geomechanics of coal-gas interactions: the role of coal ... · geomechanics. these include...

Geomechanics of Coal-Gas Interactions: The Role of Coal Permeability Evolution

by

Zhongwei Chen BEng

This thesis is presented for the degree of Doctor of Philosophy

of

The University of Western Australia

Petroleum Engineering School of Mechanical and Chemical Engineering

June 2012

School of Mechanical and Chemical Engineering

Abstract The University of Western Australia

I

Abstract

Complex interactions between stress and sorptive chemistry exert strong influence on coal

geomechanics. These include influences on gas sorption and flow, coal deformation,

porosity change and permeability modification. In this study, this chain of reactions is

labelled as ―coupled processes‖ implying that one physical process affects the initiation and

progress of another. The evolution of coal permeability is probably the most important

cross coupling to rigorously formulate the Geomechanics of coal-gas interactions.

There are an extensive suite of coal permeability models available in the literature – with

many of these models implemented into computer simulators to quantify coal-gas

interactions. The comparison of laboratory and field observations against the spectrum of

models indicate that current models have so far failed to explain the results from stress-

controlled shrinkage/swelling laboratorial tests and have only achieved some limited

success in explaining and matching in situ data. Almost all the permeability models are

derived for the coal as a porous medium, but used to explain the compound behaviours of

coal matrix and fracture. These review conclusions suggest that the impact of coal matrix-

fracture compartment interactions on the evolution of coal permeability has not yet been

understood well and further improvements are necessary. This knowledge gap defines the

goal of this study.

The issue as defined above has been addressed through an integrated approach of

experimental study, permeability model development, its implementation into a FE

simulator, and applications to field operations. Major findings are summarized as follows:

A series of gas flow-through experiments were conducted to quantify the sole influence

of the effective stress and sorption-induced strain on permeability evolution.

Experimental results demonstrate that the effective stress coefficient cannot be assumed

to be unity as generally done, and that the variations of the effective stress coefficient


Abstract The University of Western Australia

II

under the influence of different gasses play an increasingly important role in the

determination of coal permeability evolution.

Based on the experimental observations, a phenomenological permeability model was

developed to quantify coal permeability reduction under the stress controlled conditions.

This new permeability model combines the effect of swelling strain with that of the

mechanical effective stress through the concepts of nature strain and partition ratio of

total swelling strain. A directional permeability model was also introduced to define the

evolution of gas sorption-induced permeability anisotropy under the full spectrum of

mechanical conditions. The model results are consistent with the experimental data or

field observations. Both models have been implemented into a fully coupled finite

element model to recover the important non-linear responses due to effective stress and

coal matrix shrinkage/swelling effects, where mechanical influences are rigorously

coupled with the gas transport system.

The role of heterogeneity on permeability evolution was then numerically conducted

under the unconstrained swelling conditions. The heterogeneous distributions of

Young's modulus and Langmuir strain constant in the vincity of the fracture were

included in the numerical model, and a net reduction of coal permeability was achieved

from the initial no-swelling state to the final equilibrium state and a good agreement was

obtained with laboratorial data under same conditions.

A dual-porosity/dual-permeability system was incorporated to investigate the effects of

coal properties, particularly the sorption time and shape factor, on the matrix-fracture

interactions for the dual-permeability system during CBM extraction. In this numerical

model, the complex multiphysics processes were coupled together with variable

permeability through the mass exchange term given as a function of sorption time,

pressure difference between matrix and fracture systems, matrix porosity, and matrix

particle volume. The sensitivity of the reservoir behavior to each parameter was

evaluated in details.

These results have been compiled into seven papers: four of them have been published in

International Journal of Coal Geology (IJCG) and International Journal of Greenhouse

Gas Control (IJGGC), one was submitted to 2011 Asia Pacific Coalbed Methane

Symposium, two are under review. This thesis is a compilation of these papers.


Table of Contents The University of Western Australia

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Table of Contents

ABSTRACT ....................................................................................................................................... I

TABLE OF CONTENTS .............................................................................................................. III

ACKNOWLEDGEMENTS .......................................................................................................... VII

PUBLICATIONS ARISING FROM THIS THESIS .................................................................. VIII

STATEMENT OF CANDIDATE CONTRIBUTION (%) ...........................................................XI

CHAPTER 1 .................................................................................................................................. 1-1

INTRODUCTION ................................................................................................................................................................ 1-1

1.1 BACKGROUND ....................................................................................................................................................... 1-1

1.2 PERMEABILITY MODEL DEVELOPMENTS ......................................................................................................... 1-3

1.2.1 PERMEABILITY MODELS UNDER UNIAXIAL STRAIN CONDITION ............................................. 1-3

1.2.2 PERMEABILITY MODELS UNDER VARIABLE STRESS CONDITIONS ........................................... 1-8

1.2.3 ANISOTROPIC PERMEABILITY MODELS ...................................................................................... 1-11

1.2.4 DUAL-POROSITY/DUAL-PERMEABILITY MODELS ..................................................................... 1-14

1.3 SIMULATION OF COUPLED MULTIPLE PROCESSES ...................................................................................... 1-14

1.3.1 ONE-WAY COUPLING ...................................................................................................................... 1-15

1.3.2 LOOSE COUPLING............................................................................................................................. 1-16

1.3.3 FULL COUPLING ................................................................................................................................ 1-18

1.4 RESEARCH GOALS .............................................................................................................................................. 1-19

1.5 THESIS OUTLINE .................................................................................................................................................. 1-20

1.6 REFERENCES......................................................................................................................................................... 1-21

CHAPTER 2 ................................................................................................................................. 2–1

LABORATORY CHARACTERISATION OF FLUID FLOW IN COAL WITH RESPECT TO GAS TYPE AND

TEMPERATURE ................................................................................................................................................................. 2–1

2.1 INTRODUCTION .................................................................................................................................................... 2–3

2.2 EXPERIMENTAL .................................................................................................................................................... 2–4

2.2.1 EXPERIMENT APPARATUS DESCRIPTION ..................................................................................... 2–4

2.2.2 ADSORPTION MEASUREMENT ....................................................................................................... 2–5

2.2.3 PERMEABILITY MEASUREMENT ..................................................................................................... 2–6

2.2.4 CLEAT COMPRESSIBILITY ................................................................................................................ 2–7

2.2.5 SWELLING MEASUREMENT ............................................................................................................. 2–8



IV

2.2.6 YOUNG’S MODULUS AND POISSON’S RATIO .............................................................................. 2–8

2.3 RESULTS AND DISCUSSION ............................................................................................................................... 2–9

2.3.1 ADSORPTION ISOTHERM RESULTS ................................................................................................ 2–9

2.3.2 SWELLING..........................................................................................................................................2–10

2.3.3 PERMEABILITY STRESS BEHAVIOUR ............................................................................................2–11

2.3.4 CLEAT COMPRESSIBILITY ...............................................................................................................2–12

2.3.5 GEOMECHANICAL PROPERTIES ....................................................................................................2–14

2.4 CONCLUSION ......................................................................................................................................................2–16

2.5 ACKNOWLEDGEMENT ......................................................................................................................................2–16

2.6 REFERENCES ........................................................................................................................................................2–17

CHAPTER 3 .................................................................................................................................. 3-1

LINKING GAS-SORPTION INDUCED CHANGES IN COAL PERMEABILITY TO DIRECTIONAL STRAINS

THROUGH A MODULUS REDUCTION RATIO ............................................................................................................ 3-1

3.1 INTRODUCTION ..................................................................................................................................................... 3-2

3.1.1 EXPERIMENTAL OBSERVATIONS .................................................................................................... 3-3

3.1.2 PERMEABILITY MODELS ................................................................................................................... 3-4

3.1.3 THIS STUDY ......................................................................................................................................... 3-5

3.2 APPROACH ............................................................................................................................................................. 3-6

3.2.1 COAL DEFORMATION ANALYSIS .................................................................................................... 3-7

3.2.2 FLOW AND TRANSPORT ANALYSIS ................................................................................................ 3-8

3.2.3 COAL PERMEABILITY ANALYSIS ..................................................................................................... 3-9

3.2.4 COUPLED MODEL ........................................................................................................................... 3-13

3.3 UNIAXIAL STRAIN CONDITION ..................................................................................................................... 3-14

3.4 DISPLACEMENT CONTROLLED CONDITION ............................................................................................... 3-16

3.5 FIELD CASE .......................................................................................................................................................... 3-17

3.6 EVALUATION OF COUPLED PROCESSES ....................................................................................................... 3-19

3.7 CONCLUSIONS .................................................................................................................................................... 3-21

3.8 ACKNOWLEDGEMENTS .................................................................................................................................... 3-22

3.9 REFERENCES ........................................................................................................................................................ 3-22

CHAPTER 4 .................................................................................................................................. 4-1

EFFECT OF THE EFFECTIVE STRESS COEFFICIENT AND SORPTION-INDUCED STRAIN ON THE EVOLUTION

OF COAL PERMEABILITY: EXPERIMENTAL OBSERVATIONS .................................................................................. 4-1

4.1 INTRODUCTION .............................................................................................................................................. 4-2

4.1.1 EXPERIMENTS ON COAL SWELLING/SHRINKAGE AND PERMEABILITY CHANGE ................... 4-2

4.1.2 MESUREMENTS OF EFFECTIVE STRESS COEFFICIENT ................................................................. 4-3

4.1.3 OBJECTIVE OF THIS STUDY ............................................................................................................... 4-6



V

4.2 EXPERIMENTAL METHODS ................................................................................................................................. 4-6

4.2.1 EXPERIMENT APPARATUS ................................................................................................................ 4-6

4.2.2 PERMEABILITY MEASUREMENT ...................................................................................................... 4-8

4.2.3 EXPERIMENTAL PROCEDURE .......................................................................................................... 4-8

4.2.4 WORK FLOW OF DATA ANALYSIS.................................................................................................... 4-9

4.3 RESULTS AND DISCUSSION ............................................................................................................................... 4-10

4.3.1 RESULTS FOR HELIUM INJECTION ................................................................................................ 4-10

4.3.2 RESULTS FOR METHANE INJECTION............................................................................................. 4-14

4.3.3 RESULTS FOR CARBON DIOXIDE INJECTION ............................................................................... 4-16

4.4 DISCUSSION ......................................................................................................................................................... 4-19

4.5 CONCLUSIONS ..................................................................................................................................................... 4-22

4.6 ACKNOWLEDGMENTS ....................................................................................................................................... 4-23

4.7 REFERENCES......................................................................................................................................................... 4-23

CHAPTER 5 .................................................................................................................................. 5-1

INFLUENCE OF THE EFFECTIVE STRESS COEFFICIENT AND SORPTION-INDUCED STRAIN ON THE

EVOLUTION OF COAL PERMEABILITY: MODEL DEVELOPMENT AND ANALYSIS .............................................. 5-1

5.1 INTRODUCTION ..................................................................................................................................................... 5-3

5.2 PERMEABILITY MODEL DEVELOPMENT ........................................................................................................... 5-5

5.2.1 EVALUATION OF EFFECTIVE STRESS EFFECTS .............................................................................. 5-6

5.2.2 EVALUATION OF SORPTION-INDUCED STRAIN EFFECTS ........................................................... 5-9

5.2.3 DEVELOPMENT OF COAL PERMEABILITY MODEL ..................................................................... 5-11

5.2.4 PHYSICAL MEANING OF SENSITIVITY RATIO .............................................................................. 5-12

5.3 PERMEABILITY MODEL EVALUATION ............................................................................................................ 5-13

5.3.1 PERMEABILITY MODEL VERIFICATION ........................................................................................ 5-14

5.3.2 COMPARISON WITH OTHER PERMEABILITY MODELS ............................................................... 5-16

5.4 MODEL IMPLEMENTATION .............................................................................................................................. 5-20

5.4.1 MODEL DESCRIPTIONS ................................................................................................................... 5-21

5.4.2 SIMULATION RESULTS AND ANALYSIS ........................................................................................ 5-23

5.5 CONCLUSIONS ..................................................................................................................................................... 5-27

5.6 ACKNOWLEDGMENTS ....................................................................................................................................... 5-29

5.7 REFERENCES......................................................................................................................................................... 5-29

CHAPTER 6 .................................................................................................................................. 6-1

ROLES OF COAL HETEROGENEITY ON EVOLUTION OF COAL PERMEABILITY UNDER UNCONSTRAINED

BOUNDARY CONDITIONS .............................................................................................................................................. 6-1

6.1 INTRODUCTION ..................................................................................................................................................... 6-2

6.2 THEORETICAL EVALUATION OF COAL PERMEABILITY MODELS ............................................................... 6-5



VI

6.2.1 GENERAL COAL PERMEABILITY MODEL ........................................................................................ 6-5

6.2.2 EVALUATION OF PERMEABILITY MODEL UNDER TWO BOUNDARY CONDITIONS ................ 6-6

6.3 A HETEROGENEOUS MATRIX-FRACTURE INTERACTION MODEL ........................................................... 6-11

6.3.1 NUMERICAL MODEL IMPLEMENTATION .................................................................................... 6-11

6.3.2 PERFORMANCE FOR A HOMOGENEOUS COAL .......................................................................... 6-13

6.3.3 PERFORMANCE FOR A HETEROGENEOUS COAL ....................................................................... 6-17

6.4 VERIFICATION WITH EXPERIMENTAL DATA ............................................................................................... 6-24

6.5 CONCLUSIONS .................................................................................................................................................... 6-26


6.7 REFERENCES ........................................................................................................................................................ 6-27

CHAPTER 7 .................................................................................................................................. 7-1

IMPACT OF VARIOUS PARAMETERS ON THE PRODUCTION OF COALBED METHANE ................................... 7-1

7.1 INTRODUCTION ..................................................................................................................................................... 7-3

7.2 METHODOLOGY .................................................................................................................................................... 7-6

7.2.1 COAL DEFORMATION ........................................................................................................................ 7-7

7.2.2 COAL PERMEABILITY MODEL ........................................................................................................... 7-9

7.2.3 MASS EXCHANGE FUNCTION .......................................................................................................... 7-9

7.2.4 SHAPE FACTOR FOR DUAL-PERMEABILITY MODEL ................................................................ 7-11

7.2.5 GAS FLOW EQUATIONS .................................................................................................................. 7-12

7.2.6 GAS PRODUCTION RATE ............................................................................................................... 7-13

7.3 EVALUATION OF GAS PRODUCTION ............................................................................................................. 7-15

7.3.1 MODEL DESCRIPTION .................................................................................................................... 7-15

7.3.2 MODELLING STRATEGY ................................................................................................................. 7-16

7.3.3 RESERVOIR BEHAVIOUR UNDER CONSTANT TOTAL PRODUCTIVE GAS VOLUME ............. 7-17

7.3.4 RESERVOIR BEHAVIOUR UNDER VARIABLE TOTAL PRODUCTIVE GAS VOLUME ............... 7-21

7.4 FIELD APPLICATION OF DUAL-PERMEABILITY MODEL ............................................................................. 7-27

7.5 CONCLUSION ...................................................................................................................................................... 7-29


7.7 REFERENCES ........................................................................................................................................................ 7-31

CHAPTER 8 .................................................................................................................................. 8-1

CONCLUDING REMARKS ............................................................................................................................................... 8-1

8.1 MAIN FINDINGS .................................................................................................................................................... 8-1

8.2 RECOMMENDATIONS FOR FUTURE WORK ...................................................................................................... 8-4


Acknowledgements The University of Western Australia

VII

Acknowledgements

I would like to express my deep and sincere gratitude to my supervisor, Professor Jishan

Liu, who invited me to Australia and supported me persistently during the period of this

research. Prof. Liu was always there to listen and give advice, which enabled my research

work to move forward continuously. Many of the ideas in this thesis would not have taken

shape without his incisive thinking and insightful suggestions. What I learned from him will

benefit me greatly in the rest of my life.

Many thanks go to my Co-supervisors Dr. Zhejun Pan and Dr. Luke D. Connell from

CSIRO for providing experimental facilities, financial support, and critical and insightful

suggestions of some of the papers involved in this thesis. Special thanks to Professor

Derek Elsworth from Pennsylvania State University, who has given me continued and

invaluable support and help in my study and life.

I am indebted to the research group members from School of Mechanical and Chemical

Engineering for their friendship and diverse help during my study, including Dr. Jianguo

Wang, Dr. Jian Li, Dr. Jianxin Liu, Dr. Yu Wu, Mr. Hamid Ghafram Al Shahri, Mr. Dong

Chen and Ms. Hongyan Qu. I have also loved working with colleagues from CSIRO Earth

Science and Resource Engineering (Clayton), and thanks to Dr. Meng Lu, Mr. David

Down, Mr. Greg Lupton, Mr. Nick Lupton and Mr. Michael Camilleri.

I would like to acknowledge the Scholarship for International Research Fees of University

of WA (SIRF), Western Australia- CSIRO postgraduate Scholarship (WACUPS), and

CSIRO National Flagship Top-Up Scholarship for providing the financial support to me to

pursue this study.

At last, I wish to express my sincere gratitude to my father Chunfu Chen, my mother Suqin

Zou, and my wife Nana Wang, for their constant love and inspiration. Without their

support, I could not have done it.


Publications Arising from This Thesis The University of Western Australia

VIII

Publications Arising From This Thesis

Journal papers

1. Chen, Z., Liu, J., Pan, Z., Connell, L.D., Elsworth, D., 2012 Effect of the Effective

Stress Coefficient and Sorption-Induced Strain on the Evolution of Coal

Permeability: Model development and analysis. International Journal of

Greenhouse Gas Control, 8, 101-110.

2. Zheng, G, Pan, Z., Chen, Z., Tang S., 2012. Laboratory study of gas permeability

and cleat compressibility for CBM/ECBM in Chinese Coals. Energy Exploration &

Exploitation 30(3), 451-476.

3. Qu, H., Liu, J., Chen, Z., Wang, J., 2012. Complex evolution of coal permeability

during CO2 injection under variable temperatures. International Journal of

Greenhouse Gas Control 9, 281-293.

4. Wang, J., Liu, J., Kabir, A., Chen, Z., 2012. Effects of Non-Darcy Flow on the

Performance of Coal Seam Gas Wells. International Journal of Coal Geology93, 62-

74.

5. Chen, Z., Pan, Z., Liu, J., Connell, L.D., Elsworth, D., 2011. Effect of the effective

stress coefficient and sorption-induced strain on the evolution of coal permeability:

Experimental observations. International Journal of Greenhouse Gas Control 5,

1284-1293.

6. Liu, J., Chen, Z., Elsworth, D., Qu, H., Chen, D., 2011. Interactions of multiple

processes during CBM extraction: A critical review. International Journal of Coal

Geology 87, 175-189.

7. Liu, J., Chen, Z., Elsworth, D., Miao, X., Mao, X., 2011. Evolution of coal

permeability from stress-controlled to displacement-controlled swelling conditions.

Fuel 90, 2987-2997.

8. Liu, J., Wang, J., Chen, Z., Wang, S., Elsworth, D., Jiang, Y., 2011. Impact of

transition from local swelling to macro swelling on the evolution of coal

permeability. International Journal of Coal Geology 88, 31-40.

9. Wu, Y., Liu, J., Chen, Z., Elsworth, D., Pone, D., 2011. A dual poroelastic model

for CO2-enhanced coalbed methane recovery. International Journal of Coal

Geology 86, 177-189.



IX

10. Liu, J., Chen, Z., Elsworth, D., Miao, X., Mao, X., 2010. Linking gas-sorption

induced changes in coal permeability to directional strains through a modulus

reduction ratio. International Journal of Coal Geology 83, 21-30.

11. Liu, J., Chen, Z., Elsworth, D., Miao, X., Mao, X., 2010. Evaluation of stress-

controlled coal swelling processes. International Journal of Coal Geology 83, 446-

455.

12. Chen, Z., Liu, J., Elsworth, D., Connell, L.D., Pan, Z., 2010. Impact of CO2

injection and differential deformation on CO2 injectivity under in-situ stress

conditions. International Journal of Coal Geology 81, 97-108.

13. Wu, Y., Liu, J., Elsworth, D., Chen, Z., Connell, L., Pan, Z., 2010. Dual poroelastic

response of a coal seam to CO2 injection. International Journal of Greenhouse Gas

Control 4, 668-678.

Selected conferences papers

1. Pan, Z., Chen, Z., Connell, L.D., Lupton, N.. Laboratory Characterization of Fluid

Flow in Coal for Different Gasses at Different Temperatures. 2011 Asia Pacific

Coalbed Methane Symposium, Brisbane, Australia.

2. Zhou, L., Feng, Q., Chen, Z., Liu, J., 2011. Modeling and Upscaling of Binary Gas

Coal Interactions in CO2 Enhanced Coalbed Methane Recovery. International

Conference on Environment Science and Engineering (ICESE 2011), Bali Island,

Indonesia.

3. Liu, J., Chen, Z., Qu, H., 2010. Multiphysics of Coal-Gas Interactions: The

Scientific Foundation for CBM Extraction. Asia Pacific Oil & Gas Conference and

Exhibition, Brisbane, Australia. SPE-133015-PP.

4. J.G. Wang, Liu, J., Liu, J., Chen, Z., 2010. Impact of rock microstructures on the

supercritical CO2 enhanced gas Recovery. International Oil & Gas Conference and

Exhibition, Beijing China. SPE-131759-PP.

5. Chen, Z., Liu, J., Pan, Z., Connell, L.D., Elsworth, D., 2010. Relations between

Coal Permeability and Directional Strains and Their Application to San Juan Basin.

44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics

Symposium, Salt Lake City, Utah, USA. ARMA 10-245.

6. Chen, Z., Liu, J., Elsworth, D., Connell, L.D., Pan, Z., 2009. Investigation of CO2

Injection Induced Coal-Gas Interactions. The 44th U.S. Rock Mechanics

http://www.spe.org/events/apogce/documents/exhibitorcontract.pdf




X

Symposium and 5th U.S.-Canada Symposium, Asheville, North Carolina, USA.

ARMA-09-099.

7. Wu, Y., Liu, J., Elsworth, D., Chen, Z., Connell, L., Pan, Z., 2009. Dual

poroelastic responses of coal to CO2 injection. The 44th U.S. Rock Mechanics

Symposium and 5th U.S.-Canada Symposium, Asheville, North Carolina, USA. No.

ARMA-09-164.

8. Chen, Z., Liu, J., Elsworth, D., Pan, Z., Connell, L.D., 2009. In-situ numerical

testing of CO2 sequestration in coal: Effects of confining stress and injection

pressure. 2009 Asia Pacific Coalbed Methane Symposium, Xuzhou, China.

9. Wu Y., Liu J., Chen Z., Elsworth D., Connell, L.D., 2009. In-situ numerical testing

of CO2 sequestration in coal: Dual poroelastic effects. 2009 Asia Pacific Coalbed

Methane Symposium, Xuzhou, China.

10. Liu J., Chen Z., Wu Y., Elsworth D., 2009. Multiphysics of Coal-Gas Interactions.

Seventh International Symposium on Rockburst and Mine Seismicity, Dalian,

China.

11. Chen, Z., Liu, J., Connell, L., Pan, Z., 2008. Impact of Effective Stress and CH4-

CO2 Counter-Diffusion on CO2 Enhanced Coalbed Methane Recovery. 2008 Asia

Pacific Oil & Gas Conference and Exhibition, Perth, Australia. SPE 116515.




Statement of Candidate Contribution (%) The University of Western Australia

XI

Statement of Candidate Contribution (%)

This thesis contains published work and/or work prepared for publication, which has

been co-authored. The bibliographical details of the work and where it appears in the

thesis are outlined below.

Liu, J. (30%), Chen, Z. (45%), Elsworth, D. (15%), Qu, H. (5%) and Chen, D. (5%), 2011.

Interactions of multiple processes during CBM extraction: A critical review. International

Journal of Coal Geology, 87(3-4): 175-189.

Pan, Z. (30%), Chen, Z. (40%), Connell, L.D. (20%) and Lupton, N. (10%). Laboratory

Characterisation of Fluid Flow in Coal for Different Gasses at Different Temperatures,

Asia Pacific Coalbed Methane Symposium, Brisbane, Australia, in May, 2011.

Liu, J. (30%), Chen, Z. (50%), Elsworth, D. (10%), Miao, X. (5%) and Mao, X. (5%), 2010.

Linking gas-sorption induced changes in coal permeability to directional strains through a

modulus reduction ratio. International Journal of Coal Geology, 83(1): 21-30.

Chen, Z. (70%), Pan, Z. (10%), Liu, J. (10%), Connell, L.D. (5%) and Elsworth, D. (5%),

2011. Effect of the effective stress coefficient and sorption-induced strain on the evolution

of coal permeability: Experimental observations. International Journal of Greenhouse Gas

Control, 5(5): 1284-1293.

Chen, Z. (70%), Liu, J. (10%), Pan, Z. (10%), Connell, L.D. (5%) and Elsworth, D. (5%).

Effect of the effective stress coefficient and sorption-induced strain on the evolution of

coal permeability: Model development and analysis. International Journal of Greenhouse

Gas Control (Accepted).

Chen, Z. (60%), Liu, J. (20%), Elsworth, D. (5%), Wang, J. (5%), Pan, Z. (5%), and

Connell, L.D. (5%). Roles of Coal Heterogeneity on Evolution of Coal Permeability under

Unconstrained Boundary Conditions.

Chen, Z. (50%), Kabir, A. (20%), Liu, J. (15%), Wang, J. (5%), Pan, Z. (5%), and Connell,

L.D. (5%). Impact of Various Parameters on the Production of Coalbed Methane.


Chapter 1 The University of Western Australia

1-1

Chapter 1

Introduction

Jishan Liua, Zhongwei Chena, Derek Elsworthb, Hongyan Qua, Dong Chena

a School of Mechanical and Chemical Engineering, the University of Western Australia, WA 6009, Australia

b Department of Energy and Mineral Engineering, Penn State University, PA 16802-5000, USA

1.1 Background

Advances in our understanding of coal-gas interactions have changed the manner in which

we treat coalbed methane: from mitigating its dangers as a mining hazard to developing its

potential as an unconventional gas resource recovered as a useful by-product of CO2

sequestration.

As found in nature, coal is a typical dual porosity/permeability system (Harpalani and

Schraufnagel, 1990; Lu and Connell, 2007; Warren and Root, 1963) containing porous

matrix surrounded by fractures. These natural fractures form a closely-spaced, orthogonal

network called cleats. The main set of fractures, termed face cleats, is comprised of well-

developed, extensive, roughly planar fractures that run parallel to one another. Butt cleats

are orthogonal to face cleats and often terminate at them. Butt cleats are also roughly

planar but are not as well-developed or as continuous as face cleats. The cleat system

provides an essential and effective flow path for gas. Much of the measured bulk or ―seam‖

permeability is due to the cleat system, although the presence of larger scale discontinuities

such as fractures, joints, and faults can also make a significant contribution. The coal

matrix is isolated by the fracture network and is the principal medium for storage of the gas

(of the order of 98%), principally in adsorbed form and with low permeability in

comparison to the bounding cleats (Gray, 1987). The remaining gas is stored in the natural

fractures, or cleats, either as free gas or dissolved in water. The surface area of the coal on

which the methane is adsorbed is very large (20 to 200 m2/g) (Patching, 1970) and gas is

stored at near-liquid densities.

Coalbed methane has shown the enormous potential in the past several decades. Major

reserves exist in many countries and more than 90% of the estimated reserves are in



1-2

Canada, Russia, China, the United States and Australia. Kuuskraa et al. (1992) defined

global CBM reserves through a detailed study of coalbed basins around the world. This

work was further updated by different researchers before the final form was presented

(Boyer, 1994; Kuuskraa et al., 1992; Murray, 1996; Palmer, 2008; White et al., 2005).

Estimates of the global coalbed methane (CBM) reserve defined in volume of CH4 are

summarized in Table 1-1.

Table 1-1. Coalbed methane reserves around the world

Country CBM Reserves (Tcf)

Boyer (1994) Murray (1996) Kuuskraa (1998) Palmer (2008)

Canada 200-2,700 300-4,260 570-2,280 200-2,700

Russia 600-4,000 600-4,000 550-1,550 600-4,000

China 1,060-1,240 1,060-2,800 350-1,150 1,060-1,240

United States 343-414 275-650 500-1,730 400

Australia 300-500 300-500 310-410 300-500

Indonesia - - 210 -

Germany 100 100 120 (Western Europe) 100

Poland 100 100 70 100

United Kingdom 60 60 - 60

Ukraine 60 60 50 60

Kazakhstan 40 40 40 40

Southern Africa* 30 40 100 30

India 30 30 90 30

Turkey - - 50 -

Total 2,953-9,304 2,976-12,640 3,010-7,840 2,980-9,260

*Includes South Africa, Zimbabwe and Botswana.

Compared to conventional gas reservoirs, coal reservoirs have low effective porosity and

high compressibility and are dominated by gas desorption. CBM recovery triggers a series

of coal-gas interactions. For primary gas production, the reduction of gas pressure

increases effective stress which in turn closes fracture apertures and reduces the

permeability. As the gas pressure reduces below the desorption point, methane is released

from the coal matrix to the fracture network and the coal matrix shrinks. As a direct



1-3

consequence of this matrix shrinkage the fractures may dilate (zero volume change

condition) and fracture permeability correspondingly increases. Thus a rapid initial

reduction in fracture permeability (due to an increase in effective stress) is supplanted by a

slow increase in permeability (indexed to matrix shrinkage). Whether the ultimate, long-

term, permeability is greater or less than the initial permeability depends on the net

influence of these dual competing mechanisms (Chen et al., 2008; Connell, 2009; Liu et al.,

2010b,c; Shi and Durucan, 2004). Therefore, understanding the transient characteristics of

permeability evolution in fractured coals is of fundamental importance to CBM recovery.

CBM extraction induced complex interactions between stress and sorptive chemistry exert

strong influence on the transport and sorptive properties of the coal. These include

influences on gas sorption and flow, coal deformation, porosity change and permeability

modification. We label this chain of reactions as ―coupled processes‖ implying that one

physical process affects the initiation and progress of another. The individual processes, in

the absence of full consideration of cross couplings, form the basis of very well-known

disciplines such as elasticity, hydrology and heat transfer. Therefore, the inclusion of cross

couplings is the key to rigorously formulate the behaviour for coupled processes of coal-

gas interactions. The complexity of these interactions is reflected in the extensive suite of

coal permeability models available in the literature– with many of these models

implemented into computer simulators to quantify coal-gas interactions. The primary goal

of this paper is to review and evaluate the performance of these disparate models of coal

permeability evolution and to define principal physical conditions where they, and their

application in simulators, can be most successful.

1.2 Permeability model developments

1.2.1 Permeability models under uniaxial strain condition

An equation for permeability and porosity of a collection of matchsticks are discussed by

Reiss (1980) together with an equation for collections of slabs and cubes. As the coal

deposit is idealized as a collection of matchsticks, flow in the core sample is along the axis

of the matchsticks. Permeability for this geometry is given by Reiss (1980) as

32

48

1fak (1-1)

where a is cleat spacing, and f is cleat porosity.



1-4

Differentiating with respect to hydrostatic stress and combining the relationship between

coal physical properties gives (Seidle et al., 1992; Pan et al., 2010)

f

h

cE

kk

3212

(1-2)

where fc is coal cleat compressibility defined as

p

f

f

1,

h is the horizontal stress. E

and are coal Young's Modulus and Poisson's ratio respectively.

The first term in parentheses represents the contribution of changes in the coal matrix to

the stress dependence of permeability. This term is analogous to grain compaction in

conventional reservoir rocks. The second term represents the cleat volume contribution to

the stress-permeability relation, which is normally two to three orders of magnitude larger

than the coal matrix term (Reiss, 1980). Therefore, simplifying and integrating the above

equation gives

0

0

3exp hhfck

k (1-3)

where 0h is the initial horizontal stress.

Gray (1987) considered the changes in the cleat permeability to be primarily controlled by

the prevailing effective horizontal stresses that act across the cleats. Under the assumption

of uniaxial strain, the influence of matrix shrinkage on changes in coal permeability was

first incorporated into a permeability model. The horizontal stress incorporating matrix

shrinkage was expressed as

s

s

shh p

p

Epp

1100 (1-4)

where sp refers to equivalent sorption pressure.

By assuming that an individual fracture reacts as an elastic body upon a change in the

normal stress component, Gilman and Beckie (2000) proposed a simplified mathematical

model of methane movement in a coal seam taking into account the following features: a

relatively regular cleat system, adsorptive methane storage, an extremely slow mechanism

of methane release from the coal matrix into cleats and a significant change of permeability

due to desorption. Using the uniaxial strain assumption and Terzaghi formula, the effective

stress in horizontal plane, h

e , was expressed as below, which is similar to Gray's (1987)

result:



1-5

SE

ph

e

11 (1-5)

where S is the change of the adsorbate mass and is the volumetric welling/shrinkage

coefficient.

The exponential relation was used for the permeability calculation

S

Ep

EEk

k

ff

h

e

11

3exp

3exp

0

(1-6)

where fE is an analogous Young's modulus for the fracture.

Seidle and Huitt (1995) calculated the permeability increase due to matrix shrinkage alone

by assuming that coal sorption-induced strain is proportional to the amount of gas sorbed

and that the sorbed gas is related to pressure by Langmuir's equation. Their porosity and

permeability models were defined as

pP

p

pP

p

LL

L

0

0

00

21

31

(1-7)

3

0

0

00

21

31

pp

p

pp

p

k

k

LL

L

(1-8)

where L and Lp are the maximum volumetric strain and gas pressure at which the matrix

strain is half of the maximum value, respectively.

This model considered the effects of coal-matrix swelling/shrinkage only, ignoring the

impact of coal compressibility. Therefore, their model is limited to specific conditions in

which sorption-induced strain (matrix swelling or shrinkage) dwarfs pressure-induced,

elastic changes in cleat permeability (Robertson 2005).

Based on the matchstick geometry model and the relation between permeability and

porosity developed by Seidle and Huitt (1995), Shi and Durucan (2004) presented a model

for pore pressure-dependent cleat permeability for gas-desorbing, linear elastic coalbeds

under uniaxial strain conditions. In this model, it was assumed that changes in the cleat

permeability of coalbeds were controlled by the prevailing effective horizontal stresses

normal to the cleats. Variations in the effective horizontal stresses under uniaxial strain

conditions are expressed as a function of pore pressure reduction during drawdown, which

includes a cleat compression term and a matrix shrinkage term that have competing effects

on cleat permeability, as expressed below



1-6

LL

Lhhpp

p

pp

pEpp

0

000

131

(1-9)

00 3exp hhfckk (1-10)

Based on the theory of linear elasticity for strain changes, Palmer and Mansoori (1996)

developed another widely used theoretical coal permeability model as a function of pore

pressure and matrix shrinkage. In this model, the incremental pore volume strain, pd , can

be defined as a result of a simple volumetric balance between the bulk rock, the grains, and

the pores

gr

p dd

d

1 (1-11)

where rd is the incremental rock volume strain, gd and are incremental grain volume

strain and porosity, respectively.

By assuming the uniaxial strain condition, 1 , and no change in overburden stress

results in

0

0

0

0

00

11pP

p

pP

p

M

Kpp

c

LL

Lm

(1-12)

The cubic relation between porosity and permeability was used for this derivation, as

shown below

3

0

0

0

0

00

11

pP

p

pP

p

M

Kpp

c

k

k

LL

Lm

(1-13)

where '1

1

f

M

K

Mcm ,

211

1

M , ' is grain compressibility and f is a

fraction between 0 and 1.

An improved P-M model has been developed, and is summarized in Palmer et al. (2007).

The model now includes (1) cleat anisotropy and potential suppression of pressure-

dependent permeability, (2) modulus changes with depletion, and (3) undersaturated coals.

Similarly, the Advanced Resources International (ARI) group developed another

permeability model (Pekot and Reeves, 2002). This model does not have a geomechanics

framework, but instead extracts matrix strain changes from a Langmuir curve of strain

versus reservoir pressure, which is assumed to be proportional to the gas concentration

curve. The matrix shrinkage is proportional to the adsorbed gas concentration change,

multiplied by shrinkage compressibility mC (a free parameter). The ARI model has been



1-7

compared to the P-M model, and the conclusion was that the two models are essentially

equivalent in saturated coals, and where the strain versus pressure function is proportional

to the Langmuir isotherm (Palmer et al., 2007).

Following the above work, Cui and Bustin (2005) investigated quantitatively the effects of

reservoir pressure and sorption-induced volumetric strain on coal-seam permeability with

constraints from the adsorption isotherm and associated volumetric strain and derived a

stress-dependent permeability model. Initially the authors used poroelasticity to achieve the

relation between porosity change and effective stress change, as shown below

00

0

1exp pp

K p

(1-14)

where pK is the bulk modulus for pore system.

The cubic relation between permeability and porosity was used to calculate coal cleat

permeability change.

00

3

00

3exp pp

Kk

k

p

(1-15)

By assuming constant overburden stress and uniaxial strain conditions, then this

permeability model was extended to be

00

0 19

2

13

13exp ss

p

Epp

Kk

k

(1-16)

Pan and Connell (2007) developed a theoretical model for sorption-induced strain and

applied to single-component adsorption/strain experimental data. Clarkson (2008)

expanded this theoretical model to calculate the sorption-strain component of the P-M

model (Palmer et al., 2007). The expressions for sorption-induced strain and permeability

calculation are given as

s

s

s

s

ss

E

pxf

EpBRTL

21,)1ln( (1-17)

3

0

0

00

11

1

s

m

M

Kpp

c

k

k

(1-18)

wheresE is the modulus of the solid phase,

sv is Poisson's ratio for solid phase and s is

the density for solid phase. R is the gas constant (8.314J mol-1K-1), T is the temperature

(K), L is Langmuir sorption constant (mol/kg), and B is Langmuir pressure constant (Pa-1).



