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Investigation of gas sorption and mass transfer in gas shales
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Investigation of gas sorption and mass transfer in gas shales
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Content
INVESTIGATION OF GAS SORPTION AND MASS
TRANSFER IN GAS SHALES
by
Yu Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMICAL ENGINEERING)
December 2016
i
Table of Contents
Dedication ................................................................................................................................... vi
Acknowledgments ...................................................................................................................... vii
List of Tables ............................................................................................................................... ix
List of Figures ............................................................................................................................ xii
Abstract ..................................................................................................................................... xix
Chapter 1: Introduction .............................................................................................................. 1
1.1. Unconventional Oil & Gas ...................................................................................... 1
1.2. Shale Gas in the U.S. and Worldwide ..................................................................... 2
1.3. Review of Sorption Studies on Shale ...................................................................... 5
1.4. Knowledge Gaps and Objectives of This Study ...................................................... 8
1.5. References ............................................................................................................. 10
Chapter 2: Static Experiments of CH
4
and C
2
H
6
Sorption on Dry and Moist Shale ........... 14
2.1. Introduction ........................................................................................................... 14
2.2. Experimental ......................................................................................................... 16
2.2.1. Sample Preparation ............................................................................................... 16
2.2.2. Experimental Set-up .............................................................................................. 17
ii
2.2.3. System Leak Test .................................................................................................. 19
2.2.4. Volume Measurement of the Reference Cell and the Sample Cell ....................... 20
2.2.5. Sorption Measurement Procedures for CH
4
and C
2
H
6
Sorption on Dry/Moist
Shale Sample ......................................................................................................... 24
2.2.5.1. Procedure for Gas Sorption on Dry Sample .......................................................... 24
2.2.5.2. Procedure for Gas Sorption on Moist Sample ....................................................... 25
2.2.6. Analysis Approach for Gas Sorption in the Static Apparatus ............................... 26
2.3. Results ................................................................................................................... 28
2.3.1. CH
4
and C
2
H
6
Sorption on Dry Sample ................................................................ 28
2.3.2. CH
4
and C
2
H
6
Sorption on Moist Sample ............................................................. 31
2.4. Conclusion ............................................................................................................. 34
2.5. References ............................................................................................................. 34
Chapter 3: Competitive Sorption of Methane/Ethane Mixtures on Shale on a TGA
Apparatus ................................................................................................................................... 38
3.1. Introduction ........................................................................................................... 38
3.2. Experimental ......................................................................................................... 39
3.2.1. Sample Preparation ............................................................................................... 39
3.2.2. Experimental Approach ........................................................................................ 40
iii
3.3. Results ................................................................................................................... 42
3.3.1. CH
4
and C
2
H
6
Pure Component Isotherms ........................................................... 42
3.3.2. CH
4
and C
2
H
6
Pure Component Adsorption/Desorption Hysteresis .................... 46
3.3.3. Competitive Adsorption of CH
4
-C
2
H
6
Binary Mixtures ....................................... 48
3.4. Modeling ............................................................................................................... 51
3.4.1. Multicomponent Potential Theory of Adsorption (MPTA) .................................. 51
3.4.2. Modeling of CH
4
and C
2
H
6
Pure Component Isotherms ....................................... 53
3.4.3. Prediction of CH
4
-C
2
H
6
Adsorption Behavior ..................................................... 58
3.5. Discussion and Conclusion ..................................................................................... 63
3.6. References ............................................................................................................... 64
3.7. Appendix ............................................................................................................... 69
3.8. Supplementary Data .............................................................................................. 72
A. Bulk Phase Density Data and Calculations using the PR-EOS .......................... 72
B. Sorption Data ...................................................................................................... 75
Chapter 4: Sorption Kinetics and Mass Transfer in Powdered Shale Sample ................... 81
4.1. Introduction ........................................................................................................... 81
4.2. Experimental ......................................................................................................... 82
4.2.1. Sample Preparation ............................................................................................... 82
iv
4.2.2. Experimental Approach ........................................................................................ 83
4.3. Experimental Results ............................................................................................ 84
4.3.1. Sorption Isotherms and Dynamics of Pure Components ...................................... 84
4.3.2. Sorption Isotherms and Dynamics of Binary Mixtures ......................................... 86
4.4. Adsorbed Phase Density Estimates ....................................................................... 90
4.4.1. Introduction of Various Methods to Estimate Adsorbed Phase Density ............... 90
4.4.2. Langmuir Adsorption Isotherms Calculated from Various Density Models ........ 94
4.5. Sorption Dynamic Model Derivation .................................................................. 105
4.6. Modeling Results – Sorption Dynamics of Pure Components and
Binary Mixtures .................................................................................................. 110
4.6.1. CH
4
and C
2
H
6
Pure Component Sorption Dynamics ............................................ 110
4.6.2. CH
4
-C
2
H
6
Binary Mixture Sorption Dynamics ................................................... 114
4.7. Conclusion and Discussion ................................................................................. 119
4.8. References ........................................................................................................... 121
4.9. Supplementary Materials .................................................................................... 123
A. Sorption Dynamic Data Correction ................................................................ 123
B. DFT Method ................................................................................................... 126
C. LRD Method ................................................................................................... 128
D. Multicomponent Potential Theory of Adsorption (MPTA) Method .............. 131
v
E. Vapor-Liquid Equilibrium (VLE) Method ....................................................... 132
Chapter 5: Sorption Kinetics and Mass Transfer in Cubic Shale Sample .......................... 135
5.1. Introduction ......................................................................................................... 135
5.2. Experimental ....................................................................................................... 137
5.2.1. Sample Preparation ............................................................................................... 137
5.2.2. Experimental Approach ...................................................................................... 137
5.3. Experimental Results .......................................................................................... 138
5.3.1. CH
4
and C
2
H
6
Pure Component Sorption Isotherms ........................................... 138
5.3.2. CH
4
-C
2
H
6
Binary Mixture Sorption Isotherms ................................................... 139
5.3.3. CH
4
and C
2
H
6
Pure Component Sorption Dynamics .......................................... 142
5.3.4. CH
4
-C
2
H
6
Binary Mixture Sorption Dynamics ..................................................... 145
5.4. Langmuir Adsorption Isotherms Calculated from the DFT and the LRD
Approaches .......................................................................................................... 147
5.5. Conclusion and Discussion ................................................................................. 151
5.6. References ........................................................................................................... 152
5.7. Supplementary Data ............................................................................................ 154
Chapter 6: Future Work ........................................................................................................ 160
vi
Dedication
To my loving parents, Yang Wang and Chuanqin Lu
for their kindness and support in every stage of my life.
