119909 + 11988621199092 +⋯

( 6-12 )

where 1198861 1198862hellip are parameters of the polynomial and 119888119900 is equivalent to 119888 in the GAB

equation when 1198861 = 1198862 = ⋯ = 0

The modified GAB equation can be obtained by inserting Eq (6-12) into Eq (6-

10) which is derived as

162

119907

119907119898=

1198880119896119909

(1 minus 119896119909)(sum (1 + 119886119899119909119899)119899lowast1 + (1198880 minus sum (1 + 119886119899119909119899)

119899lowast1 )119896119909)

( 6-13 )

where 119899lowast is the order of polynomial in Eq (6-12) and 119899 is the index in the summation term

Eq (6-13) relates the sorption volume (119907) to the relative pressure where the former

parameter is the (119899lowast + 2)th power polynomial of the latter parameter Eq (6-13) reduces to

the GAB equation (Eq (6-10)) when 119899lowast = 0 Although the high order polynomials of 119888

reduce the error to fit the desorption isotherm it adds more freedom and uncertainty in the

determination of modeling parameters Based on the results provided in Blahovec and

Yanniotis (2008) only the modified GAB equation with 119899lowast=1 and 2 are used to fit the

nitrogen desorption isotherm and they are compared with the original GAB equation with

a constant 119888 Figure 6-12 demonstrates that the three equations were indistinguishable in

the relative pressure range of 05 minus 10 They became divergent at the very steep portion

of the desorption isotherm where the quadratic GAB equation (119899lowast = 2) delivers the best

fit to the experimental data than the cubic GAB equation (119899lowast = 1) and the GAB equation

(119899lowast = 0) Therefore the quadratic GAB equation was chosen to describe the nitrogen

desorption isotherm for raw coal sample 1F-T and 3F-T coal samples Table 6-3 lists the

fitting parameters and the corresponding fitting degree of the quadratic GAB equation

163

00 02 04 06 08 10

000

004

008

012

016

Ade

Experimental Desorption Isotherm

GAB (n=0)

Cubic GAB (n=1)

Quadratic GAB (n=2)

Qu

an

tity

Ad

so

rbed

(m

molg

)

Relative Pressure

Figure 6-12 The desorption isotherm of nitrogen at 77K on raw coal sample fitted by the

GAB equation (n=0) and the modifed GAB equation (n=1 2) The grey region is the

area under the desorption isothermal curve fitted by the quadratic GAB equation

The area under the desorption isothermal curve (119860119889119890) was evaluated by integrating

the quadratic GAB equation over the entire pressure range However an explicit expression

of the integral was not obtainable and instead numerical integration of the quadratic GAB

equation was applied with a very small interval 119909 If Eq (6-13) is simply symbolled as

119891(119909) the expression of 119860119889119890 obtained by the numerical integration can be evaluated as

119860119889119890 = int 1199071198891199091

0

= int 119907119898119891(119909)1198891199091

0

= (sum119891(119909119894) + 119891(119909119894+1)

2

1 119909

119894=0

) 119909119907119898

( 6-14 )

164

where 119909119894 = 119894 119909 are the data points that are equally extrapolated over the entire 119909 interval

of (01) 119909 is required to be a value that makes 1 119909 an integer In this study 119909 was

001 and the area under the isothermal curve was evaluated by 100 intervals

Once the values of 119860119886119889 and 119860119889119890 are computed the hysteresis index (119867119868 ) is

determined from the differential area of 119860119886119889 and 119860119889119890 with Eq (6-9) as summarized in

Table 6-3 The raw coal has the highest hysteresis index while the 1F-T coal has the lowest

hysteresis index This implies that the cryogenic treatment improves the pore connectivity

but the cyclic exposure to the cold fluid adversely acted on it An improvement in the pore

connectivity characterized by a smaller HI eliminates the transport resistance of gas

molecules within the coal matrix As a result the 1F-T coal with the smallest hysteresis

loop has the greatest methane diffusion coefficient while the raw coal with the largest

hysteresis loop incorporates the minimum methane diffusion coefficient These findings

are consistent with the diffusion coefficient measurement in our lab shown in Figure 6-7

Porosity and its size distribution are important pore structural parameters that

directly define the gas storage and transport properties of CBM reservoirs The

combination of using two adsorptive ie N2 and CO2 allowing characterizing the pore

size distribution on a complete scale from less than one nm to a few hundreds of nms As

capillary condensation is the dominant mechanism of nitrogen adsorption in meso- and

macropores the classical approach Barret Joyner and Halenda (BJH) (Barrett et al 1951)

model was applied to determine the pore size from the condensation pressure Figure 6-13

presents the pore size distribution (PSD) determined by the BJH model for raw and freeze-

thawed coal samples

165

Figure 6-13 PSD calculated from N2 sorption isotherm using the BJH model for the raw

1F-T and 3F-T coal samples

The total porosity increases after the cryogenic treatment that is mostly contributed

by the expansion of mesopore volume in the pore size of 3-5 nm The third time of F-T

cycle exerts a negligible effect on the allocation of pore volume in different pore size as

the PSD of 1F-T coal was indistinguishable from it of the 3F-T coal The low-temperature

measurements (77 K) does not give sufficient kinetic energy for the entry of N2 molecules

to micropores which is the reason why the micropore was excluded in Figure 6-13 CO2

adsorption at a higher temperature (273 K) facilitates the entry into the micropores which

allows yielding abundant information on micropore information In contrast to N2

0 20 40 60 80 100

000

001

002

003

004

0 2 4 6 8 10

000

001

002

003

004

Raw Coal

1F-T Coal

3F-T Coald

Vd

log

(w)

Po

re V

olu

me (

cm

sup3g

)

Pore Width (nm)

dV

dlo

g(w

) P

ore

Vo

lum

e (

cm

sup3g

)

Pore Width (nm)

mesopore macropore

166

adsorption pore-filling mechanism drives the CO2 adsorption in micropores The Dubinin-

Astakhov (DA) equation (Dubinin and Astakhov 1971) on the basis of Polanyirsquos work was

used to calculate micropore volume from CO2 sorption isotherm Figure 6-14 shows the

CO2 ad- and desorption isothermal curves of the raw and freeze-thawed coal samples

0000 0005 0010 0015 0020 0025 0030

00

01

02

03

04

05

06

07 Raw Coal

1F-T Coall

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Figure 6-14 CO2 sorption isotherms at 273 K of the raw 1F-T and 3F-T coal samples

As the monolayer adsorption or micropore filling is the dominant mechanism of

CO2 sorption on coal surface (Dubinin and Astakhov 1971 Dubinin and Radushkevich

1947) the adsorption and desorption isothermal curves are reversible Figure 6-14 shows

that the micropore adsorption capacity remained almost unchanged with cryogenic

treatments Correspondingly the micropore volume estimated by DA model only

experienced a slight variation between 00213 cm3g and 00203 cm3g Figure 6-15 is the

micropore size distribution analyzed by density functional theory (DFIT) The pore

167

structure of 04 to 1 nm was accurately characterized by CO2 adsorption and all samples

had two peaks with their positions at 5-7 nm and 8-9 nm The first peak shifted to the left

indicating that the cryogenic treatment caused some large micropores to break into smaller

micropores The slight decrease in micropore size explained the aforementioned decrease

in the micropore volume

4 6 8 10 12

000

004

008

012

016

Raw Coal

1F-T Coal

3F-T Coal

dV

dlo

g(W

) P

ore

Volu

me (

cm

sup3g)

Pore Width (Aring)

Figure 6-15 Micropore size distribution curves of CO2 adsorption for the raw 1F-T and

3F-T coal samples

Table 6-4 summarizes the pore volume of pores in various size fractions and the

mean pore size after the different number of freeze-thawing cycles The mesopore volume

calculated from the BJH model increases with the number of F-T cycles while the

macropore volume increases after the 1st F-T cycle but decreases after the 3rd F-T cycle

On the contrary the micropore volume decreases after the 1st F-T cycle and increases after

the 3rd F-T cycle The proportional variation of pore sizes is plotted in Figure 6-16 The

168

mesopore undergoes the greatest expansion in pore volume by 57 and 60 followed by

the increase in macropore volume by 17 and 14 and the smallest change occurs in

micropore volume by decreasing about 5 and 09 after the 1st F-T cycle and 3rd F-T

cycles respectively

Overall the cryogenic fracturing has a negligible effect on micropore volume and

its distribution The predominant change in pore size distribution is constrained in pore size

between 3 and 5 nm categorized as adsorption pores (Cai et al 2013) which illustrates the

increasing trend of adsorption capacity with the number of F-T cycles as shown in Figure

6-5 Under the application of cryogenic forces the total porosity increases from 483

cm31000g for raw coal to 640 cm31000g for 3F-T coal (see Table 6-4) with more volume

for gas molecules to transport This demonstrates the improvement of the diffusion

coefficient of the freeze-thawed coals as indicated in Figure 6-7 The decreasing trend of

diffusion coefficient when subject to multiple F-T cycles is associated with the decrease in

macropore volume and pore size due to the fatigue effect as well as the reduction in pore

connectivity characterized by the higher HI

Table 6-4 Peak pore diameter mean pore diameter total pore volume with its distribution

in different pore sizes after the different number of freeze-thawing cycles

Coal sample dmean

(nm)

Pore Volume (cm31000 g)

Vmicro Vmeso Vmacro VBJH total

Raw 665 2130 189 294 483

1F-T 614 2025 298 346 644

3F-T 602 2110 303 337 640 Vmicro micropore volume determined from CO2 sorption isotherm Vmeso Vmacro mesopore volume and

macropore volume determined from N2 sorption isotherm VBJHtotal the sum of mesopore and macropore

volumedmean average pore diameter

169

Figure 6-16 Proportional variation of pore sizes for different F-T cycles

653 Application of Micromechanical Model

The micromechanical model given in Eq (6-6) to Eq (6-8) were used to predict the

micropore dilation or the enlargement of total pore volume induced by cyclic cryogenic

fracturing Table 5 gives the required input parameters to simulate this damage process

and these values are obtained from available measurements The pore size distribution

(120588(1198860)) of the studied coal sample is given in Figure 6-13 The evaluation of frozen water

content (119860(119879 119886)) for given a pore size and freezing temperature can be referred to the

published data (Van de Veen 1987) The rest parameters in Table 6-5 have a considerable

range of values There are scare published data on coal strength parameters such as tensile

170

strength and fracture toughness because of the difficulty of obtaining accurate

measurements Following Chugh et al (1989) and in accordance with the provided

empirical relationship between tensile strength and fracture stiffness (Bhagat 1985) we

set a geologically reasonable range of values for 119870119888 as given in Table 6-5 Similar to coal

strength parameters estimates of thermal expansion coefficients of coal are fairly variable

ranging from 1 times 10minus to 11 times 10minus (NRC 1930) Besides previous works (Bell and

Jones 1989 Levine 1996) gave a distribution of the Youngs modulus and Poissons ratio

for Illinois coal such as Youngrsquos modulus (119864) and Poissonrsquos ratio (ν) Cryogenic treatment

has been reported to lower residual stresses where 120573119898 deceases with the repetition of

freezing and thawing (Kalsi et al 2010) But the measurement of residual stress is a very

time-consuming and expensive task leading to limited published data (Tavares and de

Castro 2019) As 120573119898 is largely dependent upon material heterogeneity (Beer et al 2014)

the change in 120573119898 during freezing-thawing cycles is estimated by the change in the

heterogeneity of the nanopore system of coal Qin et al (2018c) quantified the change in

the heterogeneity of coal after cryogenic treatment and the results of their work along with

the existing data on the residual stress of coal provided in Gao and Kang (2017) are used

in the modeling work

171

Table 6-5 Coal properties used in the proposed deterioration analysis

Material Property Specified Value

Youngrsquos modulus E 440 times 109 minus 612 times 1091198731198982 (Bell and

Jones 1989 Levine 1996)

Poissonrsquos ratio ν 0270 minus 0398 (Bell and Jones 1989

Levine 1996)

Fracture toughness 119870119888 for wet coal 1 times 105 minus 3 times 105Pa11989812 (Bhagat 1985

Chugh et al 1989)

Initial ratio of residual stress to crack

opening forces (120573119898) of wet coal

01 minus 02 (Gao and Kang 2017)

Thermal expansion coefficient 120572119871 1 times 10minus minus 11 times 10minus (NRC 1930)

Pore volume distribution 120588(1198860) See Figure 6-13

Frozen water content 119860(119879 119886)at minus196 1 (Van de Veen 1987)

Using the values given in Table 6-5 the effect of freezing and thawing cycles on

pore volume expansion was determined using the micromechanical model described in Eq

(6-6) - Eq (6-8) The modeled result along with the experimental result listed in Table 6-

4 are depicted in Figure 6-17 There are two model runs denoted as upper case and lower

case that predict the maximum and minimum change in pore volume with the cyclic liquid

nitrogen injections respectively The experimentally measured data points were spread

within the range of pore volume growth computed in the upper and lower case As a

common characteristic of the modeled result and experimental result it was observed that

the growth rate of pore volume and the rate of deterioration became much smaller as

freezing and thawing are repeated This was because the maximum ice crystallization

pressure (119875119888) decreased in response to the nanopore dilation as predicted by Eq (6-5)

Besides the repetition of freezing and thawing cycles reduced the residual stress and

172

enhanced the stiffness of the material (Karbhari et al 2000 Rostasy and Wiedemann

1983) which also explained why deterioration became smaller or even ceased after the first

cycle

Figure 6-14 depicts the experimental results of the change of the fractional pore

volume due to cyclic low temperature treatments In the range of very fine pores less than

2119899119898 no significant alterations of pore volume occurred Experimental evidence in the

previous study (Dabbous et al 1976) suggested that a substantial fraction of the pore space

of coal was inaccessible to water due to capillary effect As this capillary effect is more

predominant in smaller pores a limited amount of water can be sucked into micropores

and the deterioration process may not proceed under a small frost pressure (119875119908) However

a rise in pore volume along with a redistribution of the fractional pore volume occurred in

the range of mesopores and macropores (see Figure 6-11) The increase in pore volume

was well predicted by the micromechanical model In course of temperature cycles total

pore volume did not increase while fractional pore volume shifted from macropore to

mesopore (see Table 6-4) As a result mesopore volume increased with the number of F-

T cycles and macropore volume increased after the first cycle and then decreased after

subsequent cycles As more water is accessible to larger pores the deterioration is more

severe in macropore than mesopore Besides pore strength exhibits an inverse relationship

with pore radius as indicated in Eq (6-5) For this reason macropore may collapse and

break into smaller pores by fatigue under repeated application of frost-shattering forces

173

Figure 6-17 Various estimates of pore volume expansion (Upper case and Lower case)

due to cyclic liquid nitrogen injections according to the micromechanical model (solid

line) The grey area is the range of estiamtes specified by the two extreme cases The

computed results are compared with the measured pore volume expansion determined from

experimental data listed in Table 6-4 (scatter)Vpi is the intial pore volume or the pore

volume of the raw coal sample Vpf is the pore volume after freezing and thawing

corresponding to the pore volume of 1F-T sample and 3F-T sample

Porosity and its distribution govern the gas transport behavior of the coal matrix

The pore volume expansion due to liquid nitrogen injections gives more space for gas

molecules to travel and enhances the overall diffusion process of the coal matrix This

explains why the freeze-thawed (F-T) coal samples incorporated a higher diffusion

coefficient than the raw coal sample without temperature treatment as shown in Figure 6-

7 As macropore was further damaged while mesopore was slightly damaged by the

range of estimates

174

repetition of freezing and thawing the shift of fractional pore volume into the direction of

smaller pores inhibits gas diffusion in the coal matrix So the coal sample underwent

multiple freezing and thawing cycles ie 3F-T coal had lower diffusion coefficient than

the coal sample underwent a single freezing and thawing cycle ie 1F-T coal as observed

in the experiment (see Figure 6-7)

66 Experimental and Analytical Study on Fracture Structural Evolution

In this study we conducted laboratory experiments on coal cryogenic immersion

freezing to investigate its fracturing mechanism The ultrasonic method was employed to

thoroughly monitor the seismic response of coal under the cryogenic condition A

theoretical model was proposed and established to determine fracture stiffness of coal from

measured seismic velocity data Using the analytical solution for fracturing stiffness the

observed macroscopic scattered wavefield can be linked with the changes in fracture

properties which can directly inform flowability modification due to cryogenic treatment

The seismic interpretations of fracture stiffness of coal under freezing conditions can

directly predict the change in coal flowability and accessing the effectiveness of cryogenic

fracturing

661 Background of Ultrasonic Testing

Because of the importance of cleatsfractures on coal permeability active

monitoring techniques need to be employed to quantify the changes in cleat frequency and

distribution induced by cryogenic fracturing Rock mass characterization with seismic

wave monitoring provides an instant evaluation of the physical properties of the fractured

175

rock mass In the laboratory a few previous studies have been devoted to measuring the

seismic responses of various types of rocks subject to liquid nitrogen Experimental

evidence showed that the acoustic wave velocities and amplitudes decreased after

cryogenic stimulation (Cai et al 2016 Cha et al 2017 Cha et al 2014 Qin et al 2017a

Qin et al 2018a 2018b Qin et al 2016 Zhai et al 2016) Cha et al (2009) indicated

that the mechanical characteristics of fractures exert predominant effects on the elastic

wave velocity of cracked rock masses Fractures as mechanical discontinuities are potential

pathways for fluid flow that play an important role in gas production If seismic techniques

could be used to locate and characterize fractures or fracture networks then such non-

instructive geophysical techniques can probe fluid flow through fractured rock masses and

ascertain the effectiveness of formation stimulation A simple air- or fluid-filled fracture

may not be a realistic representation In fact a fracture often comprises of two rough

surfaces that do not exactly conform (Pyrak-Nolte et al 1990) They are partially in

contact and in between the contacts are the void spaces or cracks controlling fluid flow

behaviors Fracture properties such as surface roughness contact area and aperture

distributions directly govern the flowability of fractured rocks but these geometric

parameters are hard to be accurately quantified Goodman et al (1968) introduced a

concept of fracture stiffness that measures fracture closure under the stress condition to

quantify the complicated fracture topology without conducting a detailed analysis of

fracture geometry Although many studies (Hedayat et al 2014 Myer 2000 Pyrak-Nolte

et al 1990 Sayers and Han 2002 Verdon et al 2008) have estimated fracture stiffness

from elastic waves propagation within fractured media with a single artificial fracture very

176

little fracture stiffness data have been reported in the literature for naturally fractured rocks

such as coal

662 Coal Specimen Procurement

Cylindrical coal specimens of 100 mm in length and 50 mm in diameter were taken

from one CBM well in Qingshui basin Shanxi China The coal specimens were initially

cut by a rock saw and then abraded to satisfactory accuracy using a water jet The cores

were prepared in a way that the axial direction of each coal specimen is perpendicular to

its bedding plane For seismic measurements intact cores with smooth and complete

surfaces were selected Figure 6-18 is an example of a tested coal core (M-2) and basic

information on the studied coal specimens is summarized in Table 6-6 The permeability

of the virgin coal samples in Qingshui basin is ultra-low with values less than one mD

(Zhang and Kai 1997) This low permeability cannot provide economic gas flow rates

without stimulation Thus massive stimulation treatments such as hydraulic fracturing are

required in the field But the routine hydraulic fracturing in Qingshui basin does not always

give the expected gas productivity (Zhu et al 2015) As the fracturing fluid is imbibed into

the formation this elongates water drainage period and the interaction between extraneous

water and methane molecules reduces gas desorption pressure and prevents gas from being

produced Because of the associated water usage hydraulic fracturing may not be the most

effective stimulation technique for CBM exploration Cryogenic fracturing using an

anhydrous fluid that eliminates these water-related issues may substitute hydraulic

fracturing In this study we tried to study the effectiveness of cryogenic treatment through

177

the characterization of fracture stiffness which is inherently related to the change in

permeability

Figure 6-18 An intact coal specimen (M-2) before freezing

Table 6-6 Physical properties of two coal specimens used in this study

Sample Height Diameter Density Porosity Moisture Content

(mm) (mm) (gcm3)

()

M-1 9996 4989 139 0036 0

M-2 10007 5017 138 0048 058


The two coal specimens were dried in an oven with a constant temperature of 80

for 24 hrs to remove the moisture content Figure 6-19 depicts the test systems used to

investigate the velocities and attenuations of shear and compressional pulses propagated

178

through the fractured coal specimens when subjected to a low-temperature environment

Frost shattering and thermal shock are the two dominant mechanisms underlying cryogenic

fracturing To examine these mechanisms separately the measurements of transmitted

compressional and shear waves made with a dry specimen (no moisture content) would be

compared with a saturated coal specimen One of the coal specimens (M-2) was saturated

with water in a vacuum water saturation device for 12 hrs with the other one (M-1) being

a dry sample The physical properties and moisture content of the dry and saturated coal

specimens were listed in Table 6-6 Initial ultrasonic measurements of the intact coal

specimens were made with a pair of platens aligned in the axial direction The tested coal

specimens were frozen in the thermal bottle filled with LN2 for up to 60 mins and seismic

measurements were made in between the freezing process over a range of time intervals

from 5 mins to 15 mins Followed by the freezing process the coal specimens were thawed

at room temperature for a complete freezing-thawing cycle Waveforms of seismic pulses

were then collected for the treated coal specimens As coal is a highly attenuating material

the employed seismic transducers have low center frequency yielding strong penetrating

signals In this experiment the center frequency of the P-wave transducer is 50 kHz and

it of the S-wave transducer is 100 kHz

179

1 Figure IExperimental equipment and procedure

664 Seismic Theory of Wave Propagation Through Cracked Media

In this section we theoretically investigate the seismic wave transmission behavior

in the fractured rock mass and establish a mathematical expression of fracture stiffness

based on the velocity and attenuation of the propagated wave

I Fracture Model and The Meanfield Theory

A simple and effective representation of a fracture is an infinite plane interspersed

with arrays of small crack-like features (Angel and Achenbach 1985 Hudson et al 1997

Hudson et al 1996 Schoenberg and Douma 1988 Sotiropoulos and Achenbach 1988)

As illustrated in Figure 6-20 the fracture plane can be conceptualized into two distinct

180

regions where the white area corresponds to the crack region and in the grey area the two

sides of fracture are in contact

Figure 6-19 The fracture model random distribution of elliptical cracks in an otherwise

in-contact region

The seismic response of such a fracture is the same as it of an imperfect interface

or a surface of displacement discontinuity When a wave incident on the interface part of

the energy is reflected with the rest transmitted Some studies (Adler and Achenbach 1980

Baik and Thompson 1984 Gubernatis and Domany 1979) have estimated fracture

stiffness from the partitioned waves where the acoustic impedance of the reflection and

transmission waves are the required inputs However a fracture with a partial bond serves

as a poor reflector for an acoustic wave and thus the reflected wave is hard to be accurately

captured and characterized (Achenbach and Norris 1982) It is impractical to use

impedance for the determination of fracture stiffness for fractures with a complex

distribution of cracks or contact area

Incident Wave

Fracture Plane

Outgoing Wave

Scattered Wave

Undisturbed Wave

Ui(x)

ltU(x)gt = Ui(x) + Us(x)

x3

x2

x1

C Cc

F

181

This study investigates the reflection and refraction behaviors of propagating waves

as a whole which is known as the scattered wavefield For waves with wavelength large

compared with the scale of the structural discontinuity (ie the size and spacing of cracks)

the geometry of each individual crack becomes insignificant for wave propagation The

fluctuation of wave propagation induced by such ensemble of flaws can be solved with a

stochastic differential equation or by meanfield theory (Keller 1964) which takes an

average of different realizations of wavefield over a medium randomly interspersed with

scatters At long wavelength this ensemble-averaged field provides a good approximation

of the actual displacement field and retains its simplicity in computation (Hudson et al

1997 Hudson et al 1996 Keller 1964 Sato 1982 Wu 1982) Also this averaging

process over a sequence of fracture planes enables the construction of a meanwave field to

correlate with the overall properties of a rock specimen as a three-dimensional (3-D)

structure The following analysis follows Hudsonrsquos method (Hudson et al 1997) to derive

fracture stiffness from the seismic response of a fractured medium But this study proposes

the derivation in a concise manner and extends the fracture model from circular cracks to

elliptical cracks with arbitrary aspect ratio The elliptical shape closely resembles naturally

forming flaws containing locally smooth arbitrary contacting asperities For other shapes

of cracks the establishment of a meanwave field requires numerical solutions (Guan and

Norris 1992)

182

II Wave Equations and Perturbation Method

The fracture model illustrated in Figure 6-20 suggests that the boundary condition

is neither continuous nor homogenous over the entire fracture interface However a

continuous and unified boundary condition needs to be established for solving the overall

wavefield in a cracked medium In this work the meanfield theory is employed to establish

the continuity condition at the fracture plane Considering a sinusoidal or time-harmonic

plane wave incident on the fracture plane the incident displacement field (119932119920) satisfies

119906(119909 119905) = 119860119890minus119894120596119905119890119894119896119909 ( 6-15 )

where 119906 is the displacement 120596 is the angular frequency 119896 is the wavenumber and 119860 is the

amplitude of the incident wave

The generalized wave equation 119906(119909 119905) satisfies

1205972119906(119909 119905)

1205971199052= 1199072

1205972119906(119909 119905)

1205971199092 ( 6-16 )

where 119907 is the wave speed and at long wavelength it is related to the effective elastic

modulus of the cracked rock (Garbin and Knopoff 1973 1975)

A fourth-order of stiffness tensor (119862119894119895119896119897) is employed to study the two-dimensional

plane wave propagation Considering a time-harmonic wavefield with constant frequency

(120596) outlined in Eq (6-15) the displacement field becomes invariant with time The partial

differential form of wave equation given in Eq (6-16) now reduces to an ordinary

differential equation where the time-harmonic wavefield satisfies

183

1205881205962119906119894(119909) +120597

120597119909119895119862119894119895119896119897

120597119906119896(119909)

120597119909119897= 0 ( 6-17 )

When waves propagate through the cracked plane they are expected to be slowed

and attenuated These scattering effects can be reflected and quantified by linking the

outgoing or total wavefield (119932) to the incident wavefield (119932119920) The outgoing wavefield is

a superposition of the undisturbed waves (119932120782) and the scattered waves (119932119930) which are

affected by the distribution of cracks and their variations in geometry As the full details

of the scattered and total wavefield are too convoluted to be exactly analyzed the

perturbation method is employed to obtain an average solution of the displacement field

over a collection of cracks (Keller 1964) Suppose a linear stochastic operator 119872(휀) can

transform the incident wave field (119932119920) into outgoing wavefield (119932) and this transformation

can be mathematically written as

119932 = 119872(휀)119932119920 ( 6-18 )

where 휀 is a small perturbation constant implying that at long wavelength the scattering

effect induced by a small-scale crack is small

The perturbation theory (Ogilvy and Merklinger 1991) suggests that 119872(휀) can be

approximated by a power series (Keller 1964)

119872(휀) = 119871 + 휀1198711 + 119874(휀2) ( 6-19 )

119871 = 119872(0) ( 6-20 )

where the scattering operator (119872) reduces to a sure operator (119871) when 휀 = 0 1198711 is the first-

order stochastic perturbation of the sure operator (119871)

184

In Eq (6-19) only the first-order approximation of 119872(휀) is considered and the

higher-order term (119874(휀2)) is neglected for the subsequent derivation Because at long

wavelength the scattering effect induced by the interaction between cracks is negligible

when compared with it by a single crack (Budiansky and OConnell 1976) Besides such

information requires the statistic of crack distribution given the existence of a certain crack

and is hard to be obtained If more information is available the second-order term can be

added later to account for the crack-crack interactions

The application of the perturbation method allows digesting the complex solution

of the overall displacement field into the solvable part for undisturbed waves and the

perturbed part by adding a small perturbation parameter휀 to the exact solution The exact

displacement field can be solved for undisturbed waves propagating in a continuous rock

with no cracks (휀 = 0) Thus

119932120782 = 119871119932119920 ( 6-21 )

where 119932120782 is the overall wavefield of undisturbed waves

With Eq (6-19) and Eq (6-21) substituted into Eq (6-18) the total wavefield (119932) can

be related to the undisturbed wavefield (119932120782) as

119932 = 119932120782 + 휀1198711119932120782 ( 6-22 )

where for undisturbed wavefield the outgoing waves have the exact same waveform as the

incoming waves and thus 119932120782 = 119932119920

The statistical average total field or meanfield ( 119932 gt) is found by taking the

expectation of Eq (6-22) as

185

119932 gt= 119932120782 + 휀 1198711 gt 119932120782 ( 6-23 )

where angular brackets lt gt denote the expectation of the statistical variables

Clearly 119932 gt can be determined if 1198711 gt is defined Assuming the scattering effect

of individual cracks are statistically equivalent (Hudson 1980) then

1198711 gt= int 119901(119888)(119888)119865

119889119888 ( 6-24 )

where 119901(119888) is the probability density function defined for a distribution of cracks over a

fracture plane (119865)and 119888 represents the centroid of every crack The mean scattering

operator for such a collection of cracks is (119888)

With 119873 cracks per unit area the crack density function 119901(119888) is given by

119901(119888) = 119873 ( 6-25 )

and

1198711 gt= 119873int (119888)119865

119889119888 ( 6-26 )

The overall wavefield ( 119932 gt) is linked with the undisturbed wavefield (119932120782) by

the scattering operator as outlined in Eq (6-25) Boundary condition needs to be set before

obtaining the solution of the scattering operator ((119888)) Unlike a perfect separated fracture

boundary condition at a cracked plane is not uniform For the following development the

part of fracture plane (119917) containing cracks is denoted as 119914 and the rest part without cracks

is a complement set denoted as 119914119940 In the area with welded contact (119914119940) the displacement

field (119958) of waves and the seismic stress field (119957) are continuous across the fracture plane

(Kendall and Tabor 1971) providing that

186

119905119894(119909) = 0 [119906119894(119909)] = 0 119894 = 123 ( 6-27 )

where [ ] is the jump or discontinuity across the fracture interface

In 119914 the seismic stress or traction field (119957) is continuous and the displacement field

is discontinuous (Kendall and Tabor 1971 Pyrak-Nolte et al 1990) providing that

[119905119894(119909)] = 0 119894 = 123 ( 6-28 )

Dry cracks are assumed in Eq (6-28) but this can be easily extended to fluid-filled

crack by adjusting the boundary conditions as given in Hudson et al (1997) The traction

that is continuous across the fracture is assumed to be linearly correlated with the

discontinuity of displacement through the fracture stiffness matrix 119948 with dimension

stresslength (Schoenberg 1980) As illustrated Figure 6-20 1199092 are the directions

tangential to the fracture plane and 1199093 is normal to the plane If 119948 is transverse isotropic

with respect to the 1199093 axis the off-diagonal terms vanish leaving two independent stiffness

as the normal stiffness (119896119899) and shear stiffness (119896119905) Mathematically

119957 = 119948[119958] ( 6-29 )

where 119948 = [

119896119905 0 00 119896119905 00 0 119896119899

] in the unit of stress per length

Eq (6-29) is valid for every wave passing thorough the fracture plane And we need

to demonstrate that this continuity condition is also applicable to the statistical mean

wavefield ( 119932 gt) Considering a single mean crack with centroid 119888 contained in the

fracture plane the associated displacement field (119932119956(119888)) is given by

119932119956(119888) = 휀(119888)119932120782 ( 6-30 )

187

As discussed the boundary condition is not continuous over the entire fracture

plane (119917) Greenrsquos function as a function of source (Qin 2014) is applied to provide an

analytical solution of the boundary value problem where the local displacement

discontinuity serves as a source Applying boundary conditions given in Equation (13) and

Eq (6-28) the solution of 119932119956(119909) can be obtained in terms of Greenrsquos function 119866(119909 120585) as

developed in Hudson et al (1997)

119932119956(119909) = int 119905119894(119932119956(120585))[119866119894

119868(119909 120585)]119889120585119914

( 6-31 )

where 120585 = 119909 + 119888 is a general point of the mean crack with centroid119888

As there is no displacement discontinuity in the undisturbed wavefield it is

reasonable that the displacement discontinuity of total field is the same as the displacement

discontinuity of scattered field and thus

[119932119930] = [ 119932 gt] ( 6-32 )

Eq (6-31) transforms incident wavefield (119932119920) into scattered wavefield (119932119956)through

119905(119932119956) and 119905(119932119956) exhibits a linear relationship with [119932119956] given in Eq (6-29) Substitute

Eq (6-30) and Eq (6-32) into Eq (6-31) we can obtain

휀119932120782 = int 119896119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]119889120585

119914

( 6-33 )

where 119905119894(119932119930(120585)) = 119896119894119895[ 119880119895 gt (120585)] at the crack

Eq (6-32) provides an analytical expression of the mean scattering operator and

1198711 gt with Eq (6-26) substituted Considering the transformation from 119932119920into 119932 gt

given by Eq (6-23) then

188

119932 gt= 119932120782 + 119873int 119870119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]

119865

119889120585 ( 6-34 )

where119870119894119895 = int 119896119894119895119889120585119914 and [ 119932 gt] is assumed to be constant over 119914

Replace the term 119932119956 on the left-hand side (LHS) of Eq (6-31) with ( 119932 gt minus119932120782)

and compare this expression with Eq (6-34) then we are able to establish a continuity

condition for 119932 gt over the entire fracture plane 119917 which is

119905119894( 119932 gt) = 119870119894119895119905 [ 119880119895 gt] ( 6-35 )

where 119870119894119895119905 = 119873119870119894119895 = 119873int 119896119894119895119889120585119914

is the overall fracture stiffness derived from the

meanfield

Now a continuous and unified boundary condition is established for the overall

wavefield in a given cracked medium

III Fracture Stiffness of Elliptical Cracks

Eq (6-35) gives a linear correlation of displacement discontinuity field and stress

traction field for the overall mean wave field ( 119932 gt) through the fracture stiffness matrix

(119922119957) Here 119948 as well as 119922119957 are diagonal matrix with two independent components 119896119899 and

119896119905 The normal and shear component of 119957 on the elliptical crack in an otherwise traction-

free surface gives rise to the discontinuity in normal or shear displacement The normal or

shear tractions are the same as those acting on the closed area that produce the uniform

normal or shear displacement of the loaded region in the plane surface of an elastic half-

space Outside the closed area or loaded region both normal and shear tractions are zero

The total force (119875 ) integrating over the elliptical area that generates uniform normal

189

displacement of the loaded area in the surface of an elastic half-space takes the form of

(Johnson 1985)

119875 = 21205871198861198871199010 ( 6-36 )

where 119886 and 119887 are the long-axis and short-axis of the ellipse and 119886 gt 119887 1199010 is the internal

pressure

The uniform surface depression of the ellipse (1199063) due to the stress distributed over

the elliptical region is given by (Johnson 1985)

1199063 = 21 minus 1205842

1198641199010119887119825(119890) ( 6-37 )

where 1199063 is the normal displacement 120584 and 119864 are Poissonrsquos ratio and Youngrsquos modulus of

the rock matrix and 119890 is the eccentricity of the ellipse 119890 = (1 minus 11988721198862)12 119825(119890) is the

complete elliptical integral of the first kind and it is conventionally denoted as 119818(119890) Here

a different notation119825(119890) is taken to distinguish it from the notation of the fracture stiffness

matrix

By combing Eq (6-36) and Eq (6-37) 119875 can be expressed in terms of the elastic

properties as

119875 = 120587119886119864

1 minus 12058421

119825(119890)1199063 ( 6-38 )

The total force 119875 is an integration of the stress distributed over the elliptical region

and results in a unit uniform indentation of the loaded ellipse The magnitude of 119905119899 exerted

on the crack that generates unit discontinuity in normal displacement equals to half of the

190

magnitude of 119875 acting on the surface of the half-space For a random distribution of 119873

elliptical cracks 119905119899 is then given by

119905119899 =1

2119873119875[119906119899] ( 6-39 )

where 1199063 =1

2[119906119899]