1-8

1.2.2 Permeability models under variable stress conditions

Robertson and Christiansen (2006) described the derivation of a new equation that can be

used to model the permeability behaviour of a fractured, sorptive-elastic medium, such as

coal, under variable stress conditions. The model is derived for cubic geometry rather than

matchstick geometry under biaxial or hydrostatic confining pressures, and it is also

designed to handle changes in permeability caused by adsorption and desorption of gasses

from the matrix blocks.

In this model, the effective porosity of the matrix block is assumed to be zero, leaving the

fracture system to provide the only interconnected void space. The permeability model was

expressed as

00

0

00

In3

)(219)0(exp1

3exppP

pP

pP

Ppp

E

ppc

k

k

L

L

L

LL

c

cf

(1-19)

where c is the change rate in fracture compressibility.

Based on the theory of poroelasticity, a general porosity and permeability model was

developed by Zhang et al., (2008), where the expression of permeability for the pore system

is defined as

3

0

0

0

0

11

1

SSS

Sk

k

m

m

(1-20)

where s

s

vK

pS and 0

00 s

s

vK

pS .

Similarly, Connell et al. (2010) presented two new analytical permeability model

representations for standard triaxial strain and stress conditions, derived from the general

linear poroelastic constitutive law, that include the effects of triaxial strain and stress for

coal undergoing gas adsorption induced swelling. A novel approach is presented to

distinguish between the sorption strain of the coal matrix, the pores (or cleats) and the bulk

coal.

Contrary to previous models developed for field conditions, their model mainly deals with

variable stress conditions commonly used during measurement of permeability in the

laboratory.

When experimental results from these tests are interpreted, a matchstick or cubic coal

model is typically assumed with the matrix blocks completely separated from each other in

a stacked structure. Under this assumption, matrix swelling will not affect coal fracture

permeability under conditions of constant confining (total) stress, because, for a given pore



1-9

pressure, p , the coal matrix swelling will result in block swelling, rather than changes in

fracture aperture (Liu and Rutqvist, 2010, Liu, et al., 2011; Connell et al., 2010). The

effective stress is also decoupled from matrix swelling due to the complete separation

between matrix blocks caused by through-going fractures. Therefore, the permeability

should not change, but this is not consistent with laboratory observations (Harpalani and

Chen, 1997; Pini et al., 2009; Pan et al., 2010), which show dramatic reduction in

permeability with the injection of an adsorbing gas. Liu and Rutqvist (2010) believed that in

reality coal matrix blocks are not completely separated from each other by fractures but

connected by the coal-matrix bridges, and developed a new coal-permeability model, which

explicitly considers fracture–matrix interaction during coal-deformation processes based on

the internal swelling stress concept. For example, the effective stress under uniaxial strain

conditions can be calculated by the following equations

inse

EP

11 (1-21)

efC

fin eE

112

0 (1-22)

where in is the internal swelling stress, and

0f is the fracture porosity. The above

coupled equations are solved to obtain the effective stress and strain.

An alternate reasoning has been applied by Liu et al. (2010a) on this issue, considering that

the reason for the above phenomena may be the internal actions between coal fractures

and matrix have not been taken into consideration. A model capable of replicating this

apparently anomalous behaviour is developed by considering the interactions of the

fractured coal mass where cleats do not create a full separation between adjacent matrix

blocks, but where solid rock bridges are present. The role of swelling strains is

accommodated both over contact bridges that hold cleat faces apart but also over the non-

contacting span between these bridges. The effects of swelling act competitively over these

two components: increasing porosity and permeability due to swelling of the bridging

contacts but reducing porosity and permeability due to the swelling of the intervening free-

faces.

The fracture permeability was expressed as

3

0

3

0

)1(11

sv

f

m

f

f R

b

b

k

k

(1-23)



1-10

where b and b are fracture aperture and fracture aperture change, respectively. v is the

volumetric strain.

This study also considered the resultant change in coal permeability, which combined the

outcome of the reduction in fracture opening due to coal matrix swelling and effective

stress change and the decrease in effective stress due to changes in fluid pressure and

confining stress for the matrix system, as

3

000

0

3

000

0

0

111

sv

f

m

fm

fm

m

m

fm

m R

kk

k

K

pR

kk

k

k

k

(1-24)

where 0f is initial fracture porosity, and

mR is elastic modulus reduction ratio, defined as

mEE . mE is Young's modulus for coal matrix.

0mk and 0fk are initial coal matrix

permeability and coal fracture permeability respectively. Subscripts m and f refer to matrix

and fracture system respectively.

Izadi et al. (2011) proposed a mechanistic representation of coal as a collection of

unconnected cracks in an elastic swelling medium. The cracks are isolated from each other

but swelling within a homogeneous but cracked continuum results in a reduction in crack

aperture with swelling, and a concomitant reduction in permeability. In the limit, this

behaviour reduces to a change in permeability defined as a fully constrained model (zero

volume change) as,

3

0

23

0

3

0

11

L

Ls

pp

p

lb

s

lb

s

k

k (1-25)

where l is the crack length, s the cleat spacing and b0 the initial aperture and eL is the

Langmuir strain coefficient.

Ma et al. (2011) developed a model, which was based on the volumetric balance between

the bulk coal, and solid grains and pores, using the constant volume theory (Massarotto et

al., 2009). It incorporates primarily the changes in grain and cleat volumes and is, therefore,

different from the other models that lay heavy emphasis on the pore volume/cleat

compressibility. In this study, the overall matchstick strain resulting from matrix shrinkage

and decrease in pressure is given as

0

0

0 111 pp

Epp

p

pp

p

a

a

LL

L

(1-26)

The permeability change can be calculated by the following expression



1-11

a

a

a

a

k

k

1

21

3

0

0

(1-27)

where a and a are the matrix width and width change, respectively. 0 is the porosity at

virgin reservoir pressure.

1.2.3 Anisotropic Permeability Models

The permeability models as reviewed above do not reflect the directional behaviour of

permeability change. The anisotropic characteristics of a coal matrix-fracture structure

suggest that the evolution of coal permeability should be direction-dependent. With cubic

coal cores, Pomeroy and Robinson (1967) found that the flow rates of water

(corresponding to permeability) were significantly different when the confining pressures

were perpendicular to main cleats (face cleats), cross cleats (butt cleats) or bedding planes.

From field well tests, Koenig and Stubbs (1986) reported the anisotropy ratio of

permeability in the plane of bedding was as high as 17:1 in the Rock Creek coalbeds of the

Warrior Basin of the USA. Permeability anisotropy of coal was also confirmed by other

experimental results of Gash et al (1992). Using coal samples from the San Juan Basin and

under a confining stress of 6.9 MPa (1000 psi) they found that the permeability parallel to

bedding planes was 0.6~1.7 mD in the direction of the face cleat and 0.3~1.0 mD in the

direction of the butt cleat, but only 0.007 mD in the direction vertical to the bedding planes.

A few of coal permeability models have been developed to accommodate the anisotropy, as

summarized below.

Wong (2003) developed a model for deformable granular media, which quantifies the

anisotropic changes in permeability when the material experiences shear deformation. In

this study, the directions of the principal permeability magnitudes are governed by the

induced strains, so the effects of stress paths and stress levels are implicitly considered

through effective stress-strain constitutive laws. This strain-induced permeability model is

written as:

3

2

1

'''

'''

'''

30

20

10

3

2

1

abb

bab

bba

k

k

k

k

k

k

(1-28)



1-12

where, 'a and 'b are material constants and can be experimentally measured; 0ik and

ik

denote initial and current permeability, respectively; i are current principal strains.

Following the above work, Al-Yousef (2005) presented an analytical solution for the

steady-state flow problem for anisotropic permeability measurements. Gu and Chalaturnyk

(2005) developed another permeability model. In this model, coalbeds are considered as

naturally fractured reservoirs, and represented with a collection of matchsticks. The

permeability is expressed as

3

0

1

li

i

b

a

k

k (1-29)

lTilDilPilEili (1-30)

where li is the directional effective strain and each term represents in order the

mechanical deformation due to stress change, the mechanical deformation due to pressure

change, matrix shrinkage/swelling due to desorption/sorption, and thermal

contract/expansion due to temperature changes.

Recently, they extended their work by considering discontinuous coal masses as an

equivalent elastic continuum. The implementation procedure of an explicit-sequential

coupled simulation using such permeability models in industrial simulators is complex but

feasible for coupled simulation in pressure depleting CBM reservoirs (Gu and Chalaturnyk,

2010). The total change of cleat aperture is defined as

3.0/uu tan

3.0/uu

ss

ss

pm

ff

p

f

ma

ab

(1-31)

The total change of matrix block is defined as

3.0/uu tan

3.0/uu

ss

ss

pt

Lm

m

ff

t

L

pt

Lmf

t

L

ba

baa

(1-32)

The following expression is used for the permeability calculation

j

j

n

jm

jm

i

i

a

a

b

b

k

k

j

1

1

3

0

(1-33)

where a is the width of the coal matrix block, f is the change of normal strain within the

fracture (cleat), su and p

su are shear displacements of the fracture and peak shear



1-13

displacement of the fracture respectively. f is change of the shear strain of a fracture,

and m is the mobilized dilation angle. t

L is the total change of linear strain of a

composite unit including a matrix block and a fracture, and mb is the mechanical aperture

of a fracture.

Wang et al. (2009) developed a model that incorporates the anisotropic structural and

mechanical properties to describe the directional permeability of coal. In this model, the

mechanical and non-mechanical deformations of coal under confined stress conditions that

imitate coal reservoirs are taken into account. The mechanical deformation is the stress-

dominated deformation that can be described using the general stress–strain correlation

and nonmechanical deformation is sorption-induced matrix swelling/shrinkage that was

treated using a thermal expansion/contraction analogy. A strain factor, depended on coal

properties and sorption characters such as coal type and rank, and sorbent gas, was

introduced to correct the strains theoretically obtained for better interpretations of

laboratory strain data under unconstrained conditions that are widely used for tests of coal

permeability.

Liu et al. (2010b) developed a permeability model to define the evolution of gas sorption-

induced permeability anisotropy under the full spectrum of mechanical conditions

spanning prescribed in-situ stresses through constrained displacement. In the model, gas

sorption-induced coal directional permeabilities are linked into directional strains through

an elastic modulus reduction ratio, which represents the partitioning of total strain for an

equivalent porous coal medium between the fracture system and the matrix. Verification of

this model has been conducted by et al. (2010b).

The directional permeability expression is defined as follows

ji

ej

f

m

i

i R

k

k3

00

)1(31

2

1

(1-34)

where 0f is the initial fracture porosity at reference conditions, zyxji ,,, .

Recently, Pan and Connell (2011) developed an anisotropic swelling model based on the

Pan and Connell (2007) swelling model, which applies an energy balance approach where

the surface energy change caused by adsorption is equal to the elastic energy change of the

coal solid. This new model also incorporated anisotropic coal properties.



1-14

1.2.4 Dual-porosity/dual-permeability models

Dual-permeability or multiple permeability models have been developed to represent the

porosity and permeability of all constituent components (Bai et al., 1993), including the role

of sorption (Bai et al., 1997), and of multiple fluids (Douglas et al., 1991). Moreover,

several models have been applied to represent the response of permeability evolution in

deforming aquifers and reservoirs (Elsworth and Bai, 1992; Ouyang and Elsworth, 1993;

Bai et al., 1995; Liu and Elsworth, 1997), to accommodate gas flow and other mechanical

influences (Zhao et al., 2004).

Wu et al. (2010a) developed a dual poroelastic model (dual solid media – coal matrix and

fracture) for single gas under variable stress conditions. The model allows exploration of

the full range of mechanical boundary conditions from invariant stress to restrained

displacement. Wu et al. (2010b) extended their previous work (Wu et al., 2010a) to define

the evolution of gas sorption-induced anisotropic permeability. In this study, dual

permeabilities are used which is different from Gu and Chalaturnyk's work (2010). The

expression of anisotropic permeability for cleat system is defined as

ji

eisTf

fi

i

KT

K

Kk

k

3

00

1

3

1

3

1

3

11

2

1

(1-35)

where 0f is the initial fracture porosity at reference conditions, zyxji ,,, . T ,

s ,

ei refer to the change in temperature, sorption-induced strain and mechanical effective

stress. The permeability model for the matrix system is same as that of Zhang et al. (2008).

1.3 Simulation of coupled multiple processes

Gas flow within coal seams differs significantly from that of conventional reservoirs.

Detailed studies have examined the storage and transport mechanisms of gas in coal seams.

In situ and laboratory data indicate that the storage and flow of gas in coal seams is

associated with the matrix structure of coal and the absorption or desorption of gas. Coal is

a naturally fractured dual-porosity reservoir, consisting of micro-porous matrix and cleats.

Most of the gas is initially stored within micro-pores in the absorbed state. When gas

recovery begins, the gas desorbs and diffuses from the matrix to the cleats due to the

concentration gradient. The rate of gas flow through the cleats is considered to be

controlled by the permeability of the coal seam. Gas flow within coal seams is a complex



1-15

physical and chemical process coupling solid deformation, gas desorption and gas

movement. The complexities of process interactions exert a strong control on ultimate

behaviour—these include linear physical interactions, but also the development of material

nonlinearities that irreversibly alter the affected media.

According to Minkoff et al. (2003), there are three basic algorithms for the simulation of

coupled processes: one-way coupling, loose coupling, and full coupling. For one-way

coupling, separate sets of equations are solved independently over the same total time

interval. Periodically, output from one simulator is passed as input to the other; however,

information is passed in only one direction. A loose coupling resides somewhere between

full and one-way coupling. In loose coupling, different sets of equations are solved

independently (as in one-way coupling), but information is passed at designated time

intervals in both directions between the simulators. For a full coupling, a single set of

equations (generally a large system of non-linear coupled partial differential equations)

incorporating all of the relevant physics needs to be solved simultaneously.

1.3.1 One-way coupling

Coal porosity and permeability models have a variety of forms when specific conditions are

imposed. When the change in total stress is equal to zero, 0 , both coal porosity and

permeability are independent of the total stress. Similarly, when the coal sample is under

the uniaxial strain condition and the overburden load remains unchanged, they are also

independent of the total stress. In this review, studies under these assumptions are

considered as one-way coupling.

Balla (1989) developed a mathematical modelling to simulate methane flow in a borehole

coal mining system, which considered both the sorption phenomenon of methane and, as a

consequence of this, a change in the permeability of the coal. Young (1998) used the

nonequilibrium and pseudosteady state formulations to simulate coalbed methane

production performance, in which the diffusion coefficient is considered to be dependent

on the geometry of the matrix elements and time. The stress-induced changes in cleat

porosity and permeability were included, and the matrix shrinkage due to release of

adsorbed gas are also considered. Similarly, Gilman and Beckie (2000) proposed a

simplified mathematical model of methane movement in a coal seam taking into account

the following features: a relatively regular cleat system, adsorptive methane storage, an



1-16

extremely slow mechanism of methane release from the coal matrix into cleats and a

significant change of permeability due to desorption.

Considering coal to be a triple-porosity system, the implementation of a bidisperse pore-

diffusion model in a coalbed reservoir simulator was discussed by Shi and Durucan (2005),

in which the gas adsorption is assumed to take place only in the micropores with

macropores providing storage for free gas, as well as tortuous paths for gas transport

between the micropores and cleats. Recently, Ross et al. (2009) presented a 3D stochastic

reservoir model to address gas buoyancy and leakage associated with CO2 injection in

coalbeds by using geostatistical techniques and history-matching. More recently, a

mathematical model was developed by Ozdemir (2009) to predict coal bed methane (CBM)

production and carbon dioxide (CO2) sequestration in a coal seam accounting for the coal

seam properties. It was assumed that the flow in a coal seam is a two-phase flow including

a water phase and a gas phase governed by Darcy's law while the flow in the coal matrix is

a diffusional flow governed by Fick's Law, but constant absolute permeability was used in

this study.

These prior studies did not accommodate geomechanical influences related to the role of

changes in total stress on performance. Zhao and Valliappan (1995) derived the governing

equations of methane gas migration in coal seams, which considered the effect of

deformation of the medium, two-phase flow and mass/gas transfer on methane transport

processes in porous media. The permeability magnitudes for both gas and water flow were

considered to be both the same and constant. This work was extended by Valliappan and

Wohua (1996), who presented the development of a mathematical model for methane gas

migration in coal seams, mainly focusing on the coupling between the gas flow and

deformation of solid coal. Anisotropic flow and the effect of diffusion of adsorbed

methane has been considered in this study, but assumed that the porosity of the coal seam

does not change when the gas pressure varies and was constant within an individual

element.

1.3.2 Loose coupling

When coal is under variable stress conditions and the impact of coal fractures and gas

compositions is considered, coal porosity and permeability models are defined as a function

of effective stress, coal matrix-fracture interactions, and gas compositions. Under these

conditions, important non-linear responses due to the effective stress effects need to be



1-17

recovered. This can be achieved through loose coupling. In loose coupling, different sets of

equations are solved independently (as in one-way coupling), but information is passed at

designated time intervals in both directions between the simulators.

The theory describing fluid-solid coupling was first presented by Biot (1941) for a linear

solid where deformation occurred but where no updating was applied to changes in

permeabilities due to infinitesimal changes in porosity. The original Biot theory is for a

single-fluid/single-solid model consistent with single porosity behaviour. Naturally

fractured reservoirs are often modelled by the dual-porosity (overlapping continua) type of

concept developed by Barenblatt et al. (1960). Models incorporating both Biot

poroelasticity and Barenblatt dual-porosity concepts have been studied by many authors

(Duguid and Lee, 1977; Valliappan and Khalili-Naghadeh, 1990; Chen and Teufel, 1997). A

mathematical model of coupled solid-gas for gas flow in coal seams is presented by Zhao et

al. (1994), but the permeability was considered to be constant and the sorption-induced

strain was not coupled in this study. Zhao et al. (2004) extended their work by considered

permeability is a function of volumetric stress and pore pressure to emphasize the coupled

interaction laws between solid deformations and gas seepage within the coal matrix and

fractures, but the influence of sorption-induced strain on permeability change was still not

considered. Gu and Chalaturnyk (2005, 2006) utilized the dynamic change of permeability

for geomechanical and reservoir explicit-sequential coupling simulations, where the

geomechanical simulation is implemented for generalized deformation and stress change

predictions, while multiphase flow is simulated with an appropriate reservoir simulator.

Recently, Gu and Chalaturnyk (2010) established new porosity and permeability models

used for reservoir and geomechanical coupled simulation, which considered a

discontinuous coal mass (containing cleats and matrix) as an equivalent continuum elastic

medium and the anisotropic permeability of coalbeds. Matrix shrinkage/swelling due to gas

desorption/adsorption, thermal expansion due to temperature change, and mechanical

parameters, are included in their work. Similar work has also been conducted by Wang et al.

(2010). Connell (2009) conducted a coupled numerical model and used it to investigate the

applicability of these geomechanical assumptions for gas drainage from coal seams. The

modelling approach involved coupling the existing coal seam gas reservoir simulator,

SIMED II, with the geomechanical simulator, FLAC3D. While SIMED II was used to

simulate gas migration in a hypothetical coal seam and a series of production scenarios,

FLAC3D simulated the geomechanical response of the coal and the adjacent non-coal



1-18

geological formations to fluid pressure and gas content changes imported from SIMED II.

Recently, this work was extended to CO2-ECBM (Connell and Detournay, 2009). Similarly,

Zuo-Tang et al. (2009) proposed a deformation-flow coupled model to address CO2-

geosequestration enhanced coal bed methane recovery. The permeability is considered to

be a function of effective stress, but the influence of the sorption-induced strain on

permeability was not coupled. The interaction between mechanical deformation and fluid

flow in fault zones was addressed by Cappa and Rutqvist (2010), and the TOUGH–FLAC

simulator was applied to supercritical CO2 injection, geomechanics, and ground surface

deformations. Liu et al. (2010c) performed a coupled reactive flow and transport modelling

to simulate large scale CO2 injection. The governing mathematical equation employed in

TOUGHREACT to describe geochemical processes involving fluid-rock interactions.

More Recently, considering the Klinkenberg effect, Wei and Zhang (2010) developed a

two-dimensional, two-phase, triple-porosity/dual-permeability, coupled fluid-flow and

geomechanics CBM simulator for modelling gas and water production, and the coupling

effects of effective stress and micro-pore swelling/shrinkage are modelled with the coupled

fluid-flow and geomechanical deformation approach.

1.3.3 Full coupling

In order to recover important non-linear responses due to the effective stress effects,

mechanical influences need to be rigorously coupled with the gas transport system. This

can be achieved through the full coupling approach. For full coupling, a single set of

equations (generally a large system of non-linear coupled partial differential equations)

incorporating all of the relevant physics will be solved simultaneously.

A coupled mathematical model for solid deformation and gas flow is proposed and is

implemented by Zhu et al. (2007). The finite element method was used to solve the

coupled processes together with Klinkenberg effect. The empirical permeability expression

obtained by Harpalani and Schraufnagel (1990) was used. Similarly, Zhang et al. (2008)

conducted another study on coupled gas flow and coal deformation processes

incorporating the newly developed permeability model, which considers the controlling

factors of the volume occupied by the free-phase gas, the volume occupied by the

adsorbed phase gas, the coal mechanical deformation induced pore volume change, and the

sorption induced coal pore volume change. Based on Zhang et al.'s work, equivalent

poroelastic models (Liu et al., 2010a,b) were developed to simulate the interactions of

multiple processes triggered by the injection or production of single gas. Chen et al. (2009,



1-19

2010a) extended these single poroelastic models to include the flow and transport of gas

mixtures (binary gasses: CO2 and CH4). Wu et al. (2010b, 2011) extended these models

further to a dual poroelastic model (dual solid media- coal matrix and fracture) for both

single gas and binary gas systems. Based on the variable saturation model, Liu and Smirnov

(2008) solved a set of related variables regarding CO2 sequestration in coalbeds, including

capillary pressure, relative permeability, porosity, coupled adsorption model, concentration

and temperature equations. With the same assumptions, the above work was extended to

address the importance of structural deformation effects on carbon sequestration

modelling, which affects the fluid flow and leads to a faster drop of the resulting capillary

pressure and relative permeability of the gas phase (Liu and Smirnov, 2009).

In summary, to define a fully coupled computer simulator for the full mechanics of coal-

gas interactions, a single set of equations (generally a large system of non-linear coupled

partial differential equations) incorporating all of the relevant physics need to be derived.

Full coupling is often the preferred method for simulating multiple types of physics

simultaneously since it should theoretically produce the most realistic results. This could

be the best approach to represent important non-linear responses due to the effective

stress effects when mechanical influences are rigorously coupled with the gas transport

system.

1.4 Research goals

This study was undertaken with the aims of:

i. Carrying out fundamental to gain understanding of the coal properties response,

coal permeability in particular, to variations in effective stress, temperature and gas

pressure and types.

ii. Developing a permeability model to define the evolution of gas sorption-induced

permeability anisotropy.

iii. Conducting a series of gas flow-through experiments to investigate the sole

influence of effective stress and sorption-induced strain on permeability change.

iv. Developing a phenomenological permeability model to quantify coal permeability

evolution under the unconstrained conditions of variable stress by combining the

effect of swelling strain with that of the mechanical effective stress.



1-20

v. Studying the role of heterogeneity on permeability evolution under the

unconstrained swelling conditions.

vi. Conducting a finite element (FE) numerical model with the dual-porosity/dual-

permeability geometry to quantify the impact of various parameters on the CBM

production.

1.5 Thesis outline

This thesis comprises eight chapters. The seven chapters following this introductory

chapter are arranged as follows:

Fundamental experiments is conducted in Chapter 2 to gain understanding of the coal

properties response, coal permeability in particular, to variations in effective stress,

temperature, gas pressure and types. Chapter 3 presents a permeability model to define the

evolution of gas sorption-induced permeability anisotropy under the full spectrum of

mechanical conditions spanning prescribed in-situ stresses through constrained

displacement.

A series of gas flow-through experiments were conducted to quantify the sole influence of

the effective stress and sorption-induced strain on permeability evolution in Chapter 4, and

based on the observations, a phenomenological permeability model is developed in

Chapter 5 to explain this enigmatic behavior of coal permeability evolution under the

influence of gas sorption by combining the effect of swelling strain with that of the

mechanical effective stress.

In Chapter 6, the role of heterogeneity on permeability evolution is numerically conducted

under the unconstrained swelling conditions with the heterogeneous distributions of

Young's modulus and Langmuir strain constant in the vincity of the fracture. In Chapter 7,

a dual-porosity/dual-permeability system is incorporated to quantify the impact of various

parameters on the CBM production, particularly the sorption time and shape factor.

Finally, Chapter 8 summarizes the main outcomes of this research, along with suggestions

for future studies.



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2–1

Chapter 2

Laboratory Characterisation of Fluid Flow in Coal with respect to Gas Type and Temperature

Zhejun Pana, Zhongwei Chenb, Luke D. Connella, and Nicolas Luptona

a CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South 3169, Australia

b School of Mechanical and Chemical Engineering, The University of Western Australia, WA 6009, Australia

Abstract: Coalbed methane is an important unconventional gas resource. However, a

significant amount of the gas resource can not be produced through pressure drawdown

due to the nature of gas storage through adsorption. Enhanced coalbed methane via

injecting gasses such as carbon dioxide and nitrogen is regarded as a viable option to

increase the recovery of the reservoir methane. Injecting CO2 in deep, unminable coal

seams is also considered an option for CO2 sequestration to reduce greenhouse gas

emissions. Often the gas is injected at a different temperature to the seam, for instance, a

hot flue gas injection to the coal seam to enhance coalbed methane recovery. Thus a locally

temperature affected zone will form around the injection well. Although the affect of

temperature on gas adsorption in coal is well understood, how temperature affects the gas

effective permeability for different gasses and the overall gas flow behaviour has not been

extensively studied. In this work, laboratory measurements are carried out on an Australian

coal sample using Helium, N2, CH4 and CO2 at two temperatures, 35 ºC and 40 ºC. Gas

adsorption isotherms, sorption-induced coal swelling isotherms, effective gas permeability

under tri-axial conditions with respect to pore and confining pressures and geomechanical

properties are measured. The experimental results show that the amount of gas adsorbed at

35 ºC is slightly higher than that at 40 ºC for the same pressure for all three adsorptive

gasses, N2, CH4 and CO2. This is consistent with other experimental studies presented in

the literature. Adsorption induced coal swelling for the same pressures is also slightly

higher at 35 ºC and is in accord with gas adsorption behaviour. When the swelling strain is

plotted with respect to the quantity adsorbed, the measurements for the various gasses at

different temperatures tend to fall onto a single trendline, which is almost linear.

Geomechanical properties, which are different for each gas, however, do not show a

significant difference between the two temperatures. Nevertheless, the effective gas



2–2

permeability tends to be higher at higher temperature for all gasses. More importantly, cleat

compressibility, a key property in determining coal reservoir permeability behaviour, is the

least when measured with N2 and the largest using CO2 among the three gasses. This may

suggest that the cleat compressibility is correlated with adsorption and adsorption-induced

swelling. Part of the reason for this may be attributed to the impact of adsorption induced

coal swelling on the cleat porosity. In summary, these measurements show that gas species

and temperature may have a significant impact on gas flow behaviour in coal. Hence,

injected gas composition and temperature should be considered as parameters in the CO2/N2

injection strategy to optimize the ECBM process.

Keywords: coalbed methane; enhanced coalbed methane; cleat compressibility;

permeability; swelling; adsorption



2–3

2.1 Introduction

Coalbed methane (CBM) or coal seam gas (CSG) is an important unconventional gas

resource. CBM production is mainly by reservoir pressure drawdown and CBM reservoir

permeability is considered the most important reservoir property for CBM production

(Palmer, 2009; Sparks et al., 1995). However, a substantial amount of methane is expected

to be left behind by the primary recovery because reduction of reservoir pressure to lower

than 150 psia (~1 MPa) is generally not considered as practical or economic (Puri and Lee,

1990). Hence, gasses, such as N2 and CO2, can be injected as a strategy to enhance coalbed

methane recovery. Furthermore, Coal is able to adsorb more CO2 than methane at the

same pressure thus it is also considered a viable option to reduce greenhouse gas emissions.

Coal permeability is sensitive to stress (Seidle et al., 1992). Moreover, a unique

characteristic for coal reservoirs is that the coal matrix shrinks as gas desorbs from the coal

and swells as gas adsorbs into the coal matrix, leading to further permeability change under

reservoir conditions. There have been several ECBM trials so involving CO2 injection

(Fujioka et al, 2010; Reeves and Oudinot, 2005; van Bergen et al., 2006; Wong et al., 2007).

A common aspect of these trials has been that CO2 injectivity has been observed to

decrease due to permeability loss as a result of adsorption induced coal swelling. In some

trials CO2 has been injected as liquid (Wong et al., 2007). However, since liquid CO2 has a

high viscosity, which acts to reduce injection rates, it is heated and then injected as

supercritical CO2 which has a lower viscosity (Fujioka et al., 2010). Furthermore, N2 or flue

gas, a mixture of N2 and CO2, can also be injected to enhance coalbed methane recovery.

For all these injection scenarios the injected gas is likely to be at a contrasting temperature

to the seam. While the affects of temperature on gas adsorption in coal is well understood,

how temperature affects the gas effective permeability and the overall gas flow behaviour

has not been extensively studied.

A number of permeability models have been developed to include both the effective stress

and swelling/shrinkage effects. For instance, one of the widely applied models, the Shi and

Durucan (S-D) model, can be expressed as:

13100

VEPP (2-1)



2–4

where is the effective horizontal stress, 0 is the effective horizontal stress at the initial

reservoir pressure, V is the volumetric swelling/shrinkage strain (Shi and Durucan, 2004).

To relate the permeability with effective stress, the equation below is used:

03

0

fc

ekk (2-2)

where fc is referred to as the cleat volume compressibility with respect to changes in the

effective horizontal stress normal to the cleats (Shi and Durucan, 2004). From the above

equations, the coal geomechanical properties, E and , the swelling ratio, V , and the

cleat compressibility, fc , are important parameters to determine the behaviour of reservoir

permeability.

In this work, laboratory measurements were performed on an Australian coal sample from

the Woodland Hill coal seam, NSW, using Helium, N2, CH4 and CO2 at two temperatures,

35 ºC and 40 ºC. Gas adsorption isotherms, sorption-induced coal swelling isotherms, gas

effective permeability under tri-axial conditions with respect to the pore and confining

pressures and geomechanical properties are measured. Overall gas flow behaviour under

different temperatures is analysed and discussed.

2.2 Experimental

2.2.1 Experiment apparatus description

A triaxial permeability cell was used for the experimental measurement of gas adsorption

and permeability under hydrostatic conditions. Figure shows the schematic of the Triaxial

Multi-Gas Rig used for this work. Radial and axial displacements are measured at each

adsorption step to obtain swelling strain. Four displacement gauges are installed with two

to measure the axial displacement and the other two to measure the radial displacement.

The two radial displacement gauges was installed perpendicularly. The displacement gauges

are not presented in Figure 2-1 to keep this figure concise. Load and displacement tests are

performed at each pore and confining pressure to evaluate the impact of gas species and

applied stress on geomechanical properties. The load change is up to 1MPa for the load

and displacement tests. The core sample, usually 5cm in diameter and 10 to 15 cm in length,

is wrapped with a thin lead foil then a rubber sleeve before it is installed in the cell. The

thin lead foil is to prevent gas diffusion from the core to the confining fluid at high sample



2–5

pressures (Mazumder et al., 2006). The sample cell and other parts of the rig are in a

temperature controlled cabinet to maintain constant temperature during the experiment.

Figure 2-1. Schematic of the triaxial rig.

2.2.2 Adsorption measurement

Prior to gas adsorption, the void volume, voidV , in the cell is determined by injecting

known quantities of helium from a calibrated gas injection pump. Since helium adsorption

is negligible on coal, the void volume can be determined from measured values of

temperature, pressure and the helium volume injected into the cell. This helium void

volume measurement was performed at a range of pressures to investigate the consistency

of the calculated volume. The mass-balance equation, expressed in volumetric terms, is:

cellpump

voidTZ

P

TZ

P

ZT

VPV

1

1

2

2 (2-3)

where V is the volume injected from the gas injection pump, Z is the compressibility

factor of helium, T is the temperature, P is the pressure, subscripts ―cell‖ and ―pump‖ refer

Upstream

CH4

He

CO2

Downstream

CO2 Tube Heater

Upstream Cylinder Downstream

Cylinder Actuator

111.1 122.2

Vaccum

111.1 122.2

Pressure Sensor

Pressure Sensor

Pressure Sensor

Pressure Sensor

Pressure Sensor

Pressure Sensor

Pressure Sensor

Upstream Injection Pump

Confinement Pump

COMPUTER / DAQ

Water Heater

Differential Pressure transducer



2–6

to conditions in the cell and pump, respectively, and subscripts ―1‖ and ―2‖ refer to

conditions in the cell before and after injection of gas from the pump, respectively. This

void volume is used in subsequent measurements of adsorption.

The Gibbs excess adsorption (also known as the excess adsorption) is calculated directly

from the experimental quantities. For pure-gas adsorption measurements, a known

quantity, injn , of gas (e.g., methane) is injected from the gas injection pump into the cell.

Some of the injected gas will be adsorbed, and the remainder, voidn , will exist in the

equilibrium bulk (gas) phase in the cell. A mass balance is used to calculate the amount

adsorbed, Gibbs

adsn , as:

voidinj

Gibbs

ads nnn (2-4)

The amount injected can be determined from pressure, temperature and volume

measurements of the pump:

pump

injZRT

VPn

(2-5)

The amount of gas in the void volume is calculated from conditions at equilibrium in the

cell:

cell

voidvoid

ZRT

PVn

(2-6)

In Equations (2-5) and (2-6) , Z is the compressibility factor of the pure gas at the

corresponding conditions of temperature and pressure. The above steps are repeated

sequentially at higher pressures to yield a complete adsorption isotherm. Equation (2-7) is

used to calculated absolute adsorption from the measured Gibbs excess adsorption:

gasads

adsGibbsads

Absads nn

(2-7)

where ads is the adsorbed phase density, gas is the gas phase density.

Gas compressibility factors and densities for Helium, N2, CH4 and CO2 are calculated from

the NIST webbook at http://webbook.nist.gov/chemistry/fluid/.

2.2.3 Permeability measurement

The transient method of Brace et al. (1968) was used because of the shorter test durations

required compared to steady state measurements. The Brace method involves observing



2–7

the decay of a differential pressure between upstream and downstream vessels across a

sample of interest. This pressure decay is combined with the vessel volumes in the analysis

to relate the flow through the sample and thus determine the permeability (Brace et al.,

1968). The pressure decay curve can be modelled as:

t

du

du ePP

PP

0,0,

(2-8)

where du PP is the pressure difference between the up- and downstream cylinders, in the

experimental facility used for this work, measured by a differential pressure transducer;

0,0, du PP is the pressure difference between the up- and downstream cylinders at initial

stage, t is time and is described below:

du

RVV

VL

k 112

(2-9)

where k is permeability; is the gas compressibility; L is the sample length; RV is the

sample volume; uV and dV are the volume of the up- and downstream cylinders.

2.2.4 Cleat compressibility

To determine cleat compressibility, Seidle et al. (1992) derived a relationship between

permeability and stress by idealising the coal fabric as a collection of matchsticks. This

relationship is Equation (2-2) and was used in combination with the S-D permeability

model as presented in Equation (2-1). In Seidle et al.'s work (1992), water was used to

measure coal permeability and involved a series of permeability measurements at constant

confining pressure but differing pore pressure. In this current work, gasses including

helium, nitrogen, methane and CO2 were used to measure coal permeability. Pan et al.,

(2010) proved that Equation (2-2) derived by Seidle et al. (1992) is valid for the current

permeability measurement using gasses.

Cleat compressibility is defined by:

p

f

f

fP

c

1 (2-10)

where f is cleat porosity and

pP is pore pressure. Hydrostatic stress is defined by (see e.g.

Zimmerman et al., 1986):

pc mPP (2-11)



2–8

where cP is the confining pressure, and m is the effective stress coefficient.

Cleat compressibility can be estimated by fitting Equation (2-2) to permeability

measurements with respect to effective stress.

2.2.5 Swelling measurement

Swelling displacements are measured simultaneously with gas adsorption at a constant

effective stress, which is controlled by tracking the pore pressure. Volume swelling is

approximately represented by:

rarrV 21 (2-12)

where V is the volumetric swelling, 1r and 2r are the two radial strains perpendicular

to each other, ra is the axial strain, which is the average of the results by the two axial

displacement gauge.

2.2.6 Young’s modulus and Poisson’s ratio

Uniaxial stress testing was performed on the coal core. A load was applied in the axial

direction and the axial and radial displacements were monitored and used to calculate the

Young's modulus and Poisson's ratio.

ll

AFE

(2-13)

where F is the load, A is the cross-section area of the core, l is the displacement in the

axial direction, l is the length of the core. Poisson's ratio can be calculated from:

a

r

(2-14)

where r is the radial strain and a is the axial strain. Poisson's ratio can also be calculated

from the relationship between bulk modulus and Young's modulus:

213

EK (2-15)

The effect of the presence of fluid during the load-displacement test to obtain Young's

modulus and Poisson's ratio should be minimal, because the fluid is connected to a big

void volume (about 30 ml), so that the pore pressure change during load-displacement test

is negligible.



2–9

More detailed descriptions on experimental procedures can be found in Pan et al. (2010).

2.3 Results and Discussion

2.3.1 Adsorption isotherm results

Figure 2-2 presents the helium porosity and N2, CH4 and CO2 adsorption results at two

different temperatures on the coal. It can be seen from Figure 2-2(a) that the helium

porosity for this coal is about 9%, which includes the pore volume of the cleat and macro-,

meso- and micropores. The adsorption capacity increases from N2 to CH4 to CO2. At

about 8 MPa, the adsorption ratio is about 1:2:3 for the three gasses for this coal. CO2

adsorbs 1.5 times more than CH4, which is slightly lower than many coals with a ratio of

about 2 to 3 (Fitzgerald et al., 2005). CO2 absolute adsorption jumps at 10 MPa, thus not

following the Langmuir curve. The reason for this is not well understood but this

behaviour was also observed by other researchers (e.g. Hall et al., 1994). At two different

temperatures, gasses adsorb slightly more at lower temperature. However, their difference

is minor due to the small temperature difference.