vii
Acknowledgements
First and foremost, I would like to express my sincerest gratitude towards my academic advisors
Professor Kristian Jessen and Professor Theodore Tsotsis for their invaluable guidance, heartfelt
inspiration and generous support throughout my PhD studies. Without their commitment and
encouragement, it would have been impossible for me to accomplish this goal. They serve as an
example of outstanding scientists, supervisors and mentors. Their spirit will encourage me to
explore, discover and achieve in my next stage of life and beyond.
I feel grateful to Dr. Doug Hammond, Dr. Iraj Ershaghi, and Dr. Katherine Shing, who served on
my qualifying exam committee. My special thanks go to Dr. Doug Hammond, who also served
on my PhD defense committee. His invaluable suggestions and guidance are gratefully
appreciated.
Also, I am thankful to Tina Silva and Shokry Bastorous for keeping me safe from laboratory
safety violations. I want to thank Martin Olekszyk and Jason Ordonez for smoothly handling my
pay-checks and orders. The help and advice from Karen Woo and Andy Chen are also greatly
appreciated. In addition, I would like to thank Heather Alexander, Angeline Fugelso and Laura
Carlos for their support and encouragement. I feel lucky to have worked with all the staff in the
Mork Family Department of Chemical Engineering and Materials Science, who have made my
PhD study a pleasant and unforgettable experience.
My special thanks also go to my fellow PhD colleagues: Dr. Bobby Liu, Dr. Junyi Xu, Dr.
Xiaojie Yan, Dr. Wangxue Deng, Dr. Basabdatta Roychaudhuri, Devang Dasani, Zhuofan Shi,
and Zhongtang Li. They all gave me a lot of help and support along the course of my PhD
viii
studies. It was a great pleasure to have worked with them in a friendly learning environment.
Their precious friendship will be embedded in my heart forever.
I am also grateful to the Viterbi School of Engineering and the Chevron-USC University
Partnership Program, from which I received my fellowship for the first and second years of my
PhD studies. Also, I would like to thank the Research Partnership to Secure Energy for America
(RPSEA) for its financial support for the rest of my PhD studies.
Last but not least, I want to convey my deepest gratitude to my parents, Yang Wang and
Chuanqin Lu for their endless love, encouragement and support for every single day of my life.
ix
List of Tables:
Table 1.1. Estimated worldwide shale gas resources [6]. ........................................................ 3
Table 2.1. Helium expansion pressure data. ........................................................................... 21
Table 2.2. CH
4
and C
2
H
6
sorption data on dry shale at 60
o
C. ............................................... 29
Table 2.3. CH
4
and C
2
H
6
sorption data on moist shale at 60
o
C. ........................................... 31
Table 3.1. Distribution of micropore, mesopore, and macropore volumes
in the shale sample as measured by BET. .............................................................................. 48
Table 3.2. Peng-Robinson EOS model parameters. ............................................................... 56
Table 3.3. MPTA model parameters obtained from pure component
sorption isotherms at 40
o
C, 50
o
C and 60
o
C. ....................................................................... 55
Table 3.4. RMS errors for CH
4
and C
2
H
6
isotherms at 40
o
C, 50
o
C and 60
o
C. ................... 58
Table 3.5. RMS errors for CH
4
-C
2
H
6
binary mixtures at 40
o
C, 50
o
C and 60
o
C. ................ 60
Table S.1. CH
4
and C
2
H
6
pure component sorption data on shale at 40
o
C. .......................... 75
Table S.2. CH
4
and C
2
H
6
pure component sorption data on shale at 50
o
C. .......................... 76
Table S.3. CH
4
and C
2
H
6
pure component sorption data on shale at 60
o
C. .......................... 77
Table S.4. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 40
o
C. .................................................................................................................... 78
Table S.5. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 50
o
C. ................................................................................................................... 79
Table S.6. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 60
o
C. .................................................................................................................... 80
x
Table 4.1. RMS errors for single and mixed gases isotherms using
various density approaches. ................................................................................................. 100
Table 4.2. Langmuir model parameters for CH
4
and C
2
H
6
sorption on shale at 60
o
C:
ads
estimated by various estimating methods. .................................................................. 103
Table 4.3. Summary of the model parameters for CH
4
and C
2
H
6
sorption dynamics:
ads
estimated by the DFT method. .................................................................................... 111
Table 4.4. Summary of the model parameters for CH
4
and C
2
H
6
sorption dynamics:
ads
estimated by the LRD method. ................................................................................... 114
Table A.1. Specifications of the Rubotherm magnetic suspension balance. ........................ 124
Table B.1. Summary of parameters in the DFT mapping function. ..................................... 126
Table B.2. