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
Associate Professor of Energy and Mineral Engineering
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
3 and the validation results are presented in Chapter 4 Here presents an abstract of the
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 micro- pores 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
Figure 4-15 Comparison between experimental and theoretical calculated
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
Validation of Theory1
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
221 Literature Review
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
Bituminous 05119898119898 0 minus 9 10minus13
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
Figure 2-5 Flowchart of the automated computer program for effective diffusion
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
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
micropores mesopores macropores micropores mesopores macropores
61
result of different mechanisms for low-pressure and high-pressure N2 sorption 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 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
Adsorption time (seconds)
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 macro- pores 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)
Figure 4-15 Comparison between experimental and theoretical calculated diffusion
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
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)
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
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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
Figure 5-11 Matrix permeability growth during pressure depletion employed in the
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
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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
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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
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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
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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 micro- scale 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
136
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
Langmuir Isotherm for CH4 adsorption
CH4 ad-desorption excess data of 3F-T coal
Langmuir Isotherm for CH4 adsorption
Figure 6-5 Results of methane addesorption tests and the corresponding Langmuir
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
ad-desorption diffusivity of 3F-T coal
Diffu
sio
n C
oeff
icie
nt
(1e-1
3 m
2s
)
Equilibrium Pressure (MPa)
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
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
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
663 Experimental Procedures
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 )
As 119864 and 120584 can be directly inverted from elastic wave velocities data (Mavko et al
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
Figure 6-20 The workflow of seismic interpretations of fracture stiffness for coal
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
2frasl(1 minus 1198902)119825(119890) minus 119812(119890)
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
2frasl(1 minus 1198902)119825(119890) minus 119812(119890)
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
Figure 6-25 Under wet condition (M-2) variation of normal and tangential fracture
stiffness and tangentialnormal stiffness ratio with freezing time
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-
cryogenic fracturing effect on diffusion improvement and gas production enhancement
especially for high ldquosorption timerdquo CBM reservoirs
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
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
221
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
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
222
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
coupled into the commercially available simulator to predict the long-term CBM well
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
accomplished in the experimental work can be summarized as follows
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
223
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
Chapter 4 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
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
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
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
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
225
distribution and fractal dimension as inputs and it provides accurate modeling of the
variation of diffusion coefficients at different pressures and for different coals
Chapter 5 investigates the impact of the pressure-dependent diffusion coefficient
on CBM production An equivalent matrix permeability modeling is 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
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
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
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
predicting the deterioration degree of pore structure due to the repetition of freezing
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