TYPE CURVES FOR PRESSURE TRANSIENT ANALYSIS OF COMPOSITE ...

The Pennsylvania State University

The Graduate School

Department of Energy & Mineral Engineering

TYPE CURVES FOR PRESSURE TRANSIENT ANALYSIS OF COMPOSITE

DOUBLE-POROSITY GAS RESERVOIRS

A Thesis in

Energy & Mineral Engineering

by

Sachin Rana

2011 Sachin Rana

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2011

The thesis of Sachin Rana was reviewed and approved* by the following:

Turgay Ertekin

Professor of Petroleum & Natural Gas Engineering

George E.Trimble in Earth and Mineral Sciences

Thesis Advisor

Derek Elsworth

Professor of Energy and Geo-Environmental Engineering

John Yilin Wang

Assistant Professor of Petroleum & Natural Gas Engineering

R. Larry Grayson

Professor of Energy & Mineral Engineering

Graduate Program Officer of Energy & Mineral Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

Economic production from unconventional gas reservoirs require horizontal drilling with

multistage fracturing. These unconventional gas reservoirs include shale, tight gas reservoirs.

Since there has been a lot of interest in exploration and production of these systems in past few

years, it is very important to understand the pressure response of these systems. This thesis

presents forward and backward solutions for the pressure transient analysis of composite dual

porosity systems. The forward solution consists of a generalized numerical model that predicts

the wellbore pressure response by taking reservoir properties as inputs. Although the presented

numerical model has been validated to provide forward solutions for gas reservoirs with vertical

and horizontal wells, this study only focuses on gas reservoirs with horizontal wells. Backward

solution is used to calculate reservoir properties from the field data. In this study, type curves

have been presented for composite dual porosity systems with horizontal gas wells to provide the

backward solution. The presented type curves have been shown to predict correct reservoir

properties if one has field wellbore pressure response.

iv

TABLE OF CONTENTS

LIST OF FIGURES ................................................................................................................. vii

LIST OF TABLES ................................................................................................................... xiv

NOMENCLATURE ................................................................................................................ xvi

ACKNOWLEDGEMENTS ..................................................................................................... xix

Chapter 1 INTRODUCTION ................................................................................................... 1

Chapter 2 LITERATURE SURVEY ....................................................................................... 3

2.1 Well Testing In Naturally Fractured Reservoirs ........................................................ 3 2.1.1 Naturally Fractured Systems with Vertical Wells ........................................... 3 2.1.2 Naturally Fractured Systems with Horizontal Wells ....................................... 6

2.2 Well Testing in Composite Systems .......................................................................... 7 2.2.1 Composite Systems with Vertical Wells ......................................................... 7 2.2.2 Composite Systems with Horizontal Wells ..................................................... 10

2.3 Well Testing in Composite Dual Porosity Reservoirs ............................................... 11 2.3.1 Composite Dual Porosity Systems with Vertical Wells .................................. 11 2.3.2 Composite Dual Porosity Systems with Horizontal Wells .............................. 13

Chapter 3 PROBLEM STATEMENT ..................................................................................... 18

Chapter 4 NUMERICAL MODEL PRESENTATION ........................................................... 19

4.1 Model Description ...................................................................................................... 19 4.2 Mathematical Development ....................................................................................... 20

4.2.1 Implementing Warren and Root model in Diffusivity Equation ..................... 20 4.3 Numerical Approximation to partial differential equations ....................................... 22 4.4 Model Validation ....................................................................................................... 24

4.4.1 Validation for Dual Porosity Reservoirs with a Vertical Well ........................ 25 4.4.2 Validation for Composite Reservoirs with Vertical wells ............................... 27 4.4.3 Validation for Composite Naturally Fractured Reservoirs with horizontal

wells ................................................................................................................. 30

Chapter 5 SENSTIVITY ANALYSIS OF DIMENSIONLESS VARIABLES ....................... 35

5.1 Dimensionless Parameters ......................................................................................... 35 5.1.1 Inner Region Dimensionless Parameters ......................................................... 36 5.1.2 Outer Region Dimensionless Parameters ........................................................ 37 5.1.3 Other Dimensionless Parameters..................................................................... 37

5.2 Characteristic Dimensionless Derivative Plot ............................................................ 38 5.3 Effect of Constant Dimensionless Parameters ........................................................... 40 5.4 Effect of Changing Storativity Ratio of Fracture Networks (F) ................................. 41

v

5.5 Effect of Changing Mobility Ratio (M) ..................................................................... 42 5.6 Effect of Other Reservoir Parameters ........................................................................ 43

5.6.1 Conclusions drawn from sensitivity analysis of dimensionless variables: ...... 50 5.7 Scaling Correlations to Reduce the Number of Parameters ....................................... 51

5.8 Sensitivity test for correlation

1.1

1.

wr

b .......................................................... 54

Chapter 6 TYPE CURVES AND TYPE CURVE APPLICATIONS ...................................... 57

6.1 Assumptions ............................................................................................................... 57 6.2 Properties Calculated using Type Curves .................................................................. 58 6.3 Structure of generation of type curves ....................................................................... 58 6.4 Type curve analysis procedure ................................................................................... 60

6.4.1 Calculations done on the y-axis: ..................................................................... 62 6.4.2 Calculations done on the x-axis: ..................................................................... 63 6.4.3 Verification of right type curve ....................................................................... 64

6.5 Versatility of Type Curves ......................................................................................... 66 6.5.1 Pressure Drawdown Testing ............................................................................ 66 6.5.2 Pressure Build-Up Testing .............................................................................. 67

6.6 Case Studies (Examples) ............................................................................................ 68 6.6.1 Case Study 1 .................................................................................................... 68 6.6.2 Case Study 2 .................................................................................................... 75

Chapter 7 .................................................................................................................................. 83

CONCLUSIONS ..................................................................................................................... 83

Appendix A NUMERICAL MODEL DEVELOPMENT ............................................... 85

Appendix B .............................................................................................................................. 89

Simulation Data Tables ............................................................................................................ 89

Appendix C ............................................................................................................................. 95

Type Curves for Composite Dual Porosity Systems with a Horizontal Well .......................... 95

C.1 Type curves generated for F=1 .................................................................................. 95

C.1.1 1.02

y

b ........................................................................................................ 96

C.1.2 3.02

y

b ........................................................................................................ 99

C.1.3 5.02

y

b ........................................................................................................ 102

C.2 Type curves generated for F=10 ................................................................................ 105

vi

C.2.1 1.02

y

b ........................................................................................................ 105

C.2.2 3.02

y

b ........................................................................................................ 108

C.2.3 5.02

y

b ........................................................................................................ 111

C.3 Type curves generated for F=100 .............................................................................. 114

C.3.1 1.02

y

b ........................................................................................................ 114

C.3.2 3.02

y

b ........................................................................................................ 117

C.3.3 5.02

y

b ........................................................................................................ 120

Appendix D .............................................................................................................................. 123

Type Curve Match Points for Case Studies ............................................................................. 123

D.1 Case Study 1 .............................................................................................................. 123 D.1.1 Pressure Drawdown Testing ........................................................................... 123 D.1.1 Pressure Build-up Testing .............................................................................. 124

D.2 Case Study 2 .............................................................................................................. 125 D.2.1 Pressure Drawdown Testing ........................................................................... 125 D.2.2 Pressure Build-up Testing .............................................................................. 126

Appendix E .............................................................................................................................. 127

Pressure Transient Data Used for Case Studies ....................................................................... 127

E.1 Data for case study 1 ................................................................................................. 127 E.2 Data for case study 2 ................................................................................................. 133

REFERENCES ........................................................................................................................ 141

vii

LIST OF FIGURES

Figure 2-1: Schematic of Warren and Root approximation of naturally fractured

reservoirs (Warren and Root, 1963). ................................................................................ 4

Figure 2-2: Schematic of a composite reservoir with a vertical well (Satman,

Eggenschwiler et al. 1980). .............................................................................................. 8

Figure 2-3: Schematic of a horizontal well in a composite reservoir (Ambastha and

Ghaffari 1998). ................................................................................................................. 10

Figure 2-4: Schematic of composite naturally fractured system with a vertical well

(Kikani and Walkup Jr. 1991). ......................................................................................... 13

Figure 2-5: Schematic of discrete hydraulic fractures (Brohi, Pooladi-Darvish et al.

2011). ............................................................................................................................... 14

Figure 2-6: Micro-Seismic data visualization during hydraulic fracturing of horizontal

wells. ................................................................................................................................ 15

Figure 2-7: Another sample of micro-seismic data visualization collected during

hydraulic fracturing of horizontal well. ........................................................................... 16

Figure 4-1: Schematic of a dual porosity reservoir with a vertical well in the center. ........... 25

Figure 4-2: Sensitivity analysis of block dimensions. ............................................................ 26

Figure 4-3: Comparison of presented numerical solution against Warren and Root

analytical solution for dual porosity reservoirs with a vertical well. ............................... 27

Figure 4-4: Schematic of a composite reservoir with a vertical well in the center with a

circular discontinuity. ....................................................................................................... 28

Figure 4-5: Sensitivity analysis for local grid refinement around wellbore. ........................... 29

Figure 4-6: Comparison of presented numerical model with (Satman, Eggenschwiler et

al. 1980)analytical model for composite reservoir with a vertical well. .......................... 30

Figure 4-7: Schematic of a composite dual porosity reservoir with a horizontal well and

an elliptical discontinuity. ................................................................................................ 31

Figure 4-8: Sensitivity analysis of grid refinement around wellbore for a composite dual

porosity system with a horizontal well. ............................................................................ 32

Figure 4-9: Semi-log plot to compare presented numerical solution to the commercial

numerical solution. ........................................................................................................... 33

viii

Figure 4-10: Cartesian plot to compare presented numerical solution to the commercial

numerical solution. ........................................................................................................... 33

Figure 5-1: Characteristic dimensionless semi-log derivative plot for a composite dual

porosity system with a horizontal well. ............................................................................ 39

Figure 5-2: Effect of different combinations of reservoir properties while keeping the

dimensionless parameters constant. ................................................................................. 41

Figure 5-3: Effect of Storativity Ratio of Fracture Networks (F) on the dimensionless

semi-log derivative plot. .................................................................................................. 42

Figure 5-4: Effect of different mobility ratios on the dimensionless semi-log derivative

plot. .................................................................................................................................. 43

Figure 5-5: Effect of ω1 on the first transition period on dimensionless semi-log

derivative plot. ................................................................................................................. 44

Figure 5-6: Effect of different ω2 on the transition period on the dimensionless

derivative plots. ................................................................................................................ 45

Figure 5-7: Effect of changing drainage area while keeping the well length and inner

region area constant. ........................................................................................................ 46

Figure 5-8: Effect of λ1 on the first transition period. ............................................................ 47

Figure 5-9: Effect of changing minor axis (b) on the first transition period. ......................... 48

Figure 5-10: Effect of 2

1

on the first transition period......................................................... 49

Figure 5-11: The effect of scaling the y-axis by

3.0

wr

band keeping the value of

dimensionless correlation

1.1

1.

wr

b constant. ................................................................ 52

Figure 5-12: Schematic of a composite dual porosity reservoir with a horizontal well. ......... 53

Figure 5-13: Effect of different combinations of Area,x

L2, minor axis (b), 1 ,

2

1

on

the dimensionless semi log derivative plot. ..................................................................... 54

Figure 5-14: Effect of range of correlating parameter

1.1

1.

wr

b ............................................ 56

ix

Figure 6-1: Structure from which the type curves have been generated. ............................... 59

Figure 6-2: Sample type curve plot for F = 10, ω1 = 1e-2. ..................................................... 60

Figure 6-3: Sample type curve matching. ............................................................................... 61

Figure 6-4: Sample plot of pressure drawdown and build-up. ................................................ 67

Figure 6-5: Field pseudo-pressure drop derivative plot for drawdown test ............................ 70

for case study 1. ....................................................................................................................... 70

Figure 6-6: Type curve made for F=10, ,3.02

y

b,10 1

1

1.1

1

wr

bn . ................... 70

Figure 6-7: Field pseudo pressure drop derivative plot for pressure build-up test for case

study 1. ............................................................................................................................. 73

Figure 6-8: Field data derivative plot for drawdown plot for case study 2. ............................ 76

Figure 6-9: Field data derivative plot for build-up test for case study 2. ................................ 80

Figure A-1: Structure of Numerical Solution. ........................................................................ 88

Figure C-1: Type curve plotted for F=1, 1.02

y

b, 11 ,

1.1

1

wr

bn . ....................... 96


y

b,

1

1 10 ,

1.1

1

wr

bn . ................ 97


y

b,

2

1 10 ,

1.1

1

wr

bn . ................ 97


y

b,

3

1 10 ,

1.1

1

wr

bn . ................ 98


y

b,

4

1 10 ,

1.1

1

wr

bn . ................ 98


y

b, 11 ,

1.1

1

wr

bn . ...................... 99

x


y

b,

1

1 10 ,

1.1

1

wr

bn . ................ 100


y

b,

2

1 10 ,

1.1

1

wr

bn . ................ 100


y

b,

3

1 10 ,

1.1

1

wr

bn . ................ 101


y

b,

4

1 10 ,

1.1

1

wr

bn . .............. 102


y

b, 11 ,

1.1

1

wr

bn . .................... 102


y

b,

1

1 10 ,

1.1

1

wr

bn . ............. 103


y

b,

2

1 10 ,

1.1

1

wr

bn . .............. 103


y

b,

3

1 10 ,

1.1

1

wr

bn . .............. 104


y

b,

4

1 10 ,

1.1

1

wr

bn . .............. 104


y

b, 11 ,

1.1

1

wr

bn . .................. 105


y

b,

1

1 10 ,

1.1

1

wr

bn . ............ 106


y

b,

2

1 10 ,

1.1

1

wr

bn . ............ 106

xi


y

b,

3

1 10 ,

1.1

1

wr

bn . ............ 107


y

b,

4

1 10 ,

1.1

1

wr

bn . ............ 107


y

b, 11 ,

1.1

1

wr

bn . .................. 108


y

b,

1

1 10 ,

1.1

1

wr

bn . ............ 109


y

b,

2

1 10 ,

1.1

1

wr

bn . ............ 109


y

b,

3

1 10 ,

1.1

1

wr

bn . ............ 110


y

b,

4

1 10 ,

1.1

1

wr

bn . ............ 110


y

b, 11 ,

1.1

1

wr

bn . .................. 111


y

b,

1

1 10 ,

1.1

1

wr

bn . ............ 112


y

b,

2

1 10 ,

1.1

1

wr

bn . ............ 112


y

b,

3

1 10 ,

1.1

1

wr

bn . ............ 113


y

b,

4

1 10 ,

1.1

1

wr

bn . ............ 113

xii


y

b, 11 ,

1.1

1

wr

bn . ................ 114


y

b,

1

1 10 ,

1.1

1

wr

bn . .......... 115


y

b,

2

1 10 ,

1.1

1

wr

bn . .......... 115


y

b,

3

1 10 ,

1.1

1

wr

bn . .......... 116


y

b,

4

1 10 ,

1.1

1

wr

bn . .......... 116


y

b, 11 ,

1.1

1

wr

bn . ................ 117


y

b,

1

1 10 ,

1.1

1

wr

bn . .......... 118


y

b,

2

1 10 ,

1.1

1

wr

bn . .......... 118


y

b,

3

1 10 ,

1.1

1

wr

bn . .......... 119


y

b,

4

1 10 ,

1.1

1

wr

bn . .......... 119


y

b, 11 ,

1.1

1

wr

bn . ................ 120


y

b,

1

1 10 ,

1.1

1

wr

bn . .......... 121

xiii


y

b,

2

1 10 ,

1.1

1

wr

bn . .......... 121


y

b,

3

1 10 ,

1.1

1

wr

bn . .......... 122


y

b,

4

1 10 ,

1.1

1

wr

bn . .......... 122

xiv

LIST OF TABLES

Table 5-1: Data used to check applicability of dimensionless parameters for a composite

dual porosity system. ........................................................................................................ 40

Table 5-2: Calculation of the maximum and minimum values of λ1. ...................................... 55

Table 6-3: Table to calculate maximum and minimum limit of correation

1.1

1.

wr

b . ........ 55

Table 6-1: Type curve matching result from the type curve Figure 6-6 or Figure C-22. ...... 71

Table 6-2: Type curve matching result from the type curve Figure-C-17. ............................. 72

Table 6-3: Type curve matching result from the type curve Figure-C-27. ............................ 72

Table 6-4: Type curve matching result from the type curve Figure C-22.............................. 74

Table 6-6: Matching on type curve Figure C-33. .................................................................... 77