The Langmuir volume and pressure for the three gasses at the measurement temperatures

are summarised in Table 2-1. The Langmuir volumes are slightly larger for N2 and CH4 at

higher temperature. At the same time, the Langmuir pressure is larger as well. The

combined effect makes the Langmuir isotherm lower at higher temperature for the

pressure range considered, reflecting the experimental results. For CO2, the Langmuir

volume is smaller and Langmuir pressure is higher at higher temperature, thus the

Langmuir isotherm is lower at higher temperature. However, it should be noted that the

differences of the isotherm at different temperatures are only marginal and are within

experimental uncertainties.

5.0

6.0

7.0

8.0

9.0

10.0

11.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0Pressure (MPa)

He

liu

m P

oro

sit

y (

%)

increasing pressure (35°C)

Increasing pressure (40°C)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 2.0 4.0 6.0 8.0 10.0 12.0

Pressure (MPa)

Ad

so

rpti

on

(m

mo

l/g)

Pressure up (35°C)

Pressure down (35°C)

Langmuir model (35°C)

Pressure up (40°C)



2–10

(a) Helium porosity (b) N2 adsorption

(c) CH4 adsorption

(d) CO2 adsorption

Figure 2-2. The measured helium porosity and gas adsorption isotherms for the core

sample.

Table 2-1. Langmuir volumes and pressures for the measurements

Gas VL (mmol/g) PL (MPa) VL (mmol/g) PL (MPa)

35 ºC 40 ºC

N2 0.86 5.71 0.90 6.60

CH4 1.25 2.56 1.27 2.78

CO2 1.91 1.40 1.87 1.49

2.3.2 Swelling

The coal strain was measured during the adsorption measurements. For the coal studied,

swelling shows strong anisotropy with swelling strain in the axial direction (perpendicular

to the bedding) almost double that in the radial directions (parallel to the bedding). Figure

2-3 presents an example of the coal swelling strain measurements, which is for CO2 with

respect to pressure and for the two temperatures. It also shows that coal swells slightly

more at lower temperature, which is consistent with the adsorption behaviour with respect

to temperature.

An important characteristic of gas adsorption induced coal swelling is that swelling is

directly related to the adsorbed amount (Pan and Connell, 2007). Figure 2-4 presents the

swelling strain versus adsorbed amount for the three gasses. The results show that the coal

swelling strain versus adsorbed amount for the gasses at the two temperatures almost form

the same trend line, which is close to linear. There are two CO2 points that are outliers,

corresponding to the two adsorption points at higher pressure as shown in Figure 2-2(d).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10.0 12.0Pressure (MPa)

Ad

so

rpti

on

(m

mo

l/g)

Pressure up (35°C)



Pressure up (40°C)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0 12.0

Pressure (MPa)

Ad

so

rpti

on

(m

mo

l/g

)

Pressure up (35°C)



Pressure up (40°C)

Not used in Langmuir model fitting



2–11

Figure 2-3. Coal swelling with CO2 adsorption at the two measuring temperatures.

Figure 2-4. Coal swelling for the three gasses as a function of adsorbed amount at the two

measuring temperatures.

2.3.3 Permeability stress behaviour

Permeability was measured at 4 pore pressure steps for each gas up to 10 MPa. Figure 2-5

presents the permeability measurements at 1.0 MPa effective stress at 35 ºC and 40 ºC.

Permeability measurements at each pore pressure step with an effective stress of 2.0 MPa

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12Pressure (MPa)

Str

ain

(%

)

Volumetric (40°C)

Volumetric (35°C)

Axial (40°C)

Axial (35°C)

Radial (40°C)

Radial (35°C)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5Adsorption amount (mmol/g)

Vo

lum

etr

ic S

we

ll (

%)

N2 (up) 35°C

N2 (down) 35°C

CH4 (up) 35°C

CH4 (down) 35°C

CO2 (up) 35°C

CO2 (down) 35°C

N2 (up) 40°C

CH4 (up) 40°C

CO2 (up) 40°C



2–12

and 3.0 MPa were also performed and will be used to examine the permeability-stress

behaviour and obtain the cleat compressibility in the next section. Permeability decreases

slightly with pore pressure increase for all gasses at the two temperatures. The decrease of

permeability may be caused by (1) Klinkenberg effect, (2) effective stress coefficient,

and/or (3) coal swelling. The permeability at the same pore pressure for different gasses

also show differences, which are also due to the causes outlined above. However, the

decrease is not related to gas adsorption capacity and is somewhat random for this coal. In

our previous work (Pan et al., 2010), permeability was found to decrease in the sequence

from Helium, N2, CH4 to CO2.

Gas permeability measurements at 40 ºC were performed after those at 35 ºC. Helium

permeability at 40 ºC was about 4 times higher than that measured at 35 ºC. This behaviour

is inconsistent with our previous measurement. The cause for this will require further

investigation. Although the comparison of results at different temperatures may be difficult,

the results for each temperature show the same trend.

(a) 35 ºC

(b) 40 ºC

Figure 2-5. Measurements of permeability vs pore pressure at 1 MPa effective stress.

2.3.4 Cleat compressibility

At each pore pressure the permeability was measured with respect to effective stress, by

varying the confining pressure, in order to determine the permeability vs. effective stress

relationship. Figure 2-6 presents the results using N2 at the two measurement temperatures.

Measurements using the other gasses were similar and thus are not plotted. The behaviour

of permeability with respect to effective stress is close to the exponential relationship of

Equation (2-2) and this is used as a fitting function for the measurements to estimate cleat

compressibility.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12

Pore Pressure (MPa)

Pe

rme

ab

ilit

y (

md

)

He

N2

CH4

CO20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12

Pore Pressure (MPa)

Perm

eab

ilit

y (

md

)

0

0.5

1

1.5

2

2.5

3

He

liu

m P

erm

ea

bilit

y (

md

)

N2

CH4

CO2

He



2–13

(a) 35 ºC

(b) 40 ºC

Figure 2-6. Measurements of N2 permeablity with respect to effective stress/pressure and

pore pressure.

The calculated cleat compressibilities for each gas at each pore pressure and each

temperature are presented in Figure 2-7. It can be seen from this figure that the cleat

compressibility tends to increase from Helium, N2, CH4 to CO2. This may reflect the

reduction in cleat porosity due to swelling-strain. At lower porosity, cleat compressibility

tends to be higher if the cleat elasticity remains the same. The cleat compressibility results

at 40 ºC are smaller than those in 35 ºC. This may be because of the lower swelling strains

at higher temperature leading to less porosity decrease. However, since the cleat aperture

may be altered by increasing the temperature to 40 ºC, the conclusion of lower

compressibility at higher temperature will require further investigation. These cleat

compressibility results are comparable to the results from Seidle et al.'s work (1992), where

the cleat compressibility is about 0.084MPa-1 for a Warrior Basin coal sample.

(a) Helium

(b) Nitrogen

y = 0.529e-0.198x

R2 = 0.996

y = 0.485e-0.195x

R2 = 0.994

y = 0.438e-0.194x

R2 = 0.991

y = 0.404e-0.211x

R2 = 0.998

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Effective Pressure (MPa)

N2

eff

ec

tiv

e p

erm

ea

bilit

y (

md

)

0.946 MPa

2.30 MPa

5.23 MPa

10.31 MPa

y = 0.850e-0.116x

R2 = 0.995

y = 0.669e-0.123x

R2 = 0.983

y = 0.607e-0.129x

R2 = 0.988

y = 0.587e-0.152x

R2 = 0.996

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Effective Pressure (MPa)

He

eff

ec

tiv

e p

erm

ea

bilit

y (

md

)

1 MPa

4 MPa

7 MPa

10 MPa

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 2 4 6 8 10 12Pore Pressure (MPa)

Cle

at

Co

mp

res

sib

ilit

y (

MP

a-1

)

35 °C

40 °C

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6 8 10 12

Pore Pressure (MPa)

Co

mp

res

sib

ilit

y (

MP

a-1

)

35°C

40°C



2–14

(c) Methane

(d) CO2

Figure 2-7. Cleat compressibilities for the various gasses with respect to pore pressure and

temperature.

The large difference of cleat compressibility among different gasses may mean that

permeability changes with different gasses may be significantly different. This is of great

importance to the modelling of ECBM processes. Laboratory measurements are important

to provide the cleat compressibility values to be used in the reservoir simulations.

2.3.5 Geomechanical properties

Load and displacement experiments were performed at each pore pressure and confining

pressure for the gasses and temperature. Figure 2-8 presents the experimental results with

Helium as pore gas. Results with the other gasses are similar and thus not presented, but

Table 2-2 summarises the experimental results for Young's modulus, Poisson's ratio and

bulk modulus for the coal sample with respect to effective stress for each gas and

temperature. The results are averaged with respect to pore pressure.

It can be seen from Table 2-2 that the Young's modulus is smaller at 40 ºC than that

measured at 35 ºC for all gasses. Although the difference is small between the two

temperatures, this may mean that there is a degree of coal softening at higher temperature

and that softening may become important if the temperature change is significant. There is

no apparent trend for the Poisson's ratios between the two temperatures for the different

gasses, but the average among all gasses is slightly higher at lower temperature (0.288 at 35

ºC compared to 0.285 at 40 ºC). The bulk modulus is generally larger at lower temperature.

When averaging among the gasses, the bulk modulus is 1.93 GPa at 35 ºC compared to

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6 8 10 12Pore Pressure (MPa)

Co

mp

ressib

ilit

y (

MP

a-1

)

35°C

40°C

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6 8 10 12

Pore Pressure (MPa)

Co

mp

res

sib

ilit

y (

MP

a-1

)

35°C

40°C



2–15

1.86 GPa at 40 ºC. These results show that temperature has an impact on the coal

properties and the impact may become more important for large temperature differences.

Also from Table 2-2, we can see that the Young's modulus decreases with the increase of

the adsorption capacity among the three adsorbing gasses, N2, CH4 and CO2. The Young's

modulus and bulk modulus are noticeably lower when CO2 is adsorbed to the coal.

Whether adsorption causes coal softening is not clear, since the modulus with Helium

(considered to be non-adsorbing) as the pore gas is not higher than those measured when

adsorbing gasses are used. More work needs to be done to further elucidate the impact of

adsorption on geomechanical properties.

It is quite obvious that the effective stress does have a significant impact on the coal

properties. For instance when helium is the pore gas, the measured Young's modulus at 3.0

MPa effective stress is about 20% larger than that measured at 1.0 MPa effective stress.

Since coal is usually found to be a non-linear elastic rock with load-unload hysteresis, a

larger modulus at higher confining pressure is not surprising and would be important for

reservoir modelling of CBM/ECBM processes.

Figure 2-8. Young's modulus measured with respect to pore pressure at two effective

stresses and temperatures with helium as the pore fluid.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

0.0 2.0 4.0 6.0 8.0 10.0 12.0

Pore Pressure (MPa)

Yo

un

g's

Mo

du

lus

(G

Pa

)

Effective Stress 1.0 MPa (35°C)






2–16

Table 2-2. Averaged geomechanical properties

Young's Modulus (GPa) Poisson's ratio (-) Bulk Modulus (GPa)

Gas Effective

stress (MPa) 35 ºC 40 ºC 35 ºC 40 ºC 35 ºC 40 ºC

He 1.0 2.20 2.10 0.292 0.302 1.77 1.77

3.0 2.60 2.59 0.268 0.291 1.87 2.07

N2 1.0 2.31 2.18 0.300 0.275 1.94 1.61

3.0 2.75 2.68 0.289 0.282 2.17 2.05

CH4 1.0 2.26 2.25 0.291 0.273 1.82 1.65

3.0 2.73 2.70 0.294 0.289 2.21 2.13

CO2 1.0 2.17 2.11 0.279 0.271 1.64 1.54

3.0 2.54 2.50 0.288 0.295 2.00 2.04

2.4 Conclusion

This work presents the experimental measurement of adsorption isotherms, swelling,

permeability, and geomechanical properties for an Australian coal sample in three gasses,

N2, CH4 and CO2 at two temperatures, 35 ºC and 40 ºC. The results show that the gas

adsorption is less at higher temperature, which is consistent with other work reported in

the literature. Measurements of coal swelling strain with respect to adsorbed amount are

close to the same trend line irrespective of gas type and temperature. Permeability

coefficients decline with increased gas pressure. However, the permeability results do not

show a trend in relation to the gas adsorption capacity for this coal. The cleat

compressibility was estimated from the permeability measurements with respect to

effective stress and shows an increasing trend with the quantity of gas adsorbed. This may

be related to the reduction in cleat porosity due to coal swelling. Coal's elastic modulus was

found to be larger at higher effective stress, which may be caused by its non-linear elastic

behaviour. The elastic modulus is also larger at lower temperatures, which may indicate

possible coal softening at higher temperatures. However, the relationship between elastic

modulus and gas adsorption is still unclear and will require further investigation.

2.5 Acknowledgement

The authors acknowledge the financial supports from the CSIRO Advanced Coal

Technology Portfolio, the Western Australia CSIRO-University Postgraduate Research

Scholarship, and National Research Flagship Energy Transformed Top-up Scholarship.



2–17

2.6 References

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Geophysical Research, 73(6), 2225-2236.

Fitzgerald, J.E., Pan, Z., Sudibandriyo, M., Robinson, Jr. R.L., Gasem, K.A.M., and Reeves, S., 2005.

Adsorption of methane, nitrogen, carbon dioxide and their mixtures on wet Tiffany coal. Fuel. 84,

2351-2363.

Fujioka, M., Yamaguchi, S. and Nako, M., 2010. CO2-ECBM Field Tests in the Ishikari Coal Basin of Japan.

International Journal of Coal Geology, 82(3-4), 287-298.

Hall, F., Zhou, C., Gasem, K.A.M. and Robinson, Jr R.L., 1994. Adsorption of Pure Methane, Nitrogen, and

Carbon Dioxide and Their Mixtures on Wet Fruitland Coal. SPE 29194.

Mazumder, S., Karnik, A.A., and Wolf, K-H. A.A., 2006 Swelling of coal in response to CO2 sequestration

for ECBM and its effect on fracture permeability. SPE Journal, 11(3), 390-398.

Palmer, I., 2009. Permeability Changes in Coal: Analytical Modeling. International Journal of Coal Geology,

77(1-2), 119-126.

Pan, Z. and Connell, L.D., 2007. A theoretical Model for Gas Adsorption-induced Coal swelling.

International Journal of Coal Geology, 69, 243-252.

Pan, Z., Connell, L.D. and Camilleri, M., 2010. Laboratory Characterisation of Coal Reservoir Permeability

for Primary and Enhanced Coalbed Methane Recovery. International Journal of Coal Geology, 82(3-

4), 252-261.

Puri, R. and Yee, D., 1990. Enhanced Coalbed Methane Recovery, SPE Annual Technical Conference and

Exhibition. New Orleans, Louisiana.

Seidle, J.P., Jeansonne M.W. and Erickson D.J., 1992. Application of Matchstick Geometry To Stress

Dependent Permeability in Coals, SPE Rocky Mountain Regional Meeting. Casper, Wyoming.

Shi, J.Q. and Durucan, S., 2004. Drawdown Induced Changes in Permeability of Coalbeds: A New

Interpretation of the Reservoir Response to Primary Recovery. Transport in Porous Media, 56(1), 1-

16.

Sparks, D.P., McLendon, T.H., Saulsberry, J.L. and Lambert, S.W., 1995. The Effects of Stress on Coalbed

Reservoir Performance, Black Warrior Basin, U.S.A. SPE Annual Technical Conference and

Exhibition, Dallas, Texas.

Wong, S., Law, D., Deng, X., Robinson, J., Kadatz, B., Gunter, W.D., Ye, J., Feng, S., and Fan, Z., 2007.

Enhanced Coalbed Methane and CO2 Storage in Anthracitic Coals-Micro-pilot Test at South Qinshui,

Shanxi, China. International Journal of Greenhouse Gas Control, 1, 215-222.

van Bergen, F., Pagnier, H., and Krzystolik, P., 2006. Field Experiment of Enhanced Coalbed Methane– CO2

in the Upper Silesian Basin of Poland. Environmental Geosciences, 13, 201-224.

Zimmerman, R.W., Somerton, W.H., and King, S.M., 1986. Compressibility of porous rocks. Journal of

Geophysical Research, 91(B12), 12,765-12,777.



3-1

Chapter 3

Linking Gas-Sorption Induced Changes in Coal Permeability to Directional Strains through a Modulus Reduction Ratio

Jishan Liua, Zhongwei Chena, Derek Elsworthc, Xiexing Miaob, Xianbiao Maob

a School of Mechanical and Chemical Engineering, The University of Western Australia, WA, 6009, Australia

b State Key Laboratory for Geomechanics and Underground Engineering, China University of Mining and

Technology, Jiangsu 221008, China

c Department of Energy and Mineral Engineering, Penn State University, PA 16802-5000, USA

Abstract: Although coal-gas interactions have been comprehensively investigated, prior

studies have focused on one or more component processes of effective stress or sorption-

induced deformation and for resulting isotropic changes in coal permeability. In this study

a permeability model is developed to define the evolution of gas sorption-induced

permeability anisotropy under the full spectrum of mechanical conditions spanning

prescribed in-situ stresses through constrained displacement. In the model, gas sorption-

induced coal directional permeabilities are linked into directional strains through an elastic

modulus reduction ratio, Rm. This defines the ratio of coal mass elastic modulus to coal

matrix modulus (0<Rm<1) and represents the partitioning of total strain for an equivalent

porous coal medium between the fracture system and the matrix. Where bulk coal

permeability is dominated by the cleat system, the portioned fracture strains may be used to

define the evolution of the fracture permeability, and hence the response of the bulk

aggregate. The coal modulus reduction ratio provides a straightforward index to link

anisotropy in deformability characteristics to the evolution of directional permeabilities.

Constitutive models incorporating this concept are implemented in a finite element model

to represent the complex interactions of effective stress and sorption under in-situ

conditions. The validity of the model is evaluated against benchmark cases for uniaxial

swelling and for constant volume reservoirs then applied to match changes in permeability

observed in a field production test within a coalbed reservoir.

Keywords: coal permeability anisotropy; coal swelling; coal-gas interactions; numerical

modelling; modulus reduction ratio



3-2

3.1 Introduction

Knowledge of changes in coal permeability due to gas sorption-induced effective stress is

crucially important for the evaluation of both primary gas production from coalbed

reservoirs and for CO2-enhanced coalbed methane recovery (ECBM). For primary gas

production, the reduction of gas pressure increases effective stress which in turn closes the

fracture aperture and reduces the permeability. As the gas pressure reduces below the

desorption point, methane is released from the coal matrix to the fracture network and the

coal matrix shrinks. As a direct consequence of this matrix shrinkage the fractures dilate

and fracture permeability correspondingly increases. Thus, a rapid initial reduction of

fracture permeability (due to change in effective stress) is supplanted by a slow increase in

permeability (with matrix shrinkage). Whether the ultimate, long-term, permeability is

greater or less than the initial permeability depends on the net influence of these dual

competing mechanisms.

ECBM involves the injection of CO2 into a coal seam to promote the desorption of

coalbed methane (CBM) while simultaneously sequestering CO2 in the coal seam. This

process exploits the greater affinity of carbon dioxide (CO2) to adsorb onto coal relative to

methane (CH4), resulting in the net desorption of methane and its potential recovery as a

low-carbon fuel. Laboratory isotherm measurements for pure gasses have demonstrated

that coal can adsorb approximately twice (or more) as much CO2 (in moles) by volume as

methane (White et al., 2005). Correspondingly, CO2 injection with concurrent production

of methane can cause differential swelling of the coalbed particularly in the near wellbore

area. This may play an important role in determining the resulting deformation of the coal

matrix, the related permeability change and its impact on both gas diffusion to the cleats

and gas transport along the cleat network. Thus, the influence of these distinct but

connected changes in deformation, due to both effective stresses and to gas-sorption-

induced swelling, are key to unravelling the transient response to gas injection and recovery.

The complexity of the response is further increased by the overprinted effects of bedding

plane and cleat orientations, which together with directional stresses or displacement

restraints impart a further directional heterogeneity to the transient evolution of

permeability. Thus understanding the transient and anisotropic characteristics of

permeability evolution in fractured coals is of fundamental importance to the recovery of

methane from CBM reservoirs and equally important for CO2 storage suing ECBM.



3-3

3.1.1 Experimental Observations

The potential impacts of coal swelling on the performance of coalbed methane production

and in the deep geological sequestration of CO2 have been investigated through

experimental, field-scale, and numerical studies.

Experiments have investigated the sorption and related swelling characteristics of coals.

The effects of water content on swelling and sorption have been explored for CO2 uptake

at 298 K (Ceglarska-Stefanska and Czaplinski, 1993) using a gas-flame coal, a gas-coking

coal and an anthracite and indicate a reduction in swelling strain for ―dry‖ coal versus ―pre-

wetted‖ samples (Ceglarska-Stefanska and Brzoska, 1998). Rates of swelling are controlled

largely by diffusive length scales imparted by the cleats. A surrogate of this case is

powdered coals where for powdered high volatile bituminous Pennsylvanian coals the

adsorption rate decreases with increasing grain size for all experimental conditions (Busch

et al., 2004). Similarly, coal type and rank (Robertson and Christiansen, 2005; Prusty, 2007)

influences the preferential sorption behavior and the evolution of permeability with these

changes is linked to macromolecular structure (Mazumder and Wolf, 2008). Adsorption

kinetics may also be determined for various gasses (e.g. for CO2 and CH4) using confining

cells to apply desired pressures and temperatures (Charrière et al., 2010) and using X-ray

CT methods to determine the resulting sorption isotherms (Jikich et al., 2009). These

experiments have focused on the isotropic characteristics of intact or powdered coals.

Conversely, some experiments have focused on the anisotropic characteristic of coal. Water

transmission characteristics have been shown to be significantly different (Gash et al., 1993)

under confining pressures when measured perpendicular to either face cleats, butt cleats, or

bedding planes. Directional flow experiments on isotropically compressed samples have

similarly confirmed the anisotropy of permeability for gas flows (Li et al, 2004). These

results are congruent with optical measurements of coal swelling under in CO2 and other

gasses where swelling in the plane perpendicular to the bedding plane was always

substantially higher than parallel to the bedding plane (Day, et al., 2008). This

phenomenon has also been observed in the field well tests in the Warrior Basin (USA)

where the anisotropy ratio of permeability in the direction of the bedding plane was as high

as 17:1 (Koenig and Stubbs, 1986).



3-4

3.1.2 Permeability Models

Based on experimental observations, a variety of models have been formulated to quantify

the evolution of permeability during coal swelling/shrinking. The first attempts to quantify

the role of stresses on the evolution of coal-reservoir permeability assumed invariant

vertical stresses and linked changes in horizontal stress with the gas pressure and the

sorption strain (Gray, 1987). Permeability was computed as a function of reservoir

pressure-induced coal-matrix shrinkage assumed directly proportional to changes in the

equivalent sorption pressure. Since then, a number of theoretical and empirical

permeability models have been proposed. The Seidle-Huitt Model describes the evolution

of permeability assuming that all changes in permeability are caused by the sorption-

induced strain alone, neglecting the elastic strain (Seidle and Huitt, 1995). Another three of

the most widely used permeability models are the Palmer and Mansoori model (P-M

Model), the Shi and Durucan (S-D) model and the Advanced Resources International (ARI)

model (Palmer and Mansoori, 1996; Pekot and Reeves, 2002; Shi and Durucan, 2005). The

P-M model is strain-based so porosity change is defined by the change in volume strain,

and the related permeability change is calculated directly from that change in porosity. The

P-M relation is derived from the assumption of isotropic linear elasticity to determine

resulting changes in strain for an assumed invariant overburden stress. Consequently,

predicted changes in porosity are small and are permuted into permeability changes via a

cubic relationship between permeability and porosity. The S-D model is based on an

idealized bundled-matchstick geometry to represent a coalbed, and uses a stress-based

formulation to correlate changes in the effective horizontal stress caused by the volumetric

deformation together with the cleat or pore compressibilities. This stress-based model

accommodates changes in porosity and permeability through the anticipated change in

horizontal stress and includes no direct influence of swelling/shrinkage-induced strain.

Additionally, the Biot's coefficient is set to unity – requiring that the change in net stress is

equal to the difference between the overburden pressure and the change in pore pressure.

The ARI model describes the evolution of coal permeability using a semi-empirical

correlation to account for the changes of coal porosity due to pore compressibility and

matrix swelling/shrinkage (Pekot and Reeves, 2002). The ARI model is essentially

equivalent to the P-M model for saturated coal and where the strain versus stress

relationship fits the Langmuir isotherm (Palmer, 2009). However, although permeability

models incorporating sorption-induced effects have been widely studied, most of these



3-5

studies are under the assumption of either an invariant total stress or uniaxial strain

conditions. These critical and limiting assumptions have been relaxed in new models

rigorously incorporating in-situ stress conditions (Zhang et al., 2008; Palmer, 2009; Connell,

2009) and are extended to rigorously incorporate CO2-CH4 coal-gas interaction relevant to

CO2-ECBM (Connell and Detounay, 2009; Chen et al., 2010).

Despite the complexity of models applied to represent the evolution of coalbed methane

reservoirs, few accommodate feedbacks of both anisotropy and coal-gas interactions on the

evolution of permeability – including the important roles of linked stress-deformation and

gas flow and adsorption/desorption processes. The effect of stress on the evolution of

flow anisotropy in orthogonally fractured media (Sayers, 1999) and in deformable granular

media (Du et al., 2004) has been investigated although not with the influence of gas

adsorption or desorption effects. The impact of permeability anisotropy and pressure

interference on CBM gas production has been investigated specifically to seek any unique

performance feature that might distinguish between isotropic or anisotropic permeability of

the CBM reservoir or to identify the drainage geometry (Chaianansutcharit, et al., 2001).

And analytical solutions have been presented for steady-state conditions with anisotropic

permeability (Al-Yousef, 2005). More recently, an alternative approach has been proposed

to develop an improved permeability model for CO2-ECBM recovery and CO2 geo-

sequestration in coal seams, integrating the textural and mechanical properties to describe

the anisotropy of gas permeability in coal reservoirs under confined stress conditions

(Wang et al., 2009).

3.1.3 This study

In this study, a novel permeability model is developed to define the evolution of gas

sorption-induced permeability anisotropy under in-situ stress conditions. Gas sorption-

induced coal directional permeabilities are linked to directional strains through the elastic

modulus reduction ratio (the ratio of coal mass elastic modulus to coal matrix modulus)

that represents the partition of the total strain for an equivalent porous coal medium

between the fracture system and the matrix. It is assumed that only the partitioned fracture

strains are responsible for the changes in directional permeabilities. These new relations

are the key cross couplings that link effective stress-related and sorption/desorption-related

changes in permeability to fluid pressure and gas content. These constitutive relationships

are incorporated into a finite element model to represent the complex interactions of stress



3-6

and chemistry under in-situ conditions and to project their impact on rates and magnitudes

of gas recovery. The validity of the general model is evaluated against results for special

cases representing uniaxial swelling, constant volume reservoirs, and for the case of a

coalbed methane production well test. The incorporation of gas sorption-induced coal

permeability anisotropy into the multiphysics simulation of coal-gas interactions represents

a new and important contribution to this subject.

3.2 Approach

The overall approach is illustrated in Figure 3-1. The evaluation of fully coupled

deformation and gas transport in the fractured coal is conducted through four integrated

steps: (1) Coal deformation analysis; (2) Flow equivalence analysis; (3) Permeability

evolution analysis; and (4) Flow equivalence updating. These four steps are detailed in the

following sections.

Figure 3-1. Flow chart for evaluating coupled deformation and gas transport processes in

coal. Circled numbers represent steps of the analysis process.

1 Coal Deformation Analysis 2 Flow Equivalence Analysis

3 Permeability Evolution Analysis 4 Flow Equivalence Update

Coal Seam

Deformation Model

vzyx ,,,

Initial Condition

Isotropic coal permeability

0k

Discontinuous Model with

Uniform spacing s, and Aperture, b

Initial Condition

Anisotropic coal permeability

000 ,, zyx kkk

Discontinuous Model with

Uniform spacing s, and Nonuniform Aperture bx0,

by0, bz0

New Coal Permeability

zyx kkk ,,

Gas Flow & Transport

Model

sp ,

Gas Flow & Transport

Model

sp ,

1

2 2

3

4 4

2 2



3-7

3.2.1 Coal Deformation Analysis

The mechanical properties of a discontinuous medium containing orthogonal fractures and

orthotropic response can be represented by the properties of an equivalent continuous

medium (Amadei and Goodman, 1981). The following assumptions are made:

(a) The coal is a homogeneous, isotropic and elastic continuum, and the system is

isothermal.

(b) Strains are infinitesimal.

(c) Gas contained within the pores is ideal, and its viscosity is constant under isothermal

conditions.

(d) Gas flow through the coal medium is assumed to be viscous flow obeying Darcy's

law.

According to the first assumption (a), the strain-displacement relation is expressed as

)(2

1,, ijjiij uu (3-1)

where ij is the component of the total strain tensor and iu is the component of the

displacement. The equilibrium equation is defined as

0, ijij f (3-2)

where ij denotes the component of the total stress tensor and if denotes the component

of the body force.

The gas sorption-induced strain s is assumed to result in only normal strains and these

resulting strains are isotropic. The effects of gas sorption on the deformation of coal seams

can be treated analogous to the effects of temperature for elastic porous media (e.g.,

Palmer and Mansoori, 1998), stress–strain relationships for an isothermal gas adsorbing

coalbed may be written as (Shi and Durucan, 2004)

ijs

ijijkkijij pKKGG

33

)9

1

6

1(

2

1 (3-3)

where mm ERE , )1(2

E

G , )21(3

E

K , mK

K1 ,

332211 kk.



3-8

K represents the bulk modulus of coal and mK represents the bulk modulus of coal

matrixes. G is the shear modulus of coal, s is the sorption-induced strain, mE is the

Young's modulus of the matrix, mR is the modulus reduction ratio, E is the equivalent

Young's modulus of the coal-fracture assemblage and is the Poisson's ratio of the coal-

fracture assemblage. represents the Biot's coefficient, p the gas pressure in the pores

and ij is the Kronecker delta; 1 for ji and 0 for ji .

Combining Equations (3-1)–(3-3) yields the Navier-type equation expressed as

021

,,,

2

iisiii fKpeG

uG

(3-4)

where iu is the displacement in i direction, e is the volumetric strain, and

ip, is the partial

derivative of pore pressure with respect to i . Equation (3-4) is the governing equation

representing deformation of the continuum representation of the fractured coal allowing

deformations to be determined if fluid pressures, p, may be determined for both undrained

and drained response. Transient fluid pressures are recovered from the flow equation.

3.2.2 Flow and Transport Analysis

The mass balance equation for a single component gas is defined as

sgg Qt

m

)( q (3-5)

where g is the gas density, gq is the Darcy velocity vector and sQ is the gas source or

sink. m is the gas content including both free-phase and absorbed components (Saghafi, et

al., 2007) and is defined as

L

Lcgaffg

Pp

pVm

1 (3-6)

where ga is the gas density at standard conditions, c is the coal density and is

fracture porosity. LV represents the Langmuir volume constant and LP represents the

Langmuir pressure constant. According to the real gas law, the gas density is proportional

to the pore gas pressure and can be described as



3-9

pZRT

M g

g (3-7)

where gM is the molar mass of gas, R is the universal gas constant, T is the gas

temperature and Z is the correction factor that accounts for the non-ideal behaviour of

the gas which changes with R and T.

Assuming that the effect of gravity is relatively small and can be neglected, the Darcy

velocity may be defined as

pk

g

q (3-8)

where denotes the dynamic viscosity of the gas and k denotes the permeability tensor,

expressed as

zzzyzx

yyyx

xzxyxx

kkk

kkk

kkk

k yz

(3-9)

Substituting Equations (3-6)–(3-9) into Equation (3-5), we obtain

s

f

L

Lac

L

LLacff Qpp

k

tPp

pVpp

t

p

Pp

PVp

2)(1 (3-10)

where ap is atmospheric pressure (1.0 atm or 101.325 kPa). In Equation (3-10), the

permeability k is dependent on the porosity, , while is a function of the volumetric

strain, and the sorption-induced strain, s , which will be illustrated in detail in the

following sections. Therefore, Equations (3-4) and (3-10) will be coupled through the

porosity-permeability relation and pore pressure evolution.

3.2.3 Coal Permeability Analysis

In the analysis of coal permeability the fractured coal mass is treated as a discontinuous

medium comprising both matrix and fractures (cleats), as illustrated in Figure 3-2. The

individual matrix blocks are represented by cubes and may behave isotropically with regard

to swelling/shrinking, thermal expansion, and mechanical deformability. The cleats are the

three orthogonal fracture sets and may also have different apertures and mechanical

properties ascribed to the different directions.



3-10

Changes in coal permeability are determined by the redistribution of effective stresses or

strains due to changed conditions such as gas injection. Typically, stresses and strains will

evolve at different rates in the different Cartesian directions, i.e., x , y , z and

x , y , y , and result in anisotropic permeabilities, xk , yk , zk . To derive the

relationship between stresses/strains and directional permeability, several assumptions and

definitions are made:

The initial porous coal is substituted either by a discontinuous model with uniform fracture

spacing s and aperture b, as shown in Figure 3-2(a), or by a discontinuous model with

uniform spacing s and non-uniform aperture, 0xb , 0yb , and

0zb , as shown in Figure 3-

2(b).

Figure 3-2. Substitution of porous coal by discontinuous models.

The coal fracture porosity f can be determined by fracture spacing and aperture as

follows, sbsb

ssbf /3

)(

)(3

33

. For the two-dimensional case, the areal porosity for x-

or y-directions is defined as sbyfx /2 and sbxfy /2 , respectively.

Fracture and matrix deformation are both linear and fully recoverable, and deformations in

normal closure or opening are the predominant permeability alteration mode. Therefore,

coal permeability changes can be defined as a function of the variation of aperture in

corresponding directions i.e., xb , yb , and zb ; where the aperture variation partitioned

from the porous medium is realized through the elastic modulus reduction ratio, mR .

0xbs

x x

y

yz

z

0zb

0ybbs

x x

y

yz

z

(a) Initial Isotropic Permeability (b) Initial Anisotropic Permeability



3-11

The coal matrix is functionally impermeable, and the dominant fluid flow within the

fractures may be defined by an equivalent parallel plate model. This enables the strain-

dependent effective porosity and the strain-dependent permeability field to be determined

if induced strains can be adequately partitioned between fracture and solid matrix.

The schematic diagram regarding the fracture aperture change and the effective stress

alteration is shown in Figure 3-3. The aperture closure induced by the effective stress

change can be calculated by

)(m

eieiii

Es

Esbb

(3-11)

Simplifying this equation, gives

Eb

EE

Esb ei

iei

m

i

)1( (3-12)

If assuming mm EER / , Eeiei / , then the above equation can be derived

eimi

fib

Rs

b

b

1

)1( (3-13)

Because sb , Equation (3-13) can be simplified into

eimi

fib

Rs

b

b

)1( (3-14)

wheremR is the elastic modulus reduction ratio,

ei is the effective strain change in the i-

direction, s is the fracture spacing and 0ib is the initial fracture aperture along i- direction.

Figure3-3. Schematic diagram of fracture aperture interaction with effective stress.

ei

Fractured Coal Equivalent Coal Coal Matrix

b

bs/2

s/2E mE

Fracture

Matrix

Matrix

u

mu

mi uub

ib

ei

ei



3-12

Based on the above analysis, for the 3D case with three orthogonal sets of fractures, coal

directional permeability, xk , yk , and zk are defined as follows (Liu et al., 1999)

ji

ej

f

m

i

i R

k

k3

00

)1(31

2

1

(3-15)

ji

j

is

bk

12

3

0

0 (3-16)

where 0f is the initial fracture porosity at reference conditions, zyxji ,,, for 3D case

and yxji ,, for 2D case.

For the 2D case with two orthogonal sets of fractures, coal directional permeability, xk

and yk are defined as follows:

3

00

)1(21

ej

f

m

i

i R

k

k

, ji (3-17)

s

bk

j

i12

3

0

0 , ji (3-18)

Results from field and laboratory experiments indicate that coal permeability can change

significantly during adsorbable gas injection (e.g. CH4 and CO2). The injection gas pressure

tends to mechanically open coal cleats and thus enhance the permeability as the initial gas

pressure resides only in the fractures and any constrained change in total stress compresses

the matrix blocks. The subsequent gas adsorption into the coal matrix induces swelling

(volumetric strain) and has two effects: (1) it reduces effective stresses and causes an elastic

expansion of the coal due to injection gas pressure increase, overprinted by a (2) sorption-

induced swelling of the coal matrix as gas diffuses into coal matrixes. If expansion of the

cleat-matrix assemblage is constrained then fracture permeability reduces by narrowing and

even closing cleat apertures. When the coal swelling is taken into consideration, the total

effective strains in the Equation (3-15) can be replaced by the differences between the total

strain change, it , in i direction and the free swelling strain change, s , as follows

sitei 3

1 (3-19)

Thus for 3D case we have:



3-13

ji

sjt

f

m

i

i R

k

k3

00 3

1)1(31

2

1

(3-20)

For the 2D case with two orthogonal sets of fractures, coal directional permeability, xk

and yk were defined as follows:

3

00 3

1)1(21

sjt

f

m

i

i R

k

k

, ji (3-21)

3.2.4 Coupled Model

For a system containing a single gas phase the sorption-induced volumetric strain s may

be represented by a Langmuir type function (Harpalani and Schraufnagel, 1990; Cui and

Bustin, 2005; Robertson and Christiansen, 2005), defined as

pP

p

L

Ls

(3-22)

where L and LP are the Langmuir-type matrix swelling/shrinkage constants, which

represent the maximum swelling capacity and the pore pressure at which the measured

volumetric strain is equal to L5.0 , respectively.

Substituting Equation (3-22) into Equation (3-4), we rewrite the governing equation for

deformation of the coal seam as

0)(21

,2,,

2

ii

L

LLiii fp

Pp

PKpe

GuG

(3-23)

From Equation (3-14), we can determine porosity f as

smff eRa

bb

)1(3

30

0

(3-24)

Then, the partial derivative of porosity f with respect to time can be expressed as

t

p

Pp

P

t

eR

t L

LLm

f

2)()1(3

(3-25)

Substituting Equation (3-25) into Equation (3-10) yields the governing equation for gas

flow through a coal seam with the effect of gas sorption incorporated as



3-14

s

L

Lacm

L

LL

L

Lacm

L

LLacff

Qt

e

Pp

pVppR

ppk

t

p

Pp

P

Pp

pVppR

Pp

PVp

)1(3

-)(

)1(3)(

122

(3-26)

In our analysis of coal deformation, the fractured coal mass is represented as an

orthotropic fractured medium which is replaced as an equivalent continuous medium.

When we conduct flow analysis, we partition the total effective strain from the equivalent

medium between coal matrix and the fracture system. Only the partitioned strain for the

fracture system contributes to the permeability change. When the rock mass reduction ratio

is unity, i.e. 1mR then the equivalent modulus of the fractured medium is equal to that

of the coal matrix. In other words the coal mass may be considered as unfractured or the

fractures are infinitely small. Conversely, in the limit as 0mR then the coal matrix is

infinitely stiff and the observed deformational response is equivalent to that of the

fractures alone. Therefore the parameter mR1 represents the ratio of the partitioned strain

for the fracture system to the total equivalent strain. If 1mR , the partitioned strain for

the fracture system is due to that of the matrix modulus, which is essentially zero in

comparison to the anticipated response of the more compliant fractures; therefore a very

small permeability change due to the deformation of the matrix results and this is taken as

null. If 0mR then the partitioned strain is predominantly due to the fracture deformation.

Where the fractures are typically more compliant than the host from which they are derived,

then a maximum permeability change results.

The total effective strain is the difference between the total strain, as determined by the

constrained boundary conditions, and the free swelling strain. Therefore, the boundary

conditions also control the evolution of coal permeability.

In the following sections, we use three examples to illustrate these principles. These are

respectively conditions of (1) uniaxial strain; (2) constant reservoir volume; and (3) the

behaviour of a field case.

3.3 Uniaxial Strain Condition

For the case of uniaxial strain, the lateral strains, tx and ty , are equal to zero. Based

on Hooke's law, the relation between stress and strain increments are:



3-15

sezexstxexE

3

11

1

3

1 (3-27)

sezeystyeyE

3

11

1

3

1 (3-28)

sexezez

E

pp

E

13

2

1

2112

1 0 (3-29)

Substituting these equations into Equation (3-20) gives

3

0

0

3

000

13

2

1

211)1(31

2

1

3

1)1(31

2

1

s

f

m

s

f

m

y

y

x

x

E

ppR

R

k

k

k

k

(3-30)

3

00 3

1)1(31

s

f

m

z

z R

k

k

(3-31)

As shown in Equations (3-30) and (3-31), coal permeability in the x-direction is not equal

to the permeability in the z-direction. The vertical permeability, zk , is determined by the

swelling strain only while the horizontal permeability, yx kk or , is determined both by the

swelling strain and by the mechanical deformation. It is obvious that zk is not equal to xk

even if 000 zyx kkk . In order to illustrate these conclusions graphically, we use a set of

parameters in the Table 3-1 to quantify the directional permeabilities. The calculated

results are shown in Figure 3-4. If 1mR , the partitioned strain for the fracture system is

zero; therefore, no permeability change is induced: permeability ratios are equal to unity; If

0mR , the partitioned strain for the fracture system is 100%; therefore, maximum

permeability changes zx kk and are induced.

Table 3-1. Parameters used for the example calculations

Parameter Definition value

E Young's modulus 2.7GPa

Poisson's ratio 0.4

b Fracture aperture 0.0001

s Fracture spacing 0.01

L Langmuir strain constant 0.03

Lp

Langmuir pressure constant 3MPa



3-16

mR

Elastic modulus reduction ratio 0, 0.5, 1

Figure 3-4. Coal permeability as a function of matrix pore pressure under uniaxial strain

condition and the influence of gas adsorption. Permeability ratios are regulated by the

modulus ratio, mR : 1mR represents no fracture influence; 0mR represents fracture

influence only; 5.0mR combined influence of fracture and matrix deformation.

3.4 Displacement Controlled Condition

For the displacement controlled (or constant reservoir volume) case, strains in all directions,

tx , ty , and tz are equal to zero. Substituting zero value into Equation (3-20) gives

3

0000 3

1)1(31

s

f

m

z

z

y

y

x

x R

k

k

k

k

k

k

(3-32)

As shown in Equation (3-32), coal permeability ratio in the x- and y-directions is equal to

the permeability ratio in the z-direction. All permeability ratios are determined by the

swelling strain only. It is apparent that zk is equal to xk if 000 zyx kkk . In order to

illustrate these conclusions graphically, we use the parameters in the Table 3-1 to quantify

the directional permeabilities. The calculated results are shown in Figure 3-5. We also

0

0

0

1

5.0

0

x

xm

x

xm

x

xm

k

kR

k

kR

k

kR

0

0

0

1

5.0

0

z

zm

z

zm

z

zm

k

kR

k

kR

k

kR

Matrix Pore Pressure, MPa

Pe

rme

abili

ty R

atio

0.01

0.1

1

10

0 1 2 3 4 5 6 7



3-17

compare the results for the constant volume case with the ones for the uniaxial strain case

in z-direction. For the constant volume case, 100% of the swelling strain contributes to the

total effective strain; for the uniaxial strain case, only a portion of the swelling strain

contributes to the total effective strain due to the unconstrained condition in the vertical

direction. Therefore, maximum permeability changes are induced under the displacement

controlled case.

Figure 3-5. Coal permeability as a function of matrix pore pressure under constant

reservoir volume and uniaxial strain conditions. Permeability ratios are equal in all

directions and regulated by the modulus ratio, mR : 1mR represents no fracture

influence; 0mR represents fracture influence only; 5.0mR combined influence of

fracture and matrix deformation. A cut-off permeability limit is applied for 0mR .

3.5 Field Case

It is generally believed that the in-situ response of a coal gas reservoir to gas production

(injection) can be approximated either by the uniaxial deformation-permeability model (Liu

and Rutqvist 2010) or by the constant volume-permeability model (Massarotto, 2009). In

this section, we apply both coal permeability models to a field case.

Mavor and Vaughn (1997) reported coal permeability results of three wells in the Valencia

Canyon area of the San Juan Basin, and found coal permeability increased between 2.7 and

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7

1mR

Strain Uniaxial5.0mR

Strain Uniaxial0mR

VolumeConstant 5.0mR

VolumeConstant 0mR

Matrix Pore Pressure, MPa

Pe

rme

abili

ty R

atio



3-18

7.1 times as gas pressure decreased. The initial gas pressures for these wells are 5.35, 6.60

and 6.41MPa, respectively. Because of the small differences, an average value of 6.12MPa

is used in this evaluation. The following mechanical properties and matrix swelling

parameters are taken directly from Liu and Rutqvist (2010): 3.0 , 2900E MPa,

55.2Lp MPa, 0043.0L . The initial fracture porosity is 0.05% (Mavor and Vaughn,

1997). These values are representative of the San Juan Basin coalbed.

For the uniaxial strain assumption, Equations (3-30) and (3-31) were used to evaluate the

permeability changes. In this case, the vertical permeability is different from the horizontal

permeability. This means that coal shrinkage induces the permeability anisotropy. In our

analysis, we match the horizontal permeabilities with the field data. For the constant

reservoir volume assumption, Equation (3-32) was used to evaluate the permeability

changes.

For both assumed conditions, only the modulus reduction ratio,

mR , is adjustable. Best

matches were achieved when 4.0mR and 6.0mR , respectively, as shown in Figure 3-6.

For the uniaxial deformation case, 6.01 mR , representing a partitioned strain of the

equivalent porous coal medium for the fracture system of 60%. For the constant volume

case, 4.01 mR , representing a partitioned strain of the equivalent porous coal medium

for the fracture system of 40%.

1

10

100

0 1 2 3 4 5 6 7

0k

k x

00 k

k

k

k xz

000 k

k

k

k

k

k zx

Case VolumeConstant 4.01

CaseStrain Uniaxial6.01

%05.03.02900

12.60043.055.2

0

0

m

m

LL

R

R

MPaE

MPapMPap

Pore Pressure, MPa

Pe

rme

abili

ty R

atio

p

p



3-19

Figure 3-6. Matching with field data through use of the uniaxial strain and constant

reservoir volume models.

3.6 Evaluation of Coupled Processes

A field scale model is used to simulate the performance of coalbed methane production

under in-situ conditions. Input parameters for this simulation are identical to the

parameters used in Section 3.5. The simulation model geometry is 200m by 200m with a

methane production well located at the lower left corner. For the coal deformation model,

all four sides are constrained in the normal direction. For the gas transport model, the coal

is saturated initially with CH4 and the initial pressure is 6.12MPa. A condition of

atmospheric pressure is applied at the boundary representing the production well.

Simulation results are presented in Figures 3-7 through 3-9.

Figure 3-7. Spatial and temporal evolution of coal permeability ratios on a diagonal radial

traverse from the production well.

1

10

100

0 20 40 60 80 100

Pe

rme

ab

ility

ra

tio

, k/k

0

Distance from Production well (m)

dayst 100

dayst 1000

dayst 5000

dayst 10000

6.0

0043.0

55.2

12.6

3.0

2900

005.0

8.0

0

0

m

L

L

f

R

MPap

MPap

MPaE

bp

100m

10

0m



3-20

Figure 3-8. Evolution of coal permeability at a specific evaluation point.

Figure 3-9. Evolution of the cumulative gas production and pore pressure.

In this simulation, the reservoir volume remains unchanged throughout the production.

This assumption requires that global strains within the coal seam are zero. However, this

constraint does not preclude desorption-induced shrinkage (or swelling) of individual coal

blocks and complementary opening (or closing) of fractures. The direct consequence of

0

5

10

15

20

25

30

35

40

0.1 1 10 100 1000 10000

Pe

rme

ab

lity ra

tio

, k/k

0

Time (day)

6.0

0043.0

55.2

12.6

3.0

2900

005.0

8.0

0

0

m

L

L

f

R

MPap

MPap

MPaE

bp

100m

10

0m

Evaluation

Point

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

0 500 1000 1500 2000 2500 3000

Po

re p

ressu

re (M

Pa

)

Ga

s p

rod

uctio

n (×

10

5 m

3)

Time (day)

6.0

0043.0

55.2

12.6

3.0

2900

005.0

8.0

0

0

m

L

L

f

R

MPap

MPap

MPaE

bp

100m

10

0m



3-21

these internal transformations is the isotropic change in permeability defined through

Equation (3-32). As shown in this equation, the change in coal permeability is defined only

by the swelling strain. This represents the ideal case, i.e., 100% of the swelling strain

contributes to the effective stress-induced coal deformation. However, only a portion of

the effective stress-induced coal deformation contributes to the permeability change as this

is modulated through the parametermR1 . In this case, mR1 is equal to 0.4. This

means that only ~40% of the total effective stress-induced coal strain (equal to the swelling

strain) is directly responsible for the permeability growth, as shown in Figures 3-7 and 3-8.

Figure 3-9 shows the relation between the cumulative gas production and the pore pressure.

3.7 Conclusions

A novel permeability model has been developed to define the evolution of gas sorption-

induced permeability anisotropy under in-situ stress conditions. This was implemented into

a fully coupled finite element model of coal deformation and gas transport in a coal seam.

Based on the model evaluations and the analysis of coupled processes, the model

adequately and consistently reflects the conceptual assumptions:

The directional permeability of coal is determined by the mechanical boundary

conditions, the ratio of coal bulk modulus to coal matrix modulus, the initial

fracture porosity, and the magnitude of the gas-pressure-induced coal swelling

strain. The boundary conditions control the magnitudes of total strains while the

modulus reduction ratio partitions the effective strain (total strain minus the

swelling strain) between fracture and matrix.

For restraint conditions of uniaxial strain and for a constant volume reservoir,

changes in coal permeability are determined only by the gas pore pressure and the

swelling strain. In both cases, the influence of effective stress is absent in the

permeability models.

Analysis including the effect of the fully coupled processes illustrates how coal

permeability evolves both in space and in time. These evolutions are the direct

outcomes of feedbacks of coal-gas interactions on the evolution of permeability,

stress deformation, gas flow and adsorption/desorption processes.



3-22

3.8 Acknowledgements

This work was supported by the Western Australia CSIRO-University Postgraduate

Research Scholarship, National Research Flagship Energy Transformed Top-up

Scholarship, and by NIOSH under contract 200-2008-25702. These various sources of

support are gratefully acknowledged.

3.9 References

Al-Yousef, H.Y., 2005. Permeability anisotropy measurement on whole cores—Analytical solution and

application. SPE 93559.

Amadei, B., Goodman, R. E., 1981. A 3D constitutive relation for fractured rock masses, Proceedings of the

International Symposium on the Mechanical Behavior of Structured Media, Ottawa, ON, pp. 249-268.

Busch, A., Gensterblum, Y., Krooss, B.M., Littke, R., 2004. Methane and carbon dioxide adsorption–

diffusion experiments on coal: upscaling and modeling. International Journal of Coal Geology 60,

151-168.

Ceglarska-Stefanska, G. and Brzoska, K., 1998. The effect of coal metamorphism on methane desorption.

Fuel 77(6), 645-648.

Ceglarska-Stefanska, G. and Czaplinski, A., 1993. Correlation between sorption and dilatometric processes in

hard coals. Fuel 72(3), 413-417.

Chaianansutcharit, T., Chen, H.Y., Teufel, W.L., 2001. Impacts of permeability anisotropy and pressure

interference on coalbed methane (CBM) production. SPE 71069.

Charrière, D., Pokryszka, Z., Behra, P., 2010. Effect of pressure and temperature on diffusion of CO2 and

CH4 into coal from the Lorraine basin (France). International Journal of Coal Geology 81(4), 373-380.

Chen, Z.W., Liu, J.S., Elsworth, D., Connell, D.L., Pan Z.J., 2010. Impact of CO2 injection and differential

deformation on CO2 injectivity under in-situ stress conditions. International Journal of Coal Geology

81 (2), 97-108.

Cui, X., and Bustin, R. M., 2005. Volumetric strain associated with methane desorption and its impact on

coalbed gas production from deep coal seams, The American Association of Petroleum Geologists:

Bulletin 89 (9)1181-1202.

Day, S., Fry, R., Sakurovs, R., 2008. Swelling of Australian coals in supercritical CO2. International Journal of

Coal Geology 74(1), 41-52.

Du, J.C., Wong, C.K.R., Choy, E., 2004. Effects of strain-induced anisotropy in permeability on the

deformation-flow-heat transfer in porous media. The 6th North America Rock Mechanics

Symposium, Houston, Texas. ARMA/NARMS 04-613.

Harpalani, S., and Schraufnagel, A., 1990. Measurement of parameters impacting methane recovery from coal

seams, International Journal of Mining and Geological Engineering 8, 369-384.

Gash B.W., Volz, R.F., Potter, G., and Corgan, J.M., 1993. The effects of cleat orientation and confining

pressure on cleat porosity, permeability and relative permeability in coal. The 1993 International

Coalbed Methane Symposium. The University of Alabama, Tuscaloosa, Alabama, USA, paper 9321.

Gray, I., 1987. Reservoir engineering in coal seams: Part 1- the physical process of gas storage and movement

in coal seams. SPE Reservoir Engineering 2(1), 28-34. SPE-12514-PA.

Jikich, A. S., McLendon, R., Seshadri, K., Irdi, G., Smith, H.D., 2009. Carbon dioxide transport and sorption

behavior in confined coal cores for carbon sequestration. SPE Reservoir Evaluation & Engineering

12(1), 124-136.



3-23

Koenig, R.A. and Stubbs, P.B., 1986. Interference testing of a coalbed methane reservoir. The SPE

Unconventional Gas Technology Symposium. The Society of Petroleum Engineers, Richardson,

Texas, USA. SPE 15225.

Li, H.Y., Shimada, S., Zhang, M., 2004. Anisotropy of gas permeability associated with cleat pattern in a coal

seam of the Kushiro coalfield in Japan. Environmental Geology 47, 45-50.

Liu, H.-H., and Rutqvist, J., 2010. A new coal-permeability model: Internal swelling stress and fracture–matrix

interaction. Transport in Porous Media. 82, 157–171.

Liu, J.S., Elsworth, D., Brady, B.H., 1999. Linking stress-dependent effective porosity and hydraulic

conductivity fields to RMR. International Journal of Rock Mechanics and Mining Sciences 36, 581-

596.

Massarotto, P., Golding, D.S., Rudolph, V., 2009. Constant volume CBM reservoirs: An important principle.

2009 International Coalbed & Shale Gas Symposium. Paper No.0926.

Mavor, M. J. and Vaughn, J.E., 1997. Increasing absolute permeability in the San Juan basin Fruitland

formation, Proc. 1997 Int. Coalbed Methane Symp., Tuscaloosa, Alabama, 12–16 May, 33–45.

Mazumder, S., Wolf, K.-H., 2008. Differential swelling and permeability change of coal in response to CO2

injection for ECBM. International Journal of Coal Geology 74 (2), 123-138.

Palmer, I., 2009. Permeability changes in coal: analytical modeling. International Journal of Coal Geology 77,

119-126.

Palmer, I., Mansoori, J., 1996. How permeability depends on stress and pore pressure in coalbeds: a new

model. SPE-52607.

Pekot, L.J., Reeves, S.R., 2002. Modeling the effects of matrix shrinkage and differential swelling on coalbed

methane recovery and carbon sequestration. U.S. Department of Energy DE-FC26-00NT40924.

Promeroy, C. D. and Robinson, D. J., 1967. The effect of applied stresses on the permeability of a middle

rank coal to water. International journal of rock mechanics and mining sciences 4, 329-343.

Prusty, B.K., 2007. Sorption of methane and CO2 for enhanced coalbed methane recovery and carbon

dioxide sequestration. Journal of Natural Gas Chemistry 17, 29-38.

Robertson, E.P., Christiansen, R.L., 2005. Measurement of sorption-induced strain. International Coalbed

Methane Symposium, University of Alabama, Tuscaloosa. Paper 0532.

Robertson, E.P., Christiansen, R.L., 2007. Modeling laboratory permeability in coal using sorption-induced-

strain data. SPE Reservoir Evaluation & Engineering 10(3), 260–269.

Saghafi, A., Faiz, M. and Roberts, D., 2007. CO2 storage and gas diffusivity properties of coals from Sydney

Basin, Australia. International Journal of Coal Geology 70, 240-254.

Sayers, C. M., 1990. Stress-induced fluid flow anisotropy in fractured rock. Transport in Porous Media 5, 287-

297.

Seidle, J.P., Huitt, L.G., 1995. Experimental measurement of coal matrix shrinkage due to gas desorption and

implications for cleat permeability increases. SPE-30010-MS.

Shi, J.-Q., Durucan, S., 2005. A model for changes in coalbed permeability during primary and enhanced

methane recovery. SPE Reservoir Evaluation & Engineering 8(4), 291–299.

Wang, G.X., Massarottoa, P., Rudolpha, V., 2009. An improved permeability model of coal for coalbed

methane recovery and CO2 geosequestration. International Journal of Coal Geology 77, 127-136.

White, C.M., Smith, D.H., Jones, K.L., et al., 2005. Sequestration of carbon dioxide in coal with enhanced

coalbed methane recovery: a review. Energy Fuels 19(3), 659–724.

Zhang, H.B., Liu, J., Elsworth, D., 2008. How sorption-induced matrix deformation affects gas flow in coal

seams: a new FE model. International Journal of Rock Mechanics and Mining Sciences 45(8), 1226-

1236.



4-1

Chapter 4

Effect of the Effective Stress Coefficient and Sorption-Induced Strain on the Evolution of Coal Permeability: Experimental Observations

Zhongwei Chena,b, Zhejun Panb, Jishan Liua, Luke D. Connellb, Derek Elsworthc

a School of Mechanical and Chemical Engineering, The University of Western Australia, WA 6009, Australia

b CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South, Victoria 3169, Australia


Abstract: Permeability is one of the most important parameters for CO2 injection in coal

to enhance coalbed methane recovery. Laboratory characterization of coal permeability

provides useful information of in situ permeability behaviour of coal seams when

adsorbing gasses such as CO2 are injected. In this study, a series of experiments have been

conducted for coal samples using both non-adsorbing and adsorbing gasses at various

confining stresses and pore pressures. Our observations have showed that even under

controlled stress conditions, coal permeability decreases with respect to pore pressure

during the injection of adsorbing gasses. In order to find out the causes of permeability

decrease for adsorbing gasses, a non-adsorbing gas (helium) is used to determine the

effective stress coefficient. In these experiments using helium, the impact of gas sorption

can be neglected and any permeability reduction is considered as due to the variation in the

effective stress, which is controlled by the effective stress coefficient. The results show that

the effective stress coefficient is pore pressure dependent and less than unity for the coal

samples studied. The permeability reduction from helium experiments is then used to

calibrate the subsequent flow-through experiments using adsorbing gasses, CH4 and CO2.

Through this calibration, the sole effect of sorption-induced strain on permeability change

is obtained for these adsorbing gas flow-through experiments. In this paper, experimental

results and analyses are reported including how the impact of effective stress coefficient is

separated from that of the sorption-induced strain on the evolution of coal permeability.

Keywords: effective stress coefficient; swelling and shrinking; adsorption; CO2 storage



4-2

4.1 INTRODUCTION

Knowledge of changes in coal permeability due to coal matrix swelling/shrinkage strain is

crucial for the evaluation of both primary gas production from coal reservoirs and for CO2-

enhanced coalbed methane recovery (ECBM) (van Bergen et al., 2009a,b). For primary gas

production, as the gas pressure decreases below the desorption pressure, methane is

released from the coal matrix to the fracture network and the coal matrix shrinks. As a

direct consequence of this matrix shrinkage, the fractures dilate and fracture permeability

correspondingly increases. Thus, a rapid initial reduction of fracture permeability (due to

decrease of pore pressure) is supplanted by a slow increase in permeability at later

production stage (due to matrix shrinkage). Whether the ultimate, long-term permeability is

greater or less than the initial permeability depends on the net results of these two

competing mechanisms. ECBM involves the injection of CO2 into coal seams to displace

methane recovered as an energy source, while providing the additional benefit of reducing

greenhouse gas emissions by storing the CO2 underground (White et al., 2005; Liu et al.,

2010b).

4.1.1 Experiments on coal swelling/shrinkage and permeability change

Coal swelling/shrinkage due to gas adsorption/desorption is a well-known phenomenon

and is regarded as a key component for coal reservoir permeability behaviour during

primary and enhanced coalbed methane recovery (e.g. Palmer and Mansoori, 2009; Shi and

Durucan, 2004). Measurements of the effects of gas desorption on coal volumetric strain

have been performed for the injection of different gasses. The implications for changes in

cleat permeability have been evaluated using a matchstick geometry model (Harpalani and

Schraufnagel, 1990; Palmer and Mansoori, 1996; Seidle et al., 1992; Seidle and Huitt, 1995;

Shi and Durucan, 2004; St. George and Barakat, 2001). Laboratory measurements of coal

swelling with gas sorption and the causes of swelling have been investigated by various

researchers (Bustin et al., 2008; Chikatamarla et al., 2004; Cui et al., 2007; Day et al., 2008;

Levine, 1996; Moffat and Weale, 1955; Pan and Connell, 2007; Pan and Connell, 2011;

Reucroft and Patel, 1986; Reucroft and Sethuraman, 1987; Robertson and Christiansen,

2005; St. George and Barakat, 2001; Wang et al., 2010). These studies investigated the

change in volumetric strain rate as a result of gas pressure change, and suggest that

swelling/shrinkage induced deformations dominate over the effective-stress-generated

deformations at low gas pressures for both carbon dioxide and methane.



4-3

Many experiments have been completed to evaluate the separate influences of the effective

stresses and sorption-induced strains on the evolution in permeability (Harpalani and Chen,

1997; Karacan, 2003; Mazumder et al., 2006; Pan et al., 2010a; Pini et al., 2009; Robertson

and Christiansen, 2007; Siriwardane et al., 2009). Patching (1965) found that the

permeability measured using carbon dioxide was somewhat less than the permeability by air

or nitrogen. Somerton et al. (1975) investigated the permeability of fractured coal using

methane and also observed that permeability measured using methane was 20–40% lower

than that using nitrogen. Investigations of the impact of adsorption and effective stress on

permeability change have shown an increase in permeability with decreasing effective stress

on the sample using non-adsorbing gas and a reduction in permeability caused by swelling

using an adsorbing gas (Pini et al., 2009; Pan et al., 2010a). Nevertheless, an increase of

permeability even with the injection of adsorbing gasses has also been observed

(Mazumder et al., 2006; Robertson and Christiansen, 2007; Mazumder and Wolf, 2008). In

addition, Karacan (2003) used X-ray CT technology to reveal the kinetics of the

heterogeneous processes occurring in a consolidated coal maintained under constant

pressure difference as the CO2 injection pressure and confining pressure were increased to

different levels. Lin et al. (2008) conducted a set of experiments under constant effective

stress condition (the difference between confining stress and pore pressure was kept

unchanged throughout the tests) with the injections of CO2, N2, and their binary mixture.

Gas composition and pressure influence on coal porosity and permeability was also

conducted by Mavor and Gunter (2006). The permeability decreases with escalating pore

pressure has been observed. The rebound of coal permeability with decreasing the pore

pressure under constant net effective stress was observed as well (Huy et al., 2010; Liu et al.,

2010a; Pini et al., 2009; Siriwardane et al., 2009). With the assumption of effective stress

coefficient to be unity, many experiments of moisture effect on coal physical properties

and gas sorption rate and sorption capacities have also been conducted, and a general

reduction trend of these parameters with increasing moisture has been observed (Clarkson

and Bustin, 2000; Gash, 1991; Pan et al., 2010b; Ozdemir and Schroeder, 2009).

4.1.2 Mesurements of effective stress coefficient

The law of effective stress was first presented by Terzaghi (1923) to explain the

consolidation of saturated soil and the interaction between fluid and soil. Including the

effective stress coefficient, the law of effective stress is (Biot, 1941).



4-4

ijijij p ' (4-1)

where is the effective stress coefficient which ranges from 0 to 1, expressed as

mKK1 ; K is the bulk skeletal modulus of the rock media and mK is the modulus of

rock matrix (grain modulus).

The effective stress coefficient, in a cross-section in the porous material, denotes the ratio

of the area occupied by the fluid to the total area (Bear, 1972). This coefficient describes to

which extent the fluid pressure counteracts elastic deformation of porous media (Alam et

al., 2010). For the granular soil (as shown in Figure 4-1), the contact area among grains is

very small, so any cross-section can be replaced by a curved face, so the corresponding

effective stress coefficient can be approximately assumed to be 1.0. However, for porous

rock (i.e. coal) composed of crystallization or cementation as shown in Figure 4-2, the

curved face, similar to that in Figure 4-1, does not exist. Thus the effective stress

coefficient is less than 1.0 (Zhang et al., 2009).

Figure4-1. Sketch of effective stress coefficient 1 for soil and soil-like materials (Zhang

et al., 2009).

Figure4-2. Sketch of pore structure in rock or coal media (Zhang et al., 2009).

Many researchers have studied the effective stress coefficient, but almost exclusively for

rocks. For instance, Walsh (1981) showed that α= 0.9 for a rock mass containing a polished

joint, and Kranzz (1979) defined α = 0.56 for rock containing a tensile joint. A linear

Cross section

Curved face

(a) Distribution of grains

(b) Cross section (c)Curved face

Pore

Solid

Arbitrary

section

(a) Pore distribution (b) Arbitrary section



4-5

increase of effective stress coefficient with pressure difference (defined as the difference

between confining stress and pore pressure) has also been observed (Ghabezloo et al.,

2009). Nevertheless, only a few experiments have been carried out on coal, even though

the importance of this parameter has been widely realized on both the primary gas

production of coal reservoirs and the predictions of gas outburst. Experimental

measurements on different coals have suggested that effective stress coefficient is not a

constant, and is a bilinear function of volumetric stress and pore pressure of coal

(Yangsheng et al., 2003), implying a change in bulk modulus or pore modulus with pore

pressure. Similarly, a series of loading-unloading cycles applied by gas pressure were

performed on coal samples using a non-adsorbing helium gas and a relationship was

established between the effective stress and gas pressure with 71.0 (St. George and

Barakat, 2001). Nevertheless, these determinations of the effective stress coefficient are

based on the measured bulk volumetric change, which means that volumetric strain is the

only physical property required for its calculation.

However, Robin (1973) considered the variation of permeability, k , instead of volumetric

strain, and derived another effective stress equation. In Robin's work, based on the pore

volume change, the following relation was used (Nur and Byerlee, 1971):

mKKK

11111

(4-2)

where K is the bulk modulus for pore system, and is porosity. K and

mK are porous

media bulk modulus and matrix bulk modulus respectively.

If porosity is very small (e.g. %10 ), the above equation can be simplified into:

)( KK

KKK

m

m

(4-3)

Similar to derivation of Biot's coefficient (i.e. Equation (4-1)), the effective stress

coefficient for the variation of permeability (or pore volume) can be expressed as:

ij

m

ije pKK

K

1 (4-4)

Therefore, the effective stress coefficient in this case can be defined as:

KKK m 1 (4-5)

It should be noted that both volumetric strain and the variation in permeability could be

used to quantify ‗elastic properties‘ of the media. As mentioned above, α is calculated on



4-6

the observed variation in bulk volumetric strain change. However, in this work, the

measurements are based on the observed variation in permeability and the measured value,

β, is also called the effective stress coefficient.

Based on the variation in permeability, Nur and Byerlee (1971) obtained 64.0 and

97.0 for the Weber sandstone. Bernabe (1987) studied the permeability of several

crystalline rocks and found that effective stress coefficient decreases with increasing

confining pressure due to the changes in the geometry of the cracks during closure.

Measurement of effective stress coefficient for coal is rare. Nevertheless, it is important

since coal is a weak rock and the grain compressibility is larger than expected and may have

a significant impact (Palmer, 2009; Zheng, 1993).

4.1.3 Objective of this study

In this work, a series of experiments are conducted using both non-adsorbing and

adsorbing gasses all under the constant pressure difference condition, which is defined as

the differential pressure between confining stress and pore pressure. Firstly, the effective

stress coefficient is obtained from flow-though experiments with a non-adsorbing gas

(helium). In these experiments, the impact of sorption induced strain is negligible.

Furthermore, Klinkenberg effect can be neglected at high pressures. Thus, any permeability

reduction is considered to be due solely to the variation in the effective stress coefficient.

Secondly, the permeability reduction from the experiments conducted with the non-

adsorbing gas is used to calibrate the subsequent experiments using adsorbing gasses

(carbon dioxide and methane). As a result, the effect of sorption-induced strain on

permeability change is obtained.

4.2 Experimental methods

4.2.1 Experiment apparatus

The schematic of the Triaxial Multi-Gas Rig used for this work is shown in Figure 4-3. The

rig is developed to measure gas adsorption, swelling, permeability for coal cores. Gas is

injected from the upstream injection pump to the sample. After the coal sample reaches

adsorption equilibrium, which usually takes a few days to a few weeks, transient method is

applied to measure permeability. The upstream cylinder is charged to about 30kPa higher

than the sample pressure while the downstream cylinder pressure is about 30kPa lower



4-7

than the sample pressure. Then gas is allowed to flow through the sample from the

upstream cylinder to the downstream cylinder. Permeability can be calculated from the

pressure decay curve measured by a differential pressure transducer installed between the

upstream and downstream cylinders. The calculation of permeability is described in the

next section. Radial and axial displacements are measured at each adsorption step to obtain

swelling/shrinkage strain. The sample cell and other parts of the rig are in a temperature

controlled cabinet to maintain constant temperature during the experiment. The core

sample is wrapped with a thin lead foil then a rubber sleeve before it is installed in the cell.

The thin lead foil is to prevent gas diffusion from the core to the confining fluid at high

sample pressures (Pan et al., 2010a).

Figure 4-3. Schematic plot of the tri-axial multi-gas rig.

Fig 1 Schematic plot of the Triaxial Multi-Gas Rig

Upstream

CH4

He

CO2

Downstreamm

CO2 Tube

Heater Upstream Cylinder Downstrea

Cylinder

Actuator

111.

122.

Vacuum

111.1 122.2

Pressure

Sensor

Pressure

Sensor

Pressure

Sensor

Pressure

Sensor

Pressure

Sensor

Pressure

Sensor

Pressure

Sensor

Upstream Injection

Pump

Confinement

Pump

Computer / DAQ

Water

Heater

Differential

Pressure

transducer



4-8

4.2.2 Permeability measurement

We use the pressure transient method to conduct the gas flow experiments in the low

permeability samples (Brace, et al., 1968; Hsieh et al., 1981), which has been widely used

due to its shorter test durations compared to steady state measurements. Permeability is

measured after the equilibrium state is reached for changes of either confining stress or

pore pressure. This method involves observing the decay of the differential pressure

between upstream and downstream cylinders across a sample of interest. The pressure

decay is combined with the cylinder volumes in the analysis to relate the flow through the

sample and thus determine the permeability (Brace et al., 1968; Hsieh et al., 1981;

Hildenbrand et al., 2002). The pressure decay curve can be modelled as:

tm

du

du ePP

PP

0,0,

(4-6)

where du PP is the pressure difference between the up and down stream cylinders used

for permeability tests, measured by a differential pressure transducer; 0,0, du PP is the

pressure difference between the up and down stream cylinders at initial stage (ranges from

60kPa to 100kPa for this work), t is the time and m is described below:

)11

(2

du

R

g VVV

LC

km

(4-7)

where k is permeability; gC is the gas compressibility; L is the sample length; RV is the

sample volume; uV and dV are the volume of the up and downstream cylinders.

Thus permeability can be obtained from Equation (4-7) after m is obtained from the

pressure decay curve. For this study, the approximate time allowed for each pressure decay

curve varies from several minutes to a few hours before pressure difference between the

upstream and downstream cylinders reach below 2.0 kPa.

4.2.3 Experimental procedure

Two Australian coal samples from the southern Sydney basin were used. The coal samples

were bituminous coal from the Bulli seam and cored to 4.50cm×10.55 cm (core: No.01)

and 4.55am×10.10 cm (core: No.02) in diameter and length, respectively.

The coal cores were first dried in a heated vacuum oven and the weight was measured.

Then the cores was installed in the sample cell and consolidated with a few load cycles



4-9

before carrying out experiments to make sure results were repeatable. The samples were

allowed to equilibrate for a few days inside the cell in the vacuum to remove the residual

gas and to reach the desired temperature.

Three gasses were used including helium, methane and carbon dioxide. Measurements were

conducted at 45°C and 35°C for core No.01 and No.02, respectively.

For core No.01 tests, permeability was firstly measured using helium at a constant pressure

difference (defined as confining stress minus pore pressure) of 3.0MPa. The injection

pressure started from 2.1MPa and increased by 2.0MPa for each step until reaching

10.1MPa. For methane and carbon dioxide injection, the pressure difference was kept at

2.0MPa during injection until the sorption reached equilibrium. The confining pressure was

changed to achieve the pressure differences of 4.0MPa and 6.0MPa, respectively. Four

different pore pressure steps were tested for both gasses. Permeability was measured after

each confining pressure or pore pressure change. After the measurements for each gas, coal

sample was vacuumed for a couple of days to remove the residual gas. The experimental

results for core No. 01 were also documented in Pan et al.'s work (2010a) and used for the

data analysis and modelling work presented in this paper.

For core No.02 tests, constant pressure differences with 2.0 and 3.0MPa were used for

helium injection. No CH4 flow-through experiments were conducted for this sample as

methane detector in the laboratory was out of order during the measurement so methane is

not allowed to use due to safety concerns. After the completion of helium test, the core

was vacuumed and CO2 was injected. For CO2 injection, the pressure difference was kept

at 2.0MPa until the pore pressure reached equilibrium. Then the confining pressure

changed to achieve the pressure difference of 3.0 and 4.0MPa. Three pore pressures of

CO2 injection were tested for this sample. Permeability is measured after each confining

pressure or pore pressure change for both helium and CO2.

4.2.4 Work flow of data analysis

The evaluation of coal effective stress coefficient and the elimination of effective stress

effect from sorption-induced strain effect are conducted through the following six

integrated steps: (1) measure permeability using helium gas under constant pressure

difference conditions; (2) calculate the effective stress coefficient; (3) calculate the

additional effective stress value and obtain the real effective stress value; (4) evaluate the



4-10

permeability change induced by the additional effective stress for adsorbing gas injection

under constant pressure difference conditions; (5) calibrate the permeability value to

eliminate the effective stress effect; and finally (6) obtain the sorption-induced strain effect

only on permeability change.

4.3 Results and discussion

4.3.1 Results for helium injection

Permeability measurements for the helium flow-though experiments are shown in Figure 4-

4. Permeability reductions with increasing pore pressure are observed for both cases under

the condition of constant pressure difference (defined as confining stress minus pore

pressure). Helium is almost a non-adsorbing gas to coal, thus the effect of matrix swelling

on permeability change is negligible. Therefore, the reduction in permeability can be

possibly attributed by two factors: (1) the Klinkenberg (1941) effect, especially in the low

pressure range; and (2) the effective stress effect. It has been observed that the Klinkenberg

effect diminishes with increase in gas pressure, because at high pressures (e.g. >2MPa) the

mean free path of the gas molecules (diameter is about 0.98 ˚A) is far less than the aperture

of the coal cleats (3–40 μm) (Laubach et al., 1998). Correspondingly, collisions between gas

molecules are more frequent than collisions between gas molecules and the solid walls

(Han et al., 2010). Therefore, the effective stress rather than the Klinkenberg effect is

considered as the cause for the permeability reduction at high pressures.



4-11

(a) Permeability vs. pore pressure with pressure difference for core No. 01.

(b) Permeability vs. pore pressure with pressure difference for core No. 02.

Figure 4-4. Permeability measured using helium.

Coal cleat compressibility calculation

In order to calculate the effective stress change, we firstly need to know how the effective

stress determines permeability change. An exponential functional form has been widely

y = 0.9922e-0.011x

R² = 0.9665

0.7

0.8

0.9

1

0 2 4 6 8 10 12

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)

Pressure difference=3.0MPa

Flowing fluid: HeliumCore: No. 01

y = 1.1746e-0.056x

R² = 0.9887

y = 0.9366e-0.044x

R² = 0.9844

0.7

0.8

0.9

1

2 3 4 5 6 7

Pe

rme

ab

ilit

y v

alu

e, m

d

Pore pressure, MPa






4-12

used to calculate permeability with effective stress variation (Mckee et al., 1988; Seidle and

Huitt, 1995), and by combining with the influence of the effective stress coefficient on

effective stress change, the relation can be given as

pC fekk

3

0 (4-8)

where k is the permeability, 0k is the initial permeability and fC is the cleat

compressibility.

It should be noted that the original work from both Mckee et al. (1988) and Seidle and

Huitt (1995) have taken the effective stress coefficient to be unity, which means same

amount change of either pore pressure or confining stress has the same impact on effective

stress change.

From Equation (4-8) we can see that in order to calculate cleat compressibility value, the

following two steps are required: the pore pressure should be kept constant ( 0p ) and

only confining stress ( ) is allowed to change; permeability change with different

confining stress values should be measured. Then cleat compressibility values can be

obtained by regressing the permeability change with confining stress change

(

fCekk

3

0 ).

The experimental results for cleat compressibility in the presence of different gasses are

listed in Table 1.

Table 4-1. Compressibility values regarding to different gasses and pressures

Core Number Pore

pressure fC

Pore

pressure fC

Pore

pressure fC

(MPa) (MPa-1) (MPa) (MPa-1) (MPa) (MPa-1)

Helium CH4 CO2

No.01

(Pan et al., 2010a)

2.1 0.0848 0.9 0.0507 3.0 0.0606

10.1 0.0485 3.4 0.0472 6.4 0.0654

No.02

2.0 0.118 - - 1.3 0.142

3.0 0.102 - - 3.6 0.599

5.0 0.098 - - 5.0 0.687

The average compressibility values for helium, 0669.01 fC MPa-1 and 106.02 fC MPa-1

for cores No. 01 and 02, were used for the effective stress coefficient calculation.



4-13

Coal effective stress coefficient calculation:

Because the pressure difference was kept unchanged throughout the helium injection test

( p ), Equation (4-8) can be simplified to.

pC fekk

13

0 (4-9)

Substituting cleat compressibility values into the above equation, then the effective

coefficients (β) 0.945 for core No.01, and 0.842 and 0.855 for No.02 at pore pressures of

2.0 and 3.0MPa were obtained.

Therefore, the effective stress coefficient is not a constant ( 0.1 ) and it may increase

with increasing pore pressure. In this study, the incremental effective stress, due to not

unity effective stress coefficient ( 0.1 ), is called the additional effective stress, which

can be calculated by term p 1 . This stress should be added into the effective stress

term. In the following, the additional effective stress is recalculated for each case, and the

comparison for helium injection between original results and corrected magnitudes are

shown in Figure 4-5.

(a) Comparison of permeability behaviour for core No.01.

y = 1.3942e-0.146x

R² = 0.9845

y = 2.0956e-0.254x

R² = 0.9976

y = 2.1598e-0.255x

R² = 0.9977

y = 1.5127e-0.146x

R² = 0.9847

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 1 2 3 4 5 6 7

Pe

rme

ab

ilit

y (

md

)

Effective stress (MPa)

Pore Pressure=10.1MPa


Corrected Value for 2.1 Mpa

Corrected Value for 10.1 Mpa




4-14

(b) Comparison of permeability behaviour for core No.02.

Figure 4-5. Comparison of permeability behavior between constant pressure difference

method and constant effective stress method for both cores.

It shows that the constant pressure difference method (original data) underestimates the

effective stress effect on permeability change. The underestimation becomes more

significant as the pore pressure increases.

In order to compare the difference between original results and corrected results, an

arbitrary error e is introduced and defined as

%100ty valuepermeabili Corrected

ty valuepermeabili Original-ty valuepermeabili Correctede (4-10)

For core No.01, the error changes from 3.44% at 2.1MPa to 9.57% at 10.1MPa. For core

No.02, the error is 13.7% for pore pressure at 2.0MPa. Hence the difference is significant

especially at higher pore pressure and for coal with a small effective stress coefficient.

4.3.2 Results for methane injection

It is reasonable to assume that the effective stress coefficient is constant for each test, as

the pressure difference was kept constant in our experiment. Based on the calculated

effective stress coefficient, β, the permeability data are corrected by subtracting the

additional effective stress induced permeability reduction from the original data. Equation

y = 2.5788e-0.355x

R² = 0.9933

y = 2.8898e-0.355x

R² = 0.9933

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5 2 2.5 3 3.5 4 4.5

Pe

rme

ab

ilit

y (

md

)

Effective stress (MPa)

Flow fluid: HeliumCore: No.02


Corrected valuel for 2.0MPa



4-15

(4-9) is used to calculate this additional permeability change. The initial permeability for

each case is calculated from the calibrated expression for helium, as shown in Figure 4-5.

With methane injection, the comparison between original permeability data and the

corrected ones for core No.01 is shown in Figure 4-6.

(a) Comparison of permeabilities for a pressure difference of 2.0MPa.

(b) Comparison of permeabilities for a pressure difference of 4.0MPa.

y = 0.8693e-0.06x

R² = 0.9984

y = 0.8513e-0.037x

R² = 0.9873

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 2 4 6 8 10 12 14

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)


Corrected permeability value(2.0MPa)

Flowing fluid: MethaneCore: No. 01

y = 0.6172e-0.048x

R² = 0.9879

y = 0.6083e-0.026x

R² = 0.9324

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 5 10 15

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)


Corrected permeability value (4.0MPa)




4-16

(c) Comparison of permeabilities for a pressure difference of 6.0MPa.

Figure 4-6. Comparison of permeabilities recovered by the constant pressure difference

method and constant effective stress method for CH4 injection for Core No.01.

4.3.3 Results for carbon dioxide injection

Following the same procedure as for the methane injection data, the comparison between

the uncorrected permeability data and corrected data for core No.01 and core No.02 using

CO2 are shown in Figures 4-7 and 4-8, respectively. Core No.01 involves four different

pore pressure steps, namely 3.0, 6.4, 9.8 and 13.3 MPa, and core No.02 also involves three

different pore pressure steps, namely 1.3, 3.6 and 5.0MPa.

y = 0.4636e-0.046x

R² = 0.9911

y = 0.4573e-0.024x

R² = 0.9469

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 5 10 15

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)






4-17



y = 0.7686e-0.087x

R² = 0.9941

y = 0.7053e-0.049x

R² = 0.9987

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)



Flowing fluid: CO2Core: No. 01

y = 0.5665e-0.109x

R² = 0.9791

y = 0.4956e-0.054x

R² = 0.9485

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)






4-18


Figure 4-7. Comparison of permeabilities between the constant pressure difference method

and the constant effective stress method for CO2 injection for core No.01.


y = 0.4999e-0.163x

R² = 0.9728

y = 0.3822e-0.078x

R² = 0.9461

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)




y = 1.1421e-0.451x

R² = 0.9569

y = 0.8036e-0.14x

R² = 0.9981

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 1 2 3 4 5 6

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)


Corrected value for 2.0MPa




4-19



Figure 4-8. Comparison of permeabilities obtained between the constant pressure

difference method and the constant effective stress method for CO2 injection for core

No.02.

4.4 Discussion

From Figures 4-6 to 4-8, we can see that generally the original data follow an exponential

function better than the corrected one. The original data includes the effects from two

factors: effective stress and sorption-induced strain, while the corrected data includes the

y = 1.6564e-0.945x

R² = 0.8936

y = 0.5445e-0.218x

R² = 0.9956

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 1 2 3 4 5 6

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)




y = 1.1153e-0.973x

R² = 0.9434

y = 0.388e-0.275x

R² = 0.9631

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 1 2 3 4 5 6

Pe

rme

ab

ilit

y (

md

)

Pore Pressure (MPa)






4-20

sorption-induced strain only, which follows a Langmuir like equation. This comparison

supports that the corrected results represent the impact from sorption-induced strain only.

A summary of the permeability errors, as defined in Equation (4-10), for both cores No.01

and 02 are listed in Tables 4-2 and 4-3. For both coal samples, a significant difference of

permeability change vs. pore pressure is obtained between original results (β=1.0) and

corrected results. This demonstrates that the constant pressure difference method

overestimates the sorption-induced permeability change especially in the high pore pressure

range. This is because part of effective stress induced permeability reduction is considered

to be from sorption-induced strain. The overestimation is enhanced as the pore pressure

increases. For core No.01, an error as high as 27.86% is observed with CH4 injection within

the pressure range of the experiments. This error may be as large as 60% with CO2

injection. For core No.02, it is even higher than for core No.01 with CO2 injection. This

indicates that it is necessary to correct the permeability data by eliminating the influence of

the additional effective stress from the swelling strain.

Table 4-2. Summary of permeability errors under different pressure difference values with

CH4 injection

Pore pressure

(MPa)

Error (2.0MPa)

(%)

Error (4.0MPa)

(%)

Error (6.0MPa)

(%)

Core No.01

0.92 1.18 1.56 1.39

3.54 5.20 6.97 5.79

7.41 12.18 15.58 13.41

12.80 24.33 27.86 24.08

Table 4-3. Summary of permeability errors under different pressure difference values for

CO2 injection

Pore pressure

(MPa)

Error (2.0MPa)

(%)

Error (4.0MPa)

(%)

Error (6.0MPa)

(%)

Core

No.01

3.0 5.36 6.70 6.94

6.4 13.02 15.80 17.39

9.8 23.44 32.12 40.76

13.3 36.28 46.03 61.04

Pore pressure

(MPa)

Error (2.0MPa)

(%)

Error (3.0MPa)

(%)

Error (4.0MPa)

(%)

Core 1.3 12.72 12.19 9.93



4-21

No.02 3.6 43.74 54.09 57.03

5.0 73.61 94.96 94.06

In the following part, comparison of permeability change between sorption-induced and

effective stress induced is conducted to evaluate the contribution from each process.

Based on the corrected permeability data, the absolute permeability reduction value, rk ,

induced by sorption strain with different gas species is calculated to compare the

contribution of each component on permeability change, i.e. due to CO2 and CH4. rk is

defined as

fmmr kkk 0 (4-11)

where mk0

is the initial permeability at effective stress m with zero pore pressure, and fmk

represents permeability value at the final experimental pore pressure (e.g. 13.3MPa for core

No.01 and 5.0MPa No.02) at effective stress m .

The comparison is shown in Figure 4-9. This figure demonstrates that, for core No.01,

CO2 injection can cause about a 1.5 times larger permeability reduction than CH4 under the

same pressure conditions, and about 2.1 times larger than effective stress induced

permeability reduction using Helium. Similar phenomena for core No.02 are also observed,

showing CO2 could cause a 1.9 times larger permeability reduction than effective stress

induced permeability reduction using helium. Note that the first point for effective stress

induced permeability change curve of core No.02 is quite different from other points; this

may be due to the difficulty to estimate the initial coal permeability value at atmosphere

condition, which is regressed from other pore pressure measurements in this study.



4-22

Figure 4-9. Comparison of permeability change caused by compaction and swelling strain.

This set of tests shows that the effective stress induced permeability reduction is close to

that induced by CH4 injection, and the difference to CO2 injection is not as significant as

field observations. Field trails have shown dramatic permeability reduction/increase with

adsorbing gas injection/production (Fujioka et al., 2010; Mavor and Vaughn, 1998; Mavor

et al., 2004; Palmer, 2009; van Bergen et al., 2009a,b; Wong et l., 2007). This difference

could be attributed to the difference between the laboratory experimental conditions and in

situ conditions. Under in situ conditions the effect is superimposed upon the response

which is less likely in a small coal sample tested in the laboratory. The sorption-induced

strain may close these natural cleats, and in turn causes dramatic changes in permeability.

However, under experimental conditions, the cores are normally consolidated before the

experiments, which artificially close the fractures and in turn diminishes the sorption-

induced strain effects.

4.5 Conclusions

This study has demonstrated that the effective stress coefficient for coal is not equal to

unity and may be effective stress dependent. This conclusion is derived based on the

experimental observations of a series of gas flow-through experiments all under the

conditions of a constant pressure difference between confining stress and pore pressure.

First, the effective stress coefficient is obtained by the non-adsorbing gas (helium) flow-

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1 2 3 4 5 6 7

Pe

rme

ab

ilit

y r

ed

uc

tio

n (

md

)

Pressure difference (MPa)

CO2 (Core No.02)

CO2(Core No.01)

CH4 (Core No.01)

Effective stress induced (Core No.01)

Effective stress induced (Core No.02)



4-23

though experiments. In these experiments, the impact of gas sorption is negligible and any

permeability change with pore pressure is considered to be due to the variation in the

effective stress coefficient. Second, the change in permeability resulting from the non-

adsorbing gas experiments is used to calibrate the subsequent experiments using adsorbing

gasses (carbon dioxide and methane) where the sole effect of sorption-induced strain on

permeability change is obtained. This finding may be more important because even though

coal is a weak rock, the grain compressibility is larger than expected, especially at high pore

pressures. Comparison between measured permeability data and calibrated results

demonstrates that the effective stress coefficient could play an important role in the

evaluation of permeability change in adsorbing gasses.

4.6 Acknowledgments

This work was supported by WA:ERA, the Western Australia CSIRO-University

Postgraduate Research Scholarship, National Research Flagship Energy Transformed Top-

up Scholarship, and by NIOSH under contract 200-2008-25702. These supports are

gratefully acknowledged. We also wish to express our sincere thanks to Mr. Michael

Camilleri, for his assistance and guidance in carrying out these tests in this work.

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5-1

Chapter 5

Influence of the effective stress coefficient and sorption-induced strain on the evolution of coal permeability: Model development and analysis

Zhongwei Chena,b, Jishan Liua, Zhejun Panb, Luke D. Connellb, Derek Elsworthc


b CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South, Victoria 3169, Australia


ABSTRACT: A series of coal permeability experiments was conducted for coal samples

infiltrated both with non-adsorbing and adsorbing gasses - all under conditions of constant

pressure difference between the confining stress and the pore pressure. The experimental

results show that even under controlled stress conditions, coal permeability decreases with

respect to pore pressure during the injection of adsorbing gasses. This conclusion is

apparently not congruent with our conceptual understanding: when coal samples are free to

swell/shrink then no effect of swelling/shrinkage strain should be apparent on the

permeability under controlled stress conditions. In this study, we developed a

phenomenological permeability model to explain this enigmatic behavior of coal

permeability evolution under the influence of gas sorption by combining the effect of

swelling strain with that of the mechanical effective stress. For the mechanical effective

stress effect, we use the concept of natural strain to define its impact on the change in

fracture aperture; for the swelling strain effect, we introduce a partition ratio to define the

contribution of swelling strain to the fracture aperture reduction. The resulting coal

permeability model is defined as a function of both the effective stress and the swelling

strain. Compared to other commonly used models under specific boundary conditions,

such as Palmer-Mansoori (P-M), Shi-Durucan (S-D) and Cui-Bustin (C-B) models, our

model results match the experimental measurements quite well. We match the experimental

data with the model results for the correct reason, i.e., the model conditions are consistent

with the experimental conditions (both are stress-controlled), while other models only

match the data for a different reason (the model condition is uniaxial strain but the

experimental condition is stress-controlled). We have also implemented our permeability

model into a fully coupled coal deformation and gas transport finite element model to



5-2

recover the important non-linear responses due to the effective stress effects where

mechanical influences are rigorously coupled with the gas transport system.

Keywords: coal permeability; swelling strain; effective stress effect; CO2 storage



5-3

5.1 Introduction

Coal Bed Methane (CBM) is naturally occurring methane gas (CH4) in coal seams. Methane

was long considered a major problem in underground coal mining but now CBM is

recognized as a valuable resource. Australia has vast reserves of coal-bed methane (about

310 to 410 trillion m3) (White et al., 2005) and has attracted billions of dollars in foreign

investment to develop this resource. CBM recovery triggers a series of coal-gas interactions.

For gas production, the reduction of gas pressure increases effective stress which in turn

closes fracture aperture and reduces the permeability (McKee et al., 1988; Seidle and Huitt,

1995; Palmer and Mansoori, 1996). As gas pressure reduces below the desorption point,

methane is released from coal matrix to the fracture network and coal matrix shrinks. As a

direct consequence of this matrix shrinkage the fractures dilate and fracture permeability

correspondingly increases (Harpalani and Schraufnagel, 1990). Thus a rapid initial

reduction in fracture permeability (due to change in effective stress) is supplanted by a slow

increase in permeability (with matrix shrinkage). Whether the ultimate, long-term,

permeability is greater or less than the initial permeability depends on the net influence of

these dual competing mechanisms (Shi and Durucan, 2004; Chen et al., 2008; Connell,

2009). Therefore, understanding the transient characteristics of permeability evolution in

fractured coals is of fundamental importance to the CBM recovery and CO2 storage in coal,

which has dual and complementary benefits: the enhanced production of methane and

concurrent long-term storage of CO2.

A broad variety of models have evolved to represent the effects of sorption, swelling and

effective stresses on the dynamic evolution of permeability over last few decades. In the

latest review (Liu et al., 2011), these models are classified into two groups: permeability

models under conditions of uniaxial strain and permeability models under conditions of

variable stress.

Somerton et al. (1975) investigated the permeability of fractured coal to methane and

presented a correlation equation in the prediction of permeability with mean stress. Gray

(1987) considered the changes in the cleat permeability as a function of the prevailing

effective horizontal stresses, and firstly incorporated the influence of matrix shrinkage into

a permeability model. Seidle and Huitt (1995) developed a conceptual matchstick model to

explain coal permeability decrease with increasing effective stress. Other stress-based coal

permeability models include Harpalani and Chen (1997), Gilman and Beckie (2000), Shi



5-4

and Durucan (S-D) (2004), Cui and Bustin (C-B) (2005). Based on cubic geometry,

Robertson-Christiansen (2006) described the derivation of a new equation that can be used

to model the permeability behaviour of a fractured, sorptive-elastic medium, such as coal,

under variable stress conditions. Ma et al. (2010) proposed a permeability model based on

the volumetric balance between the bulk coal, solid grains and pores, using the constant

volume theory proposed by Massarotto et al. (2009).

A number of coal permeability models were developed based on strains. Mckee et al. (1988)

developed a theoretical permeability model using matrix compressibility as a fundamental

property, but did not include the effect of sorption-induced strain on permeability change.

Sawyer et al. (1990) proposed a permeability model assuming that fracture porosity (to

which permeability can be directly related) is a linear function of changes in gas pressure

and concentration. Palmer and Mansoori (P-M) (1996) presented a theoretical model for

calculating pore volume compressibility and permeability in coals as a function of effective

stress and matrix shrinkage. The P-M model was updated in Palmer et al. (2007). Similarly,

the Advanced Resources International (ARI) group developed another permeability model

(Pekot and Reeves, 2002). This model does not have a geomechanics framework, but

instead extracts matrix strain changes from a Langmuir curve type of strain versus reservoir

pressure, which is assumed to be proportional to the gas concentration curve. Zhang et al.

(2008) developed a permeability model under variable stress conditions, and was extended

to CO2-ECBM conditions by Chen et al. (2009; 2010). Connell et al. (2010) presented two

analytical permeability models for tri-axial strain and stress conditions.

Pan and Connell (2007) developed a theoretical model for sorption-induced strain and

applied to single-component adsorption/strain experimental data. Clarkson (2008)

expanded this theoretical model to calculate the sorption-strain component of the P-M

model. Pan and Connell (2011a) developed an anisotropic swelling model based on their

swelling model (Pan and Connell, 2007). The dependence of coal permeability on pore

volume compressibility was also investigated (Shi and Durucan, 2010; Tonnsen and

Miskimins, 2010).

As reviewed above, there are a large collection of coal permeability models from empirical

ones to theoretical ones. These models normally have a set of common assumptions: (1)

the overburden stress remains constant; (2) coal deforms under the uniaxial strain

condition; (3) the effective stress coefficient is assumed as one; and (4) the sorption-

induced strain is totally counteracted by the closure of the fracture aperture. These



5-5

assumptions have limited their applicability as Liu et al. (2011) concluded that current

models have so far failed to explain the results from stress-controlled shrinkage/swelling

laboratory tests and have only achieved some limited success in explaining and matching in

situ data. Liu et al. (2011) considered the main reason for these failures is the impact of coal

matrix-fracture compartment interactions has not yet been understood well and further

improvements are necessary as demonstrated in latest studies (Connell et al., 2010; Liu and

Rutqvist, 2010; Izadi et al., 2011). In this study, a coal permeability model based on coal

matrix-fracture interaction was developed and then implemented into a fully coupled coal

deformation and gas transport finite element model to recover the important non-linear

responses due to the effective stress effects.

5.2 Permeability model development

Previous work of Chen et al. (2011) has reported the findings of a series of experiments

conducted for coal samples infiltrated both with non-adsorbing and adsorbing gasses - all

under conditions of constant pressure difference between the confining stress and the pore

pressure. Observations have demonstrated that even under controlled stress conditions the

injection of adsorbing gasses actually does reduce coal permeability. The swelling strain

effect has also been separated from the effective stress effect. In this section, we combined

the swelling strain effect with the mechanical effective stress effect into a

phenomenological permeability model to explain the enigmatic behaviour of coal

permeability evolution under the influence of gas sorption.

Experimental observations have shown that swelling response to the infiltration of CO2

exhibits two key features: (1) as CO2 infiltrates coal fracture, coal matrix swells and

permeability generally reduces if the gas pressure is not very high. This occurs regardless of

the mechanical constraint applied to the cracked coal sample; (2) the permeability recovers

with increasing gas pressure as effective stress effects dominate in the absence of swelling-

induced closure.

We use the idealized model as illustrated in Figure 5-1 to represent a single fracture within

a representative elementary volume. This representation is through two steps: the effective

stress is applied first in a non-adsorbing medium as shown in Figure 5-1(a) and then only

the pore pressure in an adsorbing medium as shown in Figure 5-1(b).



5-6

(a) (b)

Figure 5-1. Illustration of two-step loading process: (1) effective stress is applied in a non-

adsorbing medium; (2) pore pressure is applied in an adsorbing medium. (a) Effective

stress effects; (b) Swelling strain effects. p is pore pressure, is overburden stress, b

is fracture aperture and s is sorption-induced bulk dimension change.

5.2.1 Evaluation of effective stress effects

In this section, we define the behaviour of coal fracture where the effective stress is applied

in a non-adsorbing medium. Coals are viewed as naturally fractured reservoirs with a matrix

that is often assumed to have a negligible permeability in comparison to the fracture system.

These fractures in coal are known as cleats with the cleat aperture sensitive to the effective

stress, and increased effective stress acting to decrease the cleat aperture and thus

permeability.

A number of empirical and theoretical expressions exist in the literature for describing the

observed relationship between effective stress and fracture aperture (Daley, 2006; Liu et al.,

2009).

Liu et al. (2009) argued that porous and fractured rock (or coal) is inherently heterogeneous

and includes both a solid phase and pores (or fractures). Thus, an accurate description of

the deformation of the rock (or coal) is important for coupled mechanical and hydrological

processes, because fluid flow occurs in pores and fractures.

To deal with this issue, it is conceptualized that the fracture system has two parts, which

are subject to the same stress, but follow different varieties of Hooke's law: the hard part

follows the engineering-strain based Hooke's law, and the soft part obeys the natural-strain

∆s

+

σ

σ

p

p

<b><b>p p p

p pp



5-7

based Hooke's law, as shown in Figure 5-2. This treatment is consistent with previous

studies (Mavko and Jizba, 1991; Berryman, 2006; Liu et al., 2009).

Figure 5-2. Conceptualization of fracture system with the hard and soft parts. They follow

engineering-strain based and natural-strain based Hooke's law, respectively. 0b is the totally

unstressed fracture aperture, eb ,0and

tb ,0 are the unstressed fracture apertures for the hard

part and the soft part, respectively.

Considering a fracture to be embedded into a core sample subject to a stress, , Hooke's

law for the hard part can be expressed as

ebe dKd , (5-1)

where eK is the bulk modulus for the hard part of fracture system, and

eb, is the

engineering strain of fracture aperture (Jaeger et al., 2007), defined as

e

eeb

b

dbd

,0

, (5-2)

where eb ,0 is the unstressed fracture aperture for the hard part.

For the soft part, the following relation is used

tbt dKd , (5-3)

wheretK is the bulk modulus for the soft part of fracture system, and

tbd , is the true or

natural strain, defined as

t

ttb

b

dbd , (5-4)

b0,t

Soft Part

b0,e

Hard Part

b0



5-8

where tb is the fracture aperture for soft part under the current state of stress. Subscripts e

and t (for ―engineering‖ and ―true‖, respectively) refer to the ―hard‖ and ―soft‖ parts in

coal fracture system.

Freed (1995) provided a historical review of the development of the concept of natural

strain and argued that the natural strain should be used for accurately describing material

deformation.

Using the condition that tt bb ,0 and

ee bb ,0 for 0 , the engineering strain and natural

strain can be integrated into the following expressions

e

eee

Kbb

1,0 (5-5)

t

ett

Kbb

exp,0 (5-6)

Based on the above analysis, the total fracture aperture, stress

b , under stressed conditions

can be given as

testressbbb (5-7)

Combining the relation of fracture aperture change for both ―hard‖ and ―soft‖ parts, as

shown in Equations (5-5) and (5-6), yields the fracture aperture change with effective stress

t

et

e

eestress K

bK

bb

exp1 ,0,0 (5-8)

eK is generally several orders larger than tK . Therefore, the above equation can be

simplified into

t

etestress K

bbb

exp,0,0 (5-9)

From Figure 5-2 we can see that te bbb ,0,00 , thus the following expression can be

obtained.

)exp(,00,0 efeestressCbbbb (5-10)

where fC is coal fracture compressibility, defined as

tf KC 1 . 0b is the initial unstressed

total fracture aperture opening.



5-9

5.2.2 Evaluation of sorption-induced strain effects

We represent the behaviour of coal permeability where fluid pressures are applied in an

adsorbing medium. When the influence of swelling strain is investigated, a common way is

to assume that the swelling strain is totally accommodated by the closure of fracture

aperture, which could dramatically overestimate the influence of the swelling strain

(Robertson, 2005; Connell et al., 2010; Liu and Rutqvist, 2010). Based on the illustration

from Figure 5-1, we believe that only part of total swelling strain contributes to fracture

aperture change and the remaining portion of the swelling strain contributes to coal bulk

deformation, and a partition factor, f , is introduced to estimate this contribution, then

the fracture aperture change can be given as

3

s

swellingsfb

(5-11)

whereswelling

b is the fracture aperture change induced by the swelling strain only, and s is

the fracture spacing, f ranges from 0 to 1.0, s is the volumetric free swelling strain

change, which can be calculated as

LL

Lpp

p

pp

p

0

0 with Langmuir type, as shown

in Figure 5-3 (Pan et al., 2011b), and 31 ssf term accounts for coal bulk

deformation. L and Lp represent maximum volumetric swelling strain and Langmuir

pressure constant, respectively.



5-10

Figure 5-3. Volumetric swelling strain with Langmuir type function matching: experimental

data is shown with dot points, and solid line is the matching curve with Langmuir type

function.

The influence of swelling process on the internal stress distribution of coal fracture is

analysed in the following section. From Figure 5-1 we can see that after coal matrix swells,

the swelling strain increases the contact area of the cleat system, and in turn closes the

fracture. The cross-section of a representative element was shown in Figure 5-4, where the

contact area of the cleat system varies with the swelling process. Because the total stress (or

confining stress) along this section is kept constant during swelling process (second stage),

this internal stress along the cross section should always be balanced with surrounding

boundary. Therefore, the stress balance expression is

fefbeb ApAp 11 (5-12)

where eb and

ef are the equivalent internal effective stress before and after swelling

respectively, and bA and

fA are the effective contact areas before and after swelling,

respectively.

Because fb AA for the swelling case, the internal effective stress actually increases with

the increase of the contact area during coal matrix swelling, as shown from Equation (5-12).

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12

Vo

lum

etr

ic s

we

llin

g s

tra

in (

%)

Pore pressure (MPa)

Experimental Data

Experimental Data

Experimental Data

Modeled Data

Modeled Data

Modeled Data

(CO2)

(CH4)

(N2)

(CO2)

(CH4)

(N2)



5-11

It was considered that the increase in effective stress is responsible for the permeability

change.

(a) (b)

Figure 5-4. Illustration of contact area change due to gas sorption: (a) pre-swelling stress

state; (b) post-swelling stress state. bA and

fA are the effective contact areas before and

after swelling, respectively.

5.2.3 Development of coal permeability model

Combining the effective stress effect (Equation (5-10)) with the swelling strain effect

(Equation (5-11)) gives the resultant fracture aperture

s

f

ef

ee fC

b

bb

b

bbb

00

,00

0

,0

0 )exp( (5-13)

where 0f is the initial fracture porosity, defined as sbf /3 00 .

For a negligibly small residual fracture aperture (0,0 bb e and 0,0 eb ), Equation (5-13)

can be simplified to

s

f

ef

fCbb

0

0 )exp( (5-14)

Simplifying the above equation yields

sfef SCbb )exp(0 (5-15)

where 0/ ff fS .

Based on the cubic law between aperture change and permeability (Mckee et al., 1988;

Seidle and Huitt, 1995), the permeability change can be expressed as

p

σ

p p p p p p p

σ



5-12

33

00

)exp( sfef SCb

b

k

k

(5-16)

Coal permeability model as shown in Equation (5-16) can be extended to the three-

dimensional case. In the analysis of coal permeability the fractured coal mass is treated as a

discontinuous medium comprising both matrix and fractures (cleats). The individual matrix

blocks are represented by cubes and may behave isotropically with regard to

swelling/shrinkage, and mechanical deformability (Liu et al., 1999). The cleats are the three

orthogonal fracture sets and may also have different apertures and mechanical properties

ascribed to the different directions. Changes in coal permeability are determined by the

redistribution of effective stresses or strains due to changed conditions such as gas

injection. Typically, stresses and strains evolve at different rates in the different Cartesian

directions, and result in anisotropic permeabilities. In simulation study, the following 3D

permeability model can be implemented into numerical models

33

00

)exp( sxfexf

xx SCb

b

k

k

(5-17a)

33

00

)exp( syfeyf

yySC

b

b

k

k

(5-17b)

33

00

)exp( szfezf

zz SCb

b

k

k

(5-17c)

where ib and

ei are the cleat opening and effective stress in i direction, respectively.

zyxi ,, .

5.2.4 Physical meaning of sensitivity ratio

The initial fracture porosity, 0f , represents the fractured extent of the coal media, and the

partition factor, f , defines the influence of both injection gas components and boundary

conditions. Different gas components with different boundary conditions may have

different partition magnitudes.

In order to explain the physical meaning of the new parameter,fS , the fracture strain

change,bs , induced by coal swelling only is defined as



5-13

sf

swelling

bs Sb

b

0

(5-18)

This can be simplified into

0ssf

bs

s

bsfS

(5-19)

where sf is the volumetric free swelling strain at the final stage, and

0s is the volumetric

free swelling strain at the initial stage.

Therefore, fS represents the ratio of fracture strain change to the incremental volumetric

swelling strain, defined as a sensitivity ratio in this study. If the boundary conditions are the

same, a largerfS value means that cleat aperture change is more sensitive to sorption-

induced strain. In our experimental tests, fS varies from 6.82 to 54.8.

5.3 Permeability model evaluation

A series of gas flow-through experiments have been carried out all under constant pressure

difference conditions (Chen et al., 2011), which were defined as the difference between

confining stress and pore pressure. First, the effective stress coefficient is measured for the

non-adsorbing gas (helium) flow-through experiments. In these experiments, the impact of

gas sorption is null and any permeability alteration is considered to be due to the variation

in the effective stress coefficient. Second, the change in permeability resulting from the

non-adsorbing gas experiments is used to calibrate the subsequent experiments using

adsorbing gasses (CO2 and CH4) where the effect of sorption-induced strain alone, on

permeability change, is obtained. The measured two sets of corrected data (core No.01 and

No.02) and another two sets of experimental data from core Anderson 01 (Robertson and

Christiansen, 2007) and core Sulcis Coal (Pini et al., 2009) were used to evaluate the newly

developed permeability model in this work. Values of the volumetric swelling parameter as

listed in Table 5-1 were taken directly from these references. These values were obtained

through matching experimental data with the Langmuir curve type of strain versus pore

pressure (Robertson and Christiansen, 2007).

Table 5-1. Parameter values obtained from experimental data matching

Parameter Core No.01 Core No.02 Anderson 01 Sulcis coal

Langmuir strain constant 0.052 0.045 0.0353 0.049



5-14

for CO2 (%)

Langmuir pressure constant

for CO2 (MPa) 5.20 3.20 3.83 7.25

Langmuir strain constant

for CH4 (%) 0.030 - 0.0168 -

Langmuir pressure constant

for CH4 (MPa) 2.96 - 6.11 -

5.3.1 Permeability model verification

Model results were compared with experimental data for cores No.01 and 02. Effects of

sorption-induced strain alone on permeability change were investigated in this comparison.

In this comparison, only the sensitivity ratio, fS , is adjustable in this match, and matching

results are shown in Figures 5-5 and 5-6.

Both data matches verify the validity of this newly developed coal permeability model. It

can be seen that the sensitivity ratio, fS , increases with the effective stress. This

observation indicates that under a higher effective stress, larger fracture opening change is

induced, and a small change of fracture aperture could cause a dramatic change in the

0bbswelling

ratio.

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14

Pe

rme

ab

ilit

y (

mD

)

Pore pressure (MPa)

Experimental Data

Experimental Data

Experimental Data

Modeled Data

Modeled Data

Modeled Data

(σe=2.0 MPa)

(σe=4.0 MPa)

(σe=6.0 MPa)

(σe=2.0 MPa)

(σe=4.0 MPa)

(σe=6.0 MPa)

Sf =6.82

Sf =7.21

Sf =6.90

Flow Fluid: CH4

Core: No.01



5-15

(b)

Figure 5-5. Comparisons of experimental permeability with the model results for core

No.01: (a) adsorbing gas CH4; (b) adsorbing gas CO2.

Figure 5-6. Comparison of experimental permeability with the modeled results for core

No.02.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14

Pe

rme

ab

ilit

y (

mD

)

Pore pressure (MPa)

Experimental Data

Experimental Data

Experimental Data

Modeled Data

Modeled Data

Modeled Data

Flow Fluid: CO2

Core: No.01(σe=2.0 MPa)

(σe=4.0 MPa)

(σe=6.0 MPa)

(σe=2.0 MPa)

(σe=4.0 MPa)

(σe=6.0 MPa)

Sf =7.25

Sf =8.70

Sf =9.93

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6

Pe

rme

ab

ilit

y (

mD

)

Pore pressure (MPa)

Experimental Data

Experimental Data

Experimental Data

Modeled Data

Modeled Data

Modeled Data

Flow Fluid: CO2

Core: No.02

(σe=2.0 MPa)

(σe=3.0 MPa)

(σe=4.0 MPa)

(σe=2.0 MPa)

(σe=3.0 MPa)

(σe=4.0 MPa)

Sf =11.32

Sf =15.44

Sf =17.03



5-16

5.3.2 Comparison with other permeability models

In this section, experimental data from core Anderson 01 (Robertson and Christiansen,

2007) and core Sulcis Coal (Pini et al., 2009) were used to compare this developed

permeability model with other widely used permeability models, including updated Palmer-

Mansoori (P-M) model, Shi-Durucan (S-D) model, Cui-Bustin (C-B) model (Shi and

Durucan, 2004; Cui and Bustin, 2005; Palmer et al., 2007). For coal Anderson 01,

experimental data for CO2 and CH4 were chosen. The confining pressure was 6.895 MPa

(1,000 psi) for all experiments, and injection pressure varied from 0.5 MPa to 5.6 MPa. For

Sulcis Coal, the confining stress was 10.0 MPa and the injection pressure increased from

0.49 to 7.75 MPa. These comparisons are to benchmark the performance of our model

against others.

Coal swelling parameters from laboratorial tests were used, as listed in Table 5-1. Because

the cleat compressibility was not given in both references, both the fracture compressibility

and the sensitivity ratio, fS , were considered as the variables for our permeability model.

For other permeability models, the physical properties of Young's modulus and Poisson's

ratio are recovered directly from the experiments (Robertson and Christiansen, 2007; Pini

et al., 2009), and fracture compressibility was considered to be variable for both S-D model

and C-B model. Fracture porosity is the matching parameter for P-M model. The best

matching parameters for each model are listed in Table 5-2, and the comparison results are

shown in Figures 5-7 and 5-8.



5-17

(a)

(b)

Figure 5-7. Comparison of experimental permeability data (Robertson and Christiansen,

2007) with the modeled ones for core Anderson 01: (a) adsorbing gas CO2; (b) adsorbing

gas CH4.

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 1 2 3 4 5 6

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Pore pressure (MPa)

Experimental Data

New Model

S-D Model

P-M model

C-B model

Flowing fluid: CO2

Core: Anderson 01

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 1 2 3 4 5 6

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Pore pressure (MPa)

Experimental Data

New Model

S-D Model

P-M model

C-B model

Flowing fluid: CH4

Core: Anderson 01



5-18

Figure 5-8. Comparison of experimental permeability data (Pini et al., 2009) with the

modeled ones for core Sulcis.

Table 5-2. Matching parameters used in the comparison of permeability models

Sample

Model name

Physical

property S-D model Our model P-M model C-B model

Anderson

01 (CO2)

ν= 0.3

E=1.38 GPa Cf=0.0142 MPa-1

Sf =35.66

βCf=0.0893 MPa-1

g=1.0

Φf0=3.18% Cf=0.029 MPa-1

Anderson

01 (CH4)

ν= 0.3

E=1.38 GPa Cf=0.0101 MPa-1

Sf =54.78

βCf=0.065 MPa-1

g=1.0

Φf0=3.55% Cf=0.165 MPa-1

Sulcis

coal

ν= 0.26

E=1.12 GPa Cf=0.01 MPa-1

Sf =36.78

βCf=7.63 MPa-1

g=1.0

Φf0=1.0% Cf=0.001MPa-1

For the compared permeability models, as shown in Figures 5-7 and 5-8, the total swelling

strain is used to calculate permeability variation. The experimental data show that sorption-

induced strain only plays a dominant role at low pressures (permeability reduction), as

shown in Figures 5-7 and 5-8, and the pore pressure induced effective stress change takes

over the dominant role (permeability increase) at higher pore pressures. Therefore, these

models are not capable of replicating this apparently anomalous behaviour if the total

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5 6 7 8

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Pore pressure (MPa)

Experimental Data

New Model

S-D Model

P-M model

C-B Model

Flowing fluid: CO2

Core: Sulcis Coal



5-19

swelling strain data are adopted since the uniaxial-strain assumption is used in these models

while the experimental conditions are not uniaxial strain.

For our model, we considered that only part of the total swelling strain contributes to the

cleat aperture change, while the remaining part contributes to coal bulk deformation. The

effect of swelling strain on permeability change is evaluated by a partition factor, as defined

in Equation (5-11). We believe that this assumption adequately reflects the mechanism for

the interaction between coal swelling strain and permeability change, and that is why this

developed model is capable of replicating this behaviour.

In order to better explain this model, the relationship between cleat porosity and the

partition ratio of total swelling strain contributing to cleat aperture change was listed and

plotted in Table 5-3 and Figure 5-9, respectively. The data were based on the fitting results

from Figures 5-5, 5-7 and 5-8, and this comparison is to show what percentage of total

swelling strain contributes to the permeability change. Although the cleat porosity term

does not directly appear in the permeability model, it is included in the fS term,

0/ ff fS , as defined in Equation (5-15).

Table 5-3. Data for porosity and partition ratio

Porosity (%)

Partition ratio ( f )

Core No.01 (CO2) Anderson 01 (CH4) Sulcis Coal

1.7 0.169 0.931 0.625

1.5 0.149 0.822 0.552

1.3 0.129 0.712 0.478

1.1 0.109 0.603 0.405

0.9 0.089 0.493 0.331

0.7 0.070 0.383 0.257

0.5 0.050 0.274 0.184

0.3 0.030 0.164 0.110

0.1 0.010 0.055 0.037

0.08 0.008 0.044 0.029

0.06 0.006 0.033 0.022

0.04 0.004 0.022 0.015

0.02 0.002 0.011 0.007



5-20

Figure 5-9. Partition ratio of total sorption-induced strain vs. cleat porosity.

As demonstrated in Figure 5-9, the partition ratio is linearly related to cleat porosity change.

Larger cleat porosity value is accompanied by a higher partition ratio, which means more

total swelling strain is absorbed by the cleat aperture system. For instance, when the cleat

porosity is 1.0%, there is 54.78% of swelling strain contributing to cleat aperture change,

but this partition ratio decreases to 5.478% when the cleat porosity reduces to 0.1%.

All three sets of matches have illustrated that using the total swelling strain to calculate the

permeability change could dramatically overestimate its contribution, which clearly

demonstrates the contribution of this work.

5.4 Model implementation

In our previous studies (Zhang et al., 2008; Chen et al., 2009 and 2010; Liu et al., 2010a,

2010b; Wu et al., 2010 and 2011), a series of single poroelastic, equivalent poroelastic, and

dual poroelastic models were developed to simulate the interactions of multiple processes

triggered by the injection or production of both single gas and binary gas. Many studies

have also been carried out by other researchers (Cui et al., 2007; Bustin et al., 2008). In

order to reproduce the typical enigmatic behaviours of coal permeability evolution with gas

injection, we applied the new developed permeability model to a coupled 3D finite element

0.001

0.01

0.1

1

0.01 0.1 1 10

Pa

rtit

ion

ra

tio

, f

(-)

Cleat porosity (%)

Anderson 01

Sulcis Coal

Core No.01

Sf= 54.78

Sf= 36.78

Sf= 9.93

(CH4)

(CO2)



5-21

numerical model to simulate the performance of CH4 injection under stress-controlled

conditions.

5.4.1 Model descriptions

This numerical model fully couples coal geomechanical deformation, gas flow, and gas

adsorption/desorption induced coal matrix swelling/shrinkage processes (see Liu et al.

(2010b) for details). The core size is 45.0 mm in diameter and 105.5 mm in length with CH4

injection at the left-hand side. Coal is initially saturated with CH4 with pressure of 0.5MPa.

A constant injection pressure boundary condition is specified from the left side with the

value of 7.0MPa, as shown in Figure 5-10. Input parameters for this simulation are listed in

Table 5-4.

This example is to investigate the sensitivity of transient permeability with CH4 injection to

different coal physical properties as well as sorption parameters, and a series of injection

conditions was simulated as listed in Table 5-3. Simulation results were presented in terms

of (1) impacts of confining stress, (2) impacts of swelling strain, (3) impacts of fracture

compressibility, (4) impacts of effective stress coefficient, and (5) impacts of sensitivity

factor. A reference point with the coordinate (80mm, 0, 0) started from the injection side

was chosen to study the evolution of coal permeability and pore pressure in terms of

different coal parameters. Simulation results were presented in Figures 5-11 through 5-15.

Figure 5-10. Numerical simulation model under controlled stress conditions. axial and

radial represent the applied stress in axial and redial directions, respectively. The symbols

on the right hand side of the figure represent the constrained deformation in horizontal

direction, but free to move in other directions.

Sample length=105.5 mm

Sample diameter=45.0mm

radial

radial

CH4

Injection axial



5-22

Table 5-4. Properties used in the simulations

Parameter Value

Coal density, kg/m3 1250

Coal Young's modulus, GPa 0.791

Poisson's ratio 0.418

Effective stress coefficient 0.945

Fracture compressibility, MPa-1 0.0669

Methane viscosity, µPa-1 11.554

Initial gas pressure, MPa 0.5

Maximum volumetric swelling strain 0.03

Maximum adsorption gas volume, m3/ton 27.0

Langmuir pressure constant, MPa 2.96

Coal matrix porosity, % 5.0

Coal fracture porosity, % 0.5

Initial permeability, md 1.0

Sensitivity factor (fS ) 30.0

Table 5-5. Sensitivity investigation of permeability and pore pressure responses to CH4

injection under different conditions

Scenario Parameter

Case 1: Impacts of hydrostatic confining stress

MPa0.10

MPa0.9

MPa0.8

Case 2: Impacts of swelling strain

%0.4

%0.3

%0.2

L

L

L

Case 3: Impacts of fracture compressibility

1

1

1

MPa0869.0

MPa0669.0

MPa0469.0

f

f

f

C

C

C

Case 4: Impacts of effective stress coefficient

0.1

914.0

8.0



5-23

Case 5: Impacts of sensitivity factor

45

30

15

f

f

f

S

S

S

5.4.2 Simulation results and analysis

Impact of confining stress: The impact of confining stress on the evolution of coal

permeability is shown in Figure 5-11. Different confining stresses represent different coal

seam depth, which can be obtained by multiplying the pressure gradient with the reservoir

depths. Assuming the gas pressure is applied on non-adsorbing coal medium, the effective

stress increases with increasing confining stress. Therefore, more reduction in coal

permeability is achieved when the confining stress is higher. However, this initial effect has

been eliminated when we plot the permeability ratio starting from 1.0 for all three cases.

Assuming the same gas pressure condition is applied to an adsorbing coal, the sensitivities

of coal permeability are regulated by the initial effective stress when coal swelling

parameters are maintained unchanged. Therefore, when the confining stress is equal to

10MPa, the reduction in coal permeability is more significant, as shown in Figure 5-11.

Figure 5-11. Evolution of coal permeability and pore pressure under different magnitudes

of confining stresses.

0

1

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40 45

Po

re p

res

su

re (

MP

a)

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Time (hour)

(σ=8.0 MPa)

(σ=9.0 MPa)

(σ=10.0 MPa)

Pressure(σ=8.0 MPa)

Pressure (σ=9.0 MPa)

Pressure (σ=10.0 MPa)

K/K0

K/K0

K/K0



5-24

Impact of swelling capacity: Studies from Robertson (2005) have shown that there is

large difference between the strains induced by adsorption of different gasses in coals, as

illustrated in Figure 5-3. Different maximum volumetric swelling strains used in this study

represent different gas adsorbed into coal. The influence of coal maximum swelling strain

on the evolution of both permeability and pore pressure were shown in Figure 5-12. In all

of these simulations, the initial effective stress is same for all cases. Therefore, we can

assume that when gas pressure is applied in an adsorbing coal, the sensitivities of coal

permeability are regulated by the maximum swelling strain only. Model results suggest that

for coal seam with a larger swelling capacity, the reduction in coal permeability is much

more significant, and this reduction in turn affects the pore pressure evolution.


of the maximum volumetric swelling strain constant.

Impact of fracture compressibility: The impact of compressibility is shown in Figure 5-

13. When the confining stress is kept as constant, both the effective stress effect and the

swelling strain effect are defined as a function of gas pressure. In this set of simulations, the

contribution of the effective stress to the enhancement in coal permeability is determined

by coal compressibility because the swelling factor, fS , is kept unchanged. When

fC is

higher, the permeability enhancement takes over the permeability reduction. Model results

are consistent with these conceptual analyses.

0

1

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30 35 40 45

Po

re p

res

su

re

(MP

a)

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Time (hour)

Pressure

Pressure

Pressure

K/K0 (εL=2.0%)

K/K0 (εL=3.0%)

K/K0 (εL=4.0%)

(εL=2.0%)

(εL=3.0%)

(εL=4.0%)



5-25


of fracture compressibility.

Impact of effective stress coefficient: Figure 5-14 shows the profiles of permeability

evolution with different effective stress coefficients. Based on the effective stress principle,

effective stress increases as the effective stress coefficient decreases. Therefore, a smaller

effective stress coefficient will result in a larger reduction in the coal permeability

When the gas pressure is applied in an adsorbing and swelling coal, the sensitivities of coal

permeability are regulated by the initial effective stress coefficient when coal swelling

parameters are maintained unchanged. Therefore, when the effective stress coefficient is

equal to 0.8, more reduction in coal permeability is observed, as demonstrated in Figure 5-

14.

0

1

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25 30 35 40 45

Po

re p

res

su

re

(MP

a)

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Time (hour)

Pressure

Pressure

Pressure

K/K0 (Cf=0.0469MPa-1)

K/K0 (Cf=0.0669MPa-1)

K/K0 (Cf=0.0869MPa-1)

(Cf=0.0469MPa-1)

(Cf=0.0669MPa-1)

(Cf=0.0869MPa-1)



5-26


of coal effective stress coefficient.

Impact of sensitivity factor: The impact of sensitivity factor is shown in Figure 5-15. As

stated before, fS represents the ratio of fracture aperture strain to swelling strain

incremental. When the confining stress is kept constant, both the effective stress effect and

the swelling strain effect are defined as a function of gas pressure. In this example, the

contribution of the swelling strain to the reduction in coal permeability is determined by

the sensitivity factor, fS . When

fS is higher, the permeability reduces more. Model results

are consistent with these conceptual analyses.

0

1

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25 30 35 40 45

Po

re p

res

su

re (

MP

a)

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Time (hour)

Pressure

Pressure

Pressure

K/K0 (α=0.8)

K/K0 (α=0.945)

K/K0 (α=1.0)

(α=0.8)

(α=0.945)

(α=1.0)



5-27


of sensitivity factor.

5.5 Conclusions

Coal permeability models are required to define the transient characteristics of permeability

evolution in fractured coals. A broad variety of models have evolved to represent the

effects of sorption, swelling and stresses on the dynamic evolution of permeability. These

models can be classified into two groups: permeability models under conditions of uniaxial

strain such as Palmer-Mansoori (P-M), Shi-Durucan (S-D) and Cui-Bustin (C-B) models,

and permeability models under conditions of variable stress such as the one developed in

this study.

Although laboratory experiments are conducted under controlled conditions of stresses,

analyses of laboratory observations are normally conducted by using permeability models

under conditions of uniaxial strain. The inconsistency between experimental conditions and

modelling conditions is the reason why permeability models under conditions of uniaxial

strain cannot match the laboratory observations well as demonstrated in this study.

Permeability models under uniaxial strain are more appropriate for the overall behaviour of

coal gas reservoirs under typical in situ conditions while models representing variable stress

0

1

2

3

4

5

6

7

8

9

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40 45

Po

re p

res

su

re

(MP

a)

Pe

rme

ab

ilit

y r

ati

o,

k/k

0

Time (hour)

K/K0 (Sf=15)

K/K0 (Sf=30)

K/K0 (Sf=45)

P (Sf=15)

P (Sf=30)

P (Sf=45)



5-28

conditions are more appropriate for behaviour examined under typical laboratory

conditions.

In this study, a phenomenological permeability model has been developed to explain why

coal permeability decreases even under the unconstrained conditions of variable stress.

Unlike permeability models under the uniaxial strain condition, this model under

conditions of variable stress is effective-stress based and can be used to recover the

important nonlinear responses due to the effective stress effects when mechanical

influences are rigorously coupled with the gas transport system. The consistency between

experimental conditions and modelling conditions is the reason why this model can match

the laboratory observations reasonably well.

Our modelling results illustrate that coal permeability profiles under the controlled stress

conditions are regulated by the following five factors: (1) confining stress. When coal

swelling parameters remain unchanged, coal permeability profiles are regulated by the

initial effective stress. Coal permeability reduces initially, recovers and then reaches 534 s

the final equilibrium magnitude. When the confining stress is higher, the final equilibrium

coal permeability is much lower than the initial permeability; (2) swelling capacity. When

confining stress conditions remain unchanged, coal permeability profiles are regulated by

coal swelling capacity. Coal permeability reduces initially, recovers and then reaches the

final equilibrium magnitude. When the swelling capacity is higher, the final equilibrium coal

permeability is much lower than the initial permeability; (3) fracture compressibility. When

the confining stress is kept as constant, both the effective stress effect and the swelling

strain effect are defined as a function of gas pressure. Under these conditions, when the

facture compressibility is higher, the permeability enhancement due to the decrease in

effective stress may take over the permeability reduction due to swelling; (4) effective stress

Coefficient. The reduction in coal permeability is larger when the effective stress coefficient

is lower because the effective stress increases as the effective stress coefficient decreases;

and (5) sensitivity factor. The sensitivity factor represents the ratio of fracture aperture

strain to swelling strain incremental. When the sensitivity factor is higher, the reduction in

coal permeability is more significant.

This study demonstrated the crucial role of the consistency between experimental

conditions and modelling conditions and the rigorous coupling between coal mechanical

deformation and gas transport in the evaluation of coal permeability observations.



5-29

5.6 Acknowledgments

This work was supported by WA:ERA, the Western Australia CSIRO-University

Postgraduate Research Scholarship, National Research Flagship Energy Transformed Top-

up Scholarship, and by NIOSH under contract 200-2008-25702. These supports are

gratefully acknowledged.

5.7 References

Berryman, J.G., 2006. Estimates and rigorous bounds on pore-fluid enhanced shear modulus in poroelastic

media with hard and soft anisotropy. International Journal of Damage Mechanics 15, 133-167.

Bustin, R.M., Cui, X., Chikatamarla, L., 2008. Impacts of volumetric strain on CO2 sequestration in coals and

enhanced CH4 recovery. AAPG Bulletin 92, 15-29.

Chen, Z., Liu, J., Connell, L., Pan, Z., Zhou, L., 2008. Impact of effective stress and CH4-CO2 counter-

diffusion on CO2 enhanced coalbed methane recovery, SPE Asia Pacific Oil and Gas Conference and

Exhibition, Perth, Australia.

Chen, Z., Liu, J., Elsworth, D., Connell, L., Pan, Z., 2009. Investigation of CO2 injection induced coal-gas

interactions, 43rd U.S. Rock Mechanics Symposium & 4th U.S. - Canada Rock Mechanics Symposium.

American Rock Mechanics Association, Asheville, North Carolina.

Chen, Z., Liu, J., Elsworth, D., Connell, L.D., Pan, Z., 2010. Impact of CO2 injection and differential


81, 97-108.

Chen, Z., Pan, Z., Liu, J., Connell, D.L., Elsworth, D., 2011. Effect of the effective stress coefficient and

sorption-Induced strain on the evolution of coal permeability: Experimental observations.

International Journal of Greenhouse Gas Control 5, 1284-1293.

Clarkson, C.R., 2008. Case study: Production data and pressure transient analysis of Horseshoe Canyon CBM

Wells, CIPC/SPE Gas Technology Symposium 2008 Joint Conference. Society of Petroleum




Connell, L.D., Lu, M., Pan, Z., 2010. An analytical coal permeability model for tri-axial strain and stress


Cui, X., Bustin, R.M., Chikatamarla, L., 2007. Adsorption-induced coal swelling and stress: Implications for

methane production and acid gas sequestration into coal seams. J. Geophys. Res. 112, B10202.

Cui, X., Bustin, R.M., 2005. Volumetric strain associated with methane desorption and its impact on coalbed

gas production from deep coal seams. AAPG Bulletin 89, 1181-1202.

Daley, T.M., Schoenberg, M.A., Rutqvist, J., Nihei, K.T., 2006. Fractured reservoirs: An analysis of coupled

elastodynamic and permeability changes from pore-pressure variation. Geophysics 71, O33-O41.

Freed, A.D., 1995. Natural strain. J. Eng. Mater. Technol. 117, 379-385.

Gilman, A., Beckie, R., 2000. Flow of coal-bed methane to a gallery. Transport in Porous Media 41, 1-16.

Gray, I., 1987. Reservoir engineering in coal seams: Part 1-The physical process of gas storage and movement

in coal seams. SPE Reservoir Engineering 2, 28-34.







5-30



Jaeger, J.C., Cook, N.G.W., Zimmerman, R.W., 2007. Fundamentals of rock mechanics, 4 ed. Oxford:

Blackwell.

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5-31

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6-1

Chapter 6

Roles of Coal Heterogeneity on Evolution of Coal Permeability under Unconstrained Boundary Conditions

Zhongwei Chena, Jishan Liua*, Derek Elsworthb, Jianguo Wanga, Zhejun Panc, Luke D.

Connellc


b Department of Energy and Mineral Engineering, Penn State University, PA 16802-5000, USA

c CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South, Victoria 3169, Australia

Abstract: Coal permeability models based constrained conditions such as constant volume

theory can successfully describe unconstrained experimental data and field observations.

However, these models have a boundary mismatch because model boundary is constrained

while experiment boundary is free displacement or unconstrained. What the mechanism is

to require such a boundary mismatch has not been well understood so far. In this study, a

full coupling approach was developed to explicitly model the interactions of coal matrix

and fracture. In this model, a matrix-fracture model is numerically investigated after

incorporating heterogeneous distributions of Young's modulus, Langmuir strain constant

in the vincity of the fracture. The impact of these local heterogeneities of coal mechanical

and swelling properties on the permeability evolution is explored. The transient

permeability evolution during gas swelling process was investigated and the difference

between the final equilibrium permeability and transient permeability was compared. It is

found that a net reduction of coal permeability is achieved from the initial no-swelling state

to the final equilibrium state. This net reduction of coal permeability increases with the

fracture (injection) pressure and is in good agreement with laboratorial data under the

unconstrained swelling conditions. Therefore, the local heterogeneity of coal in vincity of

fracture should be the mechanism of the above mismatch.

Keywords: coal permeability; gas sorption; coal swelling; coal-gas interaction; local

heterogeneity.



6-2

6.1 Introduction

The permeability of coal is a key attribute in determining coal seam methane production

and CO2 storage in coal seam reservoirs. In coal the permeability is often determined by

regular sets of fractures called cleats, with the aperture of the cleats being a key property in

the magnitude of the permeability (Connell et al., 2010). The relative roles of stress level,

gas pressure, gas composition, fracture geometry of coal, and water content are intimately

connected to the processes of gas sorption, diffusion, transport, and coal

swelling/shrinkage (Liu et al., 2011b).

Significant experimental efforts have been made to investigate coal permeability and its

evolution. Laboratory measured permeabilities of coal to adsorbing gasses, such as CH4

and CO2, are known to be lower than permeabilities to non-absorbing or lightly adsorbing

gasses such as argon and nitrogen (Durucan and Edwards, 1986; Siriwardane et al., 2009;

Somerton et al., 1975). Under constant total stress, adsorbing gas permeability decreases

with increasing pore pressure due to coal swelling (Chen et al., 2011; Mazumder and Wolf,

2008; Pan et al., 2010a; Robertson, 2005; Wang et al., 2010, 2011), and increases with

decreasing pore pressure due to matrix shrinkage (Cui and Bustin, 2005; Harpalani and

Schraufnagel, 1990; Harpalani and Chen, 1997; Seidle and Huitt, 1995). It is also

influenced by both the presence of water and the magnitude of water saturation (Han et al.,

2010, Pan et al., 2010b). One thing is in common for the above studies that they were

conducted under unconstrained boundary conditions.

A number of proposed coal permeability models have been developed to match

experimental data (Cui and Bustin, 2005; Izadi et al., 2011; Liu and Rutqvist, 2010; Liu et al.,

2011a; Palmer and Mansoori, 1996; Pekot and Reeves, 2002; Seidle and Huitt, 1995; Shi

and Durucan, 2004; Wang et al., 2009; Zhang et al., 2008). Two assumptions are applied

with the above mentioned models- uniaxial strain and constant overburden or confining

stress (Connell et al., 2010; Liu et al., 2011a). These models have been quite successful in

matching experimental data even thought the tests were conducted under stress-controlled

(unconstrained) boundary conditions. However, permeability models developed under

stress-controlled condition is incapable to match experimental data conducted under stress-

controlled conditions, particularly for the models developed with the matchstick or cubic

coal geometry. That is because matrix swelling would not affect coal permeability due to

the complete separation between matrix blocks caused by through-going fractures. In this

case, for a given fracture pore pressure, the swelling results in an increase of fracture



6-3

spacing, rather than a change in fracture aperture (Liu and Rutqvist, 2010). However, this

has not been consistent with laboratory observations that show significant effects of matrix

swelling on coal permeability under constant confining stress conditions (Chen et al., 2011;

Lin et al., 2008; Pan et al., 2010a). This behaviour remains enigmatic as the permeability of

the porous coal is determined by the effective stress only.

A few studies have been carried out on either improving current permeability models or

explaining why permeability models developed under unaixal strain condition is capable to

match experimental data conducted under stress-controlled conditions. Connell et al. (2010)

partitioned the sorption strain into bulk, pore and matrix strains in contrast to existing

approaches, and derived several different forms of the permeability models for the distinct

geometric and mechanical arrangements that can be encountered with laboratory testing.

Liu and Rutqvist (2010) believed that in reality coal matrix blocks are not completely

separated from each other by fractures but connected by the coal-matrix bridges, and

developed a new coal-permeability model for constant confining-stress conditions, which

explicitly considers fracture–matrix interaction during coal-deformation processes based on

the internal swelling stress concept. An alternative reasoning has been investigated by Liu

et al. (2010a), considering that the internal actions between coal fractures and matrix have

not been taken into consideration. Recently, Izadi et al. (2011) proposed a mechanistic

representation of coal as a collection of unconnected cracks in an elastic swelling medium,

where voids within a linear solid are surrounded by a damage zone. In the damage zone the

Langmuir swelling coefficient decreases outwards from the wall and the modulus increases

outwards from the wall. In the analysis, fluid pressures are applied uniformly throughout

the body, so it is incapable of observing the transient permeability evolution due to coal-gas

interactions during gas transport. Liu et al. (2011b) addressed the same phenomena from

different point of view, stating that coal permeability is controlled by the switching process

between local swelling and macro-swelling, and the extent of switching of coal swelling

determines coal permeability is higher or lower than initial value.

However, these studies still have three limitations can be improved: (1) they were generally

carried out on the assumption of homogeneity, where coal properties were assumed to be

same throughout the whole domain; (2) it was assumed that permeability value is only

related to pore pressure and effective stress, so with same pore pressure the permeability

value is same; (3) permeability is independent of time. These assumptions have been

conflict with many experimental observations. Maggs (1946) investigated the feature of coal



6-4

swelling, and shown that in the presence of an adsorbed film, the coal swells and a

weakening of the structure would result on adsorption. This phenomenon has also been

confirmed by Hsieh and Duda (1987). The effect of high-pressure CO2 on the

macromolecular structure of coal has been studied by Mirzaeian and Hall (2006), and

shown that the glass transition temperature of coal decreases with CO2 pressure

significantly, indicating that high-pressure CO2 diffuses through the coal matrix, causes

significant plasticization effects, and changes the macromolecular structure of coal. Similar

observation has been obtained by many other researchers (Larsen, 2004; Goodman et al.,

2005; John, 2004; Liu et al., 2010; White et al., 2005). The thermodynamics and mechanism

for this phenomenon was examined by Mirzaeian and Hall (2008). The plasticization

effects of coal adsorption have been verified by the weakening of coal mechanical strength

from experimental measurements (Ates and Barron, 1988; Ranjith et al., 2010; Viete and

Ranjith, 2006; Viete and Ranjith, 2007; Wang et al., 2011). Recently, Siriwardane et al. (2009)

found that permeability of adsorbing gas in coal is a function of gas exposure time.

The non-homogeneous feature of coal swelling has also been observed by other

approaches (Day et al., 2008; Karacan and Okandan, 2001; Karacan, 2003; Karacan, 2007)

as apparent from quantitative X-ray CT imaging and from optical methods. Gibbins et al.

(1999) examined the heterogeneity of coal samples by means of density separation and

optical and scanning electron microscopy, and found that a high degree of heterogeneity

exists between average compositions for the different density cuts within each sample,

between different particles within the same density cuts, and within the particles themselves.

Similar work has been conducted by Gathitu et al. (2009). Manovic et al. (2009) presented

the microscopic observations of coals of different rank and mineral matter content,

showing an increasing of heterogeneity with mineral matter content. Anisotropic swelling

induced by chemical heterogeneity of coal has also been observed (Douglas, 1984; French

et al., 1993; Pone et al., 2010).

As summarized above, the real behaviours of the sorption-induced swelling/shrinkage of

coal are far different from the homogeneous assumption, which is generally made for

theoretical permeability analysis. The effects of coal chemical heterogeneity and swelling

are mutual. The heterogeneity of coal brings the non-homogeneous distribution of coal

swelling strain, and meanwhile coal swelling causes the heterogeneous distribution coal

physical property (e.g. Young's modulus). In this study, it is considered that the

heterogeneities of coal physical properties and swelling strain are responsible for the



6-5

enigmatic behaviour of coal permeability reduction with adsorbing gas injection under

unconstrained conditions. To prove this, a fully coupling numerical model is conducted to

simulate the dynamic interactions between coal matrix swelling and fracture aperture

alteration, and translate these interactions to transient permeability evolution. In this

numerical model, swelling coefficient and Young's modulus are assumed to vary spatially,

and numerical predictions are then compared with observed magnitudes of permeability

change in coal. Our work is trying to explain why permeability changes with absorbing gas

injection even under stress controlled conditions.

6.2 Theoretical evaluation of coal permeability models

6.2.1 General coal permeability model

It is clear that there is a relationship between porosity, permeability and the grain-size

distribution in porous media. Chilingar (1964) defined this relationship as

2

32

)1(72

ed

k (6-1)

where k is the permeability, is porosity and ed is the effective diameter of grains. Based

on this equation, we obtain

2

0

3

00 1

1

k

k. (6-2)

When the porosity is much smaller than 1 (normally less than 10%), the second term of the

right-hand side asymptotes to unity. This yields the cubic relationship between permeability

and porosity for coal matrix

3

00

k

k (6-3)

Coal porosity can be defined as a function of the effective strain (Liu et al., 2010a, 2010b)

e

00

1 (6-4)

Substituting Equations (6-4) into (6-3) gives

3

00

1

e

k

k

(6-5)



6-6

s

s

veK

p

(6-6)

or

K

pe

(6-7)

where e is defined as the total effective volumetric strain,

v is total volumetric strain,

sKp is coal compressive strain, s is gas sorption-induced volumetric strain, and

sK

represents the bulk modulus of coal grains.

Equations (6-4) and (6-5) are models for coal porosity and permeability that are derived

based on the fundamental principles of poroelasticity. They can be applied to the evolution

of coal porosity and permeability under variable boundary conditions.

Coal porosity and permeability can be defined as a function of either effective strain (6-6)

or effective stress (6-7). However, coal porosity and permeability models may have a

variety of forms when specific conditions are imposed. Examples include:

When the change in total stress is equal to zero, 0 , both coal porosity and

permeability are independent of the total stress. Under this condition, they can be

defined as a function of gas pressure only.

Assuming coal sample is under conditions of uniaxial strain and the overburden load

remains unchanged, they can also be defined as a function of gas pressure only.

When the impact of coal fractures and gas compositions is considered, coal porosity

and permeability models can be linked to fracture parameters and gas concentrations.

6.2.2 Evaluation of permeability model under two boundary conditions

In this section, general coal permeability models are evaluated through comparing

laboratorial and in-situ measurements with theoretical solutions of the two extreme cases,

the unconstrained shrinkage/swelling model and the constrained model, as illustrated in

Figure 6-1.



6-7

(a) (b)

Figure 6-1. Schematic diagram of two extreme cases: (a) unconstrained model (free swelling

model), where constant stress conditions are applied throughout the whole process; (b)

constrained model (constant volume model), where constant volume conditions are

maintained throughout the whole process. These two cases represent the lower and upper

bounds for permeability and porosity response.

We assume that matrix blocks are completely separated from each other in coal sample.

For the unconstrained model, matrix swelling does not affect coal fracture permeability,

because for a given pore pressure coal matrix swelling results in swelling of the blocks

alone, rather than changes in fracture aperture. The ambient effective stress also exerts no

influence on matrix swelling, due to the complete separation between matrix blocks caused

by through-going fractures. However, when coal sample is completely constrained from all

directions, coal matrix swelling will be completely transferred to the reduction in fracture

apertures. For the constrained model, the entire swelling/shrinkage strain contributes to

coal permeability change provided the fractures are much more compliant than coal matrix.

Equation (6-5) is derived based on the poroelastic theory without the effect of fractures.

Therefore, the porosity should be the matrix porosity. However, when these models are

applied, the fracture porosity is actually used. Evolutions of coal permeability for both

unconstrained and constrained conditions can be defined as Equations (6-8) and (6-9),

respectively.

3

00

1

K

p

k

k

f

(6-8)

3

00

1

s

s

f K

p

k

k

(6-9)

Pre-swelling

Aperture

Post-swelling

Aperture

Pre-swelling

Aperture

Post-swelling

Aperture



6-8

Solutions of these two cases and their comparisons with typical observations are illustrated

in Figures 6-2 to 6-4. Robertson (2005) directly measured the influence of coal swelling on

permeability change with four different gasses (helium, N2, CH4 and CO2) injection. Similar

experiments have been conducted by others (Pini et al., 2009; Siriwardane et al., 2009;

Wang et al., 2010; Kiyama et al., 2011). These observations demonstrate that even under

unconstrained stress-controlled conditions the injection of adsorbing gasses reduces coal

permeability at a lower gas pressure and coal permeability may rebound at a higher gas

pressure. This observed switch in behaviour is presumably due to the dependence of coal

swelling on the gas pressure: coal swelling diminishes at high pressures. Because all of the

mentioned experimental observations were made under controlled stress conditions, they

should be equal to or close to the theoretical solution under the unconstrained swelling

condition. This has not been the case, as illustrated in Figure 6-2. These observations

indicate that although the experiments were conducted under controlled stress conditions

the experimental measurements are more closely related to those expected under constant

volume conditions. These discrepancies illustrate the obvious drawbacks of the current coal

permeability models. If a coal gas reservoir is treated as a whole, with full lateral restraint

and invariant overburden stress, its behaviour should represent components of the free

swelling/shrinkage and the constant volume models. This could explain why current coal

permeability models representing conditions of uniaxial strain condition can successfully

match some field data.

Figure 6-2. Illustration of discrepancy between model predicted coal permeability and

typical laboratory measurements for unconstrained swelling case (Liu et al., 2011a).

0k

k

110

110

p1

0p

3

00

1

K

p

k

k

f

3

00

1

s

s

f K

p

k

k

p

p

Zone of Discrepancy



6-9

The influence of coal matrix shrinkage on coal permeability has also been widely studied

and an increase with decreasing pore pressure due to matrix shrinkage has been observed

(Cui and Bustin, 2005; Harpalani and Schraufnagel, 1990; Harpalani and Chen, 1997; Seidle

and Huitt, 1995), as shown in Figure 6-3. Similar trend has obtained from field

observations. In-situ measured data show that the absolute permeability of coal gas

reservoirs increases significantly with continued gas drainage (Cherian et al., 2010; Clarkson,

2008; Sparks et al., 1995; Young et al., 1991). Comparison of field observations with both

the unconstrained and constrained models is presented in Figure 6-4. Because all of the in-

situ observations were made under unknown conditions of in-situ stress they should lie

within the bracketing behaviours. Both Figure 6-3 and Figure 6-4 demonstrate that coal gas

reservoirs behave more closely to the constrained (constant volume) case.


typical laboratory measurements for unconstrained shrinkage case (Liu et al., 2011a).

0k

k

110

110

p1

0p

3

00

1

K

p

k

k

f

3

00

1

s

s

f K

p

k

k

p

p

Zone of Discrepancy



6-10


typical in-situ observations (Liu et al., 2011a).

The analysis above has demonstrated that current coal theoretical permeability models have

so far been unsuccessful to explain the results from stress-controlled shrinkage/swelling

laboratorial tests and have only achieved some limited success in explaining and matching

in situ data. The most recent viewpoints (Izadi et al., 2011; Liu and Rutqvist, 2010) have

demonstrated that the main reason for the failure is that the impact of coal matrix–fracture

compartment interactions on the evolution of coal permeability has not been incorporated

appropriately as most of the coal permeability models are derived based on the theory of

poroelasticity.

During laboratorial tests, coal permeability is measured only when gas flow is considered

having reached equilibrium state, in which a uniform matrix swelling/shrinkage is achieved

for a homogeneous coal sample (Siriwardane et al., 2009; Wang et al., 2011). In this work,

coal permeability at this state is defined as the final equilibrium permeability. However, this

condition may never be achieved for real coal samples. A difference of the final equilibrium

(or ultimate) permeability between an ideal homogeneous coal and a real heterogeneous

coal is expected. In the following section, a simulation model is constructed to investigate

the transient permeability evolution during gas swelling process and to study the difference

between the final equilibrium permeability and transient permeability, from which the

possible reasons for permeability reduction under unconstrained conditions can be

0k

k

110

110

p1

0p

3

00

1

K

p

k

k

f

3

00

1

s

s

f K

p

k

k

p

p

Zone of Discrepancy



6-11

achieved. In this approach, the important non-linear responses of coal matrix to the

effective stress are quantified through the incorporation of heterogeneous distributions of

coal properties into complex mechanical coupling with gas transport, where swelling

coefficient and modulus vary spatially relative to the fracture void.

6.3 A heterogeneous matrix-fracture interaction model

The key to model the dynamic interactions between coal matrix swelling/shrinkage and

fracture aperture alteration is to recover important non-linear responses of coal matrix to

the effective stress. In order to achieve this goal, geomechanical influence needs to be

rigorously coupled with the gas transport process. Full coupling approach has to be chosen

to achieve this goal, from which a single set of equations (generally a large system of non-

linear coupled partial differential equations) incorporating all of the relevant physics are

solved simultaneously (Liu et al., 2011a). Full coupling is often the preferred method for

simulating multiple types of physics since it should theoretically produce the most realistic

results. Therefore, it would represent important non-linear responses due to the effective

stress effects when geomechanical influences are rigorously coupled with the gas transport

system. Over the past few years, a series of advanced modelling tools has been developed

to quantify the complex coal–gas interactions (Chen et al., 2009, Chen et al., 2010; Connell,

2009; Connell and Detournay, 2009; Gu and Chalaturnyk, 2005, 2006; Liu et al., 2010a,b,

2011b; Wu et al., 2010a,b; Zhang et al., 2008). To reproduce the typical enigmatic

behaviours of coal permeability evolution with gas injection, in the following section we

applied this full coupling approach to simulate the evolution of coal permeability under

unconstrained conditions.

6.3.1 Numerical model implementation

In this section, a simulation model was constructed to investigate the permeability change

under unconstrained conditions. It was considered that the interactions of the fractured

coal mass where cleats do not create a full separation between adjacent matrix blocks but

where solid rock bridges are present, as illustrated in Figure 6-5(a). We accommodate the

role of swelling strains both over contact bridges that hold cleat faces apart and over the

non-contacting span between these bridges. The effects of swelling act competitively over

these two components: increasing porosity and permeability due to swelling of the bridging

contacts but reducing porosity and permeability due to the swelling of the intervening free-



6-12

faces. The influence of effective stress and swelling response for a rectangular crack is

examined, and we consider a single component part removed from the array where the

appropriate boundary conditions are for uniform displacement along the boundaries. This

represents the symmetry of the displacement boundary condition mid-way between flaws

as shown in Figure 6-5(b). The change in aperture due to the combined influence of coal

sorption-induced swelling and effective stress change is calculated during gas transport

process, from which the transient permeability evolution is obtained. The cubic

relationship between permeability and aperture change was chosen to calculate permeability

ratio evolution.

The simulation model geometry is 1.0cm by 2.0cm with a fracture located at the centre of

the model. The length and opening of the fracture are 0.3cm and 0.05mm, respectively. For

coal deformation model, the right and bottom sides are constrained in the normal direction

to honor symmetry and the other two sides are stress controlled, as shown in Figure 6-5(b).

For the gas transport model (CO2 is used for this study), coal sample is initially saturated

with gas with 0.2 MPa pressure, and a constant injection pressure is specified at the

boundaries of fracture. Input parameters for this simulation are listed in Table 6-1.

(a) Multiple fracture compartment model (b) Single fracture compartment model

Figure 6-5. Numerical model for permeability change under the unconstrained boundary

condition

Table 6-1. Parameters used in the numerical model

Parameter Value

Porosity, % 5.0

Matrix permeability, m2 5.0×10-22

Viscosity, Pa*s 1.228×10-5

2.0 cm

0.3 cm

0.05 mm

1.0 cm

Co

nfi

nin

g S

tres

s

Confining Stress

Fracture

A

B



6-13

Young's modulus, GPa 3.45

Poisson's ratio 0.4

Maximum volumetric swelling strain of CO2 0.03

CO2 Langmuir sorption constant, m3/kg 0.0132

CO2 Langmuir pressure constant, MPa 3.96

Coal density, kg/m3 1500

Young's modulus softening coefficient, 0.75

Swelling strain reduction factor, 0.25

6.3.2 Performance for a homogeneous coal

Firstly coal matrix is assumed to be homogeneous, where Young's modulus and Poisson‘s

ratio are constants and the Langmuir strain constant is same throughout the domain. The

simulation scenarios are listed in Table 6-2. Typical evolution of coal permeability is shown

in Figure 6-6. As can be seen from it, coal permeability experiences a rapid reduction at the

early stage. When the modelled time is about 2000s, a switch in behaviour from

permeability reduction to recovery is observed. After this, coal permeability recovers until it

reaches the final equilibrium permeability. The final equilibrium permeability is higher than

the original one, which is inconsistent with laboratorial observations (Harpalani and

Schraufnagel, 1990; Seidle and Huitt, 1995; Levine, 1996; Robertson, 2005; Cui et al., 2007;

Karacan, 2007; Day et al., 2008; Mazumder and Wolf, 2008; Kiyama et al., 2011; Wang et

al., 2010, 2011). Permeability increase with gas injection has been obtained for all

homogenous cases as shown in Figure 6-7. Final equilibrium permeability increases with

increasing fluid pressure, Poisson‘s ratio and volume swelling strain capacity, but decreases

with rising Young‘s Modulus.

Our conceptual understanding on the modelling processes is illustrated in Figure 6-8. Prior

to the CO2 injection, the gas pressure in the fracture is equal to that in the matrix. We

define this state as the initial equilibrium, 0ppp fm . In this study, coal permeability at

this state is defined as the initial equilibrium permeability, as illustrated in Figure 6-8(a).

When CO2 is injected, the gas occupies the fracture and the gas pressure in the fracture

reaches the injection pressure almost instantly. At this stage, the maximum imbalance

between fracture pressure and matrix pressure is achieved. However, this imbalance

diminishes as the gas diffuses into coal matrix. As a consequence of the diffusion, coal



6-14

matrix swells. Initially, matrix swelling is confined in the vicinity of the fracture voids. This

localized swelling reduces the fracture aperture, and in turn reduces the fracture

permeability, as shown in Figure 6-8(b). As CO2 diffusion progresses, the swelling zone

extends further into coal matrix, and the influence of matrix swelling on the fracture

aperture weakens. As a result of the widening of the swelling zone, the fracture

permeability recovers. When the imbalance between fracture pressure and matrix pressure

diminishes completely, the final equilibrium state is achieved, as shown in Figure 6-8(c). At

the final equilibrium state, the fracture pressure is equal to the matrix pressure, i.e.,

infm ppp , where inp is the injection pressure. Coal permeability at this state is defined

as the final equilibrium permeability.

Table 6-2. List of simulation scenario for homogeneous case

Simulation

scenario

Fracture pressure

fp (MPa)

Young's

modulus 0E

Langmuir strain

constant L (%)

Poisson‘s

ratio µ(-)

Case 1 2.0, 4.0, 6.0, 8.0,

10 1.0 3.0 0.40

Case 2 10.0 0.5, 0.75,

1.0, 1.25 3.0 0.40

Case 3 10.0 1.0 1.0, 2.0,

3.0, 4.0 0.40

Case 4 10.0 1.0 3.0 0.25, 0.30,

0.35, 0.40



6-15

Figure 6-6. Numerical result of permeability evolution for the homogeneous coal.

(a)

(b)

0.2

0.4

0.6

0.8

1

1.2

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

Initial EquilibriumFinal Equilibrium

Net Permeability Increase

Transitional Period

0mpfm pp

fm pyxp ,0

Confi

nin

g S

tres

s

Confining Stress

yxpm ,

Fracture

PressureInjection fp

Constant

Constant

Case sHomogeneou

L

E

110 210 310 410 510 610

Time (S)

Co

al P

erm

eab

ilit

y R

ati

o

010

Co

al p

erm

eabil

ity r

ati

o

Time (S)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

P=2.0MPa

P=4.0MPa

P=6.0MPa

P=8.0MPa

P=10.0MPa

Initial equilibrium

Final equilibrium

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

2 4 6 8 10

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Pore pressure (MPa)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

0.50 E0

0.75 E0

E0

1.25 E0

Initial equilibrium

Final equilibrium

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

0.5 0.7 0.9 1.1 1.3

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Young's modulus ratio (E/E0)



6-16

(c)

(d)

Figure 6-7. Evolutions of both dynamic permeability and equilibrium permeability for a

homogeneous coal under the unconstrained swelling condition. (a) Influence of different

pore pressures; (b) influence of different modulus magnitudes; (c) influence of different

Langmuir strain constants; (d) influence of different Poisson‘s ratios.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

Initial equilibrium

εs=1.0%

εs=2.0%

εs=3.0%

εs=4.0%

Final equilibrium

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1 1.5 2 2.5 3 3.5 4

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Langmuir swelling strain constant (%)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

Initial equilibrium

µ=0.25

µ=0.30

µ=0.35

µ=0.40

Final equilibrium

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

0.25 0.30 0.35 0.40

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Poisson's ratio (-)



6-17

Figure 6-8. Illustration of the difference of swelling behaviours between the homogeneous

coal and the heterogeneous coal. The red dotted line represents the original location of

fracture, and blue color zone represents the gas diffusion zone.

Based on the analysis above, the final equilibrium permeability is always higher than the

initial equilibrium permeability if a uniform swelling state is achieved within coal sample.

However, laboratorial measurements show that coal equilibrium permeability is generally

much lower than the initial equilibrium permeability in low pore pressure range, may

recover but rarely exceeds the initial equilibrium permeability even at high pore pressures.

This distinct discrepancy points to that a uniform matrix swelling state has rarely been

achieved in real coal sample tests. Therefore, a difference between the ultimate

permeability for an ideal homogeneous coal and that for a real heterogeneous coal is

expected.

6.3.3 Performance for a heterogeneous coal

As summarized in the introduction part, adsorption and swelling processes have been

shown to be heterogeneous in coal (Day et al., 2008; Karacan and Okandan, 2001; Karacan,

2003; Karacan, 2007), thus characteristics for the heterogeneity may include:

Fracture Compartment

Matrix

fp fp

fm pp 0

fp

fm pp

(a) No Swelling Everywhere (b) Local Swelling (c) Uniform Swelling 0t t t

Fracture Compartment

Matrix

fp fp

fm pp 0

(d) No Swelling Everywhere (e) Local Swelling (f) Local Swelling 0t t t

fp

fm pp

Real Coal – Heterogeneous Real Coal – Heterogeneous Real Coal – Heterogeneous

Ideal Coal – Homogeneous Ideal Coal – Homogeneous Ideal Coal – Homogeneous

0ppm 0ppm

0ppm

0ppm



6-18

(1) Initial distributions of key parameters, including coal Young's modulus E ,

permeability 0k , porosity 0 , Langmuir strain constant L , and Langmuir

pressure constant Lp . If any or combination of them varies spatially, coal is

considered as initially heterogeneous.

(2) Swelling/shrinkage dependencies of these key parameters, including coal Young's

modulus E , Langmuir strain constant L , and Langmuir pressure constant Lp . If

any or combination of them varies spatially due to gas sorption, coal is considered as

swelling heterogeneous.

(3) Distribution of pore pressure p . The ultimate distribution of pore pressure is also

controlled by the boundary conditions. If this distribution is not uniform within coal

matrix, a uniform swelling within the matrix is also not achievable.

How to represent these heterogeneities more accurately is the key to reduce the difference

between the modelled coal permeability and the measured one. Based on the above review,

in the following section we represent the heterogeneity through spatial distributions of

Young's modulus and Langmuir strain constant.

6.3.3.1 Modelling scenarios for a heterogeneous matrix

We generalize changes in permeability that accompanies gas adsorption under conditions

of constant applied stress and for increments of applied gas pressure for fractures.

Specifically we explore the relations between coal transient (or dynamic) permeability and

equilibrium permeability, and how these relations are controlled both by the distributed

coal Young's modulus and Langmuir strain constant, and by the injection pressure. The

scenarios simulated in the model are listed in Table 6-3.

Table 6-3. Simulation scenarios for heterogeneous case

Simulation

scenario

Fracture pressure

fp (MPa)

Young's modulus

0E

Langmuir strain

constant L (%)

Modelling

result

Case 1 2.0, 4.0, 6.0, 8.0, 10

3.0 Figure 6-9(a)

Case 2 10.0

0.5, 0.75, 1.0, 1.25

3.0 Figure 6-9(b)

0E

0EMatrix

Fracture Wall

E

EMatrix

Fracture Wall



6-19

Case 3 10.0

1.0, 2.0, 3.0, 4.0 Figure 6-9(c)

Case 4 2.0, 4.0, 6.0, 8.0, 10 1.0

Figure 6-10(a)

Case 5 10.0 0.5, 0.75, 1.0, 1.25 Figure 6-10(b)

Case 6 10.0 1.0

1.0, 2.0, 3.0, 4.0

Figure 6-10(c)

Case 7 2.0, 4.0, 6.0, 8.0, 10

Figure 6-11(a)

Case 8 10.0

0.5, 0.75, 1.0, 1.25

Figure 6-11(b)

Case 9 10.0

1.0, 2.0, 3.0, 4.0

Figure 6-11(c)

6.3.3.2 Impacts of heterogeneous Young's modulus

In this section, the numerical results were summarized and divided into three different

groups. This division is based on the reasons for heterogeneity.

Cases 1 to 3 are for the heterogeneous coal represented by the spatial distribution of coal

Young's modulus. In all three cases, coal Young's modulus is considered to decrease

linearly from outer boundaries to the inner fracture walls, and its values at the outer

boundary and fracture wall are 0E and

0E respectively, where is defined as coal

Young's modulus softening coefficient. Langmuir strain constant is assumed as constant

throughout the whole simulated domain. As shown in Figure 6-9, the dynamic permeability

evolves from the initial rapid reduction, to recovery, and to a net reduction. The reason for

this net reduction is that as the localization extends to the outside boundary, as shown in

Figure 6-8(e), coal permeability recovers, but the even distribution of effective stress

induced strain is not achieved at the equilibrium state as the coal Young's modulus is

spatially related. More strain is expected near fracture wall as coal media is softer there than

outside shell, which is transferred to the reduction in fracture apertures, as illustrated in

Figure 6-8(f). This analysis explains why coal equilibrium permeability decreases with

increasing gas pore pressure and the Langmuir strain constants, as shown in Figures 6-9(a)

0E

0EMatrix

Fracture Wall

L

LMatrix

Fracture Wall

L

LMatrix

Fracture Wall

0E

0EMatrix

Fracture WallL

LMatrix

Fracture Wall

E

EMatrix

Fracture WallL

LMatrix

Fracture Wall

0E

0EMatrix

Fracture Wall

L

LMatrix

Fracture Wall



6-20

and 6-9(c), and increases with increasing coal Young's modulus changes, as shown in

Figure 6-9 (b).

6.3.3.3 Impacts of heterogeneous Langmuir strain constant

Cases 4 to 6 are for the heterogeneous coal represented by the spatial distribution of

Langmuir strain constant. This constant is considered to decrease linearly from the inner

fracture walls to outer boundaries, and its values at the outer boundary and fracture wall are

L and L respectively, where is swelling strain constant reduction factor. Coal

Young's modulus is assumed as constant throughout the simulations. Similar to the results

of cases 1 to 3, the dynamic permeability of these all cases also evolves from the initial

rapid reduction, to recovery, and to a net reduction, but a uniform swelling within coal

matrix is not achieved. When coal matrix swelling is localized near the fracture

compartment, this is to accommodate the swelling of a soft medium that is constrained

within a rigid outer shell. In this situation, coal sample can be considered to be constrained

from all directions, and coal matrix swelling is almost completely transferred to the

reduction in fracture apertures. Thus, most of the swelling strain contributes to coal

permeability change provided the fractures are much more compliant than coal matrix.

This analysis also explains why coal equilibrium permeability decreases at the equilibrium

state of gas transport. Modelling results are shown in Figures 6-10 (a) to (c).

6.3.3.4 Combined impacts of heterogeneous Langmuir strain and Young's modulus

Cases 7 to 9 are for the heterogeneous coal represented by the spatial distributions of both

coal Young's modulus and Langmuir strain constant. For these cases, coal Young's

modulus is considered to decrease linearly from outer boundaries to the inner fracture walls,

and its values at the outer boundary and fracture wall are 0E and 0E , respectively.

Langmuir strain constant is considered to decrease linearly from the inner fracture walls to

outer boundaries, and its values at the outer boundary and fracture wall are L and L ,

respectively. In all three cases, the dynamic permeability also evolves from the initial rapid

reduction, to recovery, and to a net reduction, but more permeability reduction was

observed, as shown in Figures 6-11 (a) to (c). For instance, the maximum equilibrium

permeability reduction is around 25%for cases 1 to 3, permeability decreases by 50% for

cases 4-6, and it further decrease by as much as 72% for cases 7 to 9, where both coal

Young's modulus and Langmuir strain constant are spatially dependent.



6-21

(a)

(b)

(c)


heterogeneous coal represented by the spatial distribution of coal Young's modulus under

the unconstrained swelling condition. (a) Influence of different pore pressures; (b)

influence of different modulus ratios; (c) influence different Langmuir strain constants.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

P=2.0MPa

P=4.0MPa

P=6.0MPa

P=8.0MPa

P=10.0MPa

Initial equilibriumFinal equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Pore pressure (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

0.50 E0

0.75 E0

E0

1.25 E0

Initial equilibrium

Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

εs=1.0%

εs=2.0%

εs=3.0%

εs=4.0%

Initial equilibrium Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.5 2 2.5 3 3.5 4

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o




6-22

(a)

(b)

(c)


heterogeneous coal represented by the spatial distribution of Langmuir strain constant

under the unconstrained swelling condition. (a) Influence of different pore pressures; (b)

influence of different modulus ratios; (c) influence of different initial values of Langmuir

strain constant.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

P=2.0MPa

P=4.0MPa

P=6.0MPa

P=8.0MPa

P=10.0MPa

Initial equilibrium

Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Pore pressure (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

Initial equilibrium

Final equilibrium

0.50 E0

0.75 E0

E0

1.25 E0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

Initial equilibrium

Final equilibrium

εs=1.0%

εs=2.0%

εs=3.0%

εs=4.0%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.5 2 2.5 3 3.5 4

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o




6-23

(a)

(b)

(c)


heterogeneous coal represented by the spatial distributions of both coal Young's modulus

and Langmuir strain constant under the unconstrained swelling condition. (a) Influence of

different pore pressures; (b) influence of different initial modulus; (c) influence of initial

Langmuir strain constants.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

P=2.0MPa

P=4.0MPa

P=6.0MPa

P=8.0MPa

P=10.0MPa

Initial equilibrium

Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o

Pore pressure (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

0.50 E0

0.75 E0

E0

1.25 E0

Initial equilibrium

Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

Dy

na

mic

pe

rme

ab

ilit

y r

ati

o (

k/k

0)

Time (s)

εs=1.0%

εs=2.0%

εs=3.0%

εs=4.0%

Initial equilibrium

Final equilibrium

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.5 2 2.5 3 3.5 4

Fin

al e

qu

ilib

riu

m p

erm

ea

bilit

y r

ati

o




6-24

6.4 Verification with experimental data

In this section, two sets of experimental data monitored under stress controlled conditions

are used to verify our assumptions. The first experimental data comparison is based on the

work of Harpalani and Chen (1997), where the confining stress was kept constant

throughout the whole measurements. CH4 was used and permeability variation with

decreasing gas pressures (from 6.2MPa to 0.62MPa) was measured. The data for the second

match is from the work of Lin et al. (2008). Permeability of dry composite coal core to CH4

was measured with constant effective stress of 300psi (2.07MPa).

Both homogeneous and heterogeneous models are used to match two sets of data. For the

heterogeneous model, it is assumed that both coal Young's modulus and Langmuir strain

constant are distributed spatially. The parameter values used in data matching are listed in

Table 6-4 and the comparison results are shown in Figures 6-12 and 6-13, respectively.

Numerical results show that coal permeability value with homogeneous assumption

increases with increasing gas pressure, which is opposite to experimental data, so it is

incapable to match experimental data, but the heterogeneous model with the spatial

distributions of both coal Young's modulus and Langmuir strain is capable to replicate the

phenomena, and matches experimental data reasonably well. Therefore, coal heterogeneity

can be the answer to permeability change even under unconstrained conditions.

Table 6-4. Parameters used for data match

Parameter Harpalani and Chen (1997) Lin et al. (2008)

Porosity, % 5.0 5.0

Matrix Permeability, m2 5.0×10-22 2.0×10-21

Viscosity, Pa*s 1.228×10-5 1.228×10-5

Young's modulus, GPa 1.45 1.45

Poisson's ratio 0.4 0.4

Maximum volumetric swelling strain 0.005 0.0087

Langmuir sorption constant, m3/kg 0.0132 0.0132

Langmuir pressure constant, MPa 3.96 1.97

Coal density, kg/m3 1500 1500

Young's modulus softening coefficient, 0.79 0.555

Swelling strain reduction ratio, 0.21 0.445



6-25

Figure 6-12. Measured variation in permeability with decreasing gas pressure (Harpalani

and Chen, 1997).

Figure 6-13. Permeability of dry composite coal core to CH4. Effective stress equals to

300psi (Lin et al., 2008).

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

Pe

rme

ab

ilit

y R

ati

o (

k/k

0)

Gas pressure (MPa)

Experimental Data

Heterogeneous Case

Homogeneous Case

Fluid: CH4

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 1 2 3 4 5 6 7

Pe

rme

ab

ilit

y R

ati

o (

k/k

0)

Gas pressure (MPa)

Experimental Data

Heterogeneous Case

Homogeneous Case

Fluid: CH4



6-26

6.5 Conclusions

The performance of current coal permeability models is evaluated against analytical

solutions for the two extreme cases of either unconstrained swelling or constrained

swelling. Constrained model predictions are apparently consistent with both typical

laboratory measurements and in-situ observations. However, this apparent consistency is

due to the mismatch between model boundary condition assumptions (constrained) and

experiment boundary condition (unconstrained). This conclusion demonstrates that current

permeability models are incapable to explain net reductions in coal permeability where

swelling is unconstrained.

With the inclusion of the heterogeneous distributions of coal physical and swelling

properties, a full coupling approach was applied to investigate coal permeability response

under the unconstrained swelling conditions. Based on our model results, the following

major conclusions were drawn:

Both homogeneous and heterogeneous models experience the swelling transition

from local swelling to macro swelling. At the initial stage of gas injection, matrix

swelling is localized within the vicinity of the fracture compartment. As the

injection continues, the swelling zone is widening further into the matrix and

becomes macro-swelling.

Coal permeability experiences a rapid reduction at the early stage, a switch in

behavior from permeability reduction to recovery is observed, and coal

permeability finally recovers until it reaches the final equilibrium permeability. For

the homogeneous model, the final equilibrium permeability is always higher than

the original value, but the opposite was obtained for the heterogeneous model.

With the heterogeneous distributions of coal physical and swelling properties, this

numerical model matches with the experimental data reasonably well, which

demonstrates that heterogeneity of coal properties can be the answer to coal

permeability reduction under unconstrained conditions.






6-27

Scholarship, ConocoPhillips, and NIOSH under contract 200-2008-25702. These sources

of support are gratefully acknowledged.

6.7 References

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6-30

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7-1

Chapter 7

Impact of Various Parameters on the Production of Coalbed Methane

Zhongwei Chena, Akim Kabirb, Jishan Liua, Jianguo Wanga, Zhejun Panc, Luke D. Connellc


b QGC, A BG Business Group, 275 George Street, Brisbane, QLD 4000, Australia

c CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South, Victoria 3169, Australia

Abstract: Coalbed methane (CBM) reservoirs are naturally fractured formations, consisting

of fractures (cleats) and coal matrixes. The interactions between fractures and matrixes

present a great challenge for the forecast of CBM productions. Although these interactions

were represented in our prior studies through dual poroelastic models, the impact of gas

mass transfer between fractures and coal matrixes on the CBM production have not been

well understood. In this study, the mass exchange between two systems is defined as a

function of gas sorption time constant at the standard condition, coal matrix porosity, and

the gas pressure difference between matrix and fracture systems. Correspondingly, the gas

diffusivity in the matrix is defined as a function of the shape factor, the gas sorption time,

and the gas pressure. These relations are integrated into a fully coupled finite element

model of coal deformation, gas flow and transport in the matrix system, and gas flow in the

fracture system. The FE model represents important non-linear impacts of the effective

stress on CBM productions that cannot be recovered where mechanical influences are not

rigorously coupled with the complex interactions between fractures and matrixes.

The FE model was applied to quantify the impact of various parameters on the CBM

production. These variables include (1) Gas Desorption Time Constant; (2) Initial Fracture

Permeability; (3) Fracture Spacing; (4) Swelling Capacity; and (5) Desorption Capacity. Our

numerical results show that the peak magnitudes of the gas production rate increase with

the initial fracture permeability, the swelling capacity and the sorption capacity, and

decrease with the gas desorption time constant and the production pressure. These results

also show dramatic increase in gas rate with decreasing magnitudes of the fracture spacing.

The comparison of the contributions of the desorbed gas, and the free phase gas from

fracture and matrix systems to the gas production rate shows that the free phase gas from



7-2

the matrix system plays the dominant role at the early stage, but diminishes when the

adsorption phase gas takes over the dominant role. The FE model is also applied to match

the production data of a single gas well production from the Fruitland Coal of San Juan

Basin. Model results match the actual production data well. This successful match

demonstrates the potential capability of the FE model for the forecast of CBM productions.

Keywords: coalbed methane; dual poroeasticity; gas production rate; numerical modeling



7-3

7.1 Introduction

Coal seams are naturally fractured reservoirs, consisting of fractures (cleats) and coal

matrixes. The micro-pores within the coal matrix are the main storage space for gas, while

the micro-fractures, fissures, fractures and faults comprise the main conduits for gas

seepage and migration. It has been widely recognised that the permeability of coalbed

methane reservoirs changes with pressure depletion during production, and the shrinkage

of coal matrix due to desorption of gas tends to open cleat fractures and increase coal

permeability, which are competitive and work in opposite directions (Harpalani and Chen,

1997; Gorucu, 2005).

Different to flow in low-permeability low-porosity formations, such as fractured crystalline

rocks, gas flow in unconsolidated materials such as coal is essentially interstitial where flow

routes may be rather tortuous. The fractured coal seam comprises both permeable fractures

and matrix blocks. Dual porosity representations (Barrenblatt et al., 1960; Warren and Root,

1963) include the response of these two principal components only – release from storage

in the porous matrix and transport in the fractured network. Conversely, dual permeability

or multiple permeability models represent the porosity and permeability of all constituent

components (Bai et al., 1993) including the role of linear sorption (Bai et al., 1997) and of

multiple fluids (Douglas et al., 1991). Traditional flow models accommodate the transport

response as overlapping continua but neglect geomechanical effects, which is an important

process that needs to be coupled in the numerical study (Connell, 2009).

Conceptualizations include analytical models for dual porosity media, their numerical

implementation and models including the component constitutive response for dual and

multi-porous (Aifantis, 1977; Bai et al., 1993; Bai and Elsworth, 2000; Elsworth and Bai,

1992) media. Such models have been applied to represent the response of permeability

evolution (Ouyang and Elsworth, 1993; Liu and Elsworth, 1997) in deforming aquifers (Bai

and Elsworth, 1994; Liu et al., 1999) and reservoirs (Bai et al., 1995). These mentioned

models were developed primarily for the flows of slightly compressible liquids without

nonlinear desorption, thus not applicable to the flow of compressible fluids such as

methane (CH4) where gasses desorption is the dominant mechanism.

Gas sorption and dissolution may cause the coal matrix to swell and/or shrink. This may

change the specific surface areas and total macro-pore volume of the coal matrix. When the

methane is produced, permeability was computed as a function of reservoir pressure and

coal matrix shrinkage. A number of theoretical and empirical permeability models have



7-4

been proposed (Chen et al., 2009; Connell, 2009; Connell and Detournay, 2009; Cui and

Bustin, 2005; Gu and Chalaturnyk, 2005; Izadi et al., 2011; Liu and Requivst, 2010; Liu et

al., 2010; Pekot and Reeves, 2002; Palmer and Mansoori, 1998; Seidle and Huitt, 1995; Shi

and Durucan, 2004; Zhang et al., 2008). These studies have contributed to our

understanding of the possible causes of fracture permeability changes, and the gas

transport in matrix system has been primarily assumed to be a diffusion process dominated

by gas desorption process. The desorption models in coal seam are classified into two

general groups: equilibrium (pressure dependent) models and non-equilibrium (pressure

and time dependent) models (King and Ertekin, 1995). In equilibrium models, it is assumed

that desorption/diffusion process occurs rapidly enough (sorption time is assumed to be

zero) that the kinetics of the process can be ignored in gas flow transport formulations;

whereas in non-equilibrium models, kinetics of desorption/diffusion is accommodated in

the transport equations using Fick′s diffusion laws (Roadifer and Moore, 2009; Ziarani et al.,

2011).

The sorption time is an important parameter to evaluate gas transport, and significant

findings have been drawn so far. Its influence on the gas production for a CBM reservoir

with a vertical well was first reported by Remener et al. (1986). Non-equilibrium sorption

phenomena for single-phase gas flow in coal seams were further investigated by Anbarci

and Ertekin (1990). Study by Ried et al. (1992) shows that sorption time is a less important

parameter for late-time production data history matching compared to other parameters

such as permeability, initial pressure and drainage area. Clarkson et al. (2006) showed that

early-time production data with non-equilibrium sorption (large sorption time, 110 days)

could be suppressed with respect to that of equilibrium sorption model. However, several

points can be improved for these studies. Firstly, the contribution of free gas from matrix

system is generally neglected, but it is unclear to what extent it affects reservoir prediction

results. Secondly, the diffusion coefficient was also assumed to be constant, which is not

consistent with other studies (Bird et al., 1960; Lama and Nguyen, 1987; Kranz et al., 1990).

Moreover, although the importance of the sorption time on coalbed methane extraction

has been widely investigated, many researchers have not taken the geomechanical process

into consideration (Palmer, 2009; Reeves and Pekot, 2001; Shi and Duracan, 2004), or

primarily assumed the absolute permeability to be a constant during both gas water

production and dry gas production (Clarkson et al., 2007; Ziarani et al., 2011), which



7-5

missed an important phenomenon of coalbed reservoir (Connell, 2009; Gu and

Chalaturnyk, 2005; Liu et al., 2011).

Another important parameter that could bring influence on gas production is the shape

factor. The shape factor concept, originally introduced by Barenblatt in 1960 and later by

Warren and Root (1963), provides an elegant and powerful upscaling method for fractured

reservoir simulation. The shape factor determines the fluid matrix blocks and the

surrounding fractures when there is a difference in pressure between two systems. An

appropriate specification of the shape factor is therefore critical for accurate modelling.

Studies have been conducted to determine the transfer shape factor for slightly

compressible fluids in the fractured reservoirs (Coats, 1989; Lim and Aziz, 1995; Kazemi,

1969; Thomas et al., 1983; Ueda et al., 1989). The functionality of the fracture pressure as a

boundary to the matrix blocks may also have a significant effect on the value of the shape

factor (Hassanzadeh and Pooladi-Darvish, 2006). Similar phenomenon has been obtained

by Ranjbar and Hassanzadeh (2011), where the effect of fracture pressure depletion regime

on the shape factor for single-phase flow of a compressible fluid was investigated and

obtained that shape factor is a function of the imposed boundary condition and its

variability with time. Recently, Heinemann and Mittermeir (2012) derived the Kazemi et al.

(1992) shape factor on the fracture-matrix dual continuum, and concluded that the Kazemi

et al. formula is exact under pseudo-steady-state conditions within the dual continuum

mathematical concept of natural fractured dual porosity systems. These expressions of the

shape factors have been used to investigate the behaviour of fracture-matrix primarily for

conventional gas reservoirs (Lim and Aziz, 1995; Mora and Wattattenbarger, 2009;

Zimmerman et al., 1993), but very little work can be found in the literature regarding its

impact on unconventional gas reservoirs (e.g. CBM) with coupled influence of

geomechanical deformation and gas desorption.

Although these interactions were represented in our prior studies through dual poroelastic

models, the impact of gas mass transfer between fractures and coal matrixes on the CBM

production have not been well understood. In this study, the mass exchange between two

systems is defined as a function of gas sorption time constant at the standard condition,

coal matrix porosity, and the gas pressure difference between matrix and fracture systems.

Correspondingly, the gas diffusivity in the matrix is defined as a function of the shape

factor, the gas sorption time, and the gas pressure. These relations are integrated into a

fully coupled finite element model of coal deformation, gas flow and transport in the



7-6

matrix system and gas flow in the fracture system. The FE model represents important

non-linear impacts of the effective stress on CBM productions that cannot be recovered

where mechanical influences are not rigorously coupled with the complex interactions

between fractures and matrixes. The FE model is then applied to quantify the impact of

various parameters on the CBM production. Afterwards model verification with field

observation is carried out followed by conclusions.

7.2 Methodology

CBM extraction induced complex interactions between stress and sorptive chemistry exert

strong influence on the transport and sorptive properties of the coal. These include

influences on gas sorption and flow, coal deformation, porosity change and permeability

modification. We label this chain of reactions as ―coupled processes‖ implying that one

physical process affects the initiation and progress of another. The individual processes, in

the absence of full consideration of cross couplings, form the basis of very well-known

disciplines such as elasticity, hydrology and heat transfer. Therefore, the inclusion of cross

couplings is the key to rigorously formulate the behavior for coupled processes of coal-gas

interactions. The complexity of these interactions is reflected in the extensive suite of

cross-coupling relations among coal deformation, gas transport in matrix, and gas flow and

transport in fracture as illustrated in Figure 7-1. These relations include:

(1) The interaction between coal deformation and gas flow in the fracture is defined by

the change of effective stress induced by pore pressure change in the fracture system,

and affects the volumetric strain of coal.

(2) The interaction between coal deformation and gas flow in the fracture is also defined

the porosity change and the permeability change induced by the coal deformation.

(3) The interaction between coal deformation and gas flow in the matrix is defined by the

change of variable diffusivity induced by pressure depletion and coal shrinkage in the

matrix, and affects the change in volumetric strain of coal.

(4) The interaction between gas flow in the fracture and gas flow in the matrix is defined

by the gas mass transfer between the two systems.



7-7

Figure 7-1. Conceptual representation of coal-gas interactions.

In order to recover important non-linear responses due to effective stress effects,

geomechanical influence has to be rigorously coupled with the gas transport system. This

can be achieved through a full coupling approach, which is often the preferred method for

simulating multiple types of physics simultaneously since it should theoretically produce the

most realistic results (Zhang et al., 2008; Liu et al. 2011; Chen et al., 2009 and 2010; Wu et

al., 2010 and 2011). For this approach, a single set of equations (generally a large system of

non-linear coupled partial differential equations) incorporating all of the relevant physics is

solved simultaneously. In this study, a mass exchange function for compressible fluid

between coal matrix and fracture systems is adapted and implemented into our simulation

framework. The derivation processes have been listed in detail in the following sections.

7.2.1 Coal deformation

For all equations, traditional conventions are used: a comma followed by subscripts

denotes differentiation with respect to spatial coordinates and repeated indices in the same

expression imply summation over the range of the indices. Based on the above

assumptions, the strain-displacement relation is expressed as

x

y

• Volumetric Change

• Porosity Change

• Permeability Change

• Volumetric Change

• Porosity Change


• Mass Exchange

• Deformation Transformation

• Porosity Change


COAL SEAM

COAL MATRIXSwelling Component

Non-swelling ComponentCOAL FRACTURE

Fig.1 Conceptual representation of coal-gas interactions



7-8

)(2

1,, ijjiij uu (7-1)

where ij is the component of the total strain tensor and iu is the component of the

displacement. The equilibrium equation is as

0, ijij f (7-2)

where ij denotes the component of the total stress tensor and if denotes the component

of the body force.

The gas sorption-induced strain s is supposed not to produce any shear strain. Its effects

on all three normal components of strain are the same. The constitutive relation (7-1) for

the deformed coal seam becomes

ijsijf

f

ijmm

ijkkijij pK

pKKGG

33

)9

1

6

1(

2

1 (7-3)

where )1(2

E

G , )21(3

E

K ,

K

Km 1 ,

s

fK

K

K

K1 . G is the shear

modulus of the fissured coal, E is Young's modulus of the fissured coal and is

Poisson's ratio of fissured coal. kk is the mean stress, mp and fp are the pore pressures in

matrix and fracture system, respectively. K and K represent the bulk modulus of coal

without fissures and with fissures, respectively. sK denotes the bulk modulus of coal grains.

m and f are defined as the fluid pressure ratio factors for matrix and fracture systems,

respectively, compatible with Biot's coefficient (Biot, 1941; Wilson and Aifantis, 1982). ij

is called Kronecker's delta, 1 for ji and 0 for ji . From Equation (7-3), one obtains

sffmmv ppK

)(1

(7-4)

where 332211 v is the volumetric strain of coal matrix and 3/kk is the

mean compressive stress. Combination of Equations (7-1), (7-3) and (7-4) yields the

following Navier-type equation:

0

21,,2,,,,

iim

Lm

LLiffimmkikkki fp

pp

pKppu

GGu

(7-5)



7-9

This equation is the governing equation for coal deformation. The two variables, mp and

fp are linked through gas flow equation.

7.2.2 Coal permeability model

In the analysis of coal permeability, the fractured coal mass is treated as a discontinuous

medium comprising both matrix and fractures (cleats). The individual matrix blocks are

represented by cubes and may behave isotropically with regard to swelling/shrinkage,

thermal expansion, and mechanical deformability (Liu et al., 1999). The cleats are the three

orthogonal fracture sets and may also have different apertures and mechanical properties

ascribed to the different directions.

In this study, the following coal permeability model is used, which includes the coupling

impact from geomechanical deformation, gas pressure change and coal shrinkage induced

strain. For the 3D case with three orthogonal sets of fractures, coal directional permeability,

xk , yk , and zk are defined as follows (see Liu et al. (2010) for detail)

3

00

)1(31

s

s

vt

f

m

K

pR

k

k

(7-6)

where 0f is the initial fracture porosity at reference conditions.

For the 2D case with two orthogonal sets of fractures, coal permeability is defined as

follows

3

00

)1(21

s

s

vt

f

m

K

pR

k

k

(7-7)

7.2.3 Mass exchange function

The concept of treating fractured reservoir as a dual-porosity medium was introduced by

Warren and Root (1963). Whenever a pressure difference exists between the matrix and

the fractures, a fluid flow between the media will occur. The volume transfer rate per unit

bulk volume between the matrix and the fracture has the form (Lim and Aziz, 1995; Heel

et al., 2008):



7-10

fmm ppV

kq

(7-8)

where was defined as the shape factor for experimental samples. V denotes the volume

of matrix blocks.

Gas transport in coal takes place under two simultaneous and parallel processes, i.e., gas

diffusion in the matrix (micro-pores) and gas flow in the fracture (macro-pores). It is

common to estimate the diffusion process using desorption time, which is the time

required to desorb 63.2% of the initial gas volume during a whole core desorption test.

The relation between desorption time, 0 , and the diffusivity,

0D , is defined as

00

0

1

D

(7-9)

The sorption time is a lumped parameter accounting both for diffusion and desorption

time. 0D is the gas diffusivity measured at 15.273aT K, and 325.101ap kPa.

The gas diffusivity under in situ condition can be related to fluid and coal properties (Bird

et al., 1960; Lama and Nguyen, 1987) by:

msrm

mkD

1

1 (7-10)

wherem is the effective matrix porosity, is gas (or fluid) compressibility (1/MPa), r is

the coal bulk compressibility (1/MPa), and s

is coal matrix compressibility (1/MPa).

For an isothermal case, the following relationship can be obtained (Kranz et al., 1990):

a

m

p

pDD 0 (7-11)

Gas compressibility is generally much higher than r , to a first approximation, it can be

simplified into:

m

mm

m

m pkkD (7-12)

Combining with Equations (7-9) and (7-11), the above equation can be expressed as:

00

0

1

a

m

a

mm

ppD

k (7-13)

Substituting Equation (7-13) into Equation (7-8) gives



7-11

0000

fm

a

mfm

a

mpp

p

Vpp

p

Vq

(7-14)

The above equation accounts for the mass exchange between matrix and fracture systems during

gas extraction.

7.2.4 Shape factor for dual-permeability model

In their 1960 landmark paper, Barenblatt et al. introduced the shape factor concept to

model the (single-phase) fluid transfer between matrix and fractures (1960). The central

idea of Barenblatt et al. was not to study the behaviour of individual matrix blocks and

their surrounding fractures, but instead to introduce two abstract interacting media: one

medium, the ―matrix‖, in which the physical matrix blocks are lumped, and one medium,

the ―fractures‖, in which the fractures are lumped.

The shape factor depends on the type of model selected for dual-porosity formulation. The

Warren and Root′s (1963), Kazemi′s (1969) and Lim and Aziz′s (1995) models are the most

widely used ones, which are formulated as follows.

Warren and Root′s model:

2

24

L

nn (7-15)

where 3 2, 1, n is sets of normal parallel fractures, and is associated with different flow

geometries (slabs, columns and cubes, respectively) and L is fracture spacing.

Kazemi′s model:

222

1114

zyx LLL (7-16)

Lim and Aziz (1995) presented analytical solutions of pressure diffusion for draining into a

constant fracture pressure (boundary condition) as

222

2 111

zyx LLL (7-17)

For equal fracture spacing, is equal to 223 L for three sets of fractures. For one and

two sets of fractures, the values for are 2 and

22 , respectively.



7-12

In this paper, comparison among three shape factors is conducted to study the difference

of the coalbed methane production behaviour to each factor value under dual-permeability

model.

7.2.5 Gas flow equations

In a dual-permeability model, there are matrix/fracture, matrix/matrix, and

fracture/fracture connections: each matrix has connections with the matrix of all its

neighbouring blocks, and each fracture node has connections with the fracture nodes of all

its neighbouring blocks (Bai et al., 1995). The dual-porosity model can be regarded as a

subset of the dual-permeability model: in a case where there are no contacts between

adjacent matrix blocks.

Based on the dual permeability system, the mass balance equation for each system is

expressed as

sgfgf

f

sgmgmm

Qqt

M

Qqt

M

(7-18)

where M is the gas content, g is the gas density, g

q is the Darcy's velocity vector, t is the

time. sQ is the gas mass exchange term that represents the net mass addition of fluid to the

fracture system from the matrix blocks, and defined as a function of Equation (7-14). mk is

the permeability for matrix system. It is assumed that the gas sorption take places in the

matrix system only. Therefore, the gas contents in the matrix and the fracture are written

out as:

Lm

mLcgamgmm

pp

pVM

(7-19)

fgffM (7-20)

where the subscript m represents for matrix, f for fracture, g for gas, and c for coal.

is porosity, ga is the gas density at standard conditions,

LV represents the Langmuir

volume constant, and Lp is the Langmuir pressure.

Based on the mass conservation law, substituting Equations (7-6), (7-14), (7-19) and (7-20)

into gas flow equations of both systems, and simplifying each one it gives



7-13

t

p

pp

p

t

pR

pp

p

pVpp

kp

tK

Rp

pp

p

pVpp

kp

tpp

pVp

m

Lm

LLv

f

fm

fm

a

mmff

f

f

sf

mff

fm

a

mmmm

mm

mL

LLcoalatmm

2

0

000

00

20

)1(3

1)1(3

(7-21)

where is the dynamic viscosity of gas.

Based on the diffusivity value at reference pressure (atmosphere pressure for this example),

substituting Equation (7-13) into Equation (7-21) yields

t

p

pp

p

t

pR

pp

p

pVpp

kp

tK

Rp

pp

p

pVpp

pp

tpp

pVp

m

Lm

LLv

f

fm

fm

a

mmff

f

f

sf

mff

fm

a

mmmm

a

mm

mL

LLcoalatmm

2

0

000

00

20

)1(3

1)1(3

1

(7-22)

As shown in Equation (7-22), gas flow in matrix is now a reverse function of desorption

time and shape factor. They affect gas transport in both matrix and fracture systems

through exchange term.

7.2.6 Gas Production Rate

For coal seam gas reservoirs, the original gas in place (OGIP) is defined as

dv

pp

pV

p

p

p

pOGIP

Li

iLcm

a

im

a

i (7-23)

where three terms on the right hand side represent free phase gas in the matrix, free phase

gas in the fracture, and adsorbed gas in the coal matrix, respectively. When the reservoir

reaches the abandoned pressure,abp , the residual gas in place (RGIP) can be defined as

dv

pp

pV

p

p

p

pRGIP

Lw

abLcf

a

abm

a

ab (7-24)

The difference between the original gas in place (OGIP) and the residual gas in place

(RGIP) gives the accumulative gas production, pG , as defined



7-14

dv

pp

pV

p

p

p

pdv

pp

pV

p

p

p

pG

Lw

abLcf

a

abm

a

ab

Li

iLcm

a

im

a

iT

(7-25)

For a closed coal seam gas reservoir, TG is constant. The gas production at time, t , is

defined as

dv

pp

pV

p

p

p

pdv

pp

pV

p

p

p

ptG

Lm

mLcf

a

f

m

a

m

Li

iLcm

a

im

a

ip

(7-26)

where mp and

fp are the pressures for matrix and fracture system at the time t, respectively.

The gas production rate is defined as

dv

dt

dp

pp

pV

dt

d

p

p

dt

dp

pdt

dp

pdt

tGdm

Lm

LLcf

a

ff

a

fm

a

mp

2

(7-27)

Therefore, the production rate is controlled primarily by the matrix pressure change rate,

dtdpm.

The matrix pressure change rate is dependent on the pressure transfer between coal matrix

and fracture induced by the gas production. Prior to the gas production, the gas pressure in

the fracture system is equal to that in the coal matrix. We define this state as the initial

equilibrium state, ifm ppp . Coal permeability at this state is defined as the initial

equilibrium permeability, 0fk . At this equilibrium state, no shrinking takes place anywhere.

When coal seam gas is extracted, the free phase gas flows out from the fracture system and

the gas pressure in the fracture reduces. At this stage, an imbalance between fracture

pressure and matrix pressure is achieved. This pressure imbalance changes as the gas in the

matrix transfers into the coal fracture. The pore pressure in the matrix decreases and as a

consequence coal matrix shrinks due to both the matrix pore pressure decrease and the gas

desorption. When the imbalance between fracture pressure and matrix pressure vanishes

completely, the final equilibrium state is achieved. At the final equilibrium state, the matrix

pressure is equal to the fracture pressure, i.e., fm pp . Coal permeability at this state is

defined as the final equilibrium permeability. When the final equilibrium state is reached,

0dtdpmindicates that no gas is produced from the reservoir.

Based on the analysis above, a number of factors could contribute to coal seam gas

production rate. These include (1) Desorption time, ; (2) Shape factor, ; (3) fk -



7-15

Fracture permeability; (4) Fracture spacing, L ; (5) Coal matrix swelling capacity, s ;(6) Gas

sorption capacity, LV ; (7) BHP– Bottom hole pressure; and (8) Matrix and fracture

porosities. The controlling effects of these factors are evaluated in the following section.

7.3 Evaluation of gas production

In this section, the sensitivity study of reservoir behaviour with respect to different

parameters is simulated, particularly the values for the shape factor and desorption time.

7.3.1 Model description

The simulation model geometry is 750m by 750m with a production well located at the

centre of the model. The wellbore diameter is 21.6 cm, and the computational model is

shown in Figure 7-2. For the coal deformation model, lateral directions were constrained.

For the gas transport model, coal sample was initially saturated with gas with certain

pressure, and a constant production pressure was applied on the wellbore boundary for all

cases. Input parameters for this simulation were listed in Table 7-2.

Figure 7-2. Vertical view of the geometry of numerical model

Φ=21.60mm

L=750.00 m



7-16

Table 7-1. Parameter values for simulation model

Parameter Value Parameter Value

Density of coal, ρc (kg/m3) 1500 Initial gas pressure, P0, (psi) 700

Matrix porosity (%) 5.0 Langmuir pressure constant ( MPa) 2.76

Fracture porosity (%) 0.5 Coal swelling capacity (%) 2.0

Fracture permeability (mD) 10 Coal matrix sorption capacity, m3/ton 6.0

Coal Young's modulus (psi) 500 Young's modulus reduction ratio (mR ) 0.1

Coal Poisson's ratio (-) 0.4 Desorption time (day) 17

Gas viscosity (Pa·s) 12.28×10-6 Fluid pressure ratio factor (m

) 0.2

Matrix size (yx LL ) (cm) 5.0 Fluid pressure ratio factor ( f ) 0.7

Shape factor ratio (0 ) 0.05 Coal seam thickness (m) 6.0

7.3.2 Modelling strategy

A series of gas extraction conditions as listed in Table 7-2 was simulated to investigate the

responses of gas production rate. Simulation results were presented in terms of (1) the

impacts of desorption time, (2) the impacts of initial fracture permeability value, (3) the

impacts of fracture spacing value (shape factor value), (4) the impacts of swelling strain

capacity, (5) the impacts of production pressure; (6) the impacts of sorption capacity; (7)

the impacts of fracture porosity; and (8) the impacts of matrix porosity. The modelling

results were shown in Figures 7-3 to 7-13.

Table 7-2. List of simulation scenarios

where * represents base case.

Case Parameter Value

1 Desorption time (day) 5; 17*; 95

2 Initial fracture permeability (mD) 1.0; 10*; 50

3 Fracture spacing (cm) 1.0; 5.0*; 10

4 Swelling strain capacity (%) 1.0; 2.0*; 3.0

5 Coal matrix sorption capacity (m3/ton) 3.0; 6.0*; 12

6 Production pressure (psi) 100; 200*; 300, 400

7 Fracture porosity (%) 0.1; 0.5*; 1.0

8 Matrix porosity (%) 1.0; 5.0*; 10



7-17

7.3.3 Reservoir behaviour under constant total productive gas volume

As shown in Equation (7-23), the total gas volume can be produced is determined by the

initial reservoir pressure, porosities, coal gas content, and the abandoned reservoir pressure

while the gas production processes are determined primarily by the matrix pressure change

rate and the fracture pressure change rate. In this section, we investigate the evolution of

gas production rate and cumulative gas production where the amount of total gas volume

can be produced is fixed. The investigated factors include cases 1 to 5, as listed in Table 7-2.

Results are shown in Figures 7-3 through 7-7.

Impacts of desorption time: For this case, three different desorption time values were

used, varying from 5 days to 50 days. A smaller desorption time, such as 5 , represents

a quicker desorption process in which the adsorbed gas on the surface of micropores

desorbs more quickly. On the other hand, a sorption time of 50 days represents a very slow

desorption/diffusion transport process of gas. As shown in Figure 7-3, the increase of

desorption time has a reverse effect on the peak of gas rate. The higher the desorption time

(slower desorption process), the smaller the peak rate, and the longer it takes to reach the

peak. For instance, it takes 134 days for 5 case to reach the peak value of 571

Mscf/day, but they require 254 days and 376 days to reach the maximum gas rate of 386

Mscf/day and 228 Mscf/day for 17 and 50 , respectively. A significant increase of

cumulative gas production with decreasing desorption time has been observed within this

simulated period, as shown in Figure 7-3(b). It is also worth to note that the difference of

production rate was observed from the beginning of the production.

(a) (b)

Figure 7-3. Evolution of gas production rate and cumulative gas production under different

magnitudes of gas desorption time

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

τ=5 days

τ=17 days

τ=50 days

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

τ=5 days

τ=17 days

τ=50 days



7-18

Impacts of initial fracture permeability: Figure 7-4 shows the evolution of gas

production rate and cumulative production under different magnitudes of initial fracture

permeability. An increase from 1.0 mD to 50 mD significantly reduces the time taken to

reach peak value and improves production performance. When the permeability is 50 mD,

gas production rate only needs 80 days to reach the peak of 817 Mscf/day, but it increases

to 245 days when the permeability is 10 mD and its peak value dwindles to 385 Mscf/day.

The peak value has yet to appear in the studying period for 1.0 mD case.

In this work, coal permeability was considered to be a variable and defined as a function of

coal deformation, pore pressure change and shrinkage strain, but it is still not very clear

how much this will affect reservoir behaviour comparing with conventional simulations

(non-coupling model or constant permeability). In the following, comparison of both gas

production rate and cumulative production between conventional simulations and the fully

coupling one is shown in Figure 7-5. It shows that the maximum gas rate and final gas

recovery from coupling simulations are higher than those from non-coupling approach for

the same initial fracture permeability. This demonstrates that coal permeability increases

with gas depletion as the shrinkage strain increases the fracture opening. The comparison

results also indicate the importance to consider the permeability as a variable and choose

more realistic permeability value.

(a) (b)


magnitudes of initial fracture permeability

0

100

200

300

400

500

600

700

800

900

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

k= 1.0 mD

k= 10 mD

k= 50 mD

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

k= 1.0 mD

k= 10 mD

k= 50 mD



7-19

(a) (b)

Figure 7-5. Comparison of evolution of gas production rate and cumulative gas production

between coupling permeability model and non-coupling one

Impacts of fracture spacing: Figure 7-6 shows that with the increase of fracture spacing,

the maximum gas production rate and total gas recovery dramatically decrease, and the

same trend for the time to reach the peak gas rate. This finding has been consistent with

other researchers (Gu and Chalaturnyk, 2005; Sawyer et al., 1987). When the fracture

spacing value is 10 cm, the gas production rate takes 380 days to reach the peak value of

158 Mscf/day, but this value jumps to 760 Mscf/day at the day of 39 when fracture spacing

is 1.0cm. The primary reason for this effect is that as the fracture spacing becomes larger,

the mass exchange between two systems become slower, as shown in Equation (7-24), and

in turn postpones gas transport capacity in matrix system.

(a) (b)


magnitudes of fracture spacing

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

Non-coupling Model

Coupling Model

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

Non-coupling Model

Coupling Model

0

100

200

300

400

500

600

700

800

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

L=1.0 cm

L=5.0 cm

L=10 cm

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

L=1.0 cm

L=5.0 cm

L=10 cm



7-20

Impacts of swelling capacity: The results from coupling simulations explore the

influence of the coal matrix swelling/shrinkage capacity (shrinkage for this case) due to

methane desorption are shown in Figure 7-7. Results suggest that for a coal seam with a

higher matrix shrinkage capacity, the maximum gas rate and final gas recovery are higher as

compared with a coal seam reservoir having a smaller value of matrix shrinkage capacity.

This is expected since a higher shrinkage capacity will cause a larger shrinkage and in turn

bring more fracture opening, thus a more significant increase in permeability as illustrated

in Equation (7-6). Different from the cases of desorption time and initial permeability value,

at the beginning (e.g. before 300 days), both gas production rate and total gas recovery for

different swelling capacities are very close. This demonstrates that the early stage

production is dominated by effective stress change due to pressure drawdown. The

increasing peak rate with increasing swelling capacity illustrates that the desorption-induced

permeability increase is playing increasingly important role on gas production.. Similar

phenomenon has been observed by other researchers even based on different permeability

models or different coal geometry models (Robertson, 2005; Shi and Durucan, 2004).

(a) (b)


magnitudes of swelling strain capacity

In summary, for all cases, total gas volume can be produced in the reservoir is kept

constant through controlling parameters, as shown in Equation (7-25). The gas production

rate is determined by the matrix gas pressure change rate, dtdpm , and the fracture gas

pressure change rate,

dtdp f . They are dependent on each other. At the initial stage of gas

production, both pressures change with production, but the fracture pressure changes

much faster than the matrix gas pressure. This creates the pressure difference, so gas

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

εL=1.0 %

εL=2.0 %

εL=3.0 %

0

100

200

300

400

500

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

εL=1.0 %

εL=2.0 %

εL=3.0 %



7-21

production rate increases as production progresses. When the fracture gas pressure

stabilizes, 0dtdp f, the gas production rate is determined primarily by the matrix gas

pressure change rate. Gas production behaves different to different values of parameters,

which is a combining outcome of different mechanism, as shown in Equations (7-5) and

(7-24).

7.3.4 Reservoir behaviour under variable total productive gas volume

This part is to investigate the evolution of both gas production rate and cumulative gas

production under variable total gas volume in the reservoir, which relies on the production

condition such as production pressure, or gas storage condition such as coal sorption

capacity, and matrix and fracture porosities. The behaviours of coal seam gas reservoir to

different controlling factors are shown in Figures 7-8 through 7-12.

Impacts of sorption capacity: The effects of sorption capacity (or Langmuir volume) on

coalbed methane production are shown in Figure 7-8. Numerical results show that both

maximum gas production rate and total gas production rise with increasing gas sorption

capacity. For instance, the peak rate increases from 342 Mscf/day for 3.0 m3/ton case to

433 Mscf/day for 12m3/ton case, and the total gas production climbs from 361 MMscf for

3.0 m3/ton case during the simulation period to 652 MMscf for the later case. The time to

reach the maximum gas rate also slightly increases with increasing gas sorption capacity.

This is due to an increase of productive gas volume with the increase of gas sorption

capacity, while the reservoir pressure is fixed. This brings a larger gas supply which can

maintain higher gas production rate, and in turn postpones the decline of gas production

rate. It is worth noting that reservoir behaviour at the early stage both gas production rate

and total gas production follow the same trend, which is similar to the case for different

swelling capacities. This demonstrates that free gas phase plays more important than

sorption gas phase at this stage, as illustrated in Equation (7-24), but the desorption gas

phase takes over the dominant role of gas production after around 300 days, as shown in

Figure 7-8.



7-22

(a) (b)


magnitudes of coal sorption capacity

Impacts of production pressure: It is a common practice to achieve the required gas

production rate through controlling bottomhole pressure (BHP). This sensitivity study is to

simulate how the reservoir behaves with respect to different BHP values. Results from

Figure 7-9 show that both gas production rate and cumulative production increase with

reducing bottomhole pressures. The maximum gas rates in the order of increasing BHP

values are 425 Mscf/day, 384 Mscf/day, 321 Mscf/day and 236 Mscf/day, respectively.

The cumulative production declines from 558 MMscf for 100p psi case to 263 MMscf

for 400p psi case. The dramatic difference in both gas rate and the cumulative gas

production against different BHP values are observed from the very early stage of

production, but only a little difference of the time takes to reach maximum value has been

observed.

(a) (b)


magnitudes of gas production pressure

0

50

100

150

200

250

300

350

400

450

500

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

VL=3 m3/ton

VL=6 m3/ton

VL=12 m3/ton

0

100

200

300

400

500

600

700

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

VL=3 m3/ton

VL=6 m3/ton

VL=12 m3/ton

0

50

100

150

200

250

300

350

400

450

500

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

P=100 psi

P=200 psi

P=300 psi

P=400 psi

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

P=100 psi

P=200 psi

P=300 psi

P=400 psi



7-23

Impacts of fracture porosity: Figure 7-10 shows the evolution of gas production

regarding different fracture porosities. Model results show that fracture porosity has

significant influence on gas production rate profile. For instance, when %0.1f , gas rate

reaches its peak of 442 Mscf/day at 130 days, the maximum gas rate reduces to 385

Mscf/day at 305 days when f decreases to 0.5%, while it rebounds to 412 Mscf/day at

508 days when fracture porosity further decreases to 0.1%, as illustrated in Figure 7-10(a).

This rebound can be explained from two competing mechanisms that fracture porosity

contributes in the flow equation: on the one hand, decreasing fracture porosity reduces free

gas phase volume, which reduces gas production rate, but on the other hand reducing

fracture porosity increases the sensitivity of coal fracture permeability, which means more

permeability increase is expected during gas depletion. This enhancement of fracture

permeability can bring increase in gas production rate, as explained for the impacts of

fracture permeability on reservoir performance in Figure 7-5. This explanation can also be

seen from Equation (7-6), and has been confirmed by other researchers as well (Harpalani

and Chen, 1995, Gu and Chalaturnyk, 2005; Palmer, 2010). The gain or lost of gas

production rate depends on the competing processes of both mechanisms. Our numerical

results show that the influence of free gas volume change plays more significant role than

fracture permeability does for both %0.1f and %5.0f case, but it has been the

opposite for %1.0f case. Meanwhile, the comparison of cumulative production for

three cases indicates that the influence of the fracture porosity is only significant for the

early stage of gas production, as plotted in Figure 7-10(b).

(a) (b)

Figure 7-10. Evolution of gas production rate and cumulative gas production under

different magnitudes of fracture porosity

0

50

100

150

200

250

300

350

400

450

500

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

Φf= 0.1%

Φf= 0.5%

Φf= 1.0%

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

Φf= 0.1%

Φf= 0.5%

Φf= 1.0%



7-24

Impacts of matrix porosity: The performance of coal gas reservoir to different

magnitudes of matrix porosity is illustrated in Figure 7-11. An increase in both gas rate and

cumulative production with increasing matrix porosity is obtained, but a slight decrease for

the time to reach peak value is also observed. When %0.1m , the maximum gas rate is

around 150 Mscf/day, and it quickly jumps to 385 Mscf/day and 565 Mscf/day for

%0.5m and %10m cases, respectively.

However, this finding is contrary to the general belief that gas is stored primarily by

sorption into the coal and accounts for 98% of the gas within a coal seam (Gray, 1987),

and as a result the influence of free gas term in matrix porosity on gas produced has been

simply dismissed in many studies, particularly for the studies using Fick‘s law (Busch et al.,

2004; Clarkson and Bustin, 1999; King et al., 1986; Mora and Wattattenbarger, 2009; Shi

and Durucan, 2008). Equation (7-24) illustrates that two ways the matrix porosity can

contribute for gas transport in matrix system: increases free gas volume and enhances mass

exchange rate between two systems. Figure 7-12 illustrates the comparison of two different

numerical models: one couples the free gas influence of matrix system, and the other does

not consider the matrix free gas influence. Our numerical results show that model without

free gas influence could significantly overestimate gas production behaviour.

Comparison of the transient contribution of the desorbed gas, and the free phase gas in

both matrix and fracture systems to gas production rate at each moment for %0.5m

and %10m cases are shown in Figure 7-13 (a, c). It needs to be pointed out that the

ratio of the contribution for each system is calculated based on the change rate of each

system at each transient moment over the gas rate for the same moment. The detailed

explanation of the transient contribution calculation is listed in Table 7-3. Figure 7-13 (a, c)

shows that the free gas in matrix porosity plays the dominant role for gas transport at the

early stage, with maximum portion of around 48% for %0.5m case, which means 48%

of gas volume production at that moment is from matrix system (the source of this gas may

from desorption, but it is stored in matrix system at that moment), and increases to 66%

for %10m case. But the transient contribution of free gas in matrix porosity declines

with gas extraction and eventually adsorption phase gas takes over the dominant role. The

desorption gas and free phase gas in matrix system contribute to 71% and 28% of the gas

production rate for %0.5m case at the final investigation period, and their values

change into 55.8% and 43.5% for %10m case.



7-25

Comparison of the transient contributions of each part on total gas accumulation for two

different matrix porosities is plotted in Figure 7-13 (b, d). It can be seen that desorption

gas plays the increasingly dominant role on total gas production. For instance, for

%0.5m case, at the beginning of gas production, adsorption gas phase and free gas of

matrix system contribute 68.3% and 28.8% of total gas production, respectively, but these

values change into 78.2% and 19.4% at the end of simulation period. Similar trend is also

observed for %10m case with 53.1% and 44.7% at the beginning, and become 65.8%

and 33.2% at the end of simulation period.

From the above analysis we can see that although a great percentage of gas is stored in

adsorbed phase in coal seam gas reservoir, the real contribution from adsorbed phase at gas

transient flow process may be much lower than expected depending on reservoirs, and it

could overestimate reservoir behaviour is free gas from matrix system is dismissed in the

numerical model. Therefore, it could be more reasonable to consider the transient

contribution from matrix porosity when gas transport process is simulated to achieve more

realistic results.

(a) (b)

Figure 7-11. Evolution of gas production rate and cumulative gas production under

different magnitudes of matrix porosity.

0

100

200

300

400

500

600

700

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

Φm= 1.0%

Φm= 5.0%

Φm= 10%

0

100

200

300

400

500

600

700

800

0 500 1000 1500 2000 2500 3000

Cu

mu

lati

ve

pro

du

cti

on

(M

Ms

cf)

Time (day)

Φm= 1.0%

Φm= 5.0%

Φm= 10%



7-26

Figure 7-12. Comparison of gas produciton between numerical models with fress gas

influence and without

Table 7-3. Explanation of the transient contribution calculation

Time Free Gas in

Matrix Free Gas in Fracture Adsorbed Gas Total

0T 0,mV 0,fV 0,adV 0V

ttT ttmV , ttfV , ttadV , ttV

tT tmV , tfV , tadV , tV

Transient

contribution

on gas rate

ttt

tmttm

VV

VV

,,

ttt

tfttf

VV

VV

,,

ttt

tadttad

VV

VV

,, -

Transient

contribution on gas

production t

tmm

VV

VV

0

,0,

t

tff

VV

VV

0

,0, t

tadad

VV

VV

0

,0, -

0

100

200

300

400

500

600

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Pro

du

cti

on

ra

te (

Ms

cf/

d)

Time (day)

Φm= 5.0%

Gas rate (with free gas)

Cu

mu

lati

ve

pro

du

cti

on

(MM

sc

f)

Gas rate (without free gas)

Production (with free gas)

Production(without free gas)



7-27

(a) (b)

(c) (d)

Figure 7-13. Transient contributions of coal matrix system, fracture system, and desorption

process to gas production rate and cumulative gas production for two difference matrix

porosity cases

7.4 Field application of dual-permeability model

The comparison between the numerical results from dual-permeability model and the field

observations is qualitatively made to check the feasibility of the developed numerical model.

This example is from a single gas phase well production from the Fruitland Coal of San

Juan Basin, which has been presented in other studies (Clarkson and McGovern, 2005;

Clarkson et al., 2006; Clarkson et al., 2007).

Critical data collected from the pilot well included reservoir pressure (500 psia), original gas

in place (OGIP) (≈3.0 Bscf), wellbore diameter (20 cm), and production data. The drainage

area for the well is 320 acres. For the history-matching process, as a lot of parameters are

not available from literature, the OGIP value was used to calculate gas contents and

adsorption isotherms. Coal physical properties, such as Young‘s modulus and Poisson‘s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000

Tra

ns

ien

t c

on

trib

utio

n o

n g

as

p

rod

uc

tio

n ra

te (

%)

Time (day)

From matrix system

From fracture system

From desorption

Φm= 5.0%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 500 1000 1500 2000 2500 3000

Tra

ns

ien

t c

on

trib

utio

n o

n g

as

p

rod

uc

tio

n (%

)

Time (day)

From matrix system


From desorption

Φm= 5.0%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000

Tra

ns

ien

t c

on

trib

utio

n o

n g

as

p

rod

uc

tio

n ra

te (

%)

Time (day)

From matrix system


From desorption

Φm= 10%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 500 1000 1500 2000 2500 3000

Tra

ns

ien

t c

on

trib

utio

n o

n g

as

p

rod

uc

tio

n (%

)

Time (day)

From matrix system


From desorption

Φm= 10%



7-28

ratio, are cited from experimental measurements from other publications (Hager and Jones,

2001; Mavor and Vaughn, 1998; Siriwardane et al., 2009). Key data associated with geologic

and reservoir simulation models are listed in Table 7-3. In this matching model, the

constrained boundary conditions are applied and the monitored flowing bottomhole

pressure is used to simulate gas production rate and to compare with field data.

Comparison between numerical results of both dual-permeability model and single

permeability mode and field observations is presented in Figures 7-14. The single

permeability model refers to the model where the free gas in the matrix system is not

accounted, and the same parameters are used for both models.

Table 7-4. Parameter values for field data match

Parameter Value Parameter Value

Density of coal, ρc (kg/m3) 1300 Initial gas pressure, P0, (psi) 500

Matrix porosity (%) 4.4 Langmuir pressure constant ( MPa) 5.67

Fracture porosity (%) 0.5 Coal swelling capacity (%) 2.0

Fracture permeability (md) 30 Coal matrix sorption capacity, m3/ton 4.2

Coal Young's modulus (psi) 521 Young's modulus reduction ratio (mR ) 0.1

Coal Poisson's ratio (-) 0.32 Desorption time (day) 8

Gas viscosity 12.28×10-6 Fluid pressure ratio factor ( m ) 0.2

Matrix size (yx LL ) (cm) 5.0 Fluid pressure ratio factor ( f ) 0.7

Shape factor (m-2) Kazemi′s model Coal seam thickness (m) 14



7-29

Figure 7-14. Comparison between numerical results and field observations.

Comparison results from Figure 7-14 show that the dual-permeability model matches the

field data reasonably well, indicating that dual-permeability is capable to replicate the real

reservoir behavior. It is worth noting that the increase in gas rate at the early stage until

reaching the peak flow rate demonstrates the enhancing contribution from the mass

exchange between matrix and fracture systems, which is dominated by gas desorption from

coal matrix. The decline of gas rate in the late stage illustrates the gas desorption rate

decreases with pressure drawdown.

7.5 Conclusion

In this study, the dual-porosity/dual-permeability geometry was incorporated to conduct a

finite element numerical model, which combines with the complex geomechanical

deformation, gas flow and transport in the matrix system, and gas flow and transport in the

fracture system together with variable permeability. In this model, the mass exchange

between matrix and fracture systems was given as a function of sorption time, matrix

porosity, matrix particle volume, and pressure difference between matrix and fracture

systems.

Based on the modelling results and data comparison with field observations, the following

conclusions can be drawn:

0

100

200

300

400

500

600

700

10

100

1000

0 1000 2000 3000 4000

Ga

s p

rod

uc

tio

n r

ate

(M

sc

f/d

)

Time (day)

Gas rate

Dual-permeability model

Single permeability model

Flowing BHP

Input BHP data

Flo

win

gb

ott

om

ho

le p

res

su

re (

Ps

ia)



7-30

(1) Sorption time has a reverse effect on the peak of gas rate. The higher the sorption

time, the lower the peak of production rate, and the longer it takes to reach the

peak. The cumulative gas production increases with decreasing sorption time

values.

(2) Results from dual-permeability model show that fracture spacing has significant

influence on gas production. Higher gas rate is observed for smaller fracture

spacing cases, while shape factor model has little influence on compressible gas

production, which has been significantly different from the behavior of

incompressible or slightly compressible reservoirs.

(3) The initial fracture permeability, gas sorption capacity (or Langmuir volume) and

swelling capacity have similar impacts on gas production performance. The gas

production enhances with increasing values of these parameters as well as the time

to reach the maximum gas rate.

(4) Comparison of the transient contributions of the desorbed gas, and the free phase

gas from both two systems to gas production rate at each moment show that the

free gas from matrix plays the dominant role for gas transport at the early stage,

but declines with gas extraction and eventually desorbed phase gas takes over the

dominant role. For instance, the desorbed gas and free phase gas in matrix system

contribute to 71% and 28% of the gas production rate for %0.5m case at the

final investigation period, and their values change into 55.8% and 43.5% for

%10m case. However, the desorbed gas phase plays the increasingly dominant

role on total gas production, but the weight decreases with increasing matrix

porosities. For example, the desorbed gas contributes 78.2% of total gas

production for %5m case at the end of simulation period, but this weight

reduces to 65.8% for %10m case.




Scholarship. These sources of support are gratefully acknowledged.



7-31

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7-32

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7-33

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7-34

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8-1

Chapter 8

Concluding Remarks

8.1 Main findings

This thesis has focused on the experimental and numerical study on geomechanics of coal-

gas interactions, particularly its effect on coal permeability evolution. This effort brings

together various aspects regarding the experimental measurements on coal properties to

different gas injections with different gas pressures and temperatures, directional

permeability model development, the roles of effective stress coefficient and sorption-

induced strain on permeability evolution under stress-controlled boundary conditions, and

the related permeability development, mechanism investigation on coal permeability

evolution and forecast of CBM production to various parameters with dual- porosity/dual-

permeability model. The major contributions and findings made in this study are

summarized below.

1. Chapter 2 presents the experimental measurement of adsorption isotherms, swelling,

permeability, and geomechanical properties for an Australian coal sample in N2, CH4

and CO2 at 35 ºC and 40 ºC. The results show that the gas adsorption is less at higher

temperature, and coal swelling strain with respect to adsorbed amount is close to the

same trend line irrespective of gas type and temperature. Permeability coefficients

decline with increased gas pressure, but it does not show a trend in relation to the gas

adsorption capacity for this coal. Elastic modulus of coal is found to be larger at higher

effective stress and lower temperatures.

2. In chapter 3, a directional permeability model was developed to define the evolution of

gas sorption-induced permeability anisotropy under in-situ stress conditions. This was

implemented into a fully coupled finite element model of coal deformation and gas

transport in a coal seam, and found that

The directional permeability of coal is determined by the mechanical boundary

conditions, the ratio of coal bulk modulus to coal matrix modulus, the initial

fracture porosity, and the magnitude of the sorption-induced coal swelling strain.

The boundary conditions control the magnitudes of total strains while the



8-2

modulus reduction ratio partitions the effective strain (total strain minus the

swelling strain) between fracture and matrix.

For restraint conditions of uniaxial strain and for a constant volume reservoir,

changes in coal permeability are determined only by the gas pore pressure and the

swelling strain.

3. Chapter 4 presented the experimental observations of a series of gas flow-through

experiments all under the conditions of a constant pressure difference between

confining stress and pore pressure. Comparison between measured permeability data

and calibrated results demonstrates that the effective stress coefficient cannot be

assumed to be unity for both two coal samples and it could play an important role in the

evaluation of permeability change in adsorbing gasses, particularly with high gas flow

pressures.

4. A phenomenological permeability model has been developed in chapter 5 to explain

why coal permeability decreases even under the unconstrained conditions of variable

stress. Based on this permeability model, numerical results illustrate that coal

permeability profiles under the controlled stress conditions are mainly regulated by the

following factors: when the confining stress or swelling capacity is higher, the final

equilibrium coal permeability is much lower than the initial permeability; the

permeability increases more when the facture compressibility or effective stress

coefficient is higher; when the sensitivity factor is higher, the reduction in coal

permeability is more significant.

5. The performance of current coal permeability models was evaluated in Chapter 6 against

analytical solutions for the two extreme cases of either unconstrained swelling or

constrained However, these models have a boundary mismatch because model

boundary is constrained while experiment boundary is free displacement or

unconstrained. In this study, a matrix-fracture model is numerically investigated after

incorporating heterogeneous distributions of Young's modulus, Langmuir strain

constant in the vincity of the fracture. Comparison between homogeneous and

heterogeneous models has also been conducted. Major findings include:

Both homogeneous and heterogeneous models experience the swelling transition

from local swelling to macro swelling. At the initial stage of gas injection, matrix

swelling is localized within the vicinity of the fracture compartment. As the



8-3

injection continues, the swelling zone is widening further into the matrix and

becomes macro-swelling.

Coal permeability experiences a rapid reduction at the early stage, a switch in

behavior from permeability reduction to recovery is observed, and coal

permeability finally recovers until it reaches the final equilibrium permeability. For

the homogeneous model, the final equilibrium permeability is always higher than

the original value, but the opposite was obtained for the heterogeneous model.

With the heterogeneous distributions of coal physical and swelling properties, this

numerical model matches with the experimental data reasonably well, which

demonstrates that heterogeneity of coal properties can be the answer to coal

permeability reduction under unconstrained conditions.

6. In Chapter 7, the dual-porosity/dual-permeability model was used to conduct a fully

coupled finite element model to quantify the impact of various parameters on the

production of coalbed methane (CBM). It couples coal deformation, gas flow and

transport in the matrix system, and gas flow in the fracture system. Based on the

modeling results, the following conclusions can be drawn:

The CBM production rate is controlled by interactions of multiple processes

triggered by the gas extraction itself. This unique feature of CBM reservoirs is

characterized through defining a number of coal properties such as permeability,

and coal-gas interactive properties such as desorption time constant, swelling and

desorption capacities, as a function of the gas extraction process.

The results of this study demonstrate important non-linear impacts of coal

property processes and coal-gas interactive property processes on CBM

productions. These impacts cannot be recovered by previous studies where

mechanical influences are not rigorously coupled with the complex interactions

between fractures and matrixes.

The results of this study also demonstrate the importance of appropriate coal

property models and coal-gas interactive property models for the successful

forecast of CBM productions because these properties are very sensitive to the gas

extraction process.

These conclusions above suggest that the optimal in-situ control of coal properties and

coal-gas interactive properties could be achieved through designing a better gas

extraction process.



8-4

8.2 Recommendations for future work

In this thesis, the comprehensive study of the geomechanics of coal-gas interactions on the

evolution of coal properties, coal permeability in particular, has been conducted, but

further investigations are important and still need research efforts in the future as outlined

below:

1. Whether adsorption causes coal softening is not clear, since the Young′s modulii with

Helium are not higher than those measured when adsorbing gasses. More work needs to

be done to further elucidate the impact of adsorption on geomechanical properties.

2. Our experimental results show the cleat compressibilities at 40 ºC are smaller than those

in 35 ºC. This may be because of the lower swelling strains at higher temperature leading

to less porosity decrease. However, since the cleat aperture may be altered by increasing

the temperature, the conclusion of lower compressibility at higher temperature will

require further investigation.

3. Due to the nature of low permeability, coal seams usually need to be hydraulically

fractured before coalbed methane recovery process begins. The fracture permeability is

not constant but changes with in situ conditions during production, so it is a challenge

to properly include and simulate the property permeability change in geomechanical

simulation.

4. Although it has been reported that the gas flow in the cleat system may be of the non-

Darcy nature, little has been known on how this non-Darcy flow affects the coalbed

methane extraction. Therefore, the non-Darcy effect is an interesting topic needs to

look at in the future.

5. Most studies on coal-gas interactions were focused on single gas component. How

multiple gasses interact with coal matrixes and fractures, especially how gas components

affect the geomechanical deformation, are not fully understood. Further research on this

issue could be important.

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