Gas Sorption and the Consequent Volumetric and Permeability ...

GAS SORPTION AND THE CONSEQUENT VOLUMETRIC AND
PERMEABILITY CHANGE OF COAL
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES
ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Wenjuan Lin
March 2010

c○ Copyright by Wenjuan Lin 2010
All Rights Reserved
ii

I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
Dr. Anthony R. Kovscek
(Principal Advisor)
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
Dr. Franklin M. Orr Jr.
(Department of Energy Resources Engineering)
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
Dr. Roland N. Horne
(Department of Energy Resources Engineering)
Approved for the Stanford University Committee on Graduate Studies.
iii

Abstract
Coalbed methane has grown in importance as an energy resource since the 1980s.
Nevertheless, effective means to release methane from the tight, fractured reser-
voirs have yet to be developed. Primary recovery by reservoir depressurization is
successful, but generally produces only about half of the gas in place. Gas (car-
bon dioxide, nitrogen, or mixtures of these components) injection is potentially
an efficient technique both to enhance coalbed methane recovery as well as se-
quester greenhouse gases (mainly carbon dioxide) in subsurface geological sites.
Due to the special features of coalbed reservoirs and the nature of gas retention
in the reservoirs, there are unique issues that need to be taken into account when
designing field operations and conducting numerical simulations of gas produc-
tion and injection in coalbed methane reservoirs. One issue of particular interest
is the permeability evolution of the reservoirs as gas is produced or injected. Two
mechanisms are believed to change permeability: (1) changing effective stress
due to the change of reservoir pressure caused by production or injection activi-
ties, and (2) strain caused by gas adsorption/desorption on the internal surfaces
of coal. In spite of the conceptual convenience of these statements, better under-
standing of the physics and sound mathematical representations of the mecha-
nisms have yet to be developed.
Experimental and numerical investigations of gas sorption on coal, and the
subsequent volumetric and permeability changes of the coal were conducted.
The goals of the study were to investigate the magnitude of permeability change
caused by gas sorption, and develop an algorithm to simulate numerically gas
v

sorption and sorption-induced permeability change. The amount of gas sorp-
tion and the subsequent volumetric and permeability change of coal samples
as a function of pore pressure and injection gas composition were measured in
the laboratory. A constant effective confining pressure (difference between the
confining pressure and pore pressure) was maintained in the process of the ex-
periments, therefore, the role of effective stress on permeability was eliminated.
Several gases, including pure CO2, pure N2, and binary mixtures of CO2 and N2 of
various composition were used as the injection gas. The coal sample was first al-
lowed to adsorb an injection gas fully at a particular pressure. The total amount
(moles) of adsorption was calculated based on a volumetric method. After ad-
sorption equilibrium was reached, gas samples were taken from the equilibrium
gaseous phase and analyzed afterwards. The composition of the gaseous phase
prior to and after the adsorption was used to calculate the composition of the
adsorbed phase based on material balance. Permeability of the sample was then
measured by flowing the injection gas through the core at varying pressure gra-
dient or varying flow rate, and an average permeability was obtained based on
Darcy’s law for compressible systems. The change of the total volume of the core
was monitored and recorded in the whole process of the experiment. Volumet-
ric strain was thereby calculated. Experimental results showed that the greater
the pressure the greater the amount of adsorption for all tested gases. At the
same pressure, the amount of adsorption was greater for CO2 than N2. For the
binary mixtures, the greater the fraction of CO2 in the injection gas, the greater
the amount of total adsorption. Volumetric strain followed the same trend as the
amount of adsorption with pressure and injection gas composition. Permeabil-
ity showed opposite behaviors, decreasing with the increase of pressure and the
percentage of CO2 in the injection gas.
The experimental adsorption, volumetric strain, and permeability data were
analyzed to investigate the numerical correlations between gas sorption, sorption-
induced volumetric strain and permeability, and pressure and injection gas com-
position. The relationship between the amount of adsorption and pressure for
vi

pure gases (CO2 and N2) were readily represented by parametric isotherm mod-
els, such as Langmuir and the N-layer BET equations. Modeling efforts of multi-
component adsorption included predicting amount of adsorption and adsorbed
phase composition based on the extended Langmuir equations and the ideal ad-
sorbed solution model. Activity coefficients of the components in the adsorbed
phase were computed based on the real adsorbed solution model and the ABC
excess Gibbs free energy model. Algorithms for modeling the CO2/N2-Coal sys-
tem were developed, and the constraints and strength of each model were dis-
cussed. The experimental volumetric strain was found to be linearly proportional
to the total amount of adsorption and independent of the injection gas compo-
sition. The permeability reduction could not be readily correlated by the models
in the literature unless the change of other coal properties (bulk modulus, axial
constrained modulus, etc.) due to gas sorption was incorporated.
The sorption, volumetric strain, and permeability data collected in this study
can be used for comparison by other researchers conducting similar studies. The
algorithms of sorption modeling and the correlations developed in this study are
readily incorporated into the simulation of enhanced coalbed methane recovery
and CO2 sequestration in coalbeds.
vii

viii

Acknowledgements
All the work presented in this dissertation was prepared with the support of the
Global Climate and Energy Project (GCEP). This support is gratefully acknowl-
edged. However, any opinions, findings, conclusions, or recommendations ex-
pressed herein are those of the author and do not necessarily reflect the views of
GCEP and its supporters.
I came to Stanford as a Master’s student in July of 2004. Here comes eventually
the big time when I am writing acknowledgement for my PhD dissertation!
First of all, I wish to express my highest appreciation to my advisor Professor
Anthony R. Kovscek. I worked with Professor Kovscek in both Master’s and PhD
programs. I came to Stanford (as well as the United States) as a fresh college grad-
uate with a Bachelor’s degree in Petroleum Engineering from China, fresh and
ambitious but knowing not much what awaited me in this totally new environ-
ment. There were truly lots of tough occasions in the past five years, in course-
work, in research, and in general life. Professor Kovscek has been always there for
me kindly and supportively. I appreciate the most that he was always patient for
my progress in research. There were times when it seemed there was no progress
for weeks. It was my greatest luck as a PhD student to have a understanding and
supportive advisor. Now I am Professor Kovscek’s most senior student, making
fewer and fewer mistakes. And it is the time for me to graduate and become a
professional. I attribute many of the merits I gained over the past five years to
Professor Kovscek!
I would like to give special thanks to people I consider as mentors and friends
along my journey to this point: Mr. Xin Li, Professor Shicheng Zhang, Dr. Tom
ix

Tang, Dr. Bunmi Owolabi, Mr. Chris West, Dr. Chris Clarkson, Dr. Hongmin
Zhang, and Dr. Xinfang Ma. They were the best teachers in specific periods in my
professional life. All that I learned from them will benefit the whole of my life!
Lots of thanks to the friends who accompanied me at Stanford in the past
five years: Hongmei Li and Deqiang Wang, Yaqing Fan, Liping Jia, Ling Liang,
Yangyang Liu, Yang Liu, Bin Jia, Jing Wang, Lisa Stright, Ays¸egül Dastan, Onur
Fidaner and many others. They are considered as dear family!
I wish to acknowledge all the professors in the Department of Energy Re-
sources Engineering for their valuable teaching and guide! I would like to thank
my classmates and lab fellows for their friendship and encouragements to be my
best self and made my staying at Stanford a wonderful experience!
Special thanks to Selçuk Fidan, for his incredible love and support in the last
few months of my staying at Stanford! The last but not the least important, I
would like to thank my family for their countless support, my father, mother and
younger brother! They are the softest part of my heart. Dearest memory to my
uncle, Gaoshan Liu, with whom I can not share my happiness now, but I owe the
deepest appreciation!
Sincere thanks to my PhD committee members: Dr. Roland N. Horne, Dr.
Franklin M. Orr Jr., Dr. Jennifer Wilcox, and Dr. Mark D. Zoback!
x

Contents
Abstract v
Acknowledgements ix
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives and Methodology . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Literature Review 9
2.1 Coal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Coalbed Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 CO2 Sequestration in Coalbeds . . . . . . . . . . . . . . . . . . . . . 16
2.4 Evolution of Coalbed Permeability . . . . . . . . . . . . . . . . . . . 19
2.5 Gas Sorption on Coal . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 The Langmuir Approach . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 The Gibbs Approach . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.3 The Potential Approach . . . . . . . . . . . . . . . . . . . . . 42
2.6 Adsorption/Desorption Hysteresis . . . . . . . . . . . . . . . . . . . 42
2.7 Research Efforts of Peers at Stanford . . . . . . . . . . . . . . . . . . 45
2.8 Goals of This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Experiments 51
3.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xi

3.2 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 The Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Experiments Measurements and Data Processing . . . . . . . . . . . 66
3.3.1 Dead Volumes, Initial Porosity and Permeability . . . . . . . 66
3.3.2 Simultaneous Sorption, Swelling/Shrinkage, and Permeabil-
ity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 Complementary Experiments using Helium . . . . . . . . . . 78
3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.1 Adsorption and Volumetric Strain . . . . . . . . . . . . . . . . 82
3.4.2 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Summary of Experimental Work . . . . . . . . . . . . . . . . . . . . . 85
4 Numerical Modeling 89
4.1 Permeability Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Sorption Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 Pure Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Multicomponent Sorption . . . . . . . . . . . . . . . . . . . . 93
4.2.3 Adsorption of Supercritical CO2/N2 Binary Gas Mixtures on
Coal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.4 Adsorption of CO2/N2 Binary Gas Mixtures on Coal . . . . . 116
4.3 Sorption-Induced Volumetric Strain and Permeability . . . . . . . . 121
4.4 Summary of Modeling Work . . . . . . . . . . . . . . . . . . . . . . . 125
5 Summary and Future Work 133
5.1 Improvements on the Current Experiments . . . . . . . . . . . . . . 134
5.2 Sorption, Permeability, and Volumetric Change of Wet Coal . . . . . 135
5.3 Sorption of Liquid CO2 on Coal . . . . . . . . . . . . . . . . . . . . . 137
A Thermodynamics of Gas Sorption on Solid 145
A.1 Mixing and Excess Gibbs Free Energy . . . . . . . . . . . . . . . . . . 145
A.2 The Standard State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
xii

B Experimental Sorption Data 149
C Adsorption Calculation Codes 157
C.1 ELM & IAS for CO2/N2-Coal System . . . . . . . . . . . . . . . . . . . 157
C.2 RAS for CO2/C2H6-NaX System . . . . . . . . . . . . . . . . . . . . . 164
C.3 IAS & RAS for CO2/N2-Coal System . . . . . . . . . . . . . . . . . . . 171
C.3.1 Binary Adsorption Calculation Based on the Ideal Adsorbed
Solution Model . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.3.2 Binary Adsorption Calculation Based on the Real Adsorbed
Solution Model . . . . . . . . . . . . . . . . . . . . . . . . . . 177
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xiv

List of Tables
2.1 Carbon dioxide storage capacity of the different geological forma-
tions (Benson, 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Parameter values for different cubic equations of state (Kovscek,
2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Exponential relation of permeability and pressure, k = k0p b (Lin
et al., 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Length, wet weight, and wet density of the coal plugs in the core. . . 59
3.3 Parameters of the composite coal core used in the experiments . . . 69
3.4 Specific volume of adsorbed phase based on different equations of
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Mean free path of CO2 at different pressures. . . . . . . . . . . . . . 81
3.6 Flow path size of typical coalbed methane reservoirs. . . . . . . . . 82
4.1 Values of pure adsorption isotherm constants. Experimental tem-
perature = 22 o C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 The values of the parameters in the modified virial equations for
pure CO2 and C2H6 adsorption on NaX, T = 20 o C. . . . . . . . . . . . 101
4.3 Experimental binary adsorption data of CO2(i)/C2H6(j) on NaX. Ex-
perimental temperature was 20 o C. . . . . . . . . . . . . . . . . . . . 102
4.4 Modeling results for adsorption of CO2(i)/C2H6(j) binary mixture
on NaX, T = 20 o C. Calculation was based on the assumptions of
an ideal gas phase, a nonideal adsorbed phase, and an ideal pure
condensed phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xv

4.5 Modeling results for adsorption of CO2(i)/C2H6(j) binary mixture
on NaX, T = 20 o C. Calculation was based on assumptions of a
nonideal gas phase, a nonideal adsorbed phase, and an ideal pure
condensed phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Pure adsorption isotherm (Langmuir) constants based on experi-
mental data of Hall et al. (1994). . . . . . . . . . . . . . . . . . . . . . 113
4.7 Adsorption of binary gas mixture of 60%CO2/40%N2 (i = CO2, j =
N2) on wet Fruitland coal, T = 115 o F . Constants for ABC excess
Gibbs free energy model: Ao = −14.55, C = 0.014. . . . . . . . . . . . 115
4.8 Adsorption of binary gas mixture of 75%CO2/25%N2 (i = CO2, j =
N2) on intact dry Powder River Basin (Montana) coal, T = 22 o C.
Ao = −22.46 and C = 1.203. . . . . . . . . . . . . . . . . . . . . . . . . 120
4.9 Helium porosity and permeability prior to the injection of different
feed gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.1 Adsorption of supercritical binary mixtures of CO2 and N2 on Fruit-
land coal (Hall et al., 1994). . . . . . . . . . . . . . . . . . . . . . . . . 150
B.2 Adsorption of supercritical binary mixtures of CO2 and N2 on Fruit-
land coal (continued) (Hall et al., 1994). . . . . . . . . . . . . . . . . 151
B.3 Adsorption of supercritical binary mixtures of CO2 and N2 on Fruit-
land coal (continued) (Hall et al., 1994). . . . . . . . . . . . . . . . . 152
B.4 Adsorption of pure CO2 on intact Montana coal and the conse-
quent volumetric strain. Experiment temperature = 22 o C. . . . . . 153
B.5 Adsorption of pure N2 on intact Montana coal and the consequent
volumetric strain. Experiment temperature = 22 o C. . . . . . . . . . 154
B.6 Adsorption of 50%CO2/50%N2 binary feed gas on intact Montana
coal sample and the consequent volumetric strain. Experiment
temperature = 22 o C. Component indices: i = CO2, j = N2. . . . . . . 155
B.7 Adsorption of 75%CO2/25%N2 binary feed gas on intact Montana
coal sample and the consequent volumetric strain. Experiment
temperature = 22 o C. Component indices: i = CO2, j = N2. . . . . . . 156
xvi

List of Figures
1.1 Historical coalbed natural gas reserves and production in the United
States (EIA, 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Coal structure and gas in coal. . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Mechanisms of permeability change of coal. . . . . . . . . . . . . . . 7
2.1 Structure of coal is similar to that of polymer. . . . . . . . . . . . . . 12
2.2 Adsorption capacity of coal of different ranks and depth (Thomas,
2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Primary recovery of coalbed methane. . . . . . . . . . . . . . . . . . 14
2.4 Historical trends in atmospheric carbon dioxide concentration and
global temperature (UNEP/GRID-Arendal Maps and Graphics Li-
brary, 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 A coalbed represented by a matchstick model. . . . . . . . . . . . . 23
2.6 Five types of adsorption isotherms, p 0 is the saturation vapor pres-
sure (after Yang, 1987). . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Plots of the BET equation with different C values (C = 1, 10, 50, 100). 30
2.8 Plots of the N-layer BET equation with different N values (C = 100,
N = 1, 5, 10, 50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.9 The Gibbs surface model (after Defay and Prigogine, 1966) . . . . . 32
2.10 Condensation and evaporation in a capillary cylinder (after Do,
2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.11 Pure gases sorption on ground Powder River Basin coal sample (Tang
et al., 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xvii

3.1 Coalbed methane fields in the United States, lower 48 states (cour-
tesy of EIA). Coal samples were from the dotted circle enclosed area. 52
3.2 Experimental setup and the coal holder for permeability measure-
ment (Lin et al., 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Permeability of a coal pack with injection of different gases (Lin
et al., 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Coal sample from Wyodak-Anderson coal zone, Powder River Basin,
Montana. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Permeability change of a composite coal core after the adsorption
of 15%CO2/85%N2 binary mixture at escalating pressures. Experi-
ments were conducted at room temperature (22 o C). Net effective
stress was kept around 400 psi. . . . . . . . . . . . . . . . . . . . . . 58
3.6 Composite core used in the experiments. . . . . . . . . . . . . . . . 60
3.7 CT images of the composite core at different conditions. Raw CT
numbers are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Experimental systems for adsorption measurement (Talu, 1998). . . 63
3.9 Experimental setup for adsorption measurement. . . . . . . . . . . 63
3.10 Schematic of the experimental apparatus of composite core. . . . . 66
3.11 Pressure versus voltage of the pressure transducers. . . . . . . . . . 68
3.12 Permeability measurement using the new apparatus. . . . . . . . . 70
3.13 The adsorption cell (valves in bold were closed while the adsorp-
tion measurement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.14 Readings of the pressure transducers and the confining pressure
pump during adsorption of binary mixture of 75%CO2/25%N2, con-
fining pressure = 800 psi. . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.15 Calibrated readings of pressure transducers during adsorption of
binary mixture of 75%CO2/25%N2, confining pressure = 800 psi. . . 73
3.16 The gas chromatograph and some gas samples. . . . . . . . . . . . . 74
3.17 Permeability measurement after adsorption of binary mixture of
75%CO2/25%N2 at 387 psi (uncalibrated), confining pressure = 811
psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xviii

3.18 Readings of the pressure transducers during desorption of binary
mixture of 75%CO2/25%N2, confining pressure = 830 psi. . . . . . . 78
3.19 Calibrated readings of pressure transducers during desorption of
binary mixture of 75%CO2/25%N2, confining pressure = 830 psi. . . 79
3.20 Pressure and volume of the confining pressure pump after opening
helium to core at 403 psi, confining pressure = 800 psi. . . . . . . . . 80
3.21 Volumetric strain at escalating pressures. Open symbols represent
the total volumetric strain, and closed symbols represent volumet-
ric strain due to adsorption only. . . . . . . . . . . . . . . . . . . . . 83
3.22 Permeability of the core with injection of different gases. . . . . . . 84
3.23 Permeability reduction of the core with injection of different gases. 86
4.1 Adsorption of pure CO2 and N2 on coal samples from Powder River
Basin, Wyoming (ground) and Montana (intact). Symbols repre-
sent the experimental data, and curves are the fitted models. . . . . 92
4.2 Simulation results of adsorption of different CO2/N2 binary mix-
tures on coal based on the extended Langmuir equations (short
dashed lines) and the IAS model (solid curves) . . . . . . . . . . . . 106
4.3 Contour of the average error in the calculated equilibrium pres-
sures and selectivity coefficients. Ao = −4.86 and C = 0.088 yielded
the minimum average error. . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 Plot of ni/ ˆ fi ∼ p for adsorption of CO2/C2H6 binary mixture on
NaX, T = 20 o C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Supercritical pure CO2 and N2 adsorption on wet Fruitland coal (T
= 115 o F ). Data from Hall et al. (1994) . . . . . . . . . . . . . . . . . . 113
4.6 Experimental data and simulation results of 60%CO2/40%N2 ad-
sorption on wet Fruitland coal. Simulations are based on the ex-
tended Langmuir equations (blue), the IAS model (magenta), and
the RAS model(red); experimental data of Hall et al. (1994) are shown
as symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xix

4.7 Plot of ψ 0 versus pressure for pure CO2 and N2. CO2/N2 binary gas
adsorption on intact dry Powder River Basin (Montana) coal sam-
ple at room temperature (22 o C) . . . . . . . . . . . . . . . . . . . . . 118
4.8 Experimental data and simulation results for CO2/N2 adsorption
on intact dry Powder River Basin (Montana) coal at 22 o C. . . . . . . 119
4.9 Sorption induced volumetric strain as a function of amount of ad-
sorption and modified surface potential. . . . . . . . . . . . . . . . . 122
4.10 Permeability reduction versus sorption induced strain. . . . . . . . 124
4.11 Permeability reduction versus total amount of adsorption. . . . . . 125
5.1 Permeability reduction of a coal pack with adsorption of pure CO2
at different temperatures, net overburden pressure = 400 psi (Lin,
2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Phase Diagram of CO2 showing temperature-pressure conditions
in CBM reservoirs of Black Warrior Basin, Alabama, USA (after Pashin
and McIntyre, 2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
xx

Chapter 1
Introduction
1.1 Background
Coalbed methane is a convenient name for gas retained in underground coal
seams. Coalbed gas contains mainly methane while other components like ethane,
propane, carbon dioxide, nitrogen, helium, and hydrogen may also be present
(British Geological Survey, 2006). Coalbed methane is generated either from a
biological process as a result of microbial action or from a thermal process as a
result of increasing temperature with depth of the coal. Almost all coal seams
contain some gas. Coal seams are often saturated with water, and methane is
held in the coal by the water pressure. The amount of gas retained by coal is a
function of pressure, temperature, pyrite content and the structure of the coal
(Thomas, 2002). Gas in coal is not a new discovery. It was usually considered un-
desirable in the coal mining industry. Explosions and outbursts associated with
coal mining is a serious problem in some part of the world. From the 1980s, how-
ever, coalbed methane began to be recognized as a possible source of clean fuel.
One cubic foot of methane has a heating capacity of approximately 1000 BT Us
(British thermal units) (Montgomery, 1986). For the same amount of heat gen-
erated, methane emits the least amount of carbon dioxide among hydrocarbons.
1

2 CHAPTER 1. INTRODUCTION
Therefore, coalbed methane is a very clean burning and low carbon dioxide pro-
ducing fuel. Additionally, deep unmineable coal seams are considered as poten-
tial geological sites for carbon dioxide sequestration along with depleted gas/oil
reservoirs, deep saline aquifers and ocean sediments (Parson and Keith, 1998).
Over the years, coalbed methane reserves and production increased dramat-
ically (Figure 1.1). In the United States, from 1989 to 2007, the proved reserve of
coalbed methane increased from 3,676 BCF (billion cubic feet) to 21,868 BCF and
the production increased from 91 BCF to 1,742 BCF. To get a sense of the magni-
tude of the coalbed methane reserve, the proved reserve for dry natural gas in the
United States is 237,726 BCF. The proved reserve of coalbed methane accounts
for about 10 % of that of the total dry natural gas.
Different from the conventional natural gas in sandstone reservoirs that exists
mainly as free gas in the pore spaces, gas in coal beds exists mainly in an adsorbed
phase on the internal surface area of the coal (Gray, 1987). The gas stays adsorbed
on the coal surface, until certain conditions are satisfied, for instance, a decrease
in the reservoir pressure or the presence of some more adsorbable gases. Cur-
rently, coalbed methane is produced mainly through primary recovery by reser-
voir depressurization. A large volume of formation water is pumped out from
the reservoir by downhole pumps, reservoir pressure decreases, methane des-
orbs from the coal surface, concentrates in the coal seams, migrates towards the
wellbore, and then flows up to the ground surface through the wellbore annulus.
Primary recovery is straightforward to implement, but it is not the most effective
way to recover the resource. Injection of nitrogen, carbon dioxide, or the binary
mixtures of the two gases into coal beds is shown to be an effective means to in-
crease the ultimate recovery of coalbed methane at laboratory scale (Tang et al.,
2005). Such gas injection is referred to as enhanced coalbed methane (ECBM)
recovery. Injecting nitrogen or carbon dioxide makes use of different recovery
mechanisms. With the injection of nitrogen, the partial pressure of methane
in the coal seams deceases, which causes methane to desorb from the coal sur-
face. While in the case of carbon dioxide injection, carbon dioxide displaces the

1.1. BACKGROUND 3
(a) Historical coalbed natural gas proved reserves.
(b) Historical coalbed natural gas production.
Figure 1.1: Historical coalbed natural gas reserves and production in the United
States (EIA, 2009).

4 CHAPTER 1. INTRODUCTION
methane on the coal surface, because coal has stronger affinity for carbon diox-
ide than methane. CO2-ECBM also provides a mechanism of greenhouse gas se-
questration.
Knowledge of gas sorption on coal is essential in estimating the original gas
in place for coalbed methane reservoirs and designing efficient ways to recover
coalbed methane and sequester carbon dioxide in coal beds.
Coalbed methane reservoirs are characterized as naturally fractured, dual poros-
ity, low permeability, and water saturated gas reservoirs (Jahediesfanjani and Civan,
2005). Coalbed methane reservoirs consist of primary and secondary storage and
mass transfer systems, Figure 1.2(a) (after Harpalani and Schraufnagel, 1990).
Figure 1.2(c) is the SEM image for a bituminous sample. The primary porosity
system (PPS) refers to the pore space in the coal matrix. PPS accounts for the
majority of the coalbed porosity and it is the major residence of the adsorbed gas
in coal beds. The secondary porosity system (SPS) refers to the natural fracture/-
cleat network in coal beds. Free gas in coalbed methane reservoirs resides mainly
in the SPS. SPS accounts for the major part of permeability for fluid flow in coal
beds. Studies of some coals from the San Juan Basin show that “the coals com-
prise closely spaced (0.3 - 1 mm apart), wide (10 - 60 mm) cleats with narrow (5
- 20 mm wide) microcleats in between, all of which are open and non-partially
mineralized” (Gamson et al., 1996).
In the process of coalbed methane recovery, the gas first desorbs from the
coal surface. The desorbed gas accumulates in the micropore spaces in the coal
matrix. With the increase of the gas concentration in the pore spaces, the gas
diffuses through the matrix to the cleat/fracture network and then flows to the
wellbore through the cleat/fracture network (Harpalani and Schraufnagel, 1990).
In summary, the gas migration in coal beds includes three stages: desorption
from the internal coal surface, diffusion through the coal matrix and Darcy flow
in the cleat/fracture network.
The permeability of coal beds changes during gas production and injection. It
is believed that two mechanisms account for the changes of coalbed permeability
(Harpalani and Zhao, 1989). One is the change of effective stress and the other is

1.1. BACKGROUND 5
(a) A simplified model of coalbed structure
and gas in coal.
(b) Microview of coalbed structure and gas in coal.
(c) SEM images of an bituminous sample: intra-particle porosity.
Magnification = 500× (left), 1000× (right) (Liu, 2009).
Figure 1.2: Coal structure and gas in coal.

6 CHAPTER 1. INTRODUCTION
sorption (adsorption/desorption) induced strain. For instance, the change in the
permeability of coal beds during reservoir depressurization is the result of two
competitive effects: as reservoir pressure decreases under a constant overburden
pressure, the effective pressure increases, permeability decreases due to the cleat
compression, i.e. closure of the fractures, Figure 1.3(a); on the other hand, as
the reservoir pressure decreases, gas desorbs from the matrix of the coal, which
causes the shrinkage of the matrix, opening of cleats, and an increase in per-
meability, Figure 1.3(b). The actual permeability evolution of coalbed methane
reservoirs is a result of the two competing effects. The change of the permeability
can be large. Therefore, it is essential to use a realistic permeability model in the
simulation of coalbed methane recovery and CO2 sequestration in coal beds to
account for this change.
1.2 Objectives and Methodology
The goal of this study was to understand the sorption behavior of gas on coal and
the consequent permeability change of coal. The initial objective was to build
a permeability model to be used in the simulation of coalbed methane recov-
ery and/or CO2 sequestration in coalbeds. To understand the sorption-induced
permeability change fully, gas sorption isotherms on coal must be studied. The
sorption behavior itself is worthy of full understanding in order to estimate the
available coalbed methane resources when a coalbed methane reservoir is first
found, and develop efficient methods for coalbed methane recovery. Knowledge
of gas adsorption on coal also gives insights to the general sorption problem in
separation and catalysis processes.
Both laboratory experiments and numerical modeling were conducted. An
experimental setup was designed to measure the amount of gas sorption, the
adsorbed phase composition, permeability, and volumetric change of a compos-
ite coal core simultaneously. Different gases including pure CO2, pure N2, and
binary mixtures of CO2 and N2 of various composition were used for injection.
Measurements were done at escalating pore pressure and constant net effective

1.2. OBJECTIVES AND METHODOLOGY 7
(a) Cleat closure and permeability decrease due to increase of effective stress.
(b) Matrix shrinkage and permeability increase due to gas desorption.
Figure 1.3: Mechanisms of permeability change of coal.
stress. At a specific pore pressure, the core was first saturated (reaching sorption
equilibrium) with the injection gas, the pressure and volume of the adsorption
cell were monitored and recorded pre- and post- adsorption, the total amount
of adsorption was calculated based on a volumetric method. Gas samples were
taken from the gas phase prior to and after adsorption. Gas composition was
analyzed using gas chromatography (GC) and used to calculate the adsorbed
phase composition. The injection gas was then allowed to flow through the core
with the flow rates and pressure gradients along the core recorded to be used
for permeability calculation. The volume of the core was monitored throughout

8 CHAPTER 1. INTRODUCTION
the whole process of the experiments, to be used in the calculation of sorption-
induced volumetric strain.
Based on the experimental measurements, appropriate adsorption isotherms
and thermodynamic models were chosen for sorption modeling. It is possible
to measure the amount of adsorption of a specific gas on a specific adsorbent at
a specific pressure and temperature. It is, however, impossible to measure ad-
sorption of all the interested gases on the adsorbent over the whole pressure and
temperature range. Therefore, it is important to have a reliable adsorption model
that enables prediction of adsorption based on limited experimental data. Also, it
is convenient to incorporate adsorption models with other models, for instance
a permeability model including a term of sorption-induced strain.
The sorption and permeability model from this study is readily used in nu-
merical simulations of gas flow in coal, and the more complex coalbed methane
recovery and CO2 sequestration.
1.3 Dissertation Outline
In this chapter, the background, objectives, and methodologies of the study were
introduced. Chapter 2 contains background information about coal structure,
coalbed methane, coalbed methane recovery and CO2 sequestration in coal beds,
and a review of some of the work in the literature on coal permeability change
and gas sorption isotherms. Chapter 3 describes the objectives of the experi-
mental study, the experimental apparatus and procedures, data processing, and a
summary of the obtained experimental data. Chapter 4 starts with sorption mod-
eling, followed by sorption induced volumetric strain and permeability model-
ing, and ends with a discussion of the experimental and modeling results. Chap-
ter 5 lists some suggestions for directions of further investigation.

Chapter 2
Literature Review
2.1 Coal Structure
It is generally agreed that coal is vegetal in origin (Haenel, 1992). Ancient swampy
plants were buried and formed peat that is believed to be the precursor of coal.
The peat, over several hundred million years, by action of moderate temperature
(about 500 K) and pressure in a geochemical stage, underwent a progressive coal-
ification process to form different types of coal, lignite, sub-bituminous, bitumi-
nous, anthracite, and graphite. The greater the degree of coalification, the higher
the rank of the coal. Coal is composed primarily of carbon and variable quan-
tities of impurities like sulfur, hydrogen, oxygen and nitrogen. The elementary
composition of coal changes with the rank. The carbon content increases from
55 wt% in peat to more than 91.5 wt% in anthracite, hydrogen decreases from 10
wt% to less than 3.75 wt%, and oxygen decreases from about 35 wt% to less than
2.5 wt% for peat versus anthracite (Ward, 1984). The other impurities, sulfur and
nitrogen, account for only a small portion of coal and their concentration change
during coalification is not significant.
The inherent constituents of coal can be divided into “macerals”, the organic
equivalent of minerals, that are mainly fossilized plant remains; and “mineral
matter”, the inorganic fraction made up of a variety of primary and secondary
minerals (Thomas, 2002). Besides organic and inorganic matter, there is also
9

10 CHAPTER 2. LITERATURE REVIEW
moisture content. The organic matter, inorganic matter, and moisture make up
the whole of coal.
The organic content of coal (macerals) are shown as distinctive areas under
microscopes. There are three major groups of macerals: vitrinite, exinite, and in-
ertinite (Thomas, 2002). Vitrinite, alternatively called huminite, originated from
woody plant materials (mainly lignin) is the most prevalent group, accounting
for 80% of the macerals in coal. The dry substance of wood consists of about 40%
cellulose, up to 35% lignin, and less than 30% hemicelluloses, among which only
lignin is strongly resistant to biochemical and chemical degradation. Exinite, al-
ternatively called liptinite, is derived from lipids and waxy plant substances; and
inertinite is originated from oxidized plant materials like char attributed to wood
fires that occurred during dry periods while peat was forming. Coal may be made
up of one major single maceral, or associations of macerals.
The inorganic fraction of coal is the minerals that are not combustible. Some-
times ash is erroneously referred to as a component of coal, whereas ash is the
mineral residue after combustion of coal. The major minerals in coal include
clay, carbonates, iron disulphides, oxides, hydroxides, sulphides, etc. (Thomas,
2002). The minerals are either detrital or authigenic in origin, and were intro-
duced into coal while peat was deposited or during the later coalification pro-
cess.
Moisture is one of the important properties of coal. Water affects gas adsorp-
tion on coal. Underground coal seams are usually soaked in formation water.
Besides the groundwater and other extraneous moisture, there is moisture held
within the coal itself, known as inherent moisture. Within coals moisture may oc-
cur in four possible forms: surface moisture held on the surface of coal particles
or macerals; hydroscopic moisture held by capillary force within the microfrac-
tures of coal; decomposition moisture incorporated in the decomposed organic
compounds of coal; and mineral moisture comprised part of the crystal structure
of hydrous silicates such as clays (Ward, 1984). Heating the sample moderately
under vacuum is an efficient way to remove the moisture from the coal.
As a result of its origin, coal is a non-volatile, insoluble, non-crystalline, highly

2.2. COALBED METHANE 11
complex mixture of organic molecules of varying size and structure (Haenel, 1992).
The characterization of coal structure is extremely difficult. Some hypothetical
average molecule models were proposed to represent specific coals. Figure 2.1(a)
shows one of these models for bituminous coal. The proposed molecular formula
is C661H561N4O74S6 and the molecular weight is 10,023 grams per mole. Shown to-
gether with the molecular model of the coal is part of the molecular structure of
lignite that is actually a natural polymer, Figure 2.1(b). In this sense, coal is actu-
ally similar to polymers.
2.2 Coalbed Methane
There is usually gas in coal seams. The composition of gas in coal is mainly
methane, with varying amounts of carbon dioxide, carbon monoxide, nitrogen
and ethane. Gases are generated in the coalification process (Thomas, 2002).
When peat forms at a relatively low temperature (below 323 K), gases are gener-
ated as the organic materials decompose. Gases generated in this stage are called
biogenic coalbed methane. In the later stages of coalification, temperature rises
as a result of increasing burial depth or direct contact with intruded igneous ma-
terial, gases are generated as the organic materials are altered and the carbon
content per unit mass increases. Gases generated in this stage are called thermo-
genic coalbed methane.
Compared with the conventional natural gas in sandstone reservoirs, a huge
amount of gas can be accommodated on the internal surface area of coal as a re-
sult of adsorption. The amount of gas retained in a coalbed methane reservoir is
a function of the rank of coal, the reservoir pressure, and reservoir temperature,
Figure 2.2. Gas in coal exists in three forms: (1) free gas in pore space and frac-
tures, (2) in an adsorbed phase on the internal surfaces of coal, and (3) adsorbed
within the molecular structure (Shi and Durucan, 2005). Adsorbed gas accounts
for 95% - 98% of the gas in coal seams. To recover the gas, certain conditions
need to be fulfilled to initiate desorption of the gas. These conditions are the de-
crease of the reservoir pressure, the presence of a more adsorbable gas such as

12 CHAPTER 2. LITERATURE REVIEW
(a) A molecular model of bituminous coal (Shinn, 1984).
(b) A partial structure for lignin (Reusch, 1999).
Figure 2.1: Structure of coal is similar to that of polymer.