Table 6-7: Matching on type curve Figure C-38. .................................................................. 78

Table 6-8: Matching on type curve Figure C-43. .................................................................. 78

Table 6-8: Matching on type curve Figure C-33. ................................................................... 80

Table B-1: Run data for validating dual porosity model against Warren & Root solution

with vertical well .............................................................................................................. 89

Table B-2: Run data for validating composite reservoir model against Eggenshwiler

solution with vertical well ................................................................................................ 90

Table B-3: Run data for validating composite dual porosity model against CMG solution ... 91

Table B-4: Simulation dataset for case study 2. ...................................................................... 93

Table D-1: Match points obtained by matching on type curve presented in Figure C-22. ..... 123





xv




Table E-1: Pressure drawdown data. ...................................................................................... 127

Table E-2: Build-up data for case study 1. ............................................................................. 130

Table E-3: Pressure drawdown data for case study 2 ............................................................. 133

Table E-4: Pressure build-up data for case study 2 ................................................................ 137

xvi

NOMENCLATURE

a = Length of major axis of inner region, ft

A = Reservoir drainage area, ft2

Ax = Cross-sectional area of grid block perpendicular to x-axis, ft2

Ay = Cross-sectional area of grid block perpendicular to y-axis, ft2

b = Length of minor axis of inner region, ft

Bg = Formation volume factor, RB/SCF

Cf = Compressibility gas in fractures, psi-1

Cm = Compressibilty of gas in matrix blocks, psi-1

F = Fracture storativity ratio of inner and outer region, fraction

Fs = Storage capacity ratio

h = Reservoir thickness, ft

k = Permeability, md

L = Horizontal half well length

M = Mobility ratio, fraction

p = Pressure, psi

qSCF/D = Production rate at surface conditions, SCF/D

qsc = Production rate at surface conditions, MMSCF/D

q* = Interporsity flow rate, SCF/D

r = Radius, ft

R = Radius to discontinuity, ft (for composite reservoirs with vertical wells)

S = Skin factor

t = time, hours

xvii

tda = Dimensionless time based on inner region area

td = Dimensionless time

T = eservoir temperature, R or F

W m = fracture spacing, ft

z = gas compressibility factor

Symbols

α = Shape factor of matrix blocks, ft-2

µ = viscosity of hydrocarbon fluid, cp

φ = porosity, fraction

λ = Dimensionless interporosity flow coefficient

ω = Dimensionless fracture storativity ratio

Ψ = Pseudo pressure, cp

psi2

= constant, 1.422*106

= diffusivity ratio, fraction

Subscrips

b = Bulk

D = dimensionless

f = Fractures

g = gas

i = Initial

m = Matrix blocks

xviii

sc = surface condition

t = total

w = Wellbore

wf = Wellbore flowing

ws = Wellbore shut down

x = x-direction

y = y-direction

z = z-direction

1 = Inner region

2 = Outer region

Superscripts

n = Known time level

n+1 = Unknown time level

k = Known iteration level

k+1 = Unknown iteration level

xix

ACKNOWLEDGEMENTS

I would like to express my deepest thanks to my thesis advisor, Dr. Turgay Ertekin for his

guidance and assistance throughout the development of this thesis without which this work would

not have been completed. Acknowledgements are also extended to Dr. Derek Elsworth and Dr.

John Yilin Wang for their interest in serving as committee members for evaluating this work.

I also would like to thank doctoral students Yogesh Bansal, Jithendra Srinivas and

Masters student Nithiwat Siripatrachai for his help at times of need and the department of Energy

and Mineral Engineering for providing scholarship during the course of my study at The

Pennsylvania State University.

I am really thankful to my father, Kirpal Singh, my mother, Vimala Rana and my

brothers Zatin Rana and Nitin Rana for being the endless support and encouragement. I would

also like to thank my friends for their emotional support at different times.

Chapter 1

INTRODUCTION

Energy demands of USA are going to increase significantly in next two decades. Major

portion of energy demands will come from oil and natural gas. The production of natural gas is

projected to exceed 30 trillion cubic feet per year in next two decades. The producers will rely on

unconventional gas reserves for the natural gas production. These conventional gas reservoirs

include tight gas sands, gas shales, coalbed methane and gas hydrates. In this study, gas shales are

primarily focused. In the past few years, there has been significant focus on the exploration and

production of shale gas reservoirs. Shale gas formations in North America hold trillions of cubic

feet of potentially recoverable natural gas.

Gas shales are usually mature petroleum source rocks where high heat and pressure have

converted the source rock material to natural gas. Gas shales are rich in organic content and

exhibit high gamma ray response. Gas shales are naturally fractured reservoirs with very low

rock permeability. The rock permeability of gas shales ranges from 10-2

to 10-6

md. Therefore,

economic production from these formations requires implementation of the latest technologies

which primarily includes horizontal well drilling with multistage hydraulic fracturing.

Hydraulic fracturing creates a zone around the wellbore which contains a large number of

interconnected fractures. In this study, we have referred to this hydraulically fractured zone as

“Crushed Zone”. The shape of this crushed zone is assumed to be of elliptical shape. Crushed

zone is considered to be surrounded by naturally fractured formation. Therefore, the resulting

system consists of two regions separated by a discontinuity of elliptical shape.

2

It is of critical interest to know the properties of the artificially fractured region in order

to implement the optimum reservoir exploitation approach. Therefore, we have presented forward

and backward solutions to perform pressure transient study on these systems.

In the forward solution, one inputs the reservoir properties and gets the wellbore pressure

response as the output. A generalized numerical model has been presented which provides

forward solution. The numerical model has been developed by extending the Warren and Root

model for dual porosity systems to composite dual porosity systems. The results of numerical

model have been validated against some analytical solutions and a commercial reservoir

simulator. The numerical model is shown to predict correct wellbore pressure response once the

reservoir properties are known.

Backward solution gives reservoir properties provided that one already has the wellbore

pressure data of the system. In this study, type curves have been presented to provide backward

solution for composite dual porosity systems. To the best of our knowledge, there are no type

curves published for such a system in the literature yet. This study provides a new approach to

construct and analyze type curves for these systems. Some case studies are included to prove the

validity of these type curves.

This study is divided into 5 chapters. Chapter 2 presents the literature survey carried out

for this study. Chapter 3 describes the numerical model and its validation. Chapter 4 presents the

sensitivity analysis of dimensionless variables. Chapter 5 presents the type curves for composite

dual porosity systems and its validation. Chapter 6 presents the conclusions and recommendations

for this study.

Chapter 2

LITERATURE SURVEY

2.1 Well Testing In Naturally Fractured Reservoirs

Most of the hydrocarbon reservoirs are naturally fractured reservoirs. These natural

fractures were formed due to the shear and tectonic forces in the formations. In general, these

systems are considered to be composed of two systems, matrix system and fracture system. The

matrix system is considered to contain very less permeability and high porosity. On the other

hand, fracture system is considered to contain higher permeability and lower porosity than matrix

systems. The hydrocarbons flow to wellbore through fractures only. Matrix blocks are considered

as an injection source to the fractures. The amount of injection from matrix to fractures depends

on the matrix geometry and pressure drop between the two systems.

2.1.1 Naturally Fractured Systems with Vertical Wells

These types of reservoirs have been studied by several investigators in the literature.

(WARREN and ROOT 1963), first presented an analytical model to simulate the pressure

response of a dual porosity system with a vertical well. They idealized the dual porosity systems

by assuming them as a set of regular cubes separated by an orthogonal set of parallel natural

fractures. The schematic of their model has been shown in Figure 2-1.

4

Warren and Root assumed the dual porosity systems to be consisting of primary and

secondary porosity. Matrix blocks contain the primary porosity and the secondary porosity is

represented by fractures. Usually, fracture porosity is very small as compared to matrix porosity

in naturally fractured reservoirs. Warren and Root showed that the dual porosity behavior can be

determined by calculating two dimensionless parameters. These parameters are named as

dimensionless fracture storativity, ω, and dimensionless interporosity flow coefficient, λ, and are

defined as:

mmff

ff

CC

C

(2.1)

2

w

f

m rk

k (2.2)

For dual porosity systems, when the wellbore pressure response is plotted on a semi-log

plot, two semi-log straight lines and a transition region is seen. When the well starts to produce

from a naturally fractured system, the flow takes place only in the fractures. This initial fracture

flow is represented by the first-semi log straight line. When a significant pressure drop occurs in

Figure 2-1: Schematic of Warren and Root approximation of naturally fractured

reservoirs (WARREN and ROOT 1963).

5

the fractures, matrix blocks start to inject hydrocarbons into the nearby fractures. The start of

hydrocarbons injection into fractures is manifested by start of transition region or deviation from

the first semi-log straight line. Start of third semi-log straight line represents that pressure of

matrix blocks and fractures are in equilibrium. Third semi log straight line continues until

boundary effects are felt.

Warren and Root showed that the time to start deviation from first semi-log straight line

depends on magnitude of λ. They also showed that the duration and path of transition region

depends on the magnitude of ω. As ω approaches unity or λ approaches to infinity, dual porosity

system reaches to homogenous or single porosity system.

Some other dual porosity models have also been presented by different investigators.

(Kazemi 1969), presented a dual porosity model in which he represented naturally fractured

system as repeating horizontal layers of matrix and fractures. Unlike Warren and Root model,

Kazemi assumed pressure gradients in the matrix blocks. The results of Warren and Root model

were found to be in very close agreement with Kazemi‟s solution.

For this study, Warren and Root model is used because it is implemented in many

commercial dual porosity simulators. Besides this, the results of Warren and Root model are in

close agreement with transient dual porosity models published later by Kazemi and De Swann.

The backward solution for dual porosity systems can be obtained by two methods. First is

by semi-log analysis as presented by Warren and Root. Second is the use of dual porosity type

curves presented later in the literature. (Onur, Satman et al. 1993), presented type curves for

naturally fractured systems. They presented derivative, integral type curves to determine dual

porosity characteristics.

6

2.1.2 Naturally Fractured Systems with Horizontal Wells

The dual porosity models published for horizontal wells are very similar to that with

vertical wells except some changes in the definition of dimensionless parameters. (de Carvalho

and Rosa 1988), described pressure transient behavior of horizontal wells in a naturally fractured

system. They used Warren and Root model to characterize the dual porosity behavior and used

definitions of dimensionless parameters as described below:

2Lk

k

f

m (2.3)

ffmm

ff

CC

C

(2.4)

In the definition of dimensionless interporosity flow coefficient, λ, horizontal half-well

length is used instead of wellbore radius. (Ng and AguiIera 1991), presented analytical solutions

for drawdown and build-up analysis of horizontal gas wells in anisotropic naturally fractured

reservoirs. They described seven flow regimes that may occur in these systems and also stated

that some of them may or may not be present depending on the reservoir and dual porosity

characteristics. (Du and Stewart 1992), presented a pressure transient study of horizontal wells in

dual porosity reservoirs. They included both pseudo-steady state and transient dual porosity

model in their study and presented type curves for both models. The definition of dual porosity

dimensionless parameters are similar to what (de Carvalho and Rosa 1988) has used in their

study. Therefore, in this study, we have used the same definitions for λ and ω as is defined in

equations (2.3) and (2.4).

7

2.2 Well Testing in Composite Systems

A composite system is a special class of single porosity systems. This system has a region

around wellbore which has different rock and fluid properties than rest of the system. The region

around the wellbore is called inner region or altered region. Rest of the system is called outer

region or unaltered region. These systems depict to many practical systems like homogenous

reservoirs undergoing steam injection, carbon dioxide injection or acidized wells. Therefore,

study of these systems is very important for reservoir engineering purposes.

2.2.1 Composite Systems with Vertical Wells

(Satman, Eggenschwiler et al. 1980), presented an analytical solution for composite

systems with a vertical well. They presented a mathematical solution for single porosity systems

undergoing steam injection due to which a discontinuity is created in the reservoir. The

discontinuity is considered to be of circular shape and the whole system is consists of concentric

circles representing different regions in the reservoir. The schematic of this system has been

shown in the Figure 2-2.

8

(Satman, Eggenschwiler et al. 1980), used Laplace transformations to solve partial

differential equations representing the flow in these systems and Stehfest algorithm to invert the

solution in real space. They plotted the wellbore pressure response on a semi-log plot and showed

that a composite system exhibits two semi-log straight lines. First semi-log straight line represents

the mobility of inner region which is followed by a transition region and eventually by a second

semi-log straight line which represents mobility of outer region. They defined two dimensionless

parameters for composite systems which are defined below.

Figure 2-2: Schematic of a composite reservoir with a vertical well (Satman, Eggenschwiler et

al. 1980).

9

Mobility Ratio:

2

2

1

1

k

k

M (2.5)

Diffusivity Ratio:

2

1

t

t

C

k

C

k

(2.6)

(Satman, Eggenschwiler et al. 1980), concluded that the permeability of inner region can

be calculated by first semi-log straight line. The pseudo steady state for the inner region occurs in

the transition period. During this pseudo steady state, a straight line is observed on a Cartesian

plot between pressure and time. Using the slope of this Cartesian straight line, volume of inner

swept region can be calculated if the porosity of inner region is known. Permeability of outer

region can be calculated from the slope of second semi-log straight line.

(Bixel and Poollen 1967), presented a type curve method to calculate inner region

permeability and radius to discontinuity. They presented type curves for different storage capacity

ratios, Fs and in each plot, different curves were presented for different mobility ratios, M. Their

definition of mobility ratio is inverse of what is described in equation (2.5) and expression for

storage capacity ratio is given below:

1

2|

C

CFs

(2.7)

(Bixel and Poollen 1967), showed that correct values of inner and outer region mobilities

can be predicted by the use of type curves. Besides this, they showed that correct values of inner

region radius can also be predicted using the presented type curves.

10

2.2.2 Composite Systems with Horizontal Wells

(Ambastha and Ghaffari 1998), presented a numerical model which simulates wellbore

pressure response of composite systems mimicking thermal recovery situations. The schematic

diagram of their model is shown in Figure2-3.

(Ambastha and Ghaffari 1998), presented the definitions of dimensionless groups

involved in the composite systems with horizontal wells. Most of the definitions are same as

conventional dimensionless terms. The most important dimensionless parameter suggested by

them was dimensionless time based on inner region area, tda which is expressed in equation (2.8).

111 .

..4637.2

baC

tket x

da

(2.8)

They plotted several dimensionless semi-log derivative plots showing the effects of

mobility ratio, storativity ratio and inner region shape and concluded that the pseudo steady state

method may be used to estimate inner region volume for steam injection through horizontal well.

Figure 2-3: Schematic of a horizontal well in a composite reservoir (Ambastha and

Ghaffari 1998).

11

The dimensionless parameters including mobility ratio, storativity ratio, inner region shape and

grid size do not affect significantly swept volume estimation. They showed that pseudo steady

state flow occurred in all the simulations. However early linear, pseudo radial and late linear flow

did not appear in any test because of specific swept volume dimensions and horizontal well

lengths assumed in their study.

2.3 Well Testing in Composite Dual Porosity Reservoirs

When a naturally fractured reservoir is hydraulically fractured around the wellbore, the

system can be approximated as a composite dual porosity system. A composite dual porosity

system consists of an inner or altered region around the wellbore which is surrounded by rest of

the system also known as outer region of unaltered region. This system is different from a

conventional composite system because both the inner and outer region are dual porosity systems.

This system matches very closely to hydraulically fractured shale gas reservoirs. It is because

shales are already naturally fractured systems and hydraulic fracturing creates a region around the

wellbore which contains a large number of interconnected fractures. In this study, we are

referring the inner hydraulically fractured region as “crushed zone”.

2.3.1 Composite Dual Porosity Systems with Vertical Wells

(Poon 1984), presented a mathematical model for composite reservoirs with uniform

fracture distribution. His system consists of a naturally fractured system undergoing steam or

carbon dioxide injection. He assumed circular discontinuity which separates inner and outer

region. He basically extended Warren and Root model for dual porosity systems to account for

composite behavior. He used Laplace transformation to solve partial differential equations

12

representing the flow in these systems and used Stehfest algorithm to numerically invert the

results into real space. He plotted wellbore pressure response on a semi-log graph and obtained

four semi-log straight lines. The slopes of first and second semi-log straight lines give mobilities

of fluid in the flooded region and outer region respectively. Third semi-log straight line represents

radial flow in the fractures of entire reservoir. Fourth semi-log straight line represents the flow in

entire reservoir i.e. matrix blocks of both inner and outer region have come in quasi steady state.

He concluded that the volume of flooded region can be calculated by obtaining a linear

relationship between pressure and time during transition region between first and second semi-log

straight lines. His model does not represent a composite dual porosity system of out interest

because his model cannot account for hydraulically fractured inner regions.