With Eq (6-35) substituted the corresponding normal fracture stiffness (119870119899) can

be determined as

119870119899 =1

2119873119875 =

1

2119873120587119886

119864

1 minus 12058421

119825(119890) ( 6-40 )

As 119864 and 120584 can be directly inverted from elastic wave velocities data (Mavko et al

2009) Eq (6-39) becomes

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41 )

In the tangential direction the total traction (119876) integrating over the loaded ellipse

that produces a uniform tangential displacement of the surface takes a form of (Johnson

1985)

119876 = 21205871198861198871199020 ( 6-42 )

where 1199020 is the tangential traction at the center of the ellipse

The corresponding tangential displacement within the ellipse is (Johnson 1985)

1199061 = 1199062 =1199020119887

119866[119825(119890) +

120584

1198902(1 minus 1198902)119825(119890) minus 119812(119890)] ( 6-43 )

where 119866 is the shear modulus of the elastic half-space 119825(119890) and 119812(119890) are the complete

elliptic integral of the first kind and second kind

191

By combining Eq (6-42) and Eq (6-43) 119876 can be expressed in terms of the elastic

properties as

119876 =2120587119886119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

11990612 ( 6-44 )

The magnitude of 119905119905 distributed over the crack that generates unit discontinuity in

tangential displacement equals the magnitude of 119876 generating frac12 tangential displacement

of the loaded ellipse on the surface of a half-space For a random distribution of 119873 elliptical

cracks 119905119905 is then given by

119905119905 =1

2119873119876[119906119905] ( 6-45 )

where 11990612 =1

2[119906119905]

With Eq (6-35) substituted the corresponding fracture stiffness (119870119905) in tangential

direction can be determined as

119870119905 =1

2119873119876 = 119873120587119886

119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

( 6-46 )


2009) Eq (6-41) becomes

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875

2frasl(1 minus 1198902)119825(119890) minus 119812(119890)

1198902]

( 6-47 )

Eq (6-41) and Eq (6-47) are the normal and shear fracture stiffness determined

from the elastic wave behavior across a flawed fracture plane containing a distribution of

elliptical cracks If 119890 = 0 and 119886 = 119887 are considered the development is then specialized to

192

circular cracks and the result of fracture stiffness has been presented in the previous work

(Hudson et al 1997) We conducted a comparison here For circular cracks 119929(0) = 1205872

and 119886 = 119887 Normal fracture stiffness (119870119899) given in Eq (6-41) becomes

119870119899 = 41198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752) ( 6-48 )

Tangential fracture stiffness (119870119905) of the embedded circular cracks takes the form of

119870119905 = 2119873120587119886120588

1198811199042

[120587 +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875

2frasl lim119890rarr0

(119825(119890) minus 119812(119890)

1198902)]

( 6-49 )

The evaluation of limerarr0

119825(e)minus119812(e)

119890 requires the application of LHospitals rule as both

the denominator and numerator of the fraction approaches zero as 119890 rarr 0

lim119890rarr0

((1 minus 1198902)119825(119890) minus 119812(119890)

1198902)

= lim119890rarr0

(minus2119890119825(119890) + (1 minus 1198902)119825prime(119890) minus 119812prime(119890)

2119890) = minus

120587

4

( 6-50 )

where 119825prime(119890) =119889119825(119890)

119889119890=

119812(119890)

119890(1minus119890 )minus119825(119890)

119890 and 119812prime(119890) =

119889119812(119890)

119889119890=119812(119890)minus119825(119890)

119890 (Polyanin and

Manzhirov 2006)

Substitute Eq (6-50) into Eq (6-49) tangential fracture stiffness (119870119905 ) of the

embedded circular crack is given by

119870119905 = 811987312058711988612058811988111990421 minus 119881119878

2 1198811198752frasl

3 minus 21198811198782 119881119875

2frasl ( 6-51 )

For cracks in circular shapes Eq (6-49) and Eq (6-51) agree with the expression

of fracture stiffness derived in Hudson et al (1997) (see Eq (54) in their work) This work

193

successfully extends the previous derivation to a more general case by taking elliptical

cracks into consideration A fundamental formulation was proposed to estimate fracture

stiffness for a fracture plane consisted of a planar distribution of small isolated areas of

cracks Both experimental and numerical evidence (Myer 2000 Petrovitch et al 2013)

suggest that stiffness captures the deformed topology and connectivity of a fracture

network and directly influences the fluid flow behavior through a fractured medium and its

faulting and failure behaviors Thus the measurement of fracture stiffness via the

ultrasonic method provides a non-destructive tool for predicting the flow capacity of a

fractured rock mass This tool was experimentally investigated in this study using seismic

data for two coal cores to characterize the change of the hydraulic properties subject to

cryogenic treatments

67 Freeze-thawing Damage to Cleat System of Coal

For the tested coal specimens P and S wave velocities were monitored and recorded

at different time intervals of the freezing process under both dry and fully saturated

conditions In the following sections results for selected freezing times are shown to

demonstrate the variation and trend of the experimental data This study aims to apply the

displacement discontinuity model given in Section 664 to characterize the change of the

fracture stiffness for two coal cores subject to cryogenic treatments using experimentally

measured seismic data

Figure 6-21 outlines the workflow Fracture stiffness derived from the theoretical

model is implicitly related to fluid flow(Pyrak-Nolte and Morris 2000) Thus the

194

estimation of fracture stiffness from seismic measurements is essential in terms of

developing a remote interpretation method for predicting the hydrodynamic response of

fractured CBM reservoirs To apply the conceptual model illustrated in Figure 6-20 we

need to initially clarify the confusion from the use of the terms crack and fracture We refer

to the bedding plane that is large relative to seismic wavelength as a fracture We refer to

open regions between areas of weld on the fracture surfaces ie cleat as cracks The

fracture zone or bedding plane consists of a complex network of cracks or cleats The

collected waveforms are modeled as the mean wavefield realized by a collection of cracks

embedded in the fractured coal specimens


specimens subject to cryogenic treatments

671 Surface Cracks

For the initial specimens the wet coal specimen (Figure 6-22(a)) was found to have

a well-developed pre-existing cleat network than the dry coal specimen (Figure 6-22(b))

195

With LN2 freezing treatment the surfaces of the frozen coal specimens were covered by

the frost due to the condensation of moisture content from the atmosphere The formation

of frost obscured surface features of the coal specimen and hided part of surface cracks

from the taken images As a result in Figure 6-22(b) not all pre-existing cracks can be

captured during the freezing process Although the accumulation of frost may hinder real-

time and accurate monitoring of the generation and propagation of surface cracks during

the freezing process it was noticeable two phenomena was simultaneous happening (1)

new cracks were generated during the treatment and (2) the cracks amalgamate to well-

extended fracture network through the pre-existing fracture propagation and new crack

coalescences for both the dry and wet coal specimens After completely thawed and

recovered back to the room temperature the surfaces of the studied coal specimens were

free of frost Besides the crack density of the thawed coal specimens was significantly

improved as well as the pre-existing cracks widened

196

Figure 6-21 Evolution of surface cracks in a complete freezing-thawing cycle for (a) dry

coal specimen (b) wet coal specimen Major cracks are marked with red lines in the images

of surface cracks taken at room temperature ie pre-existing surface cracks and surface

cracks after completely thawing

197

672 Wave Velocities

Figure 6-23 is the superimposition of waveforms recorded at different freezing

times For the ultrasonic measurements the transducer emits a pulse through the coal

specimen and a single receiver at the opposite side records the through-signal Since the

input signal was held constant throughout the freezing process the change in the amplitude

was induced by the attenuative behavior of the material The attenuation coefficient (α) is

given by

120572 = minus20

ℎ119897119900119892(119860119860119900) ( 6-52 )

where α is the attenuation coefficient in dBm ℎ is the height of the coal specimens in m

119860119900 is the initial amplitude of the incident wave and 119860 is the amplitude received at the

receiver after it has traveled a distance of ℎ

In relative to the received signals at initial condition (tf = 0 min) the attenuation

coefficients after completion of the freezing process were determined to be 144 dBm for

dry coal specimen and 150 dBm for wet coal specimen using the amplitudes of direct-

arrival or first-arrival signals as given in Figure 6-23 Overall waves propagating through

the saturated coal specimen (Figure 6-23(b)) experienced a more severe attenuation than

those propagating through the dry coal specimen (Figure 6-23(a)) Figure 6-22 suggests

that the saturated coal specimen has a higher crack density than the dry coal specimen The

rock cracks exert three effects on wave propagation that they cause the delay of the seismic

signal reduce the intensity of the seismic signal and filter out the high-frequency content

of the signal (Pyrak-Nolte 1996) For saturated specimen the acoustic waves cause relative

198

motion between the fluid and the solid matrix at high frequencies leading to the dissipation

of acoustic energy (Winkler and Murphy III 1995) Consequently the saturated coal

specimen received weaker ultrasonic signals than the dry coal specimen

Figure 6-22 Recorded waveforms of compressional waves at different freezing times for

(a) 1 dry coal specimen and (b) 2 saturated coal specimen

199

A small-time window (up to 200 μs) was applied to each received signal to separate

first wave arrival from multiple scattered waves For the dry coal specimen (Figure 6-

23(a)) there were strong correlations among these first arrival wavelets where the

waveforms collected at the freezing time of 5 min and 35 min time-shifted concerning to

the waveform collected at the freezing time of 0 min The first arrival wavelets of the

saturated coal specimen (Figure 6-23(b)) recorded at different freezing times were found

to be weakly correlated where the waveforms were broadened as the coal specimen was

being frozen In response to the thermal shock originated with the freezing treatment the

propagation of pre-existing cracks and generation of new cracks damped the high-

frequency portion of the signal and potentially distorted the shortest wave path between the

transmitter and receiver that alternate the waveform of first arrivals Because of the denser

crack pattern the first arrival wavelets of the saturated coal specimen were severely

distorted and poorly correlated The onset of first arrivals would be used in the calculation

of compressional and shear wave velocities In Figure 6-24 seismic velocities were

significantly reduced when subjected to liquid nitrogen freezing because of the provoked

thermal and frost damages The P- and S- wave velocities of the dry specimen bounced

back slightly at the freezing time of 35 min As common characteristics deterioration

usually proceeds as freezing time increases but the rate of deterioration becomes smaller

and smaller as the elapse of the freezing time Usually the deterioration ceases after

sufficient freezing time and a further supply of water imposes additional damages as it

moves through the void space (Hori and Morihiro 1998)

200

Followed by direct arrivals coda waves arrived at the receiver The coda wave

interferometry (CWI) is a powerful technique for the detection of a time-lapse in wave

propagation (Zhang et al 2013) When the scattering effect is relatively strong there will

be obvious tailing in the received wave signal

Figure 6-23 Variation of seismic velocity with freezing time for (a) dry coal specimen (b)

wet coal specimen

(a)

(b)

201

673 Fracture Stiffness

I Fracture Stiffness of Dry Coal Specimen

For dry coal specimen normal and tangential fracture stiffnesses can be derived

from Eq (6-41) and Eq (6-47) as

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41)

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )

As defined before 119873 is crack density representing the number of cracks present in

a unit area Both 119886 and 119890 are the average crack characteristics Fracture stiffness is a

function of seismic velocities and the properties of cracks The seismic velocities were

given in Figure 6-24 and we would first use the surface cracks shown in Figure 6-22 to

estimate the parameters of cracks Here we want to point out that we will use the surface

fracture characteristic to represent the bulk fracture properties This limitation can be

solved by the advanced X-ray tomography images In this study we tried to focus on the

improvement of flow capacity due to cryogenic fracturing and the surface fracture

properties can offer a good benchmark value for the bulk coal

ImageJ was used to process the images of surface cracks and it can delineate the

crack location and pattern as well as extrapolates the sizes of all the identified cracks

ImageJ can convert the image into a text file where every pixel is assigned with a

numerical entry representing its gray-scale value The estimation of fracture stiffness

202

requires the determination of crack density as well as the average length of cracks Thus

we developed a computer program built in MATLAB to automatically count the total

number of cracks and calculate the average length of cracks The detailed algorithm and

code were given in the Appendix Crack density is not amenable to direct measurement

and it is necessary to specify an algorithm of estimating this parameter The developed

program treats any crack that is not connected with another crack that has already been

counted as a new crack The only required input in this program is the threshold gray-scale

value of crack regions The determined crack-related properties are listed in Table 6-7 Due

to water invasion more cracks present in the saturated coal specimen (M-2) than the dry

coal specimen (M-1)

Table 6-7 Crack density (119873) and average half-length (119886) aperture (119887) and ellipticity (119890)

of cracks determined from the automated computer program

Sample 119873 119886 119887 119890

(1mm2) (mm) (mm) (-)

M-1 0097 10 018 098

M-2 019 10 045 090

The parameters given in Table 6-7 were evaluated for the coal specimens at room

temperature As the wavelengths of both P- and S- waves are significant with respect to the

dimension of cracks (~119898119898) crack geometry may not exert an immense effect on waves

propagated across but the crack density conveying statistics of crack distribution does

affect wave propagation and needs to be updated as coal being frozen Budiansky and

OConnell (1976) proposed workflow for the estimation of crack density as a function of

the ratio of effective modulus of cracked to a porosity-free matrix We would refer to their

203

method to interpret the evolution of crack density with the freezing time and 119873 provided

in Table 6-7 serves as a reference value for determining the properties of the porosity-free

matrix With crack properties and statistics specified normal and shear fracture stiffnesses

for the tested coal specimen can be evaluated based on measurements of compressional

and shear waves Variations of fracture stiffness with freezing time according to Eqs (6-

41) and (6-47) are shown in Figure 6-25 Overall both normal and tangential fracture

stiffnesses decreased as the coal specimen was being frozen The ratio of tangential to

normal fracture stiffness kept almost constant The coal specimen experienced significant

shrinkage when it was initially immersed in liquid nitrogen that in turn caused coal to break

and crack The increase in crack density was observed as decreases in magnitude of the

seismic velocities shown in Figure 6-24 and it resulted in the rubblization of the fracture

surface or bedding plane which decreased both normal and shear stiffnesses of the fracture

as modeled by Figure 6-25 Verdon and Wuumlstefeld (2013) provides a compilation of

stiffness ratios computed from ultrasonic measurements published in the technical

literature where 119870119899119870119905 varies over the range 0 to 3 and for most samples it has a value

between 0 and 1 as cracks are more compliant in shear than in compression (Sayers 2002)

As the presence of incompressible fluid in crack greatly enhances normal stiffness while

leaves shear stiffness unchanged 119870119899119870119905 is an effective indicator of fracture fill This

explains why 119870119899119870119905 stayed almost constant with freezing time under dry condition The

significance of shear and normal fracture stiffnesses and their ratio on seismic

characterization of fluid flow will be further discussed in the later section

204

0 10 20 30 40 50 60

0

20

40

60

80

100

120

Fra

ctu

re S

tiffness (

GP

am

)

Freezing Time (min)

Kn K

t

00

05

10

15

20

25

30 K

tK

n

Tangential to

Norm

al S

tiffness R

atio

Figure 6-24 Under dry condition (M-1) the variation of normal and tangential fracture

stiffness and tangentialnormal stiffness ratio with freezing time

II Fracture Stiffness of Saturated Coal Specimen

As discussed 119870119905119870119899 ratio was known to be dependent on the fluid content Fluid

saturated fractures exhibit much lower normal compliance (1stiffness) than those with

high gas concentration (Schoenberg 1998) The theoretical model in section two is only

valid for dry cracks In the wet case a minor modification was made to consider the

presence of incompressible fluid in the cracks which is given in Worthington and Hudson

(2000) Normal and tangential fracture stiffness can be expressed as

205

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890)+119872prime

( 6-53 )

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

+119866prime

( 6-54 )

where 119872prime and 119866prime are the constrained and shear modulus of the crack fill and is the mean

aperture of the cracks For the elliptical shape of cracks = 119887

At room temperature the cracks in the saturated coal specimen (M-2) was filled

with air and water While elastic moduli of air are very small the values of constrained

modulus (119872prime) and bulk modulus (119870prime) of water are comparable to the moduli of coal matrix

(Fine and Millero 1973) When subjected to a low-temperature environment water

contained in the tested specimen is expected to undergo a water-to-ice phase transition

The frozen water content depends on the rate of heat transfer between the coal specimen

and the surrounding

Cooling a coal specimen with liquid nitrogen can be treated as a two-step process

First heat is conducted from the sample interior to the sample surface and in the following

step heat is convected away from the sample surface to the surrounding cryogen The

freezing process can be limited either by convection or conduction Their relative

contribution to overall heat transfer is characterized by Biot number (Bi) which is

expressed as

119861119894 = ℎ119881119896119888119860 ( 6-55 )

206

where ℎ (119882

119898 119870) is the heat transfer coefficient 119896119888 (

119882

119898119870) is the thermal conductivity of the

specimen 119881(1198983) and 119860(1198982) are the volume and surface area of the specimen

The magnitude of Bi measures the relative rates of convective to conductive heat

transfer For 119861119894 1 the heat conduction within the specimen takes place faster than heat

convection from the sample surface and the freezing process is convection limited

Otherwise the freezing process is conduction limited For convection limited cooling the

average cooling rate is (Bachmann and Talmon 1984)

119889119879

119889119905= minus

119860

119881ℎ(1198790 minus 119879119888)

1

120588119862119875 ( 6-56 )

where 119889119879

119889119905(119870

119904) is the cooling rate119879119888 is the temperature of cryogen and 1198790 is the temperature

of the specimen surface 120588 (119896119892

1198983) and 119862119875 (

119869

119896119892119870) are the density and heat capacity of the

specimen

For conduction limited cooling the average cooling rate is (Jaeger and Carslaw

1959)

119889119879

119889119905= minus(

119860

119881)2

119896119888(1198790 minus 119879119888)1

120588119862119901 ( 6-57 )

Table 6-8 summarizes the required physical properties of the coal specimen to

identify the dominant heat transfer mode and determine the corresponding cooling rate

imposed by liquid nitrogen At room temperature the crack fill is composed of water and

air The volumetric fraction of water or water saturation (119904119908) of the saturated coal specimen

is 0317 which is directly determined from a combination of moisture content and void

207

volume as given in Table 6-6 Thermal properties of the wet coal specimen including

thermal conductivity and thermal capacity were experimentally measured and the heat

transfer coefficient of convection (ℎ) was inverted from the literature data on immersion

freezing by liquid nitrogen (Zasadzinski 1988) With these thermophysical parameters

specified in Table 6-8 the Biot number for the studied coal specimen is

ℎ119881

119896119888119860=(2013)(00101)

0226= 899 ( 6-58 )

Hence heat convection from the sample to the cryogen is much faster than

conduction in the sample The immersion freezing of the studied coal specimen should be

dominated by the heat conduction process In general the fracture water is very difficult

to evenly and properly freeze Here we chose to report the cooling rate and the frozen

water content at the normal freezing point of water (Bailey and Zasadzinski 1991)

According to Eq (6-57) the conduction-limited cooling rate was estimated to be 0378 Ks

It took 66 seconds to cool down the specimen to the normal freezing point of water at

273119870 The result of the thermal analysis implied that the crack fill of the frozen specimen

was a two-phase fluid ie air and ice except for the first seismic measurement made at

room temperature Considering the volumetric expansion of ice the ice occupied void

volume out of total volume increased from 0317 to 0345

208

Table 6-8 Thermophysical parameters used in modeling heat transfer in the freezing

immersion test The heat capacity (Cp) and heat conductivity (kc) of the saturated coal

specimen (M-2) were measured at room temperature of 25following the laser flash

method (ASTM E1461-01)

ℎ 119862119901 119896119888 120588 119904119908 119904119894119888119890

(Wm2K) (JkgK) (WmK) (kgm3) (-) (-)

2013 953 0226 1380 0313 0345

Under the saturated condition fracture stiffnesses can be derived from the S- and

P- wave data crack statistics and the properties of the crack infill The elastic moduli of

the crack fill were estimated as volumetric averages of elastic moduli of ice and air for the

frozen coal specimen For the first measurement they were average properties of water and

air The constrained and shear modulus of ice (Mice and Gice) are 133 and 338 GPa

(Petrenko and Whitworth 1999) of water (Mw and Gw) are 225 and 0 GPa (Rodnikova

2007) and of air (Mair and Gair) are 10times 105 and 0 Pa (Beer et al 2014) Variations of

fracture stiffness with freezing timeare shown in Figure 6-26

209



Overall both normal and tangential fracture stiffnesses exhibited decreasing trends

with freezing time except for the first measurement made at room temperature Apart from

the significant thermal contract water contained in the cracks aggravated breaking coal

when the water froze and added additional splitting forces on the pre-existing or induced

fracturescleats The resulted increase in crack density created more open region in the

fracture surface which in turn decreased both normal and shear stiffnesses of the fracture

as shown in Figure 6-26 The initial increase in fracture stiffness was due to the transition

from the liquid phase (water) to the solid state (ice) inside the cracks and hence the

stiffening of the fracture The presence of an incompressible fluid in a fracture serves to

increase 119870119899 dramatically while leaving 119870119905 unchanged such that 07 119870119905119870119899 09 when

the coal sample was dry (see Figure 6-25) and that water saturation decreased 119870119905119870119899~01

210

(see the first point of 119870119905119870119899 ratio in Figure 6-26) This is consistent with the theoretical

prediction of a menagerie of rock physics models (Liu et al 2000 Sayers and Kachanov

1995 Schoenberg 1998) Sayers and Kachanov (1995) has shown that the stiffness ratio

of gas-filled fracture is

119870119905119870119899=1 minus 120584

1 minus1205842

( 6-59 )

where ν is Poissonrsquos ratio of the uncracked rock

For coal Poissonrsquos ratio is generally in the range of 02-04 (inverted from the

seismic measurements listed in Figure 6-24) and thus a value of 07 119870119905119870119899 09 is

anticipated for dry fractures which agrees with the experimental result of this study In the

presence of fluid filling cracks Liu et al (2000) has derived the stiffness ratio to be

119870119905119870119899=

7

8 [1 +92120587

119872prime

radic1 minus 1198902119872]

( 6-60 )

In the model they ignored the shear modulus of the containing fluid For fluid-

filled cracks the estimated ratio of 01 119870119905119870119899 09 is anticipated for an ellipticity ratio

(119890) of 09 (see Table 6-7) and 119872 in the range of 1-3 GPa (inverted from the seismic

measurements listed in Figure 6-24) A value of 01 corresponds to the case of fully

saturated and a value of 09 corresponds to the case of gas drained Our 119870119905119870119899 results

under saturated condition are consistent with the theoretical prediction In Figure 6-26 the

initial increase in the value of 119870119905119870119899 was caused by the phase transition from water to ice

Figure 6-27 is a sketch to explain the different mechanical interactions operating in water

and ice-filled cracks where a saw-tooth surface simulates the natural roughness of coal

211

cracks Freezing of water in cracks leads to an inhibited shearing of asperities that increases

shear resistance of rock masses (Krautblatter et al 2013) Hence the presence of ice would

stiffen the fracture in both normal and shear direction while the presence of water cannot

sustain shear deformation and would stiffen the fracture only in normal direction This

explains why the values of 119870119905119870119899 ratio for ice-filled fracture is greater than the water-filled

values On the timescale of the applied seismic pulse (in the order of 10 120583s) the fluid will

not have time to escape the fracture in other word the cracks are hydraulically isolated

For this reason 119870119905119870119899 kept relatively unchanged with freezing time as shown in Figure 6-

26

Figure 6-26 Effect of the presence of water and ice on fracture stiffness A saw-tooth

surface represents the natural roughness of rock fractures

212

III Discussion of Hydraulic Response of Coal Specimens with Liquid Nitrogen Treatment

Under dry and saturated conditions the common behavior for coal specimens

subjected to liquid nitrogen freezing is the decreasing trend of normal and shear fracture

stiffness with the increase of freezing time Numerous work (Petrovitch et al 2013 Pyrak-

Nolte 1996 Pyrak-Nolte 2019 Pyrak-Nolte and Morris 2000) have suggested that the

fluid flow is implicitly related to the fracture stiffness because both of them depend on the

geometry the size and the distribution of the void space For lognormal Gaussian and

uniform distributions of apertures an examination of this interrelationship has been made

in Pyrak-Nolte et al (1995) and the fluid flow (119876) is related to the fracture stiffness K

through

119870 = 120575radic1198763

( 6-61 )

where 120575 is a constant dependent upon the characteristics of the flow path

This theoretical model indicates that fracture stiffness is inversely related to the

cubit root of the flow rate In addition to this theoretical model tremendous experimental

data compiled by Pyrak-Nolte (1996) and Pyrak-Nolte and Morris (2000) also indicated

that rock samples with low fracture stiffness would have a higher flowability Thus the

apparent decreases of both normal and shear fracture stiffnesses shown in Figure 6-25 and

Figure 6-26 is an indicator of the improvement in the fluid flowability due to continuous

liquid nitrogen treatment For saturated specimen the presence of ice would increase

elastic moduli of the crack fill and lead to the stiffening of the fracture As a result the

saturated specimen underwent less reduction in fracture stiffness than the dry specimen for

213

the same freezing time In terms of hydraulic property coal samples in the state of

saturation require longer freezing time to reach the same increase in flow capability as

those in the dry state

The outcome of this study confirms that the 119870119905119870119899 ratio is dependent on the fluid

content Our estimate of 119870119905119870119899 ratio for dry coal specimen has a value in the range of

07 119870119905119870119899 09 and for saturated coal specimen it has a value in the range of 01

119870119905119870119899 03 These values of 119870119905119870119899 ratio are consistent with static and dynamic

measurements of stiffness ratio from other works using different methods which are

summarized in Verdon and Wuumlstefeld (2013) Specifically Sayers (1999) found that the

dry shale samples held 047 119870119905119870119899 08 and the saturated shale samples held ratio

026 119870119905119870119899 041 where these values were inverted from ultrasonic measurements

made by Hornby et al (1994) and Johnston and Christensen (1993) Our value of 119870119905119870119899for

dry coal sample is greater than those for dry shale sample As coal is more ductile than

shale coal should have a higher value of 120584 than shale yielding a higher stiffness ratio as

dictated by Equation (45) Our measurements made for the water saturated coal specimen

are slightly lower than saturated shale specimen A key difference that might account for

this discrepancy is that while Hornby et al (1994) measurements are of clay-fluid

composite filled cracks our measurements are made for pure water saturated cracks The

constituents of solid material such as clay in the crack infill increases shear fracture

stiffness and boosts 119870119905119870119899 ratio This also explains the initial rise of 119870119905119870119899 ratio in Figure

6-26 as water evolves into ice in response to the immersion freezing by liquid nitrogen

214

Investigations of measurements on 119870119905119870119899 ratio is mainly motivated by the need to

develop the detailed discrete fracture network models for improved accuracy of flow

modeling within fractured reservoirs An accurate estimate of stiffness ratio is very useful

to interpret fluid saturating state andor presence of detrital or diagenetic material inside

the fracture Such information may be immediate relevance to fluid flow through the

reservoir and therefore to reservoir productivity The common practice is to use 119870119905119870119899

ratio of 1 when modeling gas-filled fractures (Lubbe et al 2008) The outcome of this

study suggests that a 119870119905119870119899ratio of 08 would be a more realistic estimation for air-dry

coal Inversion of ultrasonic measurements on saturated coal shows a lower value of 119870119905119870119899

in comparison with dry coal and the magnitude is sensitive to the saturation state of coal

68 Summary

Cryogenic fracturing using liquid nitrogen can be an optional choice for the

unconventional reservoir stimulation Before large-scale field implementation a

comprehensive understanding of the fracturepore alteration will be essential and required

Pore-Scale Investigation

This study analyzed the pore-scale structure evolution by cryogenic treatment for

coal and its corresponding effect on the sorption and diffusion behaviors

bull Gas sorption kinetics There are two critical parameters in long-term CBM production

which are Langmuir pressure (119875119871) and diffusion coefficient (119863) A coal reservoir with

higher values of 119875119871 and 119863 are preferred in CBM production Due to low temperature

cycles both 119875119871 and 119863 of the studied Illinois coal sample are improved This

215

experimental evidence shows the potential of applying cryogenic fracturing to improve

long-term CBM well performance

bull Experimental and modeling results of pore structural alterations Hysteresis Index

(HI) is defined for low-pressure N2 adsorption isotherm at 77K to characterize the pore

connectivity of coal particles The freeze-thawed coal samples have smaller values of

HI than the coal sample without treatment implying that cryogenic treatment improves

pore connectivity The effect of freezing and thawing on pore volume and its

distribution are studied both by experimental work and the proposed micromechanical

model Based on a hypothesis that the pore structural deterioration of coal is the dilation

of nanopores due to water freezing in them and thermal deformation a

micromechanical model is developed for simulating these microscopic processes and

predicting the deterioration degree of pore structure due to the repetition of freezing

and thawing As a common characteristic of modeled result and experimental result

the total volume of mesopore and macropore increased after cryogenic treatment but

the growth rate of pore volume became much smaller as freezing and thawing were

repeated Pores in different sizes would experience different degrees of deterioration

In the range of micropores no significant alterations of pore volume occurred with the

repetition of freezing and thawing In the range of mesopores pore volume increased

with the repetition of freezing and thawing In the range of macropores pore volume

increased after the first cycle of freezing and thawing while decreased after three

cycles of freezing and thawing

216

bull Interrelationships between pore structural characteristics and gas transport Pore

volume expansion due to liquid nitrogen injections gives more space for gas molecules

to travel and enhances the overall diffusion process of the coal matrix The effect of

cyclic cryogenic treatment on pore structure of coal varies depending on the mechanical

properties of coal For the studied coal sample as macropore were further damaged

while mesopore were slightly influenced by repeated freezing and thawing the shift of

fractional pore volume into the direction of smaller pores inhibits gas diffusion in coal

matrix Thus dependent on coal type multiple cycles of freezing and thawing may not

be as efficient as a single cycle of freezing and thawing

bull This study demonstrates that cryogenic fracturing altered the nanometer-scale pore

systems of coal Correspondingly the application of freeze-thawing treatment

benefited the desorption and transport of gas and ultimately improved CBM production

performance The outcome of this study provides a scientific justification for post-



Cleat-Scale Investigation

This study developed a method to evaluate fracture stiffness by inverting seismic

measurements for assessment of the effectiveness of cryogenic fracturing which captures

the convoluted fracture topology without conducting a detailed analysis of fracture

geometry Since fracture stiffness and fluid capability are implicitly related a theoretical

model based on the meanfield theory was established to determine fracture stiffness from

seismic measurements such that hydraulic and seismic properties are interrelated Under

217

both dry and saturated conditions we recorded the real-time seismic response of coal

specimens in the freezing process and delineated the corresponding variation in fracture

stiffness induced by cryogenic forces using the proposed model The results indicated that

ultrasonic velocity of dry and saturated coal specimens overall decrease with freezing time

because of the provoked thermal and frost damages Based on this work the following

conclusions can be drawn

bull For saturated coal specimens the frozen water content was controlled by heat

conduction process as heat convection from the coal sample to the cryogen was much

faster than conduction in the coal sample The formation of ice out of water would

stiffen the fracture in both normal and shear direction while the presence of water

would stiffen the fracture only in normal direction For this reason both normal and

shear fracture stiffness initially increased with freezing time and then decreased for

longer freezing time as more cracks and open regions were created by cryogenic forces

bull For gas-filled cracks both normal and shear fracture stiffness of the dry coal specimen

exhibited universal decreasing trends with freezing time Due to the fracture filling

gas-filled cracks have lower fracture stiffness than water and ice filled cracks The

measured fracture stiffness of dry coal specimen had values in a range of 70-120 GPam

in normal direction and 50-80 GPam in tangential direction for up to 60 minutes of

cryogenic treatment Normal and shear fracture stiffness of saturated coal specimen at

room temperature had values of 200 and 1800 GPam respectively and at cryogenic

temperature normal fracture stiffness varied between 10000 and 10500 GPam and

shear fracture stiffness varied between 2000 and 2800 GPam

218

bull The saturated coal specimen underwent less reduction in fracture stiffness than the dry

coal specimen for the same freezing time As the formation of ice provoked by

cryogenic treatment leads to the grunting of rock masses the water-filled cracks have

higher normal and shear resistance than the gas-filled cracks Our conclusions

regarding to the behavior of fracture stiffness subjected to cryogenic treatment is that

coalbed with higher water saturation are preferred in the application of cryogenic

fracturing This is because fluid filled cracks can endure larger cryogenic forces before

complete failures and the contained water aggravates breaking coal as ice pressure

builds up from volumetric expansion of water-ice phase transition and adds additional

splitting forces on the pre-existing or induced fracturescleats

bull The results of this study are confirmation that fracture stiffness ratio is dependent on

the fluid content Our estimate of 119870119905119870119899 ratio for dry coal specimen had a value in the

range of 07 119870119905119870119899 09 and for saturated coal specimen it had a value in the

range of 01 119870119905119870119899 03 The introduction of a relatively incompressible fluid into

a fracture typically increases the normal fracture stiffness but keeps the shear fracture

stiffness unchanged such that the value of 119870119905119870119899 is lowered An accurate estimate of

stiffness ratio is very useful to develop the detailed discrete fracture network models

In contrast to the common practice of using 1 as 119870119905119870119899 ratio for gas filled fractures

the 119870119905119870119899 ratio of 08 is a more realistic estimation for air-dry coal

219

Chapter 7

CONCLUSIONS

71 Overview of Completed Tasks

The work completed in this thesis explores gas sorption and diffusion behavior in

coalbed methane reservoirs with a special focus on the intrinsic relationship between

microscale pore structure and macroscale gas transport and storage mechanism This work

can be broadly separated into two parts including theoretical and experimental study The

theoretical study revisits the fundamental principles on gas sorption and diffusion in

nanoporous materials Then theoretical models are developed to predict gas adsorption

isotherm and diffusion coefficient of coal based on pore structure parameters such as pore

volume PSD surface complexity The proposed theoretical models are validated by

laboratory data obtained from gas sorption experiment The knowledge on the scale

translation from microscale structure to macroscopic gas flow in coal matrix is further

applied to forecast field production from mature CBM wells in San Juan Basin Another

application of the theoretical and experimental works is the development of cryogenic

fracturing as a substitute of traditional hydraulic fracturing in CBM reservoirs This work

investigates the damage mechanism of the injection of cool fluid into warm coal reservoirs

at pore-scale and fracture-scale that aims at an improved understanding on the effectiveness

of this relatively new fracturing technique Here we reiterate the conclusions drawn from

Chapter 2 to Chapter 6

220

72 Summary and Conclusions

In Chapter 2 a comprehensive review on gas adsorption theory and diffusion

models was accomplished This chapter presents the theoretical modeling of gas storage

and transport in nanoporous coal matrix based on pore structure information The concept

of fractal geometry is used to characterize the heterogeneity of pore structure of coal by a

single parameter fractal dimension The methane sorption behavior of coal is adequately

modeled by classical Langmuir isotherm Gas diffusion in coal is characterized by Fickrsquos

law By assuming a unimodal pore size distribution unipore model can be derived and

applied to determine diffusion coefficient from sorption rate measurements This work

establishes two theoretical models to study the intrinsic relationship between pore structure

and gas sorption and diffusion in coal as pore structure-gas sorption model and pore

structure-gas diffusion model Major findings are summarized as follows









221




sorption capacity




value of 119875119871










dynamically





222





with pressure










production profiles

Chapter 3 presents the experimental method and procedures in this study to obtain

gas sorption kinetics and pore structural characteristics of coal Major achievements





223

















Chapter 4 investigates the validity of theoretical models developed in Chapter 2





224

bull Langmuir isotherm provides adequate fits to experimentally measured sorption

isotherms of all the bituminous coal samples involved in this study Based on the FHH

method two fractal dimensions 1198631 and 1198632 referred as pore surface and structure fractal












bull The unipore model provides satisfactory accuracy to fit lab-measured sorption kinetics

and derive diffusion coefficients of coal at different gas pressures A computer program

in Appendix A is constructed to automatically and time-effectively estimate the

diffusion coefficients with regressing to experimental sorption rate data





225



Chapter 5 investigates the impact of the pressure-dependent diffusion coefficient

on CBM production An equivalent matrix permeability modeling is proposed to convert



are integrated into reservoir simulation constructed in CMG-GEM simulator History-

match to field data are made for two mature San Juan fairway wells to validate the proposed

equivalent matrix modeling in gas production forecasting Based on this work the

following conclusions can be drawn

bull Gas flow in the matrix is driven by the concentration gradient whereas in the fracture

is driven by the pressure gradient The diffusion coefficient can be converted to

equivalent permeability as gas pressure and concentration are interrelated by real gas

law

bull The diffusion coefficient is pressure-dependent in nature and in general it increases

with pressure decreases since desorption gives more pore space for gas transport

Therefore matrix permeability converted from the diffusion coefficient increases

during reservoir depletion

bull The simulation study shows that accurate modeling of matrix flow is essential to predict

CBM production For fairway wells the growth of cleat permeability during reservoir

depletion only provides good matches to field production in the early de-watering stage

226

whereas the increase in matrix permeability is the key to predict the hyperbolic decline

behavior in the long-term decline stage Even with the cleat permeability increase the

conventional constant matrix permeability simulation cannot accurately predict the

concave-up decline behavior presented in the field gas production curves

bull This study suggests that better modeling of gas transport in the matrix during reservoir

depletion will have a significant impact on the ability to predict gas flow during the

primary and enhanced recovery production process especially for coal reservoirs with

high permeability This work provides a preliminary method of coupling pressure-

dependent diffusion coefficient into commercial CBM reservoir simulators

bull The equivalent matrix permeability is a variable approach to implicitly take the


production prediction This modeling results demonstrate that the diffusivity has not

only an impact on the late stable production behavior for mature wells but also has a

considerable effect on the peak production for the well In conclusion the pressure-

dependent gas diffusion coefficient should be considered for gas production prediction

without which both peak production and elongated production tail cannot be modeled

Chapter 6 researches on the applicability of cryogenic fracturing as an alternative

of traditional hydraulic fracturing in CBM formations using the theoretical analysis

documented in Chapter 2 and experimental method depicted in Chapter 3 Waterless

fracturing using liquid nitrogen can be an optional choice for the unconventional reservoir

227

stimulation Before large-scale field implementation a comprehensive understanding of

the fracture and pore alteration is essential and required

Pore-scale investigation on the effectiveness of cryogenic fracturing focuses on

pore structure evolution induced by freeze-thawing treatment of coal and its corresponding

change in gas sorption and diffusion behaviors

bull Cyclic injections of cryogenic fluid to coal creates more pore volume with the most

predominant increase observed in mesopores between 2 nm and 50 nm by 60 based

on low-pressure N2 sorption isotherms at 77K However no significant alterations of

pore volume occur in the range of micropores when subject to the repetition of freezing

and thawing operations as characterized by low-pressure CO2 isotherms at 298 K

bull A micromechanical model is developed for simulating these microscopic processes and


and thawing This model assumes that pore structural deterioration of coal is induced

by the dilation of nanopores due to water freezing in them and thermal deformation

The results of the micromechanical model suggest that total pore volume of coal is

enlarged when subject to the frost-shattering and thermal shock forces but the growth

rate of pore volume becomes much smaller as freezing and thawing are repeated This

modeling result agrees with experimental observation where the change of pore

volume tends to be relatively small after the first cycle of freezing and thawing

bull In response to the induced pore volume expansion by liquid nitrogen injections the

overall diffusion process in coal matrix is significantly enhanced The measured

diffusion coefficient of coal increases by 30 on average due to cryogenic treatments

228

Also cryogenic fracturing homogenizes the pore structure of coal with a narrower pore

size distribution As a result desorption pressure becomes smaller after cyclic freezing

and thawing treatments Cryogenic fracturing enhances gas flow in coal matrix during

production However dependent on coal type multiple cycles of freezing and thawing

may not be as efficient as a single cycle of freezing and thawing because further frozen

damages may break large pores into smaller pores while create negligible number of

new pores that inhibits transport of gas molecules in coal matrix

bull This study demonstrates that cryogenic fracturing alters the nanometer-scale pore

systems of coal Correspondingly the application of freeze-thawing treatment benefits

gas transport in coal matrix that ultimately improves CBM production performance

The outcome of this study provides a scientific justification for post-cryogenic

fracturing effect on diffusion improvement and gas production enhancement especially

for high ldquosorption timerdquo CBM reservoirs

Fracture or cleat scale investigation of cryogenic fracturing focuses on the evolution

of fracture stiffness of coal when exposed to low-temperature environment because fracture

stiffness and fluid capability are implicitly related This study develops a theoretical

seismic model to evaluate fracture stiffness by inverting seismic measurements for

assessment of the effectiveness of cryogenic fracturing which captures the convoluted

fracture topology without conducting a detailed analysis of fracture geometry Under both

dry and saturated conditions the real-time seismic response of coal specimens in the

freezing process is recorded and analyzed by the seismic model to determine the variation

229

of fracture stiffness induced by cryogenic fracturing Based on this work the following






















230
















231

APPENDIX A USER INTERFACE IN MATLAB FOR THE ESTIMATION

OF DIFFUSION COEFFICIENT

User interface in MATLAB GUI for the estimation of effective diffusivity

An automated computer program (ldquoUniporeModel Figrdquo) was constructed in

MATLAB GUI for estimating effective diffusion coefficient of coal from sorption rate

measurements based on unipore model (Eq 2-24) In the command window of MATLAB

type lsquoopen UniporeModelfigrsquo A user interface should pop up as shown in Figure A-1 The

required input is the experimental sorption rate data (ie 119872119905

119872infin vs t) The data should be

stored in a txt file in the same directory as the lsquoUniporeModelfigrsquo and named as

lsquodiffusiontxtrsquo The next entry is the search interval of Gold Section Search method for the

apparent diffusivity (119863119890) which are marked as 119863ℎ119894119892ℎ and 119863119897119900119908 in the unit of 119904minus1The last

required input is the number of terms in the infinite summation of unipore model denoted

as 119899119898119886119909 In the infinite summation the value of each individual term decreases as the index

of the term increases Thus an entry of 50 for 119899119898119886119909 is good enough to truncate the infinite

summation

Once all the required inputs are entered in the program hit the calculate button

Then the value of apparent diffusivity (119863119890) will pop up along with the percentage error

The error of the fitting by unipore model is determined as the average sum of squared

difference which is the ratio of the result from least-square function (Eq 2-26) over the

number of sorption rate datapoints With the determined apparent diffusivity the sorption

rate data is fitted by the unipore model (Eq 2-24) A figure of the experimental sorption

232

data with the regressed curve is shown at the bottom of the window Figure A-2 is an

example of applying the lsquoUniporeModelfigrsquo to determine the apparent diffusion

coefficient

Here 119910 denotes as the sorption fraction 119909 denotes as the apparent diffusion

coefficient Subscript lsquoexprsquo is the abbreviation of experimental and lsquomodelrsquo means sorption

rate data estimated by the unipore model 119863119890119905119903119906119890 is the determined diffusion coefficient

providing the best fit to the experimental data

Figure A-1 User Interface of the Automated MATLAB Program

233

Figure A-2 Typical example of applying lsquoUniporeModelfigrsquo to determine diffusion

coefficient

MATLAB Code

function varargout = UniporeModel(varargin) MATLAB GUI code (UniporeModelfig) to determine the apparent

diffusivity Last Modified by GUIDE v25 11-Jan-2018 145013

Begin initialization code - DO NOT EDIT gui_Singleton = 1 gui_State = struct(gui_Name mfilename gui_Singleton gui_Singleton gui_OpeningFcn UniporeModel_OpeningFcn

gui_OutputFcn UniporeModel_OutputFcn gui_LayoutFcn [] gui_Callback []) if nargin ampamp ischar(varargin) gui_Stategui_Callback = str2func(varargin1) end

if nargout [varargout1nargout] = gui_mainfcn(gui_State varargin)

234

else gui_mainfcn(gui_State varargin) end End initialization code - DO NOT EDIT

--- Executes just before De_true is made visible function UniporeModel_OpeningFcn(hObject eventdata handles

varargin) This function has no output args see OutputFcn Choose default command line output for De_true handlesoutput = hObject

Update handles structure guidata(hObject handles)

UIWAIT makes De_true wait for user response (see UIRESUME) uiwait(handlesfigure1)

--- Outputs from this function are returned to the command

line function varargout = UniporeModel_OutputFcn(hObject eventdata

handles) varargout cell array for returning output args (see

VARARGOUT) hObject handle to figure eventdata reserved - to be defined in a future version of

MATLAB handles structure with handles and user data (see GUIDATA)

Get default command line output from handles structure varargout1 = handlesoutput function xhigh_Callback(hObject eventdata handles) hObject handle to xhigh (see GCBO) eventdata reserved - to be defined in a future version of


Hints get(hObjectString) returns contents of xhigh as text str2double(get(hObjectString)) returns contents of

xhigh as a double

--- Executes during object creation after setting all

properties function xhigh_CreateFcn(hObject eventdata handles) hObject handle to xhigh (see GCBO) eventdata reserved - to be defined in a future version of

MATLAB handles empty - handles not created until after all

CreateFcns called

235

Hint edit controls usually have a white background on Windows See ISPC and COMPUTER if ispc ampamp isequal(get(hObjectBackgroundColor)

get(0defaultUicontrolBackgroundColor)) set(hObjectBackgroundColorwhite) end

function xlow_Callback(hObject eventdata handles) hObject handle to xlow (see GCBO) eventdata reserved - to be defined in a future version of


Hints get(hObjectString) returns contents of xlow as text str2double(get(hObjectString)) returns contents of

xlow as a double


properties function xlow_CreateFcn(hObject eventdata handles) hObject handle to xlow (see GCBO) eventdata reserved - to be defined in a future version of


CreateFcns called



function nmax_Callback(hObject eventdata handles) hObject handle to nmax (see GCBO) eventdata reserved - to be defined in a future version of


Hints get(hObjectString) returns contents of nmax as text str2double(get(hObjectString)) returns contents of

nmax as a double


properties function nmax_CreateFcn(hObject eventdata handles) hObject handle to nmax (see GCBO) eventdata reserved - to be defined in a future version of

MATLAB

236

handles empty - handles not created until after all

CreateFcns called



--- Executes on button press in pushbutton1 function pushbutton1_Callback(hObject eventdata handles) hObject handle to pushbutton1 (see GCBO) eventdata reserved - to be defined in a future version of

MATLAB handles structure with handles and user data (see GUIDATA) xhigh=str2double(get(handlesxhighstring)) xlow=str2double(get(handlesxlowstring)) nmax=str2double(get(handlesnmaxstring)) load diffusiontxt t=diffusion(1) yexp=diffusion(2) [De_true]=GS(xhighxlowtyexpnmax) set(handlesDe_truestringDe_true)

ymodel=zeros(length(t)1) for i=1length(ymodel) F=0 for n=1nmax F=F+(1n^2)exp(-1De_truen^2(pi())^2t(i)) end ymodel(i)=1-6pi^2F end hold off scatter(tyexpfilled) hold on plot(tymodel)

xlabel(Adsoprtion Time (s)) ylabel(Fraction) legend(Experimental DataAnalytical

Solutionlocationsoutheast)

Error=sum((yexp-ymodel)^2) Error=Errorlength(yexp)100 set(handlesErrorstringError)

Golden Seaction Search Alogrithm function [De_true]=GS(xhighxlowtyexpnmax) phi=0618

237

tol=10 itr=0 while tolgt1e-7 x2=(xhigh-xlow)phi+xlow x1=xhigh-(xhigh-xlow)phi S1=obj(tyexpx1nmax) S2=obj(tyexpx2nmax)

if S1gtS2 xlow=x1 else xhigh=x2 end tol=abs(S1-S2) itr=itr+1 end De_true=(x1+x2)2

Least-squares function function [S]=obj(tyexpDenmax)

ymodel=zeros(length(yexp)1) for i=1length(ymodel) F=0 for n=1nmax F=F+(1n^2)exp(-1Den^2(pi())^2t(i)) end ymodel(i)=1-6pi^2F end Objective Function S=sum((yexp-ymodel)^2)

238

APPENDIX B MATLAB PROGRAM TO DERIVE FRACTURE DENSITY

This computer program is developed for counting the number of fractures in a rock

In this study we used the automated code to extrapolate the crack density of the tested coal

specimens from the images taken in the experiment (see Figure 6-22) The basic algorithm

of this program is that it only accounts for isolated cracks and for cracks that are in

connection it treats them as a single crack The required input of this program is a text

image obtained through any image processing method For example ImageJ is a powerful

tool to convert a colorful image into a gray-scale image and an associated matrix (ie text

image) with each member representing a pixel and its numerical value corresponding to

the darkness in grayscale Using ImageJ you can set an appropriate threshold of grayscale

value to distinguish the grids containing cracks from the whole matrix With the threshold

specified the program will first index the input matrix Figure B-1 gives an example of the

indexed matrix and the cracks are located inside the grey region Unlike the output text

image the indexed matrix only contains three different numerical values The program will

assign an index of 1 to any grid with its numerical entry greater than the threshold of cracks

and for grids next to them the index of 2 will be assigned For all other grids away from

the cracks the index of 0 will be assigned

Based on the indexed matrix the program can automatically calculate the total

number of cracks and the areal proportion of crack region Detailed description of this

program will be given as follows the routine will scan from the top raw to the bottom raw

of the indexed matrix When it encounters a grid with an index of 1 it will examine the

239

neighboring grids that have already been scanned to identify if these grids are in

communication with grids with cracks (ie girds with index of 1 or 2) If the neighborhood

contains cracks the current grid should be connected to a previous crack and the total

number of cracks will not change Otherwise if all these surrounding grids have indexes

of 0 the program will increase the number of cracks by one The source code is given at

the end of the appendix In the code A is the input text image Area_Ratio_frac represents

the areal proportion of crack region and Nf denotates the number of cracks

Figure B-1 Indexed text image for counting the number of cracks Index notation given as

follows grids with cracks are marked as 1 neighboring grids of the girds with 1 are marked

as 2 all other grids are marked as 0

MATLAB Code

load TextImagetxt A=TextImage

Step 1 Set threshold to identify the crack region Number_frac=numel(A(Altthreshold))

Area fraction of crack region Area_Ratio_frac=Number_fracnumel(A(isnan(A)==0))

Step 2 Index the matrix for counting the number of cracks

240

Assign 1 to crack region 2 to the neighboring grids of the crack region

and 0 to elsewhere

A1(A1ltthreshold)=1 A1(A1gt=threshold)=0 for i=1size(A11) for j=2size(A12) if A1(ij)==1 if A1(ij+1)==0 A1(ij+1)=2 end if A1(ij-1)==0 A1(ij-1)=2 end end end end

Step 3

Count the number of cracks (Nf) Scan from top raw (i=2) to bottom raw

(i=max(pixels in Y direction))

Nf=0 for i=2size(A11) for j=2size(A12) if A1(ij)==1

subroutine to check if nearby grids contain cracks

if A1(ij-1)gt0 || A1(i-1j)gt0 break end Nf=Nf+1 end end end

X=11size(A11) Y=11size(A12) [XXYY]=meshgrid(XY) surf(XXYYA1)

241

REFERENCE

Achenbach J D and Norris A (1982) Loss of Specular Reflection Due to Nonlinear

Crack-Face Interaction Journal of Nondestructive Evaluation 3(4) 229-239

Adler L and Achenbach J D (1980) Elastic Wave Diffraction by Elliptical Cracks

Theory and Experiment Journal of Nondestructive Evaluation 1(2) 87-99

httpdoiorg101007bf00566117

Al-Jubori A Johnston S Boyer C Lambert S W Bustos O A Pashin J C and

Wray A (2009) Coalbed Methane Clean Energy for the World Oilfield Review

21(2) 4-13

Alghamdi T M Arns C H and Eyvazzadeh R Y (2013) Correlations between Nmr-

Relaxation Response and Relative Permeability from Tomographic Reservoir-

Rock Images SPE Reservoir Evaluation amp Engineering 16(04) 369-377

httpdoiorg102118160870-PA

Anderson R B (1946) Modifications of the Brunauer Emmett and Teller Equation

Journal of American Chemical Society 68(4) 686-691

httpdoiorg101021ja01208a049

Angel Y C and Achenbach J D (1985) Reflection and Transmission of Elastic Waves

by a Periodic Array of Cracks Journal of Applied Mechanics 52(1) 33-41

Arps J J (1945) Analysis of Decline Curves Society of Petroleum Engineers 160(01)

228-247

Avnir D Farin D and Pfeifer P (1983) Chemistry in Noninteger Dimensions between

Two and Three Ii Fractal Surfaces of Adsorbents The Journal of chemical

physics 79(7) 3566-3571

Avnir D Farin D and Pfeifer P (1984) Molecular Fractal Surfaces Nature 308 261

httpdoiorg101038308261a0

Avnir D and Jaroniec M (1989) An Isotherm Equation for Adsorption on Fractal

Surfaces of Heterogeneous Porous Materials Langmuir 5(6) 1431-1433

Ayers Jr W B (2003) Coalbed Methane in the Fruitland Formation San Juan Basin

Western United States A Giant Unconventional Gas Play AAPG Memoir 85(78)

159-188

Ayers W B Kaiser W R Ambrose W A Swartz T E and Laubach S E (1990)

Geologic Evaluation of Critical Production Parameters for Coalbed Methane

Resources Part 1 San Juan Basin Annual Report August 1988-July 1989 Texas

Bureau of Economic Geology

Ayers W B J and Zellers D (1991) Geologic Controls on Fruitland Coal Occurrence

Thickness and Continuity Navajo Lake Area San Juan Basin Gas Research

Institute

Bachmann L and Talmon Y (1984) Cryomicroscopy of Liquid and Semiliquid

Specimens Direct Imaging Versus Replication Ultramicroscopy 14(3) 211-218

Baik J-M and Thompson R B (1984) Ultrasonic Scattering from Imperfect Interfaces

A Quasi-Static Model Journal of Nondestructive Evaluation 4(3-4) 177-196

242

Bailey S M and Zasadzinski J A (1991) Validation of Convection‐Limited Cooling

of Samples for Freeze‐Fracture Electron Microscopy Journal of Microscopy

163(3) 307-320

Ball P and Evans R (1989) Temperature Dependence of Gas Adsorption on a

Mesoporous Solid Capillary Criticality and Hysteresis Langmuir 5(3) 714-723

Barrett E P Joyner L G and Halenda P P (1951) The Determination of Pore Volume

and Area Distributions in Porous Substances I Computations from Nitrogen

Isotherms Journal of the American Chemical Society 73(1) 373-380


Baskaran S and Kennedy I (1999) Sorption and Desorption Kinetics of Diuron

Fluometuron Prometryn and Pyrithiobac Sodium in Soils Journal of

Environmental Science amp Health Part B 34(6) 943-963

Beer F P Johnston E R Dewolf J T and Mazurek D F (2014) Mechanics of

Materials McGraw-Hill

Bell G and Jones A (1989) Variation in Mechanical Strength with Rank of Gassy Coals

Paper presented at the Proceeding of the 1989 Coalbed Methane Symposium

Bell G J and Rakop K C (1986a) Hysteresis of MethaneCoal Sorption Isotherms

Paper presented at the SPE Annual Technical Conference and Exhibition

Bell G J and Rakop K C (1986b) Hysteresis of MethaneCoal Sorption Isotherms

Paper presented at the SPE Annual Technical Conference and Exhibition New

Orleans Louisiana httpsdoiorg10211815454-MS

Bhagat R B (1985) Mode I Fracture Toughness of Coal International Journal of Mining

Engineering 3(3) 229-236 httpdoiorg101007bf00880769

Bhandari A and Xu F (2001) Impact of Peroxidase Addition on the Sorptionminus

Desorption Behavior of Phenolic Contaminants in Surface Soils Environmental

science amp technology 35(15) 3163-3168

Bhowmik S and Dutta P (2013) Adsorption Rate Characteristics of Methane and Co2

in Coal Samples from Raniganj and Jharia Coalfields of India International

Journal of Coal Geology 113 50-59 httpsdoiorg101016jcoal201302005

Bird G (1983) Definition of Mean Free Path for Real Gases The Physics of fluids 26(11)

3222-3223

Blahovec J and Yanniotis S (2008) Gab Generalized Equation for Sorption Phenomena

Food and Bioprocess Technology 1(1) 82-90 httpdoiorg101007s11947-007-

0012-3

Boer J H D (1953) The Dynamical Character of Adsorption Oxford Clarendon Press

Braida W J Pignatello J J Lu Y Ravikovitch P I Neimark A V and Xing B

(2003) Sorption Hysteresis of Benzene in Charcoal Particles Environmental

Science amp Technology 37(2) 409-417

Brunauer S Emmett P and Teller E (1938a) Absorption of Gases in Multimolecular

Layers Journal of American Chemical Society 60 309ndash319

Brunauer S and Emmett P H (1937) The Use of Low Temperature Van Der Waals

Adsorption Isotherms in Determining the Surface Areas of Various Adsorbents


243

httpdoiorg 101021ja01291a060

Brunauer S Emmett P H and Teller E (1938b) Adsorption of Gases in Multimolecular

Layers Journal of American Chemical Society 60(2) 309-319


Brunauer S Skalny J and Bodor E E (1969) Adsorption on Nonporous Solids

Journal of Colloid and Interface Science 30(4) 546-552

httpsdoiorg1010160021-9797(69)90423-8

Budaeva A D and Zoltoev E V (2010) Porous Structure and Sorption Properties of

Nitrogen-Containing Activated Carbon Fuel 89(9) 2623-2627

Budiansky B and Oconnell R J (1976) Elastic Moduli of a Cracked Solid

International Journal of Solids and Structures 12(2) 81-97

httpsdoiorg1010160020-7683(76)90044-5

Busch A and Gensterblum Y (2011) CBM and CO2-ECBM Related Sorption Processes

in Coal A Review International Journal of Coal Geology 87(2) 49-71

Busch A Gensterblum Y and Krooss B M (2003) Methane and Co2 Sorption and

Desorption Measurements on Dry Argonne Premium Coals Pure Components and

Mixtures International Journal of Coal Geology 55(2) 205-224

httpsdoiorg101016S0166-5162(03)00113-7

Busch A Gensterblum Y Krooss B M and Littke R (2004a) Methane and Carbon

Dioxide AdsorptionndashDiffusion Experiments on Coal Upscaling and Modeling

International Journal of Coal Geology 60(2-4) 151-168

Busch A Gensterblum Y Krooss B M and Littke R (2004b) Methane and Carbon


International Journal of Coal Geology 60(2) 151-168

httpsdoiorg101016jcoal200405002

Busch A Gensterblum Y Krooss B M and Siemons N (2006) Investigation of High-

Pressure Selective AdsorptionDesorption Behaviour of CO2 and CH4 on Coals An

Experimental Study International Journal of Coal Geology 66(1-2) 53-68

Bustin R M and Clarkson C R (1998) Geological Controls on Coalbed Methane

Reservoir Capacity and Gas Content International Journal of Coal Geology 38(1-

2) 3-26

Cai C Gao F Li G Huang Z and Hou P (2016) Evaluation of Coal Damage and

Cracking Characteristics Due to Liquid Nitrogen Cooling on the Basis of the

Energy Evolution Laws Journal of Natural Gas Science and Engineering 29 30-

36

httpsdoiorg101016jjngse201512041

Cai C Li G Huang Z Shen Z and Tian S (2014a) Rock Pore Structure Damage

Due to Freeze During Liquid Nitrogen Fracturing Arabian Journal for Science and

Engineering 39(12) 9249-9257

httpdoiorg101007s13369-014-1472-1

Cai C Li G Huang Z Shen Z Tian S and Wei J (2014b) Experimental Study of

the Effect of Liquid Nitrogen Cooling on Rock Pore Structure Journal of Natural

Gas Science and Engineering 21 507-517

244


Cai Y Liu D Pan Z Yao Y Li J and Qiu Y (2013) Pore Structure and Its Impact

on Ch4 Adsorption Capacity and Flow Capability of Bituminous and

Subbituminous Coals from Northeast China Fuel 103 258-268

Cai Y Liu D Yao Y Li J and Qiu Y (2011) Geological Controls on Prediction of

Coalbed Methane of No 3 Coal Seam in Southern Qinshui Basin North China


Ceglarska-Stefańska G and Zarębska K (2002) The Competitive Sorption of Co 2 and

Ch 4 with Regard to the Release of Methane from Coal Fuel Processing

Technology 77 423-429

Cha M Alqahtani N B Yin X Kneafsey T J Yao B and Wu Y-S (2017)

Laboratory System for Studying Cryogenic Thermal Rock Fracturing for Well

Stimulation Journal of Petroleum Science and Engineering 156 780-789

httpsdoiorg101016jpetrol201706062

Cha M Cho G-C and Santamarina J C (2009) Long-Wavelength P-Wave and S-

Wave Propagation in Jointed Rock Masses Geophysics 74(5) 205-214

httpdoiorg10119013196240

Cha M Yin X Kneafsey T Johanson B Alqahtani N Miskimins J Patterson T

and Wu Y-S (2014) Cryogenic Fracturing for Reservoir Stimulation ndash

Laboratory Studies Journal of Petroleum Science and Engineering 124 436-450


Chapman S Cowling T G and Burnett D (1990) The Mathematical Theory of Non-

Uniform Gases An Account of the Kinetic Theory of Viscosity Thermal

Conduction and Diffusion in Gases In Cambridge University Press

Charriegravere D Pokryszka Z and Behra P (2010) Effect of Pressure and Temperature on

Diffusion of CO2 and CH4 into Coal from the Lorraine Basin (France)



Chen T C Yeung M R and Mori N (2004) Effect of Water Saturation on

Deterioration of Welded Tuff Due to Freeze-Thaw Action Cold Regions Science

and Technology 38(2) 127-136

httpsdoiorg101016jcoldregions200310001

Chugh Y P Atri A and Dougherty J (1989) Laboratory and Field Characterization of

Immediate Floor Strata in Illinois Basin Coal Mines Paper presented at the The

30th US Symposium on Rock Mechanics Morgantown West Virginia

Clarkson C Jordan C Ilk D and Blasingame T (2012a) Rate-Transient Analysis of

2-Phase (Gas+ Water) Cbm Wells Journal of Natural Gas Science and

Engineering 8 106-120

Clarkson C and Mcgovern J (2003) A New Tool for Unconventional Reservoir

Exploration and Development Applications Paper presented at the International

Coalbed Methane Symposium Tuscaloosa Alabama

Clarkson C R and Bustin R M (1996) Variation in Micropore Capacity and Size

Distribution with Composition in Bituminous Coal of the Western Canadian

245

Sedimentary Basin Implications for Coalbed Methane Potential Fuel 75(13)

1483-1498

httpsdoiorg1010160016-2361(96)00142-1

Clarkson C R and Bustin R M (1999a) The Effect of Pore Structure and Gas Pressure

Upon the Transport Properties of Coal A Laboratory and Modeling Study 1

Isotherms and Pore Volume Distributions Fuel 78(11) 1333-1344

httpsdoiorg101016S0016-2361(99)00055-1

Clarkson C R and Bustin R M (1999b) The Effect of Pore Structure and Gas Pressure


Adsorption Rate Modeling Fuel 78(11) 1345-1362

Clarkson C R Bustin R M and Levy J H (1997) Application of the MonoMultilayer

and Adsorption Potential Theories to Coal Methane Adsorption Isotherms at

Elevated Temperature and Pressure Carbon 35(12) 1689-1705

httpsdoiorg101016S0008-6223(97)00124-3

Clarkson C R Jensen J L and Chipperfield S (2012b) Unconventional Gas Reservoir

Evaluation What Do We Have to Consider Journal of Natural Gas Science and

Engineering 8 9-33


Clarkson C R Jordan C L Gierhart R R and Seidle J P (2008) Production Data

Analysis of Coalbed-Methane Wells SPE Reservoir Evaluation amp Engineering

11(02) 311-325


Clarkson C R Pan Z Palmer I D and Harpalani S (2010) Predicting Sorption-

Induced Strain and Permeability Increase with Depletion for Coalbed-Methane

Reservoirs Society of Petroleum Engineers Journal 15(01) 152-159


Clarkson C R Rahmanian M Kantzas A and Morad K (2011) Relative Permeability

of Cbm Reservoirs Controls on Curve Shape International Journal of Coal

Geology 88(4) 204-217


Clifford P J Berry P J and Gu H (1991) Modeling the Vertical Confinement of

Injection-Well Thermal Fractures SPE Production Engineering 6(04) 377-383


Close J C Mavor M J and Mcbane R A (1990) Importance Genesis and Recognition

of Fracture Permeability in Fruitland Coalbed Methane Reservoirs of the Northern

San Juan Basin Colorado and New Mexico Paper presented at the Annual

Technical Meeting Calgary Alberta

httpsdoiorg10211890-106

Cmg‐Gem (2015) Gem Users Guide Advanced Compositional and GHG Reservoir

Simulator

Cole M W and Saam W F (1974) Excitation Spectrum and Thermodynamic Properties

of Liquid Films in Cylindrical Pores Physical Review Letters 32(18) 985-988

httpdoiorg101103PhysRevLett32985

246

Collins R (1991) New Theory for Gas Adsorption and Transport in Coal Paper presented

at the Proceedings of the 1991 Coalbed Methane Symposium Tuscaloosa USA

Coppens M-O (1999) The Effect of Fractal Surface Roughness on Diffusion and

Reaction in Porous Catalystsndashfrom Fundamentals to Practical Applications

Catalysis Today 53(2) 225-243

Coppens M-O and Froment G F (1995) Diffusion and Reaction in a Fractal Catalyst

PoremdashI Geometrical Aspects Chemical Engineering Science 50(6) 1013-1026

httpsdoiorg1010160009-2509(94)00478-A

Crank J (1975) The Mathematics of Diffusion Oxford GB Clarendon Press

Crickmore P J and Wojciechowski B W (1977) Kinetics of Adsorption on

Energetically Heterogeneous Surfaces Journal of Chemical Society Faraday

Transactions 1 Physical Chemistry in Condensed Phases 73 1216-1223

Crosdale P J Beamish B B and Valix M (1998) Coalbed Methane Sorption Related

to Coal Composition International Journal of Coal Geology 35(1) 147-158

httpsdoiorg101016S0166-5162(97)00015-3

Cui X Bustin A and Bustin R M (2009) Measurements of Gas Permeability and

Diffusivity of Tight Reservoir Rocks Different Approaches and Their

Applications Geofluids 9(3) 208-223

Cui X Bustin R M and Dipple G (2004) Selective Transport of CO2 CH4 and N2 in

Coals Insights from Modeling of Experimental Gas Adsorption Data Fuel 83(3)

293-303

httpsdoiorg101016jfuel200309001

Cullicott C (2002) Overview of CBM Issues in the San Juan Basin Paper presented at

the Coalbed Methane Development in the Intermountain West

Dabbous M K Reznik A A Mody B G Fulton P F and Taber J J (1976) Gas-

Water Capillary Pressure in Coal at Various Overburden Pressures Society of

Petroleum Engineers Journal 16(05) 261-268


Dake L P (1983) Fundamentals of Reservoir Engineering Elsevier Science

De Boer J H (1958) The Structure and Properties of Porous Materials In D H Everett

(Ed) London Butterworths Colston Research Society

Detienne J L Creusot M Kessler N Sahuquet B and Bergerot J L (1998)

Thermally Induced Fractures A Field-Proven Analytical Model SPE Reservoir

Evaluation amp Engineering 1(01) 30-35


Ding G Novak J M Herbert S and Xing B (2002) Long-Term Tillage Effects on

Soil Metolachlor Sorption and Desorption Behavior Chemosphere 48(9) 897-904

Ding G and Rice J A (2011) Effect of Lipids on SorptionDesorption Hysteresis in

Natural Organic Matter Chemosphere 84(4) 519-526

Do D D (1998) Adsorption Analysis Equilibria and Kinetics Imperial College Press

Drillinginfo (2020) Drillinginfo Retrieved from httpsappdrillinginfocom

247

Dubinin M M and Astakhov V A (1971) Description of Adsorption Equilibria of

Vapors on Zeolites over Wide Ranges of Temperature and Pressure In Molecular

Sieve Zeolites-Ii (Vol 102 pp 69-85) American Chemical Society

Dubinin M M and Plavnik G M (1968) Microporous Structures of Carbonaceous

Adsorbents Carbon 6(2) 183-192

httpsdoiorg1010160008-6223(68)90302-3

Dubinin M M Plavnik G M and Zaverina E D (1964) Integrated Study of the Porous

Structure of Active Carbons from Carbonized Sucrose Carbon 2(3) 261-268

httpsdoiorg1010160008-6223(64)90040-5

Dubinin M M and Radushkevich L V (1947) The Equation of the Characteristic Curve

of Activated Charcoal Paper presented at the Proceedings of the Academy of

Sciences

Dutta A (2009) Multicomponent Gas Diffusion and Adsorption in Coals for Enhanced

Methane Recovery (Master Thesis) Stanford University

Dutta P Bhowmik S and Das S (2011) Methane and Carbon Dioxide Sorption on a

Set of Coals from India International Journal of Coal Geology 85(3) 289-299


Dvoyashkin M Valiullin R and Kaumlrger J (2007) Temperature Effects on Phase

Equilibrium and Diffusion in Mesopores Physical Review E 75(4) 041202

httpdoiorg101103PhysRevE75041202

EIA (2018) Coalbed Methane Proved Reserves US Department of Energy

Emmett P H and Brunauer S (1937) The Use of Low Temperature Van Der Waals

Adsorption Isotherms in Determining the Surface Area of Iron Synthetic Ammonia

Catalysts Journal of American Chemical Society 59(8) 1553-1564


Enever J Casey D and Bocking M (1999) The Role of in-Situ Stress in Coalbed

Methane Exploration In M Mastalerz M Glikson amp S D Golding (Eds)

Coalbed Methane Scientific Environmental and Economic Evaluation (pp 297-

303) Dordrecht Springer Netherlands

EPA (2013) US Economic Analysis for Existing and New Projects in the Coalbed

Methane Industry US Environmental Protection Agency

Evans Iii R Truitt J and Watson G (1961a) Superposition of Forced and Disffusive

Flow in a Large-Pore Graphite In Oak Ridge National Lab

Evans Iii R Watson G and Mason E (1961b) Gaseous Diffusion in Porous Media at

Uniform Pressure The Journal of Chemical Physics 35(6) 2076-2083

Evans R (1990) Fluids Adsorbed in Narrow Pores Phase Equilibria and Structure

Journal of Physics Condensed Matter 2(46) 8989-9007

httpdoiorg1010880953-8984246001

Everett D (1961) The Thermodynamics of Frost Damage to Porous Solids Transactions

of the Faraday Society 57 1541-1551

Everett D H and Stone F S (1958) The Structure and Properties of Porous Materials

Paper presented at the 10th Symposium of the Colston Research Society London

Butterworths

248

Fick A (1855) About diffusion Annals of Physics 170(1) 59-86

httpdoiorg doi101002andp18551700105

Fine R and Millero F (1973) Compressibility of Water as a Function of Temperature

and Pressure The Journal of Chemical Physics 59 5529-5536

httpdoiorg 10106311679903

Fitzgerald J Pan Z Sudibandriyo M Robinson Jr R Gasem K and Reeves S

(2005) Adsorption of Methane Nitrogen Carbon Dioxide and Their Mixtures on

Wet Tiffany Coal Fuel 84(18) 2351-2363

French H M (2017) The Periglacial Environment John Wiley amp Sons

Fripiat J Gatineau L and Van Damme H (1986) Multilayer Physical Adsorption on

Fractal Surfaces Langmuir 2(5) 562-567

Fu H Tang D Xu T Xu H Tao S Li S Yin Z Chen B Zhang C and Wang

L (2017) Characteristics of Pore Structure and Fractal Dimension of Low-Rank

Coal A Case Study of Lower Jurassic Xishanyao Coal in the Southern Junggar

Basin NW China Fuel 193 254-264

httpdoiorg 254-264 101016jfuel201611069

Fukuda M (1974) Rock Weathering by Freezing-Thawing Cycles Low Temperature

Science Ser A 32 243-249

Gao F and Kang H (2017) Experimental Study on the Residual Strength of Coal under

Low Confinement Rock mechanics and Rock Engineering 50(2) 285-296

httpdoiorg101007s00603-016-1120-z

Garbin H and Knopoff L (1973) The Compressional Modulus of a Material Permeated

by a Random Distribution of Circular Cracks Quarterly of Applied Mathematics

30(4) 453-464

Garbin H and Knopoff L (1975) The Shear Modulus of a Material Permeated by a

Random Distribution of Free Circular Cracks Quarterly of Applied Mathematics

33(3) 296-300

Garrido J Linares-Solano A Martin-Martinez J M Molina-Sabio M Rodriguez-

Reinoso F and Torregrosa R (1987) Use of Nitrogen VS Carbon Dioxide in the

Characterization of Activated Carbons Langmuir 3(1) 76-81

Gash B Volz R Potter G and Corgan J (1993) The Effects of Cleat Orientation and

Confining Pressure on Cleat Porosity Permeability and Relative Permeability

Paper presented at the International Coalbed Methane Symposium The University

of Alabama Tuscloosa

George J S and Barakat M (2001) The Change in Effective Stress Associated with

Shrinkage from Gas Desorption in Coal International Journal of Coal Geology

45(2) 105-113

Gierhart R R Clarkson C R and Seidle J P (2007) Spatial Variation of San Juan

Basin Fruitland Coalbed Methane Pressure Dependent Permeability Magnitude

and Functional Form Paper presented at the International Petroleum Technology

Conference Dubai UAE

httpsdoiorg102523IPTC-11333-MS

249

Gilliland E R Baddour R F Perkinson G P and Sladek K J (1974) Diffusion on

Surfaces I Effect of Concentration on the Diffusivity of Physically Adsorbed

Gases Industrial amp Engineering Chemistry Fundamentals 13(2) 95-100

Goodman R E Taylor R L and Brekke T L (1968) A Model for the Mechanics of

Jointed Rock Journal of the Soil Mechanics and Foundations Division 94(3) 637-

659

Gray I (1987) Reservoir Engineering in Coal Seams Part 1-the Physical Process of Gas

Storage and Movement in Coal Seams SPE Reservoir Engineering 2(01) 28-34

Gregg S J Sing K S W and Salzberg H (1967) Adsorption Surface Area and

Porosity Academic Press

Groen J C and Peacuterez-RamıRez J (2004) Critical Appraisal of Mesopore

Characterization by Adsorption Analysis Applied Catalysis A General 268(1)

121-125

httpsdoiorg101016japcata200403031

Grundmann S R Rodvelt G D Dials G A and Allen R E (1998) Cryogenic

Nitrogen as a Hydraulic Fracturing Fluid in the Devonian Shale Paper presented at

the SPE Eastern Regional Meeting Pittsburgh Pennsylvania

httpsdoiorg10211851067-MS

Guan L and Norris A (1992) Elastic Wave Scattering by Rectangular Cracks


httpsdoiorg1010160020-7683(92)90133-E

Gubernatis J and Domany E (1979) Rayleigh Scattering of Elastic Waves from Cracks

Journal of Applied Physics 50(2) 818-824

Han F Busch A Krooss B M Liu Z and Yang J (2013) CH4 and CO2 Sorption

Isotherms and Kinetics for Different Size Fractions of Two Coals Fuel 108 137-

142

Harpalani S and Chen G (1997) Influence of Gas Production Induced Volumetric Strain

on Permeability of Coal Geotechnical and Geological Engineering 15(4) 303-

325

Harpalani S Prusty B K and Dutta P (2006) MethaneCO2 Sorption Modeling for

Coalbed Methane Production and CO2 Sequestration Energy amp Fuels 20(4) 1591-

1599

Hedayat A Pyrak-Nolte L J and Bobet A (2014) Precursors to the Shear Failure of

Rock Discontinuities Geophysical Research Letters 41(15) 5467-5475

httpdoiorg1010022014gl060848

Hiemenz P C and Hiemenz P C (1986) Principles of Colloid and Surface Chemistry

(Vol 9) M Dekker New York

Hong C Zhang W Pan B Lv L Han Y and Zhang Q (2009) Adsorption and

Desorption Hysteresis of 4-Nitrophenol on a Hyper-Cross-Linked Polymer Resin

Nda-701 Journal of Hazardous Materials 168(2-3) 1217-1222

Hori M and Morihiro H (1998) Micromechanical Analysis on Deterioration Due to

Freezing and Thawing in Porous Brittle Materials International Journal of

Engineering Science 36(4) 511-522

250

httpsdoiorg101016S0020-7225(97)00080-3

Hornby B E Schwartz L M and Hudson J A (1994) Anisotropic Effective-Medium

Modeling of the Elastic Properties of Shales Geophysics 59(10) 1570-1583

Hudson J Liu E and Crampin S (1997) The Mean Transmission Properties of a Fault

with Imperfect Facial Contact Geophysical Journal International 129(3) 720-

726

Hudson J A (1980) Overall Properties of a Cracked Solid Mathematical Proceedings of

the Cambridge Philosophical Society 88(2) 371-384

httpdoiorg101017S0305004100057674

Hudson J A Liu E and Crampin S (1996) Transmission Properties of a Plane Fault

Geophysical Journal International 125(2) 559-566

httpdoiorg101111j1365-246X1996tb00018x

Iso I O F S (2016) Evaluation of Pore Size Distribution and Porosity of Solid Materials

by Mercury Porosimetry and Gas Adsorption mdash Part 1 Mercury Porosimetry In

BS ISO 15901-1

Jaeger J C and Carslaw H S (1959) Conduction of Heat in Solids Oxford Science

Publications

Jaroniec M (1983) Physical Adsorption on Heterogeneous Solids Advances in Colloid

and Interface Science 18(3-4) 149-225

Javadpour F Fisher D and Unsworth M (2007) Nanoscale Gas Flow in Shale Gas

Sediments Journal of Canadian Petroleum Technology 46(10)

httpsdoiorg10211807-10-06

Jeans J H (1921) The Dynamical Theory of Gases University Press

Jin K Cheng Y Liu Q Zhao W Wang L Wang F and Wu D (2016)

Experimental Investigation of Pore Structure Damage in Pulverized Coal

Implications for Methane Adsorption and Diffusion Characteristics Energy amp

Fuels 30(12) 10383-10395

Johnson K L (1985) Contact Mechanics Cambridge University Press

Johnston J E and Christensen N I (1993) Compressional to Shear Velocity Ratios in

Sedimentary Rocks International Journal of Rock Mechanics and Mining Sciences

amp Geomechanics Abstracts 30(7) 751-754

httpsdoiorg1010160148-9062(93)90018-9

Jones A H Bell G J and Schraufnagel R A (1988) A Review of the Physical and

Mechanical Properties of Coal with Implications for Coal-Bed Methane Well

Completion and Production In J E Fassett (Ed) Geological and Coal-Bed

Methane Resources of the Northern San Juan Basin Colorado and New Mexico

Rocky Mountain Association of Geologists Guide Book (pp 169ndash181)

Joubert J I Grein C T and Bienstock D (1973) Sorption of Methane in Moist Coal

Fuel 52(3) 181-185

Joubert J I Grein C T and Bienstock D (1974) Effect of Moisture on the Methane

Capacity of American Coals Fuel 53(3) 186-191

Kainourgiakis M Steriotis T A Kikkinides E Romanos G and Stubos A (2002)

Adsorption and Diffusion in Nanoporous Materials from Stochastic and Process-

251

Based Reconstruction Techniques Colloids and Surfaces A Physicochemical and

Engineering Aspects 206(1-3) 321-334

Kalsi N S Sehgal R and Sharma V S (2010) Cryogenic Treatment of Tool Materials

A Review Materials and Manufacturing Processes 25(10) 1077-1100

Kapoor A and Yang R T (1989) Surface Diffusion on Energetically Heterogeneous

Surfaces AIChE Journal 35(10) 1735-1738

httpdoiorg101002aic690351019

Karacan C Ouml (2003) Heterogeneous Sorption and Swelling in a Confined and Stressed

Coal During Co2 Injection Energy amp Fuels 17(6) 1595-1608

Karbhari V Rivera J and Dutta P (2000) Effect of Short-Term Freeze-Thaw

Cyclingon Composite Confined Concrete Journal of Composites for Construction

4(4) 191-197

Kaumlrger J Chmelik C Heinke L and Valiullin R (2010) A New View of Diffusion in

Nanoporous Materials Chemie Ingenieur Technik 82(6) 779-804

Kaumlrger J Ruthven D M and Theodorou D N (2012) Diffusion in Nanoporous

Materials John Wiley amp Sons

Kelemen S R and Kwiatek L M (2009) Physical Properties of Selected Block Argonne

Premium Bituminous Coal Related to Co2 Ch4 and N2 Adsorption International

Journal of Coal Geology 77(1) 2-9


Keller J B (1964) Stochastic Equations and Wave Propagation In Stochastic Processes

in Mathematical Physics and Engineering

Kendall K and Tabor D (1971) An Utrasonic Study of the Area of Contact between

Stationary and Sliding Surfaces Proceedings of the Royal Society of London A

Mathematical and Physical Sciences 323(1554) 321-340

Keshavarz A Sakurovs R Grigore M and Sayyafzadeh M (2017) Effect of Maceral

Composition and Coal Rank on Gas Diffusion in Australian Coals International

Journal of Coal Geology 173 65-75


Kim A G (1977) Estimating Methane Content of Bituminous Coalbeds from Adsorption

Data Pittsburgh PA US Department of the Interior Bureau of Mines

King G R (1985) Numerical Simulation of the Simultaneous Flow of Methane and Water

through Dual Porosity Coal Seams During the Degasification Process (PhD

Dissertation) Pennsylvania State University

King G R Ertekin T and Schwerer F C (1986) Numerical Simulation of the Transient

Behavior of Coal-Seam Degasification Wells SPE Formation Evaluation 1(02)

165-183


King J C Bryan J L and Clark M (2012) Factual Causation The Missing Link in

Hydraulic Fracture-Groundwater Contamination Litigation Duke Environmental

Law and Policy Forum 22 (2)

Knudsen M (1909) The Laws of Molecular Flow and the Internal Frictional Flow of

Gases through Pipes Annals of Physics 333(1) 75-130

252

httpdoi101002andp19093330106

Krautblatter M Funk D and Guumlnzel F K (2013) Why Permafrost Rocks Become

Unstable A RockndashIce-Mechanical Model in Time and Space Earth Surface

Processes and Landforms 38(8) 876-887

httpdoiorg101002esp3374

Krishna R and Wesselingh J A (1997) The Maxwell-Stefan Approach to Mass

Transfer Chemical Engineering Science 52(6) 861-911

httpsdoiorg101016S0009-2509(96)00458-7

Kumar A (2007) Methane Diffusion Characteristics of Illinois Coals (Master Thesis)

Southern Illinois University Carbondale

Lama R and Bodziony J (1998) Management of Outburst in Underground Coal Mines


Langmuir I (1918) The Adsorption of Gases on Plane Surfaces of Glass Mica and

Platinum Journal of American Chemical Society 40(9) 1361-1403

Laubach S Marrett R Olson J and Scott A (1998) Characteristics and Origins of

Coal Cleat A Review International Journal of Coal Geology 35(1) 175-207

Levine J R (1996) Model Study of the Influence of Matrix Shrinkage on Absolute

Permeability of Coal Bed Reservoirs Geological Society London Special

Publications 109(1) 197-212

Li A Ding W He J Dai P Yin S and Xie F (2016) Investigation of Pore Structure

and Fractal Characteristics of Organic-Rich Shale Reservoirs A Case Study of

Lower Cambrian Qiongzhusi Formation in Malong Block of Eastern Yunnan

Province South China Marine and Petroleum Geology 70 46-57

httpsdoiorg101016jmarpetgeo201511004

Li Z Hao Z Pang Y and Gao Y (2015) Fractal Dimensions of Coal and Their

Influence on Methane Adsorption China Coal Society 40(4) 863-869

Linstrom P J and Mallard W G (2001) The Nist Chemistry Webbook A Chemical

Data Resource on the Internet Journal of Chemical amp Engineering Data 46(5)

1059-1063

Liu E Hudson J A and Pointer T (2000) Equivalent Medium Representation of

Fractured Rock Journal of Geophysical Research Solid Earth 105(B2) 2981-

3000

Liu S and Harpalani S (2013a) A New Theoretical Approach to Model Sorption-

Induced Coal Shrinkage or Swelling AAPG bulletin 97 1033-1049

httpdoiorg10130612181212061

Liu S and Harpalani S (2013b) Permeability Prediction of Coalbed Methane Reservoirs

During Primary Depletion International Journal of Coal Geology 113 1-10


Liu X and Nie B (2016) Fractal Characteristics of Coal Samples Utilizing Image

Analysis and Gas Adsorption Fuel 182 314-322

httpdoiorg 101016jfuel201605110

Lowell S and Shields J E (1991) Powder Surface Area and Porosity London New

York Chapman amp Hall

253

Lozano-Castelloacute D Cazorla-Amoroacutes D and Linares-Solano A (2004) Usefulness of

Co2 Adsorption at 273 K for the Characterization of Porous Carbons Carbon

42(7) 1233-1242

httpsdoiorg101016jcarbon200401037

Lubbe R Sothcott J Worthington M H and Mccann C (2008) Laboratory Estimates

of Normal and Shear Fracture Compliance Geophysical Prospecting 56(2) 239-

247

httpdoiorg101111j1365-2478200700688x

Ma L Southwick L M Willis G H and Selim H M (1993) Hysteretic

Characteristics of Atrazine Adsorption-Desorption by a Sharkey Soil Weed

Science 41(4) 627-633

Mandelbrot B B (1983) The Fractal Geometry of Nature Times Books

Marecka A and Mianowski A (1998) Kinetics of Co2 and Ch4 Sorption on High Rank

Coal at Ambient Temperatures Fuel 77(14) 1691-1696

Mason E A and Malinauskas A (1983) Gas Transport in Porous Media The Dusty-

Gas Model Elsevier Science

Mason G (1982) The Effect of Pore Space Connectivity on the Hysteresis of Capillary

Condensation in AdsorptionmdashDesorption Isotherms Journal of Colloid and

Interface Science 88(1) 36-46

httpsdoiorg1010160021-9797(82)90153-9

Matsuoka N (1990) Mechanisms of Rock Breakdown by Frost Action An Experimental

Approach Cold Regions Science and Technology 17(3) 253-270

httpsdoiorg101016S0165-232X(05)80005-9

Mavko G Mukerji T and Dvorkin J (2009) The Rock Physics Handbook Tools for

Seismic Analysis of Porous Media Cambridge University Press

Mavor M Owen L and Pratt T (1990a) Measurement and Evaluation of Coal Sorption

Isotherm Data Paper presented at the SPE Annual Technical Conference and

Exhibition

Mavor M J Close J C and Mcbane R A (1990b) Formation Evaluation of

Exploration Coalbed Methane Wells Paper presented at the Annual Technical

Meeting Calgary Alberta


Maxwell J C (1881) A Treatise on Electricity and Magnetism (Vol 1) Clarendon press

Mcdaniel B Grundmann S R Kendrick W D Wilson D R and Jordan S W (1997)

Field Applications of Cryogenic Nitrogen as a Hydraulic Fracturing Fluid Paper

presented at the SPE Annual Technical Conference and Exhibition

Meyer O E (1899) The Kinetic Theory of Gases Maruschke amp Berendt

Mistler T E Correll G R and Mingle J O (1970) Gaseous Diffusion in the Transition

Pressure Range AIChE Journal 16(1) 32-37


Moore R L Loftin D F and Palmer I D (2011) History Matching and Permeability

Increases of Mature Coalbed Methane Wells in San Juan Basin Paper presented at

254

the SPE Asia Pacific Oil and Gas Conference and Exhibition Jakarta Indonesia


Mora C A Wattenbarger R A and Mcketta S (2007) Comparison of Computation

Methods for Cbm Performance Paper presented at the Canadian International

Petroleum Conference Calgary Alberta


Morse D G and Demir I (2007) Resource Assessment and Production Testing for

Coalbed Methane in Illinois Illinois State Geological Survey Illinois Department

of Natural Resources

Mullen M J (1989) Coalbed Methane Resource Evaluation from Wireline Logs in the

Northeastern San Juan Basin A Case Study Paper presented at the Low

Permeability Reservoirs Symposium Denver Colorado


Myer L R (2000) Fractures as Collections of Cracks International Journal of Rock

Mechanics and Mining Sciences 37(1) 231-243

httpsdoiorg101016S1365-1609(99)00103-3

Nandi S P and Walker P L (1975) Activated Diffusion of Methane from Coals at

Elevated Pressures Fuel 54(2) 81-86

httpsdoiorg1010160016-2361(75)90061-7

Naveen P Asif M Ojha K Panigrahi D C and Vuthaluru H B (2017) Sorption

Kinetics of CH4 and CO2 Diffusion in Coal Theoretical and Experimental Study

Energy amp Fuels 31(7) 6825-6837

httpdoiorg101021acsenergyfuels7b00721

Neimark A V Ravikovitch P I and Vishnyakov A (2000) Adsorption Hysteresis in

Nanopores Physical Review E 62(2) 1493-1496

httpdoiorg101103PhysRevE62R1493

Nemat-Nasser S and Hori M (2013) Micromechanics Overall Properties of

Heterogeneous Materials North Holland

Nie B He X and Wang B (2000) Mechanism and Modes of Gas Diffusion in Coal

Seams China Safety Science Journal 10(6) 24-28

Nielsen L F (1998) On Strength of Porous Material Simple Systems and Densified

Systems Materials and Structures 31(10) 651-661

NRC (1930) International Critical Tables of Numerical Data Physics Chemistry and

Technology National Research Concil

Ogilvy J A (1991) Theory of Wave Scattering from Random Rough Surfaces CRC Press

Okazaki M Tamon H and Toei R (1981) Interpretation of Surface Flow Phenomenon

of Adsorbed Gases by Hopping Model AIChE Journal 27(2) 262-270


Okolo G N Everson R C Neomagus H W Roberts M J and Sakurovs R (2015)

Comparing the Porosity and Surface Areas of Coal as Measured by Gas Adsorption

Mercury Intrusion and Saxs Techniques Fuel 141 293-304

255

Okotie V U and Moore R L (2011) Well-Production Challenges and Solutions in a

Mature Very-Low-Pressure Coalbed-Methane Reservoir SPE Production 26(02)

149-161

Okuszko K E Gault B W and Mattar L (2007) Production Decline Performance of

Cbm Wells Paper presented at the Canadian International Petroleum Conference

Calgary Alberta


Oulton T D (1948) The Pore SizendashSurface Area Distribution of a Cracking Catalyst The

Journal of Physical and Colloid Chemistry 52(8) 1296-1314

httpdoiorg101021j150464a003

Ozdemir E Morsi B I and Schroeder K (2003) Importance of Volume Effects to

Adsorption Isotherms of Carbon Dioxide on Coals Langmuir 19(23) 9764-9773

Palmer I and Mansoori J (1998) How Permeability Depends on Stress and Pore

Pressure in Coalbeds A New Model SPE Reservoir Evaluation amp Engineering

1(06) 539-544


Palmer I Mavor M and Gunter B (2007) Permeability Changes in Coal Seams During

Production and Injection Paper presented at the International Coalbed Methane

Symposium University of Alabama Tuscaloosa Alabama

Pan Z Connell L D Camilleri M and Connelly L (2010) Effects of Matrix Moisture

on Gas Diffusion and Flow in Coal Fuel 89(11) 3207-3217

Pashin J C (2010) Variable Gas Saturation in Coalbed Methane Reservoirs of the Black

Warrior Basin Implications for Exploration and Production International Journal

of Coal Geology 82(3) 135-146

Paul G W and Young G B C (1990) Chpater 6 Simulating Coalbed Methane

Reservoirs In A Guide to Coalbed Methane Reservoir Engineering Chicago Gas

Research Insititute

Peng D Y and Robinson D B (1976) A New Two-Constant Equation of State

Industrial amp Engineering Chemistry Fundamentals 15(1) 59-64

Perkins T K and Gonzalez J A (1984) Changes in Earth Stresses around a Wellbore

Caused by Radially Symmetrical Pressure and Temperature Gradients Society of



Perkins T K and Gonzalez J A (1985) The Effect of Thermoelastic Stresses on

Injection Well Fracturing Society of Petroleum Engineers Journal 25(01) 78-88


Petrenko V F and Whitworth R W (1999) Physics of Ice Oxford University Press

Petrovitch C L Nolte D D and Pyrak-Nolte L J (2013) Scaling of Fluid Flow Versus

Fracture Stiffness Geophysical Research Letters 40(10) 2076-2080

httpdoiorg 101002grl50479

Pfeifer P and Avnir D (1983) Chemistry in Noninteger Dimensions between Two and

Three I Fractal Theory of Heterogeneous Surfaces The Journal of Chemical

Physics 79(7) 3558-3565

256

Philibert J (2005) One and a Half Century of Diffusion Fick Einstein before and

Beyond Diffusion Fundamentals 2(1) 1-10

Pickett G (1945) Modification of the Brunauer-Emmett-Teller Theory of Multimolecular

Adsorption Journal of American Chemical society 67(11) 1958-1962


Pillalamarry M Harpalani S and Liu S (2011) Gas Diffusion Behavior of Coal and Its

Impact on Production from Coalbed Methane Reservoirs International Journal of

Coal Geology 86(4) 342-348

Pollard W G and Present R D (1948) On Gaseous Self-Diffusion in Long Capillary

Tubes Physical Review 73(7) 762-774

httpdoiorg101103PhysRev73762

Polyanin A D and Manzhirov A V (2006) Handbook of Mathematics for Engineers

and Scientists Chapman and Hall

Pone J D N Halleck P M and Mathews J P (2009) Sorption Capacity and Sorption

Kinetic Measurements of CO2 and CH4 in Confined and Unconfined Bituminous

Coal Energy amp Fuels 23(9) 4688-4695

httpdoiorg 101021ef9003158

Press W H Teukolsky S A Vetterling W T and Flannery B P (1992) Numerical

Recipes in Fortran 77 the Art of Scientific Computing In Cambridge University

Press

Pyrak-Nolte L (1996) The Seismic Response of Fractures and the Interrelations among

Fracture Properties Paper presented at the International Journal of Rock

Mechanics and Mining Sciences amp Geomechanics Abstracts

Pyrak-Nolte L J (2019a) Chapter 14 - Fracture Specific Stiffness The Critical Link

between the Scaling Behavior of Hydro-Mechanical Coupling in Fractures and

Seismic Monitoring In P Newell amp A G Ilgen (Eds) Science of Carbon Storage

in Deep Saline Formations (pp 311-335) Elsevier

Pyrak-Nolte L J and Morris J P (2000) Single Fractures under Normal Stress The

Relation between Fracture Specific Stiffness and Fluid Flow International Journal

of Rock Mechanics and Mining Sciences 37(1) 245-262

httpsdoiorg101016S1365-1609(99)00104-5

Pyrak-Nolte L J Myer L R and Cook N G W (1990) Transmission of Seismic

Waves across Single Natural Fractures Journal of Geophysical Research 95(B6)

httpdoiorg101029JB095iB06p08617

Pyrak-Nolte L J Nolte D D and Cook N G W (1995) Hierarchical Cascades and

the Single Fracture In Fractals in Petroleum Geology and Earth Processes (pp

143-178) Boston MA Springer US

Pyun S I and Rhee C K (2004) An Investigation of Fractal Characteristics of

Mesoporous Carbon Electrodes with Various Pore Structures Electrochimica Acta

49(24) 4171-4180

httpsdoiorg101016jelectacta200404012

Qi H Ma J and Wong P-Z (2002) Adsorption Isotherms of Fractal Surfaces Colloids

and Surfaces A Physicochemical and Engineering Aspects 206(1) 401-407

257

httpsdoiorg101016S0927-7757(02)00063-8

Qin L Zhai C Liu S and Xu J (2017a) Factors Controlling the Mechanical Properties

Degradation and Permeability of Coal Subjected to Liquid Nitrogen Freeze-Thaw

Sci Rep 7(1) 3675

httpdoiorg101038s41598-017-04019-7

Qin L Zhai C Liu S and Xu J (2018a) Infrared Thermal Image and Heat Transfer

Characteristics of Coal Injected with Liquid Nitrogen under Triaxial Loading for

Coalbed Methane Recovery International Journal of Heat and Mass Transfer 118

1231-1242

httpdoiorg101016jijheatmasstransfer201711051

Qin L Zhai C Liu S and Xu J (2018b) Mechanical Behavior and Fracture Spatial

Propagation of Coal Injected with Liquid Nitrogen under Triaxial Stress Applied

for Coalbed Methane Recovery Engineering Geology 233 1-10

httpdoiorg101016jenggeo201711019

Qin L Zhai C Liu S Xu J Tang Z and Yu G (2016) Failure Mechanism of Coal

after Cryogenic Freezing with Cyclic Liquid Nitrogen and Its Influences on

Coalbed Methane Exploitation Energy amp Fuels 30(10) 8567-8578


Qin L Zhai C Liu S Xu J Wu S and Dong R (2018c) Fractal Dimensions of Low

Rank Coal Subjected to Liquid Nitrogen Freeze-Thaw Based on Nuclear Magnetic

Resonance Applied for Coalbed Methane Recovery Powder Technology 325 11-

20

httpdoiorg101016jpowtec201711027

Qin L Zhai C Liu S Xu J Yu G and Sun Y (2017b) Changes in the Petrophysical

Properties of Coal Subjected to Liquid Nitrogen Freeze-Thaw ndash a Nuclear Magnetic

Resonance Investigation Fuel 194 102-114

httpdoiorg101016jfuel201701005

Qin Q H (2014) Greenrsquos Functions of Magneto-Electro-Elastic Plate under Thermal

Loading Encyclopedia of Thermal Stresses 38 (2014) 2096-2103

Ran Y Xing B Rao P S C and Fu J (2004) Importance of Adsorption (Hole-Filling)

Mechanism for Hydrophobic Organic Contaminants on an Aquifer Kerogen Isolate

Environmental science amp technology 38(16) 4340-4348

Ravikovitch P I Domhnaill S C O Neimark A V Schueth F and Unger K K

(1995) Capillary Hysteresis in Nanopores Theoretical and Experimental Studies

of Nitrogen Adsorption on Mcm-41 Langmuir 11(12) 4765-4772

httpdoiorg101021la00012a030

Rayleigh L (1892) Lvi On the Influence of Obstacles Arranged in Rectangular Order

Upon the Properties of a Medium The London Edinburgh and Dublin

Philosophical Magazine and Journal of Science 34(211) 481-502

Reeves S and Pekot L (2001) Advanced Reservoir Modeling in Desorption-Controlled

Reservoirs Paper presented at the SPE Rocky Mountain Petroleum Technology

Conference Keystone Colorado


258

Remner D J Ertekin T Sung W and King G R (1986) A Parametric Study of the

Effects of Coal Seam Properties on Gas Drainage Efficiency SPE Reservoir

Engineering 1(06) 633-646


Rodnikova M N (2007) A New Approach to the Mechanism of Solvophobic Interactions

Journal of Molecular Liquids 136(3) 211-213

httpsdoiorg101016jmolliq200708003

Ross J W and Good R J (1956) Adsorption and Surface Diffusion of N-Butane on

Spheron 6 (2700deg) Carbon Black The Journal of Physical Chemistry 60(9) 1167-

1171

Rostasy F Schneider U and Wiedemann G (1979) Behaviour of Mortar and Concrete

at Extremely Low Temperatures Cement and Concrete Research 9(3) 365-376

Rostasy F and Wiedemann G (1983) Strength and Deformability of Concrete after Low

Temperature Cycles Paper presented at 2nd International Conference on

Cryogenic Concrete

Ruppel T C Grein C T and Bienstock D (1974) Adsorption of Methane on Dry Coal

at Elevated Pressure Fuel 53(3) 152-162

httpsdoiorg1010160016-2361(74)90002-7

Rushing J A Perego A D and Blasingame T A (2008) Applicability of the Arps

Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas

Recovery of Coalbed Methane Wells Paper presented at the CIPCSPE Gas

Technology Symposium 2008 Joint Conference Calgary Alberta Canada


Salmachi A and Haghighi M (2012) Temperature Effect on Methane Sorption and

Diffusion in Coal Application for Thermal Recovery from Coal Seam Gas

Reservoirs The APPEA Journal 52(1) 291-300

httpsdoiorg101071AJ11021

Sato H (1982) Amplitude Attenuation of Impulsive Waves in Random Media Based on

Travel Time Corrected Mean Wave Formalism The Journal of the Acoustical

Society of America 71(3) 559-564

Saulsberry J L Schafer P S Schraufnagel R A and Institute G R (1996) A Guide

to Coalbed Methane Reservoir Engineering Gas Research Institute

Sawyer W K Zuber M D Kuuskraa V A and Horner D M (1987) Using Reservoir

Simulation and Field Data to Define Mechanisms Controlling Coalbed Methane

Production Paper presented at the Coalbed Methane Symposium Tuscaloosa

Alabama

Sayers C M (1999) Stress-Dependent Seismic Anisotropy of Shales Geophysics 64(1)

93-98

Sayers C M (2002) Stress-Dependent Elastic Anisotropy of Sandstones Geophysical

Prospecting 50(1) 85-95

httpdoiorg101046j1365-2478200200289x

259

Sayers C M and Han D-H (2002) The Effect of Pore Fluid on the Stress-Dependent

Elastic Wave Velocities in Sandstones In Seg Technical Program Expanded

Abstracts 2002 (pp 1842-1845) Society of Exploration Geophysicists

Sayers C M and Kachanov M (1995) Microcrack-Induced Elastic Wave Anisotropy of

Brittle Rocks Journal of Geophysical Research Solid Earth 100(B3) 4149-4156

httpdoiorg10102994jb03134

Schoenberg M (1980) Elastic Wave Behavior across Linear Slip Interfaces The Journal

of the Acoustical Society of America 68(5) 1516-1521

Schoenberg M (1998) Acoustic Characterization of Underground Fractures SEG

Technical Program Expanded Abstracts 1624-1627


Schoenberg M and Douma J (1988) Elastic Wave Propagation in Media with Parallel

Fractures and Aligned Cracks Geophysical Prospecting 36(6) 571-590

Scholes P L and Johnston D (1993) Coalbed Methane Applications of Wireline Logs

In Hydrocarbons from Coal (pp 287-302) AAPG Speical Volumes

Scott A R (2002) Hydrogeologic Factors Affecting Gas Content Distribution in Coal

Beds International Journal of Coal Geology 50(1-4) 363-387

Scott D S and Dullien F a L (1962) Diffusion of Ideal Gases in Capillaries and Porous

Solids AIChE Journal 8(1) 113-117


Sherwood J N (1969 1971) Diffusion Processes Proceedings of the Thomas Graham

Memorial Symposium University of Strathclyde Paper presented at the Thomas

Graham Memorial Symposium Glasgow

Shi J Q and Durucan S (2003a) Gas Storage and Flow in Coalbed Reservoirs

Implementation of a Bidisperse Pore Model for Gas Diffusion in Coal Matrix Paper

presented at the SPE Annual Technical Conference and Exhibition Denver

Colorado


Shi J Q and Durucan S (2005) A Model for Changes in Coalbed Permeability During

Primary and Enhanced Methane Recovery SPE Reservoir Evaluation amp



Shi J Q and Durucan S (2010) Exponential Growth in San Juan Basin Fruitland

Coalbed Permeability with Reservoir Drawdown Model Match and New Insights

SPE Reservoir Evaluation amp Engineering 13(06) 914-925


Shi J Q and Durucan S (2003b) A Bidisperse Pore Diffusion Model for Methane

Displacement Desorption in Coal by CO2 Injection Fuel 82(10) 1219-1229

httpsdoiorg101016S0016-2361(03)00010-3

Sing K S (1985) Reporting Physisorption Data for GasSolid Systems with Special

Reference to the Determination of Surface Area and Porosity (Recommendations

1984) Pure and Applied Chemistry 57(4) 603-619

260

Sladek K J Gilliland E R and Baddour R F (1974) Diffusion on Surfaces Ii

Correlation of Diffusivities of Physically and Chemically Adsorbed Species


Smith D M and Williams F L (1984) Diffusional Effects in the Recovery of Methane

from Coalbeds Society of Petroleum Engineers Journal 24(05) 529-535

Sneddon I (1946) The Distribution of Stress in the Neighbourhood of a Crack in an

Elastic Solid Paper presented at the Proceedings of the Royal Society of London A

Mathematical Physical and Engineering Sciences

Solano A L C S M Monge J A and D C A (1998) Further Advances in the

Characterization of Microporous Carbons by Physical Adsorption of Gases Tanso

The Carbon Society of Japan 1998(185) 316-325

http107209tanso1998316

Somerton W H Soumlylemezoḡlu I and Dudley R (1975) Effect of Stress on

Permeability of Coal Paper presented at the 16th Annual Logging Symposium New

Orleans Louisiana

Sotiropoulos D and Achenbach J D (1988) Reflection of Elastic Waves by a

Distribution of Coplanar Cracks The Journal of the Acoustical Society of America

84(2) 752-759

Standard Test Method for Total Moisture in Coal (2017) In ASTM D3302-07a West

Conshohocken PA ASTM International

Steckelmacher W (1986) Knudsen Flow 75 Years On The Current State of the Art for

Flow of Rarefied Gases in Tubes and Systems Reports on Progress in Physics

49(10) 1083

Strange J H Webber J B W and Schmidt S D (1996) Pore Size Distribution

Mapping Magnetic Resonance Imaging 14(7) 803-805

httpsdoiorg101016S0730-725X(96)00167-1

Tamon H Okazaki M and Toei R (1981) Flow Mechanism of Adsorbate through

Porous Media in Presence of Capillary Condensation AIChE Journal 27(2) 271-

277


Tavares S M O and De Castro P M S T (2019) Residual Stress In Damage Tolerance

of Metallic Aircraft Structures Materials and Numerical Modelling (pp 67-90)

Springer

Tedesco S (2003) Coalbed Methane Potential and Activity of the Illinois Basin Paper

presented at the AAPG Annual Meeting Salt Lake City Utah

Thararoop P Karpyn Z T and Ertekin T (2012) Development of a Multi-Mechanistic

Dual-Porosity Dual-Permeability Numerical Flow Model for Coalbed Methane

Reservoirs Journal of Natural Gas Science and Engineering 8 121-131


Thommes M Morlay C Ahmad R and Joly J (2011) Assessing Surface Chemistry

and Pore Structure of Active Carbons by a Combination of Physisorption (H2O Ar

N2 CO2) XPS and TPD-MS Adsorption 17(3) 653-661

261

Thommes M Smarsly B Groenewolt M Ravikovitch P I and Neimark A V (2006)

Adsorption Hysteresis of Nitrogen and Argon in Pore Networks and

Characterization of Novel Micro- and Mesoporous Silicas Langmuir 22(2)

httpdoiorg756-764 101021la051686h

Timmermann E Chirife J and Iglesias H (2001) Water Sorption Isotherms of Foods

and Foodstuffs Bet or Gab Parameters Journal of Food Engineering 48(1) 19-

31

Treworgy C G Korose C P and Wiscombe C L (2000) Availability of the Herrin

Coal for Mining in Illinois Illinois State Geological Survey Department of Natural

Resources

US Department of the Interior San Juan Field Office (1999) Coalbed Methane

Development in the Northern San Juan Basin of Colorado

Valiullin R Kortunov P Kaumlrger J and Timoshenko V (2004) Concentration-

Dependent Self-Diffusion of Liquids in Nanopores A Nuclear Magnetic

Resonance Study The Journal of chemical physics 120(24) 11804-11814

Van De Veen V (1987) Properties of Concrete at Very Low Temperatures A Survey of

the Literature Delft University of Technology Civil Engineering and Geosciences

Verdon J P Angus D A Kendall J M and Hall S A (2008) The Effect of

Microstructure and Nonlinear Stress on Anisotropic Seismic Velocities

Geophysics 73(4) 41-51 httpdoiorg10119012931680

Verdon J P and Wuumlstefeld A (2013) Measurement of the NormalTangential Fracture

Compliance Ratio (ZnZt) During Hydraulic Fracture Stimulation Using S-Wave

Splitting Data Geophysical Prospecting 61 461-477

Vishnyakov A and Neimark A V (2003) Nucleation of Liquid Bridges and Bubbles in

Nanoscale Capillaries The Journal of Chemical Physics 119(18) 9755-9764


Walker P L (1981) Microporosity in Coal Its Characterization and Its Implications for

Coal Utilization Philosophical Transactions of the Royal Society of London Series

A Mathematical and Physical Sciences 300(1453) 65-81

Wang B Qin Y Shen J Zhang Q and Wang G (2018a) Pore Structure

Characteristics of Low- and Medium-Rank Coals and Their Differential Adsorption

and Desorption Effects Journal of Petroleum Science and Engineering

httpdoiorg101016jpetrol201802014

Wang G Ren T Qi Q Lin J Liu Q and Zhang J (2017) Determining the Diffusion

Coefficient of Gas Diffusion in Coal Development of Numerical Solution Fuel

196 47-58


Wang K Wang G Ren T and Cheng Y (2014) Methane and Co2 Sorption Hysteresis

on Coal A Critical Review International Journal of Coal Geology 132 60-80


Wang Y and Liu S (2016) Estimation of Pressure-Dependent Diffusive Permeability of

Coal Using Methane Diffusion Coefficient Laboratory Measurements and

Modeling Energy amp Fuels 30(11) 8968-8976

262


Wang Y Liu S and Zhao Y (2018b) Modeling of Permeability for Ultra-Tight Coal

and Shale Matrix A Multi-Mechanistic Flow Approach Fuel 232 60-70


Wang Y Zhu Y Liu S and Zhang R (2016) Pore Characterization and Its Impact on

Methane Adsorption Capacity for Organic-Rich Marine Shales Fuel 181 227-

237


Warren J and Root P J (1963) The Behavior of Naturally Fractured Reservoirs Society

of Petroleum Engineers Journal 3(03) 245-255

Wei X Wang G Massarotto P and Golding S (2006) A Case Study on the Numerical

Simulation of Enhanced Coalbed Methane Recovery Paper presented at the SPE

Asia Pacific Oil amp Gas Conference and Exhibition

Weissberg H L (1963) Effective Diffusion Coefficient in Porous Media Journal of

Applied Physics 34(9) 2636-2639

Welty J R Rorrer G L and Foster D G (2014) Fundamentals of Momentum Heat

and Mass Transfer John Wiley amp Sons

Wheatcraft S W and Tyler S W (1988) An Explanation of Scale-Dependent

Dispersivity in Heterogeneous Aquifers Using Concepts of Fractal Geometry

Water Resources Research 24(4) 566-578

httpdoiorg101029WR024i004p00566

Wheeler A (1955) Reaction Rates and Selectivity in Catalyst Pores In Catalysis

Fundamental Principles (Part 2) (Vol 2 pp 105-165) Reinhold Publishing

Corporation

Whitaker S (1986) Flow in Porous Media I A Theoretical Derivation of Darcys Law

Transport in Porous Media 1(1) 3-25

httpdoiorg101007BF01036523

Winkler K W amp Murphy III W F (1995) Acoustic velocity and attenuation in porous

rocks Rock physics and phase relations A Handbook of physical constants 20-34

Worthington M and Hudson J (2000) Fault Properties from Seismic Q Geophysical

Journal International 143(3) 937-944

Wu K Li X Wang C Chen Z and Yu W (2014) Apparent Permeability for Gas

Flow in Shale Reservoirs Coupling Effects of Gas Diffusion and Desorption Paper

presented at the Unconventional Resources Technology Conference Denver

Colorado 25-27 August 2014

Wu R S (1982) Mean Field Attenuation and Amplitude Attenuation Due to Wave

Scattering Wave Motion 4(3) 305-316

Wu W and Sun H (2010) SorptionndashDesorption Hysteresis of PhenanthrenendashEffect of

Nanopores Solute Concentration and Salinity Chemosphere 81(7) 961-967

Xu J Zhai C Liu S Qin L and Wu S (2017) Pore Variation of Three Different

Metamorphic Coals by Multiple Freezing-Thawing Cycles of Liquid Co 2 Injection

for Coalbed Methane Recovery Fuel 208 41-51


263

Yang R T (2013) Gas Separation by Adsorption Processes Butterworth-Heinemann

Yao Y Liu D Tang D Tang S and Huang W (2008) Fractal Characterization of

Adsorption-Pores of Coals from North China An Investigation on Ch4 Adsorption

Capacity of Coals International Journal of Coal Geology 73(1) 27-42

httpdoiorg101016jcoal200707003

Yao Y Liu D Tang D Tang S Huang W Liu Z and Che Y (2009) Fractal

Characterization of Seepage-Pores of Coals from China An Investigation on

Permeability of Coals Computers amp Geosciences 35(6) 1159-1166

httpsdoiorg101016jcageo200809005

Yee D Seidle J P and Hanson W B (1993) Gas Sorption on Coal and Measurement

of Gas Content Chapter 9 In Hydrocarbons from Coal

httpsdoiorg101306St38577C9

Young G B C Mcelhiney J E Paul G W and Mcbane R A (1991) An Analysis of

Fruitland Coalbed Methane Production Cedar Hill Field Northern San Juan Basin

Paper presented at the SPE Annual Technical Conference and Exhibition Dallas

Texas httpsdoiorg10211822913-MS

Yushu Wu X Y Timothy J Kneafsey Jennifer L Miskimins Lei Wang Minsu Cha

Bowen Yao Taylor W Patterson Naif B Alqahtani (2013) Development of Non-

Contaminating Cryogenic Fracturing Technology for Shale and Tight Gas

Reservoirs Project Number 10122-20 Research Partnership to Secure Energy for

America (RPSEA)

Zasadzinski J A (1988) A New Heat Transfer Model to Predict Cooling Rates for Rapid

Freezing Fixation Journal of Microscopy 150(2) 137-149

httpdoiorg101111j1365-28181988tb04604x

Zhai C Qin L Liu S Xu J Tang Z and Wu S (2016) Pore Structure in Coal Pore

Evolution after Cryogenic Freezing with Cyclic Liquid Nitrogen Injection and Its

Implication on Coalbed Methane Extraction Energy amp Fuels 30(7) 6009-6020


Zhai C Wu S Liu S Qin L and Xu J (2017) Experimental Study on Coal Pore

Structure Deterioration under FreezendashThaw Cycles Environmental Earth Sciences

76(15) 507

httpdoiorg101007s12665-017-6829-9

Zhang H Y and Kai L (1997) A Research of Some Problems of the Exploration and

Exploitation of Gas-Bearing Coal Seam in Shanxi Province Pittsburgh Coal

Conference Pittsburgh PA (United States)

Zhang R and Liu S (2017) Experimental and Theoretical Characterization of Methane

and Co 2 Sorption Hysteresis in Coals Based on Langmuir Desorption

International Journal of Coal Geology 171 49-60 10


Zhang Y Abraham O Tournat V Le Duff A Lascoup B Loukili A Grondin F

and Durand O (2013) Validation of a Thermal Bias Control Technique for Coda

Wave Interferometry (CWI) Ultrasonics 53(3) 658-664

httpsdoiorg101016jultras201208003

264

Zhao J Xu H Tang D Mathews J P Li S and Tao S (2016) A Comparative

Evaluation of Coal Specific Surface Area by CO2 and N2 Adsorption and Its

Influence on CH4 Adsorption Capacity at Different Pore Sizes Fuel 183 420-431

Zhao W Cheng Y Yuan M and An F (2014) Effect of Adsorption Contact Time on

Coking Coal Particle Desorption Characteristics Energy amp Fuels 28(4) 2287-

2296

Zheng Q Yu B Wang S and Luo L (2012) A Diffusivity Model for Gas Diffusion

through Fractal Porous Media Chemical Engineering Science 68(1) 650-655

Zhu H and Selim H (2000) Hysteretic Behavior of Metolachlor Adsorption-Desorption

in Soils Soil Science 165(8) 632-645

Zhu Q Zuo Y and Yang Y (2015) How to Solve the Technical Problems in Cbm

Development A Case Study of a Cbm Gas Reservoir in the Southern Qinshui

Basin Natural Gas Industry B 2(2-3) 277-281

Ziarani A S Aguilera R and Clarkson C R (2011) Investigating the Effect of Sorption

Time on Coalbed Methane Recovery through Numerical Simulation Fuel 90(7)

2428-2444


Zimmerman R W Chen G Hadgu T and Bodvarsson G S (1993) A Numerical

Dual-Porosity Model with Semianalytical Treatment of FractureMatrix Flow


httpdoiorg10102993wr00749

VITA

Yun Yang

EDUCATION


bull PhD in Energy and Mineral Engineering 2017-2020

bull MS in Petroleum and Natural Gas Engineering 2016-2017

The University of Tulsa

bull BS in Petroleum Engineering with minor in Mathematics 2012-2015

RESEARCH EXPERIENCES

Research Assistant The Pennsylvania State University

bull Gas Transport in Porous Media 2017-2020

bull Experimental Sorption Kinetics


bull Flowback Analysis 2016-2017

JOURNNAL PUBLICATIONS

bull Yang Y Liu S Zhao W amp Wang L (2019) Intrinsic relationship between

Langmuir sorption volume and pressure for coal Experimental and thermodynamic

modeling study Fuel 241 105-117

bull Yang Y amp Liu S (2019) Estimation and modeling of pressure-dependent gas

diffusion coefficient for coal A fractal theory-based approach Fuel 253 588-606

bull Yang Y amp Liu S (2020) Laboratory study of cryogenic treatment-induced pore-

scale structural alterations of Illinois coal and their implications on gas sorption and

diffusion behaviors Journal of Petroleum Science and Engineering 194 107507

bull Yang Y amp Liu S Fracture stiffness evaluation with waterless cryogenic treatment

and its implication in fluid flowability of treated coal International Journal of Rock

Mechanics and Mining Sciences (Under Review)

bull Yang Y amp Liu S Modeling of gas production behavior of mature San Juan coalbed

methane reservoir role of the forgotten dynamic gas diffusivity International Journal

of Coal Geology (Under Review)

ii

The dissertation of Yun Yang was reviewed and approved by the following

Shimin Liu


Dissertation Advisor

Chair of Committee

Derek Elsworth

Professor of Energy and Mineral Engineering

Sekhar Bhattacharyya


Chair of Mining Engineering Program

Chris Marone

Professor of Geosciences

Mort Webester

Professor of Energy Engineering

Associate Department Head for Graduate Education

iii

ABSTRACT





















iv























v























vi























vii























viii























ix























x






xi


LIST OF TABLES xx



11 Background 1






23 Summary 41









xii








56 Summary 127













xiii







REFERENCE 241

xiv

LIST OF FIGURES







surfaces 16



24















setup 49


China 58



xv



isotherm curves 65






(VL PL) 72













respectively 83






pressure 85




xvi






105








Po=1542 psi 113







production data 119













xvii



































xviii












samples 166






















xix











xx

LIST OF TABLES





this study 46


Helium 51






248 414 607 and 807 MPa) 63








offset wells 109


growth 113


wells 118





xxi












cycles 168









xxii

ACKNOWLEDGEMENTS



















Chapter 1

INTRODUCTION

11 Background

















2


















3








0 5 10 15 20 25 30

0

50

100

150

Pro

duct

ion r

ate

(M

cfd

ay)

time (yrs)

Desorption from



Butt cleat

Face cleat

Darcyrsquos

flow

Log (nm) 012gt3

Dominated by




Well information



4























5























6














application






7




















8





Theory Experiment

Pore structure-Gas




Experiment

Adsorption

Capacity

Adsorption

Rate

Transport

RateHeterogeneity

Pore structure-

Sorption Model

Pore structure-

Diffusion Model


9

Chapter 2

THEORETICAL MODEL



















10























11























12











ln (V

V0) = 119860 ln (ln (

P0119875)) + 119864 ( 2-1 )





(Liu and Nie 2016)

119863119891 = 119860 + 3 ( 2-2 )

and

119863119891 = 3119860 + 3 ( 2-3 )

13




1198810) vs ln (ln (

P0
















119863 = 2

(a)

2 119863 3

(b)

14





119881 =

119875

119875 + 119875119871119881119871

( 2-4 )


plots of 119875119881 vs 119875















15



as

119877119905 =1198891205791119889119905

= 119896119886 119875(1 minus 1205791)119908+1 minus 119896119889 1205791

119898+1 ( 2-5 )







119877119905 =119889120579119905119889119905

= 119896119886 120583(1 minus 1205791)119898+1 minus 119896119889 1205791

119898+1 ( 2-6 )



120583 =119875

(2120587119872119877119879)12 ( 2-7 )







16





1205791 =

(119896119886 prime

119896119889 )120584

119875

1 + (119896119886 prime

119896119889 )120584

119875

( 2-8 )

where 120584 = 1(119898 + 1) and 119896119886 prime= 119896119886 (2120587119872119877119879)

minus12120584



119886 =1

119875119871= (

119896119886 prime

119896119889 )

120584

( 2-9 )





17




120584 = 1 minus (119863119891 minus 2

2) ( 2-10 )





by

119886 = (119896119886 prime

119896119889 )

119862120584

( 2-11 )




119896119886 prime = 1198921198730(2120587119872119877119879)minus12119862120584 ( 2-12 )




factor (119896119861)

119896119889 = 119891119890minus119876119896119861119879 ( 2-13 )

18




119886 = (119896119886 prime

119896119889 )

119862120584

=1

radic2120587119872119877119879(119892

119891119881119871119890

119876119896119861119879)119862120584

( 2-14 )





119886 =1

radic2120587119872119877119879(119881119871119890

119876119896119861119879)119862120584

( 2-15 )











19


surface (119876119904119905) as

119876 = 119870119876119904119905119862120584

( 2-16 )










119886 =1

119875119871= (119881119871)

119862120584119865(119879) ( 2-17 )

119865(119879) =1

radic2120587119872119877119879119890minus119870119876119904119905(119896119861119879) ( 2-18 )


condition





20









119881119871 = 119878(120590)1198631198912 + 119861 ( 2-19 )












ln(119875119871) = minus119862 ln(119881119871120584) + ln(119865(119879)) ( 2-20 )

21









ln(119875119871) = 119862 ln((119878(120590)1198631198912 + 119861)120584) + 119865(119879) ( 2-21 )













22






















23























24


















25












diffusion regime











26






27





119898119898

Pressure

MPa

Range of

119863 1198982119904 Nandi and Walker

(1975) 1975 US coals

Anthracite to

HVB 0315 119898119898

114minus 252

10minus13

minus 10minus14

Smith and


Fruitland San

Juan Basin Sub 19119898119898 57

10minus13

minus 10minus14

Mavor et al

(1990a) 1990

Fruitland San


Marecka and


Semi-

anthracite 125 062 02 0032119898119898 0-01

10minus10

minus 10minus15

Clarkson and

Bustin (1999b) 1999

Lower

Cretaceous

Gates

Formation

Canada


minus 10minus13

Busch et al

(2004b) 2004

Silesian Basin


Cui et al (2004)

(further reworked

by (Pillalamarry et

al 2011) )


10minus13

minus 10minus14


10minus13

minus 10minus15


Kentucky

Coalfield



(2010) 2010

Lorraine


Pillalamarry et al


10minus13

minus 10minus14

Salmachi and


Australian

coal seam HVB 0294119898119898

0014minus 4678

10minus12

Bhowmik and

Dutta (2013) 2013

Raniganj

Coalfield

Jharia

Coalfield

Gondwana

Basin of India

Sub to HVB 01245119898119898 036minus 461

10minus12

minus 10minus13


10minus11

minus 10minus12

Wang and Liu

(2016) 2016

San Juan

Basin and

Pittsburgh


minus 10minus14

Naveen et al

(2017) 2017

Jharia

Coalfield

Gondwana

Basin of India

HVB 023119898119898 0 minus 7 10minus13

28





described as

119863

1199032120597

120597119903(1199032

120597119862

120597119903) =

120597119862

120597119905

( 2-22 )


time














29











119862(119903 119905 gt 0) = 1198620 ( 2-23 )





119872119905119872infin

= 1 minus6

1205872sum

1

1198992119890119909119901(minus119863119890119899

21205872119905)

infin

119899=1

( 2-24 )

119863119890 = 1198631199031198902 ( 2-25 )



diffusivity

30









119878 =sum((119872119905119872infin)119890119909119901

minus (119872119905119872infin)119898119900119889119890119897

)

2

( 2-26 )

where (119872119905

119872infin)119890119909119901

and (119872119905

119872infin)119898119900119889119890119897


sorption fraction




((119872119905







31







in MATLAB GUI

32



33















(Knudsen 1909)

119863119870 =1

3119889119888 ( 2-27 )



119888 = radic8119877119879120587119872 ( 2-28 )


34







119863119870119901119898 =

120601

120591119863119870

( 2-29 )













35








119865 = 119873휀119863119891 = 119888119900119899119904119905119886119899119905 ( 2-30 )


휀



119871(휀) = 119873휀 ( 2-31 )



119871

119903

36

119871(휀) = 119865휀1minus119863119891 ( 2-32 )



119865 = 119871119904119863119891 ( 2-33 )


119871(휀) = 119871119904119863119891휀1minus119863119891 ( 2-34 )





120591 = 119871

119903=119871119904119863119891휀1minus119863119891

119871119904= (

휀

119871119904)1minus119863119891

( 2-35 )





120591 = (휀

119871119904)2minus119863119891

( 2-36 )




37

119863119870119901119898 =1

3120601 (119871119904휀)2minus119863119891

119863119870 =2radic21206011198891198770511987905

31205870511987205(119871119904휀)2minus119863119891

( 2-37 )


Froment 1995)














120582 =

5

8

120583

119875radic119877119879120587

2119872

( 2-38 )




38







2014)

119863119861 =1

3120582119888

( 2-39 )





1

119863119901=1

119863119870+1

119863119861 ( 2-40 )








39


(2014)

1

119863119901= 119908119870

1

119863119870119901119898+ (1 minus 119908119870)

1

119863119861 ( 2-41 )







119908119870 = 119891(119870119899) =


( 2-42 )











40














41




media

23 Summary











Pressure

minus



Pressure

minus

(a)(b)

Considering

tortuosity factor


42












sorption capacity




value of 119875119871







43




dynamically









with pressure









44


production profiles

45

Chapter 3

EXPERIMENTAL WORK


















46






study

Nos Coal sample

Moisture

content

()

Ash

content

()

Volatile

matter

()

Fixed

carbon

()

Ro max

()

1 Xiuwu-21 149 2911 1037 6303 402

2 Luling-9 125 754 3217 6104 089

3 Luling-10 137 1027 3817 5119 083

4 Sijiangzhuang-15 203 3542 1223 549 311

47






















48













simultaneously

49



331 Void Volume









Sample Cell

Reference

Cell

Pressure Transducer

1

23

4

Water Bath

(Constant T)

Data Acquisition

System


(c)


Reference cell

ValvePressure

transducer

Water bath

Sample cell

Pressuretransducer

50





119875119881 = 119885119899119877119879 ( 3-1 )
















119860119881 = 119861 ( 3-2 )

51

119860 =

[ 119875119904 minus

(119885)119875119904(119885)119875119903

119875119903 119875119904

119875119903119894

(119885)119875119903119894minus

119875119904119894

(119885)119875119904119894

119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894]

( 3-3 )

119861 = [

0119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894] 119881119904119886119898 ( 3-4 )

119881 = [119881119903119881119904] ( 3-5 )


is set to be zero



void (cm

3)

1 27582 31818 26631 27611 2 27665 31788 26660 27666 3 27689 31782 26648 27688

Average 27645 31796 26647 27655







52




119899119886119889119904119894 = 119899119900

119894 minus 119899119906119899119886119889119904119894 ( 3-6 )




119899119900119894 =

119875119904119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119894minus1119877119879+

119875119903119894119881119903

(119885)119875119903119894119877119879 ( 3-7 )


119899119906119899119886119889119904119894 =

119875119904119894(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894119877119879 ( 3-8 )



119899119886119889119904119894 = 119899119886119889119904

119894 + 119899119886119889119904119894minus1 ( 3-9 )





sample

53






















54


model








119872119905119872infin

=1

119899119904119886119905119894 119877119879

(119875119904119890119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119890119894minus1+119875119903119894119881119903

(119885)119875119903119894minus119875119904119894(119905)(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894) ( 3-10 )










34 Summary




55




















56

Chapter 4


















(3542 )

57






















58











59

















Coal

sample

Mean PD

(nm)

Pore Volume (cm3100 g) 119889119898119894119899

(nm)

119889119898119886119909

(nm)

BET σ


Xiuwu-21 761 1178 00247 0703 0451 1741 83759 485

Luling-9 1249 0395 000330 0172 0220 1880 115440 081

Luling-10 1505 0393 000372 0149 0240 1870 112430 089

Sijiangzhu

ang-15 46 2772 00537 0456 2262 1565 132447 511

60






P0





Luling-10

( )10 50 100 500 1000

00000

00005

00010

00015

00020

00025

00030

00035

00040

00045

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

10 50 100 500 1000

0000

0001

0002

0003

0004

0005

0006

0007

0008

0009

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Luling-9

( )10 50 100 500 1000

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

0020d

Vd

log

(W

) P

ore

Vo

lum

e (

cm

3g

)

Pore Width

Xiuwu-21

( )

10 50 100 500 1000

000

002

004

006

008

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Sijiazhuang-15

( )



61





















62









Xiuwu-21 -0868 2132 0981 -0313 2687 0983 -0772 2229 0967

Luling-9 -0445 2555 0980 -0439 2561 0998 -0505 2495 0989

Luling-10 -0426 2574 0971 -0468 2532 0997 -0504 2496 0975

Sijiangzhuang

-15-0452 2547 0972 -0677 2324 0983 -0425 2575 0932

63













results



and 807 MPa)



Xiuwu-21 -0772 2229 0967


1787 2199 2506 2800 3029 3199

Luling-9 -0505 2495 0989 4128 6472 8587 10924 12948 14576

Luling-10 -0504 2496 09754078 6395 8486 10798 12800 14409

Sijiangzhuang-

15-0537 2463 0932

5606 9444 13111 17336 21114 24223

64














65


curves








66


















Xiuwu-21 2736 069 0984 1

Luling-9 1674 134 0987 2

Luling-10 1388 123 0986 8


67





















68




and Luling-10











0 20000 40000 60000 80000 100000

00

02

04

06

08

10

Adsorp

tion F

raction


Exp

Unipore Model

0 20000 40000 60000 80000 10000003

04

05

06

07

08

09

10

11

Adsorp

tion F

raction


Exp

Unipore Model

Xiuwu-21 Luling-10

69













70


pressure









times 10minus12

0 2 4 6 8

0

2

4

6

8

10

Measure

d D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

71























72


PL)











Rsup2 = 0138

0

10

20

30

40

15 17 19 21 23 25 27

VL m

3 to

n

D1

Rsup2 = 01642

0

10

20

30

40

50

15 17 19 21 23 25 27

VL m

3 to

n

D2

Rsup2 = 06301

0

04

08

12

16

15 17 19 21 23 25 27

PL M

Pa

D1

Rsup2 = 00137

0

04

08

12

16

15 17 19 21 23 25 27P

L M

Pa

D2

(a) (b)

(c) (d)

73







119881119871 = 119878(120590)11986312 + 119861 ( 4-1 )














74
















of the pore surface


D12 X3 = (Sσ11986312 + 119861)ν

Xiuwu-21 223 089 1874 581 293

Luling-9 250 075 833 077 205

Luing-10 250 075 723 087 206

Sijiangzhuang-15 257 071 1217 818 250

75























76







y = -06973x + 16643

Rsup2 = 09324

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

ln(X1)ln(1198831)

ln(119875119871)

ln 119875119871 = minus07ln (119881119871ν) + 17

1198772 = 093

y = 24372x + 133

Rsup2 = 09804

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

VL

m3

ton

X2 106 m2ton

VL m3tminus1

119883210 (m2 tminus1)

119881119871 = 24 1205901198632 + 133

1198772 = 098119881119871 = 24120590

1198631198912 + 133

1198772 = 098

77















y = -0723x + 17268

Rsup2 = 09834

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

X3

ln(119875119871)

ln 1198833

ln 119875119871 = minus07 ln 24 1205901198632 + 133

120584

+17

1198772 = 098

119891

78





















79


















Pressure [MPa] 055 138 248 414 607 807


Coefficient

[times10101198982119904]


Xiuwu-21 10477 6760 4227 5494 2388 4822 1469 4315 1042 3990 820 3777

Luling-9 4187 1922 1689 1226 954 924 587 726 416 613 328 544

Luling-10 3847 2154 1552 1373 877 1035 539 813 383 686 301 610

Sijiazhuang-15 26248 5102 10589 3029 5982 2181 3679 1650 2611 1355 2056 1181

80







0 2 4 6 8

0

5

10

15

20

25

30

DB

DKpm

Diffu

sio

n C

oeff

icie

nt

(m2s

)

Pressure (MPa)

times 10minus9

81



(P))











Horizontal

pure Knudsen

diffusion

combined

diffusion

pure bulk diffusion

119875lowast

Non-linear Linear

times 1012

0 2 4 6 8 10

0

2

4

6

8

10

Re

cip

rocal D

iffu

sio

n C

oeff

icie

nt

(sm

2)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

82












diffusion as

119863119901 = 119863119861(1 minus eminus1119870119899) ( 4-2 )









83











00 01 02 03 04 0500

02

04

06

08

10

Wk

Kn

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

pure bulk

combined

pure Knudsen

84























85

0 2 4 6 8 10

0

2

4

6

8

10


Experim

enta

l D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

0

2

4

6

8

This Work

Wheeler (1955)

Theore

tical D

iffu

sio

n C

oeffic

ient (m

2s

)






0 2 4 6 80

2

4

6

8

10

Exp

erim

enta

l D

iffu

sio

n C

oe

ffic

ien

t (m

2s

)


055 MPa

138 MPa

248 MPa

414 MPa

607 MPa

807 MPa

1198772 = 0782

1198772 = 09801198772 = 0992

1198772 = 0963

1198772 = 0926

1198772 = 0997

times10minus12

times10minus9

86





















87

47 Summary




















88

















dominates






89





90

Chapter 5





















91



















the decline stage

92












93























94






















95























96























97





















98




119902120588|119903+119889119903 minus 119902120588|119903 = 4120587119903

2119889119903120601120597120588

120597119905+ 41205871199032119889119903(1 minus 120601)

120597119902119886119889119904120597119905

( 5-1 )





120597(119902120588)

120597119903= 41205871199032120601

120597120588

120597119905+ 41205871199032(1 minus 120601)

120597119902119886119889119904120597119905

( 5-2 )




99

119902 =

41205871199032119896119898120583

120597119875

120597119903

( 5-3 )



1

1199032120597

120597119903(1199032119896119898120583

120588120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-4 )



120597119903


119869 = 119863

120597120588

120597119903

( 5-5 )




119902 =

119860

120588119869

( 5-6 )


41205871199032120601


2) with Eq (5-3) as

1

1199032120597

120597119903(1199032119863120601

120597120588

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-7 )


100

119888119892 = minus

1

119881

120597119881

120597119875=1

120588

120597120588

120597119875

( 5-8 )


1

1199032120597

120597119903(1199032119863120601119888119892120588

120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-9 )




119896119898 = 120601119888119892120583119863 ( 5-10 )














101




boundaries

















density log devices

102

















API)





gas content



temperature




103

















104



105












106












in Figure 5-1



107





















108




109


wells



Pressure

gradient

Reservoir

Pressure


1 3205 40 1181988 140 0478 1552

2 3440 26 1211995 157 0432 1508

3 3414 72 5291994 150 0458 1562

4 3495 34 12311993 155 0442 1527










119896119891

119896119891119900= (

120601119891

120601119891119900)

3

= [1 +119862119898120601119891119900

(119875 minus 119875119900) +1

120601119891119900(119870

119872minus 1) 휀]

3

( 5-11 )


119862119898 =

1

119872minus (

119870

119872+ 119891 minus 1) 119888119903

( 5-12 )

where 119896119891


pressure of 119875 120601119891


110


between 0 and 1



휀 =

3119881119871120588119904119877119879

119864119860119881119900int

1

119875119871 + 119875119889119875

119875

1198751

( 5-13 )

















111



120601119898 =

119881119901

119881119901 + 119881119892119903119886119894119899

( 5-14 )



















112






















113


production stage


s VL P

L E EEA c


(psi-1

) (F) 14 674 292 290E+05 03 5 201E-06 07 107


psi


114












115























116














117























118







Depth (ft) 3460 3460




















Skin Factor -2 -2


119


data

0 400 800 1200 1600

0

20

40

60

80

100

Matr

ix P

erm

ea

bili

ty M

ultip

lier

Pressure (psi)

Well A

Well B


matching process



120























121



CMG




122























123























124



















stage

125












126























127

56 Summary





















128




production curves






simulators









129

Chapter 6


61 Introduction









flow of hydrocarbon










130






















131











Gonzalez 1984)











132







631 Cleat-Scale



133










632 Pore-Scale












134






















135




















studied

136






















137









structure













138























139









140










sample

141


thawing operation

0 minDumping

Freeze

1 min 10 min

30 min 20 min

Freeze Freeze

Freeze

FreezeFreeze

40 min

Freeze

50 min

Freeze

Freeze


(a)

60 min


Thaw

10 min 20 min

40 min 30 min

Thaw

Thaw

ThawThaw

50 min

Thaw

60 min

Thaw

240 min

Finish 1 F-T cycle

Thaw

(b)

142





















143


center









[119906(119909)] =

4(1 minus 1205842)

120587119864119875119908 (ln |

119886

119909| + radic120587(1 minus (119909119886)2))

( 6-1 )




ie 119886(1198860) = 119886119899minus1(1198860)



144

[119906(119886)] = int [119906(119909)]

119886

minus119886

119889119909 =2radic120587(1 minus 1205842)

119864119875119908119886

( 6-2 )



휀119888 = int

[119906(119886)]

119886119889120588(1198860)

120588(119886119898119886119909)

120588(119886119898119894119899)

=2radic120587(1 minus 1205842)

119864int 119875119908119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-3 )











Thus

119875119908 = 119860(119879 119886)120573119898119875119888 ( 6-4 )





145






119875119888 = 119870119888radic1119886 ( 6-5 )




휀119888 = 2radic120587119860(119879 119886)120573119898

(1 minus 1205842)119870119888119864

int radic1119886119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-6 )




휀119905 = 120573119898120572119871 119879 ( 6-7 )


a negative value




휀 = 휀119905 + 휀119888 ( 6-8 )

146





651 Gas Kinetics
















147























148















149

0 2 4 6 8 10

0

5

10

15

Ad

so

rption

Cap

acity (

mlg

)













Coal

Sample

119881119871 ml g

119875119871 MPa

R2

Raw 1446 091 0998 5

1F-T 1643 099 (79) 0998 5

3F-T 1665 112 (232) 0997 9

150










151












0 2 4 6 8 10

2

4

6

8




Diffu

sio

n C

oeff

icie

nt

(1e-1

3 m

2s

)


Improve by

1876

Improve by

3018

152





[MPa] [1e-13

m2s]

[1e-13

m2s] [1e-13

m2s]

Adsorption 049 157 186 1832 174 1056 103 189 240 2659 219 1550 209 269 326 2111 296 986 352 316 374 1859 344 895 559 377 462 2251 408 816 842 535 564 544 553 333

Desorption 052 189 258 3680 218 1562 106 243 321 3226 290 1919 205 310 414 3363 353 1386 338 357 475 3313 433 2114 535 563 648 1511 591 501












153























154





6-7











gas molecules




155






















156









samples

00 02 04 06 08 10

000

005

010

015

020 Raw Coal

1F-T Coal

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure


slit shaped pores

157





samples















158























159





2000)

119867119868 =

119860119889119890 minus 119860119886119889119860119886119889

( 6-9 )


respectively













160



119907

119907119898=

119888119896119909

(1 minus 119896119909)(1 + (119888 minus 1)119896119909)

( 6-10 )




00 02 04 06 08 10

000

004

008

012

016


BET

GAB

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Aad




the GAB equation



161



119860119886119889 = int 1199071

0119889119909 =

119907119898















119888 = 119888119900

1

1 + 1198861119909 + 11988621199092 +⋯

( 6-12 )


equation when 1198861 = 1198862 = ⋯ = 0



162

119907

119907119898=

1198880119896119909


119899lowast1 )119896119909)

( 6-13 )
















163

00 02 04 06 08 10

000

004

008

012

016

Ade


GAB (n=0)

Cubic GAB (n=1)

Quadratic GAB (n=2)

Qu

an

tity

Ad

so

rbed

(m

molg

)

Relative Pressure









119860119889119890 = int 1199071198891199091

0

= int 119907119898119891(119909)1198891199091

0

= (sum119891(119909119894) + 119891(119909119894+1)

2

1 119909

119894=0

) 119909119907119898

( 6-14 )

164






















thawed coal samples

165











0 20 40 60 80 100

000

001

002

003

004

0 2 4 6 8 10

000

001

002

003

004

Raw Coal

1F-T Coal

3F-T Coald

Vd

log

(w)

Po

re V

olu

me (

cm

sup3g

)

Pore Width (nm)

dV

dlo

g(w

) P

ore

Vo

lum

e (

cm

sup3g

)

Pore Width (nm)

mesopore macropore

166





0000 0005 0010 0015 0020 0025 0030

00

01

02

03

04

05

06

07 Raw Coal

1F-T Coall

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure









167






4 6 8 10 12

000

004

008

012

016

Raw Coal

1F-T Coal

3F-T Coal

dV

dlo

g(W

) P

ore

Volu

me (

cm

sup3g)

Pore Width (Aring)


3F-T coal samples







168




cycles respectively














Coal sample dmean

(nm)



Raw 665 2130 189 294 483

1F-T 614 2025 298 346 644




169











170


















171






Levine 1996)


Chugh et al 1989)



















172



cycle


















173














range of estimates

174
















fracturing






175























176


such as coal




















177


permeability




(mm) (mm) (gcm3)

()

M-1 9996 4989 139 0036 0

M-2 10007 5017 138 0048 058





178


















179











180




in-contact region











Incident Wave

Fracture Plane

Outgoing Wave

Scattered Wave

Undisturbed Wave

Ui(x)


x3

x2

x1

C Cc

F

181



















Norris 1992)

182








119906(119909 119905) = 119860119890minus119894120596119905119890119894119896119909 ( 6-15 )




1205972119906(119909 119905)

1205971199052= 1199072

1205972119906(119909 119905)

1205971199092 ( 6-16 )








183

1205881205962119906119894(119909) +120597

120597119909119895119862119894119895119896119897

120597119906119896(119909)

120597119909119897= 0 ( 6-17 )











119932 = 119872(휀)119932119920 ( 6-18 )





119872(휀) = 119871 + 휀1198711 + 119874(휀2) ( 6-19 )

119871 = 119872(0) ( 6-20 )



184













119932120782 = 119871119932119920 ( 6-21 )




119932 = 119932120782 + 휀1198711119932120782 ( 6-22 )





185

119932 gt= 119932120782 + 휀 1198711 gt 119932120782 ( 6-23 )




1198711 gt= int 119901(119888)(119888)119865

119889119888 ( 6-24 )





119901(119888) = 119873 ( 6-25 )

and

1198711 gt= 119873int (119888)119865

119889119888 ( 6-26 )









186

119905119894(119909) = 0 [119906119894(119909)] = 0 119894 = 123 ( 6-27 )




[119905119894(119909)] = 0 119894 = 123 ( 6-28 )









119957 = 119948[119958] ( 6-29 )

where 119948 = [

119896119905 0 00 119896119905 00 0 119896119899






119932119956(119888) = 휀(119888)119932120782 ( 6-30 )

187







119932119956(119909) = int 119905119894(119932119956(120585))[119866119894

119868(119909 120585)]119889120585119914

( 6-31 )





[119932119930] = [ 119932 gt] ( 6-32 )




휀119932120782 = int 119896119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]119889120585

119914

( 6-33 )





188

119932 gt= 119932120782 + 119873int 119870119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]

119865

119889120585 ( 6-34 )





119905119894( 119932 gt) = 119870119894119895119905 [ 119880119895 gt] ( 6-35 )

where 119870119894119895119905 = 119873119870119894119895 = 119873int 119896119894119895119889120585119914


meanfield













189


(Johnson 1985)

119875 = 21205871198861198871199010 ( 6-36 )


pressure



1199063 = 21 minus 1205842

1198641199010119887119825(119890) ( 6-37 )





matrix


properties as

119875 = 120587119886119864

1 minus 12058421

119825(119890)1199063 ( 6-38 )




190



119905119899 =1

2119873119875[119906119899] ( 6-39 )

where 1199063 =1

2[119906119899]


be determined as

119870119899 =1

2119873119875 =

1

2119873120587119886

119864

1 minus 12058421

119825(119890) ( 6-40 )


2009) Eq (6-39) becomes

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41 )



1985)

119876 = 21205871198861198871199020 ( 6-42 )



1199061 = 1199062 =1199020119887

119866[119825(119890) +

120584

1198902(1 minus 1198902)119825(119890) minus 119812(119890)] ( 6-43 )



191


properties as

119876 =2120587119886119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

11990612 ( 6-44 )





119905119905 =1

2119873119876[119906119905] ( 6-45 )

where 11990612 =1

2[119906119905]



119870119905 =1

2119873119876 = 119873120587119886

119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

( 6-46 )


2009) Eq (6-41) becomes

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )




192




119870119899 = 41198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752) ( 6-48 )


119870119905 = 2119873120587119886120588

1198811199042

[120587 +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


(119825(119890) minus 119812(119890)

1198902)]

( 6-49 )


119825(e)minus119812(e)



lim119890rarr0

((1 minus 1198902)119825(119890) minus 119812(119890)

1198902)

= lim119890rarr0


2119890) = minus

120587

4

( 6-50 )

where 119825prime(119890) =119889119825(119890)

119889119890=

119812(119890)

119890(1minus119890 )minus119825(119890)

119890 and 119812prime(119890) =

119889119812(119890)

119889119890=119812(119890)minus119825(119890)


Manzhirov 2006)



119870119905 = 811987312058711988612058811988111990421 minus 119881119878

2 1198811198752frasl

3 minus 21198811198782 119881119875

2frasl ( 6-51 )



193






















194












671 Surface Cracks



195














196





197

672 Wave Velocities






given by

120572 = minus20

ℎ119897119900119892(119860119860119900) ( 6-52 )














198






199






















200






wet coal specimen

(a)

(b)

201





119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41)

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )














202










coal specimen (M-1)



Sample 119873 119886 119887 119890

(1mm2) (mm) (mm) (-)

M-1 0097 10 018 098

M-2 019 10 045 090








203






















204

0 10 20 30 40 50 60

0

20

40

60

80

100

120

Fra

ctu

re S

tiffness (

GP

am

)

Freezing Time (min)

Kn K

t

00

05

10

15

20

25

30 K

tK

n

Tangential to

Norm

al S

tiffness R

atio










205

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890)+119872prime

( 6-53 )

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

+119866prime

( 6-54 )









and the surrounding






expressed as

119861119894 = ℎ119881119896119888119860 ( 6-55 )

206

where ℎ (119882


119882








119889119879

119889119905= minus

119860

119881ℎ(1198790 minus 119879119888)

1

120588119862119875 ( 6-56 )

where 119889119879

119889119905(119870



1198983) and 119862119875 (

119869


specimen


1959)

119889119879

119889119905= minus(

119860

119881)2

119896119888(1198790 minus 119879119888)1

120588119862119901 ( 6-57 )






207






ℎ119881

119896119888119860=(2013)(00101)

0226= 899 ( 6-58 )












208





ℎ 119862119901 119896119888 120588 119904119908 119904119894119888119890

(Wm2K) (JkgK) (WmK) (kgm3) (-) (-)

2013 953 0226 1380 0313 0345









209














210





119870119905119870119899=1 minus 120584

1 minus1205842

( 6-59 )






119870119905119870119899=

7

8 [1 +92120587

119872prime

radic1 minus 1198902119872]

( 6-60 )










211









26



212










through

119870 = 120575radic1198763

( 6-61 )











213






















214











68 Summary











215






















216























217























218




















219

Chapter 7

CONCLUSIONS



















220





















221




sorption capacity




value of 119875119871










dynamically





222





with pressure










production profiles







223






















224























225














law








226





















227























228






















229























230
















231
















summation







232



coefficient






233


coefficient

MATLAB Code






234














xhigh as a double




CreateFcns called

235






xlow as a double




CreateFcns called






nmax as a double



MATLAB

236


CreateFcns called










237





238






















239











MATLAB Code





240


and 0 to elsewhere


Step 3







241

REFERENCE








21(2) 4-13











228-247



physics 79(7) 3566-3571







159-188







Institute





242



163(3) 307-320




























3222-3223



0012-3










243







httpsdoiorg1010160021-9797(69)90423-8





httpsdoiorg1010160020-7683(76)90044-5






httpsdoiorg101016S0166-5162(03)00113-7













2) 3-26




36




Engineering 39(12) 9249-9257

httpdoiorg101007s13369-014-1472-1




244












































245


1483-1498

httpsdoiorg1010160016-2361(96)00142-1




httpsdoiorg101016S0016-2361(99)00055-1







httpsdoiorg101016S0008-6223(97)00124-3



Engineering 8 9-33




11(02) 311-325








Geology 88(4) 204-217











Simulator




246








httpsdoiorg1010160009-2509(94)00478-A







httpsdoiorg101016S0166-5162(97)00015-3






293-303





















247






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Sciences


























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Resources


























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2296










2428-2444






VITA

Yun Yang

EDUCATION



























iii

ABSTRACT





















iv























v























vi























vii























viii























ix























x






xi


LIST OF TABLES xx



11 Background 1






23 Summary 41









xii








56 Summary 127













xiii







REFERENCE 241

xiv

LIST OF FIGURES







surfaces 16



24















setup 49


China 58



xv



isotherm curves 65






(VL PL) 72













respectively 83






pressure 85




xvi






105








Po=1542 psi 113







production data 119













xvii



































xviii












samples 166






















xix











xx

LIST OF TABLES





this study 46


Helium 51






248 414 607 and 807 MPa) 63








offset wells 109


growth 113


wells 118





xxi












cycles 168









xxii

ACKNOWLEDGEMENTS



















Chapter 1

INTRODUCTION

11 Background

















2


















3








0 5 10 15 20 25 30

0

50

100

150

Pro

duct

ion r

ate

(M

cfd

ay)

time (yrs)

Desorption from



Butt cleat

Face cleat

Darcyrsquos

flow

Log (nm) 012gt3

Dominated by




Well information



4























5























6














application






7




















8





Theory Experiment

Pore structure-Gas




Experiment

Adsorption

Capacity

Adsorption

Rate

Transport

RateHeterogeneity

Pore structure-

Sorption Model

Pore structure-

Diffusion Model


9

Chapter 2

THEORETICAL MODEL



















10























11























12











ln (V

V0) = 119860 ln (ln (

P0119875)) + 119864 ( 2-1 )





(Liu and Nie 2016)

119863119891 = 119860 + 3 ( 2-2 )

and

119863119891 = 3119860 + 3 ( 2-3 )

13




1198810) vs ln (ln (

P0
















119863 = 2

(a)

2 119863 3

(b)

14





119881 =

119875

119875 + 119875119871119881119871

( 2-4 )


plots of 119875119881 vs 119875















15



as

119877119905 =1198891205791119889119905

= 119896119886 119875(1 minus 1205791)119908+1 minus 119896119889 1205791

119898+1 ( 2-5 )







119877119905 =119889120579119905119889119905

= 119896119886 120583(1 minus 1205791)119898+1 minus 119896119889 1205791

119898+1 ( 2-6 )



120583 =119875

(2120587119872119877119879)12 ( 2-7 )







16





1205791 =

(119896119886 prime

119896119889 )120584

119875

1 + (119896119886 prime

119896119889 )120584

119875

( 2-8 )

where 120584 = 1(119898 + 1) and 119896119886 prime= 119896119886 (2120587119872119877119879)

minus12120584



119886 =1

119875119871= (

119896119886 prime

119896119889 )

120584

( 2-9 )





17




120584 = 1 minus (119863119891 minus 2

2) ( 2-10 )





by

119886 = (119896119886 prime

119896119889 )

119862120584

( 2-11 )




119896119886 prime = 1198921198730(2120587119872119877119879)minus12119862120584 ( 2-12 )




factor (119896119861)

119896119889 = 119891119890minus119876119896119861119879 ( 2-13 )

18




119886 = (119896119886 prime

119896119889 )

119862120584

=1

radic2120587119872119877119879(119892

119891119881119871119890

119876119896119861119879)119862120584

( 2-14 )





119886 =1

radic2120587119872119877119879(119881119871119890

119876119896119861119879)119862120584

( 2-15 )











19


surface (119876119904119905) as

119876 = 119870119876119904119905119862120584

( 2-16 )










119886 =1

119875119871= (119881119871)

119862120584119865(119879) ( 2-17 )

119865(119879) =1

radic2120587119872119877119879119890minus119870119876119904119905(119896119861119879) ( 2-18 )


condition





20









119881119871 = 119878(120590)1198631198912 + 119861 ( 2-19 )












ln(119875119871) = minus119862 ln(119881119871120584) + ln(119865(119879)) ( 2-20 )

21









ln(119875119871) = 119862 ln((119878(120590)1198631198912 + 119861)120584) + 119865(119879) ( 2-21 )













22






















23























24


















25












diffusion regime











26






27





119898119898

Pressure

MPa

Range of

119863 1198982119904 Nandi and Walker

(1975) 1975 US coals

Anthracite to

HVB 0315 119898119898

114minus 252

10minus13

minus 10minus14

Smith and


Fruitland San

Juan Basin Sub 19119898119898 57

10minus13

minus 10minus14

Mavor et al

(1990a) 1990

Fruitland San


Marecka and


Semi-

anthracite 125 062 02 0032119898119898 0-01

10minus10

minus 10minus15

Clarkson and

Bustin (1999b) 1999

Lower

Cretaceous

Gates

Formation

Canada


minus 10minus13

Busch et al

(2004b) 2004

Silesian Basin


Cui et al (2004)

(further reworked

by (Pillalamarry et

al 2011) )


10minus13

minus 10minus14


10minus13

minus 10minus15


Kentucky

Coalfield



(2010) 2010

Lorraine


Pillalamarry et al


10minus13

minus 10minus14

Salmachi and


Australian

coal seam HVB 0294119898119898

0014minus 4678

10minus12

Bhowmik and

Dutta (2013) 2013

Raniganj

Coalfield

Jharia

Coalfield

Gondwana

Basin of India

Sub to HVB 01245119898119898 036minus 461

10minus12

minus 10minus13


10minus11

minus 10minus12

Wang and Liu

(2016) 2016

San Juan

Basin and

Pittsburgh


minus 10minus14

Naveen et al

(2017) 2017

Jharia

Coalfield

Gondwana

Basin of India

HVB 023119898119898 0 minus 7 10minus13

28





described as

119863

1199032120597

120597119903(1199032

120597119862

120597119903) =

120597119862

120597119905

( 2-22 )


time














29











119862(119903 119905 gt 0) = 1198620 ( 2-23 )





119872119905119872infin

= 1 minus6

1205872sum

1

1198992119890119909119901(minus119863119890119899

21205872119905)

infin

119899=1

( 2-24 )

119863119890 = 1198631199031198902 ( 2-25 )



diffusivity

30









119878 =sum((119872119905119872infin)119890119909119901

minus (119872119905119872infin)119898119900119889119890119897

)

2

( 2-26 )

where (119872119905

119872infin)119890119909119901

and (119872119905

119872infin)119898119900119889119890119897


sorption fraction




((119872119905







31







in MATLAB GUI

32



33















(Knudsen 1909)

119863119870 =1

3119889119888 ( 2-27 )



119888 = radic8119877119879120587119872 ( 2-28 )


34







119863119870119901119898 =

120601

120591119863119870

( 2-29 )













35








119865 = 119873휀119863119891 = 119888119900119899119904119905119886119899119905 ( 2-30 )


휀



119871(휀) = 119873휀 ( 2-31 )



119871

119903

36

119871(휀) = 119865휀1minus119863119891 ( 2-32 )



119865 = 119871119904119863119891 ( 2-33 )


119871(휀) = 119871119904119863119891휀1minus119863119891 ( 2-34 )





120591 = 119871

119903=119871119904119863119891휀1minus119863119891

119871119904= (

휀

119871119904)1minus119863119891

( 2-35 )





120591 = (휀

119871119904)2minus119863119891

( 2-36 )




37

119863119870119901119898 =1

3120601 (119871119904휀)2minus119863119891

119863119870 =2radic21206011198891198770511987905

31205870511987205(119871119904휀)2minus119863119891

( 2-37 )


Froment 1995)














120582 =

5

8

120583

119875radic119877119879120587

2119872

( 2-38 )




38







2014)

119863119861 =1

3120582119888

( 2-39 )





1

119863119901=1

119863119870+1

119863119861 ( 2-40 )








39


(2014)

1

119863119901= 119908119870

1

119863119870119901119898+ (1 minus 119908119870)

1

119863119861 ( 2-41 )







119908119870 = 119891(119870119899) =


( 2-42 )











40














41




media

23 Summary











Pressure

minus



Pressure

minus

(a)(b)

Considering

tortuosity factor


42












sorption capacity




value of 119875119871







43




dynamically









with pressure









44


production profiles

45

Chapter 3

EXPERIMENTAL WORK


















46






study

Nos Coal sample

Moisture

content

()

Ash

content

()

Volatile

matter

()

Fixed

carbon

()

Ro max

()

1 Xiuwu-21 149 2911 1037 6303 402

2 Luling-9 125 754 3217 6104 089

3 Luling-10 137 1027 3817 5119 083

4 Sijiangzhuang-15 203 3542 1223 549 311

47






















48













simultaneously

49



331 Void Volume









Sample Cell

Reference

Cell

Pressure Transducer

1

23

4

Water Bath

(Constant T)

Data Acquisition

System


(c)


Reference cell

ValvePressure

transducer

Water bath

Sample cell

Pressuretransducer

50





119875119881 = 119885119899119877119879 ( 3-1 )
















119860119881 = 119861 ( 3-2 )

51

119860 =

[ 119875119904 minus

(119885)119875119904(119885)119875119903

119875119903 119875119904

119875119903119894

(119885)119875119903119894minus

119875119904119894

(119885)119875119904119894

119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894]

( 3-3 )

119861 = [

0119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894] 119881119904119886119898 ( 3-4 )

119881 = [119881119903119881119904] ( 3-5 )


is set to be zero



void (cm

3)

1 27582 31818 26631 27611 2 27665 31788 26660 27666 3 27689 31782 26648 27688

Average 27645 31796 26647 27655







52




119899119886119889119904119894 = 119899119900

119894 minus 119899119906119899119886119889119904119894 ( 3-6 )




119899119900119894 =

119875119904119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119894minus1119877119879+

119875119903119894119881119903

(119885)119875119903119894119877119879 ( 3-7 )


119899119906119899119886119889119904119894 =

119875119904119894(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894119877119879 ( 3-8 )



119899119886119889119904119894 = 119899119886119889119904

119894 + 119899119886119889119904119894minus1 ( 3-9 )





sample

53






















54


model








119872119905119872infin

=1

119899119904119886119905119894 119877119879

(119875119904119890119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119890119894minus1+119875119903119894119881119903

(119885)119875119903119894minus119875119904119894(119905)(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894) ( 3-10 )










34 Summary




55




















56

Chapter 4


















(3542 )

57






















58











59

















Coal

sample

Mean PD

(nm)

Pore Volume (cm3100 g) 119889119898119894119899

(nm)

119889119898119886119909

(nm)

BET σ


Xiuwu-21 761 1178 00247 0703 0451 1741 83759 485

Luling-9 1249 0395 000330 0172 0220 1880 115440 081

Luling-10 1505 0393 000372 0149 0240 1870 112430 089

Sijiangzhu

ang-15 46 2772 00537 0456 2262 1565 132447 511

60






P0





Luling-10

( )10 50 100 500 1000

00000

00005

00010

00015

00020

00025

00030

00035

00040

00045

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

10 50 100 500 1000

0000

0001

0002

0003

0004

0005

0006

0007

0008

0009

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Luling-9

( )10 50 100 500 1000

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

0020d

Vd

log

(W

) P

ore

Vo

lum

e (

cm

3g

)

Pore Width

Xiuwu-21

( )

10 50 100 500 1000

000

002

004

006

008

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Sijiazhuang-15

( )



61





















62









Xiuwu-21 -0868 2132 0981 -0313 2687 0983 -0772 2229 0967

Luling-9 -0445 2555 0980 -0439 2561 0998 -0505 2495 0989

Luling-10 -0426 2574 0971 -0468 2532 0997 -0504 2496 0975

Sijiangzhuang

-15-0452 2547 0972 -0677 2324 0983 -0425 2575 0932

63













results



and 807 MPa)



Xiuwu-21 -0772 2229 0967


1787 2199 2506 2800 3029 3199

Luling-9 -0505 2495 0989 4128 6472 8587 10924 12948 14576

Luling-10 -0504 2496 09754078 6395 8486 10798 12800 14409

Sijiangzhuang-

15-0537 2463 0932

5606 9444 13111 17336 21114 24223

64














65


curves








66


















Xiuwu-21 2736 069 0984 1

Luling-9 1674 134 0987 2

Luling-10 1388 123 0986 8


67





















68




and Luling-10











0 20000 40000 60000 80000 100000

00

02

04

06

08

10

Adsorp

tion F

raction


Exp

Unipore Model

0 20000 40000 60000 80000 10000003

04

05

06

07

08

09

10

11

Adsorp

tion F

raction


Exp

Unipore Model

Xiuwu-21 Luling-10

69













70


pressure









times 10minus12

0 2 4 6 8

0

2

4

6

8

10

Measure

d D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

71























72


PL)











Rsup2 = 0138

0

10

20

30

40

15 17 19 21 23 25 27

VL m

3 to

n

D1

Rsup2 = 01642

0

10

20

30

40

50

15 17 19 21 23 25 27

VL m

3 to

n

D2

Rsup2 = 06301

0

04

08

12

16

15 17 19 21 23 25 27

PL M

Pa

D1

Rsup2 = 00137

0

04

08

12

16

15 17 19 21 23 25 27P

L M

Pa

D2

(a) (b)

(c) (d)

73







119881119871 = 119878(120590)11986312 + 119861 ( 4-1 )














74
















of the pore surface


D12 X3 = (Sσ11986312 + 119861)ν

Xiuwu-21 223 089 1874 581 293

Luling-9 250 075 833 077 205

Luing-10 250 075 723 087 206

Sijiangzhuang-15 257 071 1217 818 250

75























76







y = -06973x + 16643

Rsup2 = 09324

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

ln(X1)ln(1198831)

ln(119875119871)

ln 119875119871 = minus07ln (119881119871ν) + 17

1198772 = 093

y = 24372x + 133

Rsup2 = 09804

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

VL

m3

ton

X2 106 m2ton

VL m3tminus1

119883210 (m2 tminus1)

119881119871 = 24 1205901198632 + 133

1198772 = 098119881119871 = 24120590

1198631198912 + 133

1198772 = 098

77















y = -0723x + 17268

Rsup2 = 09834

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

X3

ln(119875119871)

ln 1198833

ln 119875119871 = minus07 ln 24 1205901198632 + 133

120584

+17

1198772 = 098

119891

78





















79


















Pressure [MPa] 055 138 248 414 607 807


Coefficient

[times10101198982119904]


Xiuwu-21 10477 6760 4227 5494 2388 4822 1469 4315 1042 3990 820 3777

Luling-9 4187 1922 1689 1226 954 924 587 726 416 613 328 544

Luling-10 3847 2154 1552 1373 877 1035 539 813 383 686 301 610

Sijiazhuang-15 26248 5102 10589 3029 5982 2181 3679 1650 2611 1355 2056 1181

80







0 2 4 6 8

0

5

10

15

20

25

30

DB

DKpm

Diffu

sio

n C

oeff

icie

nt

(m2s

)

Pressure (MPa)

times 10minus9

81



(P))











Horizontal

pure Knudsen

diffusion

combined

diffusion

pure bulk diffusion

119875lowast

Non-linear Linear

times 1012

0 2 4 6 8 10

0

2

4

6

8

10

Re

cip

rocal D

iffu

sio

n C

oeff

icie

nt

(sm

2)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

82












diffusion as

119863119901 = 119863119861(1 minus eminus1119870119899) ( 4-2 )









83











00 01 02 03 04 0500

02

04

06

08

10

Wk

Kn

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

pure bulk

combined

pure Knudsen

84























85

0 2 4 6 8 10

0

2

4

6

8

10


Experim

enta

l D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

0

2

4

6

8

This Work

Wheeler (1955)

Theore

tical D

iffu

sio

n C

oeffic

ient (m

2s

)






0 2 4 6 80

2

4

6

8

10

Exp

erim

enta

l D

iffu

sio

n C

oe

ffic

ien

t (m

2s

)


055 MPa

138 MPa

248 MPa

414 MPa

607 MPa

807 MPa

1198772 = 0782

1198772 = 09801198772 = 0992

1198772 = 0963

1198772 = 0926

1198772 = 0997

times10minus12

times10minus9

86





















87

47 Summary




















88

















dominates






89





90

Chapter 5





















91



















the decline stage

92












93























94






















95























96























97





















98




119902120588|119903+119889119903 minus 119902120588|119903 = 4120587119903

2119889119903120601120597120588

120597119905+ 41205871199032119889119903(1 minus 120601)

120597119902119886119889119904120597119905

( 5-1 )





120597(119902120588)

120597119903= 41205871199032120601

120597120588

120597119905+ 41205871199032(1 minus 120601)

120597119902119886119889119904120597119905

( 5-2 )




99

119902 =

41205871199032119896119898120583

120597119875

120597119903

( 5-3 )



1

1199032120597

120597119903(1199032119896119898120583

120588120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-4 )



120597119903


119869 = 119863

120597120588

120597119903

( 5-5 )




119902 =

119860

120588119869

( 5-6 )


41205871199032120601


2) with Eq (5-3) as

1

1199032120597

120597119903(1199032119863120601

120597120588

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-7 )


100

119888119892 = minus

1

119881

120597119881

120597119875=1

120588

120597120588

120597119875

( 5-8 )


1

1199032120597

120597119903(1199032119863120601119888119892120588

120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-9 )




119896119898 = 120601119888119892120583119863 ( 5-10 )














101




boundaries

















density log devices

102

















API)





gas content



temperature




103

















104



105












106












in Figure 5-1



107





















108




109


wells



Pressure

gradient

Reservoir

Pressure


1 3205 40 1181988 140 0478 1552

2 3440 26 1211995 157 0432 1508

3 3414 72 5291994 150 0458 1562

4 3495 34 12311993 155 0442 1527










119896119891

119896119891119900= (

120601119891

120601119891119900)

3

= [1 +119862119898120601119891119900

(119875 minus 119875119900) +1

120601119891119900(119870

119872minus 1) 휀]

3

( 5-11 )


119862119898 =

1

119872minus (

119870

119872+ 119891 minus 1) 119888119903

( 5-12 )

where 119896119891


pressure of 119875 120601119891


110


between 0 and 1



휀 =

3119881119871120588119904119877119879

119864119860119881119900int

1

119875119871 + 119875119889119875

119875

1198751

( 5-13 )

















111



120601119898 =

119881119901

119881119901 + 119881119892119903119886119894119899

( 5-14 )



















112






















113


production stage


s VL P

L E EEA c


(psi-1

) (F) 14 674 292 290E+05 03 5 201E-06 07 107


psi


114












115























116














117























118







Depth (ft) 3460 3460




















Skin Factor -2 -2


119


data

0 400 800 1200 1600

0

20

40

60

80

100

Matr

ix P

erm

ea

bili

ty M

ultip

lier

Pressure (psi)

Well A

Well B


matching process



120























121



CMG




122























123























124



















stage

125












126























127

56 Summary





















128




production curves






simulators









129

Chapter 6


61 Introduction









flow of hydrocarbon










130






















131











Gonzalez 1984)











132







631 Cleat-Scale



133










632 Pore-Scale












134






















135




















studied

136






















137









structure













138























139









140










sample

141


thawing operation

0 minDumping

Freeze

1 min 10 min

30 min 20 min

Freeze Freeze

Freeze

FreezeFreeze

40 min

Freeze

50 min

Freeze

Freeze


(a)

60 min


Thaw

10 min 20 min

40 min 30 min

Thaw

Thaw

ThawThaw

50 min

Thaw

60 min

Thaw

240 min

Finish 1 F-T cycle

Thaw

(b)

142





















143


center









[119906(119909)] =

4(1 minus 1205842)

120587119864119875119908 (ln |

119886

119909| + radic120587(1 minus (119909119886)2))

( 6-1 )




ie 119886(1198860) = 119886119899minus1(1198860)



144

[119906(119886)] = int [119906(119909)]

119886

minus119886

119889119909 =2radic120587(1 minus 1205842)

119864119875119908119886

( 6-2 )



휀119888 = int

[119906(119886)]

119886119889120588(1198860)

120588(119886119898119886119909)

120588(119886119898119894119899)

=2radic120587(1 minus 1205842)

119864int 119875119908119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-3 )











Thus

119875119908 = 119860(119879 119886)120573119898119875119888 ( 6-4 )





145






119875119888 = 119870119888radic1119886 ( 6-5 )




휀119888 = 2radic120587119860(119879 119886)120573119898

(1 minus 1205842)119870119888119864

int radic1119886119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-6 )




휀119905 = 120573119898120572119871 119879 ( 6-7 )


a negative value




휀 = 휀119905 + 휀119888 ( 6-8 )

146





651 Gas Kinetics
















147























148















149

0 2 4 6 8 10

0

5

10

15

Ad

so

rption

Cap

acity (

mlg

)













Coal

Sample

119881119871 ml g

119875119871 MPa

R2

Raw 1446 091 0998 5

1F-T 1643 099 (79) 0998 5

3F-T 1665 112 (232) 0997 9

150










151












0 2 4 6 8 10

2

4

6

8




Diffu

sio

n C

oeff

icie

nt

(1e-1

3 m

2s

)


Improve by

1876

Improve by

3018

152





[MPa] [1e-13

m2s]

[1e-13

m2s] [1e-13

m2s]

Adsorption 049 157 186 1832 174 1056 103 189 240 2659 219 1550 209 269 326 2111 296 986 352 316 374 1859 344 895 559 377 462 2251 408 816 842 535 564 544 553 333

Desorption 052 189 258 3680 218 1562 106 243 321 3226 290 1919 205 310 414 3363 353 1386 338 357 475 3313 433 2114 535 563 648 1511 591 501












153























154





6-7











gas molecules




155






















156









samples

00 02 04 06 08 10

000

005

010

015

020 Raw Coal

1F-T Coal

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure


slit shaped pores

157





samples















158























159





2000)

119867119868 =

119860119889119890 minus 119860119886119889119860119886119889

( 6-9 )


respectively













160



119907

119907119898=

119888119896119909

(1 minus 119896119909)(1 + (119888 minus 1)119896119909)

( 6-10 )




00 02 04 06 08 10

000

004

008

012

016


BET

GAB

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Aad




the GAB equation



161



119860119886119889 = int 1199071

0119889119909 =

119907119898















119888 = 119888119900

1

1 + 1198861119909 + 11988621199092 +⋯

( 6-12 )


equation when 1198861 = 1198862 = ⋯ = 0



162

119907

119907119898=

1198880119896119909


119899lowast1 )119896119909)

( 6-13 )
















163

00 02 04 06 08 10

000

004

008

012

016

Ade


GAB (n=0)

Cubic GAB (n=1)

Quadratic GAB (n=2)

Qu

an

tity

Ad

so

rbed

(m

molg

)

Relative Pressure









119860119889119890 = int 1199071198891199091

0

= int 119907119898119891(119909)1198891199091

0

= (sum119891(119909119894) + 119891(119909119894+1)

2

1 119909

119894=0

) 119909119907119898

( 6-14 )

164






















thawed coal samples

165











0 20 40 60 80 100

000

001

002

003

004

0 2 4 6 8 10

000

001

002

003

004

Raw Coal

1F-T Coal

3F-T Coald

Vd

log

(w)

Po

re V

olu

me (

cm

sup3g

)

Pore Width (nm)

dV

dlo

g(w

) P

ore

Vo

lum

e (

cm

sup3g

)

Pore Width (nm)

mesopore macropore

166





0000 0005 0010 0015 0020 0025 0030

00

01

02

03

04

05

06

07 Raw Coal

1F-T Coall

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure









167






4 6 8 10 12

000

004

008

012

016

Raw Coal

1F-T Coal

3F-T Coal

dV

dlo

g(W

) P

ore

Volu

me (

cm

sup3g)

Pore Width (Aring)


3F-T coal samples







168




cycles respectively














Coal sample dmean

(nm)



Raw 665 2130 189 294 483

1F-T 614 2025 298 346 644




169











170


















171






Levine 1996)


Chugh et al 1989)



















172



cycle


















173














range of estimates

174
















fracturing






175























176


such as coal




















177


permeability




(mm) (mm) (gcm3)

()

M-1 9996 4989 139 0036 0

M-2 10007 5017 138 0048 058





178


















179











180




in-contact region











Incident Wave

Fracture Plane

Outgoing Wave

Scattered Wave

Undisturbed Wave

Ui(x)


x3

x2

x1

C Cc

F

181



















Norris 1992)

182








119906(119909 119905) = 119860119890minus119894120596119905119890119894119896119909 ( 6-15 )




1205972119906(119909 119905)

1205971199052= 1199072

1205972119906(119909 119905)

1205971199092 ( 6-16 )








183

1205881205962119906119894(119909) +120597

120597119909119895119862119894119895119896119897

120597119906119896(119909)

120597119909119897= 0 ( 6-17 )











119932 = 119872(휀)119932119920 ( 6-18 )





119872(휀) = 119871 + 휀1198711 + 119874(휀2) ( 6-19 )

119871 = 119872(0) ( 6-20 )



184













119932120782 = 119871119932119920 ( 6-21 )




119932 = 119932120782 + 휀1198711119932120782 ( 6-22 )





185

119932 gt= 119932120782 + 휀 1198711 gt 119932120782 ( 6-23 )




1198711 gt= int 119901(119888)(119888)119865

119889119888 ( 6-24 )





119901(119888) = 119873 ( 6-25 )

and

1198711 gt= 119873int (119888)119865

119889119888 ( 6-26 )









186

119905119894(119909) = 0 [119906119894(119909)] = 0 119894 = 123 ( 6-27 )




[119905119894(119909)] = 0 119894 = 123 ( 6-28 )









119957 = 119948[119958] ( 6-29 )

where 119948 = [

119896119905 0 00 119896119905 00 0 119896119899






119932119956(119888) = 휀(119888)119932120782 ( 6-30 )

187







119932119956(119909) = int 119905119894(119932119956(120585))[119866119894

119868(119909 120585)]119889120585119914

( 6-31 )





[119932119930] = [ 119932 gt] ( 6-32 )




휀119932120782 = int 119896119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]119889120585

119914

( 6-33 )





188

119932 gt= 119932120782 + 119873int 119870119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]

119865

119889120585 ( 6-34 )





119905119894( 119932 gt) = 119870119894119895119905 [ 119880119895 gt] ( 6-35 )

where 119870119894119895119905 = 119873119870119894119895 = 119873int 119896119894119895119889120585119914


meanfield













189


(Johnson 1985)

119875 = 21205871198861198871199010 ( 6-36 )


pressure



1199063 = 21 minus 1205842

1198641199010119887119825(119890) ( 6-37 )





matrix


properties as

119875 = 120587119886119864

1 minus 12058421

119825(119890)1199063 ( 6-38 )




190



119905119899 =1

2119873119875[119906119899] ( 6-39 )

where 1199063 =1

2[119906119899]


be determined as

119870119899 =1

2119873119875 =

1

2119873120587119886

119864

1 minus 12058421

119825(119890) ( 6-40 )


2009) Eq (6-39) becomes

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41 )



1985)

119876 = 21205871198861198871199020 ( 6-42 )



1199061 = 1199062 =1199020119887

119866[119825(119890) +

120584

1198902(1 minus 1198902)119825(119890) minus 119812(119890)] ( 6-43 )



191


properties as

119876 =2120587119886119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

11990612 ( 6-44 )





119905119905 =1

2119873119876[119906119905] ( 6-45 )

where 11990612 =1

2[119906119905]



119870119905 =1

2119873119876 = 119873120587119886

119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

( 6-46 )


2009) Eq (6-41) becomes

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )




192




119870119899 = 41198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752) ( 6-48 )


119870119905 = 2119873120587119886120588

1198811199042

[120587 +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


(119825(119890) minus 119812(119890)

1198902)]

( 6-49 )


119825(e)minus119812(e)



lim119890rarr0

((1 minus 1198902)119825(119890) minus 119812(119890)

1198902)

= lim119890rarr0


2119890) = minus

120587

4

( 6-50 )

where 119825prime(119890) =119889119825(119890)

119889119890=

119812(119890)

119890(1minus119890 )minus119825(119890)

119890 and 119812prime(119890) =

119889119812(119890)

119889119890=119812(119890)minus119825(119890)


Manzhirov 2006)



119870119905 = 811987312058711988612058811988111990421 minus 119881119878

2 1198811198752frasl

3 minus 21198811198782 119881119875

2frasl ( 6-51 )



193






















194












671 Surface Cracks



195














196





197

672 Wave Velocities






given by

120572 = minus20

ℎ119897119900119892(119860119860119900) ( 6-52 )














198






199






















200






wet coal specimen

(a)

(b)

201





119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41)

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )














202










coal specimen (M-1)



Sample 119873 119886 119887 119890

(1mm2) (mm) (mm) (-)

M-1 0097 10 018 098

M-2 019 10 045 090








203






















204

0 10 20 30 40 50 60

0

20

40

60

80

100

120

Fra

ctu

re S

tiffness (

GP

am

)

Freezing Time (min)

Kn K

t

00

05

10

15

20

25

30 K

tK

n

Tangential to

Norm

al S

tiffness R

atio










205

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890)+119872prime

( 6-53 )

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

+119866prime

( 6-54 )









and the surrounding






expressed as

119861119894 = ℎ119881119896119888119860 ( 6-55 )

206

where ℎ (119882


119882








119889119879

119889119905= minus

119860

119881ℎ(1198790 minus 119879119888)

1

120588119862119875 ( 6-56 )

where 119889119879

119889119905(119870



1198983) and 119862119875 (

119869


specimen


1959)

119889119879

119889119905= minus(

119860

119881)2

119896119888(1198790 minus 119879119888)1

120588119862119901 ( 6-57 )






207






ℎ119881

119896119888119860=(2013)(00101)

0226= 899 ( 6-58 )












208





ℎ 119862119901 119896119888 120588 119904119908 119904119894119888119890

(Wm2K) (JkgK) (WmK) (kgm3) (-) (-)

2013 953 0226 1380 0313 0345









209














210





119870119905119870119899=1 minus 120584

1 minus1205842

( 6-59 )






119870119905119870119899=

7

8 [1 +92120587

119872prime

radic1 minus 1198902119872]

( 6-60 )










211









26



212










through

119870 = 120575radic1198763

( 6-61 )











213






















214











68 Summary











215






















216























217























218




















219

Chapter 7

CONCLUSIONS



















220





















221




sorption capacity




value of 119875119871










dynamically





222





with pressure










production profiles







223






















224























225














law








226





















227























228






















229























230
















231
















summation







232



coefficient






233


coefficient

MATLAB Code






234














xhigh as a double




CreateFcns called

235






xlow as a double




CreateFcns called






nmax as a double



MATLAB

236


CreateFcns called










237





238






















239











MATLAB Code





240


and 0 to elsewhere


Step 3







241

REFERENCE








21(2) 4-13











228-247



physics 79(7) 3566-3571







159-188







Institute





242



163(3) 307-320




























3222-3223



0012-3










243







httpsdoiorg1010160021-9797(69)90423-8





httpsdoiorg1010160020-7683(76)90044-5






httpsdoiorg101016S0166-5162(03)00113-7













2) 3-26




36




Engineering 39(12) 9249-9257

httpdoiorg101007s13369-014-1472-1




244












































245


1483-1498

httpsdoiorg1010160016-2361(96)00142-1




httpsdoiorg101016S0016-2361(99)00055-1







httpsdoiorg101016S0008-6223(97)00124-3



Engineering 8 9-33




11(02) 311-325








Geology 88(4) 204-217











Simulator




246








httpsdoiorg1010160009-2509(94)00478-A







httpsdoiorg101016S0166-5162(97)00015-3






293-303





















247






httpsdoiorg1010160008-6223(68)90302-3



httpsdoiorg1010160008-6223(64)90040-5



Sciences


























httpdoiorg1010880953-8984246001





Butterworths

248





















httpdoiorg101007s00603-016-1120-z



30(4) 453-464



33(3) 296-300










45(2) 105-113






249






659







121-125








httpsdoiorg1010160020-7683(92)90133-E





142



325



1599












250

httpsdoiorg101016S0020-7225(97)00080-3





726



httpdoiorg101017S0305004100057674






BS ISO 15901-1


Publications










Fuels 30(12) 10383-10395





httpsdoiorg1010160148-9062(93)90018-9







Fuel 52(3) 181-185





251












4(4) 191-197

























165-183







252

httpdoi101002andp19093330106







httpsdoiorg101016S0009-2509(96)00458-7





















1059-1063



3000



httpdoiorg10130612181212061









253



42(7) 1233-1242




247

httpdoiorg101111j1365-2478200700688x



Science 41(4) 627-633









httpsdoiorg1010160021-9797(82)90153-9



httpsdoiorg101016S0165-232X(05)80005-9





Exhibition















254
















httpsdoiorg101016S1365-1609(99)00103-3



httpsdoiorg1010160016-2361(75)90061-7























255



149-161



Calgary Alberta









1(06) 539-544












Research Insititute
















Physics 79(7) 3558-3565

256




















Press











httpsdoiorg101016S1365-1609(99)00104-5









49(24) 4171-4180




257

httpsdoiorg101016S0927-7757(02)00063-8



Sci Rep 7(1) 3675

httpdoiorg101038s41598-017-04019-7




1231-1242













20






















258










1171





Cryogenic Concrete



httpsdoiorg1010160016-2361(74)90002-7


















Alabama


93-98



httpdoiorg101046j1365-2478200200289x

259



























Colorado












httpsdoiorg101016S0016-2361(03)00010-3




260















Orleans Louisiana



84(2) 752-759





49(10) 1083



httpsdoiorg101016S0730-725X(96)00167-1



277




Springer










261




httpdoiorg756-764 101021la051686h



31



Resources


























196 47-58








262







237

















Corporation




















263





















America (RPSEA)



httpdoiorg101111j1365-28181988tb04604x







76(15) 507

httpdoiorg101007s12665-017-6829-9












264






2296










2428-2444






VITA

Yun Yang

EDUCATION



























iv























v























vi























vii























viii























ix























x






xi


LIST OF TABLES xx



11 Background 1






23 Summary 41









xii








56 Summary 127













xiii







REFERENCE 241

xiv

LIST OF FIGURES







surfaces 16



24















setup 49


China 58



xv



isotherm curves 65






(VL PL) 72













respectively 83






pressure 85




xvi






105








Po=1542 psi 113







production data 119













xvii



































xviii












samples 166






















xix











xx

LIST OF TABLES





this study 46


Helium 51






248 414 607 and 807 MPa) 63








offset wells 109


growth 113


wells 118





xxi












cycles 168









xxii

ACKNOWLEDGEMENTS



















Chapter 1

INTRODUCTION

11 Background

















2


















3








0 5 10 15 20 25 30

0

50

100

150

Pro

duct

ion r

ate

(M

cfd

ay)

time (yrs)

Desorption from



Butt cleat

Face cleat

Darcyrsquos

flow

Log (nm) 012gt3

Dominated by




Well information



4























5























6














application






7




















8





Theory Experiment

Pore structure-Gas




Experiment

Adsorption

Capacity

Adsorption

Rate

Transport

RateHeterogeneity

Pore structure-

Sorption Model

Pore structure-

Diffusion Model


9

Chapter 2

THEORETICAL MODEL



















10























11























12











ln (V

V0) = 119860 ln (ln (

P0119875)) + 119864 ( 2-1 )





(Liu and Nie 2016)

119863119891 = 119860 + 3 ( 2-2 )

and

119863119891 = 3119860 + 3 ( 2-3 )

13




1198810) vs ln (ln (

P0
















119863 = 2

(a)

2 119863 3

(b)

14





119881 =

119875

119875 + 119875119871119881119871

( 2-4 )


plots of 119875119881 vs 119875















15



as

119877119905 =1198891205791119889119905

= 119896119886 119875(1 minus 1205791)119908+1 minus 119896119889 1205791

119898+1 ( 2-5 )







119877119905 =119889120579119905119889119905

= 119896119886 120583(1 minus 1205791)119898+1 minus 119896119889 1205791

119898+1 ( 2-6 )



120583 =119875

(2120587119872119877119879)12 ( 2-7 )







16





1205791 =

(119896119886 prime

119896119889 )120584

119875

1 + (119896119886 prime

119896119889 )120584

119875

( 2-8 )

where 120584 = 1(119898 + 1) and 119896119886 prime= 119896119886 (2120587119872119877119879)

minus12120584



119886 =1

119875119871= (

119896119886 prime

119896119889 )

120584

( 2-9 )





17




120584 = 1 minus (119863119891 minus 2

2) ( 2-10 )





by

119886 = (119896119886 prime

119896119889 )

119862120584

( 2-11 )




119896119886 prime = 1198921198730(2120587119872119877119879)minus12119862120584 ( 2-12 )




factor (119896119861)

119896119889 = 119891119890minus119876119896119861119879 ( 2-13 )

18




119886 = (119896119886 prime

119896119889 )

119862120584

=1

radic2120587119872119877119879(119892

119891119881119871119890

119876119896119861119879)119862120584

( 2-14 )





119886 =1

radic2120587119872119877119879(119881119871119890

119876119896119861119879)119862120584

( 2-15 )











19


surface (119876119904119905) as

119876 = 119870119876119904119905119862120584

( 2-16 )










119886 =1

119875119871= (119881119871)

119862120584119865(119879) ( 2-17 )

119865(119879) =1

radic2120587119872119877119879119890minus119870119876119904119905(119896119861119879) ( 2-18 )


condition





20









119881119871 = 119878(120590)1198631198912 + 119861 ( 2-19 )












ln(119875119871) = minus119862 ln(119881119871120584) + ln(119865(119879)) ( 2-20 )

21









ln(119875119871) = 119862 ln((119878(120590)1198631198912 + 119861)120584) + 119865(119879) ( 2-21 )













22






















23























24


















25












diffusion regime











26






27





119898119898

Pressure

MPa

Range of

119863 1198982119904 Nandi and Walker

(1975) 1975 US coals

Anthracite to

HVB 0315 119898119898

114minus 252

10minus13

minus 10minus14

Smith and


Fruitland San

Juan Basin Sub 19119898119898 57

10minus13

minus 10minus14

Mavor et al

(1990a) 1990

Fruitland San


Marecka and


Semi-

anthracite 125 062 02 0032119898119898 0-01

10minus10

minus 10minus15

Clarkson and

Bustin (1999b) 1999

Lower

Cretaceous

Gates

Formation

Canada


minus 10minus13

Busch et al

(2004b) 2004

Silesian Basin


Cui et al (2004)

(further reworked

by (Pillalamarry et

al 2011) )


10minus13

minus 10minus14


10minus13

minus 10minus15


Kentucky

Coalfield



(2010) 2010

Lorraine


Pillalamarry et al


10minus13

minus 10minus14

Salmachi and


Australian

coal seam HVB 0294119898119898

0014minus 4678

10minus12

Bhowmik and

Dutta (2013) 2013

Raniganj

Coalfield

Jharia

Coalfield

Gondwana

Basin of India

Sub to HVB 01245119898119898 036minus 461

10minus12

minus 10minus13


10minus11

minus 10minus12

Wang and Liu

(2016) 2016

San Juan

Basin and

Pittsburgh


minus 10minus14

Naveen et al

(2017) 2017

Jharia

Coalfield

Gondwana

Basin of India

HVB 023119898119898 0 minus 7 10minus13

28





described as

119863

1199032120597

120597119903(1199032

120597119862

120597119903) =

120597119862

120597119905

( 2-22 )


time














29











119862(119903 119905 gt 0) = 1198620 ( 2-23 )





119872119905119872infin

= 1 minus6

1205872sum

1

1198992119890119909119901(minus119863119890119899

21205872119905)

infin

119899=1

( 2-24 )

119863119890 = 1198631199031198902 ( 2-25 )



diffusivity

30









119878 =sum((119872119905119872infin)119890119909119901

minus (119872119905119872infin)119898119900119889119890119897

)

2

( 2-26 )

where (119872119905

119872infin)119890119909119901

and (119872119905

119872infin)119898119900119889119890119897


sorption fraction




((119872119905







31







in MATLAB GUI

32



33















(Knudsen 1909)

119863119870 =1

3119889119888 ( 2-27 )



119888 = radic8119877119879120587119872 ( 2-28 )


34







119863119870119901119898 =

120601

120591119863119870

( 2-29 )













35








119865 = 119873휀119863119891 = 119888119900119899119904119905119886119899119905 ( 2-30 )


휀



119871(휀) = 119873휀 ( 2-31 )



119871

119903

36

119871(휀) = 119865휀1minus119863119891 ( 2-32 )



119865 = 119871119904119863119891 ( 2-33 )


119871(휀) = 119871119904119863119891휀1minus119863119891 ( 2-34 )





120591 = 119871

119903=119871119904119863119891휀1minus119863119891

119871119904= (

휀

119871119904)1minus119863119891

( 2-35 )





120591 = (휀

119871119904)2minus119863119891

( 2-36 )




37

119863119870119901119898 =1

3120601 (119871119904휀)2minus119863119891

119863119870 =2radic21206011198891198770511987905

31205870511987205(119871119904휀)2minus119863119891

( 2-37 )


Froment 1995)














120582 =

5

8

120583

119875radic119877119879120587

2119872

( 2-38 )




38







2014)

119863119861 =1

3120582119888

( 2-39 )





1

119863119901=1

119863119870+1

119863119861 ( 2-40 )








39


(2014)

1

119863119901= 119908119870

1

119863119870119901119898+ (1 minus 119908119870)

1

119863119861 ( 2-41 )







119908119870 = 119891(119870119899) =


( 2-42 )











40














41




media

23 Summary











Pressure

minus



Pressure

minus

(a)(b)

Considering

tortuosity factor


42












sorption capacity




value of 119875119871







43




dynamically









with pressure









44


production profiles

45

Chapter 3

EXPERIMENTAL WORK


















46






study

Nos Coal sample

Moisture

content

()

Ash

content

()

Volatile

matter

()

Fixed

carbon

()

Ro max

()

1 Xiuwu-21 149 2911 1037 6303 402

2 Luling-9 125 754 3217 6104 089

3 Luling-10 137 1027 3817 5119 083

4 Sijiangzhuang-15 203 3542 1223 549 311

47






















48













simultaneously

49



331 Void Volume









Sample Cell

Reference

Cell

Pressure Transducer

1

23

4

Water Bath

(Constant T)

Data Acquisition

System


(c)


Reference cell

ValvePressure

transducer

Water bath

Sample cell

Pressuretransducer

50





119875119881 = 119885119899119877119879 ( 3-1 )
















119860119881 = 119861 ( 3-2 )

51

119860 =

[ 119875119904 minus

(119885)119875119904(119885)119875119903

119875119903 119875119904

119875119903119894

(119885)119875119903119894minus

119875119904119894

(119885)119875119904119894

119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894]

( 3-3 )

119861 = [

0119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894] 119881119904119886119898 ( 3-4 )

119881 = [119881119903119881119904] ( 3-5 )


is set to be zero



void (cm

3)

1 27582 31818 26631 27611 2 27665 31788 26660 27666 3 27689 31782 26648 27688

Average 27645 31796 26647 27655







52




119899119886119889119904119894 = 119899119900

119894 minus 119899119906119899119886119889119904119894 ( 3-6 )




119899119900119894 =

119875119904119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119894minus1119877119879+

119875119903119894119881119903

(119885)119875119903119894119877119879 ( 3-7 )


119899119906119899119886119889119904119894 =

119875119904119894(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894119877119879 ( 3-8 )



119899119886119889119904119894 = 119899119886119889119904

119894 + 119899119886119889119904119894minus1 ( 3-9 )





sample

53






















54


model








119872119905119872infin

=1

119899119904119886119905119894 119877119879

(119875119904119890119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119890119894minus1+119875119903119894119881119903

(119885)119875119903119894minus119875119904119894(119905)(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894) ( 3-10 )










34 Summary




55




















56

Chapter 4


















(3542 )

57






















58











59

















Coal

sample

Mean PD

(nm)

Pore Volume (cm3100 g) 119889119898119894119899

(nm)

119889119898119886119909

(nm)

BET σ


Xiuwu-21 761 1178 00247 0703 0451 1741 83759 485

Luling-9 1249 0395 000330 0172 0220 1880 115440 081

Luling-10 1505 0393 000372 0149 0240 1870 112430 089

Sijiangzhu

ang-15 46 2772 00537 0456 2262 1565 132447 511

60






P0





Luling-10

( )10 50 100 500 1000

00000

00005

00010

00015

00020

00025

00030

00035

00040

00045

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

10 50 100 500 1000

0000

0001

0002

0003

0004

0005

0006

0007

0008

0009

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Luling-9

( )10 50 100 500 1000

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

0020d

Vd

log

(W

) P

ore

Vo

lum

e (

cm

3g

)

Pore Width

Xiuwu-21

( )

10 50 100 500 1000

000

002

004

006

008

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Sijiazhuang-15

( )



61





















62









Xiuwu-21 -0868 2132 0981 -0313 2687 0983 -0772 2229 0967

Luling-9 -0445 2555 0980 -0439 2561 0998 -0505 2495 0989

Luling-10 -0426 2574 0971 -0468 2532 0997 -0504 2496 0975

Sijiangzhuang

-15-0452 2547 0972 -0677 2324 0983 -0425 2575 0932

63













results



and 807 MPa)



Xiuwu-21 -0772 2229 0967


1787 2199 2506 2800 3029 3199

Luling-9 -0505 2495 0989 4128 6472 8587 10924 12948 14576

Luling-10 -0504 2496 09754078 6395 8486 10798 12800 14409

Sijiangzhuang-

15-0537 2463 0932

5606 9444 13111 17336 21114 24223

64














65


curves








66


















Xiuwu-21 2736 069 0984 1

Luling-9 1674 134 0987 2

Luling-10 1388 123 0986 8


67





















68




and Luling-10











0 20000 40000 60000 80000 100000

00

02

04

06

08

10

Adsorp

tion F

raction


Exp

Unipore Model

0 20000 40000 60000 80000 10000003

04

05

06

07

08

09

10

11

Adsorp

tion F

raction


Exp

Unipore Model

Xiuwu-21 Luling-10

69













70


pressure









times 10minus12

0 2 4 6 8

0

2

4

6

8

10

Measure

d D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

71























72


PL)











Rsup2 = 0138

0

10

20

30

40

15 17 19 21 23 25 27

VL m

3 to

n

D1

Rsup2 = 01642

0

10

20

30

40

50

15 17 19 21 23 25 27

VL m

3 to

n

D2

Rsup2 = 06301

0

04

08

12

16

15 17 19 21 23 25 27

PL M

Pa

D1

Rsup2 = 00137

0

04

08

12

16

15 17 19 21 23 25 27P

L M

Pa

D2

(a) (b)

(c) (d)

73







119881119871 = 119878(120590)11986312 + 119861 ( 4-1 )














74
















of the pore surface


D12 X3 = (Sσ11986312 + 119861)ν

Xiuwu-21 223 089 1874 581 293

Luling-9 250 075 833 077 205

Luing-10 250 075 723 087 206

Sijiangzhuang-15 257 071 1217 818 250

75























76







y = -06973x + 16643

Rsup2 = 09324

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

ln(X1)ln(1198831)

ln(119875119871)

ln 119875119871 = minus07ln (119881119871ν) + 17

1198772 = 093

y = 24372x + 133

Rsup2 = 09804

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

VL

m3

ton

X2 106 m2ton

VL m3tminus1

119883210 (m2 tminus1)

119881119871 = 24 1205901198632 + 133

1198772 = 098119881119871 = 24120590

1198631198912 + 133

1198772 = 098

77















y = -0723x + 17268

Rsup2 = 09834

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

X3

ln(119875119871)

ln 1198833

ln 119875119871 = minus07 ln 24 1205901198632 + 133

120584

+17

1198772 = 098

119891

78





















79


















Pressure [MPa] 055 138 248 414 607 807


Coefficient

[times10101198982119904]


Xiuwu-21 10477 6760 4227 5494 2388 4822 1469 4315 1042 3990 820 3777

Luling-9 4187 1922 1689 1226 954 924 587 726 416 613 328 544

Luling-10 3847 2154 1552 1373 877 1035 539 813 383 686 301 610

Sijiazhuang-15 26248 5102 10589 3029 5982 2181 3679 1650 2611 1355 2056 1181

80







0 2 4 6 8

0

5

10

15

20

25

30

DB

DKpm

Diffu

sio

n C

oeff

icie

nt

(m2s

)

Pressure (MPa)

times 10minus9

81



(P))











Horizontal

pure Knudsen

diffusion

combined

diffusion

pure bulk diffusion

119875lowast

Non-linear Linear

times 1012

0 2 4 6 8 10

0

2

4

6

8

10

Re

cip

rocal D

iffu

sio

n C

oeff

icie

nt

(sm

2)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

82












diffusion as

119863119901 = 119863119861(1 minus eminus1119870119899) ( 4-2 )









83











00 01 02 03 04 0500

02

04

06

08

10

Wk

Kn

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

pure bulk

combined

pure Knudsen

84























85

0 2 4 6 8 10

0

2

4

6

8

10


Experim

enta

l D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

0

2

4

6

8

This Work

Wheeler (1955)

Theore

tical D

iffu

sio

n C

oeffic

ient (m

2s

)






0 2 4 6 80

2

4

6

8

10

Exp

erim

enta

l D

iffu

sio

n C

oe

ffic

ien

t (m

2s

)


055 MPa

138 MPa

248 MPa

414 MPa

607 MPa

807 MPa

1198772 = 0782

1198772 = 09801198772 = 0992

1198772 = 0963

1198772 = 0926

1198772 = 0997

times10minus12

times10minus9

86





















87

47 Summary




















88

















dominates






89





90

Chapter 5





















91



















the decline stage

92












93























94






















95























96























97





















98




119902120588|119903+119889119903 minus 119902120588|119903 = 4120587119903

2119889119903120601120597120588

120597119905+ 41205871199032119889119903(1 minus 120601)

120597119902119886119889119904120597119905

( 5-1 )





120597(119902120588)

120597119903= 41205871199032120601

120597120588

120597119905+ 41205871199032(1 minus 120601)

120597119902119886119889119904120597119905

( 5-2 )




99

119902 =

41205871199032119896119898120583

120597119875

120597119903

( 5-3 )



1

1199032120597

120597119903(1199032119896119898120583

120588120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-4 )



120597119903


119869 = 119863

120597120588

120597119903

( 5-5 )




119902 =

119860

120588119869

( 5-6 )


41205871199032120601


2) with Eq (5-3) as

1

1199032120597

120597119903(1199032119863120601

120597120588

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-7 )


100

119888119892 = minus

1

119881

120597119881

120597119875=1

120588

120597120588

120597119875

( 5-8 )


1

1199032120597

120597119903(1199032119863120601119888119892120588

120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-9 )




119896119898 = 120601119888119892120583119863 ( 5-10 )














101




boundaries

















density log devices

102

















API)





gas content



temperature




103

















104



105












106












in Figure 5-1



107





















108




109


wells



Pressure

gradient

Reservoir

Pressure


1 3205 40 1181988 140 0478 1552

2 3440 26 1211995 157 0432 1508

3 3414 72 5291994 150 0458 1562

4 3495 34 12311993 155 0442 1527










119896119891

119896119891119900= (

120601119891

120601119891119900)

3

= [1 +119862119898120601119891119900

(119875 minus 119875119900) +1

120601119891119900(119870

119872minus 1) 휀]

3

( 5-11 )


119862119898 =

1

119872minus (

119870

119872+ 119891 minus 1) 119888119903

( 5-12 )

where 119896119891


pressure of 119875 120601119891


110


between 0 and 1



휀 =

3119881119871120588119904119877119879

119864119860119881119900int

1

119875119871 + 119875119889119875

119875

1198751

( 5-13 )

















111



120601119898 =

119881119901

119881119901 + 119881119892119903119886119894119899

( 5-14 )



















112






















113


production stage


s VL P

L E EEA c


(psi-1

) (F) 14 674 292 290E+05 03 5 201E-06 07 107


psi


114












115























116














117























118







Depth (ft) 3460 3460




















Skin Factor -2 -2


119


data

0 400 800 1200 1600

0

20

40

60

80

100

Matr

ix P

erm

ea

bili

ty M

ultip

lier

Pressure (psi)

Well A

Well B


matching process



120























121



CMG




122























123























124



















stage

125












126























127

56 Summary





















128




production curves






simulators









129

Chapter 6


61 Introduction









flow of hydrocarbon










130






















131











Gonzalez 1984)











132







631 Cleat-Scale



133










632 Pore-Scale












134






















135




















studied

136






















137









structure













138























139









140










sample

141


thawing operation

0 minDumping

Freeze

1 min 10 min

30 min 20 min

Freeze Freeze

Freeze

FreezeFreeze

40 min

Freeze

50 min

Freeze

Freeze


(a)

60 min


Thaw

10 min 20 min

40 min 30 min

Thaw

Thaw

ThawThaw

50 min

Thaw

60 min

Thaw

240 min

Finish 1 F-T cycle

Thaw

(b)

142





















143


center









[119906(119909)] =

4(1 minus 1205842)

120587119864119875119908 (ln |

119886

119909| + radic120587(1 minus (119909119886)2))

( 6-1 )




ie 119886(1198860) = 119886119899minus1(1198860)



144

[119906(119886)] = int [119906(119909)]

119886

minus119886

119889119909 =2radic120587(1 minus 1205842)

119864119875119908119886

( 6-2 )



휀119888 = int

[119906(119886)]

119886119889120588(1198860)

120588(119886119898119886119909)

120588(119886119898119894119899)

=2radic120587(1 minus 1205842)

119864int 119875119908119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-3 )











Thus

119875119908 = 119860(119879 119886)120573119898119875119888 ( 6-4 )





145






119875119888 = 119870119888radic1119886 ( 6-5 )




휀119888 = 2radic120587119860(119879 119886)120573119898

(1 minus 1205842)119870119888119864

int radic1119886119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-6 )




휀119905 = 120573119898120572119871 119879 ( 6-7 )


a negative value




휀 = 휀119905 + 휀119888 ( 6-8 )

146





651 Gas Kinetics
















147























148















149

0 2 4 6 8 10

0

5

10

15

Ad

so

rption

Cap

acity (

mlg

)













Coal

Sample

119881119871 ml g

119875119871 MPa

R2

Raw 1446 091 0998 5

1F-T 1643 099 (79) 0998 5

3F-T 1665 112 (232) 0997 9

150










151












0 2 4 6 8 10

2

4

6

8




Diffu

sio

n C

oeff

icie

nt

(1e-1

3 m

2s

)


Improve by

1876

Improve by

3018

152





[MPa] [1e-13

m2s]

[1e-13

m2s] [1e-13

m2s]

Adsorption 049 157 186 1832 174 1056 103 189 240 2659 219 1550 209 269 326 2111 296 986 352 316 374 1859 344 895 559 377 462 2251 408 816 842 535 564 544 553 333

Desorption 052 189 258 3680 218 1562 106 243 321 3226 290 1919 205 310 414 3363 353 1386 338 357 475 3313 433 2114 535 563 648 1511 591 501












153























154





6-7











gas molecules




155






















156









samples

00 02 04 06 08 10

000

005

010

015

020 Raw Coal

1F-T Coal

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure


slit shaped pores

157





samples















158























159





2000)

119867119868 =

119860119889119890 minus 119860119886119889119860119886119889

( 6-9 )


respectively













160



119907

119907119898=

119888119896119909

(1 minus 119896119909)(1 + (119888 minus 1)119896119909)

( 6-10 )




00 02 04 06 08 10

000

004

008

012

016


BET

GAB

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Aad




the GAB equation



161



119860119886119889 = int 1199071

0119889119909 =

119907119898















119888 = 119888119900

1

1 + 1198861119909 + 11988621199092 +⋯

( 6-12 )


equation when 1198861 = 1198862 = ⋯ = 0



162

119907

119907119898=

1198880119896119909


119899lowast1 )119896119909)

( 6-13 )
















163

00 02 04 06 08 10

000

004

008

012

016

Ade


GAB (n=0)

Cubic GAB (n=1)

Quadratic GAB (n=2)

Qu

an

tity

Ad

so

rbed

(m

molg

)

Relative Pressure









119860119889119890 = int 1199071198891199091

0

= int 119907119898119891(119909)1198891199091

0

= (sum119891(119909119894) + 119891(119909119894+1)

2

1 119909

119894=0

) 119909119907119898

( 6-14 )

164






















thawed coal samples

165











0 20 40 60 80 100

000

001

002

003

004

0 2 4 6 8 10

000

001

002

003

004

Raw Coal

1F-T Coal

3F-T Coald

Vd

log

(w)

Po

re V

olu

me (

cm

sup3g

)

Pore Width (nm)

dV

dlo

g(w

) P

ore

Vo

lum

e (

cm

sup3g

)

Pore Width (nm)

mesopore macropore

166





0000 0005 0010 0015 0020 0025 0030

00

01

02

03

04

05

06

07 Raw Coal

1F-T Coall

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure









167






4 6 8 10 12

000

004

008

012

016

Raw Coal

1F-T Coal

3F-T Coal

dV

dlo

g(W

) P

ore

Volu

me (

cm

sup3g)

Pore Width (Aring)


3F-T coal samples







168




cycles respectively














Coal sample dmean

(nm)



Raw 665 2130 189 294 483

1F-T 614 2025 298 346 644




169











170


















171






Levine 1996)


Chugh et al 1989)



















172



cycle


















173














range of estimates

174
















fracturing






175























176


such as coal




















177


permeability




(mm) (mm) (gcm3)

()

M-1 9996 4989 139 0036 0

M-2 10007 5017 138 0048 058





178


















179











180




in-contact region











Incident Wave

Fracture Plane

Outgoing Wave

Scattered Wave

Undisturbed Wave

Ui(x)


x3

x2

x1

C Cc

F

181



















Norris 1992)

182








119906(119909 119905) = 119860119890minus119894120596119905119890119894119896119909 ( 6-15 )




1205972119906(119909 119905)

1205971199052= 1199072

1205972119906(119909 119905)

1205971199092 ( 6-16 )








183

1205881205962119906119894(119909) +120597

120597119909119895119862119894119895119896119897

120597119906119896(119909)

120597119909119897= 0 ( 6-17 )











119932 = 119872(휀)119932119920 ( 6-18 )





119872(휀) = 119871 + 휀1198711 + 119874(휀2) ( 6-19 )

119871 = 119872(0) ( 6-20 )



184













119932120782 = 119871119932119920 ( 6-21 )




119932 = 119932120782 + 휀1198711119932120782 ( 6-22 )





185

119932 gt= 119932120782 + 휀 1198711 gt 119932120782 ( 6-23 )




1198711 gt= int 119901(119888)(119888)119865

119889119888 ( 6-24 )





119901(119888) = 119873 ( 6-25 )

and

1198711 gt= 119873int (119888)119865

119889119888 ( 6-26 )









186

119905119894(119909) = 0 [119906119894(119909)] = 0 119894 = 123 ( 6-27 )




[119905119894(119909)] = 0 119894 = 123 ( 6-28 )









119957 = 119948[119958] ( 6-29 )

where 119948 = [

119896119905 0 00 119896119905 00 0 119896119899






119932119956(119888) = 휀(119888)119932120782 ( 6-30 )

187







119932119956(119909) = int 119905119894(119932119956(120585))[119866119894

119868(119909 120585)]119889120585119914

( 6-31 )





[119932119930] = [ 119932 gt] ( 6-32 )




휀119932120782 = int 119896119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]119889120585

119914

( 6-33 )





188

119932 gt= 119932120782 + 119873int 119870119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]

119865

119889120585 ( 6-34 )





119905119894( 119932 gt) = 119870119894119895119905 [ 119880119895 gt] ( 6-35 )

where 119870119894119895119905 = 119873119870119894119895 = 119873int 119896119894119895119889120585119914


meanfield













189


(Johnson 1985)

119875 = 21205871198861198871199010 ( 6-36 )


pressure



1199063 = 21 minus 1205842

1198641199010119887119825(119890) ( 6-37 )





matrix


properties as

119875 = 120587119886119864

1 minus 12058421

119825(119890)1199063 ( 6-38 )




190



119905119899 =1

2119873119875[119906119899] ( 6-39 )

where 1199063 =1

2[119906119899]


be determined as

119870119899 =1

2119873119875 =

1

2119873120587119886

119864

1 minus 12058421

119825(119890) ( 6-40 )


2009) Eq (6-39) becomes

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41 )



1985)

119876 = 21205871198861198871199020 ( 6-42 )



1199061 = 1199062 =1199020119887

119866[119825(119890) +

120584

1198902(1 minus 1198902)119825(119890) minus 119812(119890)] ( 6-43 )



191


properties as

119876 =2120587119886119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

11990612 ( 6-44 )





119905119905 =1

2119873119876[119906119905] ( 6-45 )

where 11990612 =1

2[119906119905]



119870119905 =1

2119873119876 = 119873120587119886

119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

( 6-46 )


2009) Eq (6-41) becomes

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )




192




119870119899 = 41198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752) ( 6-48 )


119870119905 = 2119873120587119886120588

1198811199042

[120587 +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


(119825(119890) minus 119812(119890)

1198902)]

( 6-49 )


119825(e)minus119812(e)



lim119890rarr0

((1 minus 1198902)119825(119890) minus 119812(119890)

1198902)

= lim119890rarr0


2119890) = minus

120587

4

( 6-50 )

where 119825prime(119890) =119889119825(119890)

119889119890=

119812(119890)

119890(1minus119890 )minus119825(119890)

119890 and 119812prime(119890) =

119889119812(119890)

119889119890=119812(119890)minus119825(119890)


Manzhirov 2006)



119870119905 = 811987312058711988612058811988111990421 minus 119881119878

2 1198811198752frasl

3 minus 21198811198782 119881119875

2frasl ( 6-51 )



193






















194












671 Surface Cracks



195














196





197

672 Wave Velocities






given by

120572 = minus20

ℎ119897119900119892(119860119860119900) ( 6-52 )














198






199






















200






wet coal specimen

(a)

(b)

201





119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41)

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )














202










coal specimen (M-1)



Sample 119873 119886 119887 119890

(1mm2) (mm) (mm) (-)

M-1 0097 10 018 098

M-2 019 10 045 090








203






















204

0 10 20 30 40 50 60

0

20

40

60

80

100

120

Fra

ctu

re S

tiffness (

GP

am

)

Freezing Time (min)

Kn K

t

00

05

10

15

20

25

30 K

tK

n

Tangential to

Norm

al S

tiffness R

atio










205

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890)+119872prime

( 6-53 )

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

+119866prime

( 6-54 )









and the surrounding






expressed as

119861119894 = ℎ119881119896119888119860 ( 6-55 )

206

where ℎ (119882


119882








119889119879

119889119905= minus

119860

119881ℎ(1198790 minus 119879119888)

1

120588119862119875 ( 6-56 )

where 119889119879

119889119905(119870



1198983) and 119862119875 (

119869


specimen


1959)

119889119879

119889119905= minus(

119860

119881)2

119896119888(1198790 minus 119879119888)1

120588119862119901 ( 6-57 )






207






ℎ119881

119896119888119860=(2013)(00101)

0226= 899 ( 6-58 )












208





ℎ 119862119901 119896119888 120588 119904119908 119904119894119888119890

(Wm2K) (JkgK) (WmK) (kgm3) (-) (-)

2013 953 0226 1380 0313 0345









209














210





119870119905119870119899=1 minus 120584

1 minus1205842

( 6-59 )






119870119905119870119899=

7

8 [1 +92120587

119872prime

radic1 minus 1198902119872]

( 6-60 )










211









26



212










through

119870 = 120575radic1198763

( 6-61 )











213






















214











68 Summary











215






















216























217























218




















219

Chapter 7

CONCLUSIONS



















220





















221




sorption capacity




value of 119875119871










dynamically





222





with pressure










production profiles







223






















224























225














law








226





















227























228






















229























230
















231
















summation







232



coefficient






233


coefficient

MATLAB Code






234














xhigh as a double




CreateFcns called

235






xlow as a double




CreateFcns called






nmax as a double



MATLAB

236


CreateFcns called










237





238






















239











MATLAB Code





240


and 0 to elsewhere


Step 3







241

REFERENCE








21(2) 4-13











228-247



physics 79(7) 3566-3571







159-188







Institute





242



163(3) 307-320




























3222-3223



0012-3










243







httpsdoiorg1010160021-9797(69)90423-8





httpsdoiorg1010160020-7683(76)90044-5






httpsdoiorg101016S0166-5162(03)00113-7













2) 3-26




36




Engineering 39(12) 9249-9257

httpdoiorg101007s13369-014-1472-1




244












































245


1483-1498

httpsdoiorg1010160016-2361(96)00142-1




httpsdoiorg101016S0016-2361(99)00055-1







httpsdoiorg101016S0008-6223(97)00124-3



Engineering 8 9-33




11(02) 311-325








Geology 88(4) 204-217











Simulator




246








httpsdoiorg1010160009-2509(94)00478-A







httpsdoiorg101016S0166-5162(97)00015-3






293-303





















247






httpsdoiorg1010160008-6223(68)90302-3



httpsdoiorg1010160008-6223(64)90040-5



Sciences


























httpdoiorg1010880953-8984246001





Butterworths

248





















httpdoiorg101007s00603-016-1120-z



30(4) 453-464



33(3) 296-300










45(2) 105-113






249






659







121-125








httpsdoiorg1010160020-7683(92)90133-E





142



325



1599












250

httpsdoiorg101016S0020-7225(97)00080-3





726



httpdoiorg101017S0305004100057674






BS ISO 15901-1


Publications










Fuels 30(12) 10383-10395





httpsdoiorg1010160148-9062(93)90018-9







Fuel 52(3) 181-185





251












4(4) 191-197

























165-183







252

httpdoi101002andp19093330106







httpsdoiorg101016S0009-2509(96)00458-7





















1059-1063



3000



httpdoiorg10130612181212061









253



42(7) 1233-1242




247

httpdoiorg101111j1365-2478200700688x



Science 41(4) 627-633









httpsdoiorg1010160021-9797(82)90153-9



httpsdoiorg101016S0165-232X(05)80005-9





Exhibition















254
















httpsdoiorg101016S1365-1609(99)00103-3



httpsdoiorg1010160016-2361(75)90061-7























255



149-161



Calgary Alberta









1(06) 539-544












Research Insititute
















Physics 79(7) 3558-3565

256




















Press











httpsdoiorg101016S1365-1609(99)00104-5









49(24) 4171-4180




257

httpsdoiorg101016S0927-7757(02)00063-8



Sci Rep 7(1) 3675

httpdoiorg101038s41598-017-04019-7




1231-1242













20






















258










1171





Cryogenic Concrete



httpsdoiorg1010160016-2361(74)90002-7


















Alabama


93-98



httpdoiorg101046j1365-2478200200289x

259



























Colorado












httpsdoiorg101016S0016-2361(03)00010-3




260















Orleans Louisiana



84(2) 752-759





49(10) 1083



httpsdoiorg101016S0730-725X(96)00167-1



277




Springer










261




httpdoiorg756-764 101021la051686h



31



Resources


























196 47-58








262







237

















Corporation




















263





















America (RPSEA)



httpdoiorg101111j1365-28181988tb04604x







76(15) 507

httpdoiorg101007s12665-017-6829-9












264






2296










2428-2444






VITA

Yun Yang

EDUCATION



























v























vi























vii























viii























ix























x






xi


LIST OF TABLES xx



11 Background 1






23 Summary 41









xii








56 Summary 127













xiii







REFERENCE 241

xiv

LIST OF FIGURES







surfaces 16



24















setup 49


China 58



xv



isotherm curves 65






(VL PL) 72













respectively 83






pressure 85




xvi






105








Po=1542 psi 113







production data 119













xvii



































xviii












samples 166






















xix











xx

LIST OF TABLES





this study 46


Helium 51






248 414 607 and 807 MPa) 63








offset wells 109


growth 113


wells 118





xxi












cycles 168









xxii

ACKNOWLEDGEMENTS



















Chapter 1

INTRODUCTION

11 Background

















2


















3








0 5 10 15 20 25 30

0

50

100

150

Pro

duct

ion r

ate

(M

cfd

ay)

time (yrs)

Desorption from



Butt cleat

Face cleat

Darcyrsquos

flow

Log (nm) 012gt3

Dominated by




Well information



4























5























6














application






7




















8





Theory Experiment

Pore structure-Gas




Experiment

Adsorption

Capacity

Adsorption

Rate

Transport

RateHeterogeneity

Pore structure-

Sorption Model

Pore structure-

Diffusion Model


9

Chapter 2

THEORETICAL MODEL



















10























11























12











ln (V

V0) = 119860 ln (ln (

P0119875)) + 119864 ( 2-1 )





(Liu and Nie 2016)

119863119891 = 119860 + 3 ( 2-2 )

and

119863119891 = 3119860 + 3 ( 2-3 )

13




1198810) vs ln (ln (

P0
















119863 = 2

(a)

2 119863 3

(b)

14





119881 =

119875

119875 + 119875119871119881119871

( 2-4 )


plots of 119875119881 vs 119875















15



as

119877119905 =1198891205791119889119905

= 119896119886 119875(1 minus 1205791)119908+1 minus 119896119889 1205791

119898+1 ( 2-5 )







119877119905 =119889120579119905119889119905

= 119896119886 120583(1 minus 1205791)119898+1 minus 119896119889 1205791

119898+1 ( 2-6 )



120583 =119875

(2120587119872119877119879)12 ( 2-7 )







16





1205791 =

(119896119886 prime

119896119889 )120584

119875

1 + (119896119886 prime

119896119889 )120584

119875

( 2-8 )

where 120584 = 1(119898 + 1) and 119896119886 prime= 119896119886 (2120587119872119877119879)

minus12120584



119886 =1

119875119871= (

119896119886 prime

119896119889 )

120584

( 2-9 )





17




120584 = 1 minus (119863119891 minus 2

2) ( 2-10 )





by

119886 = (119896119886 prime

119896119889 )

119862120584

( 2-11 )




119896119886 prime = 1198921198730(2120587119872119877119879)minus12119862120584 ( 2-12 )




factor (119896119861)

119896119889 = 119891119890minus119876119896119861119879 ( 2-13 )

18




119886 = (119896119886 prime

119896119889 )

119862120584

=1

radic2120587119872119877119879(119892

119891119881119871119890

119876119896119861119879)119862120584

( 2-14 )





119886 =1

radic2120587119872119877119879(119881119871119890

119876119896119861119879)119862120584

( 2-15 )











19


surface (119876119904119905) as

119876 = 119870119876119904119905119862120584

( 2-16 )










119886 =1

119875119871= (119881119871)

119862120584119865(119879) ( 2-17 )

119865(119879) =1

radic2120587119872119877119879119890minus119870119876119904119905(119896119861119879) ( 2-18 )


condition





20









119881119871 = 119878(120590)1198631198912 + 119861 ( 2-19 )












ln(119875119871) = minus119862 ln(119881119871120584) + ln(119865(119879)) ( 2-20 )

21









ln(119875119871) = 119862 ln((119878(120590)1198631198912 + 119861)120584) + 119865(119879) ( 2-21 )













22






















23























24


















25












diffusion regime











26






27





119898119898

Pressure

MPa

Range of

119863 1198982119904 Nandi and Walker

(1975) 1975 US coals

Anthracite to

HVB 0315 119898119898

114minus 252

10minus13

minus 10minus14

Smith and


Fruitland San

Juan Basin Sub 19119898119898 57

10minus13

minus 10minus14

Mavor et al

(1990a) 1990

Fruitland San


Marecka and


Semi-

anthracite 125 062 02 0032119898119898 0-01

10minus10

minus 10minus15

Clarkson and

Bustin (1999b) 1999

Lower

Cretaceous

Gates

Formation

Canada


minus 10minus13

Busch et al

(2004b) 2004

Silesian Basin


Cui et al (2004)

(further reworked

by (Pillalamarry et

al 2011) )


10minus13

minus 10minus14


10minus13

minus 10minus15


Kentucky

Coalfield



(2010) 2010

Lorraine


Pillalamarry et al


10minus13

minus 10minus14

Salmachi and


Australian

coal seam HVB 0294119898119898

0014minus 4678

10minus12

Bhowmik and

Dutta (2013) 2013

Raniganj

Coalfield

Jharia

Coalfield

Gondwana

Basin of India

Sub to HVB 01245119898119898 036minus 461

10minus12

minus 10minus13


10minus11

minus 10minus12

Wang and Liu

(2016) 2016

San Juan

Basin and

Pittsburgh


minus 10minus14

Naveen et al

(2017) 2017

Jharia

Coalfield

Gondwana

Basin of India

HVB 023119898119898 0 minus 7 10minus13

28





described as

119863

1199032120597

120597119903(1199032

120597119862

120597119903) =

120597119862

120597119905

( 2-22 )


time














29











119862(119903 119905 gt 0) = 1198620 ( 2-23 )





119872119905119872infin

= 1 minus6

1205872sum

1

1198992119890119909119901(minus119863119890119899

21205872119905)

infin

119899=1

( 2-24 )

119863119890 = 1198631199031198902 ( 2-25 )



diffusivity

30









119878 =sum((119872119905119872infin)119890119909119901

minus (119872119905119872infin)119898119900119889119890119897

)

2

( 2-26 )

where (119872119905

119872infin)119890119909119901

and (119872119905

119872infin)119898119900119889119890119897


sorption fraction




((119872119905







31







in MATLAB GUI

32



33















(Knudsen 1909)

119863119870 =1

3119889119888 ( 2-27 )



119888 = radic8119877119879120587119872 ( 2-28 )


34







119863119870119901119898 =

120601

120591119863119870

( 2-29 )













35








119865 = 119873휀119863119891 = 119888119900119899119904119905119886119899119905 ( 2-30 )


휀



119871(휀) = 119873휀 ( 2-31 )



119871

119903

36

119871(휀) = 119865휀1minus119863119891 ( 2-32 )



119865 = 119871119904119863119891 ( 2-33 )


119871(휀) = 119871119904119863119891휀1minus119863119891 ( 2-34 )





120591 = 119871

119903=119871119904119863119891휀1minus119863119891

119871119904= (

휀

119871119904)1minus119863119891

( 2-35 )





120591 = (휀

119871119904)2minus119863119891

( 2-36 )




37

119863119870119901119898 =1

3120601 (119871119904휀)2minus119863119891

119863119870 =2radic21206011198891198770511987905

31205870511987205(119871119904휀)2minus119863119891

( 2-37 )


Froment 1995)














120582 =

5

8

120583

119875radic119877119879120587

2119872

( 2-38 )




38







2014)

119863119861 =1

3120582119888

( 2-39 )





1

119863119901=1

119863119870+1

119863119861 ( 2-40 )








39


(2014)

1

119863119901= 119908119870

1

119863119870119901119898+ (1 minus 119908119870)

1

119863119861 ( 2-41 )







119908119870 = 119891(119870119899) =


( 2-42 )











40














41




media

23 Summary











Pressure

minus



Pressure

minus

(a)(b)

Considering

tortuosity factor


42












sorption capacity




value of 119875119871







43




dynamically









with pressure









44


production profiles

45

Chapter 3

EXPERIMENTAL WORK


















46






study

Nos Coal sample

Moisture

content

()

Ash

content

()

Volatile

matter

()

Fixed

carbon

()

Ro max

()

1 Xiuwu-21 149 2911 1037 6303 402

2 Luling-9 125 754 3217 6104 089

3 Luling-10 137 1027 3817 5119 083

4 Sijiangzhuang-15 203 3542 1223 549 311

47






















48













simultaneously

49



331 Void Volume









Sample Cell

Reference

Cell

Pressure Transducer

1

23

4

Water Bath

(Constant T)

Data Acquisition

System


(c)


Reference cell

ValvePressure

transducer

Water bath

Sample cell

Pressuretransducer

50





119875119881 = 119885119899119877119879 ( 3-1 )
















119860119881 = 119861 ( 3-2 )

51

119860 =

[ 119875119904 minus

(119885)119875119904(119885)119875119903

119875119903 119875119904

119875119903119894

(119885)119875119903119894minus

119875119904119894

(119885)119875119904119894

119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894]

( 3-3 )

119861 = [

0119875119904119894minus1

(119885)119875119904119894minus1minus

119875119904119894

(119885)119875119904119894] 119881119904119886119898 ( 3-4 )

119881 = [119881119903119881119904] ( 3-5 )


is set to be zero



void (cm

3)

1 27582 31818 26631 27611 2 27665 31788 26660 27666 3 27689 31782 26648 27688

Average 27645 31796 26647 27655







52




119899119886119889119904119894 = 119899119900

119894 minus 119899119906119899119886119889119904119894 ( 3-6 )




119899119900119894 =

119875119904119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119894minus1119877119879+

119875119903119894119881119903

(119885)119875119903119894119877119879 ( 3-7 )


119899119906119899119886119889119904119894 =

119875119904119894(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894119877119879 ( 3-8 )



119899119886119889119904119894 = 119899119886119889119904

119894 + 119899119886119889119904119894minus1 ( 3-9 )





sample

53






















54


model








119872119905119872infin

=1

119899119904119886119905119894 119877119879

(119875119904119890119894minus1(119881119904 minus 119881119904119886119898)

(119885)119875119904119890119894minus1+119875119903119894119881119903

(119885)119875119903119894minus119875119904119894(119905)(119881119903 + 119881119904 minus 119881119904119886119898)

(119885)119875119904119894) ( 3-10 )










34 Summary




55




















56

Chapter 4


















(3542 )

57






















58











59

















Coal

sample

Mean PD

(nm)

Pore Volume (cm3100 g) 119889119898119894119899

(nm)

119889119898119886119909

(nm)

BET σ


Xiuwu-21 761 1178 00247 0703 0451 1741 83759 485

Luling-9 1249 0395 000330 0172 0220 1880 115440 081

Luling-10 1505 0393 000372 0149 0240 1870 112430 089

Sijiangzhu

ang-15 46 2772 00537 0456 2262 1565 132447 511

60






P0





Luling-10

( )10 50 100 500 1000

00000

00005

00010

00015

00020

00025

00030

00035

00040

00045

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

10 50 100 500 1000

0000

0001

0002

0003

0004

0005

0006

0007

0008

0009

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Luling-9

( )10 50 100 500 1000

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

0020d

Vd

log

(W

) P

ore

Vo

lum

e (

cm

3g

)

Pore Width

Xiuwu-21

( )

10 50 100 500 1000

000

002

004

006

008

dV

dlo

g (

W)

Po

re V

olu

me

(cm

3g

)

Pore Width

Sijiazhuang-15

( )



61





















62









Xiuwu-21 -0868 2132 0981 -0313 2687 0983 -0772 2229 0967

Luling-9 -0445 2555 0980 -0439 2561 0998 -0505 2495 0989

Luling-10 -0426 2574 0971 -0468 2532 0997 -0504 2496 0975

Sijiangzhuang

-15-0452 2547 0972 -0677 2324 0983 -0425 2575 0932

63













results



and 807 MPa)



Xiuwu-21 -0772 2229 0967


1787 2199 2506 2800 3029 3199

Luling-9 -0505 2495 0989 4128 6472 8587 10924 12948 14576

Luling-10 -0504 2496 09754078 6395 8486 10798 12800 14409

Sijiangzhuang-

15-0537 2463 0932

5606 9444 13111 17336 21114 24223

64














65


curves








66


















Xiuwu-21 2736 069 0984 1

Luling-9 1674 134 0987 2

Luling-10 1388 123 0986 8


67





















68




and Luling-10











0 20000 40000 60000 80000 100000

00

02

04

06

08

10

Adsorp

tion F

raction


Exp

Unipore Model

0 20000 40000 60000 80000 10000003

04

05

06

07

08

09

10

11

Adsorp

tion F

raction


Exp

Unipore Model

Xiuwu-21 Luling-10

69













70


pressure









times 10minus12

0 2 4 6 8

0

2

4

6

8

10

Measure

d D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

71























72


PL)











Rsup2 = 0138

0

10

20

30

40

15 17 19 21 23 25 27

VL m

3 to

n

D1

Rsup2 = 01642

0

10

20

30

40

50

15 17 19 21 23 25 27

VL m

3 to

n

D2

Rsup2 = 06301

0

04

08

12

16

15 17 19 21 23 25 27

PL M

Pa

D1

Rsup2 = 00137

0

04

08

12

16

15 17 19 21 23 25 27P

L M

Pa

D2

(a) (b)

(c) (d)

73







119881119871 = 119878(120590)11986312 + 119861 ( 4-1 )














74
















of the pore surface


D12 X3 = (Sσ11986312 + 119861)ν

Xiuwu-21 223 089 1874 581 293

Luling-9 250 075 833 077 205

Luing-10 250 075 723 087 206

Sijiangzhuang-15 257 071 1217 818 250

75























76







y = -06973x + 16643

Rsup2 = 09324

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

ln(X1)ln(1198831)

ln(119875119871)

ln 119875119871 = minus07ln (119881119871ν) + 17

1198772 = 093

y = 24372x + 133

Rsup2 = 09804

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

VL

m3

ton

X2 106 m2ton

VL m3tminus1

119883210 (m2 tminus1)

119881119871 = 24 1205901198632 + 133

1198772 = 098119881119871 = 24120590

1198631198912 + 133

1198772 = 098

77















y = -0723x + 17268

Rsup2 = 09834

-06

-04

-02

0

02

04

1 15 2 25 3 35

ln(P

L)

X3

ln(119875119871)

ln 1198833

ln 119875119871 = minus07 ln 24 1205901198632 + 133

120584

+17

1198772 = 098

119891

78





















79


















Pressure [MPa] 055 138 248 414 607 807


Coefficient

[times10101198982119904]


Xiuwu-21 10477 6760 4227 5494 2388 4822 1469 4315 1042 3990 820 3777

Luling-9 4187 1922 1689 1226 954 924 587 726 416 613 328 544

Luling-10 3847 2154 1552 1373 877 1035 539 813 383 686 301 610

Sijiazhuang-15 26248 5102 10589 3029 5982 2181 3679 1650 2611 1355 2056 1181

80







0 2 4 6 8

0

5

10

15

20

25

30

DB

DKpm

Diffu

sio

n C

oeff

icie

nt

(m2s

)

Pressure (MPa)

times 10minus9

81



(P))











Horizontal

pure Knudsen

diffusion

combined

diffusion

pure bulk diffusion

119875lowast

Non-linear Linear

times 1012

0 2 4 6 8 10

0

2

4

6

8

10

Re

cip

rocal D

iffu

sio

n C

oeff

icie

nt

(sm

2)

Pressure (MPa)

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

82












diffusion as

119863119901 = 119863119861(1 minus eminus1119870119899) ( 4-2 )









83











00 01 02 03 04 0500

02

04

06

08

10

Wk

Kn

Xiuwu-21

Luling-9

Luling-10

Sijiazhuang-15

pure bulk

combined

pure Knudsen

84























85

0 2 4 6 8 10

0

2

4

6

8

10


Experim

enta

l D

iffu

sio

n C

oeffic

ient (m

2s

)

Pressure (MPa)

0

2

4

6

8

This Work

Wheeler (1955)

Theore

tical D

iffu

sio

n C

oeffic

ient (m

2s

)






0 2 4 6 80

2

4

6

8

10

Exp

erim

enta

l D

iffu

sio

n C

oe

ffic

ien

t (m

2s

)


055 MPa

138 MPa

248 MPa

414 MPa

607 MPa

807 MPa

1198772 = 0782

1198772 = 09801198772 = 0992

1198772 = 0963

1198772 = 0926

1198772 = 0997

times10minus12

times10minus9

86





















87

47 Summary




















88

















dominates






89





90

Chapter 5





















91



















the decline stage

92












93























94






















95























96























97





















98




119902120588|119903+119889119903 minus 119902120588|119903 = 4120587119903

2119889119903120601120597120588

120597119905+ 41205871199032119889119903(1 minus 120601)

120597119902119886119889119904120597119905

( 5-1 )





120597(119902120588)

120597119903= 41205871199032120601

120597120588

120597119905+ 41205871199032(1 minus 120601)

120597119902119886119889119904120597119905

( 5-2 )




99

119902 =

41205871199032119896119898120583

120597119875

120597119903

( 5-3 )



1

1199032120597

120597119903(1199032119896119898120583

120588120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-4 )



120597119903


119869 = 119863

120597120588

120597119903

( 5-5 )




119902 =

119860

120588119869

( 5-6 )


41205871199032120601


2) with Eq (5-3) as

1

1199032120597

120597119903(1199032119863120601

120597120588

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-7 )


100

119888119892 = minus

1

119881

120597119881

120597119875=1

120588

120597120588

120597119875

( 5-8 )


1

1199032120597

120597119903(1199032119863120601119888119892120588

120597119875

120597119903) = 120601

120597120588

120597119905+ (1 minus 120601)

120597119902119886119889119904120597119905

( 5-9 )




119896119898 = 120601119888119892120583119863 ( 5-10 )














101




boundaries

















density log devices

102

















API)





gas content



temperature




103

















104



105












106












in Figure 5-1



107





















108




109


wells



Pressure

gradient

Reservoir

Pressure


1 3205 40 1181988 140 0478 1552

2 3440 26 1211995 157 0432 1508

3 3414 72 5291994 150 0458 1562

4 3495 34 12311993 155 0442 1527










119896119891

119896119891119900= (

120601119891

120601119891119900)

3

= [1 +119862119898120601119891119900

(119875 minus 119875119900) +1

120601119891119900(119870

119872minus 1) 휀]

3

( 5-11 )


119862119898 =

1

119872minus (

119870

119872+ 119891 minus 1) 119888119903

( 5-12 )

where 119896119891


pressure of 119875 120601119891


110


between 0 and 1



휀 =

3119881119871120588119904119877119879

119864119860119881119900int

1

119875119871 + 119875119889119875

119875

1198751

( 5-13 )

















111



120601119898 =

119881119901

119881119901 + 119881119892119903119886119894119899

( 5-14 )



















112






















113


production stage


s VL P

L E EEA c


(psi-1

) (F) 14 674 292 290E+05 03 5 201E-06 07 107


psi


114












115























116














117























118







Depth (ft) 3460 3460




















Skin Factor -2 -2


119


data

0 400 800 1200 1600

0

20

40

60

80

100

Matr

ix P

erm

ea

bili

ty M

ultip

lier

Pressure (psi)

Well A

Well B


matching process



120























121



CMG




122























123























124



















stage

125












126























127

56 Summary





















128




production curves






simulators









129

Chapter 6


61 Introduction









flow of hydrocarbon










130






















131











Gonzalez 1984)











132







631 Cleat-Scale



133










632 Pore-Scale












134






















135




















studied

136






















137









structure













138























139









140










sample

141


thawing operation

0 minDumping

Freeze

1 min 10 min

30 min 20 min

Freeze Freeze

Freeze

FreezeFreeze

40 min

Freeze

50 min

Freeze

Freeze


(a)

60 min


Thaw

10 min 20 min

40 min 30 min

Thaw

Thaw

ThawThaw

50 min

Thaw

60 min

Thaw

240 min

Finish 1 F-T cycle

Thaw

(b)

142





















143


center









[119906(119909)] =

4(1 minus 1205842)

120587119864119875119908 (ln |

119886

119909| + radic120587(1 minus (119909119886)2))

( 6-1 )




ie 119886(1198860) = 119886119899minus1(1198860)



144

[119906(119886)] = int [119906(119909)]

119886

minus119886

119889119909 =2radic120587(1 minus 1205842)

119864119875119908119886

( 6-2 )



휀119888 = int

[119906(119886)]

119886119889120588(1198860)

120588(119886119898119886119909)

120588(119886119898119894119899)

=2radic120587(1 minus 1205842)

119864int 119875119908119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-3 )











Thus

119875119908 = 119860(119879 119886)120573119898119875119888 ( 6-4 )





145






119875119888 = 119870119888radic1119886 ( 6-5 )




휀119888 = 2radic120587119860(119879 119886)120573119898

(1 minus 1205842)119870119888119864

int radic1119886119889120588(1198860)120588(119886119898119886119909)

120588(119886119898119894119899)

( 6-6 )




휀119905 = 120573119898120572119871 119879 ( 6-7 )


a negative value




휀 = 휀119905 + 휀119888 ( 6-8 )

146





651 Gas Kinetics
















147























148















149

0 2 4 6 8 10

0

5

10

15

Ad

so

rption

Cap

acity (

mlg

)













Coal

Sample

119881119871 ml g

119875119871 MPa

R2

Raw 1446 091 0998 5

1F-T 1643 099 (79) 0998 5

3F-T 1665 112 (232) 0997 9

150










151












0 2 4 6 8 10

2

4

6

8




Diffu

sio

n C

oeff

icie

nt

(1e-1

3 m

2s

)


Improve by

1876

Improve by

3018

152





[MPa] [1e-13

m2s]

[1e-13

m2s] [1e-13

m2s]

Adsorption 049 157 186 1832 174 1056 103 189 240 2659 219 1550 209 269 326 2111 296 986 352 316 374 1859 344 895 559 377 462 2251 408 816 842 535 564 544 553 333

Desorption 052 189 258 3680 218 1562 106 243 321 3226 290 1919 205 310 414 3363 353 1386 338 357 475 3313 433 2114 535 563 648 1511 591 501












153























154





6-7











gas molecules




155






















156









samples

00 02 04 06 08 10

000

005

010

015

020 Raw Coal

1F-T Coal

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure


slit shaped pores

157





samples















158























159





2000)

119867119868 =

119860119889119890 minus 119860119886119889119860119886119889

( 6-9 )


respectively













160



119907

119907119898=

119888119896119909

(1 minus 119896119909)(1 + (119888 minus 1)119896119909)

( 6-10 )




00 02 04 06 08 10

000

004

008

012

016


BET

GAB

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure

Aad




the GAB equation



161



119860119886119889 = int 1199071

0119889119909 =

119907119898















119888 = 119888119900

1

1 + 1198861119909 + 11988621199092 +⋯

( 6-12 )


equation when 1198861 = 1198862 = ⋯ = 0



162

119907

119907119898=

1198880119896119909


119899lowast1 )119896119909)

( 6-13 )
















163

00 02 04 06 08 10

000

004

008

012

016

Ade


GAB (n=0)

Cubic GAB (n=1)

Quadratic GAB (n=2)

Qu

an

tity

Ad

so

rbed

(m

molg

)

Relative Pressure









119860119889119890 = int 1199071198891199091

0

= int 119907119898119891(119909)1198891199091

0

= (sum119891(119909119894) + 119891(119909119894+1)

2

1 119909

119894=0

) 119909119907119898

( 6-14 )

164






















thawed coal samples

165











0 20 40 60 80 100

000

001

002

003

004

0 2 4 6 8 10

000

001

002

003

004

Raw Coal

1F-T Coal

3F-T Coald

Vd

log

(w)

Po

re V

olu

me (

cm

sup3g

)

Pore Width (nm)

dV

dlo

g(w

) P

ore

Vo

lum

e (

cm

sup3g

)

Pore Width (nm)

mesopore macropore

166





0000 0005 0010 0015 0020 0025 0030

00

01

02

03

04

05

06

07 Raw Coal

1F-T Coall

3F-T Coal

Quantity

Adsorb

ed (

mm

olg)

Relative Pressure









167






4 6 8 10 12

000

004

008

012

016

Raw Coal

1F-T Coal

3F-T Coal

dV

dlo

g(W

) P

ore

Volu

me (

cm

sup3g)

Pore Width (Aring)


3F-T coal samples







168




cycles respectively














Coal sample dmean

(nm)



Raw 665 2130 189 294 483

1F-T 614 2025 298 346 644




169











170


















171






Levine 1996)


Chugh et al 1989)



















172



cycle


















173














range of estimates

174
















fracturing






175























176


such as coal




















177


permeability




(mm) (mm) (gcm3)

()

M-1 9996 4989 139 0036 0

M-2 10007 5017 138 0048 058





178


















179











180




in-contact region











Incident Wave

Fracture Plane

Outgoing Wave

Scattered Wave

Undisturbed Wave

Ui(x)


x3

x2

x1

C Cc

F

181



















Norris 1992)

182








119906(119909 119905) = 119860119890minus119894120596119905119890119894119896119909 ( 6-15 )




1205972119906(119909 119905)

1205971199052= 1199072

1205972119906(119909 119905)

1205971199092 ( 6-16 )








183

1205881205962119906119894(119909) +120597

120597119909119895119862119894119895119896119897

120597119906119896(119909)

120597119909119897= 0 ( 6-17 )











119932 = 119872(휀)119932119920 ( 6-18 )





119872(휀) = 119871 + 휀1198711 + 119874(휀2) ( 6-19 )

119871 = 119872(0) ( 6-20 )



184













119932120782 = 119871119932119920 ( 6-21 )




119932 = 119932120782 + 휀1198711119932120782 ( 6-22 )





185

119932 gt= 119932120782 + 휀 1198711 gt 119932120782 ( 6-23 )




1198711 gt= int 119901(119888)(119888)119865

119889119888 ( 6-24 )





119901(119888) = 119873 ( 6-25 )

and

1198711 gt= 119873int (119888)119865

119889119888 ( 6-26 )









186

119905119894(119909) = 0 [119906119894(119909)] = 0 119894 = 123 ( 6-27 )




[119905119894(119909)] = 0 119894 = 123 ( 6-28 )









119957 = 119948[119958] ( 6-29 )

where 119948 = [

119896119905 0 00 119896119905 00 0 119896119899






119932119956(119888) = 휀(119888)119932120782 ( 6-30 )

187







119932119956(119909) = int 119905119894(119932119956(120585))[119866119894

119868(119909 120585)]119889120585119914

( 6-31 )





[119932119930] = [ 119932 gt] ( 6-32 )




휀119932120782 = int 119896119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]119889120585

119914

( 6-33 )





188

119932 gt= 119932120782 + 119873int 119870119894119895[ 119880119895 gt (120585)][119866119894119868(119909 120585)]

119865

119889120585 ( 6-34 )





119905119894( 119932 gt) = 119870119894119895119905 [ 119880119895 gt] ( 6-35 )

where 119870119894119895119905 = 119873119870119894119895 = 119873int 119896119894119895119889120585119914


meanfield













189


(Johnson 1985)

119875 = 21205871198861198871199010 ( 6-36 )


pressure



1199063 = 21 minus 1205842

1198641199010119887119825(119890) ( 6-37 )





matrix


properties as

119875 = 120587119886119864

1 minus 12058421

119825(119890)1199063 ( 6-38 )




190



119905119899 =1

2119873119875[119906119899] ( 6-39 )

where 1199063 =1

2[119906119899]


be determined as

119870119899 =1

2119873119875 =

1

2119873120587119886

119864

1 minus 12058421

119825(119890) ( 6-40 )


2009) Eq (6-39) becomes

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41 )



1985)

119876 = 21205871198861198871199020 ( 6-42 )



1199061 = 1199062 =1199020119887

119866[119825(119890) +

120584

1198902(1 minus 1198902)119825(119890) minus 119812(119890)] ( 6-43 )



191


properties as

119876 =2120587119886119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

11990612 ( 6-44 )





119905119905 =1

2119873119876[119906119905] ( 6-45 )

where 11990612 =1

2[119906119905]



119870119905 =1

2119873119876 = 119873120587119886

119866

[119825(119890) +1205841198902(1 minus 1198902)119825(119890) minus 119812(119890)]

( 6-46 )


2009) Eq (6-41) becomes

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )




192




119870119899 = 41198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752) ( 6-48 )


119870119905 = 2119873120587119886120588

1198811199042

[120587 +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


(119825(119890) minus 119812(119890)

1198902)]

( 6-49 )


119825(e)minus119812(e)



lim119890rarr0

((1 minus 1198902)119825(119890) minus 119812(119890)

1198902)

= lim119890rarr0


2119890) = minus

120587

4

( 6-50 )

where 119825prime(119890) =119889119825(119890)

119889119890=

119812(119890)

119890(1minus119890 )minus119825(119890)

119890 and 119812prime(119890) =

119889119812(119890)

119889119890=119812(119890)minus119825(119890)


Manzhirov 2006)



119870119905 = 811987312058711988612058811988111990421 minus 119881119878

2 1198811198752frasl

3 minus 21198811198782 119881119875

2frasl ( 6-51 )



193






















194












671 Surface Cracks



195














196





197

672 Wave Velocities






given by

120572 = minus20

ℎ119897119900119892(119860119860119900) ( 6-52 )














198






199






















200






wet coal specimen

(a)

(b)

201





119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890) ( 6-41)

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

( 6-47 )














202










coal specimen (M-1)



Sample 119873 119886 119887 119890

(1mm2) (mm) (mm) (-)

M-1 0097 10 018 098

M-2 019 10 045 090








203






















204

0 10 20 30 40 50 60

0

20

40

60

80

100

120

Fra

ctu

re S

tiffness (

GP

am

)

Freezing Time (min)

Kn K

t

00

05

10

15

20

25

30 K

tK

n

Tangential to

Norm

al S

tiffness R

atio










205

119870119899 = 21198731205871198861205881198811199042 (1 minus

1198811199042

1198811198752)

1

119825(119890)+119872prime

( 6-53 )

119870119905 = 2119873120587119886

1205881198811199042

[2119825(119890) +1 minus 2119881119878

2 1198811198752frasl

1 minus 1198811198782 119881119875


1198902]

+119866prime

( 6-54 )









and the surrounding






expressed as

119861119894 = ℎ119881119896119888119860 ( 6-55 )

206

where ℎ (119882


119882








119889119879

119889119905= minus

119860

119881ℎ(1198790 minus 119879119888)

1

120588119862119875 ( 6-56 )

where 119889119879

119889119905(119870



1198983) and 119862119875 (

119869


specimen


1959)

119889119879

119889119905= minus(

119860

119881)2

119896119888(1198790 minus 119879119888)1

120588119862119901 ( 6-57 )






207






ℎ119881

119896119888119860=(2013)(00101)

0226= 899 ( 6-58 )












208





ℎ 119862119901 119896119888 120588 119904119908 119904119894119888119890

(Wm2K) (JkgK) (WmK) (kgm3) (-) (-)

2013 953 0226 1380 0313 0345









209














210





119870119905119870119899=1 minus 120584

1 minus1205842

( 6-59 )






119870119905119870119899=

7

8 [1 +92120587

119872prime

radic1 minus 1198902119872]

( 6-60 )










211









26



212










through

119870 = 120575radic1198763

( 6-61 )











213






















214











68 Summary











215






















216























217























218




















219

Chapter 7

CONCLUSIONS



















220





















221




sorption capacity




value of 119875119871










dynamically





222





with pressure










production profiles







223






















224























225














law








226





















227























228






















229























230
















231
















summation







232



coefficient






233


coefficient

MATLAB Code






234














xhigh as a double




CreateFcns called

235






xlow as a double




CreateFcns called






nmax as a double



MATLAB

236


CreateFcns called










237





238






















239











MATLAB Code





240


and 0 to elsewhere


Step 3







241

REFERENCE








21(2) 4-13











228-247



physics 79(7) 3566-3571







159-188







Institute





242



163(3) 307-320




























3222-3223



0012-3










243







httpsdoiorg1010160021-9797(69)90423-8





httpsdoiorg1010160020-7683(76)90044-5






httpsdoiorg101016S0166-5162(03)00113-7













2) 3-26




36




Engineering 39(12) 9249-9257

httpdoiorg101007s13369-014-1472-1




244












































245


1483-1498

httpsdoiorg1010160016-2361(96)00142-1




httpsdoiorg101016S0016-2361(99)00055-1







httpsdoiorg101016S0008-6223(97)00124-3



Engineering 8 9-33




11(02) 311-325








Geology 88(4) 204-217











Simulator




246








httpsdoiorg1010160009-2509(94)00478-A







httpsdoiorg101016S0166-5162(97)00015-3






293-303





















247






httpsdoiorg1010160008-6223(68)90302-3



httpsdoiorg1010160008-6223(64)90040-5



Sciences


























httpdoiorg1010880953-8984246001





Butterworths

248





















httpdoiorg101007s00603-016-1120-z



30(4) 453-464



33(3) 296-300










45(2) 105-113






249






659







121-125








httpsdoiorg1010160020-7683(92)90133-E





142



325



1599












250

httpsdoiorg101016S0020-7225(97)00080-3





726



httpdoiorg101017S0305004100057674






BS ISO 15901-1


Publications










Fuels 30(12) 10383-10395





httpsdoiorg1010160148-9062(93)90018-9







Fuel 52(3) 181-185





251












4(4) 191-197

























165-183







252

httpdoi101002andp19093330106







httpsdoiorg101016S0009-2509(96)00458-7





















1059-1063



3000



httpdoiorg10130612181212061









253



42(7) 1233-1242




247

httpdoiorg101111j1365-2478200700688x



Science 41(4) 627-633









httpsdoiorg1010160021-9797(82)90153-9



httpsdoiorg101016S0165-232X(05)80005-9





Exhibition















254
















httpsdoiorg101016S1365-1609(99)00103-3



httpsdoiorg1010160016-2361(75)90061-7























255



149-161



Calgary Alberta









1(06) 539-544












Research Insititute
















Physics 79(7) 3558-3565

256




















Press











httpsdoiorg101016S1365-1609(99)00104-5









49(24) 4171-4180




257

httpsdoiorg101016S0927-7757(02)00063-8



Sci Rep 7(1) 3675

httpdoiorg101038s41598-017-04019-7




1231-1242













20






















258










1171





Cryogenic Concrete



httpsdoiorg1010160016-2361(74)90002-7


















Alabama


93-98



httpdoiorg101046j1365-2478200200289x

259



























Colorado












httpsdoiorg101016S0016-2361(03)00010-3




260















Orleans Louisiana



84(2) 752-759





49(10) 1083



httpsdoiorg101016S0730-725X(96)00167-1



277




Springer










261




httpdoiorg756-764 101021la051686h



31



Resources


























196 47-58








262







237

















Corporation




















263





















America (RPSEA)



httpdoiorg101111j1365-28181988tb04604x







76(15) 507

httpdoiorg101007s12665-017-6829-9












264






2296










2428-2444






VITA

Yun Yang

EDUCATION



























MODELING OF GAS SORPTION AND DIFFUSION BEHAVIOR …

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