2.2. COALBED METHANE 13
Figure 2.2: Adsorption capacity of coal of different ranks and depth (Thomas,
2002).
CO2, or a reduction in the methane partial pressure. These mechanisms led to
the ideas of primary recovery of coalbed methane by depressurization, and en-
hanced coalbed methane recovery by injecting N2 and/or CO2.
Most of the CBM production in the world to date uses the primary recovery
method. Production wells are completed open hole. Casing is set to the top of
the target coal bed and the underlying target zone is under-reamed and cleaned
out with a fresh-water flush. During production, downhole submersible pumps
are used to move formation water up the tubing. With the pumping out of water,
the reservoir pressure decreases. Methane desorbs from the coal surface, diffuses

14 CHAPTER 2. LITERATURE REVIEW
to the cleats/fracture network and flows to the wellbore. The gas then flows up
the wellbore annulus to the surface facilities. Coal beds usually have low perme-
ability ranging from 0.1 to 30 millidarcies (md) (McKee et al., 1989), hence, wells
are usually fractured hydraulically to attain sufficient permeability for fluid flow
in the near wellbore region.
Figure 2.3: Primary recovery of coalbed methane.
Coalbed methane wells do not usually produce gas at a rate as large as that of
conventional natural gas wells. Most coalbed methane wells in the United States
produce at a rate of 100 to 500 thousand cubic feet (MCF) per day, while conven-
tional gas wells produce approximately 1,700 MCF per day (Stevens et al., 1998a).
Primary recovery by depressurization typically recovers less than half of the re-
source underground (Stevens et al., 1998b). Sizeable gas resources are left behind
after primary recovery. The San Juan basin alone is estimated as having as much

2.2. COALBED METHANE 15
as 10 TCF of natural gas left behind after primary recovery (Stevens et al., 1998b).
Horizontal and multilateral wells may yield higher gas production, especially for
thin coal beds (Maricic et al., 2005). Besides the low recovery ratio, however, pri-
mary recovery also poses environmental and operational issues. These include
disposal of the large amount of produced water, a drop in the level of aquifers
providing drinking water near coalbed methane fields, and the potential con-
tamination of aquifers, etc. (Rawn-Schatzinger, 2003).
To increase the gas production rate, as well as solve some of the other prob-
lems associated with primary recovery, substitution gas injection is under inves-
tigation as a potential enhanced coalbed methane (ECBM) recovery method. Two
popular variants of enhanced coalbed methane recovery are inert gas stripping
using nitrogen and displacement resorption employing carbon dioxide. Labora-
tory experiments showed that much greater recovery ratio is achieved, and less
water is produced in the gas injection ECBM processes (Tang et al., 2005).
As injection gas, nitrogen is chosen because it is abundant. When N2 is in-
jected into coal seams, it acts as an inert gas because it is only slightly adsorbable
on coal. Methane desorbs as its partial pressure in the pore spaces decreases in
the presence of N2. The desorbed methane is then carried away by the continu-
ous N2 flow, which in turn enhances the desorption of methane. The produced
gas mixture is separated in the surface facilities. Injecting N2 typically gives ear-
lier incremental coalbed methane recovery (Zhu et al., 2002). Simulation and
early demonstration projects showed that nitrogen injection is capable of recov-
ering 90% or more of the gas in place, and the average incremental capital and
operating cost is about $ 1.00/MCF (Stevens et al., 1998b).
On the other hand, carbon dioxide may be preferred as an injection gas be-
cause of its effectiveness in displacing methane on the coal surface and the ad-
ditional benefit of sequestering a greenhouse gas in the subsurface. When CO2 is
injected, it displaces the methane on the coal surface because coal has a stronger
affinity for CO2. The desorbed methane is produced and the CO2 is retained.
Carbon dioxide adsorbs strongly to coal surfaces thereby impeding premature

16 CHAPTER 2. LITERATURE REVIEW
breakthrough (Tang et al., 2005). Even though no large-scale field CO2-ECBM im-
plementation has occurred, pilot projects were conducted (Stevens et al., 1998b;
Mavor et al., 2004). At the world’s first CO2-ECBM pilot in the San Juan Basin, by
continuous injection of carbon dioxide, the optimal gas production was as high
as 150% of the primary recovery methods with negligible breakthrough of car-
bon dioxide. In three years, the four-injection-well pilot injected over 2 BCF of
carbon dioxide. The project was found to be profitable at wellhead prices above
$1.75/MCF. As much as 13 TCF of additional methane resource potential is added
within the San Juan Basin if ECBM is implemented over the whole basin (Stevens
et al., 1998b). Pilot tests on two Alberta wells showed that carbon dioxide in-
jectivity was acceptable in spite of the low permeability of the coal beds (Mavor
et al., 2004). Part of the reason may be that injection opened preexisting fractures
and created new fractures. In spite of the big potential of CO2-ECBM, opera-
tional strategies are still to be refined. The continuous carbon dioxide injection
and simultaneous production process in the San Juan Basin pilot was successful
with significant enhanced coal bed methane recovery and minimal carbon diox-
ide breakthrough, whereas the injection-soak-production process employed in
the Alberta wells was less successful. More pilot studies have been conducted in
many other places of the worlds (van Bergen et al., 2006; Wong et al., 2007; Shi
et al., 2008). Enhanced coalbed methane recovery is very likely to be obtained as
long as there was sufficient injectivity.
2.3 CO2 Sequestration in Coalbeds
Nowadays, global warming is considered to be a significant environmental issue.
It took a long time and lots of investigation for people to realize that increas-
ing anthropogenic CO2 emissions are probably one of the most important rea-
sons causing the dramatic increase in atmospheric CO2 concentration and global
temperature (Weart, 2008). In the 1820s, scientists revealed that gases in the at-
mosphere might trap the heat the Earth received from the Sun. Then in 1859,
John Tyndall identified that water vapor (H2O) and carbon dioxide (CO2) could

2.3. CO2 SEQUESTRATION IN COALBEDS 17
trap heat rays. In 1896, Svante Arrhenius made a calculation and reached the
conclusion that doubling the CO2 concentration in the atmosphere would raise
the Earth’s temperature about 5-6 o C. Even though the model of Arrhenius was
too simple to capture all the elements in the climate system of the Earth, his es-
timation was not very far off from the value people believe nowadays. People,
however, were not concerned about anthropogenic greenhouse gas emissions
in the 19th and 20th century for several reasons: (1) the level of the CO2 emis-
sion caused by human activities was very low; (2) it was believed that sea water
would absorb five sixths of the additional gas; and (3) warming might not be a
bad thing, because it means more vegetation coverage on the earth and more
pleasant climate for some parts of the world. Several findings helped people to
realize the greenhouse effect of CO2 should not be ignored. In 1955, Hans Suess
detected fossil carbon in the atmosphere. Hans Suess and Roger Revelle found
that the ocean could take up most of the anthropogenic CO2 emissions, but over
a long run of many thousand years. In 1973, Keeling pointed out that at some
time plants would reach their limit in taking up carbon and there would even-
tually be so much CO2 in the ocean surface that the oceans would not be able
to absorb gas as rapidly. From the 1980s, scientists cut samples from cores from
Greenland and Antarctic ice and revealed that the historical level of atmospheric
CO2 had gone up and down closely related with temperature, Figure 2.4. Now
people can calculate the effect of CO2 on radiation, gas dissolution in sea water,
and other physical phenomena. It is well accepted that greenhouse gas emission
from human activities causes global warming.
Carbon dioxide emissions from the consumption and flaring of fossil fuels is
about 30 Gt (gigatonnes) globally and 7 Gt in the United States in the year 2006
(EIA, 2006). The CO2 level in the atmosphere is estimated at 383 ppm in 2007.
It will continue to rise as global population and activity increase, particularly in
developing countries such as China and India. One way to reduce atmospheric
CO2 concentration is to reduce CO2 emission by improving efficiency and switch-
ing from coal and crude oil to energy sources with a lower carbon footprint such
as natural gas and renewable energy (Freund et al., 2005). Another option is to

18 CHAPTER 2. LITERATURE REVIEW
Figure 2.4: Historical trends in atmospheric carbon dioxide concentration and
global temperature (UNEP/GRID-Arendal Maps and Graphics Library, 2007).
capture CO2 from anthropogenic sources and store it in some geological sites, for
instance, depleted oil/gas reservoirs, deep saline aquifers, unmineable coal beds,
and ocean sediments. Table 2.1 provides worldwide CO2 storage capacity of the
different geological sites (Benson, 2005). Deep saline aquifers have the largest ca-
pacity. A lot of research is going on to investigate the mechanisms of CO2 storage
in deep saline aquifers. The storage capacity of unmineable coal beds is uncer-
tain due to the structural complexity of coal beds and the lack of knowledge of
gas adsorption on coal. The CO2 storage capacity in coal beds is usually esti-
mated based on the figure of recoverable coalbed methane resources. A recent
study estimated the CO2 storage capacity of coal beds in the United States to be

2.4. EVOLUTION OF COALBED PERMEABILITY 19
about 90 Gt, among which 25-30 Gt can be sequestered at a profit (with enhanced
coalbed methane recovery), and 80-85 Gt can be sequestered at a cost of less than
$5/tonne (exclusive of the costs of capture and transportation) (Reeves, 2003).
The enhanced coalbed methane recovery potential associated with this seques-
tration is estimated to be over 150 Tcf. By comparison, the total CBM recoverable
resources are currently estimated to be about 170 Tcf.
Table 2.1: Carbon dioxide storage capacity of the different geological formations
(Benson, 2005).
Geological Formation Capacity
(Gt CO2)
Depleted gas and oil reservoirs 675-900
Deep saline aquifers >1000
Unmineable coalbeds 3-200
2.4 Evolution of Coalbed Permeability
Coalbed methane reservoirs are characterized by two distinctive porosity sys-
tems: a network of natural fractures formed mainly by shrinkage of the source
plant materials during the coalification process, and the matrix blocks contain-
ing a highly heterogeneous porous structure, Figure 1.2(a). The natural fractures,
also known as cleats, are further divided into two categories: the face cleats that
are continuous throughout the reservoir and the butt cleats that are usually in a
perpendicular direction and terminate at the face cleats. The face cleats and the
butt cleats divide the coal into matrix blocks that contain pores varying in size
from a few nanometers (nm) to over a micrometer (µm) (Shi and Durucan, 2005).
According to International Union of Pure and Applied Chemistry (IUPAC), pores
are categorized into macropores (>50 nm), transient or mesopores (between 2
and 50 nm) and micropores (

20 CHAPTER 2. LITERATURE REVIEW
Coalbed methane reservoirs are in equilibrium until some processes, such
as coalbed methane recovery or gas injection, are initiated. For instance, in the
process of coalbed methane primary recovery by depressurization, with the de-
crease of reservoir pressure (the overburden pressure is assumed remaining unal-
tered during drainage), the effective stress increases, which causes fracture com-
paction and a decrease in the coal permeability; meanwhile, when the reservoir
pressure decreases, the adsorbed gas in coal desorbs which causes the shrinkage
of the coal matrix, the opening of the fractures and an increase in the permeabil-
ity. The reservoir permeability may decrease at the beginning of depressurization
due to the predominant effect of increasing effective stress; at the late stage of de-
pressurization, however, when the pressure is low enough and a large amount of
gas desorbs from the coal, the permeability may increase because the effect of
matrix shrinkage dominates (Harpalani and Zhao, 1989).
Somerton et al. (1975) investigated the effect of stress on coal permeability of
three bituminous coals. They found that the permeability of the coal specimens
were strongly stress-dependent. The permeability of the coals under investiga-
tion varied over a wide range from about 0.1 md to nearly 100 md, among which
higher permeability specimens showed permeability reduction of one order of
magnitude and lower permeability specimens showed permeability reduction of
over two orders of magnitude within the tested stress range. The magnitude of
the permeability reduction of the low permeability specimens is compatible with
that for other rocks. A exponential correlation between permeability and effec-
tive stress was presented:
k = Ae Bσ
where, σ is the effective stress, and A and B are constants.
(2.1)
Both permeability increase and reduction have been observed in the field.
Observations of flow and pressure variations in the coal seams in the coal mines
in the Bowen basin area of Queensland, Australia, indicated that there was large
increase in permeability during drainage into boreholes or mine openings; whereas,
decreasing permeability during drainage was observed in some coal mines in

2.4. EVOLUTION OF COALBED PERMEABILITY 21
Japan (Gray, 1987).
Harpalani and Zhao (1989) measured the permeability of cylindrical speci-
mens and found that under a constant confining pressure, permeability of coal
decreases with decreasing pore pressure, however, the permeability starts in-
creasing once the pressure falls below the desorption pressure. Harpalani and
Schraufnagel (1990) measured the volumetric change of their coal samples inde-
pendent of the permeability measurement. The experiments were conducted in
the absence of a pressure difference between the pore pressure and the confin-
ing pressure to eliminate the volumetric change of the void space. Harpalani and
Schraufnagel (1990) found that the volume of coal matrix decreased linearly with
increasing gas pressure when helium was used; whereas, the volume increased
linearly with pressure when the experiments were repeated using methane, which
showed the sorption-induced swelling/shrinkage clearly.
Harpalani and Chen (1997) conducted a series of experiments to study the ef-
fects of sorption-induced swelling/shrinkage and gas slippage on the permeabil-
ity of coal. When the pore sizes are very small, gas permeability increases with
increasing pore pressure due to slip flow of gas at pore walls (Tanikawa and Shi-
mamoto, 2006). This is called the “Klinkenberg effect”. When the pore pressure
changes while the effective stress is kept constant, the coal permeability change is
actually caused by two effects, the Klinkenberg effect and the swelling/shrinkage
of the coal matrix. Harpalani and Chen (1997) separated the two effects during
pore pressure decrease by first using helium to test the Klinkenberg effect. The
Klinkenberg effect is represented by the following equation:
k = k0
�
1 + b
pm
�
(2.2)
where, k0 is the absolute permeability of the medium, pm is the mean gas pres-
sure, and b is the Klinkenberg slip factor. Methane was then used to study the

22 CHAPTER 2. LITERATURE REVIEW
permeability change due to shrinkage of the coal matrix which was found lin-
early proportional to the volumetric strain:
∆kshrinkage = α
� ∆Vmatrix
Vmatrix
�
(2.3)
In Equation 2.3, α is a constant depending on the characteristics of the coal type,
and ∆Vmatrix/Vmatrix is the volumetric strain (matrix shrinkage) caused by desorp-
tion of methane from the coal:
∆Vmatrix
Vmatrix
= βVdes
(2.4)
where, β is a constant depending on the characteristics of the coal type, Vdes is the
volume of gas desorbed that can be calculated based on some sorption isotherm.
The role of effective stress and/or sorption-induced matrix swelling/shrink-
age on the permeability of coal was also investigated by many other researchers
and more complex permeability models were built (Seidle and Huitt, 1995; Palmer
and Mansoori, 1998; Robertson and Christiansen, 2005). Many of the permeabil-
ity models are based on the Reiss cubic correlation between permeability and
porosity (Reiss, 1980):
k
k0
�
φ
=
φ0
� 3
(2.5)
The porosity ratio, thereby the permeability ratio, is a function of pressure, ad-
sorbed gas species, and the coal properties(e.g. the initial porosity and mechani-
cal modulus of the coal). The following discussion lists several of the permeability
models.
Seidle-Huitt Model (Seidle and Huitt, 1995): Permeability is expressed as a
function of initial porosity, the Langmuir strain constants, and pressure, only tak-
ing into account permeability change caused by sorption-induced strain.
k
k0
=
� �
1 + 1 + 2
� �
Bp0
CmatrixVm −
φ0
1 + Bp0
Bp
��3 1 + Bp
(2.6)

2.4. EVOLUTION OF COALBED PERMEABILITY 23
where, k is the permeability at pressure p, k0 is the initial permeability at pres-
sure p0, φ0 is the initial porosity, cmatrix is the matrix swelling coefficient, Vm is the
amount of adsorption at infinite pressure, and B is the Langmuir constant. There
are several assumptions behind this model:
1. Swelling is proportional to the amount of gas adsorbed,
εmatrix = cmatrixVads
where,εmatrix is the strain due to matrix swelling.
2. The adsorbed gas is related to the pressure by the Langmuir equation:
Vads = VmBp
1 + Bp
(2.7)
(2.8)
3. The coalbed is represented by a matchstick geometry as shown in Figure
2.5.
Figure 2.5: A coalbed represented by a matchstick model.

24 CHAPTER 2. LITERATURE REVIEW
Palmer-Mansoori Model (Palmer and Mansoori, 1998): Permeability is ex-
pressed as a function of Poisson’s ratio, Young’s modulus, net stress, initial poros-
ity, the Langmuir strain constants, and the pressure. The equation includes both
stress effects and matrix shrinkage:
k
k0
=
�
1 + Cp (p − p0) + ε∞
φ0
� K
M
� � ��3 bp bp0
− 1 −
1 + bp 1 + bp0
(2.9)
where, Cp is the natural fracture pore volume compressibility, ε∞ is volumetric
strain at infinite pressure, K is the bulk modulus, and M is the constrained axial
modulus. The second term in Equation 2.9 is the mechanical strain due to change
of pressure:
εp = Cp (p − p0) (2.10)
And the last term reflects the effect of sorption-induced strain.
In a more general form, Equation 2.9 is written as (Clarkson et al., 2008)
k
k0
=
�
1 + Cp (p − p0) + 1
φ0
� K
M
� �3 − 1 (∆ε)
(2.11)
where, ∆ε is the total volumetric strain due to gas sorption on coal. Different
models can be used to calculate the volumetric strain. One such model relates
strain (due to adsorption and pressure compression) to surface potential of ad-
sorption (Pan and Connell, 2007):
ε = − Φρs
f (x, νs) −
Es
p
(1 − 2νs) (2.12)
Es
where, Φ is surface potential of sorption that can be calculated based on gas ad-
sorption isotherms, ρs is the density of the solid adsorbent, Es is the Young’s mod-
ulus of the solid adsorbent, νs is the Poissons ratio, and
f (x, νs) = [2 (1 − νs) − (1 + νs) cx] [3 − 5νs − 4 (1 − 2νs) cx]
(3 − 5νs) (2 − 3cx)
(2.13)

2.5. GAS SORPTION ON COAL 25
where, c = 1.2, and x = a/l. Parameter a is the cylindrical radius in the selected
pore structure model, and parameter l is the length in the selected pore structure
model.
2.5 Gas Sorption on Coal
One of the reasons that coalbed methane resources are different from conven-
tional natural gas resources is that sorption of gas is involved when producing
coalbed methane. About 90% of gas in coal seams is in the form of adsorbed gas.
When estimating the original resources in place at the preliminary stage of a field
development, the conventional volumetric method used for sandstone oil and
gas reservoirs can not be simply implemented for coalbed methane reservoirs.
There is large error in the estimated original gas in place if the adsorbed gas is not
taken into account. Also, in order to design effective enhanced coalbed methane
recovery methods, it is essential to have knowledge of competitive adsorption of
gases on coal.
The amount of adsorption of a specific adsorbate on a specific adsorbent at
adsorption equilibrium is a function of temperature and pressure (Yang, 1987):
V = F (p, T ) (2.14)
At constant temperature, the amount of adsorption is only a function of pres-
sure. Competitive adsorption of different gases in coal (or any other adsorptive
solid/liquid) is usually described by sorption isotherms. A sorption isotherm is
a relationship between the amount (volume or moles) of adsorption of a specific
adsorbate on a unit (mass) of a specific adsorbent and the pressure at a constant
temperature. The majority of the isotherms observed are classified into five types
as shown in Figure 2.6. Type I isotherm, also called the Langmuir type isotherm,
is typical of adsorption in microporous solids. With the increase of pressure, the
amount of adsorption increases, and there is a maximum amount of adsorption
that would be reached at relatively high pressure when the solid surface is fully

26 CHAPTER 2. LITERATURE REVIEW
covered by a monolayer of adsorbed molecules. Type II (BET adsorption mech-
anism) and type III isotherms do not have the constraint of monolayer adsorp-
tion. Type IV and V isotherms are similar to type II and III, whereas, they have
a maximum amount of adsorption due to assumption of finite pore volume of
the porous media (Do, 2008).
Figure 2.6: Five types of adsorption isotherms, p 0 is the saturation vapor pressure
(after Yang, 1987).
In the literature, many numerical models have been developed to represent
gas adsorption isotherms. All of them fall into three different categories: the
Langmuir approach, the Gibbs approach, and the potential theory approach (Yang,
1987), as described next.

2.5. GAS SORPTION ON COAL 27
2.5.1 The Langmuir Approach
Langmuir Equations (Langmuir, 1916)
The Langmuir approach is based on the assumption that adsorption reaches dy-
namic equilibrium when the rate of adsorption (condensation) equals the rate
of desorption(evaporation). It also assumes that each site of adsorption can ac-
commodate only one adsorbate molecule/atom. Therefore, there is a maximum
amount of adsorption is reached when all the adsorptive sites are occupied by a
monolayer of adsorbate molecules/atoms. Langmuir adsorption falls into type I
of the adsorption isotherms shown in Figure 2.6.
The rate of adsorption per unit area of adsorbent is αv (1 − θ), where α is the
sticking probability, v is the collision frequency of adsorbate molecules/atoms
striking the adsorbent surface, and θ is the percentage of the available adsorbent
surface which has already been occupied by adsorbate molecules/atoms. Ac-
cording to the kinetic theory of gases, the collision frequency of gas molecules
striking a solid surface is
v =
p
(2πMκT ) 1/2
(2.15)
where, M is the mass of one gas molecule/atom, and κ is the Boltzmann constant.
The rate of desorption is βθe −Ed/RT , where β is the rate constant of desorption,
Ed is the activation energy of desorption and R is the universal gas constant.
tion:
At adsorption equilibrium, the rate of desorption equals the rate of adsorp-
βθe −Ed/RT p
= α
(1 − θ) (2.16)
1/2
(2πMκT )
where, β is the rate constant for desorption, θ is fractional coverage of adsorption,
Ed is activation energy of desorption, R is the universal gas constant, and α is
sticking probability. Therefore,
θ = Bp
1 + Bp
(2.17)

28 CHAPTER 2. LITERATURE REVIEW
where,
B =
is called the Langmuir constant.
α
eEd/RT
1/2
β (2πMκT )
The Fractional Coverage of adsorption, θ, is defined as
θ ≡ n
m
(2.18)
(2.19)
where, n is the amount (moles) of adsorption on one unit (mass) of adsorbent
at pressure p, and m is the maximum amount of (monolayer) adsorption on one
unit (mass) of adsorbent at infinite pressure. Sometimes, the moles of adsorption
are used instead of the volume. Once the Langmuir constant and the maximum
amount of adsorption are given, a Langmuir-type adsorption is completely spec-
ified.
For pure gas adsorption, the moles of adsorption are
n = mBp
1 + Bp
(2.20)
where, n is the amount of adsorption in moles, and m is the maximum amount of
adsorption in moles. For gas mixtures:
ni = miBipi
1 + NC �
Bjpj
j=1
(2.21)
where, i and j are component indices, and NC is the number of components in
the adsorbate gas. Equation 2.20 and Equation 2.21 are respectively the so-called
Langmuir Equation and Extended Langmuir Equation (Langmuir, 1916).
Besides the Langmuir equation, there are some other formulas to represent
type I adsorption isotherm, for instance, the modified virial equations (Siper-
stein and Myers, 2001):
p = n
� �
m
exp
H m − n
� C1n + C2n 2 + C3n 3 + C4n 4�
(2.22)

2.5. GAS SORPTION ON COAL 29
where, n is the amount of adsorption in moles; C1, C2, C3, C4 are constants; and
H is the Henry’s constant defined as (Talu et al., 1995)
n
H ≡ lim
p→0 p
(2.23)
In Equation 2.22, the term � �
m enforces Langmuir behavior at high pres-
m−n
sure, while the virial expansion terms modify the low-pressure adsorption.
The BET Model (Brunauer et al., 1938)
Monolayer adsorption is probably a good assumption for adsorption at low pres-
sures. For strongly absorbed gas (e.g. CO2 on coal), however, it may not be mono-
layer adsorption (type II isotherms). The BET model gives a quantitative descrip-
tion of adsorption that is not limited to one adsorption layer on the adsorbent
surface:
n =
mCp
(p 0 − p) [1 + (C − 1) (p/p 0 )]
(2.24)
where, p 0 is the saturation pressure of the adsorbate at the temperature of ad-
sorption, p/p 0 is called the relative pressure or the reduced pressure. Equation 2.24
is the so-called BET equation. Similar to the Langmuir equation, the BET equa-
tion also has two fitting parameters: m is the maximum monolayer adsorption
in moles, and C is a parameter that controls the shape the adsorption isotherm
at low pressure range, Figure 2.7. Parameter C is usually greater than unity, indi-
cating that the heat of adsorption of the first adsorption layer is greater than the
heat of liquefaction.
Based on Equation 2.24, when p = p 0 , n → ∞. Therefore, the BET model does
not have a limit on the maximum amount of adsorption, indicating that there
can be infinite layers of molecules built up on the surface. When the adsorption
space (pore volume) is finite, however, there is a maximum number of layers that
can be built on top of the surface, resulting in the n-layer BET equation:
n
m
CpR 1 − (N + 1) pR
=
1 − pR
N + NpR N+1
1 + (C − 1) pR − CpR N+1
(2.25)

30 CHAPTER 2. LITERATURE REVIEW
Figure 2.7: Plots of the BET equation with different C values (C = 1, 10, 50, 100).
where, pR = p/p 0 is relative pressure, and N is the allowed number of adsorp-
tion layers. Figure 2.8 plots reduced pressure versus n/m when there are differ-
ent number of adsorption layers. When N = 1, Equation 2.25 reduces to the
Langmuir equation. When pressure approaches the vapor pressure, a maximum
amount of adsorption is reached:
n
lim
pR→1 m
= N (N + 1) C
2 (NC + 1)
(2.26)
The maximum total amount of adsorption depends on the number of adsorption
layers allowed on the adsorbent surface.
Many other modified forms of the BET equation were developed in the litera-
ture to represent different types of adsorption isotherm (Do, 2008). For pure ad-
sorption, different models will work equally well by adjusting the fitting parame-
ters. It is the modeling of multicomponent adsorption that is more challenging.
Fitting-parameter formulas can still be used to represent the total amount of ad-
sorption. The adsorbed phase composition, however, is not readily predicted by

2.5. GAS SORPTION ON COAL 31
Figure 2.8: Plots of the N-layer BET equation with different N values (C = 100, N
= 1, 5, 10, 50).
fitting-parameter formulas. Thermodynamics has to be taken into account. The
Gibbs approach is by far the most well developed approach for modeling multi-
component adsorption.
2.5.2 The Gibbs Approach
The Gibbs approach is based on surface thermodynamics and is analogous to the
classical thermodynamics of vapor-liquid equilibrium (VLE). The fundamental
assumption of the Gibbs approach is that at adsorption equilibrium, the chemi-
cal potential in the adsorbed phase is equal to the chemical potential in the gas
phase. Several concepts were introduced to facilitate the development of the
model.

32 CHAPTER 2. LITERATURE REVIEW
Gibbs Surface
A real system of two phases consists of two homogeneous bulk phases in which
the concentration of component i is constant throughout the individual phase,
and an interphase in which the concentration of component i is different from
either of the two bulk phases. The concentration of a component varies within
the interface and merges into the values in the two bulk phases at the extremities
of the surface layers, Figure 2.9. Gibbs defined a surface that has zero thickness
and divides the system into two volumes (Defay and Prigogine, 1966). Each of the
two volumes represents a bulk phase. The dividing surface is complementary,
such that the two volumes plus the dividing surface is identical to the real system
in material, energy, and mechanics, Figure 2.9.
Figure 2.9: The Gibbs surface model (after Defay and Prigogine, 1966)
If the total moles of component i in the system is ni, the moles of component
i in the two bulk phases are n ′ i and n ′′
i respectively, according to material balance,
the moles of component i in the interphase is
n σ i = ni − n ′ i − n ′′
i
(2.27)

2.5. GAS SORPTION ON COAL 33
For a system of adsorption, we have the bulk adsorbate phase (gas or liq-
uid) and the bulk adsorbent phase (solid); the adsorbed phase is a condensed
phase that has different physical properties from the bulk adsorbate and adsor-
bent phase. At adsorption equilibrium, the concentration of a component in the
bulk phases are constant values respectively. The adsorbed phase is actually an
interphase that occupies certain volume in which the concentration of the com-
ponent varies and merges into the values in the bulk adsorbate and adsorbent
phase. However, for simplicity, we treat the adsorbed phase as a homogeneous
phase in which the concentration of a component is constant throughout the
phase, and there is a sharp change from the adsorbed phase to the bulk adsor-
bate and adsorbent phase. The summation of the homogeneous adsorbate phase
and the assumed homogenous adsorbed phase is equivalent to the real homoge-
neous bulk adsorbate phase plus the heterogenous adsorbed phase.
Phase Equilibrium
The Gibbs approach employs the well established vapor-liquid equilibrium (VLE)
thermodynamics for vapor-adsorbate equilibrium (VAE). The fundamental rule
is that at equilibrium the chemical potential of a component in the gaseous phase
and the chemical potential of the component in the adsorbed phase equal:
µi (T, p, y1...) = µi (T, Π, x1...) (2.28)
The left hand side of Equation 2.28 is the chemical potential of component
i in the gas phase which is a function of the temperature, the pressure, and the
gas phase composition. And the right hand side of the equation is the chemi-
cal potential of component i in the adsorbed phase which is a function of the
temperature, the spreading pressure, and the adsorbed phase composition. The
concept of spreading pressure is discussed in detail later in this section.
Chemical potential was first defined by Gibbs: “If to any homogeneous mass

34 CHAPTER 2. LITERATURE REVIEW
we suppose an infinitesimal quantity of any substance to be added, the mass re-
maining homogeneous and its entropy and volume remaining unchanged, the
increase of the energy of the mass divided by the quantity of the substance added
is the potential for that substance in the mass considered” (Gibbs, 1906). The
“potential” mentioned in the above statement is the chemical potential. And the
“energy” can be the internal, Helmholz, enthalpy, or Gibbs free energies. Chemi-
cal potential is the partial molar Gibbs energy:
µi =
� � T ∂(n G)
∂ni
T,Π,nj�=i
≡ ¯ Gi
(2.29)
Even though it is convenient to use the concept of chemical potential to define
VLE and VAE, chemical potential does not have an equivalence in the physical
world. Fugacity is used as an auxiliary function to describe chemical potential
(Prausnitz et al., 1999):
µi − µ 0 i = RT ln fi
f 0 i
(2.30)
where, µ 0 i and f 0 i are the chemical potential and fugacity of component i at some
reference (standard) state. The choice of the reference is arbitrary, however, µ 0 i
and f 0 i are not independent, when one is chosen, the other is fixed.
At VAE, Equation 2.28 becomes
ˆf V i = ˆ f A i (2.31)
where, ˆ f V i is the partial molar fugacity of component i in the vapor phase, and
ˆf A i is the partial molar fugacity of component i in the adsorbed phase. The hat
over fi indicates it is the fugacity of component i in a mixture. One important as-
sumption underlying Equation 2.31 is that the reference state of the vapor phase
and the reference state of the adsorbed phase must be consistent (see discussion
in the appendix).

2.5. GAS SORPTION ON COAL 35
Fugacity has the units of pressure, and is sometimes referred to as the “cor-
rected pressure”. For a component in an ideal mixture, the fugacity equals the
partial pressure of the component. Taking nonideality into account, fugacity of
component i in gas phase is (Prausnitz et al., 1999)
ˆf V i = pyi ˆϕi (2.32)
where, ˆϕi is the fugacity coefficient accounts for the nonideality of the vapor
phase. For an ideal gas, ˆϕi = 1; the deviation of ˆϕi from unity is an indication
of nonideality of a gas.
Fugacity coefficients can be calculated based on an equation of state for gas.
The fugacity coefficient for a pure gas is (Walas, 1985)
ln ϕ =
� p
0
Z − 1
dp (2.33)
p
where, Z is the compressibility factor. The partial fugacity coefficient for a gas
mixture is
where,
ln ˆϕi =
� p
0
¯Zi = p
RT ¯ Vi
¯Zi − 1
dp (2.34)
p
(2.35)
Cubic equations of state (EOS) are often used to calculate the compressibility
factors:
p = RT
V − b −
a
V 2 + ubV + wb2 (2.36)
where, the four parameters, u, w, a, and b, depend on the actual equation of state.
Table 2.2 listed the explicit values of the parameters for different cubic EOS.
using
When the van der Waals EOS is chosen, fugacity coefficients are calculated
ln ˆϕi = bi
� �
− ln Z 1 −
V − b b
��
−
V
2√aai RT V
(2.37)

36 CHAPTER 2. LITERATURE REVIEW
Table 2.2: Parameter values for different cubic equations of state (Kovscek, 2005).
Cubic EOS u w b a
van der Waals 0 0 RTc
8pc
Redlich-Kwong 1 0 0.08664RTc
pc
Soave 1 0 0.08664RTc
pc
Peng-Robinson 2 -1 0.07780RTc
pc
a =
27R2T 2 c
64pc
0.42748R2T ( c 5/2)
pcT ( 1/2)
0.42748R 2 T 2 c
pc
0.45724R 2 T 2 c
pc
�
1 + fw
�
1 − T 1/2
r
�� 2
where fw = 0.48 + 1.574w − 0.176w2 � �
1 + fw 1 − T 1/2
��2 r
where fw = 0.37464 + 1.54226w − 0.26992w 2
�� √
yi ai
b = � yibi
� 2
(2.38)
(2.39)
When the three other cubic EOS are chosen, fugacity coefficients are calcu-
lated using
ln ˆϕi = bi
b (Z − 1)−ln (Z − B∗ A
)+
∗
B∗√u2 − 4w
bi
b =
δi = 2a1/2 i
a
� bi
b
Tc,ipc,i
� xiTc,i/pc,i
�
j
− δi
�
xja 1/2 � �
j 1 − kij
A ∗ = ap
R 2 T 2
ln 2Z + B∗ � u + √ u 2 − 4w �
2Z − B ∗ � u + √ u 2 − 4w �
(2.40)
(2.41)
(2.42)
(2.43)

2.5. GAS SORPTION ON COAL 37
B ∗ = bp
R 2 T 2
(2.44)
The fugacity of the same component in the adsorbed phase, ˆ f A i , is calculated
based on the definition of activity (Prausnitz et al., 1999):
Therefore,
ai ≡ γixi ≡ ˆ fi
f 0 i
(2.45)
ˆf A i = f 0 i γixi (2.46)
where, f 0 i is the fugacity of component i in the standard state which is chosen to
be the pure component (condensed phase) fugacity of component i at the same
specified spreading pressure of the adsorbed mixture; xi is the mole fraction of
component i in the adsorbed phase; and γi is the activity coefficient accounting
for the nonideality of the adsorbed mixture.
Activity coefficients are calculated based on Equation A.11:
RT ln γi =
� �
T ex ∂ n G ��
∂ni
T,Π,nj�=i
(2.47)
and a proper model for the excess Gibbs free energy. The ABC equation is an
example of such a model (Siperstein and Myers, 2001).
G ex � −Cψ
= (A + BT ) x1x2 1 − e �
(2.48)
where, A, B, and C are constants that are independent of temperature, loading,
and the composition; and ψ is related to the surface potential of adsorption. This
variable is discussed in detail later in this section. The ABC equation fulfills the
requirements of thermodynamic consistency and reduction to an ideal adsorbed
solution at the limit of zero loading. Based on the ABC equation for the excess

38 CHAPTER 2. LITERATURE REVIEW
molar Gibbs free energy, the activity coefficient of component i in a binary system
consisting of components i and j is
RT ln γi = (A + BT ) � 1 − e −Cψ� x 2 j
(2.49)
A complementary equation to calculate the activity coefficients for a binary
system is based on the concept of regular solution (Prausnitz et al., 1999):
RT ln γi = viΛ 2 j (χi − χj) (2.50)
where, vi is the molar volume of component i, Λj is the volume fraction of com-
ponent j:
Λj ≡ xjvj
NC �
xivi
i=1
and, χi is the solubility parameter of component i defined by
� �1/2 ∆vapu
χi =
v i
(2.51)
(2.52)
where, ∆vapu is the energy of complete vaporization (isothermal vaporization of
saturated liquid to the ideal-gas state, i.e., infinite volume).
Substituting Equation 2.32 and Equation 2.46 into Equation 2.31, at adsorp-
tion equilibrium we have
pyi ˆϕi = f 0 i γixi
(2.53)
This is the fundamental iso-fugacity equation used in multicomponent adsorp-
tion calculation, often referred to as the Real Adsorbed Solution (RAS) model.
At low to moderate pressures, the gas phase and the adsorbed phase are con-
sidered ideal. Equation (2.53) becomes
pyi = p 0 i xi
(2.54)

2.5. GAS SORPTION ON COAL 39
This is analogous to Raoult’s law for vapor-liquid equilibrium. In the equation, p 0 i
is the equilibrium gas phase pressure of pure component i at the temperature and
spreading pressure of the mixture. Equation 2.54 is the so-called Ideal Adsorbed
Solution (IAS) model.
Surface Potential
The standard state of Equation 2.53 and Equation 2.54 is fixed by a parameter
called surface potential. Surface potential is actually the chemical potential of
the solid adsorbent that has the adsorbed phase on its surface relative to its pure
state (without adsorption) at the same temperature.
Surface potential is related with spreading pressure by Φ = −ΠA. Analogous
to the fundamental equations in classical thermodynamics, the differential form
of the total internal energy and total Gibbs free energy of the adsorbed phase are
d(n T U) = T d(n T NC �
S) − ΠdA +
i=1
d(n T G) = −(n T NC �
S)dT + AdΠ +
i=1
µidni
µidni
(2.55)
(2.56)
Compared with the corresponding classical thermodynamic equations, spread-
ing pressure is used in place of pressure, and surface area is used in place of vol-
ume. Based on Equation 2.55, spreading pressure defines the reduction of the
surface tension at the solid-gas interface upon adsorption.
Integrating Equation 2.55 from a state of zero mass to a state of finite mass at
constant temperature, pressure and composition, we get the total internal energy
of the adsorbed phase:
n T U = T (n T NC �
S) − ΠA + µini
i=1
(2.57)

40 CHAPTER 2. LITERATURE REVIEW
Differentiating Equation 2.57 gives the general differential form of the total inter-
nal energy:
d(n T U) = T d(n T S) + (n T NC �
S)dT − ΠdA − AdΠ +
Comparing Equation 2.55 and Equation 2.58, we obtain
i=1
NC �
µidni + nidµi
i=1
(2.58)
(n T NC �
S)dT − AdΠ + nidµi = 0 (2.59)
Equation 2.59 is one form of the Gibbs-Duhem Equation. At constant tempera-
ture:
i=1
NC �
−AdΠ + nidµi = 0 (2.60)
i=1
Equation 2.60 is the Gibbs Adsorption Isotherm for multicomponent systems. Sub-
stituting the relation between fugacity and chemical potential as in Equation 2.30
into Equation 2.60:
Integrate to obtain
For pure adsorption
or approximated by
NC �
AdΠ = RT nid ln fi
AΠ
RT =
i=1
NC �
� p
i=1
0
AΠ0 i
RT =
� p0 i
0
AΠ0 i
RT =
� p0 i
0
(2.61)
� �
n
df (2.62)
f i
� �
n
df (2.63)
f i
� �
n
dp (2.64)
p i
for an ideal solution. In Equation 2.62 - 2.64, n is the amount of adsorption, f is
the gas-phase fugacity, p is the pressure. The integrals can be evaluated from ex-
perimental observations of n versus p combined with an equation of state based

2.5. GAS SORPTION ON COAL 41
calculation of the gas fugacity. Variable, ψ = ΠA
RT
Φ = − , is a modified version
RT
of the surface potential. It has the same unit of surface loading (adsorption),
mol/kg.
The condition under which Equation 2.53 can be implemented is that the fu-
gacity of the pure components, f 0 i , are evaluated at the spreading pressure (or
surface potential) of the mixture adsorption. Therefore, for a binary system at
adsorption equilibrium:
Total Amount of Adsorption
Π 0 i = Π = Π 0 j
ψ 0 i = ψ = ψ 0 j
(2.65)
(2.66)
In order to describe a multicomponent adsorption system fully, we must have an
equation of the amounts of adsorption. Based on Equation 2.56, surface area is
the partial derivative of the Gibbs free energy in terms of the spreading pressure
at constant pressure and amount of adsorption of all the components:
� � T ∂(n G)
A =
∂Π
T,ni
(2.67)
The molar Gibbs energy upon mixing at constant T and Π as in Equation A.12 is
G mix (T, Π, x1, ...) = RT � xi ln γixi
Therefore, the molar surface area of adsorbate upon mixing is
a mix (T, Π, x1, ...) = RT � xi
Also, based on the definition of a mixing property
a mix ≡ a − � xia 0 i
� �
∂ ln γi
∂Π T,xi
(2.68)
(2.69)
(2.70)

42 CHAPTER 2. LITERATURE REVIEW
Dividing the left-hand side and the right-hand side of Equation 2.70 by the total
surface area A:
a mix
A
= 1
nt
− � xi
n 0 i
(2.71)
Substituting Equation 2.69 into Equation 2.71, we obtain the equation for total
amount of adsorption:
1
nt
= � xi
n 0 i
+ � xi
For an ideal adsorbed solution, a mix = 0, and
1
nt
= � xi
n 0 i
� �
∂ ln γi
∂ψ T,xi
(2.72)
(2.73)
All the concepts and equations described in this section are the fundamentals
of the Gibbs approach. In Chapter 4, we are going to discuss how to use them to
model binary gas adsorption on coal.
2.5.3 The Potential Approach
In this theory, it is assumed that the adsorbate molecules surround the adsorbent
surface at some concentration gradient due to a potential field. The isotherms
derived from the potential theory are useful in interpreting adsorption by capil-
lary condensation, or pore filling (Yang, 1987). This approach is not within the
scope of interest of this study.
2.6 Adsorption/Desorption Hysteresis
As shown in adsorption isotherm type IV and V in Figure 2.6, when decreas-
ing the pressure, the desorption curve may not follow the same path down the
adsorption curve (irreversible). At some pressure range, the desorption curve is
above the adsorption curve, which forms a special shape in the isotherm plot.
This is called the hysteresis loop. Adsorption/desorption hysteresis was observed

2.6. ADSORPTION/DESORPTION HYSTERESIS 43
in laboratory measurement of gas adsorption/desorption on coal by several re-
searchers (Bell and Rakop, 1986; Harpalani and McPherson, 1986; Tang et al.,
2005). Seri-Levy and Avnir (1993) also demonstrated that surface geometry het-
erogeneity induces adsorption/desorption hysteresis using Monte Carlo simula-
tion.
1967):
There are several possible causes of adsorption/desorption hysteresis (Bond,
1. Surface impurities that may be displaced in the course of experiment. These
impurities may influence the angle of wetting which in turn affects sorption
in small pores.
2. There may be some irreversible sorbate phase change in an adsorption/des-
orption cycle.
3. Swelling of adsorbent during adsorption may be irreversible.
4. In small pores, over a certain pressure range, there may be irreversible cap-
illary condensation-evaporation processes.
Hysteresis due to impurities and adsorbate phase change can be eliminated
by repeating the adsorption-desorption cycles and carefully designing experi-
ments in a safe temperature range. Hysteresis caused by irreversible swelling is
often of negligible significance. In general, the most interesting hysteresis is that
associated with capillary condensation in small pores.
Due to capillary force, the vapor pressure of liquid inside small pores is less
than that on a flat surface. The pressure at which a vapor condenses in a pore of
radius r follows (Do, 2008)
�
p
= exp −
p0 2σvc
RT
1
rm
�
(2.74)
where, p 0 is the vapor pressure of the bulk phase, σ is the surface tension, vc is the
condensate molar volume, and rm is the mean radius of the vapor-condensate

44 CHAPTER 2. LITERATURE REVIEW
interface defined as
2
rm
= 1
r1
+ 1
r2
where, r1 and r2 are two principle radii of the curved interface.
(2.75)
For adsorption, condensate is formed via surface layering, Figure 2.10(a), the
radius of curvature rm = 2r. Therefore,
�
pads
= exp −
p0 σvc
�
1
RT r
(a) Adsorption.
(b) Desorption.
(2.76)
Figure 2.10: Condensation and evaporation in a capillary cylinder (after Do,
2008).
For desorption, the sorbate evaporates from the condensed phase meniscus
that has a hemispherical shape, Figure 2.10(b), rm = r/ cos θ, where θ is the con-
tact angle. When the wetting angle of the surface is zero, rm = r. Therefore,
�
pdes
= exp −
p0 2σvc
�
1
RT r
(2.77)
For the same amount of uptake, the pressure along the path of adsorption

2.7. RESEARCH EFFORTS OF PEERS AT STANFORD 45
is different from that along the path of desorption. The differences in the rela-
tive pressure of filling and emptying the small pores is the basis of hysteresis in
mesoporous solids. The shape of the hysteresis loop gives information about the
mesopore size distribution.
2.7 Research Efforts of Peers at Stanford
Laboratory and numerical investigations were conducted at the School of Earth
Sciences at Stanford University to study:
1. Gas adsorption/desorption in coal: (1) adsorption isotherms, and (2) ad-
sorption/desorption hysteresis (Tang et al., 2005; Jessen et al., 2007; Liu,
2009).
2. Gas diffusion in coal (Dutta, 2009).
3. Gas displacement flow in coal (Tang et al., 2005; Jessen et al., 2007).
4. Change of coal properties (permeability, static and dynamic bulk modulus)
due to change of effective stress (Hagin and Zoback, 2010).
5. Change of coal properties (permeability and wettability) due to gas adsorp-
tion/desorption (Lin et al., 2008; Chaturvedi, 2006).
Gas Sorption and Displacement Flow in Coal
Tang et al. (2005) conducted gas sorption and displacement measurements on
a cylindrical ground coal pack at room temperature (22 o C). Adsorption and des-
orption of pure CO2, N2, and CH4 was measured based on a gravimetric method
at first escalating and then decreasing pore pressures. Experimental results are
plotted in Figure 2.11. Observations based on experimental results were
1. The amount of adsorption increased with the increase of pressure.
2. At the same pressure, the amount of adsorption was the greatest for CO2,
followed by CH4 and N2.

46 CHAPTER 2. LITERATURE REVIEW
3. Adsorption/desorption hysteresis was observed for all of the three gases.
4. The Langmuir equations represented the experimental amount of adsorp-
tion very well for all of the three gases.
Figure 2.11: Pure gases sorption on ground Powder River Basin coal sample (Tang
et al., 2005).
Displacement experiments were conducted under various conditions. The
coal pack was always first vacuum evacuated and then saturated with CH4 before
injecting pure CO2, pure N2, or binary mixtures of the two. Gas was injected at a
constant volumetric rate at different controlled system pressures. Pressure along
the coal pack and gas composition at the outlet of the coal pack were measured
throughout the tests. Observations of the displacement experiments included
1. The recovery factors were greater than 94% in all cases.

2.7. RESEARCH EFFORTS OF PEERS AT STANFORD 47
2. When injecting pure CO2 to the CH4 saturated coal pack, piston-like dis-
placement was observed. At breakthrough, the effluent CO2 concentration
increased sharply from zero to 100%. The higher the system pressure, the
sharper the increase of the effluent CO2 concentration, indicating greater
pressure shortened the time for CO2 to replace CH4 on the coal surface.
3. The higher the concentration of CO2 in the injection gas, the slower initial
recovery, later breakthrough, and the less gas needed to strip all the recov-
erable CH4 from the core. On the other hand, the more N2 in the injection
gas, the faster initial recovery, earlier breakthrough, and more gases needed
to strip all the recoverable CH4.
Experiments of Tang et al. (2005) also showed that the permeability of the coal
pack was a function of gas species and the system pressure when certain gases
were used. The permeability of the coal pack to CO2 at low pressure was smaller
than that to N2 which was in turn smaller than that to helium. It was also ob-
served that the CO2 permeability of the coal pack decreased about 50% when the
system pressure increased from 0.69 to 4.1 MPa (100 to 600 psi). The N2 perme-
ability of the coal pack was more or less constant with increased system pres-
sure. The authors also conduced experiments on water saturated coal samples
which were not mentioned in their paper. The maximum amount of adsorption
dropped more than 35% for CH4 and N2, and about 18% for CO2.
Efforts were made to model the experimental displacement results (Tang et al.,
2005; Jessen et al., 2007) based on the extended Langmuir equations and the ideal
adsorbed solution (IAS) model with/without hysteresis. Modeling based on the
extended Langmuir equations predicted the effluent concentration profiles very
well for the binary systems (pure CO2 and N2 displacing CH4). However, it was
not able to predict the experimental effluent concentration profile for the ternary
systems (CO2/N2 mixtures displacing CH4). The use of IAS model improved the
simulation results, and the incorporation of hysteresis helped the situation fur-
ther more. More complicated models, however, might have to be used in order to
obtain satisfactory simulation results for the ternary system.

48 CHAPTER 2. LITERATURE REVIEW
Coal Wettability
Chaturvedi (2006) conducted numerical and experimental investigation of
the wettability of coal at microscopic, core, and reservoir scales. Microscopic
wettability is defined by contact angles. Contact angles values vary with the pH
value of the solution in coal. For the specific coal water system studied by the au-
thor, the calculated contact angle values went through a maximum at pH around
4 and became small at low and high pH, suggesting an alteration of coal wet-
tability with pH and therefore with CO2 dissolution in the systems. The actual
value and location of the maximum contact angle varied with coal systems. Core-
scale spontaneous imbibition measurement results were used to calculate the
macroscopic contact angle. Results suggested similarity in the behavior of coal
scale wettability with pore-scale wettability. The macroscopic contact angle val-
ues were small at low and high pH, and went through a maximum at neutral pH.
Relative permeability curves of air-water flow in coal were obtained based on CT
scanning monitored imbibition of different pH solutions. Results showed that
the intersection points of the wetting and non-wetting phase relative permeabil-
ity curves varied with pH. The results also suggested that the coal-water-air sys-
tem was most water wet at high pH and least water wet at neutral pH. The very
low values of both krw and krnw for a large range of saturations, however, sug-
gested a mixed-wet nature of Powder River Basin coal.
Gas Diffusion-Sorption in Coal
Dutta (2009) developed a coupled adsorption-diffusion model for multicom-
ponent gas-coal systems. The diffusion model was based on Fick’s diffusion model
integrated with Maxwell-Stefan (MS) diffusion theory, taking into account the
interactions between multicomponent gas molecules. The Fickian diffusivities
were functions of MS diffusivities and composition. Multicomponent gas sorp-
tion was modeled using the extended Langmuir and Ideal Adsorbate Solution
(IAS) models. In general, multicomponent, countercurrent gas diffusion was rel-
atively rapid.

2.7. RESEARCH EFFORTS OF PEERS AT STANFORD 49
Gas Sorption in Coal in a Microscopic Scope
Liu (2009) proposed multiscale experimental and computational procedures
to obtain a thorough understanding of the mechanism of gas adsorption on coal.
1. She built realistic coal structure models taking into account both structural
and chemical heterogeneity by conducting experimental studies of the sur-
faces and pore structures of coal samples of different ranks using scanning
electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), and
Fourier transform infrared spectroscopy (FT-IR).
2. Calculated molecular level interactions (interactions between adsorbate molecules
and interactions between adsorbate molecules and pore surfaces) based on
quantum chemistry approaches.
3. Simulated the amount of adsorption as well as pressure, average density, in-
ternal energy, and other thermodynamic properties (Helmholtz free energy,
entropy, etc.) based on the calculated chemical potential, and the system
temperature and volume.
4. Conducted larger-scale simulations based on the parameters provided by
the molecular-scale modeling to investigate macroscale phenomena such
as sorption-induced matrix swelling/shrinkage and adsorption/desoprtion
hysteresis.
5. Conducted experimental studies to validate the modeling results.
This would be the first tool to incorporate a full multiscale approach from molec-
ular to reservoir scale that will aid in understanding interactions between gases
in coal and coal.
Coal Compressibility and Permeability
Hagin and Zoback (2010) conducted mechanical studies on powdered coal
samples (60 mesh) from the Powder River Basin at room temperature, 22 o C, and
found that

50 CHAPTER 2. LITERATURE REVIEW
1. The physical properties of coal samples changed significantly when exposed
to CO2.
2. At a given effective stress, the static bulk modulus decreased with adsorp-
tion of CO2. Saturating samples with CO2 at a pore pressure of 1 MPa caused
the static bulk modulus to decrease by a factor of 2.
3. The dynamic bulk modulus increased with adsorption of CO2 at a given ef-
fective stress. Increasing the CO2 pore pressure from 1 - 4 MPa resulted in a
10% increase in the modulus.
4. Permeability decreases due to changes in effective stress were approximately
equal to decreases caused by coal swelling in response to adsorption of CO2.
5. Samples saturated with helium simply deformed elastically, whereas, sig-
nificant viscoplastic deformation was observed in samples saturated with
CO2.
2.8 Goals of This Study
In this study, we conducted laboratory measurements and modeled numerically
gas adsorption in coal, sorption induced volumetric strain, and sorption induced
permeability change of the coal. The study was aimed at
1. Developing an experimental approach to measure sorption, volumetric strain,
and permeability simultaneously.
2. Adding pure and binary adsorption data of CO2/N2 on coal to the literature.
3. Investigating/developing algorithms of binary adsorption calculation.
4. Modeling sorption induced volumetric strain.
5. Modeling sorption induced permeability change.

Chapter 3
Experiments
3.1 Previous Work
The initial interest of the study was to investigate the permeability change of coal
with the injection of different gases and gas mixtures (Lin, 2006). We were in-
terested in investigating sorption induced permeability change. Core-flood ex-
periments were conducted to measure the permeability of a coal pack to differ-
ent gases under equilibrium conditions at varying pore pressures and constant
net effective stress. The net effective stress is the difference between the confin-
ing pressure and the pore pressure. Different pore pressures resulted in different
amounts of gas adsorption, while the constant net effective stress eliminated the
effect of effective stress on the permeability of coal.
The coal samples were from the Powder River Basin, Wyoming, Figure 3.1.
The depth of the coalbed where the coal samples originated is between 900 to
1200 feet with an average in-situ reservoir pressure of 2480-3450 kPa (360 to 500
psi) and temperature of 28-32 o C. The coal samples were delivered to Stanford
immersed in formation water within a core barrel. After receipt, the coal sample
was dried and ground to a particle size of 60 mesh (diameter ≈ 0.25 mm). The
ground coal was thereafter stored in an air-tight vessel under continuous vacuum
to avoid oxidation. For the experiments, the ground coal was packed tightly into
a viton rubber sleeve surrounded by a perforated aluminium sleeve to form a
51

52 CHAPTER 3. EXPERIMENTS
Figure 3.1: Coalbed methane fields in the United States, lower 48 states (courtesy
of EIA). Coal samples were from the dotted circle enclosed area.
semiconsolidated porous medium. The porosity and permeability of the pack
were assumed to be uniformly distributed. Two aluminum end plugs both with a
through hole in the middle and connections of fittings and tubes to form a flow
path into the coal pack were fitted to the two ends of the coal holder, Figure 3.2(c).
The coal holder was then placed into a second aluminum sleeve with caps at the
two ends, Figure 3.2. Between the coal holder and the second aluminum sleeve
was an annulus that was charged with gas or liquid to provide confining pressure
to the inner coal holder. Flow experiments were then conducted with the core
using the apparatus shown in Figure 3.2(a).
Prior to the injection of a specific gas, the coal pack was always vacuum evac-
uated to remove all the moisture and gas contents. The coal pack was then con-
nected to the injection gas source. The pore pressure within the coal pack was
controlled by the outlet pressure from the gas source upstream and a back-pressure
regulator downstream. The confining pressure was set according to the pore

3.1. PREVIOUS WORK 53
(a) Experimental setup for permeability measurement
(b) Inner coal holder and confining
pressure sleeve.
(c) Assembled coal holder
Figure 3.2: Experimental setup and the coal holder for permeability measurement
(Lin et al., 2008).

54 CHAPTER 3. EXPERIMENTS
pressure to make sure the net effective stress was around 400 psi. The core re-
mained connected to the injection gas source overnight (over 12 hours) until
adsorption equilibrium was reached. Permeability measurement was then con-
ducted. The valve at the inlet of the core was closed. The pressure at the inlet side
of the core was increased to build a pressure difference between the upstream
and downstream ends of the core. The inlet valve to the core was then opened
to allow gas to flow. The pressure difference across the core was read from the
pressure transducer. The stable gas flow rate was measured using a bubble flow
meter, and the permeability of the coal was then calculated based on the Darcy’s
equation for gas, Equation 3.8.
Different gases, including pure CH4, N2, and CO2, and binary mixtures of the
two of various composition, were used as injection gas. For each gas, experi-
ments were conducted at several escalating pore pressures. All the experiments
were conducted at room temperature (22 o C) and a constant net effective stress
of about 400 psi. The experimental results of permeability versus pressure are
plotted in Figure 3.3. Some of the observations from the experiments were
1. Permeability of coal reduced as the pore pressure of a given gas increased.
2. The magnitude of the permeability reduction varied with the type and com-
position of gas.
3. At any pore pressure, pure CO2 caused the most significant permeability
reduction, followed by CH4 and then N2. For the binary mixtures, the more
CO2 in the injection gas, the lower the permeability.
The net confining pressure of the system was kept constant for all the exper-
iments, therefore, the stress induced permeability change was minimized. The
observed permeability reduction of the coal pack is believed to be related to gas
adsorption, because the permeability reduction (Figure 3.3) shared the same fea-
ture as gas adsorption isotherms (Figure 2.11) on coal:
1. All of the injection gases adsorb to coal.

3.1. PREVIOUS WORK 55
Figure 3.3: Permeability of a coal pack with injection of different gases (Lin et al.,
2008).
2. The amount of adsorption varies with the species of the injection gas and
the pore pressure.
3. For a specific gas species, the greater the pore pressure, the more the ad-
sorption.
4. At any pressure, the amount of adsorption is the highest for pure CO2 , fol-
lowed by CH4 and then N2.
The permeability and the pressure was seen to follow an exponential relation.
The values of the exponent for different injection gases are listed in Table 3.1.
The correlation between the amount of adsorption and the magnitude of per-
meability reduction appears straightforward. A numerical permeability model
representing this correlation is not readily established. However, experimental
data of gas, especially multicomponent gas, adsorption on coal are rare and the
mechanism of sorption-induced permeability change is not completely clear. To
investigate the interplay of gas sorption on coal, volumetric swelling of coal, and

56 CHAPTER 3. EXPERIMENTS
Table 3.1: Exponential relation of permeability and pressure, k = k0p b (Lin et al.,
2008).
Injection Gas b Minimum k/k0 Average Error
100% N2 -0.0982 0.5159 5.21%
100% CH4 -0.1624 0.3304 6.29%
100% CO2 -0.7468 0.0047 47.1%
25%CO2/75%N2 -0.1367 0.4001 3.16%
50%CO2/50%N2 -0.1564 0.3546 2.31%
75%CO2/25%N2 -0.1666 0.3321 1.38%
coal permeability, and add to the pool of experimental data on these three areas,
new experiments were designed to measure sorption, volumetric change, and
permeability simultaneously.
3.2 Experimental Apparatus
3.2.1 The Core
To obtain more realistic experimental conditions, composite coal cores were used
in current study. Coal samples were collected at a mine face of the Wyodak-
Anderson coal zone, Powder River Basin, Montana, Figure 3.1. As collected, the
samples were in big chunks, Figure 3.4. The samples were heavily wrapped with
foam plastic tapes and air transported to Stanford in boxes. Upon receipt, the
samples were unpacked and washed using distilled water, and afterwards kept in
large containers filled with deaerated distilled water to avoid any oxidization of
the coal surfaces. To make composite cores, cylindrical coal plugs were drilled
from the big chunks. A thin-wall diamond core bit (inner diameter = 1 inch) was
used for coring. Water was used as a cooling agent while drilling. Core recovery
was generally poor due to the fragile nature of the coal. The length of individual
coal plugs varied from less than 1 inch to about 3 inches. Fractures were clearly

3.2. EXPERIMENTAL APPARATUS 57
observable in all of the coal plugs, Figure 3.6(a). Several pieces of such coal plugs
were assembled together to make a core of about one foot long. Before assembly,
the end faces of the coal plugs were sanded to ensure that they were flush when
fitted together.
Figure 3.4: Coal sample from Wyodak-Anderson coal zone, Powder River Basin,
Montana.
Two cores were made and discarded before a third one, that fulfilled the ex-
perimental requirements, was obtained. The first core was made from five pieces
of coal plugs. The total length of the core was 10.31 inches. The total wet weight
of the coal was 179.84 grams. The average wet density of the coal was 1.35 g/cm 3 .
The wet density value is consistent with the range of the wet density of bitumi-
nous coal, 1.20-1.80 g/cm 3 (Thomas, 2002). The initial helium porosity and per-
meability of this core was 9.24% and 2.33 md under effective overburden pres-
sure of 400 psi. Gas sorption, sorption-induced volumetric and permeability
changes measurements were conducted using feed gases 15%CO2/85%N2 and
50%CO2/50%N2. Binary mixture of 15%CO2/85%N2 was used because it was close
to flue gas composition. Figure 3.5 shows the permeability change of the core
with the adsorption of 15%CO2/85%N2 at escalating system pressures. If flue gas
would be injected directly (after removing poisonous components) to coalbeds

58 CHAPTER 3. EXPERIMENTS
for the purpose of ECBM or CO2 sequestration, the permeability reduction of the
coalbeds might not be very significant. The core was discarded after adsorption
of 50%CO2/50%N2 at 350 psi, because the permeability of the core was too low
(0.38 md) to be accurately measured using our experimental apparatus.
Figure 3.5: Permeability change of a composite coal core after the adsorption of
15%CO2/85%N2 binary mixture at escalating pressures. Experiments were conducted
at room temperature (22 o C). Net effective stress was kept around 400
psi.
The second composite core was made using seven pieces of coal plugs. The
total length of the core was 11.34 inches. The total wet weight of the coal was
239.56 grams. The average wet density was 1.64 g/cm 3 which is at the high end of
the range of the wet density of bituminous coals. The coal was tight and no gas
flow through the core was detected when we tried to measure the initial perme-
ability of the core using helium. The core was discarded after the initial perme-
ability test. The coal plugs drilled from chunks of coals were not representative
for the actually coal properties in the coalbeds, because mostly only the tight
coals yielded successful coring.

3.2. EXPERIMENTAL APPARATUS 59
A third core was then made and used in our experiments. It consisted eight
coal plugs. The length and weight of the wet coal plugs were measured, and
the wet density was calculated, Table 3.2. The total length of the core was 10.86
inches, and the average wet density of the coal was 1.21 g/cm 3 .
Table 3.2: Length, wet weight, and wet density of the coal plugs in the core.
Plug # Length Wet Weight Volume Wet Density
(inches) (g) (cm 3 ) (g/cm 3 )
#1 1.12 17.35 14.35 1.21
#2 1.18 18.36 15.19 1.21
#3 1.09 16.93 14.04 1.21
#4 2.31 35.67 29.74 1.20
#5 1.16 18.35 14.95 1.23
#6 1.50 23.29 19.25 1.21
#7 1.51 23.66 19.47 1.22
#8 0.99 14.98 12.71 1.18
Total/Ave. 10.86 168.59 139.71 1.21
The coal plugs together with two stainless steel end plugs were assembled end
to end, coated with silica gel, and then wrapped with a piece of aluminum foil,
Figure 3.6(b). The aluminum foil was used to prevent gas migration by diffusion
to the confining fluid in the annulus. The composite core was then placed in a
teflon heat-shrink tube. Teflon was used instead of viton tube because the con-
fining pressure was supplied with water, viton would degrade in the presence
of water and CO2. Heat was applied to the teflon tube which shrank and fitted
tightly onto the composite core, Figure 3.6(c) and Figure 3.6(d). The core was
then dried for several days to allow the silica gel to set. After that the core was
vacuumed under heat (at around 60 o C) for several days to make sure all the mois-
ture and gas contents in the core were removed. At the end, the core was placed
within an aluminum sleeve with end caps that served as a pressure vessel to allow
for application of confining pressure in the annulus space. The core was scanned

60 CHAPTER 3. EXPERIMENTS
using X-ray computed tomography (CT) prior to experiments. The core was CT
scanned when it was first made, after removing the moisture content, and after
confining pressure was applied, Figure 3.7. In the image, black is low density
and white indicates greater density. Fractures along the core were observed from
the CT images. The dry coal has lower density compared with when it was wet,
and the fractures were closed with the application of the confining pressure. The
post-experiment images of the core are shown together with the pre-experiments
images for comparison. The post-experiment images will be discussed in the ex-
periment result section later in this chapter.
(a) A piece of coal plug. (b) Assembled coal plugs.
(c) Heating the shrinkable tube. (d) The core.
Figure 3.6: Composite core used in the experiments.
After being made, the core was connected in an apparatus similar to that
of the permeability measurement in Figure 3.2(a) for preliminary permeability

3.2. EXPERIMENTAL APPARATUS 61
Figure 3.7: CT images of the composite core at different conditions. Raw CT numbers
are shown.

62 CHAPTER 3. EXPERIMENTS
measurement. Only the well consolidated coals were successfully cored; there-
fore, the permeability of a core made from the drilled coal plugs may be much
smaller than the actual coal seam where the coal samples were collected. A good
core should have sufficient lateral permeability to allow for gas flowing through
it at a measurable rate. Also, the aluminum foil and silica gel wrapped over the
core should fit on the core evenly to make sure that there is no bypass along the
lateral side of the core.
3.2.2 Experimental Design
Besides the core, several other components were included in the experimental
setup to fulfill the purposes of the experiments - to measure the total amount
of adsorption, the adsorbed phase composition, the volumetric change, and the
permeability change of the core simultaneously.
Sorption Measurement
There are two common ways to measure the amount of adsorption: the volumet-
ric method and the gravitational method, as shown in Figure 3.8. The volumet-
ric method was employed in our experiment using an apparatus similar to the
schematics shown in Figure 3.9.
The first step was to measure the dead volumes of the system using an non-
adsorbing gas, such as helium. A known amount (moles) of helium was charged
into the gas container, the pressure of the system, p1, is read. The volume of the
part of the system before the inlet valve to the core and the inlet valve to the gas
sampling portion was calculated using the following real gas law:
VDV,1 = nHeZ1RT
p1
(3.1)
The inlet valve to the core was then opened, and the pressure of the system, pt,

3.2. EXPERIMENTAL APPARATUS 63
Figure 3.8: Experimental systems for adsorption measurement (Talu, 1998).
Figure 3.9: Experimental setup for adsorption measurement.
was read. The total dead volume of the system was calculated as
VDV,t = nHeZtRT
pt
(3.2)

64 CHAPTER 3. EXPERIMENTS
To measure the amount of adsorption, an injection gas of known composi-
tion was charged into the gas container. The initial pressure, pini, was read, and
then the inlet valves to the core were opened to allow gas to enter the pore vol-
ume. Pressure, pequ, was read after adsorption equilibrium was reached. The total
amount of adsorption was calculated using the following equation:
nt,ads =
� �
piniVDV,1
−
ZiniRT
� �
pequVDV,t
ZequRT
(3.3)
where, VDV,1 and VDV,t are the dead volumes of the system measured using he-
lium, Zini and Zequ are the compressibility factor (function of the temperature,
pressure, and gas composition) of the gas prior to adsorption and after equilib-
rium being reached.
Gas samples were taken from the gas phase prior to opening the gas to the
core and after adsorption equilibrium was reached, and analyzed using gas chro-
matography (GC). The composition of the adsorbed phase was calculated based
on material balance:
ni,ads =
Volumetric Strain Measurement
� �
piniVDV,1
yi,ini −
ZiniRT
�
xi = ni,ads
i
ni,ads
� �
pequVDV,t
yi,equ
ZequRT
(3.4)
(3.5)
Volumetric strain is the ratio of the change in volume (deformation) to the origi-
nal volume of a medium (Robertson and Christiansen, 2005). To obtain the volu-
metric strain of the core due to adsorption, the volume change of the core in the
process of adsorption was measured. A high precision syringe pump was used
to provide the confining pressure to the core, as in Figure 3.9. The pump ran
in constant pressure mode, and the volume of water in the pump was recorded

3.2. EXPERIMENTAL APPARATUS 65
throughout the process of adsorption. If the volume of the core changed, some
water in the confining annulus flowed into or out of the confining annulus. The
volumetric change of the core due to gas sorption was captured by the change of
the volume of water in the confining annulus. Volumetric strain due to sorption
was then calculated as:
ε = ∆V
Vcore
(3.6)
where, Vcore is the volume of the core, and ∆V is the amount of the core volume
change.
Permeability Measurement
Schemes of permeability measurement similar to those used by Lin et al. (2008)
were employed. After the core reached adsorption equilibrium, a pressure dif-
ference was built across the core, and then gas flowed through the core under
the pressure gradient. The corresponding pressure gradients and gas flow rates
were measured. Application of the compressible form of Darcy’s Law yielded the
permeability of the coal:
kg = 2µgLcoreqgp2
Acore (p 2 1 − p 2 2)
(3.7)
where, p1 is the inlet pressure, p2 is the outlet pressure, qg is the gas flow rate
measured at the outlet of the core, µg is the gas viscosity, Lcore is the length of
the core, and Acore is the cross-sectional area of the core.
Apparatus
We built an experimental apparatus capable of conducting all three measure-
ments: sorption, volumetric strain, and permeability as shown in Figure 3.10.
The piston accumulators driven by high precision syringe pumps at both ends
of the setup were used to store injection gas and allow gas to flow through the
core at either constant volumetric flow rates or constant pressure gradients. A
third syringe pump was used to provide confining pressure and keep track of the
volumetric change of the core. Three pressure transducers were used to monitor

66 CHAPTER 3. EXPERIMENTS
Figure 3.10: Schematic of the experimental apparatus of composite core.
the pressure of different compartments of the system. The pressure, volume and
flow rate of the pumps, as well as the readings of the pressure transducers were
recorded using appropriate data acquisition devices and software on a computer.
Two gas bombs of fixed volume were used as reference cells. The valves were all
stainless steel high-pressure ball valves. Also, the flow lines were all 316 stainless
steel tubing (outer diameter = 0.125 inches, wall thickness = 0.028 inches).
3.3 Experiments Measurements and Data Processing
3.3.1 Dead Volumes, Initial Porosity and Permeability
After assembly, the whole apparatus was first vacuumed. The confining pressure
was set to run in constant pressure mode at 500 psi. At the end of the vacuuming,
valves along the flow lines were all closed. The next step was to determine the

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 67
dead volumes of the different compartments of the system, and the initial poros-
ity and permeability of the core. It was not suitable to determine the porosity
of coal by the conventional way of mercury injection, because the surface ten-
sion of mercury is much too high to permit mercury entry into the micropores of
coal which probably account for the majority of the coal porosity (Harpalani and
McPherson, 1986). A gas expansion method was used instead. The gas used for
this purpose should have little adsorption on coal, for example, helium. Another
good reason to use helium is that the helium molecules are small in size which
ensures they enter the micropores in coal.
Gas cylinder #1 in Figure 3.10 was detached from the system and weighed.
The weight of the vacuumed gas cylinder is W0. The gas cylinder was then charged
with helium, and the weight of the helium-filled gas cylinder was measured (W1).
The amount of helium in the gas cylinder was then calculated as
nhelium = W1 − W0
Mhelium
where, Mhelium is the molar weight of helium, Mhelium = 4.0026g/mol.
(3.8)
The helium-filled cylinder was reconnected to the system. The valves be-
tween the gas cylinder and pressure transducer #1 were first opened. Pressure in
the gas cylinder was read from the pressure transducer and recorded. The signals
from the pressure transducers are actually voltage that is converted to pressure
values. The pressure transducers were calibrated with one of the pumps prior
to the experiments. Figure 3.11 shows the correlations between pressure trans-
ducer voltage reading and pressure. The valves downstream of the helium-filled
cylinder were opened sequentially and the pressure change was recorded. The
volume of each portion of the system was then calculated based on the pressures
and the real gas law as in Equation 3.1. The compressibility factors were obtained
by simulations using the software CMG WINPROP (2008).
Table 3.3 summarizes the parameters of the composite coal core used in the
experiments. The pore volume of the core was obtained by subtracting the vol-
ume of the tube in-between the two valves leading to the core from the dead

68 CHAPTER 3. EXPERIMENTS
Figure 3.11: Pressure versus voltage of the pressure transducers.
volume between the two valves that was determined during the dead volume
measurement. The total core volume of the coal was calculated based on the
dimensions (cross-sectional area and total length) of the core. The dry weight of
the coal is estimated based on the wet weight of the coal, the pore volume, and
the density of water.
At the end of the dead volume measurement, the valves leading to the core
and the valves at the two ends of the bypass were closed. The system was thus
separated into two portions by the core and the bypass. A pressure difference
was established between the two portions by the two pumps connected with the
piston accumulators. The pressure difference was about 20 psi and the average
pressure equaled to the average pore pressure at the end of the dead volume mea-
surement. The valves to the core were then opened and helium was allowed to
flow through the core, Figure 3.12. The gas flow rates were captured by the pumps
and recorded by the computer. The pressure differences between the inlet and
outlet of the core were recorded using the pressure transducers. The recording

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 69
Table 3.3: Parameters of the composite coal core used in the experiments
Length (cm) 27.57
Cross-sectional area (cm 2 ) 5.07
Core volume (cm 3 ) 139.71
Dry weight (g) 151.77
Pore volume* (cm 3 ) 16.82
Initial porosity* 0.12
Initial permeability* (md) 18
Note: * measured under pore pressure of 65 psi, and confining pressure of 465 psi.
frequency of the flow rates was set to be the same as that of the pressures. Per-
meability was calculated based on the pressure difference across the core and the
flow rates. An average permeability was calculated over a time interval at which
the pressure gradients and flow rates were constant. The initial helium perme-
ability of the core was about 18 md.
3.3.2 Simultaneous Sorption, Swelling/Shrinkage, and Perme-
ability Measurements
Different gases were used as the injection gas, including, in the order of being
tested,
1. binary mixture of 50%CO2/50%N2
2. pure CO2
3. pure N2
4. pure helium
5. binary mixture of 75%CO2/25%N2
6. binary mixture of 25%CO2/75%N2

70 CHAPTER 3. EXPERIMENTS
Figure 3.12: Permeability measurement using the new apparatus.
For the pure gases, bottled (99.0 percent) gases were used. For CO2/N2 binary
mixtures, either bottled gases from commercial manufacturers were used; or they
were made in the laboratory according to partial pressure and mass.
Prior to adsorption measurement of each injection gas, the whole system was
vacuumed for a few days to make sure all the moisture and gas contents in the
core and other parts of the system were removed. Prior to adsorption measure-
ments, the core was first isolated from the rest of the system by closing the valves
leading to the core. The injection gas was then charged into the system. A sample
of the injection gas was taken, and analyzed afterwards using gas chromatogra-
phy. The starting pressure of adsorption was set by running the pumps driving
the piston accumulators at the desired pressure. After the desired starting pres-
sure was set, the valves leading to the gas portion of the piston accumulators were
closed to leave the portion of the system as shown in Figure 3.13 as the adsorp-
tion cell.

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 71
Figure 3.13: The adsorption cell (valves in bold were closed while the adsorption
measurement).
To start the adsorption process, the valves at the two ends of the core were
opened to let the gas in. Adsorption was allowed to reach equilibrium over a time
range of about 20 hours. Pressure within the adsorption cell was monitored using
pressure transducer #2 and #3. Adsorption equilibrium was considered to have
been reached when the pressure within the adsorption cell was stabilized. For
example, Figure 3.14 shows the pressure readings of the three pressure transduc-
ers during adsorption measurement of binary mixture of 75%CO2/25%N2. The
starting pressure in the adsorption cell was about 403 psi. For the first 19 hours,
the valves leading to the core were closed, there should not have been variation
in the pressure readings because the adsorption had not been started yet. It was
not the case, however, as shown in the figure. We found that the pressure vari-
ation was a result of the laboratory temperature variation during a day. To deal
with this issue, an isolated fixed volume whose pressure was monitored using
pressure transducer #1 was used as the temperature reference cell. The pressure

72 CHAPTER 3. EXPERIMENTS
readings of pressure transducer #2 and #3 were calibrated against that of pres-
sure transducer #1 to eliminate the effect of changing lab temperature on pres-
sure readings. Figure 3.15 shows the calibrated pressure readings. The equilib-
rium pressure is about 393 psi. From the figure it is also observed that adsorption
reached equilibrium within a relatively short time period (a couple of hours). In
all our experiments, about 20 hours were allowed for the adsorption measure-
ment to ensure that sorption equilibrium was reached. After the system reached
adsorption equilibrium, the valves to the core were closed, and a sample of the
equilibrium gas was taken from the adsorption cell, and analyzed afterwards us-
ing gas chromatography, Figure 3.16.
Figure 3.14: Readings of the pressure transducers and the confining pressure
pump during adsorption of binary mixture of 75%CO2/25%N2, confining pressure
= 800 psi.
The total amount of adsorption was calculated based on the known dead vol-
ume of the adsorption cell, the initial pressure, and the equilibrium pressure us-
ing Equation 3.3. For binary adsorption, the amount of adsorption for each com-
ponent and the adsorbed phase composition were calculated using Equation 3.4

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 73
Figure 3.15: Calibrated readings of pressure transducers during adsorption of binary
mixture of 75%CO2/25%N2, confining pressure = 800 psi.
and Equation 3.6. Equation 3.3 does not take into account of the volume of the
adsorbed phase. The value obtained using Equation 3.3 is called the measured
adsorption, also called the apparent Gibbs adsorption (Hall et al., 1994). Taking
account of the volume occupied by the adsorbed phase, the absolute adsorption
was found as
nads,absolute = nads,measured
1 − ρads
ρgas
(3.9)
where, nads,measured and nads,absolute are respectively the measured adsorption and
the absolute adsorption in mol/kg, ρads is the specific volume of the adsorbed
phase at adsorption equilibrium in cm 3 /mol, and ρgas is the specific volume of the
equilibrium gas phase in cm 3 /mol. The adsorbed specific volume can be taken
as the covolume of a chosen equation of state, parameter b in Equation 2.36. The
volumes of the EOS covolumes are listed in Table 3.4. In this study, we assumed

74 CHAPTER 3. EXPERIMENTS
(a) gas chromatography.
(b) gas samples.
Figure 3.16: The gas chromatograph and some gas samples.

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 75
Table 3.4: Specific volume of adsorbed phase based on different equations of
state.
Gas pC TC bvdW bSRK bP R
(bar) (K) (cm 3 /mol) (cm 3 /mol) (cm 3 /mol)
[1pt] CO2 73.8 304.1 42.8257 29.6834 26.6547
N2 33.9 126.2 38.6905 26.8172 24.0810
the density of the adsorbed phase equals to the hard-sphere term, b, in the Soave-
Redlich-Kwong cubic equation of state:
ρads,i = 0.08664RTc,i
pc,i
(3.10)
where, R is the universal gas constant, Tc,i and pc,i is the critical temperature and
pressure of component i respectively. The equilibrium gas phase density is ob-
tained based on the real gas law:
ρgas,i = ZequRT
pequ
(3.11)
Experimental adsorption data for all the injection gases (in the sequence of
experiments: 50%CO2/50%N2, 100%CO2, 100%N2, 75%CO2/25%N2, and 25%CO2/75%N2)
are tabulated in Appendix B.
Also plotted in Figure 3.14 was the volume reading of pump B that provided
the confining pressure to the core. The pump ran at a constant pressure of 800 psi
throughout the adsorption measurement. The increase in the pump volume after
the gas was opened to the core was because of the combination of two reasons:
(1) the decrease in net effective stress due to the increase in pore pressure, and
(2) the sorption induced volumetric swelling of the core. The first effect was cor-
rected by complementary measurements using helium at the same conditions

76 CHAPTER 3. EXPERIMENTS
that is discussed later in this chapter. The pump volume then decreased. We be-
lieved that this was due to minor leakage of the pump and change of the labora-
tory temperature, because it follows the same trend as the pump readings prior to
adsorption. We calculated the volumetric strain based on the maximum volume
change observed and subtracted the effect of the changing net effective stress.
After the adsorption measurement at a given pressure, permeability of the
core was measured in the same way as the initial permeability: the valves be-
tween the adsorption cell and the gas portions of the piston accumulators were
opened, while the valves leading to the core and the bypass were closed; a pres-
sure difference (about 20 psi) was set between the two portions of the system
separated by the core and the bypass by running pump A and pump C at dif-
ferent pressures, while the average pressure was equal to the pore pressure at
the adsorption equilibrium so there would be no additional adsorption during
the permeability measurement; the valves at the two ends of the core were then
opened to let gas flow through the core under the set pressure gradient, Fig-
ure 3.12. Figure 3.17 shows the pressure readings at the two ends of the core
and the calculated permeability of the core after adsorption of binary mixture of
75%CO2/25%N2 at 387 psi. The average pore pressure was kept at 387 psi dur-
ing the permeability measurement. The uncalibrated equilibrium pressure was
used because the pressure transducer readings while conducting the permeabil-
ity measurement was not calibrated for the laboratory temperature. Permeability
corresponding to each set of pressure gradient and gas flow rate was calculated
using Equation 3.8. The viscosities of the gas at the laboratory temperature and
the average pore pressure was obtained using CBMW inP rop(2008.10). An av-
erage permeability was taken for the time interval when the flow rate readings
from the pump well reflected the flow rates of gas, such as 1.6-3.2 hour in Figure
3.17. The average permeability of the core after adsorption of binary mixture of
75%CO2/25%N2 at 387 psi is about 1.12 md. During the measurement, the con-
fining pressure was kept at 811 psi using pump B, thus the net effective stress was
about 423 psi. The net effective stress was always 423 psi during all permeability
measurement.

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 77
Figure 3.17: Permeability measurement after adsorption of binary mixture of
75%CO2/25%N2 at 387 psi (uncalibrated), confining pressure = 811 psi.
After the permeability measurement, the valves at the two ends of the core
were closed and the valves to the bypass were opened. The system pressure was
set to a new value using the pumps and piston accumulators. The adsorption
and permeability measurements were repeated. For each injection gas, 5-7 pore
pressures were tested. At each pressure, an additional amount of adsorption was
calculated, and the total amount of adsorption at the pressure was the sum of the
additional adsorption and the amount of adsorption at the previous pressure.
When the adsorption path was finished, the system pressure was decreased
in steps to obtain the desorption data. Figure 3.18 and Figure 3.19 show the
pressure readings of the adsorption cell while conducting desorption measure-
ment at equilibrium pressure of 403 psi after adsorption of binary mixture of
75%CO2/25%N2. Also shown in Figure 3.18 is the laboratory temperature mea-
sured using a thermocouple. The pressure readings of pressure transducer #1
was indeed a good indication of the laboratory temperature.

78 CHAPTER 3. EXPERIMENTS
Figure 3.18: Readings of the pressure transducers during desorption of binary
mixture of 75%CO2/25%N2, confining pressure = 830 psi.
3.3.3 Complementary Experiments using Helium
During adsorption measurement, when gas in the adsorption cell is opened to
the core, the pore pressure in the core changes momentarily. Thus the net effec-
tive stress (confining pressure minus the pore pressure) changes. Besides sorp-
tion, the change in net effective stress also contributes to the change in the vol-
ume of the water in the confining annulus as well as in pump B. To eliminate the
effect of the changing net effective stress and obtain the volumetric change of
the core merely due to sorption, helium was used as injection gas. The same “ad-
sorption”, volumetric swelling, and permeability measurements were conducted
at escalating pore pressures. Figure 3.20 shows the pressure in the adsorption cell
and the volume of water in the confining pressure pump when injecting helium
to the core at a starting pressure of 403 psi. Comparing Figure 3.20 with Figure
3.14, the pressure drop and pump volume increase in Figure 3.14 were due to the
effect of changing pore pressure as well as adsorption, while the pressure drop

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 79
Figure 3.19: Calibrated readings of pressure transducers during desorption of binary
mixture of 75%CO2/25%N2, confining pressure = 830 psi.
and pump volume increase in Figure 3.20 were due to the effect of changing pore
pressure only. To obtain the changes due to adsorption only, the effect of chang-
ing pore pressure was subtracted.
As mentioned in Chapter 2, when the pore sizes of a porous medium are very
small, gas permeability increases with increasing pore pressure due to slip flow
of gas at pore walls (Harpalani and Chen, 1997; Tanikawa and Shimamoto, 2006).
The permeability measurement at escalating pore pressures using helium served
as an investigation of the Klinkenberg effect.
The average distance traveled by a molecule between successive collisions is
called the mean free path (Bird et al., 1960).
λ =
1
√ 2πd 2 N
(3.12)
where, λ is the mean free path; d is the molecular diameter, and N is the number
of molecules per unit volume. The kinetic diameter of the molecules of N2 and
CO2 are 3.64 ˚A and 3.3 ˚A respectively (Adamson and Gast, 1997). Based on the

80 CHAPTER 3. EXPERIMENTS
Figure 3.20: Pressure and volume of the confining pressure pump after opening
helium to core at 403 psi, confining pressure = 800 psi.
real gas law,
n
N = NA
V
= NA
p
ZRT
(3.13)
where, NA = 6.02 × 10 23 , NA the Avogadro constant gives the number of atoms
or molecules in 1 mole of pure substance. The values of the mean free path of
N2 and CO2 calculated based on Equation 3.12 and 3.13 are in the magnitude of
several to tens of nanometers over the pressure ranges we are interested in (1000
- 5000 kPa), Table 3.3.3. The mean free path of helium would be even smaller
than those of N2 and CO2.
For gas flow in porous media, when the mean free path of the gas molecule is
of the same magnitude as that of the flow path, the gas molecules have apprecia-
ble interaction (i.e. collision) with the flow path surface, which contributes to an
additional flow component besides the Darcian flow. To get a sense of the size of
the flow path in the core we used and in typical coalbed methane reservoirs, we

3.3. EXPERIMENTS MEASUREMENTS AND DATA PROCESSING 81
Table 3.5: Mean free path of CO2 at different pressures.
Pressure Pressure Z n λ
(kPa) (psi) (molecules/m 3 ) (m)
1 0.1 1 2.45E + 23 8.43E-06
501 72.7 0.9713 1.27E + 26 1.63E-08
1001 145.2 0.9418 2.61E+26 7.93E-09
1501 217.7 0.9113 4.04E+26 5.12E-09
2001 290.2 0.8795 5.58E+26 3.70E-09
2501 362.7 0.8463 7.25E+26 2.85E-09
3001 435.3 0.8113 9.07E+26 2.28E-09
3501 507.8 0.7742 1.11E+27 1.86E-09
4001 580.3 0.7343 1.34E+27 1.55E-09
4501 652.8 0.6907 1.60E+27 1.29E-09
5001 725.3 0.6416 1.91E+27 1.08E-09
5501 797.9 0.5832 2.31E+27 8.93E-10
6001 870.4 0.5040 2.92E+27 7.08E-10
did a calculation based on the Carmen-Kozeny equation (Lake, 1989):
k = 1 φ
72τ
3D2 p
(1 − φ) 2
(3.14)
where, k is the permeability; τ is the tortuosity of the permeable medium, it is
usually between 2 and 5 for the reservoir rocks we are interested in (Lake, 1989);
φ is the porosity; and Dp is the pore (flow path) size. Table 3.3.3 shows that
the diameters of the flow path in typical coalbed methane reservoirs are of the
magnitude of tens to hundreds of micrometers. Therefore, the likelihood of the
Klinkenberg effect, i.e., gas slippage, in low rank coals of moderate permeabili-
ties ranging from 1 to 100 md appears to be unlikely. The estimate of the size of
the flow path based on the Carmen-Kozeny equation is truly a rough estimate.
The cleat apertures under in situ confining pressure are estimated to be between

82 CHAPTER 3. EXPERIMENTS
1E −4 and 1E −8 m, varying with the rank of the coal, lithotype and bed thickness
(Harpalani and Chen, 1997). The Klinkenberg effect may be a concern for some
lower-permeability coalbed methane reservoir rocks.
Table 3.6: Flow path size of typical coalbed methane reservoirs.
k k τ φ DP
(md) (m 2 ) (m)
3.4 Experimental Results
18 1.78E-14 1 0.12 2.39E-05
18 1.78E-14 5 0.12 5.35E-05
0.1 9.87E-17 5 0.12 3.99E-06
30 2.96E-14 5 0.12 6.91E-05
100 9.87E-14 5 0.12 1.26E-04
3.4.1 Adsorption and Volumetric Strain
The experimental results for sorption and the consequent volumetric change of
the core when injecting different gases at different pressures are tabulated in Ta-
ble B.4 through Table B.7. Obviously, the amount of adsorption increased with
the increase of pressure, and decreased with the decrease of pressure. No obvious
adsorption/desorption hysteresis was observed. At the same pore pressure, the
amount of adsorption for CO2 was higher than that of N2. The more CO2 in the in-
jection gas, the greater the total amount of adsorption. Selectivity coefficients of
CO2 to N2 were greater than one, but still in the order of one. The selectivity fac-
tors obtained in this study were smaller than those obtained by Hall et al. (1994)
for supercritical CO2/N2 adsorption on their wet ground coal samples. The selec-
tivity coefficients obtained by Hall et al. (1994) and us were all decreased with the
increase of pressure.

3.4. EXPERIMENTAL RESULTS 83
Figure 3.21 shows volumetric strain with the injection of different gases at es-
calating pressures. The total volumetric strain due to momentary effective pres-
sure decrease and adsorption are plotted together with the volumetric strains
caused by adsorption only. The latter are only slightly smaller. Compared with
the results of Hagin and Zoback (2010), our volumetric strain values were slightly
higher. The volumetric strain with the injection of CO2 obtained by Hagin and
Zoback (2010) was less than 1% at a pore pressure of 1 MPa (about 145 psi) and
effective pressure of 5 MPa (about 725 psi). The volumetric strain with the injec-
tion of CO2 at 1 MPa was about 1.5% under an effective pressure of 400 psi for our
system.
Figure 3.21: Volumetric strain at escalating pressures. Open symbols represent
the total volumetric strain, and closed symbols represent volumetric strain due
to adsorption only.

84 CHAPTER 3. EXPERIMENTS
3.4.2 Permeability
Figure 3.22 plots the permeability of the core after adsorption of different feed
gases versus pore pressure. The values of permeability at zero pressure for each
gas composition are the helium permeability of the core prior to the injection of
the gas. The helium permeability of the core decreased in the process of the ex-
periments, which may indicate that the sorption induced permeability reduction
is not fully recoverable. One possible explanation for that is adsorption hystere-
sis. Gas adsorbs in coal with escalating pressure, when the pressure is reduced
(even to vacuum) not all of the adsorbed gas desorbs.
Figure 3.22: Permeability of the core with injection of different gases.
In order to compare the magnitude of permeability reduction for different in-
jection gases. We plotted the permeability reduction in Figure 3.23. Permeability
reduction is defined as
Rk = k
k0
(3.15)

3.5. SUMMARY OF EXPERIMENTAL WORK 85
where, k0 is the initial permeability prior to adsorption of a specific feed gas, and
k is the permeability of the core after adsorption of the feed gas at pressure p.
Permeability reduction less than unity indicates permeability decreases. Based
on the experimental permeability data, the following observations are made
1. Under constant net effective stress, during flow of CO2, N2, or the binary
mixture of them, the permeability of the core decreases with escalating pore
pressure.
2. The more CO2 in the feed gas, the more the permeability reduction.
3. Permeability of the core rebounds when decreasing the pore pressure under
constant net effective stress. It does not, however, recover to the permeabil-
ity value prior to adsorption.
4. Under constant effective stress, helium permeability of the core increased
slightly with the increase of pore pressure. This means that Klinkenberg
effect is negligible for the core we used in our experiments.
Density of the core increased as shown in the CT images, Figure 3.7, which
might be an indication that the core went through irreversible physical damages
(crushing) in the process of the experiments.
3.5 Summary of Experimental Work
Gas sorption on coal and the consequent volumetric and permeability changes
were measured simultaneously. The following is a summary of the experimental
apparatus:
1. The core: composite core made from coal plugs drilled from intact coal
samples, moisture-free, wrapped in heat shrink tube.
2. Experimental conditions: room temperature (22 o C), constant effective over-
burden pressure inserted by water supplied by a syringe pump.

86 CHAPTER 3. EXPERIMENTS
Figure 3.23: Permeability reduction of the core with injection of different gases.
3. Test gases: pure CO2, N2, or CO2/ N2 binary mixtures at various composi-
tion.
4. Gas sorption, sorption induced volumetric strain, and permeability of the
core were measured simultaneously.
Experimental data of 25%CO2/75%N2 feed gas are not available due to exper-
iment failure. The system was set up in a way that dynamic sorption, volume
of the core, and permeability of the core could be measured simultaneously. A
dynamic volume including the gas portions of the two piston accumulators and
all the volumes along the flow lines were to be used. It was a dynamic volume
in the sense that the volume was changed by moving the pistons in the piston
accumulators by the pumps. Feed gas was to be first injected into the dynamic
volume exclusive of the core. The two pumps connected with the piston accumu-
lators were used to set up the equilibrium pressure. The valves to the bypass were
then to be closed, and a moderate (about 20 psi) pressure difference was to be set

3.5. SUMMARY OF EXPERIMENTAL WORK 87
between the two portions separated by the bypass by running the pumps con-
nected to the piston accumulators at constant pressures. The valves to the core
were then opened, and the feed gas flows through the core from the high pres-
sure portion to the low pressure portion until pressure equilibrium was reached
or the piston of the piston accumulator at the high pressure portion reached the
end of the cylinder. The initial and equilibrium volume of the dynamic volume
were obtained by keeping track of the volume of liquid in the two pumps con-
nected to the piston accumulators, and adsorption was calculated based on the
initial and equilibrium volumes and pressures of the system. In the whole pro-
cess of sorption measurement, not only the volume of core was tracked, but also
the permeability of the core was measured by recording the pressure differences
at the two ends of the core using the pressure transducers and the corresponding
gas flow rates using the pumps connected with the piston accumulators. Dy-
namic measurements gave the dynamic permeability of the core through the
process of sorption. Experiments following these procedures did not work out
perfectly, because it is hard to track the dynamic volume accurately. This is the
reason we adopted the method of static sorption measurement after trying the
dynamic approaches and failing. The dynamic approach was tested again us-
ing 25%CO2/75%N2 feed gas after the other measurements. Still no satisfactory
experimental data were obtained.

88 CHAPTER 3. EXPERIMENTS

Chapter 4
Numerical Modeling
4.1 Permeability Modeling
The permeability of coal is not invariant; it changes when the effective stress
changes or gas adsorbs or desorbs from the coal. As discussed in Chapter 2, many
numerical models were proposed to represent the evolution of the permeability
of coal. The Palmer-Mansoori model, Equation 2.11, is one of the examples:
k
k0
=
�
1 + cp (p − p0) + 1
φ0
� K
M
� �3 − 1 ε
(4.1)
The second term on the right side of the equation, cp (p − p0), accounts for the
permeability change caused by change of effective stress while the overburden
pressure is constant; and the third term containing the sorption induced strain,
ε, accounts for the permeability change due to gas sorption. In this study, we
focus on sorption induced permeability change of coal.
In Equation 4.1, k0, φ0, K, M, and cp are the initial permeability, initial porosity,
the bulk modulus, the constrained axial modulus, and the pore compressibility
of the coal. They are coal properties that are measured in the lab (Palmer and
Mansoori, 1998):
89

90 CHAPTER 4. NUMERICAL MODELING
M =
(1 − ν) E
(1 + ν) (1 − 2ν)
K = M
3
� �
1 + ν
1 − ν
(4.2)
(4.3)
where, ν is the Poisson’s ratio, and E is the Young’s modulus. The value of K and
M may also change with the change of effective stress and sorption (Hagin and
Zoback, 2010).
Sorption induced strain is believed to be related with the amount of adsorp-
tion, following either a linear relation such as Equation 2.4 and Equation 2.7:
∆ε = c∆Vsorption
Or a more complex relation as Equation 2.12:
(4.4)
ε = − Φρs
f (x, νs) −
Es
p
(1 − 2νs) (4.5)
Es
Surface potential, Φ, is a function of the amount of adsorption, Equation 2.62:
ψ = − Φ
RT =
NC �
� fi
i=1
0
� �
n
df (4.6)
f i
Therefore, to calculate the sorption induced strain, the first step is to obtain the
amount of adsorption.
4.2 Sorption Modeling
4.2.1 Pure Adsorption
As discussed in Chapter 2, for pure gases, the amount of adsorption is well repre-
sented by parametric equations, for instance,

4.2. SORPTION MODELING 91
The Langmuir equation:
The N-layer BET equation:
And, the modified virial equation:
n = mBp
1 + Bp
� � �
mCpR 1 − (N + 1) pR
n =
1 − pR
N + NpR N+1
1 + (C − 1) pR − CpR N+1
�
p = n
� �
m
exp
H m − n
� C1n + C2n 2 + C3n 3 + C4n 4�
(4.7)
(4.8)
(4.9)
In those three equations, n is the amount of adsorption; p is the equilibrium pres-
sure; m is the maximum amount of adsorption at infinite pressure in the Lang-
muir equation and the modified virial equation, and the maximum monolayer
adsorption in the N-layer BET equation; B is the Langmuir constant; H is the
Henry’s constant; pR = p/p sat is the reduced pressure; N is the total number of
adsorption layers allowed on the adsorbent surface; and C, C1, C2, C3, and C4 are
constants respectively.
Table 4.1: Values of pure adsorption isotherm constants. Experimental temperature
= 22 o C.
Gas Coal Sample Model B m C N Error
(kP a −1 ) (mol/kg) %
CO2 Montana intact coal N-BET 1.1360 27.090 5.164 0.11
CO2 Wyoming ground coal N-BET 1.2680 9.148 2.168 0.18
CO2 Wyoming ground coal Langmuir 0.00066 2.3533 0.35
N2 Montana intact coal Langmuir 0.00031 0.5056 0.39
N2 Wyoming ground coal Langmuir 0.00037 0.3321 0.54
Previous experiments of gas sorption on ground coal sample from Powder

92 CHAPTER 4. NUMERICAL MODELING
Figure 4.1: Adsorption of pure CO2 and N2 on coal samples from Powder River
Basin, Wyoming (ground) and Montana (intact). Symbols represent the experimental
data, and curves are the fitted models.
River Basin, Wyoming showed that the Langmuir equations are adequate to rep-
resent the adsorption of pure CO2, N2, and CH4 (Tang et al., 2005). For our exper-
imental data of gas sorption on intact Powder River Basin (Montana) coal sam-
ples, the Langmuir equation represented the adsorption of pure N2 very well;
however, it could not represent the adsorption of pure CO2, Figure 4.1. Experi-
mental data of Hall et al. (1994) of supercritical CO2 adsorption on wet Fruitland
coal also shows highly non-Langmuir behavior at high pressures. The N-layer
BET equation turned out to be able to represent the adsorption of pure CO2 on
our intact Powder River Basin (Montana) coal sample very well. The values of
the parameters of the corresponding pure adsorption isotherms are listed in Ta-
ble 4.1. For the adsorption of pure CO2 on the ground Wyoming coal, both the
N-BET equation and the Langmuir equation represented the experimental data

4.2. SORPTION MODELING 93
fairly well. The N-layer BET equation yielded a slightly better fit than the Lang-
muir equation. The modified virial equation was not a good choice for the ad-
sorption of pure CO2 and N2 on the tested coal samples.
Pure gas adsorption is usually well represented by one or another of the pro-
posed models in the literature (Clarkson and Bustin, 2000; Siperstein and Myers,
2001; Tang et al., 2005). Multicomponent adsorption, however, is not readily rep-
resented by simple parametric equations.
4.2.2 Multicomponent Sorption
In this part of the study, we investigated two approaches to model multicompo-
nent adsorption:
1. calculating multicomponent adsorption based on pure adsorption isotherms
only, and
2. modeling multicomponent adsorption based on pure adsorption isotherms
and experimental binary adsorption data.
Extended Langmuir (ELM) Equations
If the pure adsorption for each element component fits the Langmuir equation,
the most straightforward approach to calculate multicomponent adsorption is
to use the extended Langmuir equations, Equation 2.21. As long as the parame-
ters of the pure adsorption isotherm for each element component are known, the
amount of adsorption for each component, the total amount of adsorption, and
the equilibrium adsorbed phase composition of multicomponent adsorption can
be calculated easily:
where, partial pressure pi = pyi.
ni = miBipi
1 + NC �
Bjpj
j=1
(4.10)

94 CHAPTER 4. NUMERICAL MODELING
NC �
nt = ni
i=1
xi = ni
The separation factor (also called the selectivity coefficient) is defined as
nt
Sij ≡ xi/yi
xj/yj
(4.11)
(4.12)
(4.13)
It is an indication of the adsorbability of two components. When Si,j > 1, com-
ponent i is more adsorbed than component j. The separation factor based on the
extended Langmuir equations is
Sij,Langmuir = miBi
mjBj
(4.14)
The separation factors based on the extended Langmuir equations are invariant
with pressure and the composition (concentration of each component in the gas
mixture) of the adsorbate.
Ideal Adsorbed Solution (IAS) Model
Anther approach to conduct multicomponent adsorption calculation is based on
the IAS model, Equation 2.54. To implement this approach, only pure compo-
nent adsorption data (isotherms) are needed. The assumption of the IAS model
is that the gas and the adsorbed phase are ideal. For a binary system consisting of
components i and j, the isofugacity equation at equilibrium for each component
is
yip = xip 0 i
yjp = xjp 0 j
(4.15)
(4.16)

4.2. SORPTION MODELING 95
where, p 0 i and p 0 i , analogous to the vapor pressure of the pure substances in the
Raoult’s law, are the pure component adsorption pressures at the same surface
potential and temperature of the multicomponent adsorption; xi and xj are the
mole fraction of component of i and j in the adsorbed phase.
xi + xj = 1 (4.17)
In order to implement the isofugacity equations, the standard state of the gas
and adsorbed phase must be consistent, i.e., Equation 2.66 must be fulfilled:
ψ 0 i = ψ = ψ 0 j
(4.18)
where, ψ is the modified surface potential of multicomponent adsorption, ψ 0 i and
ψ 0 j are the corresponding modified surface potentials of pure adsorption. Accord-
ing to Equation 2.64, ψ 0 i and ψ 0 j can be calculated based on the pure adsorption
isotherms:
ψ 0 � p0 i
i =
0
n · d ln p (4.19)
If the Langmuir equation is used for the pure adsorption isotherm,
ψ 0 i = mi ln � 1 + Bip 0 i
�
(4.20)
If the N-layer BET equation is used for the pure adsorption isotherm (obtained
using Mathematica),
ψ 0 i = mi ln
�
−1 + pR 0 i − CpR 0 i + CpR 0N+1 i
−1 + pR 0 i
where, pR 0 i = p 0 i /p sat
i .
If the modified virial equation is used for the pure adsorption isotherm,
ψ 0 i = 1
2 C1in 02 2
i +
3 C2in 03 3
i +
4 C3in 04 4
i +
5 C4in 0 �
5
i − miln 1 − n0 �
i
mi
�
(4.21)
(4.22)