(Prado and Da Prat 1987), presented a mathematical model for homogenous systems with

hydraulically fractured vertical wells. Their model also does not represent a composite dual

porosity system of our interest because the outer region is considered to be single porosity. Their

model is useful in determining the extent and flow parameters of swept and unswept regions.

They concluded that no general type curves could be created for this system due to a large

number of dimensionless parameters are involved in this system.

Prado and Da Prat, 1987, presented an analytical solution in Laplace space for composite

dual porosity systems with vertical wells and with circular discontinuity between hydraulically

fractured and naturally fractured regions. The schematic of their system is presented in Figure 2-

4. (Kikani and Walkup Jr. 1991)

They based their analysis on dimensionless semi-log pressure derivative curves. They

found that 2

1 * DR is a correlating parameter for pressure derivative plots with dimensionless

time based on front radius. This is a very important finding because it significantly decreases the

number of independent dimensionless parameters for this system.

13

2.3.2 Composite Dual Porosity Systems with Horizontal Wells

According to our knowledge, there has not been much work done on analyzing the

behavior of composite naturally fractured systems with a horizontal well. There is only one paper

in the literature that describes a similar system. (Brohi, Pooladi-Darvish et al. 2011), considered a

multistage hydraulically fractured horizontal well in a homogenous system. Hence, their system is

not truly a representative of composite dual porosity system the outer region is still a homogenous

system. The schematic of his system is shown in Figure 2-5.

Figure 2-4: Schematic of composite naturally fractured system with a vertical well

(Kikani and Walkup Jr. 1991).

14

In the literature, investigators have assumed two approximate models to represent a

multistage hydraulically fractured horizontal well. First model is the one in which these hydraulic

fractures are assumed to be a set of planer set of fractures. Second model is the one in which a

complex set of interconnected fractures are assumed around the wellbore.

The first model is used by several investigators in the literature. (Brohi, Pooladi-Darvish

et al. 2011), used the first method to model the inner hydraulically fractured region. Besides

them, (Medeiros, Ozkan et al. 2007), and (Medeiros, Kurtoglu et al. 2007), also used the same

approximation to model the hydraulic fractures in their analytical model.

Figure 2-5: Schematic of discrete hydraulic fractures (Brohi, Pooladi-Darvish et al.

2011).

15

In this study, we are using the second model which assumes a region of complex network

of interconnected fractures. The reason of using this approach is that at a number of times, the

micro seismic data obtained during hydraulic fracturing reveals that the hydraulic fractures may

not grow as a set of discrete planes. Instead hydraulic fracturing operations may create a network

of interconnected fractures. In the Figures 2-6 and 2-7, some samples of micro seismic data are

shown which supports this argument.

Figure 2-6: Micro-Seismic data visualization during hydraulic fracturing of horizontal

wells.

16

From Figures 2-6 and 2-7, it is concluded that hydraulic fractures might not propagate as

straight discrete fractures but instead the fractures may propagate in any direction depending on

the stress distribution and reservoir lithology surrounding the wellbore. Hence, the second

approach is valid. In this study, we are referring the inner hydraulically fractured zone as

“crushed zone”.

In this study, the shape of curshed zone is assumed to be of elliptical shape. This

assumption is supported by some previous studies in the literature. (Obuto and Ertekin 1987),

presented a study of composite systems with horizontal wells and they assumed the inner region

to be of elliptical shape. (Behr, Mtchedlishvili et al. 2006), also used presented elliptical shape

1 Figure 2-7 is taken from website: http://content.edgar-

online.com/edgar_conv_img/2010/09/09/0001340282-10-000037_EXROSEUPDATEDPRES33.JPG.

Figure 2-7: Another sample of micro-seismic data visualization collected during

hydraulic fracturing of horizontal well1.

17

inner damaged region formed due to hydraulic fracturing. Therefore, the assumption of elliptical

crushed zone is also valid for our purposes.

18

Chapter 3

PROBLEM STATEMENT

Upon the completion of multistage hydraulic fracturing of horizontal gas wells in

naturally fractured systems, it is very important to know the properties of crushed zone in order to

predict the pressure response of these systems. This information is critical to design an optimum

production strategy. Therefore, the objectives of this study are:

1) To develop a numerical model which can predict the wellbore pressure response for a

composite dual porosity system once the properties of the reservoir are known. This

approach is called as forward solution.

2) To perform the sensitivity analysis of all the dimensionless parameters involved in a

composite dual porosity system.

3) To develop type curves for predicting crushed zone properties from the field data.

These properties include bk f ,,, 111 . This approach is called as backward solution.

4) To validate type curves by presenting different case studies.

Chapter 4

NUMERICAL MODEL PRESENTATION

This chapter presents a numerical model which has been validated to predict correct

values of wellbore pressure response of a composite dual porosity system. This chapter provides

forward solution to perform pressure transient analysis. The numerical model takes reservoir

properties as inputs. The numerical model accommodates both vertical and horizontal wells.

4.1 Model Description

A square shaped reservoir has been assumed with constant thickness for this study. The

reservoir is divided into small blocks of constant dimensions Δx and Δy. The well is placed in the

centre of the reservoir. The well is set to produce at a constant flow rate. The assumptions of this

model are described below.

Assumptions:

1) Flow is taking place only in horizontal plane i.e. only in the x-y plane.

2) Reservoir contains a single phase compressible gas.

3) Gravitational effects and compressibility of the formation are neglected.

4) Wellbore storage and skin effects are neglected.

5) The initial pressure of fracture and matrix blocks is the same.

20

4.2 Mathematical Development

In this section, mathematical material balance equations describing the flow in a small

block of dimensions Δx and Δy have been described. For an anisotropic reservoir, combining the

continuity equation with Darcy‟s law and neglecting the gravitational effects, the two

dimensional diffusivity equation is given by equation (4.1).

g

bDSCF

gg

yy

gg

xx

Bt

Vqy

y

p

B

kA

yx

x

p

B

kA

x

615.5/

(4.1)

where,

sc

scg

Tp

TzpB

**615.5

**

(4.2)

Multiplying equation (4.1) by sc

sc

T

Tp

615.5 and rearranging, we get

z

p

t

VQqy

y

p

z

pkA

yx

x

p

z

pkA

x

bDSCF

g

yy

g

xx 615.5

*/

(4.3)

where,

sc

sc

T

TpQ

615.5

(4.4)

4.2.1 Implementing Warren and Root model in Diffusivity Equation

Naturally fractured reservoirs are made up of matrix blocks and fractures. Therefore, the

diffusivity equation (4.3) is written for both matrix blocks and fractures. Matrix blocks serve as

injection source for fractures. Therefore, the flow rate that appears in the diffusivity equation

written for a matrix block is actually the rate at which the hydrocarbons are injected into

fractures. The diffusivity equation for matrix blocks is shown in equation (4.5)

21

For Matrix:

m

mm

bm

m

m

g

yym

m

m

g

xx

z

p

t

VQqy

y

p

z

pkA

yx

x

p

z

pkA

x

615.5.*

(4.5)

Since, Warren and Root neglected any pressure transients in the matrix blocks, hence

0

y

p

x

p mm

(4.6)

Therefore, from the matrix flow equation we get

m

mm

b

z

p

t

VQq

615.5.*

(4.7)

For Fractures:

Since, matrix blocks are constantly bringing in fluids to fractures, the diffusivity equation

of fractures contain this injection term as a source term. This is apparent in the equation (4.8).

f

f

f

b

DSCF

f

f

f

g

yyf

f

f

g

xx

z

p

t

V

QqQqyy

p

z

pKA

yx

x

p

z

pKA

x

615.5

.. /

*

(4.8)

Substituting for q*.Q from equation (4.7) into equation (4.8) and rearranging, we get

m

m

m

b

f

f

f

b

DSCF

f

f

f

g

yyf

f

f

g

xx

z

p

t

V

z

p

t

V

Qqyy

p

z

pKA

yx

x

p

z

pKA

x

615.5615.5

./

(4.9)

Equation (4.9) is solved in every reservoir block to predict the pressure response of the

dual porosity system. Equation (4.10) relates the interporosity flow rate with pressure drop

between matrix and fractures using Darcy‟s law:

22

gm

fm

g

mb

B

PPkVq

)(*

(4.10)

where,

2)(

60

mW

(4.11)

The equations (4.9) and (4.10) are solved simultaneously in every block of reservoir to

obtain pressure change due to fluid flow in each block. The initial pressure of matrix blocks and

fractures is assumed to be equal. The initial condition is given by equation (4.12).

imf pyxpyxp )0,,()0,,( (4.12)

Since no flow boundaries are assumed in both x and y directions, the pressure derivatives

go to zero. The boundary condition is given by equation (4.13).

0

y

p

x

p ff

(4.13)

4.3 Numerical Approximation to partial differential equations

In this section, numerical approximation to the partial differential equation (4.9) is

described. Finite difference approximation to this equation is given below:

(4.14)

n

jim

n

jim

n

jim

n

jim

ji

mb

n

jif

n

jif

n

jif

n

jif

ji

fb

n

jiDSCF

n

jif

n

jif

n

jifgf

fyfyn

jif

n

jif

n

jifgf

fyfy

n

jif

n

jif

n

jifgf

fxfxn

jif

n

jif

n

jifgf

fxfx

z

p

z

p

t

V

z

p

z

p

t

V

qQppzy

pkApp

zy

pkA

ppzx

pkApp

zx

pkA

,

,

1

,

1

,

,,

,

1

,

1

,

,

1

,/

1

,2

1

1

,

1

,2

1

1

,

1

,2

1

1

,2

1

1

2

1,

1

,

1

2

1,

1

,

1

1,

1

2

1,

.615.5.615.5

.).(.

.).(

.

.

).(.

.).(

.

.

23

Equation (4.14) can also be written as:

0...615.5

...615.5

...615.5

...615.5

.

).(.

).(.

).(.

).(.

,,

1

,

1

,

,

,

1

,

1

,

1

,/

1

,1

1

,

1

,1

1

,

21

,1

21

,

,2

11

,1

1

,

1

,1

1

,

21

,

21

,1

,2

1

1

1,

1

,

1

1,

1

,

21

1,

21

,

2

1,

1

1,

1

,

1

1,

1

,

21

,

21

1,

2

1,

n

jif

n

jim

mbn

jim

n

jim

mb

n

jif

n

jif

fbn

jif

n

jif

fbn

jiDSCF

n

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jiyn

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jiy

n

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jixn

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jix

pzt

Vp

zt

V

pzt

Vp

zt

VqQ

zz

ppT

zz

ppT

zz

ppT

zz

ppT

(4.15)

In the equation (4.15),

jijixfjixjijixfjix

jixfjixfjixjix

jixxkAxkA

kkAAT

,1,1,1,,,

1,,1,,

1,....

..2

(4.16)

1, jixT is called the transmissibility term obtained by using harmonic average between

blocks (i,j) and (i,j+1). Similarly, other transmissibility terms appearing in equation (4.15) are

calculated.

The equation (4.15) is a non-linear equation in two variables: pf and pm. It is because

viscosity (µgf) and gas compressibility factor (zf and zm) are also function of pressures. The

variable pf is solved implicitly and is always kept at current time and iteration level. The

variable, pf is updated at each iteration level using the equation (4.17).

dtn

n

n

f

n

m

n

f

n

m epppp.

111 1

2

).(

(4.17)

24

The derivation for the equation (4.17) is given in the appendix A. The value of pm is

substituted into equation (4.15) to couple the equations (4.9) and (4.10).

Modified Newton Raphson method is used to solve this equation over the entire reservoir.

Description of Newton Raphson method is shown in appendix A. The wellbore pressure is

determined using the Peaceman‟s Equation as shown below:

sr

r

pp

B

hkkq

w

e

n

jiwf

n

jif

n

jig

n

jig

jizfjiyf

DSCF ji

ln

.)(2

1

,

1

,

1

,

1

,

5.0

,,

/ ,

(4.18)

Since, we have neglected skin factor. Therefore, s=0 and rearranging, the value of

sandface pressure is calculated using the equation (4.19) as shown below.

w

e

jizfjiyf

n

jig

n

jig

DSCF

n

jif

n

jiwfr

r

hkk

Bqpp

jiln.

)(2.

5.0

,,

1

,

1

,

/

1

,

1

, ,

(4.19)

Hence, the wellbore pressure response is obtained for a composite dual porosity system.

This is the forward solution for this system because this numerical model takes the reservoir

properties as inputs and predicts the wellbore pressure response of the system.

4.4 Model Validation

In this section, the presented numerical model is compared against analytical and

numerical solutions. The comparision is made against analytical solutions of Warren and Root for

dual porosity systems with a vertical well and (Satman, Eggenschwiler et al. 1980), solution for

composite systems with a vertical well. For the case of composite dual porosity system with a

horizontal well, the numerical model is tested with a commercial reservoir simulator.

25

4.4.1 Validation for Dual Porosity Reservoirs with a Vertical Well

In this section, the numerical model is tested with the widely used Warren and Root

ananlytical solution. The schematic of dual porosity system with vertical well is shown in Figure

4-1.

The reservoir is considered to be a square and the block dimensions are equal i.e.

Δx=Δy. First, a block size sensitivity analysis is performed to reduce the numerical error. The

result is shown below in Figure 4-2. The dataset for this run is shown in the Table B-1.

Figure 4-1: Schematic of a dual porosity reservoir with a vertical well in the center.

26

From Figure 4-2, it is concluded that there is not much improvement in the pressure

response if Δx<110 ft. Therefore, block dimensions are kept equal to 110 ft for future runs in

order to minimize the numerical error. The comparison of results of the numerical model and

Warren and Root model for the same dataset is given Figure 4-3.

Figure 4-2: Sensitivity analysis of block dimensions.

27

From Figure 4-3, it was found that the maximum error percentage was 2.11%. Hence, it

is concluded that the presented numerical model is working fine for dual porosity systems.

4.4.2 Validation for Composite Reservoirs with Vertical wells

In this section, the presented numerical model is tested with the analytical solution of

(Satman, Eggenschwiler et al. 1980). for composite systems with a vertical well in center and a

circular discontinuity on the either side of which rock and fluid properties are different. The

schematic of composite system is presented in Figure 4-4.

Figure 4-3: Comparison of presented numerical solution against Warren and Root

analytical solution for dual porosity reservoirs with a vertical well.

28

From Figure 4-2, it was concluded that block dimensions has to be less than or equal to

110 ft to reduce numerical error. Therefore, in this section the same block dimension (110 ft) is

used but with local grid refinement around the wellbore. Grid refinement is done to make the

discontinuity of circular shape. In Figure 4-5, the sensitivity analysis of grid refinement is shown.

Figure 4-4: Schematic of a composite reservoir with a vertical well in the center with a

circular discontinuity.

29

A comparison of the presented numerical model with analytical solution of (Satman,

Eggenschwiler et al. 1980) was made and the result is shown in the Figure 4-6.

Figure 4-5: Sensitivity analysis for local grid refinement around wellbore.

30

From Figure 4-6, the maximum percentage deviation between two models is 0.2%.

Therefore, it is concluded that the presented numerical solution is predicting correct results for

composite systems with vertical wells and circular discontinuity.

4.4.3 Validation for Composite Naturally Fractured Reservoirs with horizontal wells

In this section, the presented numerical model is tested with a commercial numerical

model to validate for composite dual porosity systems with a horizontal well. The discontinuity is

assumed to be of elliptical shape. The schematic of this system is shown in Figure 4-7.

Figure 4-6: Comparison of presented numerical model with (Satman, Eggenschwiler et

al. 1980)analytical model for composite reservoir with a vertical well.

31

Local grid refinement has been implemented to obtain the inner elliptical region. In this

study, the inner region is called crushed zone. The equation of ellipse has been used to select the

blocks which lie in the inner region. Block size dimension is kept equal to 110 ft and blocks near

the wellbore are refined to obtain inner elliptical region. A sensitivity analysis of the grid

refinement is conducted and the results are shown in Figure 4-8.

Figure 4-7: Schematic of a composite dual porosity reservoir with a horizontal well and

an elliptical discontinuity.

32

From Figure 4-8, it is concluded that there is not much improvement in the pressure

response in increasing the grid refinement. Since, it takes longer time to run for extremely refined

grids, inner grids are refined by 2 for this study.

A comparison of the presented numerical model is made with a commercial model for the

data presented in Table B-3 in the appendix B. Result of this comparison is presented on a semi-

log plot in Figure 4-9 and on Cartesian plot in Figure 4-10.

Figure 4-8: Sensitivity analysis of grid refinement around wellbore for a composite dual

porosity system with a horizontal well.

33

Figure 4-9: Semi-log plot to compare presented numerical solution to the commercial

numerical solution.

Figure 4-10: Cartesian plot to compare presented numerical solution to the commercial

numerical solution.

34

From Figures 4-9 and 4-10, it is clear that the maximum percentage error is 0.26 %.

Hence, it is concluded that the presented numerical model is predicting correct wellbore pressure

response for composite dual porosity systems with a horizontal well. The presented numerical

model provides forward solution in well testing which means that it requires reservoir properties

as inputs and generates the wellbore pressure response as output. This numerical model can also

be used to history match the field data and in this way it can be used backward solution.

However, in this study the presented numerical model is used just as a tool to provide forward

solution.

Chapter 5

SENSTIVITY ANALYSIS OF DIMENSIONLESS VARIABLES

A composite dual porosity system can be expressed using a number of dimensionless

variables. Each dimensionless variable has a specific effect on the pressure response of a

composite dual porosity system. In this chapter, all the dimensionless parameters are defined and

their characteristic effect is discussed. Dimensionless semi-log derivative plots are presented for

different cases. Based on the sensitivity analysis of these dimensionless parameters, some

correlations are found to reduce ease the design and analysis of these systems.

5.1 Dimensionless Parameters

Since, a gas reservoir is considered is this study, therefore, in order to define the

dimensionless parameters, variance of compressibility and viscosity has to be linearized using Ψ-

approach as shown is equation (5.1)

.

P

dpz

p

0

*2

(5.1)

In equation (5.1), Ψ is called pseudo pressure. Dimensionless pseudo-pressure drop is

defined in equation (5.2).

sc

wfif

DTq

hk

).(.

(5.2)

36

Dimensionless time in field units is:

2

4

*)(

**10*637.2

LCC

tkt

mmmfff

f

D

(5.3)

In equation (5.3), gas viscosity and compressibility are kept at initial conditions. Since,

initial pressure in matrix blocks and fractures is same, dimensionless time can be defined as:

2

4

..).(

**10*637.2

LC

tkt

iimf

f

D

(5.4)

In order to plot derivative plots for a composite dual porosity system, dimensionless time

based on the extent of the inner region is:

baC

tkt

iimf

f

DA...).(

**10*637.2 4

(5.5)

Semi-log dimensionless pseudo-pressure derivative is defined as:

tTq

hk

t

wfi

sc

f

D

D

ln.

..

ln

(5.6)

5.1.1 Inner Region Dimensionless Parameters

The interporosity flow coefficient is defined as:

)( 2

1

111 L

k

k

f

m

(5.7)

The dimensioness fracture storativity ratio:

1111

11

1

mmff

ff

CC

C

(5.8)

37

Here, gas compressibility is kept at initial condition. Since initial pressure in matrix

blocks and fractures is same, hence for simplicity the dimensionless fracture storativity ratio is

defined as:

11

1

1

mf

f

(5.9)

5.1.2 Outer Region Dimensionless Parameters

Dimensionless interporosity flow coefficient is defined as:

)( 2

2

222 L

k

k

f

m

(5.10)

Dimensionless fracture storativity ratio is defined as:

22

2

2

mf

f

(5.11)

5.1.3 Other Dimensionless Parameters

Mobility Ratio:

2

2

1

1

f

f

k

k

M

(5.12)

For simplicity, the viscosity of gas in inner and outer region is assumed to be equal.

Hence, the definition of mobility ratio that we are using in this study is:

38

2

1

f

f

k

kM

(5.13)

Storativity ratio of the fracture networks is defined as:

2

1

f

fF

(5.14)

5.2 Characteristic Dimensionless Derivative Plot

For this study, we have presented derivative plots because it is easier to identify and

interpret dual porosity signatures on derivative plots than conventional pressure drop plots. In this

section, a characteristic dimensionless semi-log derivative plot is presented. This plot describes

the wellbore pressure response of a typical composite dual porosity system with a horizontal well.

The characteristic plot is shown in Figure 5-1.

39

Based on Figure 5-1, the following flow regimes have been described below:

1) First log-log straight line from point M to N represents that the flow is occurring only

in the fractures of inner region and system is in infinite acting flow regime.

2) The first transition period from point N to O represents that the matrix blocks of

crushed zone have started injecting hydrocarbons into the nearby fractures. The path

of this transition region depends on the size and shape of matrix blocks.

3) The second log-log straight line from point O to P represents that the pressure in

matrix and fracture blocks has reached in equilibrium.

4) Second transition as shown from point P to Q starts when the matrix blocks of outer

region starts to inject into the nearby natural fractures.

Figure 5-1: Characteristic dimensionless semi-log derivative plot for a composite dual

porosity system with a horizontal well.

40

5) Third log-log straight line as shown after point Q represents that the pressure in the

outer region matrix and fracture blocks have come in equilibrium. This straight line

will continue until the boundary effects are felt.

5.3 Effect of Constant Dimensionless Parameters

In this section, it is checked if the defined dimensionless parameters are correct or not. In

other words, if the different combinations of reservoir properties are used keeping the values of

dimensionless parameters constant and the dimensionless curves lie on top of each other, then

that means the defined dimensionless parameters are correct. The datasets used for this run is

shown in Table 5-1.

Table 5-1: Data used to check applicability of dimensionless parameters for a composite

dual porosity system.

Property Dataset 1 Dataset 2 Dataset 3

kf1 (md) 20 10 5

km1 (md) 5.93e-6 2.96e-6 1.48e-6

Φf1 5.0e-3 3.0e-3 2.0e-3

h (ft) 250 100 50

T (F) 130 200 250

2*L 1500 1500 1500

2*b 900 900 900

Area (acres) 300 300 300

41

Figure 5-2 shows the wellbore pressure response of these datasets on a dimensionless

semi-log derivative plot.

From Figure 6-1, it is concluded that the defined dimensionless parameters are correct

because the dimensionless derivative curves lie on top of each other for different values of

reservoir properties.

5.4 Effect of Changing Storativity Ratio of Fracture Networks (F)

In this section, the effect of changing storativity ratio of fracture networks of inner and

outer region is illustrated. Storativity ratio of fracture network is defined by equation (5.14).

Figure 5-3 describes the effect of F on dimensionless semi-log derivative plot.

Figure 5-2: Effect of different combinations of reservoir properties while keeping the

dimensionless parameters constant.

42

From Figure 5-3, it is concluded that the storativity ratio of fracture network, F does not

affect dual porosity signature of inner region as it goes higher than 100. For F>100,

dimensionless semi log derivative plots lie on top of each other. Therefore, plotting dimensionless

type curves for 1<F<100 is sufficient to characterize composite dual porosity systems.

5.5 Effect of Changing Mobility Ratio (M)

Mobility ratio is defined by equation (5.13) as ratio of inner region fracture permeability

to outer region fracture permeability. Figure 5-4 shows the effect of changing magnitudes of

mobility ratio on dimensionless semi-log derivative plot.

Figure 5-3: Effect of Storativity Ratio of Fracture Networks (F) on the dimensionless

semi-log derivative plot.

43

From Figure 5-3, it is concluded that mobility ratio does not affect semi-log derivative

plot if M is greater than unity. Therefore for our purposes, mobility ratio is not included as a

parameter since crushed zone fracture permeability will always be greater than outer region

fracture permeability which means M is always greater than unity.

5.6 Effect of Other Reservoir Parameters

In this section, the effect of all the other parameters is shown on dimensionless semi-log

derivative plots. The effect of inner region dimensionless fracture storativity ratio, ω1 is shown in

Figure 5-5.

Figure 5-4: Effect of different mobility ratios on the dimensionless semi-log derivative

plot.

44

From Figure 5-5, it is concluded that shape and duration of first transition region or

signature of the inner dual porosity region depends on inner region dimensionless fracture

storativity ratio, ω1. As ω1 approaches unity, the inner region becomes a single porosity system.

The lower the value of ω1, the larger is the transition region between the first and second log-log

straight lines. The effect of outer region dimensionless fracture storativity ratio, ω2 is shown in

Figure 5-6.

Figure 5-5: Effect of ω1 on the first transition period on dimensionless semi-log

derivative plot.

45

From Figure 5-6, it is clear that inner region dual porosity signature or deviation from

first log-log straight line is independent of outer region dimensionless fracture storativity ratio,

ω2.

Figure 5-6: Effect of different ω2 on the transition period on the dimensionless

derivative plots.

46

From Figure 5-7, it is concluded that extent of the drainage area affects the inner region

dual porosity signature. This is because when the reservoir area is changed, the well length and

minor axis length also has to be changed accordingly in order to keep the dimensionless plots on

top of each other.

The effect of inner region dimensionless interporosity flow coefficient, λ1 on derivative

plot is shown in Figure 5-8.

Figure 5-7: Effect of changing drainage area while keeping the well length and inner

region area constant.

47

From Figure 5-8, it is concluded that time for the occurrence of first transition zone

depends on the magnitude of inner region interporosity flow coefficient, λ1. As the magnitude of

λ1 increases, time to start the first transition zone decreases and vice versa.

Figure 5-8: Effect of λ1 on the first transition period.

48

From Figure 5-9, it is concluded that the extent of inner region or the extent of minor axis

also affects the time for the appearance of the first transition region. The time for first transition

period seems to be decreasing with the increasing values of the minor axis. Therefore, from

Figures 5-8 and 5-9, it is evident that the time at which the inner matrix blocks starts to inject in

the inner region fractures depends on two parameters which are inner region interporosity flow

coefficient, λ1 and minor axis length, b.

Figure 5-9: Effect of changing minor axis (b) on the first transition period.

49

From Figure 5-10, it is clear that ratio of inner region interporosity coefficient to outer

region interporosity coefficient affects the time at which first transition region begins. The

following conclusions can be drawn:

1) If 12

1

, the time at which first transition region begins is dependent on magnitudes

of both λ1 and λ2.

2) If 12

1

, the time at which first transition region begins does not depend on the

magnitude of λ2.

It is very difficult to make any generalized type curve for this system if 12

1

because

the time at which first transition region starts depends on magnitudes of both λ1 and λ2.

Figure 5-10: Effect of 2

1

on the first transition period.

50

Therefore, in this study, we have assumed that the ratio 2

1

is always greater than unity

for these systems.

5.6.1 Conclusions drawn from sensitivity analysis of dimensionless variables:

1) The time at which first transition region begins depends on the magnitude of inner

region‟s dimensionless interporosity flow coefficient, λ1 and extent of minor axis, b.

2) Mobility ratio does not affect inner region dual porosity signature on derivative plot.

3) Storativity ratio of fracture networks affects the shape of first transition region if it

varies between 1 and 100. If F>100, the dimensionless derivative type curve coincide

with the curve plotted for F=100.

4) For a constant drainage area, well length and minor axis length, the shape and

duration of first transition region depends on the magnitude of ω1 and is independent

of magnitude of ω2.

5) Drainage area affects the shape first transition region. For changing drainage area, the

well length and minor axis length should be changed accordingly in order to keep the

shape of first transition region same.

6) The assumption 12

1

enables us to exclude λ2 from the number of dimensionless

parameters which affect the shape and time for first transition region.

51

5.7 Scaling Correlations to Reduce the Number of Parameters

In this section, some dimensionless correlations are described to reduce the number of

dimensionless parameters in order to plot type curves. Finding correlations is important because a

composite dual porosity system has too many dimensionless parameters to plot any generalized

type curves. Use of correlations reduce the number of type curves significantly and make the use

of type curves more feasible.

From conclusions in section 5.6.1, it is clear that the time at which first transition region

starts depends on λ1 and b. Therefore, by using different combinations of these two parameters it

was found that if the product

1.1

1.

wr

b is kept constant, and y-axis is scaled by

3.0

wr

b, then the

first transition region will coincide for different values of λ1 and the minor axis length, b. The

validity of this dimensionless correlation is based on the assumption that horizontal wellbore

radius, rw remains constant for all the cases. The applicability of this correlation is illustrated in

Figure 5-11.

52

For the result shown in Figure 5-11, the well length and drainage area are kept constant.

In practice, the drainage area and well length will very for different reservoirs. Therefore, it is

necessary to find out some scaling factors and also some other dimensionless correlations to take

into account the changes in drainage area and well length. A schematic of a typical composite

dual porosity reservoir is shown in Figure 5-12.

Figure 5-11: The effect of scaling the y-axis by

3.0

wr

band keeping the value of

dimensionless correlation

1.1

1.

wr

b constant.

53

In order to account for changing drainage area, the well length and minor axis length has

to change accordingly. To scale the composite dual porosity system, the following dimensionless

parameters are defined:

consty

b

2 (5.15)

constx

L

2 (5.16)

Between these two dimensionless variables, x

L2is used to scale x-y axes and

y

b2 is kept

constant for a given plot. By doing this, dimensionless semi-log derivative response will be

independent of changing reservoir area and well lengths. This is illustrated in Figure 5-13.

Figure 5-12: Schematic of a composite dual porosity reservoir with a horizontal well.

54

For the results in the Figure 5-13, the storativity ratio of fracture networks, F and the

parameter y

b2is kept constant. From this Figure, it is concluded that the dual porosity signature

of inner region coincide for different combinations of drainage area, well lengths, minor axis

length.

5.8 Sensitivity test for correlation

1.1

1.

wr

b

From Figure 5-11, it is clear that the correlation

1.1

1.

wr

b will be used to plot type

curves. Here, the range of values for this parameter is presented that will be used in the analysis.

Figure 5-13: Effect of different combinations of Area,x

L2, minor axis (b), 1 ,

2

1

on

the dimensionless semi log derivative plot.

55

From Table 5-2, the maximum and minimum values of the inner region interporosity

flow coefficient are obtained. The range of values for correlating parameter

1.1

1.

wr

b are

calculated in Table 5-3 which is shown below:

It is important to observe the effects of these range of values on a dimensionless semi-log

derivative plot with the scaled x-y axes as described in section 5.7. This is shown in Figure 5-14.

Table 5-2: Calculation of the maximum and minimum values of λ1.

Fracture

Spacing

km1 kf1 L 1 1

(approx.)

Maximum 10 1e-3 20 500 3.3e+6 1e+6

Minimum 0.1 1e-7 4 500 7.5e-4 1e-3

Table 6-3: Table to calculate maximum and minimum limit of correation

1.1

1.

wr

b .

λ1 b rw 1.1

1.

wr

b

(Approx)

Maximum 1e+6 1000 0.25 1e+10

Minimum 1e-3 150 0.25 1e+0

56

Normally, pressure transient tests are run for elongated periods of time for tight gas or

shale gas reservoirs. Therefore, the numerical simulator is run for 150 days to generate pressure

transient data. From Figure 5-14, it is clear that if5

1.1

1 10*1.

wr

b , then the dual porosity

signature occurs so early that it cannot be captured by pressure measuring devices. For example,

in case of 5

1.1

1 10*1.

wr

b , the inner region dual porosity signature develops before 1 minute

of the start of test. Therefore, if one goes above this limit, the dual porosity signature cannot be

seen. Therefore, we decided to limit our study between the limits: 5

1.1

1 10*1.1

wr

b .

Figure 5-14: Effect of range of correlating parameter

1.1

1.

wr

b .

Chapter 6

TYPE CURVES AND TYPE CURVE APPLICATIONS

Type curves are powerful tools for solving reservoir engineering inverse problems. Type

curves have been in use for a long time in well test analysis. Type curves are system specific and

exhibit different behaviors for different systems. They are easy to use and very much reliable

particularly when the pressure transient test is not run long enough to interpret the data. In this

chapter, type curves are presented for composite dual porosity systems with a horizontal gas well.

Some case studies are shown later in this chapter to illustrate calculation of inner region or

crushed zone properties using these type curves.

6.1 Assumptions

1) Reservoir contains only a single phase compressible gas.

2) Inner region or crushed zone is of elliptical shape with major axis, a and minor axis, b.

3) Horizontal half well length, L = Major axis length, a

4) Inner region interporosity flow coefficient, λ1 is always greater than or equal to outer

region interporosity flow coefficient, λ2. In other words, 12

1

. The basis of this

assumption is also explained in the Figure 5-10.

5) Outer region properties are already known from pressure transient testing before

hydraulic fracturing. These properties include ω2 and λ2.

6) The reservoir is a square and the reservoir drainage area is known. In other words, the

length of reservoir is known.

58

7) The length of horizontal well is known.

8) The storativity ratio of fracture networks, F is known.