96 CHAPTER 4. NUMERICAL MODELING
Solving the nine equations, Equation 4.15, 4.16, 4.17, 4.18 (two equations),
(two) equations for ψ 0 i and ψ 0 j , combined with (two) equations for pure adsorp-
tion isotherms jointly yields the value of the nine unknowns xi, xj, p 0 i , p 0 j, n 0 i , n 0 j, ψ,
ψ 0 i , and ψ 0 j . The only inputs to be supplied are the equilibrium pressure, the gas
phase composition, and the pure component adsorption isotherms. One of the
approaches to solve the problem is as following (Do, 2008):
1. Estimate the surface potential ψ as the mole-fraction weighted average of
the pure adsorption surface potential for each component
NC �
ψ =
i=1
yiψ 0 i
� � 0
pi using the equilibrium system pressure as an initial guess for p 0 i .
(4.23)
2. Back calculate p 0 i and p 0 j for each component at ψ 0 i = ψ and ψ 0 j = ψ. After
obtaining p 0 i , p 0 j, xi and xj are calculated based on Equation 4.15 and 4.16,
where p, yi, and yj are inputs.
3. Form an objective function employing Equation 4.17, and use Newton-Raphson
iteration to update the estimate of ψ.
NC �
F (ψ) = xi − 1 = 0 (4.24)
i=1
If the Langmuir Equations are used for the pure adsorption isotherms, the
derivative is evaluated as
F ′ NC �
(ψ) = −
i=1
pyi
(p 0 i (ψ))2
dp o i
dψ
4. Repeat Step 2 and Step 3 until convergence.
NC �
= −
i=1
pyi
p 0 i (ψ) no i
(4.25)
5. After convergence, the values of p 0 i and p 0 j are known. The parameters for

4.2. SORPTION MODELING 97
the adsorbed phase are calculated as
and
1
nt
xi = pyi
p 0 i
= � xi
n 0 i
ni = ntxi
Example: CO2/N2 Binary Gas Adsorption on Wyoming Coal
(4.26)
(4.27)
(4.28)
For the purpose of comparing the results calculated based on the IAS model
and the extended Langmuir equations, we consider the adsorption of CO2/N2
binary gas on the coal sample of Tang et al. (2005). Adsorption isotherms for
pure CO2 and N2 are well represented by the Langmuir equations, Figure 4.1.
The values of the parameters in the Langmuir equations are listed in Table 4.1.
The total amount of adsorption, the composition of the adsorbed phase, and
the separation factors are calculated based on the extended Langmuir equations
and the IAS model (refer to the MatLab code in Appendix C.1). The adsorption
of four different binary mixture compositions, 25%CO2/75%N2, 50%CO2/50%N2,
75%CO2/25%N2, and 85%CO2/15%N2, at escalating pressures were modeled. The
calculated total amount of adsorption, mole fraction of CO2 in the adsorbed phase,
selectivity coefficients of CO2 to N2 are plotted in Figure 4.2.
Figure 4.2(a) shows that at low pressures (below 700 kPa) the total amount of
adsorption calculated based on the extended Langmuir equations and the IAS
model are very close. At high pressure, the values of total amount of adsorption
calculated based on the IAS model are higher than those calculated based on the
extended Langmuir equations. The ideality assumption may not be appropriate
at high pressures. Another observation is that the more N2 in the gas mixtures, the
more discrepancy in the results based on the two models. This may indicate that
N2 shows more nonideal behavior in the adsorbed phase. From Figure 4.2(b) and

98 CHAPTER 4. NUMERICAL MODELING
4.2(c), the mole fraction of CO2 in the adsorbed phase for a particular gas com-
position calculated based on the extended Langmuir equations does not change
with pressure; and the selectivity factors of CO2 to N2 calculated based on the ex-
tended Langmuir equations are constants for all gas compositions and pressures.
The mole fraction of CO2 in the adsorbed phase and the selectivity factors of CO2
to N2 calculated based on the IAS model are functions of both gas composition
and pressure. With the increase of pressure, however, the selectivity factors of
CO2 to N2 for a particular gas composition decrease with the increase of pressure
based on the experimental data of Hall et al. (1994). It seems that neither the ex-
tended Langmuir equations nor the IAS model represents the multicomponent
adsorption perfectly.
Real Adsorbed Solution (RAS) Model
For a real system (nonideal gas and adsorbed phase), the isofugacity equation at
equilibrium is
pyi ˆϕi = f 0 i γixi
There are three nonideal elements in the equation:
(4.29)
1. Nonideality of the gas phase indicated by nonunity fugacity coefficients ˆϕi,
2. Nonideality of the adsorbed phase indicated by nonunity activity coeffi-
cients γi, and
3. Nonideality of the pure condensed phase indicated by the pure condensed
phase fugacity f 0 i .
One, two, or all three of the nonideal elements can be taken into account in mul-
ticomponent sorption calculation to achieve different level of accuracy.
Based on Appendix A.2, p 0 i can be used as an approximation of f 0 i . For the
simplicity of calculation, the following equation
pyi ˆϕi = p 0 i γixi
(4.30)

4.2. SORPTION MODELING 99
is used instead of Equation 4.29.
The fugacity coefficients are functions of pressure, temperature, and the gas
phase composition. They are readily calculated using Equation 2.33 or 2.34 com-
bined with a chosen equation of state for gas, as discussed in Chapter 2.
To calculate the activity coefficients, the relation between partial molar excess
Gibbs energy and activity coefficient, Equation A.11,
¯G ex
i = RT ln γi (4.31)
is combined with a numerical model for the molar excess Gibbs energy, for in-
stance, the ABC model (Siperstein and Myers, 2001)
Therefore,
G ex = (A + BT ) xixj
� 1 − e −Cψ �
� −Cψ
RT ln γi = Ao 1 − e � x 2 j
(4.32)
(4.33)
where, A, B, and C are constants. For adsorption at constant temperature, Ao =
A + BT . Ao is also a constant.
The total amount of adsorption, taking nonideality into account, is calculated
using Equation 2.72:
1
nt
= � xi
n 0 i
+ � xi
� �
∂ ln γi
∂ψ T,xi
(4.34)
Substituting Equation 4.33 into Equation 4.34, an equation for the excess recip-
rocal loading is obtained:
� �ex 1
≡
n
1
−
nt
� xi
n0 i
= C
−Cψ
Aoxixje
RT
(4.35)
Similar to the IAS model, in order to implement the isofugacity equations as
the equilibrium criteria, Equation 4.18 must be fulfilled. Equation 2.63 combined

100 CHAPTER 4. NUMERICAL MODELING
with the chosen pure adsorption isotherm models is used to calculate the corre-
sponding ψ 0 i :
ψ 0 � p0 i
i =
0
� �
n
df (4.36)
f i
where f is the gas phase fugacity. For simplicity, Equation 4.19 is usually used
instead. The actual equation for ψ 0 i may take different forms depending on the
pure adsorption isotherms, Equation 4.20 through Equation 4.22.
With experimental binary adsorption data, p, nt, yi, yj, xi, and xj, eleven equa-
tions, two of Equation 4.30 (one for each component), two of Equation 4.33 (one
for each component), Equation 4.35, Equation 4.18 (two equations), two equa-
tions for ψ 0 i , and two equations of the pure adsorption isotherms, are solved
jointly to obtain the eleven unknowns: Ao, C, ψ, ψ 0 i , ψ 0 j , p 0 i , p 0 j, n 0 i , n 0 j, γi, and γj.
An Alternative Way to Calculate ψ: Theoretically, when the amount of ad-
sorption for each component is obtained from experiment, ψ is calculated di-
rectly using Equation 2.62. For a binary system:
ψ =
� p
0
� �
n
df +
ˆf
i
� p
0
� �
n
df (4.37)
ˆf
where, ˆ fi and ˆ fj are fugacity of component i and j in the gas phase, they are cal-
culated using Equation 2.32 combined with a chosen equation of state for the gas
phase, as discussed in Chapter 2. Due to the complexity of the relation between
the pressure and the gas phase fugacity in a multicomponent system, the inte-
grals of Equation 4.37 are obtained numerically from the area below the curves
in ni/ ˆ fi ∼ pi plots. To obtain an accurate estimate of the integral, the shape of the
curve down to the limit of zero pressure needs to be known; whereas, adsorption
j

4.2. SORPTION MODELING 101
data at near zero pressures are rarely available.
Example: CO2/C2H6 Binary Gas Adsorption on NaX
To test the algorithms for multicomponent adsorption calculation, we imple-
mented them using experimental data of Siperstein and Myers (2001) for adsorp-
tion of CO2/C2H6 binary gas mixture on NaX (a FAU-structural-type zeolite).
The pure adsorption isotherms were represented by the modified virial equa-
tions, Equation 4.9. Table 4.2 lists the values of the parameters in the modified
virial equation for pure CO2 and C2H6 on NaX. The experimental binary adsorp-
tion data including the total amount of adsorption, the equilibrium adsorbed and
gas phase composition, and the selectivity coefficients of CO2 to N2 at a constant
temperature and various equilibrium pressures are listed in Table 4.3.
Table 4.2: The values of the parameters in the modified virial equations for pure
CO2 and C2H6 adsorption on NaX, T = 20 o C.
Gas H C1 C2 C3 C4 m Error
mol/(kg · kP a) (mol/kg) %
CO2 27.253 1.2338 -0.1241 0.0038 0 6.4674 3.0
C2H6 0.1545 -0.267 -0.0499 0.0192 0 3.8937 0.2
For this system, the eleven equations to be solved are now discussed.
1. The isofugacity equations (two equations for two components):
pyi = p 0 i γixi
(4.38)
Siperstein and Myers (2001) adopted a semireal model, took into account
only the nonideality of the adsorbed phase. The fugacity coefficients were
assumed to be unity, thus the gas phase was assumed to be ideal. Also, p 0 i

102 CHAPTER 4. NUMERICAL MODELING
Table 4.3: Experimental binary adsorption data of CO2(i)/C2H6(j) on NaX. Experimental
temperature was 20 o C.
pequ nt xi yi Sij
(kP a) (mol/kg)
6.38 3.532 1.000 1.000
10.62 3.717 0.949 0.633 10.81
12.61 3.924 0.953 0.666 10.09
18.22 4.087 0.913 0.498 10.55
21.43 4.274 0.919 0.528 10.13
28.65 4.406 0.889 0.431 10.54
and p 0 j were used in place of f 0 i and f 0 j , therefore, the standard state pure
condensed phases were also considered to be ideal.
2. The activity coefficient equations (two equations for two components):
� −Cψ
RT ln γi = Ao 1 − e � x 2 j
(4.39)
3. The pure component adsorption isotherm equations (two equations for two
components):
p 0 i = n0 �
i mi
Hi mi − n0 � �
exp C1in
i
0 i + C2in 02 i + C3in 03 i + C4in 04 i
�
4. The modified surface potential equations (four equations):
ψ 0 i = ψ = ψ 0 j
ψ 0 i = 1
2 C1in 02 2
i +
3 C2in 03 3
i +
4 C3in 04 4
i +
5 C4in 0 �
5
i − miln 1 − n0 �
i
mi
(4.40)
(4.41)
(4.42)

4.2. SORPTION MODELING 103
5. The nonideal reciprocal total amount of adsorption equation (one equa-
tion):
1
nt
= � xi
n 0 i
+ C
−Cψ
Aoxixje
RT
(4.43)
Siperstein and Myers (2001) implemented the following algorithm to solve the
equations:
1. Determine the values of Ao and C by minimizing the error in the calculated
equilibrium pressures and selectivity coefficients.
1) Guess the ranges of Ao and C: start with a wide range and narrow it down
by trial and error. Specify an incremental to get multiple [Ao C] sets
within the ranges.
2) For a specific set of [Ao C]:
i Solve Equation 4.41 (two equations), 4.42 (two equations for two com-
ponents), and 4.43 jointly to obtain the values of ψ, ψ 0 i , ψ 0 j , n 0 i , and
n 0 j by minimizing the difference between the calculated and experi-
mental total amount of adsorption. The bisection method was used
to solve for ψ.
i) Define the initial bracket: give an initial guess of the range of the
value of ψ.
ii) For a given ψ value, the value of ψ 0 i and ψ 0 j are known based on
Equation 4.41. Calculate n 0 i and n 0 j based on Equation 4.42: bi-
section method was again employed; the initial ranges of n 0 i and
n 0 j are ni < n 0 i < mi, nj < n 0 j < mj.
iii) Substitute the values of ψ, n 0 i and n 0 j into Equation 4.43 to calcu-
late the reciprocal total amount of adsorption 1/nt.
iv) Calculate the error between the calculated and experimental re-
ciprocal total amount of adsorption, narrow the range of ψ until
a desired minimal error is obtained .
ii Substitute the value of ψ into Equation 4.39 to calculate the activity

104 CHAPTER 4. NUMERICAL MODELING
coefficients, γi and γj. Substitute the values of n 0 i and n 0 j into Equa-
tion 4.40 to calculate p 0 i and p 0 j.
iii Substitute the values of γi, γj, p 0 i and p 0 j into Equation 4.38 for the
two component and combined with relation yi + yj = 1 to solve for
the equilibrium pressure p, and the gas phase composition yi and yj.
After obtaining the values of yi, and yj, calculate the selectivity coef-
ficients Si,j based on Equation 4.13.
iv Calculate the error between the calculated p, Sij and the experimen-
tal p, Sij.
3) Repeat Step i-iv for another set of [Ao C].
4) Plot the contour of the errors in the calculated p and Sij over the ranges
of Ao and C. The values of Ao and C are those yield the minimum error
in the calculated p and Sij.
2. After the value of Ao and C are determined, follow the same procedures as
in Step i-iv to obtain the final values of ψ, ψ 0 i , ψ 0 j , n 0 i , n 0 j, p 0 i , p 0 j, γi, and γj.
The range of the value of ψ is not known intuitively. An alternative way to
conduct the calculation is to iterate on n 0 i instead of ψ in Step i, because the range
of n 0 i is well defined based on the physics of adsorption, ni < n 0 i < mi:
i) Given a value of n 0 i , ψ 0 i is calculated based on Equation 4.42. The values of ψ
and ψ 0 j are then known based on Equation 4.41.
ii) Calculate n 0 j by substituting the value of ψ 0 j into Equation 4.42: nj < n 0 j < mj,
minimize the error in the calculated ψ 0 j by updating n 0 j using the bisection
method.
iii) Substitute the value of ψ, n 0 i , and n 0 j into Equation 4.43 to calculate the recip-
rocal total amount of adsorption.
iv) Calculate the error in the calculated reciprocal total amount of adsorption,
update the value of n 0 i using the bisection method until a desired minimal
error is obtained.

4.2. SORPTION MODELING 105
MatLab programs were written to realize the algorithms, Appendix C.2. It
turned out that the results based on the two approaches were the same within
the machine error, even though it was faster to conduct the calculation by iterat-
ing on n 0 i instead of ψ in Step i. The range of Ao was narrowed down to −5 to −3,
and C between 0 and 0.25. An incremental value of 0.001 was used which yielded
501 × 2001 [Ao C] sets. Figure 4.3 shows the contour of the average error in the
calculated equilibrium pressures and selectivity coefficients. The average error
was defined as
where,
epS,ave =
�
e 2 p + e 2 S
length (pequ)
ep = |pequ,cal − pequ|
pequ
eS = |Sij,cal − Sij|
Sij
(4.44)
(4.45)
(4.46)
The values of Ao and C were chosen as those that yielded the minimum average
error. For the CO2/C2H6-NaX system under investigation, the minimum average
error is 0.0541 (5.41%) when Ao = −4.86 and C = 0.088. These values are close
to those found by Siperstein and Myers (2001): Ao = −4.36 and C = 0.11. Their
definition of errors was not stated in the paper. The values of the other calcu-
lated parameters are listed in Table 4.4. The values of the activity coefficients for
both components are less than unity. The activity coefficients for the less adsorp-
tive component (C2H6) have a more significant deviation from unity compared
with the more adsorptive component (CO2), indicating the less adsorptive com-
ponent shows more nonideality in the adsorbed phase. An activity coefficient
less than unity is called negative deviation from the Raoult’s law (ideal solution).
A negative deviation implies that the substance becomes less volatile. The ac-
tivity coefficient for the less adsorptive component is more negative than that of
the more adsorptive component. This might mean that the adsorbate covered
surface tends to adsorb more of the less adsorptive component.

106 CHAPTER 4. NUMERICAL MODELING
(a) Total amount of adsorption.
(b) Mole fraction of CO2 in the adsorbed phase.
(c) Selectivity coefficients.
Figure 4.2: Simulation results of adsorption of different CO2/N2 binary mixtures
on coal based on the extended Langmuir equations (short dashed lines) and the
IAS model (solid curves)

4.2. SORPTION MODELING 107
Figure 4.3: Contour of the average error in the calculated equilibrium pressures
and selectivity coefficients. Ao = −4.86 and C = 0.088 yielded the minimum
average error.

Table 4.4: Modeling results for adsorption of CO2(i)/C2H6(j) binary mixture on NaX, T = 20 o C. Calculation
was based on the assumptions of an ideal gas phase, a nonideal adsorbed phase, and an ideal pure condensed
phase.
pequ pequ,cal nt nt,cal Sij Sij,cal γi γj ψ n 0 i n 0 j p 0 i p 0 j
(kP a) (kP a) (mol/kg) (mol/kg) (mol/kg) (mol/kg) (kP a) (kP a)
10.62 9.68 3.717 3.717 10.81 10.50 0.9969 0.3471 10.16 3.6701 3.6623 6.5382 197.2557
12.61 11.73 3.924 3.924 10.09 9.89 0.9973 0.3251 11.06 3.8877 3.7089 8.2980 251.8454
18.22 17.72 4.087 4.087 10.55 10.67 0.9904 0.3449 11.68 4.0334 3.7356 9.7135 297.7868
21.43 21.03 4.274 4.274 10.13 10.27 0.9913 0.3246 12.60 4.2394 3.7680 12.1175 379.9876
28.65 29.25 4.406 4.406 10.54 11.04 0.9833 0.3391 13.24 4.3785 3.7868 14.0652 450.4083

4.2. SORPTION MODELING 109
If the gas phase is assumed to be nonideal, the gas-phase fugacity coefficients
can be calculated based on the method described in Chapter 2. Table 4.5 shows
the results based on the assumptions of a nonideal gas phase, a nonideal ad-
sorbed phase, and an ideal pure condensed phase. The fugacity coefficients only
have a nominal deviation from unity. The values of Ao, C (Ao = −4.865, C = 0.088)
and other parameters care very close to those calculated based on the assump-
tion of ideal gas phase. At the pressure range of the experiments, and the as-
sumption of an ideal gas phase is reasonable.
Table 4.5: Modeling results for adsorption of CO2(i)/C2H6(j) binary mixture on
NaX, T = 20 o C. Calculation was based on assumptions of a nonideal gas phase, a
nonideal adsorbed phase, and an ideal pure condensed phase.
pequ ˆϕi ˆϕj γi γj
ˆfi
ˆfj ni/ ˆ fi nj/ ˆ fj
(kPa) (kPa) (kPa) (mol/kg-kPa) (mol/kg-kPa)
10.62 0.9995 0.9992 0.9969 0.3468 6.7188 3.8944 0.5257 0.0487
12.61 0.9994 0.9990 0.9973 0.3248 8.3928 4.2077 0.4203 0.0438
18.22 0.9991 0.9986 0.9904 0.3446 9.0652 9.1335 0.4125 0.0389
21.43 0.9989 0.9983 0.9913 0.3243 11.3027 10.0982 0.3301 0.0343
28.65 0.9986 0.9978 0.9833 0.3388 12.3303 16.2656 0.3185 0.0301
Figure 4.4 shows the ni/ ˆ fi ∼ p plot for the system, ˆ fi = pyi ˆϕi. The value of
ψ equals to the summation of the areas below the curve for each component.
Besides the five points for each component calculated based on the experimental
data, the value of ni/ ˆ fi at the limit of zero pressure equals to Henry’s constant
according to Equation 2.23. As shown in Figure 4.4, the areas below the curves are
mostly determined by the shape of the curves at the low pressure ranges (close to
zero). In order to use Equation 4.37 to calculate the value of ψ, substantial binary
experimental data at the very low pressure range should be collected, which is
not an easy task in the laboratory.

110 CHAPTER 4. NUMERICAL MODELING
Figure 4.4: Plot of ni/ ˆ fi ∼ p for adsorption of CO2/C2H6 binary mixture on NaX, T
= 20 o C.
4.2.3 Adsorption of Supercritical CO2/N2 Binary Gas Mixtures on
Coal
Hall et al. (1994) conducted adsorption experiments of supercritical pure CH4,
N2, and CO2 and their binary mixtures on wet Fruitland coal (medium volatile
bituminous) at 115 o F (46 o C). The authors used various two-dimensional equa-
tions of state, the Langmuir and loading ratio corrections, and the IAS model to
calculate the total amount of adsorption. The models represented the adsorption
of pure CH4 and N2 over the whole experimental pressure range (up to around
12500 kPa or 1800 psi), pure CO2 up to moderate pressure (around 8300 kPa or
1200 psi) fairly well. The accuracy of different models varies. The models failed to
yield accurate total amount of adsorption for binary adsorption at high pressures.
The authors did not examine the ideality of the adsorbed phase by interpreting
activity coefficients.

4.2. SORPTION MODELING 111
We implemented the algorithms of adsorption calculation based on the ex-
tended Langmuir equations, the ideal adsorbed solution model, and the real ad-
sorbed solution model on the experimental data of Hall et al. (1994). The pure gas
adsorption data are plotted in Figure 4.5. Notice that when the moisture level in
the coal was above the equilibrium moisture content, the amount of adsorption
was not affected by the amount of the moisture content. Langmuir equations
were used to represent the adsorption of pure N2 and the adsorption of pure CO2
up to 8300 kPa (1200 psi). The extra-high pure CO2 adsorption data at high pres-
sures (multilayer adsorption regime) were disregarded. The values of the Lang-
muir equation constants are listed in Table 4.6.
The adsorption data for CO2/N2 binary mixtures of various composition are
tabulated in Table B.1, B.2, and B.3 for the convenience of future reference. Tak-
ing the adsorption of 60%CO2/40%N2 as an example, adsorption calculations
based on the ELM equations and IAS model were the same as in the example
of CO2/N2 binary gas adsorption on Wyoming coal in Section 4.2.2. Calculation
based on the RAS adsorption model and the ABC excess Gibbs free energy model
was similar to that in the example of CO2/C2H6 binary adsorption on NaX in Sec-
tion 4.2.2. Due to the simplicity of the form of the equation for ψ 0 i (Equation 4.20),
the calculation of n 0 i for a given value of ψ 0 i was more straightforward than that in
the system of CO2/C2H6-NaX.
p 0 i = e ψ0 i
mi − 1
Bi
n 0 i = miBip 0 i
1 + Bip 0 i
(4.47)
(4.48)
The calculated total amount of adsorption, mole fraction of CO2 in the ad-
sorbed phase, and the selectivity coefficients of CO2 to N2 are plotted together
with the experimental data in Figure 4.6. And Table 4.7 shows the simulation re-
sults based on the RAS adsorption model and the ABC excess Gibbs free energy
model.

As shown in Figure 4.6(a), calculations based on the IAS model and ELM mod-
els were capable of predicting the total amount of adsorption for the binary mix-
ture at low pressures (less than 2000 kPa); whereas, they failed to yield accurate
results at high pressures. The calculations predicted neither the adsorbed phase
composition nor the values of the selectivity coefficients accurately, Figure 4.6(b)
and 4.6(c). Experimental data indicated that for a particular feed gas, with the
increase of pressure, the values of the separation coefficients decrease. Calcula-
tions based on the IAS and ELM models, however, predicted increasing or con-
stant selectivity coefficients with increasing pressure. Calculation based on the
RAS model, on the other hand, predicted decreasing separation coefficients with
increasing pressure. The values of the separation coefficients obtained based on
the RAS model combined with the ABC model of the excess Gibbs free energy,
however, were not close to the experimental values. As described in the algorithm
in the previous section, the total amount of adsorption and the mole fraction of
the adsorbed phase were inputs in modeling based on the RAS model.
Based on the simulation results in Table 4.7, the deviation of fugacity coeffi-
cients from unity becomes greater with the increase of pressure for both CO2 and
N2 in the gas phase. The higher the pressure, the more nonideal the components
in the gas phase. Compared with CO2, N2 is closer to ideality. This is because N2
has a lower critical temperature than CO2. The deviation of the values of the ac-
tivity coefficients from unity also increases with the increase of pressure for both
CO2 and N2 in the adsorbed phase. Both CO2 and N2 have a negative deviation.
A negative deviation implies that the substance becomes less volatile. The de-
viations are larger for N2 than for CO2, which means that N2 become relatively
easier to be adsorbed to the mostly CO2-covered surface. This result agrees with
the experimental observation that with the increase of pressure the selectivity
coefficients of CO2 over N2 decreased. The less adsorptive component showed
more nonideality in the adsorbed phase. This result is consistent with that for
the CO2/C2H6-NaX system of Siperstein and Myers (2001). The value of variable
ψ increases with pressure, so does the total amount of adsorption.
112

4.2. SORPTION MODELING 113
Figure 4.5: Supercritical pure CO2 and N2 adsorption on wet Fruitland coal (T =
115 o F ). Data from Hall et al. (1994)
Table 4.6: Pure adsorption isotherm (Langmuir) constants based on experimental
data of Hall et al. (1994).
Gas m B
(mol/kg) (1/kPa)
CO2 1.5775 0.00066
N2 1.0242 0.00008

114 CHAPTER 4. NUMERICAL MODELING
(a) Total amount of adsorption.
(b) Mole fraction of CO2 in the adsorbed phase.
(c) Selectivity coefficients.
Figure 4.6: Experimental data and simulation results of 60%CO2/40%N2 adsorption
on wet Fruitland coal. Simulations are based on the extended Langmuir
equations (blue), the IAS model (magenta), and the RAS model(red); experimental
data of Hall et al. (1994) are shown as symbols.

Table 4.7: Adsorption of binary gas mixture of 60%CO2/40%N2 (i = CO2, j = N2) on wet Fruitland coal, T = 115
o F . Constants for ABC excess Gibbs free energy model: Ao = −14.55, C = 0.014.
pEquil pEquil,cal nt,ads nt,ads,cal Sij Sij,cal ˆϕi ˆϕj γi γj ψ
(kPa) (kPa) (mol/kg) (mol/kg) (mol/kg)
724.64 591.91 0.2694 0.2694 26.37 12.64 0.9773 1.0021 0.9999 0.9807 0.2961
1375.50 1120.40 0.4282 0.4282 20.70 12.66 0.9561 1.0049 0.9998 0.9677 0.5019
2712.40 2108.58 0.6409 0.6409 17.58 12.54 0.9126 1.0123 0.9997 0.9469 0.8293
4143.06 3283.94 0.7940 0.7940 15.30 12.31 0.8670 1.0226 0.9996 0.9293 1.1179
5554.42 4545.82 0.8983 0.8983 13.51 12.00 0.8235 1.0349 0.9994 0.9156 1.3519
6950.60 6079.25 0.9799 0.9799 11.78 11.68 0.7821 1.0492 0.9993 0.9040 1.5658
8323.35 7426.98 1.0376 1.0374 11.17 11.27 0.7433 1.0650 0.9992 0.8943 1.7360
9698.17 9969.54 1.1049 1.1049 9.73 11.03 0.7064 1.0827 0.9989 0.8829 1.9692
11063.33 11886.91 1.1420 1.1420 8.99 10.62 0.6719 1.1020 0.9987 0.8758 2.1152
4.2. SORPTION MODELING 115

116 CHAPTER 4. NUMERICAL MODELING
4.2.4 Adsorption of CO2/N2 Binary Gas Mixtures on Coal
As described in Chapter 3, sorption of pure CO2 and N2, and the binary mixtures
of the two were measured in the laboratory on intact dry Powder River Basin
(Montana) coal samples at room temperature (22 o C). Algorithms based on the
IAS and RAS model were implemented to model the adsorption of the binary mix-
tures of CO2/N2. The difference between our system and the system of CO2/C2H6
binary gas adsorption on NaX (Siperstein and Myers, 2001) and the system of su-
percritical CO2/N2 binary mixture adsorption on wet Fruitland coal (Hall et al.,
1994) is the pure adsorption isotherms. For CO2/C2H6 binary gas adsorption
on NaX, the modified virial equations were used for the the pure gas adsorption
isotherms for both components. For the system of supercritical CO2/N2 on Fruit-
land coal, Langmuir equations were used as the pure gas adsorption isotherms
for both components. For our system of CO2/N2 adsorption on intact Powder
River Basin (Montana) coal, the Langmuir equation was used to represent the
pure adsorption of N2, whereas, the adsorption of pure CO2 was represented
by the N-layer BET equation, Figure 4.1. Therefore, ELM equations were not a
choice for our system.
It is straightforward to implement the IAS model as described in the previ-
ous section. MatLab code of calculation based on the IAS model was included
in Appendix C.3.1. The differences between this calculation and the calculation
shown in Appendix C.1 were the pure adsorption isotherms, thus the equations
of the modified surface potential ψ 0 i , and the algorithm of calculating n 0 i given
a value of ψ 0 i . For calculation based on the RAS model, we took into account of
the nonideality of the gas phase by calculating the fugacity coefficients and the
nonideality of the adsorbed phased by calculating the activity coefficients. The
pure component condensed phase was considered ideal, i.e., p 0 i instead of f 0 i was
used in the isofugacity equations, Equation 4.30. The simulation results based on
the IAS and RAS models are plotted together with the experimental data in Figure
4.8. And Table 4.8 shows the results of the calculation based on the RAS model
and the ABC excess Gibbs free energy model.

Similar observations were made as for the system of supercritical CO2/N2 ad-
sorption on wet Fruitland coal. At high pressures, the values of the total amount
of adsorption calculated based on the IAS model differed significantly from the
experimental data, Figure 4.8(a). The calculated adsorbed phase composition
and selectivity coefficients based on the IAS model were also very different from
the experimental values, Figure 4.8(b) and Figure 4.8(c). Whereas, the calculation
based on the RAS model combined with the ABC model of the excess Gibbs free
energy yielded accurate predictions for the separation coefficients, Figure 4.8(c).
As shown in Table 4.8, fugacity coefficients deviated from unity with the increase
of pressure for both CO2 and N2 in the gas phase, indicating nonideal gas phase
at high pressures. The deviations of fugacity coefficients of CO2 from unity was
greater than those for N2, indicating that CO2 was more nonideal than N2 in the
gas phase. The calculated values of the activity coefficients also deviated from
unity with increasing pressure for both CO2 and N2 in the adsorbed phase. Again,
both CO2 and N2 had negative deviations, and the deviations were larger for N2
than for CO2. The less adsorptive N2 was more nonideal than CO2 in the adsorbed
phase.
There were significant differences between the calculated activity coefficients
of CO2 and N2. For a real multicomponent adsorption system, based on Equation
4.38:
γi
γj
= p0 j
p 0 i
ˆϕi
ˆϕj
1
Sij
(4.49)
Based on the experimental data, the separation coefficients of CO2 to N2 were in
the magnitude of one. The values of the fugacity coefficients were calculated to
be close to unity. The ratios of activity coefficients depends on the ratio of p 0 j and
p 0 i . The values of p 0 j and p 0 i were calculated based on the value of modified surface
potential ψ 0 CO2 and ψ0 N2 . Figure 4.7 is a plot of ψ0 CO2 and ψ0 N2
versus pressure based
on Equations 4.21 and 4.20. Due to the significant difference in the amount of
adsorption of pure CO2 and N2 on the coal samples, for a specific binary adsorp-
tion ψ, there was big difference between the corresponding p0 CO2 and p0N2 , Table
4.8.
117

118 CHAPTER 4. NUMERICAL MODELING
Figure 4.7: Plot of ψ 0 versus pressure for pure CO2 and N2. CO2/N2 binary gas adsorption
on intact dry Powder River Basin (Montana) coal sample at room temperature
(22 o C)

4.2. SORPTION MODELING 119
(a) Total amount of adsorption.
(b) Mole fraction of CO2 in the adsorbed phase.
(c) Selectivity coefficients.
Figure 4.8: Experimental data and simulation results for CO2/N2 adsorption on
intact dry Powder River Basin (Montana) coal at 22 o C.

Table 4.8: Adsorption of binary gas mixture of 75%CO2/25%N2 (i = CO2, j = N2) on intact dry Powder River
Basin (Montana) coal, T = 22 o C. Ao = −22.46 and C = 1.203.
pEquil pEquil,cal nt,ads nt,ads,cal Sij Sij,cal ˆϕi ˆϕj γi γj ψ p 0 i p 0 j
(kPa) (kPa) (mol/kg) (mol/kg) (mol/kg) (kPa) (kPa)
483.05 114.37 0.4545 0.4545 2.5754 2.9829 0.9763 1.0050 0.9418 0.0681 0.4080 94.26 4003.86
941.28 144.39 0.6003 0.6003 2.0632 2.0838 0.9539 1.0105 0.9075 0.0430 0.5357 130.62 6081.52
1310.16 173.98 0.7212 0.7212 1.7396 1.6917 0.9360 1.0155 0.8705 0.0314 0.6542 167.72 8537.86
1654.31 205.62 0.8193 0.8193 1.5366 1.4739 0.9194 1.0206 0.8328 0.0244 0.7744 209.00 11695.32
1999.10 252.61 0.9021 0.9021 1.4169 1.1496 0.9027 1.0265 0.8047 0.0163 0.9481 275.68 17814.59
2670.29 833.70 1.0471 1.0471 1.3353 0.9983 0.8709 1.0383 0.7277 0.0042 2.1091 998.08 205823.04
3664.73 1838.66 1.1603 1.1603 1.2168 1.6368 0.8233 1.0639 0.7344 0.0028 2.9111 1842.48 1018205.44
120 CHAPTER 4. NUMERICAL MODELING

4.3. SORPTION-INDUCED VOLUMETRIC STRAIN AND PERMEABILITY 121
4.3 Sorption-Induced Volumetric Strain and Perme-
ability
As mentioned in Chapter 2 and the beginning of this chapter, sorption-induced
volumetric strain is usually considered as a function of the amount of adsorption
or the surface potential which is in turn a function of the amount of adsorption,
Equation 4.4 and 4.5. As discussed in Chapter 3, sorption-induced volumetric
strain of a composite coal core was measured with the injection of pure CO2,
N2, and binary mixtures of the two gases at different pore pressures and a con-
stant effective confining pressure. The corresponding total amount of adsorption
was measured simultaneously. Figure 4.9(a) plots the experimental sorption-
induced volumetric strain versus the experimental total amount of adsorption.
The sorption-induced volumetric strain and the total amount of adsorption roughly
follows a linear correlation as proposed in Equation 4.4. In Section 4.2.4, we cal-
culated the modified surface potential,ψ, based on the RAS adsorption model
and the ABC excess free Gibbs energy model. Figure 4.9(b) plots the experimen-
tal sorption-induced volumetric strain the calculated modified surface poten-
tial. Since not all of the parameters in Equation 4.5 were known, the feasibility
of the equations was not verified. The modified surface potential is a function
of the total amount of adsorption as well as the adsorbed phased composition.
Given the shape of the plot, the adsorbed phase composition with the injection
of 50%CO2/50%N2 was considered as inaccurate experimental data.
The sorption induced permeability change is believed to be a consequence
of the sorption induced volumetric strain. Based on Equation 4.1, permeability
change due to sorption only (under constant effective stress) may be expressed
as
k
k0
=
�
1 + 1
φ0
� K
M
� �m − 1 ε
(4.50)
In this equation, φ0 is the initial porosity of the core, M is the bulk modulus,and
K is the constrained axial modulus, and m is a constant. These parameters are
usually considered invariants, and the sorption induced volumetric strain ε and

122 CHAPTER 4. NUMERICAL MODELING
(a) Experimental sorption-induced volumetric strain versus experimental
total amount of adsorption.
(b) Experimental sorption-induced volumetric strain versus calculated
modified surface potential, ψ.
Figure 4.9: Sorption induced volumetric strain as a function of amount of adsorption
and modified surface potential.

4.3. SORPTION-INDUCED VOLUMETRIC STRAIN AND PERMEABILITY 123
�
(k/k0) 1/m �
− 1 have a linear relationship. This is, however, not the case based on
our experimental results.
The initial helium porosity of the composite coal core used in the experiments
was about 12% when the core was first made. After injecting one gas into the core
at first escalating and then de-escalating pore pressures, the core was vacuum
evacuated prior to the injection of another test gas. The helium porosity and
permeability of the vacuumed core were measured prior to injection. It turned
out that the helium porosity and permeability of the core was not restored to
the original values by vacuum escalation, Table 4.9. The helium permeability of
the core decreased indicating permeability reduction hysteresis as discussed in
Chapter 3. Overall, the helium porosity of the core increased. The increase in the
core helium porosity is counter-intuitive. One explanation could be that the core
was physically damaged (crushed) in the process of experiments.
Table 4.9: Helium porosity and permeability prior to the injection of different
feed gases.
Feed Gas φ0 k0
(md)
50%CO2/50%N2 12.04% 18.00
Pure CO2 20.98% 12.08
Pure N2 16.95% 4.83
75%CO2/25%N2 17.65% 3.98
The bulk modulus and the constrained axial modulus of coal could also change
with adsorption (Hagin and Zoback, 2010). Therefore, the sorption induced volu-
metric strain and the sorption induced permeability reduction most likely would
not follow a linear relationship.

124 CHAPTER 4. NUMERICAL MODELING
Figure 4.10 shows a plot of term
�
(k/k0) 1/3 �
− 1 versus the experimental volu-
metric strain ε after adsorption of pure CO2, pure N2, and 75%CO2/25%N2. Vari-
able k is the permeability of the core after adsorption of a feed gas at a certain
pressure; k0 is the permeability of the core to helium prior to the injection of the
feed gas; the exponent m in the permeability-porosity correlation was taken as 3
which gives the Reiss cubic correlation, Equation 2.5. The data points for all of
the feed gases fell into a unique slope line except for these of pure CO2. The slope
changes with the choice of the value of the exponent m.
Figure 4.10: Permeability reduction versus sorption induced strain.
We also plotted the permeability reduction versus the experimental total amount
of adsorption in Figure 4.11. Same observations were made that all the data
points but the ones for pure CO2 fell into a unique slope line. It seems that with
the adsorption of pure CO2, the mechanical properties of coal might change dra-
matically even at moderate pressures.

4.4. SUMMARY OF MODELING WORK 125
Figure 4.11: Permeability reduction versus total amount of adsorption.
4.4 Summary of Modeling Work
After measuring gas sorption and the consequent volumetric and permeability
change of a composite coal core in the laboratory, efforts were made to model
the experimental results. Our modeling efforts included
1. modeling of isothermal gas (pure CO2 and N2, and the binary mixtures of
the two) sorption on coal,
2. modeling of sorption-induced volumetric strain, and
3. modeling of sorption-induced permeability change of coal.
Sorption modeling included verifying several of the adsorption isotherms pro-
posed in the literature making use of our experimental data. We found that pure
gas adsorption isotherms were represented by parametric equations very well,
such as the Langmuir equations and the N-layer BET equations. The Langmuir
equation was capable of representing the adsorption of gases such as N2 whose

126 CHAPTER 4. NUMERICAL MODELING
adsorption might occur mainly in a monolayer on the coal surface. Models that
are capable of modeling multiple layer adsorption, such as the N-layer BET equa-
tions, were better choice for the adsorption of strongly adsorbing gases, such as
CO2, at relatively high pressures.
Multicomponent adsorption was not readily represented by simple paramet-
ric equations. Multicomponent adsorption calculation (including the calculation
of the total amount of adsorption, the amount of adsorption for each compo-
nent, the composition of the adsorbed phase, and the activity coefficients of the
components in the adsorbed phase) was conducted following several different
algorithms:
1. Calculation based on the extended Langmuir equations,
2. Calculation based on the ideal adsorbed solution (IAS) model,
3. Calculation based on the real adsorbed solution (RAS) model combined
with some excess Gibbs free energy model.
Calculations based on the extended Langmuir equations and the IAS model
are straightforward. No experimental binary adsorption data are needed. The in-
puts for the calculations are the feed gas composition, the equilibrium pressure,
and the pure component gas adsorption isotherms. Implementing the extended
Langmuir equations and the IAS model to the system of the adsorption of gas
CO2/N2 on dry Powder River Basin coal (Tang et al., 2005) and the adsorption of
supercritical CO2/N2 on wet Fruitland coal (Hall et al., 1994), the following obser-
vations were made
1. At low pressures (below 700 kPa) the total amount of adsorption calculated
based on the extended Langmuir equations and the IAS model are very
close. Both the extended Langmuir equations and the IAS model predicted
the total amount of adsorption very well at the low pressure range. At high
pressure, the values of total amount of adsorption calculated based on the
IAS model are higher than those calculated based on the extended Lang-
muir equations. The ideality assumption may not be appropriate at high

4.4. SUMMARY OF MODELING WORK 127
pressures.
2. The more N2 in the gas mixtures, the more discrepancy in the results based
on the two models. This may indicate that N2 shows more nonideal behav-
ior in the adsorbed phase.
3. The selectivity factors calculated based on the extended Langmuir equa-
tions are invariant for all gas compositions and pressures. The selectivity
factors calculated based on the IAS model are functions of both gas com-
position and pressure. For a particular injection gas, the selectivity factors
increase with the increase of pressure. Whereas, the selectivity factors de-
crease with the increase of pressure based on the experimental data of Hall
et al. (1994).
4. Neither the extended Langmuir equations nor the IAS model represents the
multicomponent adsorption perfectly.
There are constraints of using the extended Langmuir equations and the IAS model.
The extended Langmuir equations can only be to calculate the amount of ad-
sorption for each species, the total amount of adsorption, and the composition
of the adsorbed phase when all of the pure compositional gases have a Langmuir
adsorption isotherm. For calculation base on the IAS model, the assumption of
ideal gas and adsorbed phase may not be appropriate especially at relatively high
pressures, for instance, the pressures encountered in coalbed methane reser-
voirs.
Multicomponent adsorption calculations can also be conducted based on the
RAS model, if experimental binary adsorption data (total amount of adsorption
and the composition of the adsorbed phase) are available. The inputs for calcu-
lations based on the RAS model are the composition of the feed gas, the equilib-
rium pressure, the total multicomponent amount of adsorption, the composition
of the adsorbed phase, and the pure adsorption isotherms of each compositional
gas. The calculations were based on the isofugacity equations. Three elements of
nonideality are included in the original isofugacity equations, Equation 4.29:

128 CHAPTER 4. NUMERICAL MODELING
1. Nonideality of the components in the gas phase indicated by nonunity fu-
gacity coefficients ˆϕi,
2. Nonideality of the components in the adsorbed phase indicated by nonunity
activity coefficients γi, and
3. Nonideality of the pure condensed phase indicated by the pure condensed
phase fugacity f 0 i .
One, two, or all three of the nonideal elements can be taken into account in mul-
ticomponent sorption calculation to achieve different level of accuracy. In our
calculations, two of the three elements of nonideality were considered: nonide-
ality of the components in the gas phase indicated by nonunity fugacity coeffi-
cients and the nonideality of the components in the adsorbed phase indicated by
nonunity activity coefficients. The calculations included calculating the fugac-
ity coefficients of the components in the gas phase based on a chosen equation
of state (EOS) and solving a eleven-equation-eleven-unknown system to obtain
the values of activity coefficients as described in Section 4.2.2. Lots of effort was
made to find efficient approaches to solve the eleven-equation-eleven-unknown
system. It was found that the choice of algorithms depends largely on the form of
the pure adsorption isotherms. In this chapter, we investigated with three cases:
1. When Langmuir equations are used to represent the pure adsorption isotherm
for both components, due to the simple form of the Langmuir equations
and the corresponding equations for ψ 0 i and ψ 0 j , variable ψ is easily solved by
minimizing the error between the calculated and experimental total amount
of adsorption by updating the value of ψ using Newton iterations.
2. When modified virial equations are used to represent pure adsorption isotherm
for both components, variable ψ is solved by minimizing the error between
the calculated and experimental total amount of adsorption by updating
the value of ψ or n 0 i using the bisection method. The results obtained by
the two approaches were very close, whereas updating n 0 i was considered a

4.4. SUMMARY OF MODELING WORK 129
better approaches because (1) the range of the value of n 0 i was well defined,
ni < n 0 i < mi, and (2) the convergence was obtained faster.
3. When the N-layer BET equation is used to represent the pure adsorption
isotherm of one component, and the Langmuir equation is used to repre-
sent pure adsorption isotherm of the other component, variable ψ is solved
by minimizing the error between the calculated and experimental total amount
of adsorption by updating the value of ψ using Newton iterations.
Factors that affect the efficiency and accuracy of multicomponent adsorption
calculations include:
1. the algorithm,
2. the accuracy of the pure adsorption isotherms, and
3. the accuracy of the experimental binary adsorption data.
The following conclusions were drawn from our sorption modeling study:
1. Pure component adsorption can be represented by parametric equations.
Langmuir equations are adequate to represent the adsorption of pure N2
at various state on various coal samples. N-layer BET equation may be a
better choice to represent the adsorption of pure CO2 especially in the high
pressure domain.
2. The extended Langmuir equations can be implemented to predict multi-
component adsorption (the amount of adsorption of each component, the
total amount of adsorption, the adsorbed phase composition, and the se-
lectivity coefficients) when all of the pure component adsorption isotherms
are well represented by the Langmuir equations. The injection gas com-
position, the equilibrium pressure, and the pure component adsorption
isotherms are the only inputs needed. No experimental binary adsorption
data are needed. The extended Langmuir equations yielded good predic-
tions for the total amount of adsorption at low pressures. However, they

130 CHAPTER 4. NUMERICAL MODELING
failed to give predictions of the selectivity coefficients that are consistent
with the experimental data.
3. The ideal adsorbed solution model can be implemented to predict multi-
component adsorption (the adsorbed phase composition, the total amount
of adsorption, the amount of adsorption for each component, and the se-
lectivity coefficients) when the gas and the adsorbed phase are considered
as ideal. Again, the only inputs needed are the injection gas composition,
the equilibrium pressure, and the pure component adsorption isotherms.
No experimental binary adsorption data are needed. The total amount of
adsorption predicted based on the IAS model was close to those based on
the extended Langmuir equations. They were good at the low pressure range.
The higher the pressure the more discrepancy between the predictions based
on the IAS model and the extended Langmuir equations, and the more dis-
crepancy between he predictions and the experimental data. This result
indicating non-Langmuir adsorption for one or all of the component gases
at high pressures and non-ideal gas and/or adsorbed phase at high pres-
sures. Another observations is that the more N2 in the gas phase the greater
the discrepancy between the predictions based on the extended Langmuir
equations and the IAS model, which might be an indication that N2 is more
non-ideal in the adsorbed phase. The adsorbed phase composition and
selectivity coefficients predictions based on the IAs model were no longer
constant for different pressures and gas compositions, but they were still
not consistent with the experimental data.
4. The real adsorbed solution model combined with the ABC excess Gibbs free
energy model was implemented in the attempt of obtaining the values of
activity coefficients for the CO2/N2-Coal system. The activity coefficients
of both CO2 and N2 moved away from unity as the adsorption pressure in-
creased. The activity coefficients of N2 showed greater deviation for unity
which might indicate the it would be relatively easier for N2 to be adsorbed

4.4. SUMMARY OF MODELING WORK 131
on the CO2 covered surface than CO2 itself. This is consistent with the ex-
perimental observation of decreasing selectivity coefficients of CO2 over N2
with the increase of pressure.
Sorption-induced volumetric strain was found following a linear relationship
with the total amount of adsorption. Based on our experimental data, sorption
induced volumetric strain is only a function of the total amount of adsorption
and independent of the adsorbed gas composition.
The attempt of describing the sorption-induced permeability evolution using
the Palmer-Mansoori permeability equation, Equation 4.50, turned out to be of
limited success.

132 CHAPTER 4. NUMERICAL MODELING

Chapter 5
Summary and Future Work
In this study, we accomplished:
1. Developed an experimental approach to measure sorption, volumetric strain,
and permeability simultaneously. A composite coal core was used.
2. Modeled pure CO2 adsorption over the whole pressure range using N-layer
BET equation.
3. Added pure and binary adsorption data of CO2/N2 on coal to the literature.
4. Investigated/developed algorithms of binary adsorption calculation based
on the extended Lamgniur eqautions, the ideal adsorbed model and the
realm adsorbed model. Obtained values of activity coefficients of CO2/N2
binary adsorption on coal.
5. Confirmed the linear relationship between sorption induced volumetric strain
and amount of adsorption.
6. Investigated the validation of the sorption-induced term of the Palmer-Mansoori
permeability equation, and verified the necessity of considering change of
the mechanical properties of coal due to gas sorption.
Further investigation in many aspects are considered to be interesting and
meaningful to fully understand gas sorption in coal, and the sorption induced
133

134 CHAPTER 5. SUMMARY AND FUTURE WORK
strain, permeability, and many other coal properties, for instance, the mechani-
cal properties of the coal. A few future work are proposed here.
5.1 Improvements on the Current Experiments
To our knowledge, the current experiments are the first attempt to measure si-
multaneously the amount of adsorption, the composition of the adsorbed phase,
and the consequent volumetric and permeability change of coal. The whole ap-
paratus worked out quite well to fulfill the purposes of the experiments. Improve-
ments, however, could be made to have better control over the system and obtain
more accurate data. The following are several improvements that can be consid-
ered if similar experiments were to be conducted in the future:
1. Put the system in a controlled-temperature environment, for instance, an
air temperature bath. Not only the pressure and volume readings but also
adsorption itself are affected by temperature (Boxho et al., 1980; Yang and
Saunders, 1985). Also, it is desirable to conduct experiments at tempera-
tures different from the laboratory temperature. Previously, experiments
were conducted in the laboratory to measure the permeability of a coal
pack after adsorption of pure CO2 at different temperatures, Figure 5.1. The
permeability of the core decreased with the increase of pore pressure at
both low and high temperatures, whereas, the permeability decreased to
a smaller extent at higher temperature. This might be because the amount
of gas adsorption decreases with the increase of temperature.
2. Use a high accuracy inline gas chromatographic analysis device. In the cur-
rent experiments, gas samples were taken using gas sampling bags, and
later analyzed by manual injection into a gas chromograph (GC). For gas
samples containing N2 and the column we used in the GC, the percentage
readings of of N2 can be inaccurate if there is O2 contamination in the gas
samples. It is better to have an inline gas chromatographic analysis device

5.2. SORPTION, PERMEABILITY, AND VOLUMETRIC CHANGE OF WET COAL135
Figure 5.1: Permeability reduction of a coal pack with adsorption of pure CO2 at
different temperatures, net overburden pressure = 400 psi (Lin, 2006).
to reduce the chance of gas contamination. Given the small amount of ad-
sorption on the available sample in the system, high accuracy in gas com-
position measurement is desirable.
3. Measure the mechanical properties of the coal separately or along with the
other measurements.
5.2 Sorption, Permeability, and Volumetric Change
of Wet Coal
Moisture content in coal below the equilibrium moisture content (EMC) affects
the adsorption behavior of coal (Joubert et al., 1974). At a certain pressure the
higher the moisture content, the less the amount of adsorption. Mohammad
et al. (2009) did experiments of pure CO2 adsorption on dry and wet Argonne

136 CHAPTER 5. SUMMARY AND FUTURE WORK
coals, they found that (1) the amount of adsorption of on wet coals is lower than
that on the dry coals, and (2) maximum adsorption occurs at a higher pressure
for the wet coals compared with the dry ones. It is believed that the moisture
content in coal affects its gas adsorption capacity because water molecules clus-
ter around the adsorptive sites on the surface and block a significant fraction
of the surface to gas adsorption (Muller and Hung, 2000). Simulation results of
Muller and Hung (2000) showed that this effect can be significant, the amount of
methane adsorption reduces more than 50% in some case. On the other hand,
water content beyond the EMC of the coal is considered having no significant
effect on adsorption (Arri and Yee, 1992; Hall et al., 1994).
One of the major challenges in conducting experiments on wet coal is to con-
trol the moisture content level of the samples throughout experiments. Adsorp-
tion experiments were conducted on wet coal samples by several researchers
(Joubert et al., 1974; Levy et al., 1997; Clarkson and Bustin, 2000; Fitzgerald et al.,
2005; Mohammad et al., 2009). Similar procedures were followed to prepare the
samples according to the standard American Society for Testing and Materials
(ASTM) procedure D1412 (ASTM, 2000): coal samples are placed in a vacuumed
desiccator in which the relative humidity is maintained at 96%-97% by water
vapor in equilibrium with a saturated solution of K2SO4 at 30 o C, the sample is
weighed periodically until a constant sample weight is achieved, the sample is
considered to reach the EMC. In the process of adsorption experiments, the sys-
tem pressure is maintained above the vapor pressure of water at the experimental
temperature to minimize moisture loss from the sample surface. The equilibrium
moisture content of the sample is determined by placing part of the sample in a
continuous vacuum at higher temperature (> 30 o C); the weight of the sample is
again monitored until a stable weight is achieved, the weight loss from the sam-
ple is used to calculated the EMC. Joubert et al. (1974); Mohammad et al. (2009)
found that the use of vacuum in the desiccator can result in condensation prob-
lems which negate the experiment, and the problem can be solved by filling the
desiccator with inert nitrogen.

5.3. SORPTION OF LIQUID CO2 ON COAL 137
5.3 Sorption of Liquid CO2 on Coal
Temperatures of coalbed methane reservoirs typically range from 21 o C to 65 o C
(Suarez-Ruiz and Crelling, 2008). The geothermal gradient of the coalbed methane
reservoirs in the Black Warrior Basin, Alabama, USA ranges from 10.9 to 36.2
o C/km, and the hydrostatic pressure gradient ranges from normal (9.73 MP a/km)
to extremely underpressured (< 1.13 MP a/km) (Pashin and McIntyre, 2003). The
critical point for CO2 is at a temperature of 31.1 o C and 7.4 MP a. Under normal
hydrostatic reservoir pressure, beyond a depth of 756 m CO2 is at supercritical
state. The liquidus for CO2 lies outside of the realm of typical reservoir condi-
tions for CBM, Figure 5.2. Liquid CO2 can present in the coal beds when injecting
CO2 into shallow coal beds whose temperature is lower than 31.1 o C. Experi-
ments on adsorption of gaseous and supercritical CO2 were conducted by some
researchers (Hall et al., 1994; Tang et al., 2005; Lin et al., 2008), whereas, adsorp-
tion of liquid CO2 is less investigated. Implementing the current experimental
setup and schemes, experiments can be conducted up to higher pressures (>6
MPa) for pure CO2.

138 CHAPTER 5. SUMMARY AND FUTURE WORK
Figure 5.2: Phase Diagram of CO2 showing temperature-pressure conditions in
CBM reservoirs of Black Warrior Basin, Alabama, USA (after Pashin and McIntyre,
2003).

Nomenclature
Roman Symbols
A area
a activity
B Langmuir constant
b Klinkenberg slip factor
Cp
pore compressibility
Cmatrix matrix swelling coefficient
d diameter
Dp
pore diameter
E Young’s modulus
Ed
activation energy of desorption
F objective function
f fugacity
G Gibbs free energy
H Henry’s constant
K bulk modulus
139

140 NOMENCLATURE
k permeability
k0
ki,j
L length
initial permeability
interaction factor
M constrained axial modulus
m maximum monolayer adsorption (moles)
N Number
n moles
n T total moles
NA
NC
ni
Avogadro constant
number of components
moles of component i
p pressure
p 0 saturation pressure
pm
pR
mean gas pressure
relative pressure
q volumetric flow rate
R universal gas constant
r radius
Rk
rm
permeability reduction
mean radius

S entropy
T temperature
U internal energy
u molar internal energy
u, w, a, b equation of state constants
V volume
v molar volume
vc
condensate molar volume
Vmatrix volume of coal matrix
W weight
x mole fraction in the adsorbed phase
y mole fraction in the gas phase
Z gas compressibility factor
Greek Symbols
χ solubility parameter
∆ change
γ activity coefficient
κ Boltzmann constant
Λ volume fraction
λ mean free path
µ chemical potential
141

142 NOMENCLATURE
ν Poissons ratio
Φ surface potential
φ porosity
φ0
initial porosity
Π spreading pressure
ψ modified surface potential
ρ density
σ effective stress
τ tortuosity
θ fractional coverage of adsorption
ε Strain
ϕ fugacity coefficient
Superscripts
0 values at standard condition
A adsorbed phase
mix mixing
T total
V vapor phase
Subscripts
0 initial values
ads adsorption

C component
c critical
des desorption
equ equilibrium
g gas
i, j component indices
ini initial
t total
Other Symbols
∞ infinity
N average number of molecular per unit volume
Acronyms
CBM coalbed methane
DV dead volume
ECBM enhanced coalbed methane recovery
ELM extended Langmuir equations
EOS equation of state
IAS ideal adsorbed solution
RAS real adsorbed solution
V AE vapor-adsorbate equilibrium
V LE vapor-liquid equilibrium
143

144 NOMENCLATURE

Appendix A
Thermodynamics of Gas Sorption on
Solid
In this section, we clarify some of the thermodynamic concepts and definitions
used in the analysis of vapor-adsorbate equilibrium, including mixing properties,
excess properties, and the standard state.
A.1 Mixing and Excess Gibbs Free Energy
The mixing property is the difference between a property of a mixture and the
weighed sum of the property of the pure constitutes at the same conditions (Walas,
1985):
M mix = M real − � xiMi
(A.1)
The excess property is the difference between the actual value of a property and
the value it would be if the solution were ideal solution at the same conditions:
where,
M ex = M real − M ideal
M ideal = � xiMi + � xi ˜ Mi
145
(A.2)
(A.3)

146 APPENDIX A. THERMODYNAMICS OF GAS SORPTION ON SOLID
For all properties except entropy and those defined in terms of entropy (for in-
stance, surface area and the Gibbs free energy), ˜ Mi = 0. For the Gibbs free energy,
˜Gi = RT ln xi.
The excess Gibbs free energy for a multicomponent adsorbed solution is
At constant temperature,
for a single component system, and
for a multicomponent system.
G ex = G real − G ideal
(A.4)
dG = RT d ln f (A.5)
d ¯ Gi = RT d ln ˆ fi
Therefore, the partial molar excess Gibbs energy, ¯ G ex
i =
expressed as
¯G ex
i = RT ln
� ˆfi(real)
ˆfi(ideal)
�
� ∂(n T G ex )
∂ni
�
T,p,nj�=i
(A.6)
, can be
(A.7)
Based on the definition of activity, Equation 2.45, the fugacity of component i in
a real solution is
ˆfi(real) = γixi ˆ f 0 i
(A.8)
where, ˆ f 0 i is the fugacity of component i in the standard state which is at the same
temperature as that of the mixture and at some arbitrary but specified pressure
and composition.
And fugacity of an ideal solution is (Prausnitz et al., 1999)
ˆfi(ideal) = xiℜi
(A.9)

A.2. THE STANDARD STATE 147
where, ℜi is a proportionality constant that is a function of temperature and pres-
sure but independent of the composition xi. Setting ℜi = f 0 i , we have
ˆfi(ideal) = xi ˆ f 0 i
(A.10)
Substitute Equation A.8 and Equation A.10 into Equation A.7, a relation between
partial molar excess Gibbs energy and activity coefficient is obtained
¯G ex
i = RT ln γi (A.11)
Based on Equation A.1, A.2, A.3, and A.11, a relation between partial molar Gibbs
free energy upon mixing activity coefficient is obtained
A.2 The Standard State
In Lewis’ definition of fugacity (Prausnitz et al., 1999)
¯G mix
i = RT ln γixi (A.12)
µi − µ 0 i = RT ln fi
f 0 i
(A.13)
µ 0 i and f 0 i is the chemical potential and fugacity of component i at some reference
state. The choice of the reference is arbitrary, however, in order to take advantage
of the iso-fugacity relationship, Equation 2.53 and Equation 2.54,
pyi ˆϕi = f 0 i γixi
(A.14)
at phase equilibrium, the standard state of the vapor (gas) and condensed (liq-
uid/adsorbed) phase must be consistent. The factor in common is the saturation
pressure (at the system temperature) of pure component i at which the fugacity

148 APPENDIX A. THERMODYNAMICS OF GAS SORPTION ON SOLID
of the vapor and condensed phase are the same, thus,
f 0 i = f sat
i
= p sat
i ϕ sat
i
(A.15)
At low to moderate pressures, ϕ sat
i is close to unity. To a first approximation, in
the case of VLE, f 0 i equals to p sat
i , the vapor pressure of the pure saturated liquid at
the temperature of the system; whereas, in the case of VAE, f 0 i equals to p 0 i which
is the equilibrium gas phase pressure at the temperature and spreading pressure
of the mixture (Myers and Prausnitz, 1965).

Appendix B
Experimental Sorption Data
Table B.1, B.2, and B.3 include the experimental sorption data of Hall et al. (1994).
The experiments were conducted on wet Fruitland coal (medium volatile bitu-
minous) sample from the San Juan Basin. The coal sample was ground to 50-230
meshes (63-300 µm). The feed gases were supercritical pure CH4, N2, CO2, and
their binary mixtures. The experimental temperature was maintained at 115 o F
(46 o C) by constant temperature air baths. No confining pressure was applied to
the sample during the experiments. The amount of adsorption was determined
gravimetrically based on mass balance.
Table B.4, B.5, B.6, and B.7 include the experimental sorption and swelling
data of our current study. The experiments were conducted on dry intact Powder
River basin coal samples. The feed gases were gas-phase pure N2, CO2, and their
binary mixtures. The experiments were conducted at room temperature (22 o C).
Confining pressure was applied to the sample during the experiments. The con-
fining pressure was changed according to the pore pressure to maintain a con-
stant net effective stress on the core. The amount of adsorption was determined
volumetrically based on mass balance.
149

150 APPENDIX B. EXPERIMENTAL SORPTION DATA
Table B.1: Adsorption of supercritical binary mixtures of CO2 and N2 on Fruitland
coal (Hall et al., 1994).
pequ nt,ads xCO2 yCO2 SCO2,N2
(kP a) (mol/kg)
(Feed Gas Composition: 80%CO2/20%N2)
837.02 0.4913 0.8467 0.6071 3.58
1427.21 0.5837 0.9765 0.6495 22.45
2712.40 0.7942 0.9771 0.6937 18.83
4145.13 0.9359 0.9681 0.7244 11.53
5579.93 1.0442 0.9672 0.7398 10.39
6909.93 1.1264 0.9588 0.7527 7.65
8319.90 1.1886 0.9675 0.7584 9.49
9608.53 1.2481 0.9615 0.7655 7.64
10993.69 1.2708 0.9522 0.7726 5.86
12356.09 1.4225 0.9561 0.7738 6.36

Table B.2: Adsorption of supercritical binary mixtures of CO2 and N2 on Fruitland
coal (continued) (Hall et al., 1994).
pequ nt,ads xCO2 yCO2 SCO2,N2
(kP a) (mol/kg)
(Feed Gas Composition: 60%CO2/40%N2)
724.64 0.2694 0.9280 0.3283 26.37
1375.50 0.4282 0.9248 0.3727 20.70
2712.40 0.6409 0.9284 0.4244 17.58
4143.06 0.7940 0.9282 0.4580 15.30
5554.42 0.8983 0.9262 0.4815 13.51
6950.60 0.9799 0.9215 0.4992 11.78
8323.35 1.0376 0.9214 0.5120 11.17
9698.17 1.1049 0.9141 0.5225 9.73
11063.33 1.1420 0.9107 0.5313 8.99
12401.60 1.1950 0.9038 0.5384 8.05
(Feed Gas Composition: 40%CO2/60%N2)
730.15 0.1795 0.8635 0.1483 36.33
1399.64 0.3027 0.8523 0.1823 25.89
2721.36 0.4876 0.8491 0.2189 20.07
4124.44 0.6242 0.8539 0.2443 18.08
5527.53 0.7280 0.8516 0.2643 15.98
6918.20 0.8100 0.8506 0.2791 14.71
8299.91 0.8770 0.8426 0.2923 12.97
9663.00 0.9350 0.8374 0.3022 11.89
11034.37 0.9870 0.8318 0.3105 10.98
12398.84 1.0420 0.8205 0.3178 9.81
151

152 APPENDIX B. EXPERIMENTAL SORPTION DATA
Table B.3: Adsorption of supercritical binary mixtures of CO2 and N2 on Fruitland
coal (continued) (Hall et al., 1994).
pequ nt,ads xCO2 yCO2 SCO2,N2
(kP a) (mol/kg)
(Feed Gas Composition: 30%CO2/70%N2)
720.50 0.1458 0.7682 0.0919 32.74
1446.52 0.2578 0.7564 0.1183 23.14
2728.26 0.4039 0.7675 0.1438 19.66
4133.41 0.5280 0.7708 0.1626 17.32
5588.89 0.6310 0.7718 0.1774 15.68
6947.16 0.7090 0.7729 0.1881 14.69
8308.18 0.7850 0.7707 0.1962 13.77
9689.89 0.8370 0.7658 0.2052 12.67
11069.53 0.8900 0.7584 0.2124 11.64
12417.46 0.9530 0.7492 0.2175 10.75
(Feed Gas Composition: 20%CO2/80%N2)
721.88 0.1070 0.6551 0.0519 34.70
1429.28 0.1959 0.6432 0.0658 25.59
2793.07 0.3280 0.6463 0.0825 20.32
4140.99 0.4280 0.6495 0.0945 17.76
5549.59 0.5240 0.6431 0.1043 15.48
6927.85 0.5940 0.6448 0.1119 14.41
8309.56 0.6570 0.6438 0.1180 13.51
9670.59 0.7190 0.6412 0.1227 12.78
11044.71 0.7670 0.6349 0.1280 11.85
12396.08 0.8220 0.6277 0.1317 11.12

Table B.4: Adsorption of pure CO2 on intact Montana coal and the consequent volumetric strain. Experiment
temperature = 22 o C.
pEquil pNetConfining nt,ads,Exp ZEquil ρgas,Equil nt,ads,Abs nt,ads,Err ∆V/Vcore ∆Vads/Vcore
(kP a) (kP a) (mol/kg) (cm 3 /mol) (mol/kg)
0 0 0
168.55 2934.09 0.5782 0.9905 14421.01 0.5794 0.21% 0.82% 0.80%
422.75 2886.73 0.7667 0.9759 5664.99 0.7707 0.53% 1.22% 1.20%
685.82 2899.45 0.9568 0.9605 3436.88 0.9652 0.87% 1.55% 1.52%
1095.32 2903.64 1.1567 0.9361 2097.28 1.1733 1.44% 1.95% 1.92%
1564.41 2917.18 1.3706 0.9073 1423.24 1.3998 2.13% 2.42% 2.37%
2075.98 2888.25 1.5763 0.8746 1033.87 1.6229 2.96% 2.85% 2.81%
2620.50 2895.30 1.7998 0.8381 784.85 1.8705 3.93% 3.37% 3.31%
3597.98 2952.04 2.1811 0.7667 522.93 2.3123 6.02% 3.76% 3.67%
4603.99 2980.24 2.5920 0.6811 363.04 2.8228 8.90% 4.47% 4.37%
5317.77 2955.94 2.8518 0.6061 279.70 3.1903 11.87% 4.57% 4.50%
4945.67 3121.20 2.6576 0.6473 321.19 2.9282 10.18%
3745.69 3149.07 2.1974 0.7551 494.71 2.3377 6.38%
2415.12 3100.69 1.7078 0.8521 865.82 1.7684 3.55%
1099.13 3382.47 1.1482 0.9359 2089.58 1.1648 1.44%
153

Table B.5: Adsorption of pure N2 on intact Montana coal and the consequent volumetric strain. Experiment
temperature = 22 o C.
pEquil pNetConfining nt,ads,Exp ZEquil ρgas,Equil nt,ads,Abs nt,ads,Err ∆V/Vcore ∆Vads/Vcore
(kP a) (kP a) (mol/kg) (cm 3 /mol) (mol/kg)
0 0 0
168.55 2934.09 0.5782 0.9905 14421.01 0.5794 0.21% 0.82% 0.41%
422.75 2886.73 0.7667 0.9759 5664.99 0.7707 0.53% 1.22% 0.80%
685.82 2899.45 0.9568 0.9605 3436.88 0.9652 0.87% 1.55% 1.10%
1095.32 2903.64 1.1567 0.9361 2097.28 1.1733 1.44% 1.95% 1.47%
1564.41 2917.18 1.3706 0.9073 1423.24 1.3998 2.13% 2.42% 1.89%
2075.98 2888.25 1.5763 0.8746 1033.87 1.6229 2.96% 2.85% 2.28%
2620.50 2895.30 1.7998 0.8381 784.85 1.8705 3.93% 3.37% 2.74%
3597.98 2952.04 2.1811 0.7667 522.93 2.3123 6.02% 4.28% 3.56%
4603.99 2980.24 2.5920 0.6811 363.04 2.8228 8.90% 5.38% 4.57%
5317.77 2955.94 2.8518 0.6061 279.70 3.1903 11.87% 6.19% 5.30%
4945.67 3121.20 2.6576 0.6473 321.19 2.9282 10.18%
3745.69 3149.07 2.1974 0.7551 494.71 2.3377 6.38%
2415.12 3100.69 1.7078 0.8521 865.82 1.7684 3.55%
1099.13 3382.47 1.1482 0.9359 2089.58 1.1648 1.44%
154 APPENDIX B. EXPERIMENTAL SORPTION DATA

Table B.6: Adsorption of 50%CO2/50%N2 binary feed gas on intact Montana coal sample and the consequent
volumetric strain. Experiment temperature = 22 o C. Component indices: i = CO2, j = N2.
pEquil nt,ads,Abs xi yi Si,j pNetConfining ∆V/Vcore ∆Vads/Vcore
(kP a) (mol/kg) (kP a)
0 0 0 0 0
512.96 0.2598 0.7244 0.4967 2.6632 2934.42 0.35% 0.31%
803.09 0.3453 0.7195 0.4810 2.7670 2851.13 0.49% 0.42%
1183.43 0.4266 0.6955 0.4752 2.5233 2815.52 0.64% 0.54%
1660.13 0.5135 0.6520 0.4715 2.1004 2821.46 0.81% 0.66%
2354.26 0.5461 0.5985 0.4712 1.6727 2816.81 0.98% 0.77%
3073.74 0.5754 0.5686 0.4653 1.5148 2786.80 1.11% 0.84%
3738.71 0.6335 0.5681 0.4568 1.5638 2811.31 1.26% 0.93%
4775.14 0.6813 0.5650 0.4425 1.6364 2809.09 1.46% 1.03%
5927.58 0.8536 0.5392 0.4321 1.5376 2918.39 1.71% 1.16%
155

Table B.7: Adsorption of 75%CO2/25%N2 binary feed gas on intact Montana coal sample and the consequent
volumetric strain. Experiment temperature = 22 o C. Component indices: i = CO2, j = N2.
pEquil nt,ads,Abs xi yi Si,j pNetConfining ∆V/Vcore ∆Vads/Vcore
(kP a) (mol/kg) (kP a)
0 0
483.05 0.4545 0.8700 0.7221 2.5754 2964.33 0.39% 0.35%
941.28 0.6003 0.8506 0.7340 2.0632 2850.83 0.62% 0.54%
1310.16 0.7212 0.8332 0.7417 1.7396 2826.70 0.91% 0.79%
1654.31 0.8193 0.8184 0.7457 1.5366 2827.28 1.04% 0.90%
1999.10 0.9021 0.8132 0.7544 1.4169 2827.23 1.17% 0.99%
2670.29 1.0471 0.8058 0.7565 1.3353 2845.51 1.48% 1.23%
3664.73 1.1603 0.8135 0.7818 1.2168 2885.29 1.91% 1.57%
3622.26 1.1038 0.8123 0.7792 1.2261 2789.86
2776.22 1.0141 0.8111 0.7678 1.2984 2946.43
2155.65 0.9053 0.8099 0.7779 1.2159 2980.95
1513.60 0.7397 0.8087 0.7757 1.2221 2967.99
893.75 0.4934 0.8075 0.7741 1.2239 2967.32
508.40 0.2821 0.8063 0.7788 1.1821 2973.46
156 APPENDIX B. EXPERIMENTAL SORPTION DATA

Appendix C
Adsorption Calculation Codes
C.1 ELM & IAS for CO2/N2-Coal System
The following are the MatLab codes for binary adsorption calculation based on
the extended Langmuir equations and the ideal adsorbed solution model. The
system under consideration was CO2/N2 binary mixture adsorption on coal. Pure
adsorption isotherm (Langmuir) inputs were based on experimental results of
Tang et al. (2005). The total amount of adsorption, the amount of adsorption for
each component, the adsorbed phased composition, and the selectivity coeffi-
cient of CO2 over N2 were calculated for different injection gas compositions and
equilibrium pressures.
157

158 APPENDIX C. ADSORPTION CALCULATION CODES
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The code c a l c u l a t e s adsorption of binary mixture of component i and j
% based on the Extended Langmuir (ELM) equations and the i d e a l adsrobed
% s o l u t i o n ( IAS ) modle
% i = CO2, j = N2
% Pure ads . isotherms based on experimental data of Tang e t . al , 2005
% L a t e s t modification on Jan . 29 , 2010
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Clear Up Workspace f o r a New Run
c l o s e a l l
c l e a r a l l
c l c
%% Inputs : Gas Phase Composition , Pressure , etc .
y i A l l = [ 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , 0 . 8 5 ] ; % Mole f r a c t i o n of comp . i in gas phase
Ngas = length ( y i A l l ) ; % Num. of d i f f e r e n t types of i n j . gases
p E q u i l A l l = 1 : 50 : 6001; % Equilibrium pressures , kPa
p E q u i l A l l = pEquilAll ’ ;
Np = length ( p E q u i l A l l ) ; % Number of pressure points
%% Inputs : Pure Adsorption Isotherms
% CO2 : Langmuir Equation
Bi = 0 . 0 0 0 6 6 ; % Langmuir constant , 1/kPa
mi = 2 . 3 5 3 3 ; % Maximum amount of adsorption , mol/ kg
n P u r e i A l l = ( mi . ∗ Bi . ∗ p E q u i l A l l ) . / (1 + Bi . ∗ p E q u i l A l l ) ;
% N2 : Langmuir Equation
Bj = 0 . 0 0 0 3 7 ; % Langmuir constant , 1/kPa
mj = 0 . 3 3 2 1 ; % Maximum amount of adsorption , mol/ kg
n P u r e j A l l = ( mj . ∗ Bj . ∗ p E q u i l A l l ) . / (1 + Bj . ∗ p E q u i l A l l ) ;
%% P l o t of \ p s i 0 i and \ p s i 0 j
p0CO2 = 1 : 50 : 6001; % Pressure range f o r \ psi CO2 ˆ0 plot , kPa
p0CO2 = p0CO2 ’ ;
psi0CO2 = mi . ∗ log ( 1 + Bi . ∗ p0CO2 ) ;

C.1. ELM & IAS FOR CO2/N2-COAL SYSTEM 159
p0N2 = 1 : 10000 : 600000001; % Pressure range f o r \ psi CO2 ˆ0 plot , kPa
p0N2 = p0N2 ’ ;
psi0N2 = mj . ∗ log ( 1 + Bj . ∗ p0N2 ) ;
f i g u r e
hl1 = l i n e ( p0N2 , psi0N2 , ’ Color ’ , ’b ’ , ’ LineWidth ’ , 2 ) ;
ax1 = gca ;
s e t ( ax1 , ’ XColor ’ , ’b ’ , ’ YColor ’ , ’b ’ )
x l i m i t s = get ( ax1 , ’XLim ’ ) ;
y l i m i t s = get ( ax1 , ’ YLim ’ ) ;
xinc = ( x l i m i t s (2) − x l i m i t s ( 1 ) ) / 1 0 ;
yinc = ( y l i m i t s (2) − y l i m i t s ( 1 ) ) / 1 0 ;
s e t ( ax1 , ’ XTick ’ , [ x l i m i t s ( 1 ) : xinc : x l i m i t s ( 2 ) ] , . . .
’ YTick ’ , [ y l i m i t s ( 1 ) : yinc : y l i m i t s ( 2 ) ] )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ \ p s i {N2} ˆ0 ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
grid on
ax2 = axes ( ’ P o s i t i o n ’ , get ( ax1 , ’ P o s i t i o n ’ ) , . . .
’ XAxisLocation ’ , ’ top ’ , . . .
’ YAxisLocation ’ , ’ r i g h t ’ , . . .
’ Color ’ , ’ none ’ , . . .
’ XColor ’ , ’ r ’ , ’ YColor ’ , ’ r ’ ) ;
hl2 = l i n e (p0CO2 , psi0CO2 , ’ Color ’ , ’ r ’ , ’ Parent ’ , ax2 , ’ LineWidth ’ , 2 ) ;
x l i m i t s = get ( ax2 , ’XLim ’ ) ;
y l i m i t s = get ( ax2 , ’ YLim ’ ) ;
xinc = ( x l i m i t s (2) − x l i m i t s ( 1 ) ) / 1 0 ;
yinc = ( y l i m i t s (2) − y l i m i t s ( 1 ) ) / 1 0 ;
s e t ( ax2 , ’ XTick ’ , [ x l i m i t s ( 1 ) : xinc : x l i m i t s ( 2 ) ] , . . .
’ YTick ’ , [ y l i m i t s ( 1 ) : yinc : y l i m i t s ( 2 ) ] )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ \ p s i {CO2} ˆ0 ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
s e t ( gcf , ’ Color ’ , [ 1 1 1 ] )
%% Binary Adsorption C a l c u l a t i o n Based on the IAS Model

160 APPENDIX C. ADSORPTION CALCULATION CODES
f o r InxGas = 1 : Ngas % I t e r a t e f o r a l l i n j . gas compositions
y i = y i A l l ( InxGas ) ;
y j = 1 − y i ;
f o r InxP = 1 : Np % I t e r a t e f o r a l l equilibrium pressure points
pEquil = p E q u i l A l l ( InxP ) ;
%% Binary Adsorption C a l c u l a t i o n Based on ELM equations
niLan = ( mi∗ Bi ∗ y i ∗ pEquil ) / (1 + Bi ∗ y i ∗ pEquil + Bj ∗ y j ∗ pEquil ) ;
njLan = ( mj∗ Bj ∗ y j ∗ pEquil ) / (1 + Bi ∗ y i ∗ pEquil + Bj ∗ y j ∗ pEquil ) ;
ntLan = niLan + njLan ; % Total amount of adsorption , mol/ kg
xiLan = niLan / ntLan ; % Mole f r a c t i o n of comp . i in ads . phase
xjLan = 1 − xiLan ;
SijLan = ( xiLan / y i ) / ( xjLan / y j ) ; % S e l e c t i v i t y f a c t o r
% R e s u l t s based on ELM
niLanAll ( InxP , InxGas ) = niLan ;
njLanAll ( InxP , InxGas ) = njLan ;
ntLanAll ( InxP , InxGas ) = ntLan ;
x i L a n A l l ( InxP , InxGas ) = xiLan ;
x j L a n A l l ( InxP , InxGas ) = xjLan ;
S i j L a n A l l ( InxP , InxGas ) = SijLan ;
%% Binary Adsorption C a l c u l a t i o n Based on the IAS Model
%% C a l c u l a t i o n of \ p s i
% I n i t i a l Guess of \ p s i :
% Mole−f r a c t i o n weighted average of \ p s i 0 i and \ p s i 0 j
p0i = pEquil ;
p0j = pEquil ;
p s i 0 i = mi ∗ log ( 1 + Bi ∗ p0i ) ;
p s i 0 j = mj ∗ log ( 1 + Bj ∗ p0j ) ;
p s i = y i ∗ p s i 0 i + y j ∗ p s i 0 j ;
% Error Tolerance , etc .
xSumErrTol = 1e−6;
xSumErr = 1 ;
psiIterNum = 1 ;
while abs ( xSumErr ) > xSumErrTol

C.1. ELM & IAS FOR CO2/N2-COAL SYSTEM 161
end
% C a l c u l a t e p0i and pj0 based on the estimated \ p s i
p0i = ( exp ( p s i /mi ) − 1 ) / Bi ;
p0j = ( exp ( p s i /mj ) − 1 ) / Bj ;
% Objective function : F = ( x i + x j ) − 1
x i = pEquil ∗ y i / p0i ;
x j = pEquil ∗ y j / p0j ;
xSumErr = ( x i + x j ) − 1 ;
% D e r i v a t i v e of o b j e c t i v e function in terms of \ p s i
n0i = ( mi∗ Bi ∗ p0i ) / (1+ Bi ∗ p0i ) ;
n0j = ( mj∗ Bj ∗ p0j ) / (1+ Bj ∗ p0j ) ;
dFSumxdpsi = −(( pEquil ∗ y i ) / ( p0i ∗ n0i ) + ( pEquil ∗ y j ) / ( p0j ∗ n0j ) ) ;
% Newton update of v a r i a b l e \ p s i
p s i = p s i − xSumErr/dFSumxdpsi ;
psiIterNum = psiIterNum +1;
%% C a l c u l a t i o n based on \ p s i
% p0i and p0j
p0i = ( exp ( p s i /mi ) − 1 ) / Bi ;
p0j = ( exp ( p s i /mj ) − 1 ) / Bj ;
%x i and x j
x i = pEquil ∗ y i / p0i ;
x j = pEquil ∗ y j / p0j ;
% p s i 0 i and p s i 0 j
p s i 0 i = mi ∗ log (1+ Bi ∗ p0i ) ;
p s i 0 j = mj ∗ log (1+ Bj ∗ p0j ) ;
% n0i and n0j
n0i = ( mi∗ Bi ∗ p0i ) / (1+ Bi ∗ p0i ) ;
n0j = ( mj∗ Bj ∗ p0j ) / (1+ Bj ∗ p0j ) ;
% nt , ni , nj ans S i j
nt = 1 / ( x i / n0i + x j / n0j ) ;
ni = nt ∗ x i ;
nj = nt ∗ x j ;
S i j = ( x i / y i ) / ( x j / y j ) ;
% R e s u l t s based on IAS model
n i A l l ( InxP , InxGas ) = ni ;

162 APPENDIX C. ADSORPTION CALCULATION CODES
end
end
n j A l l ( InxP , InxGas ) = nj ;
n t A l l ( InxP , InxGas ) = nt ;
x i A l l ( InxP , InxGas ) = x i ;
x j A l l ( InxP , InxGas ) = x j ;
S i j A l l ( InxP , InxGas ) = S i j ;
p s i A l l ( InxP , InxGas ) = p s i ;
p s i 0 i A l l ( InxP , InxGas ) = p s i 0 i ;
p s i 0 j A l l ( InxP , InxGas ) = p s i 0 j ;
%% P l o t s of R e s u l t s
% t o t a l amount of adsorption v . s . equilibrium pressure
f i g u r e
plot ( pEquilAll , nPureiAll , ’b−− ’ , pEquilAll , nPurejAll , ’ r−− ’ , . . .
hold on
’ LineWidth ’ , 2 . 5 )
plot ( pEquilAll , n t A l l ( : , 1 ) , ’ r ’ , . . .
pEquilAll , n t A l l ( : , 2 ) , ’ g ’ , . . .
pEquilAll , n t A l l ( : , 3 ) , ’m’ , . . .
pEquilAll , n t A l l ( : , 4 ) , ’b ’ , . . .
pEquilAll , ntLanAll ( : , 1 ) , ’ : r ’ , . . .
pEquilAll , ntLanAll ( : , 2 ) , ’ : g ’ , . . .
pEquilAll , ntLanAll ( : , 3 ) , ’ :m’ , . . .
pEquilAll , ntLanAll ( : , 4 ) , ’ : b ’ , . . .
’ LineWidth ’ , 2 )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ Total Amount of Adsorption , mol/ kg ’ , . . .
grid on
’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
% mole f r a c t i o n of comp . i in the ads . phase v . s . equilibrium pressure
f i g u r e
plot ( pEquilAll , x i A l l ( : , 1 ) , ’ r ’ , . . .
pEquilAll , x i A l l ( : , 2 ) , ’ g ’ , . . .

C.1. ELM & IAS FOR CO2/N2-COAL SYSTEM 163
pEquilAll , x i A l l ( : , 3 ) , ’m’ , . . .
pEquilAll , x i A l l ( : , 4 ) , ’b ’ , . . .
pEquilAll , x i L a n A l l ( : , 1 ) , ’ : r ’ , . . .
pEquilAll , x i L a n A l l ( : , 2 ) , ’ : g ’ , . . .
pEquilAll , x i L a n A l l ( : , 3 ) , ’ :m’ , . . .
pEquilAll , x i L a n A l l ( : , 4 ) , ’ : b ’ , . . .
’ LineWidth ’ , 2 )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ x {CO2} ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
grid on
% s e l e c t i v i t y f a c t o r s v . s . equilibrium pressure
f i g u r e
plot ( pEquilAll , S i j A l l ( : , 1 ) , ’ r ’ , . . .
pEquilAll , S i j A l l ( : , 2 ) , ’ g ’ , . . .
pEquilAll , S i j A l l ( : , 3 ) , ’m’ , . . .
pEquilAll , S i j A l l ( : , 4 ) , ’b ’ , . . .
pEquilAll , S i j L a n A l l ( : , 1 ) , ’ : r ’ , . . .
pEquilAll , S i j L a n A l l ( : , 2 ) , ’ : g ’ , . . .
pEquilAll , S i j L a n A l l ( : , 3 ) , ’ :m’ , . . .
pEquilAll , S i j L a n A l l ( : , 4 ) , ’ : b ’ , . . .
’ LineWidth ’ , 2 )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ S {CO2, N2} ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
grid on
% f i g u r e
% point1 = 0 : 0 . 5 : 5 ;
% point2 = 0 : 0 . 5 : 5 ;
% p l o t ( point1 , point2 , ’−−k ’ )
% hold on
% p l o t ( p s i 0 i A l l ( : , 1 ) , p s i 0 j A l l ( : , 1 ) , ’∗ r ’ , . . .
% p s i 0 i A l l ( : , 2 ) , p s i 0 j A l l ( : , 2 ) , ’ ˆ g ’ , . . .
% p s i 0 i A l l ( : , 3 ) , p s i 0 j A l l ( : , 3 ) , ’dm’ , . . .
% p s i 0 i A l l ( : , 4 ) , p s i 0 j A l l ( : , 4 ) , ’ sb ’ , . . .
% ’ LineWidth ’ , 2 )
% x l a b e l ( ’ \ p s i ˆ0 i ( IAS ) ’ )

164 APPENDIX C. ADSORPTION CALCULATION CODES
% y l a b e l ( ’ \ p s i ˆ0 j ( IAS ) ’ )
C.2 RAS for CO2/C2H6-NaX System
The following are the MatLab codes for binary adsorption calculation based on
the real adsorbed solution model for a CO2/C2H6-NaX system. The calculation
was based on the experimental data and algorithms of Siperstein and Myers (2001).
The main function
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The code c a l c u l a t e s adsorption of binary mixture of component i and j
% based on the algorithm described by S i p e r s t e i n and Myers (2001)
% i = CO2, j = C2H6
% Pure adsorption isotherms are represented by modified v i r i a l equations
% Lates t modi f i c a t ion on Jan . 30 , 2010
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Clear Up Workspace f o r a New Run
c l o s e a l l ;
c l e a r a l l ;
c l c ;
%% InputsPure Gas Adsorption Isotherms
% CO2 : the modified V i r i a l equation
Hi = 2 7 . 2 5 3 ; % mol / ( kg∗kPa )
C1i = 1 . 2 3 3 8 ;
C2i = −0.1241;
C3i = 0 . 0 0 3 8 ;
C4i = 0 . 0 ;
mi = 6 . 4 6 7 4 ; % mol/ kg
% C2H6 : the modified V i r i a l equation
Hj = 0 . 1 5 4 5 ; % mol / ( kg∗kPa )
C1j = −0.2670;
C2j = −0.0499;
C3j = 0 . 0 1 9 2 ;

C.2. RAS FOR CO2/C2H6-NAX SYSTEM 165
C4j = 0 ;
mj = 3 . 8 9 3 7 ; % mol/ kg
%% Inputs : Experimental Binary Adsorption Data
T = 294; % Experiment temperature , deg K
RABC = 8.314472 ∗ 10ˆ( −3); % Universal gas constant used in
% the ABC excess Gibbs f r e e energy model , k J K −1 mol −1
REOS = 8.314472 ∗ 1 0 ˆ 3 ; % Universal gas constant used in gas EOS ,
load CO2 C2H6 NaX . t x t ;
% f o r f u g a c i t y c o e f f i c i e n t c a l c u l a t i o n , cm3 kPa K −1 mol −1
p E q u i l A l l = CO2 C2H6 NaX ( : , 1 ) ; % Equilibrium pressure , kPa
Np = length ( p E q u i l A l l ) ; % Number of equilibrium pressure points
ntAdsAll = CO2 C2H6 NaX ( : , 2 ) ; % Total amount of adsorption , mol/ kg
x i A l l = CO2 C2H6 NaX ( : , 3 ) ; % Mole f r a c t i o n of comp . i in the ads . phase
y i A l l = CO2 C2H6 NaX ( : , 4 ) ; % Mole f r a c t i o n of comp . i in the gas phase
S i j A l l = CO2 C2H6 NaX ( : , 8 ) ; % S e l e c t i v i t y c o e f f i c i e n t s of i to j
x j A l l = 1 − x i A l l ; % Mole f r a c t i o n of comp . j in the ads . phase
y j A l l = 1 − y i A l l ; % Mole f r a c t i o n of comp . j in the gas phase
n i A d s A l l = ntAdsAll . ∗ x i A l l ; % Amount of adsorption of comp . i
n j A d s A l l = ntAdsAll . ∗ x j A l l ; % Amount of adsorption of comp . j
%% C a l c u l a t i o n of Ao and C : Minimizing the Error between the Calculated
%% and Experimental Pressures and S e l e c t i v i t y C o e f f i c i e n t s
% Range of the values of Ao and C
[ Ao Vector , C Vector ] = meshgrid(−5 : 0.001 : −3, 0 : 0.001 : 0 . 2 5 ) ;
[CNum, AoNum] = s i z e ( Ao Vector ) ;
% C a l c u l a t i o n based on a s p e c i f i c s e t of [ Ao C]
f o r i = 1 : CNum;
f o r j = 1 : AoNum;
Ao = Ao Vector ( i , j ) ;
C = C Vector ( i , j ) ;

166 APPENDIX C. ADSORPTION CALCULATION CODES
end
end
CalAoC % D irect to code CalAoC .m
T o t a l E r r o r ( i , j ) = sum(sum( pSError ) ) / length ( pSError )
% Error contour and values of Ao and C
f i g u r e
[ S , h ] = contour ( C Vector , Ao Vector , T o t a l E r r o r ) ;
s e t ( h , ’ ShowText ’ , ’on ’ , ’ TextStep ’ , get ( h , ’ LevelStep ’ ) ∗ 2 ) ;
colormap cool ;
[ MinVal1 , ColInd ] = min(min( T o t a l E r r o r ) ) ;
[ MinVal2 , RowInd ] = min(min( TotalError ’ ) ) ;
Ao = Ao Vector ( RowInd , ColInd )
C = C Vector ( RowInd , ColInd )
%% C a l c u l a t i o n a f t e r Obtained the Values of Ao and C
CalAoC % D irect to code CalAoC .m
ResultTable = [ pEquilAll , pEquilCalAll ’ , ntAdsAll , ntAdsCal ’ , . . .
S i j A l l , S i j C a l A l l ’ , gamaiAll ’ , gamajAll ’ , P s i A l l ’ , . . .
n 0 i A l l ’ , n 0 j A l l ’ , p 0 i A l l ’ , p 0 j A l l ’ ]
Code block: CalAoC.m:
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This code i s the c a l c u l a t i o n based on a s p e c i f i c s e t of [ Ao C]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
f o r InxP = 1 : Np; % I t e r a t e over every equilibrium pressure point
% Experimental data
pEquil = p E q u i l A l l ( InxP ) ; % Equilibrium pressure , kPa
ntAds = ntAdsAll ( InxP ) ; % Total amount of adsorption , mol/ kg
x i = x i A l l ( InxP ) ; % Mole f r a c t i o n of comp . i in ads . phase
y i = y i A l l ( InxP ) ; % Mole f r a c t i o n of comp . i in gas phase
x j = x j A l l ( InxP ) ; % Mole f r a c t i o n of comp . j in ads . phase

C.2. RAS FOR CO2/C2H6-NAX SYSTEM 167
y j = y j A l l ( InxP ) ; % Mole f r a c t i o n of comp . j in gas phase
niAds = n i A d s A l l ( InxP ) ; % Amount of adsorption of comp . i
njAds = n j A d s A l l ( InxP ) ; % Amount of adsorption of comp . j
S i j = S i j A l l ( InxP ) ; % S e l e c t i v i t y c o e f f i c i e n t s of i to j
%% C a l c u l a t e P s i using B i s e c t i o n Method : miminizing ( ntExp − ntCal )
% Error t o l e r a n c e
ntErrTol = 1e−10;
% I n i t i a l ” bracket ”
P s i 1 = 1e−6;
P s i 2 = pEquil ;
P s i 0 = ( P s i 1 + P s i 2 ) / 2 ;
% Maximum number of i t e r a t i o n
PsiIterMax = log2 ( abs ( P s i 1 − P s i 2 ) / ntErrTol ) ;
PsiNumIter = 1 ; % Number of i t e r a t i o n s
f o r PsiNumIter = 1 : PsiIterMax
% C a l c u l a t i o n based on the value of P s i 1
n0i 1 = n0 ( Psi 1 , mi , C1i , C2i , C3i , C4i ) ;
n 0 j 1 = n0 ( Psi 1 , mj , C1j , C2j , C3j , C4j ) ;
E r r n t 1 = ntAds − 1 / . . .
( x i / n0i 1 + x j / n 0 j 1 + (C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ P s i 1 ) ) ;
% C a l c u l a t i o n based on the value of P s i 2
n0i 2 = n0 ( Psi 2 , mi , C1i , C2i , C3i , C4i ) ;
n 0 j 2 = n0 ( Psi 2 , mj , C1j , C2j , C3j , C4j ) ;
E r r n t 2 = ntAds − 1 / . . .
( x i / n0i 2 + x j / n 0 j 2 + (C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ P s i 2 ) ) ;
% C a l c u l a t i o n based on the value of P s i 0
n0i 0 = n0 ( Psi 0 , mi , C1i , C2i , C3i , C4i ) ;
n 0 j 0 = n0 ( Psi 0 , mj , C1j , C2j , C3j , C4j ) ;
E r r n t 0 = ntAds − 1 / . . .
( x i / n0i 0 + x j / n 0 j 0 + (C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ P s i 0 ) ) ;
% Update the value of P s i
i f abs ( E r r n t 0 ) < ntErrTol ;
P s i = P s i 0 ;
e l s e i f E r r n t 1 ∗ E r r n t 0 < 0 ;
P s i 2 = P s i 0 ;

168 APPENDIX C. ADSORPTION CALCULATION CODES
end
end
P s i 0 = ( P s i 1 + P s i 2 ) / 2 ;
P s i = P s i 0 ;
e l s e E r r n t 0 ∗ E r r n t 2 < 0 ;
end
P s i 1 = P s i 0 ;
P s i 0 = ( P s i 1 + P s i 2 ) / 2 ;
P s i = P s i 0 ;
PsiNumIter = PsiNumIter +1;
%% C a l c u a l t i o n a f t r e Obtained the Value of P s i
% n0i , n0j
n0i = n0 ( Psi , mi , C1i , C2i , C3i , C4i ) ;
n0j = n0 ( Psi , mj , C1j , C2j , C3j , C4j ) ;
% \ p s i ˆ0 i and \ p s i ˆ0 j
P s i 0 i = ( 1 / 2 ) ∗ C1i∗ n0i ˆ2 + ( 2 / 3 ) ∗ C2i∗ n0i ˆ3 . . .
+ ( 3 / 4 ) ∗ C3i∗ n0i ˆ4 + ( 4 / 5 ) ∗ C4i∗ n0i ˆ5 − mi∗ log (1− n0i /mi ) ;
P s i 0 j = ( 1 / 2 ) ∗ C1j ∗ n0j ˆ2 + ( 2 / 3 ) ∗ C2j ∗ n0j ˆ3 . . .
% nt
+ ( 3 / 4 ) ∗ C3j ∗ n0j ˆ4 + ( 4 / 5 ) ∗ C4j ∗ n0j ˆ5 − mj∗ log (1− n0j /mj ) ;
ntAdsCal = 1 / ( x i / n0i + x j / n0j + (C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ P s i ) ) ;
% p0i , p0j
p0i = ( n0i /Hi ) ∗ ( mi / ( mi−n0i ) ) . . .
∗ exp ( C1i∗ n0i + C2i∗ n0i ˆ2 + C3i∗ n0i ˆ3 + C4i∗ n0i ˆ4 ) ;
p0j =( n0j /Hj ) ∗ ( mj / ( mj−n0j ) ) . . .
∗ exp ( C1j ∗ n0j + C2j ∗ n0j ˆ2 + C3j ∗ n0j ˆ3 + C4j ∗ n0j ˆ4 ) ;
% A c t i v i t y C o e f f i c i e n t s
gamai = exp ( Ao∗( 1−exp(−C∗ P s i ) )∗ x j ˆ 2 / (RABC∗T ) ) ;
gamaj = exp ( Ao∗( 1−exp(−C∗ P s i ) )∗ x i ˆ 2 / (RABC∗T ) ) ;
% Equilibrium Pressure and S e l e c t i v i t y C o e f f i c i e n t
pEquilCal = p0i ∗gamai∗ x i + p0j ∗gamaj∗ x j ;
S i j C a l = ( p0j ∗gamaj ) / ( p0i ∗gamai ) ;
%% Normalized E r r o r s
pError ( InxP ) = abs ( pEquilCal − pEquil ) / pEquil ;
SError ( InxP ) = abs ( S i j C a l − S i j ) / S i j ;

C.2. RAS FOR CO2/C2H6-NAX SYSTEM 169
end
pSError ( InxP ) = s q r t ( pError ( InxP ) ˆ 2 + SError ( InxP ) ˆ 2 ) ;
%% Simulation R e s u l t s
p E q u i l C a l A l l ( InxP ) = pEquilCal ;
S i j C a l A l l ( InxP ) = S i j C a l ;
gamaiAll ( InxP ) = gamai ;
gamajAll ( InxP ) = gamaj ;
P s i A l l ( InxP ) = P s i ;
P s i 0 i A l l ( InxP ) = P s i 0 i ;
P s i 0 j A l l ( InxP ) = P s i 0 j ;
n 0 i A l l ( InxP ) = n0i ;
n 0 j A l l ( InxP ) = n0j ;
ntAdsCal ( InxP ) = ntAdsCal ;
p 0 i A l l ( InxP ) = p0i ;
p 0 j A l l ( InxP ) = p0j ;
Function n0.m: calculate the value of n 0 i based on known value of ψ 0 i
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This code i s used to c a l c u l a t e n0 given value of Psi0
% using b i s e t i o n method .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
function f = n0 ( Psi , m, C1 , C2 , C3 , C4 )
% I n i t i a l range of n0
n0 1 = 1e−6;
n0 2 = m − (1 e −6); % nAds < n0 < m
n0 0 = ( n0 1 + n0 2 ) / 2 ;
% Error t o l e r a n c e
P s i E r r T o l = 1e−10;
% Maximun nuber of i t e r a t i o n
n0IterMax = log2 ( abs ( n0 1 − n0 2 ) / P s i E r r T o l ) ;
% Number of i t e r a t i o n s
n0NumIter = 1 ;
f o r n0NumIter = 1 : n0IterMax

170 APPENDIX C. ADSORPTION CALCULATION CODES
end
% C a l c u l a t i o n # 1 : n0 = n0 1
P s i 0 1 = ( 1 / 2 ) ∗ C1 ∗ n0 1 ˆ2 + ( 2 / 3 ) ∗ C2 ∗ n0 1 ˆ3 . . .
+ ( 3 / 4 ) ∗ C3 ∗ n0 1 ˆ4 + ( 4 / 5 ) ∗ C4 ∗ n0 1 ˆ5 . . .
− m ∗ log ( 1 − n0 1 /m ) ;
E r r P s i 0 1 = P s i 0 1 − P s i ;
% C a l c u l a t i o n # 2 : n0 = n0 2
P s i 0 2 = ( 1 / 2 ) ∗ C1 ∗ n0 2 ˆ2 + ( 2 / 3 ) ∗ C2 ∗ n0 2 ˆ3 . . .
+ ( 3 / 4 ) ∗ C3 ∗ n0 2 ˆ4 + ( 4 / 5 ) ∗ C4 ∗ n0 2 ˆ5 . . .
− m ∗ log ( 1 − n0 2 /m ) ;
E r r P s i 0 2 = P s i 0 2 − P s i ;
% C a l c u l a t i o n # 3 : n0 = n0 0
P s i 0 0 = ( 1 / 2 ) ∗ C1 ∗ n0 0 ˆ2 + ( 2 / 3 ) ∗ C2 ∗ n0 0 ˆ3 . . .
+ ( 3 / 4 ) ∗ C3 ∗ n0 0 ˆ4 + ( 4 / 5 ) ∗ C4 ∗ n0 0 ˆ5 . . .
− m ∗ log ( 1 − n0 0 /m ) ;
E r r P s i 0 0 = P s i 0 0 − P s i ;
% Update the value of n0
i f abs ( E r r P s i 0 0 ) < P s i E r r T o l ;
n0 = n0 0 ;
e l s e i f E r r P s i 0 1 ∗ E r r P s i 0 0 < 0 ;
end
n0 2 = n0 0 ;
n0 0 = ( n0 1 + n0 2 ) / 2 ;
n0 = n0 0 ;
e l s e E r r P s i 0 2 ∗ E r r P s i 0 0 < 0 ;
end
n0 1 = n0 0 ;
n0 0 = ( n0 1 + n0 2 ) / 2 ;
n0 = n0 0 ;
n0NumIter = n0NumIter +1;
f = n0 ; % Return r e s u l t of n0

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 171
C.3 IAS & RAS for CO2/N2-Coal System
The following are the MatLab codes for binary adsorption calculation based on
the ideal and real adsorbed solution models. The system under consideration
was 75%CO2/25%N2 binary gas mixture adsorption on dry intact PRB (Montana)
coal. Pure adsorption isotherm (N-layer BET & Langmuir) inputs were based on
experimental results of the current study.
C.3.1 Binary Adsorption Calculation Based on the Ideal Adsorbed
Solution Model
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This code c a l c u l a t e s adsorption of binary mixtures of i and j
% based on the IAS Modle
% i = CO2, j = N2 , adsorbent = dry i n t a c t PRB ( Montana ) coal
% Adsorption isotherm of pure CO2 : N−l a y e r BET equation
% Adsorption isotherm of pure N2 : Langmuir equation
% Pressure in the unit of kPa , temperature in the unit of dge K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Clear Up the Work Space f o r a New Run
c l o s e a l l
c l e a r a l l
c l c
%% Inputs : Pure Adsorption Isotherms
pPure = 1 : 5 0 : 6 0 0 1 ; % Pressure range f o r pure ads . isotherm plots , kPa
% CO2 : N−l a y e r BET equation
Ci = 2 7 . 0 9 ; % Constant
mi = 1 . 1 3 6 ; % Maximum monolayer adsorption , mol/ kg
Ni = 5 . 1 6 4 ; % Total allowed number of adsorption l a y e r s
p S a t i = 6 0 0 3 . 0 6 ; % S a t u r a t i o n pressure of CO2 at room temperature , kPa
pRi = pPure . / p S a t i ; % Reduced pressure

172 APPENDIX C. ADSORPTION CALCULATION CODES
n P u r e i A l l = ( ( mi . ∗ Ci . ∗ pRi )./(1 − pRi ) ) . ∗ . . .
( (1 − ( Ni + 1 ) . ∗ pRi . ˆ Ni + Ni . ∗ pRi . ˆ ( Ni + 1 ) ) . / . . .
(1 + ( Ci −1).∗ pRi − Ci . ∗ pRi . ˆ ( Ni + 1 ) ) ) ;
% N2 : Langmuir equation
Bj = 0 . 0 0 0 3 1 ; % Langmuir constant , 1/kPa
mj = 0 . 5 0 5 6 ; % Maximum amount of ads . at i n f i n i t e pressure , mol/ kg
n P u r e j A l l = ( mj . ∗ Bj . ∗ pPure ) . / (1 + Bj . ∗ pPure ) ;
%% P l o t of \ p s i 0 i and \ p s i 0 j in the Same Figure
pCO2 = 1 : 50 : 6001; % Pressure range f o r \ p s i {CO2}ˆ0 plot , kPa
pR0i = pCO2 . / p S a t i ; % Reduced pressure
p s i 0 i = mi . ∗ log ( (−1+pR0i−Ci . ∗ pR0i+Ci . ∗ pR0i . ˆ ( Ni + 1 ) ) . / (−1+pR0i ) ) ;
pN2 = 1 : 1000 : 120000001; % Pressure range f o r \ p s i {N2}ˆ0 plot , kPa
p s i 0 j = mj . ∗ log ( 1 + Bj . ∗ pN2 ) ;
f i g u r e
% p l o t of \ p s i {N2}ˆ0
hl1 = l i n e (pN2 , p s i 0 j , ’ Color ’ , ’b ’ , ’ LineWidth ’ , 2 ) ;
ax1 = gca ;
s e t ( ax1 , ’ XColor ’ , ’b ’ , ’ YColor ’ , ’b ’ )
x l i m i t s = get ( ax1 , ’XLim ’ ) ;
y l i m i t s = get ( ax1 , ’ YLim ’ ) ;
xinc = ( x l i m i t s (2) − x l i m i t s ( 1 ) ) / 1 0 ;
yinc = ( y l i m i t s (2) − y l i m i t s ( 1 ) ) / 1 0 ;
s e t ( ax1 , ’ XTick ’ , [ x l i m i t s ( 1 ) : xinc : x l i m i t s ( 2 ) ] , . . .
’ YTick ’ , [ y l i m i t s ( 1 ) : yinc : y l i m i t s ( 2 ) ] )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ \ p s i {N2} ˆ0 ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
grid on
ax2 = axes ( ’ P o s i t i o n ’ , get ( ax1 , ’ P o s i t i o n ’ ) , . . .
’ XAxisLocation ’ , ’ top ’ , . . .
’ YAxisLocation ’ , ’ r i g h t ’ , . . .
’ Color ’ , ’ none ’ , . . .
’ XColor ’ , ’ r ’ , ’ YColor ’ , ’ r ’ ) ;

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 173
% P l o t of \ p s i {CO2}ˆ0
hl2 = l i n e (pCO2, p s i 0 i , ’ Color ’ , ’ r ’ , ’ Parent ’ , ax2 , ’ LineWidth ’ , 2 ) ;
x l i m i t s = get ( ax2 , ’XLim ’ ) ;
y l i m i t s = get ( ax2 , ’ YLim ’ ) ;
xinc = ( x l i m i t s (2) − x l i m i t s ( 1 ) ) / 1 0 ;
yinc = ( y l i m i t s (2) − y l i m i t s ( 1 ) ) / 1 0 ;
s e t ( ax2 , ’ XTick ’ , [ x l i m i t s ( 1 ) : xinc : x l i m i t s ( 2 ) ] , . . .
’ YTick ’ , [ y l i m i t s ( 1 ) : yinc : y l i m i t s ( 2 ) ] )
x l a b e l ( ’ Equilibrium Pressure , kPa ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
y l a b e l ( ’ \ p s i {CO2} ˆ0 ’ , ’ f o n t s i z e ’ , 1 6 , ’ fontweight ’ , ’b ’ )
s e t ( gcf , ’ Color ’ , [ 1 1 1 ] )
%% Inputs : Experimental Binary Adsorption Data
T = 273.15 + 2 2 ; % Experiment temperature , deg K
RABC = 8.314472 ∗ 10ˆ( −3); % Universal gas constant
% used in the ABC excess Gibbs f r e e energy model , k J K −1 mol −1
load CO275Coal . t x t ;
p E q u i l A l l = CO275Coal ( : , 1 ) ; % Equilibrium pressure , kPa
ntAdsAll = CO275Coal ( : , 2 ) ; % Total amount of adsorption , mol/ kg
x i A l l = CO275Coal ( : , 3 ) ; % Mole f r a c t i o n of comp . i in ads . phase
y i A l l = CO275Coal ( : , 4 ) ; % Mole f r a c t i o n of comp . i in gas phase
S i j A l l = CO275Coal ( : , 5 ) ; % Experimental separation c o e f f i c i e n t
x j A l l = 1 − x i A l l ; % Mole f r a c t i o n of comp . j in ads . phase
y j A l l = 1 − y i A l l ; % Mole f r a c t i o n of comp . j in gas phase
n i A d s A l l = ntAdsAll . ∗ x i A l l ; % Amount of adsorption of comp . i , mol/ kg
n j A d s A l l = ntAdsAll . ∗ x j A l l ; % Amount of adsorption of comp . j , mol/ kg
%% Binary Adsorption C a l c u l a t i o n Based on the IAS Model
% Error Tolerance f o r D i f f e r e n t V a r i a b l e s
p s i 0 i E r r T o l = 1e−5; % Modified s u r f a c e p o t e n t i a l
xSumErrTol = 1e−6; % xSum = x i + x j
f o r CountP = 1 : length ( p E q u i l A l l )

174 APPENDIX C. ADSORPTION CALCULATION CODES
% Experimental−Related Parameters
pEquil = p E q u i l A l l ( CountP ) ; % Equilibrium pressure , kPa
ntAds = ntAdsAll ( CountP ) ; % Total amount of adsorption , mol/ kg
x i = x i A l l ( CountP ) ; % Mole f r a c t i o n of i in adsorbed phase
y i = y i A l l ( CountP ) ; % Mole f r a c t i o n of i in gas phase
x j = x j A l l ( CountP ) ; % Mole f r a c t i o n of j in adsorbed phase
y j = y j A l l ( CountP ) ; % Mole f r a c t i o n of j in gas phase
niAds = n i A d s A l l ( CountP ) ; % Amount of adsorption of comp . i , mol/ kg
njAds = n j A d s A l l ( CountP ) ; % Amount of adsorption of comp . j , mol/ kg
S i j = S i j A l l ( CountP ) ; % Separation c o e f f i c i e n t
%% Solve f o r \ p s i
% I n i t i a l Guesses :
% \ p s i equals to mole−f r a c t i o n weighted average of \ p s i 0 i and \ p s i 0 j
p0i = pEquil ;
p0j = pEquil ;
pR0i = p0i / p S a t i ; % Reduced pressure
p s i 0 i = mi∗ log ((−1+ pR0i−Ci∗pR0i+Ci∗pR0i ˆ ( Ni +1))/( −1+ pR0i ) ) ;
p s i 0 j = mj ∗ log (1+ Bj ∗ p0j ) ;
p s i = y i ∗ p s i 0 i + y j ∗ p s i 0 j ;
xSumErr = 1 ;
psiIterNum = 1 ;
while abs ( xSumErr ) > xSumErrTol
% C a l c u l a t e p0i based on the estimated \ p s i : Newton i t e r a t i o n s
p s i 0 i E r r = abs ( p s i 0 i − p s i ) ;
psi0iIterNum = 1 ;
while abs ( p s i 0 i E r r ) > p s i 0 i E r r T o l
pR0i = p0i / p S a t i ;
p s i 0 i E r r = mi∗ log ((−1+ pR0i−Ci ∗ pR0i+Ci∗pR0i ˆ ( Ni + 1 ) ) . . .
/(−1+pR0i ) ) − p s i ;
dFpsi0idpR0i = ( ( mi∗ Ci )/(1 − pR0i ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i ˆ Ni + Ni∗ pR0i ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i − Ci∗pR0i ˆ ( Ni + 1 ) ) ) ;

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 175
end
end
p0i = p0i − p s i 0 i E r r / dFpsi0idpR0i ;
psi0iIterNum = psi0iIterNum + 1 ;
% C a l c u l a t e p0j based on the estimated \ p s i
p0j = ( exp ( p s i /mj ) − 1) / Bj ;
% Update the value of \ p s i by minimizing e r r o r function :
% F = ( x i + x j ) − 1
x i C a l = pEquil ∗ y i / p0i ;
x j C a l = pEquil ∗ y j / p0j ;
xSumErr = ( x i C a l + x j C a l ) − 1 ;
% D e r i v a t i v e dFdpsi
pR0i = p0i / p S a t i ;
n0i = ( ( mi∗ Ci∗pR0i )/(1 − pR0i ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i ˆ Ni + Ni∗ pR0i ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i − Ci∗pR0i ˆ ( Ni + 1 ) ) ) ;
n0j = ( mj∗ Bj ∗ p0j ) / (1+ Bj ∗ p0j ) ;
dFSumxdpsi = −(( pEquil ∗ y i ) / ( p0i ∗ n0i ) + ( pEquil ∗ y j ) / ( p0j ∗ n0j ) ) ;
% Newton update of \ p s i
p s i = p s i − xSumErr/dFSumxdpsi ;
psiIterNum = psiIterNum +1;
%% C a l c u l a t i o n based on the c a l c u l a t e d \ p s i
% C a l c u l a t i o n of p0i : Newton i t e r a t i o n s
p s i 0 i E r r = abs ( p s i 0 i − p s i ) ;
psi0iIterNum = 1 ;
while abs ( p s i 0 i E r r ) > p s i 0 i E r r T o l
end
pR0i = p0i / p S a t i ;
p s i 0 i E r r = mi∗ log ((−1+ pR0i−Ci∗pR0i+Ci∗pR0i ˆ ( Ni + 1 ) ) . . .
/(−1+pR0i ) ) − p s i ;
dFpsi0idpR0i = ( ( mi∗ Ci )/(1 − pR0i ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i ˆ Ni + Ni∗ pR0i ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i − Ci∗pR0i ˆ ( Ni + 1 ) ) ) ;
p0i = p0i − p s i 0 i E r r / dFpsi0idpR0i ;
psi0iIterNum = psi0iIterNum + 1 ;
% C a l c u l a t i o n of p0j

176 APPENDIX C. ADSORPTION CALCULATION CODES
end
p0j = ( exp ( p s i /mj ) − 1) / Bj ;
% C a l c u a l t i o n of x i and x j
x i I A S = pEquil ∗ y i / p0i ;
x j I A S = pEquil ∗ y j / p0j ;
% C a l c u a l t i o n of p s i 0 i and \ p s i 0 j
pR0i = p0i / p S a t i ;
p s i 0 i = mi∗ log ((−1+ pR0i−Ci∗pR0i+Ci∗pR0i ˆ ( Ni +1))/( −1+ pR0i ) ) ;
p s i 0 j = mj ∗ log (1+ Bj ∗ p0j ) ;
% C a l c u l a t i o n of n0i and n0j
n0i = ( ( mi∗ Ci∗pR0i )/(1 − pR0i ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i ˆ Ni + Ni∗ pR0i ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i − Ci∗pR0i ˆ ( Ni + 1 ) ) ) ;
n0j = ( mj∗ Bj ∗ p0j ) / (1+ Bj ∗ p0j ) ;
% C a l c u l a t i o n of t o t a l amount of adsorption
ntIAS = 1 / ( x i I A S / n0i + x j I A S / n0j ) ;
% C a l c u l a t i o n of amount of adsotpion of comp . i and j
niIAS = ntIAS ∗ x i I A S ;
njIAS = ntIAS ∗ x j I A S ;
% C a l c u l a t i o n of s e l e c t i v i t y c o e f f i c i e n t of comp . i to comp . j
S i j I A S = ( x i I A S / y i ) / ( x j I A S / y j ) ;
%% R e s u l t s
n i I A S A l l ( CountP ) = niIAS ;
n j I A S A l l ( CountP ) = njIAS ;
n t I A S A l l ( CountP ) = ntIAS ;
x i I A S A l l ( CountP ) = x i I A S ;
x j I A S A l l ( CountP ) = x j I A S ;
S i j I A S A l l ( CountP ) = S i j I A S ;
p s i I A S A l l ( CountP ) = p s i ;
p s i 0 i I A S A l l ( CountP ) = p s i 0 i ;
p s i 0 j I A S A l l ( CountP ) = p s i 0 j ;

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 177
C.3.2 Binary Adsorption Calculation Based on the Real Adsorbed
Solution Model
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This code c a l c u l a t e binary adsorption of compnent i and component j
% based on the r e a l adsorbed s o l u t i o n model and the
% ABC excess Gibbs f r e e energy model .
% Component i n d i c e s : i = CO2, j = N2 .
% The algorithm i s s i m i l a r to t h a t implemented by S i p e r s t e i n and Myers
% ( 2 0 0 1 ) . The d i f f e r e n c e are the pure adsorption isotherms , thus the
% equations f o r \ p s i i ˆ0 and the way to c a l c u l a t e \ p s i .
% Last modified on Feb . 7 , 2010 by Wenjuan Lin
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Clean Up the Workspace f o r a New Run
c l o s e a l l
c l e a r a l l
c l c
%% Clear Up the Workspace f o r a New Run
c l o s e a l l
c l e a r a l l
c l c
%% Inputs : Pure Adsorption Isotherms
pPure = 1 : 5 0 : 6 0 0 1 ; % Pressure range f o r pure ads . isotherm plots , kPa
% CO2 : N−l a y e r BET equation
Ci = 2 7 . 0 9 ; % Constant
mi = 1 . 1 3 6 ; % Maximum monolayer adsorption , mol/ kg
Ni = 5 . 1 6 4 ; % Total allowed number of adsorption l a y e r s
p S a t i = 6 0 0 3 . 0 6 ; % S a t u r a t i o n pressure of CO2 at room temperature , kPa
pRi = pPure . / p S a t i ; % Reduced pressure
n P u r e i A l l = ( ( mi . ∗ Ci . ∗ pRi )./(1 − pRi ) ) . ∗ . . .
( (1 − ( Ni + 1 ) . ∗ pRi . ˆ Ni + Ni . ∗ pRi . ˆ ( Ni + 1 ) ) . / . . .
(1 + ( Ci −1).∗ pRi − Ci . ∗ pRi . ˆ ( Ni + 1 ) ) ) ;
% N2 : Langmuir equation
Bj = 0 . 0 0 0 3 1 ; % Langmuir constant , 1/kPa

178 APPENDIX C. ADSORPTION CALCULATION CODES
mj = 0 . 5 0 5 6 ; % Maximum amount of ads . at i n f i n i t e pressure , mol/ kg
n P u r e j A l l = ( mj . ∗ Bj . ∗ pPure ) . / (1 + Bj . ∗ pPure ) ;
%% Binary Adsorption Experimental Data
T = 273.15 + 2 2 ; % Experiment temperature , deg K
RABC = 8.314472 ∗ 10ˆ( −3); % Universal gas constant
% used in the ABC excess Gibbs f r e e energy model , k J K −1 mol −1
REOS = 8.314472 ∗ 1 0 ˆ 3 ; % Universal gas constant used in EOS
% f o r f u g a c i t y c o e f f i c i e n t s c a l c u l a t i o n , m l k P a K −1 mol −1
load CO275Coal . t x t ; % Experimental binary adsorption data f i l e
p E q u i l A l l = CO275Coal ( : , 1 ) ; % Equilibrium pressure , kPa
ntAdsAll = CO275Coal ( : , 2 ) ; % Total amount of adsorption , mol/ kg
x i A l l = CO275Coal ( : , 3 ) ; % Mole f r a c t i o n of comp . i in ads . phase
y i A l l = CO275Coal ( : , 4 ) ; % Mole f r a c t i o n of comp . i in gas phase
S i j A l l = CO275Coal ( : , 5 ) ; % Experimental separation c o e f f i c i e n t s
x j A l l = 1 − x i A l l ; % Mole f r a c t i o n of comp . j in ads . phase
y j A l l = 1 − y i A l l ; % Mole f r a c t i o n of comp . j in gas phase
n i A d s A l l = ntAdsAll . ∗ x i A l l ; % Amount of adsorption of comp . i , mol/ kg
n j A d s A l l = ntAdsAll . ∗ x j A l l ; % Amount of adsorption of comp . j , mol/ kg
%% Fugacity C o e f f i c i e n t C a l c u a l t i o n
Nc = 2 ; % Number of components
% Thermodynamic p r o p e r t i e s of the pure components
% CO2 N2
pc = [7380 , 3 3 9 0 ] ; % C r i t i c a l pressure , kPa
Tc = [ 3 0 4 . 1 , 1 2 6 . 2 ] ; % C r i t i c a l temperature , K
w = [ 0 . 2 3 9 , 0 . 0 3 9 ] ; % Accentric f a c t o r
EOS = ’SRK ’ ; % Equation of S t a t e used in f u g a c i t y c o e f f i c i e n t c a l c u l a t i o n
% Fugacity c o e f f i c i e n t c a l c u l a t i o n
f o r CountP = 1 : length ( p E q u i l A l l ) ;
pEquil = p E q u i l A l l ( CountP ) ; % Equilibrium pressure , kPa
y i = y i A l l ( CountP ) ; % Mole f r a c t i o n of comp . i in gas phase
y j = y j A l l ( CountP ) ; % Mole f r a c t i o n of comp . j in gas phase
yEquil = [ yi , y j ] ; % Gas phase composition
f u g a c o e f f = f e i ( pEquil , T , REOS , Nc , pc , Tc , w, EOS , yEquil ) ;
f e i i A l l ( CountP ) = f u g a c o e f f ( 1 ) ; % Fugacity c o e f f i c i e n t s of comp . i
f e i j A l l ( CountP ) = f u g a c o e f f ( 2 ) ; % Fugacity c o e f f i c i e n t s of comp . j

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 179
end
%% C a l c u l a t e Ao and C by Minimizing the Error between the Calculated and
%% the Experimental Pressure and Separation C o e f f i c i e n t s
% Range of the value of Ao and C
[ Ao Vector , C Vector ] = meshgrid(−25 : 0.01 : −20, 0 : 0.01 : 2 ) ;
[CNum, AoNum] = s i z e ( Ao Vector ) ;
f o r i = 1 : CNum;
end
f o r j = 1 : AoNum;
end
Ao = Ao Vector ( i , j ) ;
C = C Vector ( i , j ) ;
CalAoCNBETLanPi0 % Direct to code CalAoCNBETLanPi0 .m
T o t a l E r r o r ( i , j ) = sum(sum( pSError ) ) / length ( pSError ) ;
%% Contour P l o t of Error in Calculated Equilibrium Pressure and S e l e c t i v i t y
%% C o e f f i c i e n t s
f i g u r e
[ S , h ] = contour ( C Vector , Ao Vector , T o t a l E r r o r ) ;
s e t ( h , ’ ShowText ’ , ’on ’ , ’ TextStep ’ , get ( h , ’ LevelStep ’ ) ∗ 2 ) ;
colormap cool ;
%% Calcuation based on the Chosen Values of Ao and C
[ MinVal1 , ColInd ] = min(min( T o t a l E r r o r ) ) ;
[ MinVal2 , RowInd ] = min(min( TotalError ’ ) ) ;
min(min( T o t a l E r r o r ) )
Ao = Ao Vector ( RowInd , ColInd )
C = C Vector ( RowInd , ColInd )
CalAoCNBETLanPi0 % Direct to code CalAoCNBETLanPi0 .m
%% Result Output
ResultTable = [ pEquilAll , pEquilCalAll ’ , ntAdsAll , n t C a l A l l ’ , . . .
S i j A l l , S i j C a l A l l ’ , f e i i A l l ’ , f e i j A l l ’ , gamaiAll ’ , gamajAll ’ , . . .
p s i A l l ’ , p s i 0 i A l l ’ , p s i 0 j A l l ’ , n 0 i A l l ’ , n 0 j A l l ’ , p 0 i A l l ’ , p 0 j A l l ’ ]
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

180 APPENDIX C. ADSORPTION CALCULATION CODES
% CalAoCNBETLanPi0 .m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f o r CountP = 1 : length ( p E q u i l A l l ) ;
% Experimental Data
pEquil = p E q u i l A l l ( CountP ) ; % Equilibrium pressure , kPa
ntAds = ntAdsAll ( CountP ) ; % Total amount of adsorption , mol/ kg
x i = x i A l l ( CountP ) ; % Mole f r a c t i o n of comp . i in ads . phase
y i = y i A l l ( CountP ) ; % Mole f r a c t i o n of comp . i in gas phase
x j = x j A l l ( CountP ) ; % Mole f r a c t i o n of comp . j in ads . phase
y j = y j A l l ( CountP ) ; % Mole f r a c t i o n of comp . j in gas phase
niAds = n i A d s A l l ( CountP ) ; % Amount of adsorption of comp . i , mol/ kg
njAds = n j A d s A l l ( CountP ) ; % Amount of adsorption of comp . j , mol/ kg
S i j = S i j A l l ( CountP ) ; % S e l e c t i v i t y c o e f f i c i e n t
% Fugacity C o e f f i c i e n t s
f e i i = f e i i A l l ( CountP ) ; % Fugacity c o e f f i c i e n t of comp . i
f e i j = f e i j A l l ( CountP ) ; % Fugacity c o e f f i c i e n t of comp . j
%% C a l c u l a t e p0i using B i s e c t i o n Method .
%% Target function : f = p s i 0 i − p s i 0 j
% I n i t i a l Guesses
p 0 i 1 = 1 ;
p 0 i 2 = p S a t i ;
p 0 i 0 = ( p 0 i 1 + p 0 i 2 ) / 2 ;
% Error Tolerance in Calculated and Experimental nt
ntErrTol = 1e−6; % mol/ kg
% Maximum Number of I t e r a t i o n
IterMax = log2 ( abs ( p 0 i 1 − p 0 i 2 ) / ntErrTol ) ;
NumIter = 1 ;
f o r NumIter = 1 : IterMax
% Reduced Pressures
pR0i 1 = p 0 i 1 / p S a t i ;
pR0i 2 = p 0 i 2 / p S a t i ;
pR0i 0 = p 0 i 0 / p S a t i ;
% C a l c u l a t i o n # 1 : p0i = p 0 i 1

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 181
n0i 1 = ( ( mi∗ Ci∗ pR0i 1 )/(1 − pR0i 1 ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i 1 ˆ Ni + Ni∗ pR0i 1 ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i 1 − Ci∗ pR0i 1 ˆ ( Ni + 1 ) ) ) ;
p s i 0 i 1 = mi∗ log ( (−1+pR0i 1−Ci∗ pR0i 1+Ci ∗ pR0i 1 ˆ ( Ni + 1 ) ) . . .
/(−1+ pR0i 1 ) ) ;
p s i 0 j 1 = p s i 0 i 1 ;
p 0 j 1 = ( exp ( p s i 0 j 1 /mj ) − 1 ) / Bj ;
n 0 j 1 = ( mj∗ Bj ∗ p 0 j 1 ) / (1 + Bj ∗ p 0 j 1 ) ;
p s i 1 = p s i 0 i 1 ;
ntCal 1 = 1 / ( x i / n0i 1 + x j / n 0 j 1 + . . .
(C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ p s i 1 ) ) ;
n t E r r 1 = ntCal 1 − ntAds ;
% C a l c u l a t i o n # 2 : p0i = p 0 i 2
n0i 2 = ( ( mi∗ Ci∗ pR0i 2 )/(1 − pR0i 2 ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i 2 ˆ Ni + Ni∗ pR0i 2 ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i 2 − Ci∗ pR0i 2 ˆ ( Ni + 1 ) ) ) ;
p s i 0 i 2 = mi∗ log ( (−1+pR0i 2−Ci∗ pR0i 2+Ci ∗ pR0i 2 ˆ ( Ni + 1 ) ) . . .
/(−1+ pR0i 2 ) ) ;
p s i 0 j 2 = p s i 0 i 2 ;
p 0 j 2 = ( exp ( p s i 0 j 2 /mj ) − 1 ) / Bj ;
n 0 j 2 = ( mj∗ Bj ∗ p 0 j 2 ) / (1 + Bj ∗ p 0 j 2 ) ;
p s i 2 = p s i 0 i 2 ;
ntCal 2 = 1 / ( x i / n0i 2 + x j / n 0 j 2 + . . .
(C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ p s i 2 ) ) ;
n t E r r 2 = ntCal 2 − ntAds ;
% C a l c u l a t i o n # 3 : p0i = p 0 i 0
n0i 0 = ( ( mi∗ Ci∗ pR0i 0 )/(1 − pR0i 0 ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i 0 ˆ Ni + Ni∗ pR0i 0 ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i 0 − Ci∗ pR0i 0 ˆ ( Ni + 1 ) ) ) ;
p s i 0 i 0 = mi∗ log ( (−1+pR0i 0−Ci∗ pR0i 0+Ci ∗ pR0i 0 ˆ ( Ni + 1 ) ) . . .
/(−1+ pR0i 0 ) ) ;
p s i 0 j 0 = p s i 0 i 0 ;
p 0 j 0 = ( exp ( p s i 0 j 0 /mj ) − 1 ) / Bj ;
n 0 j 0 = ( mj∗ Bj ∗ p 0 j 0 ) / (1 + Bj ∗ p 0 j 0 ) ;
p s i 0 = p s i 0 i 0 ;
ntCal 0 = 1 / ( x i / n0i 0 + x j / n 0 j 0 + . . .
(C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ p s i 0 ) ) ;

182 APPENDIX C. ADSORPTION CALCULATION CODES
end
n t E r r 0 = ntCal 0 − ntAds ;
% B i s e c t i o n a l Update of p0i
i f abs ( n t E r r 0 ) < ntErrTol ;
p0i = p 0 i 0 ;
e l s e i f n t E r r 1 ∗ n t E r r 0 < 0 ;
end
p 0 i 2 = p 0 i 0 ;
p 0 i 0 = ( p 0 i 1 + p 0 i 2 ) / 2 ;
p0i = p 0 i 0 ;
e l s e n t E r r 0 ∗ n t E r r 2 < 0 ;
end
p 0 i 1 = p 0 i 0 ;
p 0 i 0 = ( p 0 i 1 + p 0 i 2 ) / 2 ;
p0i = p 0 i 0 ;
NumIter = NumIter +1;
%% C a l c u l a t i o n Based on the c a l c u l a t e d pi0
pR0i = p0i / p S a t i ;
n0i = ( ( mi∗ Ci∗pR0i )/(1 − pR0i ) ) ∗ . . .
( (1 − ( Ni +1)∗ pR0i ˆ Ni + Ni∗ pR0i ˆ ( Ni + 1 ) ) / . . .
(1 + ( Ci −1)∗ pR0i − Ci∗pR0i ˆ ( Ni + 1 ) ) ) ;
p s i 0 i = mi∗ log ( (−1+pR0i−Ci∗pR0i+Ci∗pR0i ˆ ( Ni + 1 ) ) . . .
/(−1+pR0i ) ) ;
p s i 0 j = p s i 0 i ;
p0j = ( exp ( p s i 0 j /mj ) − 1 ) / Bj ;
n0j = ( mj∗ Bj ∗ p0j ) / (1 + Bj ∗ p0j ) ;
p s i = p s i 0 i ;
% Calculated t o t a l Amount of Adsorption , mol/ kg
ntCal = 1 / ( x i / n0i + x j / n0j + . . .
(C/ (RABC∗T ) ) ∗ Ao∗ x i ∗ x j ∗exp(−C∗ p s i ) ) ;
ntErr = ntCal − ntAds ;
% A c t i v i t y C o e f f i c i e n t s
gamai = exp ( Ao ∗ ( 1−exp(−C∗ p s i ) )∗ x j ˆ2 / (RABC∗T ) ) ;
gamaj = exp ( Ao ∗ ( 1−exp(−C∗ p s i ) )∗ x i ˆ2/ (RABC∗T ) ) ;
% Calculated Equilibrium Pressure and Separation C o e f f i c i e n t
pEquilCal = ( p0i ∗gamai∗ x i ) / f e i i + ( p0j ∗gamaj∗ x j ) / f e i j ;

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 183
end
S i j C a l = ( f e i i ∗ p0j ∗gamaj ) / ( f e i j ∗ p0i ∗gamai ) ;
% Normalized Error
pError ( CountP ) = abs ( pEquilCal − pEquil ) / pEquil ;
SError ( CountP ) = abs ( S i j C a l − S i j ) / S i j ;
pSError ( CountP ) = s q r t ( pError ( CountP ) ˆ 2 + SError ( CountP ) ˆ 2 ) ;
%% Simulation R e s u l t s
p E q u i l C a l A l l ( CountP ) = pEquilCal ;
S i j C a l A l l ( CountP ) = S i j C a l ;
gamaiAll ( CountP ) = gamai ;
gamajAll ( CountP ) = gamaj ;
p s i A l l ( CountP ) = p s i ;
p s i 0 i A l l ( CountP ) = p s i 0 i ;
p s i 0 j A l l ( CountP ) = p s i 0 j ;
n 0 i A l l ( CountP ) = n0i ;
n 0 j A l l ( CountP ) = n0j ;
p 0 i A l l ( CountP ) = p0i ;
p 0 j A l l ( CountP ) = p0j ;
n t C a l A l l ( CountP ) = ntCal ;
Fugacity Coefficient Calculation
function f = f e i ( p , T , R , numcomponent , pc , Tc , w, EOS , y )
%% EOS parameters
% Notes :
% I n t e r a c t i o n parameters f o r the van der Waals and Redlich−Kwong EOS are ,
% by d e f i n i t i o n , zero . I n t e r a c t i o n parameter f o r CO2−N2 i s zero
% ( Table 7 . 7 of l e c t u r e notes of PE251 of Kovscek .
% Reduced Temperature ( must be c a l c u l a t e d with absolute temp . , degree K)
Tr = T . / Tc ;
switch EOS
case ’vdW ’ % van der Waals EOS
u = 0 ;
ww = 0 ;
kk = [ 0 . 0 0 0 0 0.0000

184 APPENDIX C. ADSORPTION CALCULATION CODES
end
% a and b
0.0000 0 . 0 0 0 0 ] ;
a = 2 7 . ∗ ( R ˆ 2 ) . ∗ ( Tc . ˆ 2 ) . / ( 6 4 . ∗ pc ) ;
b = R . ∗ Tc . / ( 8 . ∗ pc ) ;
case ’PR ’ % Peng−Robinson EOS
u = 2 ;
ww = −1;
kk = [ 0 . 0 0 0 0 0.0000
% a and b
0.0000 0 . 0 0 0 0 ] ;
fw = 0.37464 + 1 . 5 4 2 2 6 .∗w − 0 . 2 6 9 9 2 . ∗ (w . ˆ 2 ) ;
a = ( 0 . 4 5 7 2 4 . ∗ (R ˆ 2 ) . ∗ ( Tc . ˆ 2 ) . / pc ) . ∗ ( ( 1 + fw .∗(1 − Tr . ˆ 0 . 5 ) ) . ˆ 2 ) ;
b = 0 . 0 7 7 8 0 .∗R . ∗ Tc . / pc ;
case ’RK ’ % Redlich−Kwong EOS
u = 1 ;
ww = 0 ;
kk = [ 0 . 0 0 0 0 0.0000
% a and b
0.0000 0 . 0 0 0 0 ] ;
a = 0 . 4 2 7 4 8 . ∗ (R ˆ 2 ) . ∗ ( Tc . ˆ ( 5 / 2 ) ) . / ( pc . ∗ ( T ˆ 0 . 5 ) ) ;
b = 0 . 0 8 6 6 4 .∗R . ∗ Tc . / pc ;
case ’SRK ’ % Soave−Redlich−Kwong EOS
u = 1 ;
ww = 0 ;
kk = [ 0 . 0 0 0 0 0.0000
% a and b
0.0000 0 . 0 0 0 0 ] ;
fw = 0.48 + 1 . 5 7 4 . ∗w − 0 . 1 7 6 . ∗ (w . ˆ 2 ) ;
a = ( 0 . 4 2 7 4 8 . ∗ (R ˆ 2 ) . ∗ ( Tc . ˆ 2 ) . / pc ) . ∗ ( ( 1 + fw .∗(1 − Tr . ˆ 0 . 5 ) ) . ˆ 2 ) ;
b = 0 . 0 8 6 6 4 .∗R . ∗ Tc . / pc ;
% C a l c u l a t e am and bm according to mixture r u l e
f o r i = 1 : numcomponent
f o r j = 1 : numcomponent
end
aaVapor ( j ) = y ( i )∗ y ( j ) ∗ ( s q r t ( a ( i )∗ a ( j ) ) ) ∗ ( 1−kk ( i , j ) ) ;

C.3. IAS & RAS FOR CO2/N2-COAL SYSTEM 185
end
aaVap ( i ) = sum( aaVapor ) ;
bbVap ( i ) = y ( i )∗b( i ) ;
aVap = sum( aaVap ) ;
bVap = sum( bbVap ) ;
% C a l c u l a t e the c o m p r e s s i b i l i t y f a c t o r of the vapor phase
Avap = aVap∗p / (Rˆ2∗T ˆ 2 ) ;
Bvap = bVap∗p / (R∗T ) ;
C1vap = 1 ;
C2vap = −(1+Bvap−u∗Bvap ) ;
C3vap = ( Avap+ww∗Bvapˆ2−u∗Bvap−u∗Bvap ˆ 2 ) ;
C4vap = −Avap∗Bvap−ww∗Bvapˆ2−ww∗Bvap ˆ 3 ;
compVap = [ C1vap , C2vap , C3vap , C4vap ] ;
ZZvap = roots (compVap ) ;
f o r i = 1 : s i z e ( ZZvap )
end
im ( i ) = imag ( ZZvap ( i ) ) ;
i f im ( i ) ˜= 0
end
ZZvap ( i ) = 0 ;
Zvap = max( ZZvap ) ;
% C a l c u l a t e vapor volume
Vvap = Zvap∗R∗T/p ;
% C a l c u l a t e f u g a c i t y c o e f f i c i e n t s
switch ’EOS ’
case ’vdW ’
f e i v = exp ( log (R∗T / ( p∗( Vvap−bVap ) ) ) + b . / ( Vvap−bVap ) − . . .
otherwise
2 . ∗ ( s q r t ( aVap . ∗ a ) ) . / ( R∗T∗Vvap ) ) ;
bibVap = ( Tc . / pc ) . / (sum( y . ∗ Tc . / pc ) ) ;
sVap = 2 . ∗ ( a . / aVap ) . ˆ 0 . 5 ;
f e i v = exp ( bibVap . ∗ ( Zvap−1) − log ( Zvap−Bvap ) + . . .
( Avap . ∗ ( bibVap −sVap ) . / ( Bvap ∗ ( s q r t ( uˆ2−4∗ww) ) ) ) . ∗ . . .
log ( (2∗ Zvap+Bvap ∗ ( u+ s q r t ( uˆ2−4∗ww) ) ) / . . .

186 APPENDIX C. ADSORPTION CALCULATION CODES
end
(2∗ Zvap+Bvap ∗ ( u−s q r t ( uˆ2−4∗ww) ) ) ) ) ;
% Output r e s u l t s of the f u g a c i t y c o e f f i c i e n t s
f = f e i v ;

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