6.2 Properties Calculated using Type Curves

From a pressure transient data, the following properties can be calculated using the

presented type curves:

1) Inner region fracture permeability, kf1

2) Minor axis length of inner elliptical region, b

3) Inner region dimensionless interporosity flow coefficient, λ1

4) Inner region dimensionless fracture storativity ratio, ω1

6.3 Structure of generation of type curves

For a composite dual porosity system, a total of 45 type curves are presented in this

study. One may get confused by these many type curves and it may be difficult to find out the one

curve which gives correct results for a particular field. Therefore, it is important to understand the

structure in which these type curves are plotted. This structure can be comprehended easily with

the help of Figure 6-1.

59

From Figure 6-1, it is clear that every plot has six type curves. Each curve represents a

unique value of correlation

1.1

1.

wr

b . Each plot is made for a constant value of ω1. Furthermore,

each plot is made for a constant value of dimensionless parameter y

b2and storativity ratio of

fracture networks, F. A sample type curve plot is shown in Figure 6-2. It can be easily seen that

Figure 6-1: Structure from which the type curves have been generated.

60

this plot is made for a constant value of F=10, 2

1 10*1,3.02 y

b. Each curve on this plot

represents a unique value of

1.1

1.

wr

b .

6.4 Type curve analysis procedure

Properties that should be already known before using the type curves:

1) Storativity Ratio of inner region fractures to outer region fractures, F

2) Horizontal wellbore length, 2L (ft)

3) Reservoir Thickness, h (ft)

4) Total porosity of inner region, i.e (φf+ φm)

5) Flow rate at which the system is being produced, qsc (MMSCF/D)

6) eservoir Temperature, T ( R)

Figure 6-2: Sample type curve plot for F = 10, ω1 = 1e-2.

61

7) Wellbore radius, rw. For this study we have assumed the wellbore radius to be

constant, rw = 0.25 ft

8) Initial pressure of reservoir, Pi (psi)

9) Gas properties including molecular weight, MW, critical temperature, Tc and critical

pressure, Pc

When the field data is matched on the type curve, the match points are recorded. One

sample type curve matching is shown in Figure 6-3.

In Figure 6-3, it is demonstrated that once the field data has been matched on a type

curve, any point on the field data curve may be chosen. In Figure 6-3, that point is represented by

the “match point”. The x and y coordinates of that match point is read on the field data plot and

the type curve plot. Once, these match point coordinates have been recorded, the following

calculations are done.

Figure 6-3: Sample type curve matching.

62

6.4.1 Calculations done on the y-axis:

t

D

D

w

w ytx

L

r

b

L

r

ln.

2..

7.03.03.1

(6.1)

In equation (6.1), rw, L and x is known. Therefore, we can group them in one constant,

P1:

7.03.03.1

1

2.

1.

x

L

rL

rP

w

w

(6.2)

Hence the equation (6.1) becomes:

t

D

D yt

bP

ln.. 3.0

1

(6.3)

t

wf

sc

fy

tqT

hkbP

ln.

..

....

13.0

1

(6.4)

Here, one knows the value of t

wf

ln

from the match point in Figure 6-3.

f

wfy

t

ln

(6.5)

tf

sc

fyy

qT

hkbP .

..

....

13.0

1

(6.6)

1

1

3.0 1.

....

Ph

qT

y

ykb sc

f

tf

(6.7)

63

6.4.2 Calculations done on the x-axis:

tDA

w xtx

L

L

r

.

2.

2.0

(6.8)

In equation (6.8), rw, L and x is known. Therefore, we can group them in one constant,

P2:

2.0

2

2.

x

L

L

rP w

(6.9)

Therefore, equation (6.8) can be written as:

tDA xtP .2 (6.10)

t

iimf

fx

baC

tkP

....

..10*637.2.

11

1

4

2

(6.11)

In equation (6.11), t is the x-coordinate of the match point on field data plot. In Figure 6-

3, fxt

2

4

111

.10*637.2

....

P

aC

x

x

b

k iimf

f

tf

(6.12)

By dividing equations (6.7) by (6.12), one can solve for b as follows:

3.1

1

11

2

4

1

....

.10*637.2.

1.

...

f

t

iimf

sc

f

t

x

x

aC

P

Ph

qT

y

yb

(6.13)

Now, since one knows the value of b, one can calculate the inner region fracture

permeability, kf1 by either equation (6.7) or (6.12). Here, we are using the equation (6.11) to

calculate kf1 as follows:

2

4

11

1.10*637.2

.....

P

aC

x

xbk

iimf

f

tf

(6.14)

64

Till now, one has calculated inner region minor axis length, b and inner region fracture

permeability, kf1. The properties that still need to be calculated are inner region interporosity flow

coefficient, λ1 and inner region dimensionless fracture storativity ratio, ω1.

Since each curve is plotted for a fixed value of correlation

1.1

1.

wr

b and one has already

calculated the value of minor axis length, b from equation (6.13), the value of λ1 can be calculated

as follows:

1.1

1 .

wr

bconst

(6.15)

Since each plot is made for a constant value of inner dimensionless storativity ratio, ω1.

Therefore, one knows the value of ω1.

6.4.3 Verification of right type curve

A field data can match more than one type curve. Each type curve gives different value

for b and kf1. So, how to decide which type is the representative of that field data. For this, we

have presented an approach below in this section. Following this approach, one can determine

which type curve should be used as the representative type curve for that system.

Approach:

Since, the type curves are plotted for a constant value ofy

b2.and one already knows the

value of reservoir width, y, whenever the field data is matched on a type curve and the value of

minor axis length, b is calculated using equation (6.13), it is compared with the value of b got

from the constant y

b2 for which the type curve is plotted.

65

cy

b

2

(6.16)

Just to differentiate, we are calling the „b‟ in the above expression as „b_corr’.

cy

corrb

_2

(6.17)

2._y

ccorrb (6.18)

In equation (6.18), we know the value of „c‟ from the type curve on which the field data

is matched. It is because each type curve plot is made for a constant value ofy

b2. For the type

curve which gives, bcorrb _ is the one which gives the correct properties of system. All the

values calculated from other type curves have to be discarded because those are not correct.

Sometimes, more than one type curves may give bcorrb _ . In those cases, the ones

which give impractical values for inner region permeabilities, kf1 have to be discarded because

those ones are not the representative for that field data.

Therefore, the type curve matching and analysis is a three-step process:

1) Check for the results which give corrbb _ and select them.

2) Among these selected results, check for the ones which give unreasonable values of

kf1 and discard them. These unreasonable values may be in the order of 103 to 10

5.

3) After this, one will be left with only one result which satisfies both of the checks.

Now, one can calculate rest of the inner region parameters from that unique type

curve.

66

6.5 Versatility of Type Curves

The type curves presented in this thesis can be used to analyze the pressure drawdown

and build up test data.

6.5.1 Pressure Drawdown Testing

Pressure drawdown testing starts when the well starts to produce. The field pressure

drawdown data is collected and is converted to the pseudo pressure semi log derivative, i.e.

t

wf

ln

and is plotted against time. Refer to Figure 6-4 to see the definition of wf . This field

data is then matched on the type curves and exactly same procedure is followed as described in

Section 6.4.

67

6.5.2 Pressure Build-Up Testing

A pressure build-up test starts as soon as the production well is shut down. The procedure

for type curve matching is exactly the same as is followed in the pressure drawdown testing. For a

pressure build up test, the superposition principle is used to calculate the pseudo pressure drop.

The field derivative plot is plotted between )ln( t

ws

vs. t for the build-up test. Refer to

Figure 6-4 for the definition of ΔΨws. In the literature, equivalent time is used for x-axis for

pressure build up tests which is defined as:

Figure 6-4: Sample plot of pressure drawdown and build-up.

68

p

e

t

t

tt

1

(6.19)

If, ptt *1.0 , then approximate the equivalent time can be defined as:

tte (6.20)

where,

pttt (6.21)

6.6 Case Studies (Examples)

In this section, some examples or case studies are shown to validate the presented type

curves and also to illustrate the approach to use type curves. Pressure transient data is generated

using the presented numerical model for validating type curves.

6.6.1 Case Study 1

The same reservoir input that was used in Table B-3 has been used to validate the type

curves in this section. . The aim is to overlay the pressure transient data on the type curves and

extract the properties of inner region. . If these properties are equal or in close agreement with the

input data we used in Table B-3, this means that the type curves are correct. In this section,

pressure drawdown and build-up analysis is shown.

69

Properties known to us before using the type curves:

5.0

6097.2951

100

4829.885

4829.885

10

67.2

10

11

2

2

2

fm

ftyx

acresArea

fta

ftL

F

6.6.1.1 Pressure Drawdown Test

The data to generate the derivative plot is given in Table E-1. The pseudo pressure drop

derivative plot for this drawdown test is shown in Figure 6-5.

The graph in Figure 6-5 is to be matched on type curves to obtain crushed zone

properties. Since one already knows the storativity of fracture networks, F=10, one goes to the

type curves generated for F=10 and select those plots for which the field data is matching the type

curves. One example of matching the field data point is given in Figure 6-6.

70

Figure 6-5: Field pseudo-pressure drop derivative plot for drawdown test

for case study 1.

Figure 6-6: Type curve made for F=10, ,3.02

y

b,10 1

1

1.1

1

wr

bn .

71

Type curve used in Figure 6-6 is also presented in Figure C-22. This is just one example

but in the same way, field data points are matched on other type curves on which it could match.

In this case, field data plot matches to two other type curves presented in Figures C-17 and Figure

C-27 and the match point coordinates are recorded to perform calculations described in sections

6.4.1, 6.4.2 and6.4.3 to obtain inner region properties.

The field data has to be matched with every type curve on which it could match. Even if

sometimes a type curve slightly matches the field data, it is recommended that one should record

the values of match points. Hence, just to be on the safe side, one should match field data to as

many type curves as one thinks that they can be matched.

By doing this and following the same procedure as described in the sections 6.4.1, 6.4.2

and6.4.3, it is highly possible that one will be able to get the correct values of the inner region

reservoir parameters. For this particular field data, the results for all the possible type curve

matching are shown in Tables 6-1, 6-2 and 6-3. For the results in Tables 6-1, 6-2, 6-3, the

coordinates of match points are given in Tables D-1, D-2, D-3 respectively.

Table 6-1: Type curve matching result from the type curve Figure 6-6 or Figure C-22.

y

b2

1.1

1.

wr

b

b b_corr Kf1 λ1 ω1

1e+0 335.5711 442.7414 1.0175e+005 3.6594e-004 0.1000

1e+1 413.5312 442.7414 1.1946e+004 0.0029 0.1000

0.3 1e+2 398.9948 442.7414 905.5928 0.0300 0.1000

1e+3 366.7681 442.7414 68.1094 3.2878 0.1000

1e+4 442.2810 442.7414 10.6429 2.5684 0.1000

72

As can be seen, from Tables 6-1, 6-2, 6-3, the closest match between b and b_corr has been

achieved in Table 6-1. Here one should make a second check once he gets more than one result

with corrbb _ . In this particular test, two results came up with closely matching values of b

and b_corr. These two results have been highlighted with blue and yellow in Table 6-1. As

described in section 6.4.3, when this type of conflict occurs, one should check the value of inner

region permeability, kf1. In Table 6-1, the result highlighted in blue color gives a value of kf1 =

1.19*104 md which is not reasonable. Therefore, one should discard that result and is left with

Table 6-2: Type curve matching result from the type curve Figure-C-17.

y

b2

1.1

1.

wr

b


0.1 1e+2 314.3062 147.5805 453.9667 0.0390 0.1000

1e+3 340.8403 147.5805 63.2946 0.3564 0.1000

1e+4 387.4347 147.5805 12.7906 0.3095 0.1000

Table 6-3: Type curve matching result from the type curve Figure-C-27.

y

b2

1.1

1.

wr

b


1e+0 420.8526 737.9024 1.1289e+005 2.8261e-004 0.1000

1e+1 2.9262e+003 737.9024 6.6416e+004 3.3481e-004 0.1000

0.5 1e+2 437.7899 737.9024 1.1743e+003 0.0271 0.1000

1e+3 512.6482 737.9024 95.1996 0.2275 0.1000

1e+4 82.2384 737.9024 2.0362 17.0275 0.1000

73

only one result highlighted in yellow color which gives corrbb _ and also plausible value of

kf1. Hence, one can conclude that this is the correct result for this field data.

6.6.1.2 Pressure Build-up Test

Pressure build-up data obtained from the numerical simulator is shown in Table E-2. For

this case study, as is also mentioned in the Table B-3, the reservoir is produced for 150 days and

the build-up time is 15 days. Since this satisfies the condition, ptt *1.0 , we can use Δt as

the equivalent time for the build-up test data. The pressure build-up data is plotted on log-log plot

of )ln( t

ws

vs. t and matched on the type curve. The derivative field data plot for build-up

test is shown in Figure 6-7 for this case study.

Figure 6-7: Field pseudo pressure drop derivative plot for pressure build-up test for case

study 1.

74

The calculations are done in the same way as described in sections 6.4.1, 6.4.2 and 6.4.3

to obtain inner region properties.

In this study, we are using pressure build-up test just to verify the result that was

obtained from pressure drawdown test. From pressure drawdown test, we already know that the

type curve which is plotted for 3.02

y

b,

1

1 10 is the correct type curve for this system. So

one can go directly to the same type curve, i.e. in this case, Figure 6-5 and match on the same

curve, which represents4

1.1

1 10.

wr

b . Coordinate value of match point for this case is given

in Table D-4. The result of calculation is given in Table 6-4.

For this case study, the comparison of pressure drawdown and build-up test with the

input data are shown in Table 6-5.

Table 6-4: Type curve matching result from the type curve Figure C-22.

y

b2

1.1

1.

wr

b

b b_corr Kf1 λ1

0.3 1e+4 581.0779 442.7414 11.3261 1.9819

75

From the Table 6-5, it is concluded that the presented type curves give accurate results

for pressure drawdown testing. However, the reservoir properties obtained by matching build-up

test data on the type curves may not be correct.

6.6.2 Case Study 2

In this case study, a shale gas reservoir is taken with a multi stage fractured horizontal

well but the fractures did not propagate deep into the formation. This case can occur in the field

because the formation might be too tight. Therefore, length of minor axis of the crushed zone is

very small. The input data for the numerical simulator is given in Table B-4.

Table 6-5: Comparison of pressure drawdown and build-up test results with input data

for case study 1.

Property Input Data Pressure

Drawdown

Result

Pressure

Build Up

Result

Absolute %

Error from

Drawdown

Test

Absolute %

Error from

Build Up

Test

λ1 2.67 2.5684 1.9819 3.8 25.77

ω1 1e-1 1e-1 1e-1 0 0

b (ft) 442.7414 442.2810 581.0779 0.103 31.24

Kf1 (md) 10 10.6429 11.3261 6.429 13.261

76

6.6.2.1 Pressure Drawdown Test

For this test, the well is set to produce for 100 days. Wellbore pressure response is

obtained by inputting the reservoir properties to the numerical simulator. Pressure transient data

generated for this case study is shown in Table E-3. Wellbore pressure is converted to pseudo

pressure using Ψ-approach and the pseudo pressure derivative is plotted against time on a log-log

plot for type curve matching. The field data derivative plot for drawdown test for this case study

is shown in Figure 6-8.

Figure 6-8: Field data derivative plot for drawdown plot for case study 2.

77

Properties known before using type curves:

5.0

1033.2087

100

4829.885

4829.885

10

10*31.1

100

2

2

2

2

fm

ftyx

acresArea

fta

ftL

F

In this case, field data derivative plot matches type curves made for ω1 = 10-2

from initial

screening or glance and one already knows the value of storatvity of fracture networks of inner

and outer region, F=100. Therefore, field data derivative curve shown in Figure 6-8 is matched on

the type curves C-33, C-38, C-43. The results of type curve matching are given in Tables 6-6, 6-7,

6-8 and the coordinate values of match points are given in Tables D-5, D-6, D-7 respectively.

Table 6-6: Matching on type curve Figure C-33.

y

b2

1.1

1.

wr

b


1e+0 43.4263 104.3552 7.5749e+003 3.4372 1*10-2

1e+1 74.0141 104.3552 1.6138e+003 1.9120 1*10-2

0.1 1e+2 74.0141 104.3552 161.3797 1.9120 1*10-2

1e+3 104.6440 104.3552 15.5152 1.3063 1*10-2

1e+4 121.8830 104.3552 3.2422 11.0458 1*10-2

78


y

b2

1.1

1.

wr

b


1e+0 78.7145 313.0655 1.3730e+004 0.0018 1*10-2

1e+1 86.7991 313.0655 1.5898e+003 0.0160 1*10-2

0.3 1e+2 88.4231 313.0655 158.0935 0.1572 1*10-2

1e+3 101.1042 313.0655 17.6357 1.3634 1*10-2

1e+4 94.1175 313.0655 2.0521 14.6790 1*10-2


y

b2

1.1

1.

wr

b


1e+0 107.5767 521.7759 1.9234e+004 0.0013 1*10-2

1e+1 92.3467 521.7759 2.0135e+003 0.0150 1*10-2

0.5 1e+2 105.6010 521.7759 193.4112 0.1293 1*10-2

1e+3 95.2925 521.7759 19.9464 0.1448 1*10-2

1e+4 109.6396 521.7759 2.3906 12.4099 1*10-2

1e+5 162.4128 521.7759 0.5099 80.5474 1*10-2

79

From Tables 6-6, 6-7 and 6-8, it is clear that there is one unique type curve which gave

corrbb _ and is also highlighted in yellow color. Rest of the type curves did not satisfy this

condition. Hence, for this reservoir, inner region properties are obtained using the type curves and

the comparison with the input data is shown in Table 6-9.

6.6.2.2 Pressure Build-up Test

The reservoir is set to produce for 100 days. After 100 days, the well is shut in for 10

days for build-up test. Since this satisfies the condition, ptt *1.0 , we can use Δt as the

equivalent time for the build-up test data. Pressure build-up data generated for this case study is

shown in Table E-4. The pressure build-up data is plotted on log-log plot of )ln( t

ws

vs. t

and matched on the type curve. The field data derivative plot for build-up test is given in Figure

6-9.

80

From drawdown test, it is known that type curve in Figure C-33 is the correct type curve

for this system. Therefore, for build-up test, one just go that same type curve for matching.

Coordinate value of match point for this case is given in Table D-8 and the result is shown in

Table 6-9.

Figure 6-9: Field data derivative plot for build-up test for case study 2.


y

b2

1.1

1.

wr

b


0.1 1e+3 136.7962 104.3552 17.8962 0.9729 1*10-2

81

For this case study, the comparison of pressure drawdown and build-up test with the

input data are shown in Table 6-9.

From Table 6-9, it is concluded that the presented type curves give results with low

percentage error for pressure drawdown testing. However, the percentage error for the results

obtained in the build-up test is on the higher side. This result is consistent with the results of case

study 1 as the type curves yielded correct values of inner region reservoir properties for

drawdown test but not for build-up test.

From the two case studies presented in this chapter, type curves are successfully verified

to give correct results of inner region reservoir properties for pressure drawdown and build-up

test. However, the error obtained in the results is higher for the build-up test as compared to

drawdown test. It is because the definition of the equivalent time that is used for the build-up test

is applicable rigorously for the radial flow and infinite acting period. But in our system, we

neither have radial flow nor infinite acting system. Therefore, the build tests should not be

Table 6-9: Comparison of Pressure Drawdown and Build Up Result for case study 2.

Properties Input Data Pressure

Drawdown

Result

Pressure

Build-Up

Result

Absolute %

Error from

Drawdown

Test

Absolute %

Error from

Build-Up

Test

λ1 1.31 1.3063 0.9729 0.28 24.96

ω1 10-2

10-2

10-2

0 0

b (ft 104.3552 104.6440 136.7962 0.276 31.08

Kf1 (md) 15 15.5152 17.8962 3.43 19.3

82

analyzed by using the presented type curves. However, one can use it as a verification tool for the

pressure drawdown testing results.

83

Chapter 7

CONCLUSIONS

In the current research, the pressure transient response of the composite dual porosity gas

systems with horizontal wells has been studied. The main focus of this research was to develop a

forward and inverse solution for the pressure transient analysis of these systems. Besides this, a

sensitivity analysis was also done to observe the individual effects of dimensionless parameters

involved in these systems. From the current study of composite naturally fractured systems, the

following conclusions can be drawn:

1) Forward solution for composite dual porosity systems can be obtained with high

accuracy using the numerical model presented in this study. The presented numerical

model can also be used for history matching the pressure transient data.

2) The pressure transient behavior of these systems can be studied by the use of

dimensionless semi-log derivative plots. The dual porosity of inner and outer region

can be easily seen on derivative plots as compared to pressure drop plots.

3) All the dimensionless parameters involved in this system have their characteristic

effect on the pressure response of the system. However, if all the dimensionless

parameters are kept constant, the dimensionless semi-log derivative plots lie on top of

each other.

4) The shape of dual porosity signature or transition zone of inner and outer region is

dependent on ω1 and ω2 respectively.

84

5) If 12

1

, it is not possible to make generalized type curves for this system because

λ2 starts to interfere with the effect of λ1. So, in this study, for the sake of producing

type curves for these systems, we have assumed that 12

1

is always true.

6) The type curves provide the inverse solution for well testing in these systems. Two

case studies have been shown to verify presented type curves.

7) The type curves give reasonably accurate values for the inner region reservoir

properties for the pressure drawdown test. However, the buildup tests may generate

results with lower accuracy.

85

Appendix A

NUMERICAL MODEL DEVELOPMENT

Starting from the equation (4.15) which is obtained by using numerical approximation to

the partial differential equation (4.9) and combining equation (4.7) with (4.10) we get an equation

which relates the interporosity flow rate, q* with pressure drop in matrix and fracture blocks. This

is shown in equation (A.1).

gm

fm

g

m

b

m

m

m

b

B

PPkVQ

z

P

t

Vq

)(.

615.5

*

(A.1)

Substituting the values of Bg and Q from equations (4.2) and(4.4), we get

fm

m

m

gf

mm

m

mmm ppz

pk

t

p

z

pC

...

615.5

(A.2)

Here Cm is the compressibility of gas in the matrix blocks and by definition:

m

m

mm

mp

z

zpC

11

(A.3)

The equation (A.2) is simplified and written as:

fm

gf

mmmm ppk

t

pC

..

615.5

(A.4)

Here, putting

615.51

mmCn

(A.5)

gf

mkn

2 (A.6)

Therefore, the equation (A.4) can be written as:

86

).(21 fmm ppnt

pn

(A.7)

Rearranging the equation (A.7) and integrating,

1

1

2

1

.)(

n

n

n

n fm

m dtn

n

pp

p (A.8)

dtn

npp

n

nfm

1

21)ln(

(A-9)

dtn

n

PP

PPn

f

n

m

n

f

n

m

1

2

1

11

ln

(A.10)

dtn

n

n

f

n

m

n

f

n

me

pp

pp1

2

1

11

(A.11)

dtn

n

n

f

n

m

n

f

n

m epppp 1

2

111 ).(

(A.12)

Substituting the value of pm in the equation (4.15),

0...615.5

)).(.(..615.5

...615.5

...615.5

.

).(.

).(.

).(.

).(.

,

,

1

21

,,

1

,

1

,

,

,

1

,

1

,

1

,/

1

,1

1

,

1

,1

1

,

21

,1

21

,

,2

11

,1

1

,

1

,1

1

,

21

,

21

,1

,2

1

1

1,

1

,

1

1,

1

,

21

1,

21

,

2

1,

1

1,

1

,

1

1,

1

,

21

,

21

1,

2

1,

n

jim

n

jim

mbdt

n

nn

jif

n

jim

n

jif

n

jim

mb

n

jif

n

jif

fbn

jif

n

jif

fbn

jiDSCF

n

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jiyn

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jiy

n

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jixn

jif

n

jif

n

jigf

n

jigf

n

jif

n

jif

jix

pzt

Veppp

zt

V

pzt

Vp

zt

VqQ

zz

ppT

zz

ppT

zz

ppT

zz

ppT

(A.13)

87

The left hand side of the equation (A.13) is called as residual, Ri,j. Applying Newton Raphson

method to the above equation, we get,

kn

ji

k

n

jif

n

jikn

jif

k

n

jif

n

jikn

jif

k

n

jif

n

jikn

jif

k

n

jif

n

jikn

jif

k

n

jif

n

jikn

jif

Rp

Rp

p

Rp

p

Rp

p

Rp

p

Rp

1

,1

,

1

,11

,1

,1

1

,11

,1

1

,1

1

,11

,11

1,

1

,11

1,1

1,

1

,11

1,

..

...

(A.14)

In the equation (A.14) k and n represents iteration and time level respectively.

kn

ji

kn

jif

kn

ji

kn

jif

kn

ji

kn

jif

kn

ji

kn

jif

kn

ji

kn

jif

kn

ji

RpC

pSpNpEpW

1

,

11

,

1

.

11

,1

1

.

11

,1

1

.

11

1,

1

.

11

1,

1

.

.

....

(A.15)

where,

kn

jiW 1

.

=

k

n

jif

n

ji

p

R

1

1,

1

,.

(A.16)

Similarly, by comparing equation (A.15) and (A.14), the equations for other coefficients can be

written. The term

fp

R

is calculated by numerical differentiation using forward difference

approximation.

For example, to calculate the term,

k

n

jif

n

ji

P

R

1

1,

1

,, we have used the formula given below:

88

ep

PPPPPRPPPPepPR

P

Rkn

jif

kn

jif

kn

jif

kn

jif

kn

jif

n

ji

kn

jif

kn

jif

kn

jif

kn

jif

kn

jif

n

ji

k

n

jif

n

ji

1

,

1

,1

1

,1

1

1,

1

1,

1

,

1

,

1

,1

1

,1

1

1,

1

1,

1

,

1

1,

1

,,,,,,,,,

(A.17)

Similarly, the other derivatives are calculated numerically. Here ep = 0.0001. The terms kn

jiW 1

.

,

kn

jiS 1

.

,

kn

jiE 1

.

,

kn

jiN 1

.

,

kn

jiC 1

.

are put in Jacobian matrix. The structure of numerical solution is

shown in Figure A-1.

The schematic numerical solution shown in Figure A-1 can also be written as given below.

Or J(k+1)

. ΔPf(k+1)

= - R(k)

(A.18)

Conjugate Gradient method was used to solve this equation. The pressure of fractures in each

block i.e. jifP

, is updated at after every iteration level by:

11

,,,

l

f

k

f

k

f jijijiPPP

(A.19)

When the convergence is reached then, the final fracture pressure for time level (n+1) is

calculated by using equation (A.19) and pressure of matrix in each block 1

,

k

m jiP is calculated by

using equation (A-3).

Figure A-1: Structure of Numerical Solution.

Appendix B

Simulation Data Tables

Table B-1: Run data for validating dual porosity model against Warren & Root solution

with vertical well

Property Value

Fracture Permeability (kf) 1md

Matrix Permeability (km) 0.0001 md

Fracture Porosity (φ ) 1%

Matrix Porosity (φm) 30%

Reservoir Thickness (h) 100 t

Fracture Spacing 20 ft

Production Rate (qSCF/D) 10 MMscf/day

Reservoir Temperature (T) 130 F

Wellbore Radius (r ) 0.25 ft

Initial Pressure (Pi) 5000 psi

Production time 100 days

Molecular Weight (gm) 16

Critical Temperature (R) 343

Critical Pressure (psi) 666

Surface Temperature (F) 60

90

Table B-2: Run data for validating composite reservoir model against Eggenshwiler

solution with vertical well

Property Value

Altered Region Permeability 5 md

Unaltered Permeability (km) 0.05 md

Altered egion Porosity (φf) 20%

Unaltered egion Porosity (φm) 10%

Reservoir Thickness (h) 100 ft

Production Rate (q) 1 MMscf/day


Wellbore Radius (rw) 0.25 ft


Production time 200 days





91

Table B-3: Run data for validating composite dual porosity model against CMG solution

Property Value

Outer Region Fracture Permeability (kf) 0.01 md

Outer Region Matrix Permeability (km) 5.68e-7 md

Outer egion Fracture Porosity (φf) 0.5%

Outer Region Matrix Porosity (φm) 0.495%

Outer Region Fracture Spacing 31.62277 ft

Inner Region Fracture Permeability (kf) 10 md

Inner Region Matrix Permeability (km) 5.68e-7 md

Inner egion Fracture Porosity (φf) 5%

Inner egion Matrix Porosity (φm) 45%

Inner Region Fracture Spacing 1 ft

Length of Horizontal well (ft) 1770.9658

Major Axis, 2*a (ft) 1770.9658

Minor Axis, 2*b (ft) 885

Rock Compressibility (psi-1

) 0


Production Rate (q) 1 MMScf/day




Production time (days) 150 days

Build Up time (days) 15 days


92




λ1 2.67

λ2 2.67

ω1 1e-1

ω2 1e-2

F 10

93

Table B-4: Simulation dataset for case study 2.

Property Value

Outer Region Fracture Permeability (kf) 0.15 md

Outer Region Matrix Permeability (km) 1.88*10-8

md

Outer Region Fracture Porosity (φf) 0.005%

Outer egion Matrix Porosity (φm) 50%

Outer Region Fracture Spacing 10 ft

Inner Region Fracture Permeability (kf) 15 md

Inner Region Matrix Permeability (km) 1.88*10-8

md

Inner egion Fracture Porosity (φf) 0.5%

Inner egion Matrix Porosity (φm) 49.5%

Inner Region Fracture Spacing 0.1 ft

Length of Horizontal wells (ft) 834.8413

Major Axis, 2*a (ft) 834.8413

Minor Axis, 2*b (ft) 209

Rock Compressibility (psi-1

) 0


Production Rate (q) 1 MMscf/day




Production time (days) 100 days

Build Up time (days) 10 days


94




λ1 1.31

λ2 1.31*10-2

ω1 10-3

ω2 10-4

F 100

95

Appendix C

Type Curves for Composite Dual Porosity Systems with a Horizontal Well

In this appendix, the type curves for composite dual porosity systems with a horizontal

well are presented. Type curves are arranged in three categories. These three categories represents

three different values of storativity ratio of fracture networks i.e. F = 1, 10, 100. For each value of

storativity ratio of fracture networks, there are three subcategories which represents y

b2= 0.1,

0.3, 0.5. For each value of y

b2, five different graphs are plotted for constant inner region fracture

storativity ratio i.e. ω1= 1, 10-1

, 10-2

, 10-3

, 10-4

C.1 Type curves generated for F=1

96

C.1.1 1.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

97


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

98


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

99

C.1.2 3.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

100


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

101


y

b, 3

1 10 ,

1.1

1

wr

bn .

102

C.1.3 5.02

y

b


y

b, 4

1 10 ,

1.1

1

wr

bn .


y

b, 11 ,

1.1

1

wr

bn .

103


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

104


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

105


C.2.1 1.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

106


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

107


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

108

C.2.2 3.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

109


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

110


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

111

C.2.3 5.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

112


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

113


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

114


C.3.1 1.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

115


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

116


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

117

C.3.2 3.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

118


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

119


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

120

C.3.3 5.02

y

b


y

b, 11 ,

1.1

1

wr

bn .

121


y

b, 1

1 10 ,

1.1

1

wr

bn .


y

b, 2

1 10 ,

1.1

1

wr

bn .

122


y

b, 3

1 10 ,

1.1

1

wr

bn .


y

b, 4

1 10 ,

1.1

1

wr

bn .

123

Appendix D

Type Curve Match Points for Case Studies

D.1 Case Study 1

D.1.1 Pressure Drawdown Testing

Table D-1: Match points obtained by matching on type curve presented in Figure C-22.

y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+00 1.60E-02 1.40E-04 104

10-2

1.00E+01 2.1E-03 1.40E-05 104 10

-2

0.3 1.00E+02 1.40E-04 1.20E-06 104 10

-2

1.00E+03 1.20E-04 1.00E-05 105 10

0

1.00E+04 1.00E-02 1.00E-05 5.5*107 9*10

0


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+02 7.00E-05 7.00E-05 104 10

0

0.1 1.00E+03 1.00E+04 1.00E-04 105 10

1

1.00E+04 1.20E-05 1.20E-05 105 10

1

124

D.1.1 Pressure Build-up Testing


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+00 1.80E-02 1.20E-04 104

10-2

1.00E+01 2.00E-03 1.40E-05 104 10

-2

0.5 1.00E+02 2.00E-04 1.40E-06 104 10

-2

1.00E+03 1.80E-04 9.00E-06 105 10

0

1.00E+04 2.00E-06 1.20E-06 105 10

0


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

ws

ln

Δt

0.3 1.00E+04 2.10E-05 9.00E-06 105 10

1

125

D.2 Case Study 2

D.2.1 Pressure Drawdown Testing


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+00 3.20E-06 3.40E-07 104

10-2

1.00E+01 1.20E-03 4.00E-04 104 10

-2

0.1 1.00E+02 3.00E-04 5.00E-05 104 10

-2

1.00E+03 3.20E-06 3.20E-07 104 10

-2

1.00E+04 7.00E-06 6.10E-07 105 10

-1


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+00 2.60E-03 4.00E-04 104

10-2

1.00E+01 3.10E-04 4.20E-05 104 10

-2

0.3 1.00E+02 3.10E-05 4.10E-06 104 10

-2

1.00E+03 3.60E-06 4.00E-07 104 10

-2

1.00E+04 4.10E-06 5.00E-07 105 10

-1

126

D.2.2 Pressure Build-up Testing


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

wf

ln

t

1.00E+00 4.00E-03 4.10E-04 104

10-2

1.00E+01 4.00E-04 5.00E-05 104 10

-2

0.5 1.00E+02 4.00E-05 4.20E-06 104 10

-2

1.00E+03 4.00E-06 4.80E-07 104 10

-2

1.00E+04 4.00E-07 4.50E-08 104 10

-2

1.00E+05 1.20E-06 7.20E-07 105

100


y

b2

1.1

1

wr

b

D

D

w

w

tx

L

r

b

L

r

ln.

2..

7.03.03.1

DA

w tx

L

L

r.

2.

2.0

t

ws

ln

Δt

0.1 1.00E+03 4.00E-06 3.00E-07 104 10

-2

127

Appendix E

Pressure Transient Data Used for Case Studies

E.1 Data for case study 1

Table E-1: Pressure drawdown data.

t(hrs) pwf(psi) Ψwf ΔΨwf

t

wf

ln

0 5000 1.35E+09 2.50E+0 none

0.04176 4999.39 1.35E+09 250390 none

0.08352 4999.34 1.35E+09 270679 29271.1

0.12528 4999.3 1.35E+09 288160 43113

0.16704 4999.26 1.35E+09 303522 53401

0.2088 4999.23 1. 5E+09 317275 6162 .2

0.25056 4999.2 1.35E+09 329791 68647.5

0.29232 4999.17 1.35E+09 341346 74963.9

0.33408 4999.14 1.35E+09 352146 80878.1

0.37584 4999.12 1.35E+09 362341 86561.1

0.4176 4999.09 1.35E+09 372046 92104.8

0.45936 4999.07 1.35E+09 381344 97554.2

0.50112 4999.05 1.35E+09 390299 102927

0.54288 4999.03 1.35E+09 398962 108227

0.58464 4999.01 1.35E+09 407370 113451

0.6264 4998.99 1.35E+09 415552 118592

0.66816 4998.97 1.35E+09 423532 123643

0.70992 4998.95 1.35E+09 431328 128599

0.75168 4998.93 1.35E+09 438956 133456

0.79344 4998.91 1.35E+09 446429 138212

0.8352 4998.89 1.35E+09 453757 142866

0.87696 4998.88 1.35E+09 460949 147418

0.91872 4998.86 1.35E+09 468014 151870

0.96048 4998.84 1.35E+09 474959 156223

1.00224 4998.83 1.35E+09 481789 160482

1.044 4998.81 1.35E+09 488510 164649

128

1.08576 4998.79 1.35E+09 495128 168727

1.12752 4998.78 1.35E+09 501646 172720

1.16928 4998.76 1.35E+09 508070 176632

1.21104 4998.75 1.35E+09 514403 180466

1.2528 4998.73 1.35E+09 520648 184226

1.29456 4998.72 1.35E+09 526810 187916

1.33632 4998.7 1.35E+09 532891 191538

1.37808 4998.69 1.35E+09 538894 195095

1.41984 4998.67 1.35E+09 544823 198591

1.4616 4998.66 1.35E+09 550679 202028

1.50336 4998.64 1.35E+09 556466 205409

1.54512 4998.63 1.35E+09 562185 208736

1.58688 4998.62 1.35E+09 567839 212013

1.62864 4998.6 1.35E+09 573430 215240

1.6704 4998.59 1.35E+09 578960 218420

1.71216 4998.58 1.35E+09 584431 221555

1.75392 4998.56 1.35E+09 589844 224646

1.79568 4998.55 1.35E+09 595202 227696

1.83744 4998.54 1.35E+09 600506 230706

1.8792 4998.52 1.35E+09 605757 233677

1.92096 4998.51 1.35E+09 610957 236611

1.96272 4998.5 1.35E+09 616108 239509

2.00448 4998.49 1.35E+09 621211 242372

2.04624 4998.47 1.35E+09 626267 245202

2.088 4998.46 1.35E+09 631277 247999

2.12976 4998.45 1.35E+09 636243 250765

2.17152 4998.44 1.35E+09 641166 253500

2.21328 4998.42 1.35E+09 646046 256206

2.25504 4998.41 1.35E+09 650885 258883

2.2968 4998.4 1.35E+09 655684 261532

2.33856 4998.39 1.35E+09 660443 264154

2.38032 4998.38 1.35E+09 665165 266750

2.42208 4998.37 1.35E+09 669849 269319

2.46384 4998.36 1.35E+09 674496 271864

2.5056 4998.34 1.35E+09 679108 274384

2.54736 4998.33 1.35E+09 683684 276880

2.58912 4998.32 1.35E+09 688227 279353

15.1589 4996.44 1.35E+09 1460970 590841

15.2006 4996.43 1.35E+09 1462590 591189

16.7875 4996.29 1.35E+09 1521900 602717

129

16.8293 4996.29 1.35E+09 1523400 602979

16.871 4996.28 1.35E+09 1524900 603238

16.9128 4996.28 1.35E+09 1526390 603495

16.9546 4996.28 1.35E+09 1527880 603750

16.9963 4996.27 1.35E+09 1529360 604004

17.0381 4996.27 1.35E+09 1530850 604255

17.7062 4996.21 1.35E+09 1554170 608009

750.569 4980.55 1.34E+09 7978150 6041990

756.569 4980.43 1.34E+09 8026640 6089670

762.569 4980.31 1.34E+09 8075120 6137340

768.569 4980.2 1.34E+09 8123600 6185000

774.569 4980.08 1.34E+09 8172060 6232650

780.569 4979.96 1.34E+09 8220520 6280300

786.569 4979.84 1.34E+09 8268980 6327930

792.569 4979.72 1.34E+09 8317430 6375560

798.569 4979.6 1.34E+09 8365870 6423190

804.569 4979.49 1.34E+09 8414310 6470800

810.569 4979.37 1.34E+09 8462740 6518410

816.569 4979.25 1.34E+09 8511160 6566010

822.569 4979.13 1.34E+09 8559580 6613600

828.569 4979.01 1.34E+09 8607990 6661190

834.569 4978.9 1.34E+09 8656400 6708760

840.569 4978.78 1.34E+09 8704800 6756330

1182.57 4972.08 1.34E+09 11454400 9454340

1188.57 4971.96 1.34E+09 11502500 9501430

1194.57 4971.84 1.34E+09 11550500 9548510

1200.57 4971.72 1.34E+09 11598600 9595590

1206.57 4971.61 1.34E+09 11646700 9642650

1212.57 4971.49 1.34E+09 11694800 9689700

1218.57 4971.37 1.34E+09 11742800 9736750

1224.57 4971.25 1.34E+09 11790900 9783790

1686.57 4962.27 1.33E+09 15476400 13379300

1692.57 4962.15 1.33E+09 15524100 13425600

1698.57 4962.04 1.33E+09 15571800 13472000

1704.57 4961.92 1.33E+09 15619400 13518300

1710.57 4961.8 1.33E+09 15667100 13564600

1716.57 4961.69 1.33E+09 15714800 13610900

3540.57 4926.88 1.32E+09 29991300 27291400

3546.57 4926.77 1.32E+09 30037500 27335100

3552.57 4926.66 1.32E+09 30083800 27378800

130

3558.57 4926.55 1.32E+09 30130100 27422500

3564.57 4926.43 1.32E+09 30176400 27466200

3570.57 4926.32 1.32E+09 30222600 27509900

3576.57 4926.21 1.32E+09 30268900 27553600

3582.57 4926.09 1.32E+09 30315200 27597300

3588.57 4925.98 1.32E+09 30361400 27640900

3594.57 4925.87 1.32E+09 30407700 27684600

Table E-2: Build-up data for case study 1.

t(hrs) pttt pwf Ψws ΔΨws

t

ws

ln

3594.57 0 4925.87 1318930000 0 none

3600.61 6.04 4927.21 1.32E+09 250000 none

3600.65 6.08 4927.26 1.32E+09 270000 28582.9

3600.69 6.12 4927.3 1. 2E+09 290000 42016.8

3600.74 6.17 4927.34 1.32E+09 310000 51922.4

3600. 8 6.21 4927.37 1.32E+09 320000 59770.9

3600.82 6.25 4927.39 1.32E+09 330000 66404.2

3600.86 6.29 4927.42 1.32E+09 340000 72328.3

3600.9 6.33 4927.44 1.32E+09 350000 77844.1

3600.94 6.37 4927.46 1.32E+09 360000 83124.5

3600.99 6.42 4927.48 1.32E+09 370000 88263.6

3601.03 6.46 4927.5 1.32E+09 380000 93308.1

3601.07 6.5 4927.52 1.32E+09 390000 98277.5

3601.11 6.54 4927.54 1.32E+09 400000 103176

3601.15 6.58 4927.55 1.32E+09 410000 108000

3601.2 6.63 4927.57 1.32E+09 410000 112745

3601.24 6.67 4927.58 1.32E+09 420000 117403

3601.28 6.71 4927.6 1.32E+09 430000 121968

3601.32 6.75 4927.61 1.32E+09 440000 126438

3601.36 6.79 4927.62 1.32E+09 440000 130808

3601.4 6.83 4927.63 1.32E+09 450000 135079

3601.45 6.88 4927.65 1.32E+09 460000 139250

3601.49 6.92 4927.66 1.32E+09 460000 143322

3601.53 6.96 4927.67 1.32E+09 470000 147298

3601.57 7 4927.68 1.32E+09 480000 151180

131

3601.61 7.04 4927.69 1.32E+09 480000 154971

3601.65 7.08 4927.7 1.32E+09 490000 158674

3601.7 7.13 4927.71 1.32E+09 490000 162294

3601.74 7.17 4927.72 1.32E+09 500000 165833

3601.78 7.21 4927.73 1.32E+09 510000 169294

3604.12 9.55 4928.11 1.32E+09 760000 299197

3604.16 9.59 4928.11 1.32E+09 760000 300868

3604.2 9.63 4928.12 1.32E+09 760000 302523

3604.24 9.67 4928.12 1.32E+09 770000 304164

3604.29 9.72 4928.13 1.32E+09 770000 305791

3608.21 13.64 4928.56 1.32E+09 1030000 409899

3608.25 13.68 4928.57 1.32E+09 1030000 410614

3608.29 13.72 4928.57 1.32E+09 1030000 411322

3608.34 13.77 4928.58 1.32E+09 1040000 412023

3608.38 13.81 4928.58 1.32E+09 1040000 412719

3608.42 13.85 4928.59 1.32E+09 1040000 413408

3608.46 13.89 4928.59 1.32E+09 1040000 414091

3608.5 13.93 4928.59 1.32E+09 1040000 414769

3608.54 13.97 4928.6 1.32E+09 1050000 415440

3608.59 14.02 4928.6 1.32E+09 1050000 416105

3608.63 14.06 4928.61 1.32E+09 1050000 416764

3663.02 68.45 4930.38 1.32E+09 1830000 158894

3663.52 68.95 4930.39 1.32E+09 1840000 156799

3664.02 69.45 4930.39 1.32E+09 1840000 154745

3664.52 69.95 4930.39 1.32E+09 1840000 152732

3665.02 70.45 4930.39 1.32E+09 1840000 150760

3665.52 70.95 4930.4 1.32E+09 1840000 148827

3666.01 71.44 4930.4 1.32E+09 1840000 146934

3666.51 71.94 4930.4 1.32E+09 1840000 145079

3667.01 72.44 4930.4 1.32E+09 1840000 143263

3667.51 72.94 4930.41 1.32E+09 1840000 141485

3668.01 73.44 4930.41 1.32E+09 1850000 139744

3668.51 73.94 4930.41 1.32E+09 1850000 138040

3669.01 74.44 4930.41 1.32E+09 1850000 136372

3669.51 74.94 4930.41 1.32E+09 1850000 134741

3670.01 75.44 4930.42 1.32E+09 1850000 133144

3670.51 75.94 4930.42 1.32E+09 1850000 131583

3671.01 76.44 4930.42 1.32E+09 1850000 130055

3671.51 76.94 4930.42 1.32E+09 1850000 128561

3672 77.43 4930.42 1.32E+09 1850000 127101

132

3672.5 77.93 4930.42 1.32E+09 1850000 125673

3754.07 159.5 4930.54 1.32E+09 1920000 87610.4

3755.07 160.5 4930.54 1.32E+09 1920000 87949

3756.07 161.5 4930.55 1.32E+09 1920000 88290.8

3757.07 162.5 4930.55 1.32E+09 1920000 88635.6

3758.07 163.5 4930.55 1.32E+09 1920000 88983.3

3759.07 164.5 4930.55 1.32E+09 1920000 89333.7

3760.07 165.5 4930.55 1.32E+09 1930000 89686.6

3761.07 166.5 4930.55 1.32E+09 1930000 90041.9

3762.07 167.5 4930.55 1.32E+09 1930000 90399.5

3763.08 168.51 4930.55 1.32E+09 1930000 90759.2

3764.08 169.51 4930.55 1.32E+09 1930000 91120.9

3765.08 170.51 4930.55 1.32E+09 1930000 91484.5

3766.08 171.51 4930.55 1.32E+09 1930000 91849.9

3767.08 172.51 4930.56 1.32E+09 1930000 92217

3768.08 173.51 4930.56 1.32E+09 1930000 92585.7

3769.08 174.51 4930.56 1.32E+09 1930000 92955.8

3770.08 175.51 4930.56 1.32E+09 1930000 93327.3

3771.08 176.51 4930.56 1.32E+09 1930000 93700.2

3772.08 177.51 4930.56 1.32E+09 1930000 94074.2

3773.08 178.51 4930.56 1.32E+09 1930000 94449.4

3774.08 179.51 4930.56 1.32E+09 1930000 94825.6

3919.14 324.57 4930.69 1.32E+09 2000000 148159

3925.14 330.57 4930.7 1.32E+09 2010000 150391

3931.14 336.57 4930.71 1.32E+09 2010000 152598

3937.14 342.57 4930.71 1.32E+09 2010000 154784

3943.14 348.57 4930.72 1.32E+09 2020000 156953

3949.14 354.57 4930.72 1.32E+09 2020000 159109

3955.14 360.57 4930.73 1.32E+09 2020000 161255

133

E.2 Data for case study 2

Table E-3: Pressure drawdown data for case study 2

t (hrs) pwf Ψwf ΔΨwf t

wf

ln

0 6000 1.73 +09 none none

0.004176 5998.93 1729630000 460000 none

0.008352 5998.74 729550000 540000 115415.6

0.012528 5998.57 1729480000 610000 172641.2

0.638928 5990.14 1725840000 4250000 1524995

0.643104 5990.12 1 25820000 4270000 3069989

0.64728 5990.09 1725810000 4280000 1544995

0.651456 5990.07 1725800000 4290000 1554995

0.655632 5990.04 1725790000 4300000 1564995

0.659808 5990.02 1725780000 4310000 1574995

0.663984 5989.99 1725770000 4320000 1584995

0.66816 5989.97 1725760000 4330000 1594995

0.672336 5989.95 1725750000 4340000 1604995

0.676512 5989.92 1725740000 4350000 1614995

0.680688 5989.9 1725730000 4360000 1624995

0.684864 5989.88 1725720000 4370000 1634995

0.68904 5989.86 1725710000 4380000 1644995

0.693216 5989.83 1725700000 4390000 1654995

0.697392 5989.81 1725690000 4400000 1664995

0.701568 5989.79 1725680000 4410000 1674995

0.705744 5989.77 1725670000 4420000 1684995

24.5028 5981.03 1721900000 8190000 1062783

25.002 5980.92 1721860000 8230000 1983298

25.5012 5980.81 1721810000 8280000 2529124

26.0004 5980.69 1721750000 8340000 3094951

26.4996 5980.57 1721700000 8390000 2629127

26.9988 5980.45 1721650000 8440000 2679129

27.498 5980.32 1721600000 8490000 2729130

27.9972 5980.2 1721540000 8550000 3334958

28.4964 5980.08 1721490000 8600000 2829133

134

28.9956 5979.96 1721440000 8650000 2879134

29.4948 5979.83 1721390000 8700000 2929136

29.994 5979.71 1721330000 8760000 3574964

30.4932 5979.59 1721280000 8810000 3029138

30.9924 5979.47 1721230000 8860000 3079139

31.4916 5979.34 1721170000 8920000 3754968

31.9908 5979.22 1721120000 8970000 3179141

32.49 5979.1 1721070000 9020000 3229142

32.9892 5978.98 1721020000 9070000 3279143

33.4884 5978.85 1720960000 9130000 3994973

33.9876 5978.73 1720910000 9180000 3379145

34.4868 5978.61 1720860000 9230000 3429146

34.986 5978.49 1720800000 9290000 4174976

35.4852 5978.36 1720750000 9340000 3529148

35.9844 5978.24 1720700000 9390000 3579149

36.4836 5978.12 1720650000 9440000 3629149

36.9828 5978 1720590000 9500000 4414980

37.482 5977.87 1720540000 9550000 3729151

37.9812 5977.75 1720490000 9600000 3779152

38.4804 5977.63 1720430000 9660000 4594983

38.9796 5977.51 1720380000 9710000 3879153

39.4788 5977.38 1720330000 9760000 3929154

39.978 5977.26 1720280000 9810000 3979154

40.4772 5977.14 1720220000 9870000 4834986

40.9764 5977.02 1720170000 9920000 4079156

41.4756 5976.89 1720120000 9970000 4129156

41.9748 5976.77 1720060000 10030000 5014988

42.474 5976.65 1720010000 10080000 4229157

42.9732 5976.53 1719960000 10130000 4279158

43.4724 5976.4 1719910000 10180000 4329159

43.9716 5976.28 1719850000 10240000 5254991

44.4708 5976.16 1719800000 10290000 4429160

44.97 5976.04 1719750000 10340000 4479160

45.4692 5975.91 1719690000 10400000 5434993

45.9684 5975.79 1719640000 10450000 4579161

46.4676 5975.67 1719590000 10500000 4629162

46.9668 5975.55 1719540000 10550000 4679162

47.466 5975.43 1719480000 10610000 5674995

47.9652 5975.3 1719430000 10660000 4779163

48.4644 5975.18 1719380000 10710000 4829164

135

48.9636 5975.06 1719320000 10770000 5854997

49.4628 5974.94 1719270000 10820000 4929164

49.962 5974.81 1719220000 10870000 4979165

50.4612 5974.69 1719170000 10920000 5029165

50.9604 5974.57 1719110000 10980000 6094999

51.4596 5974.45 1719060000 11030000 5129166

51.9588 5974.33 1719010000 11080000 5179167

52.458 5974.2 1718960000 11130000 5229167

52.9572 5974.08 1718900000 11190000 6335001

53.4564 5973.96 1718850000 11240000 5329168

53.9556 5973.84 1718800000 11290000 5379168

54.4548 5973.71 1718740000 11350000 6515002

54.954 5973.59 1718690000 11400000 5479169

55.4532 5973.47 1718640000 11450000 5529169

55.9524 5973.35 1718590000 11500000 5579169

56.4516 5973.23 1718530000 11560000 6755004

56.9508 5973.1 1718480000 11610000 5679170

57.45 5972.98 1718430000 11660000 5729170

57.9492 5972.86 1718380000 11710000 5779171

58.4484 5972.74 1718320000 11770000 6995005

58.9476 5972.62 1718270000 11820000 5879171

59.4468 5972.49 1718220000 11870000 5929172

59.946 5972.37 1718160000 11930000 7175006

60.4452 5972.25 1718110000 11980000 6029172

60.9444 5972.13 1718060000 12030000 6079172

61.4436 5972.01 1718010000 12080000 6129173

61.9428 5971.88 1717950000 12140000 7415008

62.442 5971.76 1717900000 12190000 6229173

62.9412 5971.64 1717850000 12240000 6279174

63.4404 5971.52 1717800000 12290000 6329174

63.9396 5971.4 1717740000 12350000 7655009

64.4388 5971.27 1717690000 12400000 6429174

64.938 5971.15 1717640000 12450000 6479175

65.4372 5971.03 1717590000 12500000 6529175

65.9364 5970.91 1717530000 12560000 7895010

66.4356 5970.79 1717480000 12610000 6629175

66.9348 5970.66 1717430000 12660000 6679176

67.434 5970.54 1717370000 12720000 8075011

67.9332 5970.42 1717320000 12770000 6779176

68.4324 5970.3 1717270000 12820000 6829176

136

68.9316 5970.18 1717220000 12870000 6879176

69.4308 5970.05 1717160000 12930000 8315012

576.56 5850.26 1665450000 64640000 57355477

582.56 5848.87 1664850000 65240000 57955482

588.56 5847.49 1664250000 65840000 58555488

594.56 5846.1 1663650000 66440000 59155493

600.56 5844.72 1663060000 67030000 58759573

606.56 5843.33 1662460000 67630000 60355503

612.56 5841.95 1661860000 68230000 60955508

618.56 5840.57 1661270000 68820000 60529587

624.56 5839.19 1660670000 69420000 62155517

630.56 5837.81 1660070000 70020000 62755522

636.56 5836.43 1659480000 70610000 62299601

642.56 5835.05 1658880000 71210000 63955531

648.56 5833.67 1658290000 71800000 63479610

654.56 5832.3 1657690000 72400000 65155540

660.56 5830.92 1657100000 72990000 64659618

666.56 5829.55 1656510000 73580000 65249622

672.56 5828.17 1655910000 74180000 66955552

678.56 5826.8 1655320000 74770000 66429630

684.56 5825.43 1654730000 75360000 67019634

690.56 5824.05 1654140000 75950000 67609638

696.56 5822.68 1653540000 76550000 69355567

702.56 5821.31 1652950000 77140000 68789645

708.56 5819.94 1652360000 77730000 69379649

714.56 5818.58 1651770000 78320000 69969652

720.56 5817.21 1651180000 78910000 70559656

726.56 5815.84 1650590000 79500000 71149659

732.56 5814.48 1650000000 80090000 71739662

738.56 5813.11 1649410000 80680000 72329666

2058.56 5532.37 1528210000 201880000 1.78E+08

2064.56 5531.18 1527700000 202390000 1.75E+08

2070.56 5529.98 1527190000 202900000 1.76E+08

2076.56 5528.79 1526670000 203420000 1.8E+08

2082.56 5527.6 1526160000 203930000 1.77E+08

2088.56 5526.41 1525640000 204450000 1.81E+08

2316.56 5481.96 1506630000 223460000 1.85E+08

2322.56 5480.83 1506150000 223940000 1.86E+08

2328.56 5479.69 1505680000 224410000 1.82E+08

2334.56 5478.57 1505200000 224890000 1.87E+08

137

2340.56 5477.44 1504720000 225370000 1.87E+08

2346.56 5476.31 1504250000 225840000 1.84E+08

2352.56 5475.18 1503770000 226320000 1.88E+08

2358.56 5474.05 1503300000 226790000 1.85E+08

2364.56 5472.93 1502820000 227270000 1.89E+08

2370.56 5471.8 1502350000 227740000 1.85E+08

2376.56 5470.68 1501870000 228220000 1.9E+08

2382.56 5469.55 1501400000 228690000 1.86E+08

2388.56 5468.43 1500930000 229160000 1.87E+08

2394.56 5467.3 1500450000 229640000 1.91E+08

Table E-4: Pressure build-up data for case study 2

t (hrs) pttt

pws Ψws ΔΨws )ln( t

ws

2394.56 0 5467.86 1.50E+09 0 none

2400.56 6 5468.88 1.50E+09 452057 none

2400.57 6.01 5469 1.50E+09 525805 106395

2400.57 6.01 5469.08 1.50E+09 588836 155454

2400.58 6 02 5469.16 1.50 +09 646554 200632

2400.58 6.02 54 9.23 1.50E+09 701090 244397

2400.59 6.03 5469.3 1.50E+09 753327 286512

2400.59 6.03 5469.37 1.50E+09 803680 326643

2400.65 6.09 5470.39 1.50E+09 1416230 750164

2400.66 6.1 5470.46 1.50E+09 1450520 771288

2400.66 6.1 5470.53 1.50E+09 1484220 791884

2401.12 6.56 5475.62 1.50E+09 3668480 1589660

2401.13 6.57 5475.64 1.50E+09 3680200 1588600

2401.13 6.57 5475.67 1.50E+09 3691830 1587450

2401.14 6.58 5475.7 1.50E+09 3703370 1586230

2401.14 6.58 5475.73 1.50E+09 3714810 1584930

2401.15 6.59 5475.75 1.50E+09 3726160 1583550

2401.15 6.59 5475.78 1.50E+09 3737420 1582100

2401.15 6.59 5475.81 1.50E+09 3748590 1580580

2401.16 6.6 5475.83 1.50E+09 3759670 1578990

2401.16 6.6 5475.86 1.50E+09 3770670 1577320

2401.17 6.61 5475.89 1.50E+09 3781570 1575590

2401.17 6.61 5475.91 1.50E+09 3792390 1573790

2401.17 6.61 5475.94 1.50E+09 3803120 1571920

2402.36 7.8 5478.84 1.51E+09 5038780 584613

138

2402.37 7.81 5478.85 1.51E+09 5040130 582404

2403.97 9.41 5479.35 1.51E+09 5276230 251231

2403.97 9.41 5479.35 1.51E+09 5276540 251031

2403.98 9.42 5479.35 1.51E+09 5276850 250832

2403.98 9.42 5479.35 1.51E+09 5277150 250635

2403.98 9.42 5479.36 1.51E+09 5277460 250439

2403.99 9.43 5479.36 1.51E+09 5277760 250244

2403.99 9.43 5479.36 1.51E+09 5278070 250050

2404 9.44 5479.36 1.51E+09 5278370 249857

2404 9.44 5479.36 1.51E+09 5278670 249666

2404.01 9.45 5479.36 1.51E+09 5278980 249476

2404.01 9.45 5479.36 1.51E+09 5279280 249287

2404.01 9.45 5479.36 1.51E+09 5279580 249099

2404.02 9.46 5479.36 1.51E+09 5279880 248912

2404.02 9.46 5479.36 1.51E+09 5280180 248726

2404.03 9.47 5479.36 1.51E+09 5280480 248542

2404.03 9.47 5479.36 1.51E+09 5280780 248358

2404.03 9.47 5479.36 1.51E+09 5281080 248176

2404.04 9.48 5479.36 1.51E+09 5281380 247995

2404.04 9.48 5479.36 1.51E+09 5281670 247815

2404.05 9.49 5479.36 1.51E+09 5281970 247636

2404.05 9.49 5479.36 1.51E+09 5282270 247458

2404.06 9.5 5479.37 1.51E+09 5282560 247281

2404.06 9.5 5479.37 1.51E+09 5282860 247106

2404.06 9.5 5479.37 1.51E+09 5283150 246931

2404.07 9.51 5479.37 1.51E+09 5283440 246758

2404.07 9.51 5479.37 1.51E+09 5283740 246585

2404.08 9.52 5479.37 1.51E+09 5284030 246414

2404.08 9.52 5479.37 1.51E+09 5284320 246243

2404.08 9.52 5479.37 1.51E+09 5284610 246074

2404.09 9.53 5479.37 1.51E+09 5284910 245906

2404.09 9.53 5479.37 1.51E+09 5285200 245739

2404.1 9.54 5479.37 1.51E+09 5285490 245572

2404.1 9.54 5479.37 1.51E+09 5285780 245407

2404.11 9.55 5479.37 1.51E+09 5286070 245243

2414.28 19.72 5479.98 1.51E+09 5578340 241708

2414.28 19.72 5479.98 1.51E+09 5578420 241759

2414.29 19.73 5479.98 1.51E+09 5578490 241810

2414.29 19.73 5479.98 1.51E+09 5578560 241862

2414.3 19.74 5479.98 1.51E+09 5578640 241913

139

2414.3 19.74 5479.98 1.51E+09 5578710 241965

2414.56 20 5479.99 1.51E+09 5583330 245244

2414.57 20.01 5479.99 1.51E+09 5583410 245297

2414.57 20.01 5479.99 1.51E+09 5583480 245349

2414.58 20.02 5479.99 1.51E+09 5583550 245402

2414.58 20.02 5480 1.51E+09 5583620 245455

2450.52 55.96 5481.35 1.51E+09 6173560 811838

2451.02 56.46 5481.37 1.51E+09 6181710 819839

2451.52 56.96 5481.38 1.51E+09 6189860 827839

2452.02 57.46 5481.4 1.51E+09 6198010 835836

2452.52 57.96 5481.42 1.51E+09 6206150 843832

2453.02 58.46 5481.44 1.51E+09 6214300 851825

2453.52 58.96 5481.46 1.51E+09 6222440 859817

2454.02 59.46 5481.48 1.51E+09 6230590 867807

2454.52 59.96 5481.5 1.51E+09 6238730 875795

2455.02 60.46 5481.52 1.51E+09 6246870 883782

2455.51 60.95 5481.54 1.51E+09 6255000 891766

2456.01 61.45 5481.55 1.51E+09 6263140 899749

2456.51 61.95 5481.57 1.51E+09 6271270 907730

2457.01 62.45 5481.59 1.51E+09 6279410 915710

2457.51 62.95 5481.61 1.51E+09 6287540 923687

2458.01 63.45 5481.63 1.51E+09 6295670 931663

2458.51 63.95 5481.65 1.51E+09 6303800 939638

2459.01 64.45 5481.67 1.51E+09 6311930 947611

2459.51 64.95 5481.69 1.51E+09 6320060 955582

2460.01 65.45 5481.71 1.51E+09 6328180 963551

2460.51 65.95 5481.72 1.51E+09 6336310 971519

2461.01 66.45 5481.74 1.51E+09 6344430 979486

2461.5 66.94 5481.76 1.51E+09 6352550 987450

2462 67.44 5481.78 1.51E+09 6360670 995414

2462.5 67.94 5481.8 1.51E+09 6368790 1003380

2463 68.44 5481.82 1.51E+09 6376910 1011340

2463.5 68.94 5481.84 1.51E+09 6385020 1019290

2464 69.44 5481.86 1.51E+09 6393140 1027250

2464.5 69.94 5481.88 1.51E+09 6401250 1035210

2465 70.44 5481.89 1.51E+09 6409370 1043160

2465.5 70.94 5481.91 1.51E+09 6417480 1051120

2466 71.44 5481.93 1.51E+09 6425590 1059080

2466.5 71.94 5481.95 1.51E+09 6433700 1067030

2467 72.44 5481.97 1.51E+09 6441800 1074970

140

2467.5 72.94 5481.99 1.51E+09 6449910 1082920

2467.99 73.43 5482.01 1.51E+09 6458020 1090860

2468.49 73.93 5482.03 1.51E+09 6466120 1098810

2468.99 74.43 5482.05 1.51E+09 6474220 1106750

2469.49 74.93 5482.06 1.51E+09 6482330 1114690

2469.99 75.43 5482.08 1.51E+09 6490430 1122630

2470.49 75.93 5482.1 1.51E+09 6498530 1130570

2470.99 76.43 5482.12 1.51E+09 6506620 1138510

2471.49 76.93 5482.14 1.51E+09 6514720 1146450

2471.99 77.43 5482.16 1.51E+09 6522820 1154380

2472.49 77.93 5482.18 1.51E+09 6530910 1162320

2472.99 78.43 5482.2 1.51E+09 6539010 1170250

2473.99 79.43 5482.23 1.51E+09 6552690 996969

2474.99 80.43 5482.26 1.51E+09 6568050 1134540

2475.99 81.43 5482.3 1.51E+09 6583950 1190560

2476.99 82.43 5482.34 1.51E+09 6600040 1220360

2477.99 83.43 5482.38 1.51E+09 6616180 1241340

2478.99 84.43 5482.41 1.51E+09 6632360 1259250

2479.99 85.43 5482.45 1.51E+09 6648540 1276060

2480.99 86.43 5482.49 1.51E+09 6664720 1292420

2481.99 87.43 5482.53 1.51E+09 6680900 1308600

2482.99 88.43 5482.57 1.51E+09 6697080 1324660

2484 89.44 5482.6 1.51E+09 6713260 1340650

2485 90.44 5482.64 1.51E+09 6729440 1356610

2486 91.44 5482.68 1.51E+09 6745610 1372540

2487 92.44 5482.72 1.51E+09 6761780 1388440

2488 93.44 5482.76 1.51E+09 6777940 1404320

2489 94.44 5482.79 1.51E+09 6794110 1420190

2490 95.44 5482.83 1.51E+09 6810270 1436040

2491 96.44 5482.87 1.51E+09 6826420 1451890

2492 97.44 5482.91 1.51E+09 6842570 1467730

2493 98.44 5482.94 1.51E+09 6858720 1483570

2494 99.44 5482.98 1.51E+09 6874870 1499400

2495 100.44 5483.02 1.51E+09 6891010 1515220

2496 101.44 5483.06 1.51E+09 6907150 1531040

2497.01 102.45 5483.1 1.51E+09 6923290 1546860

2498.01 103.45 5483.13 1.51E+09 6939420 1562670

141

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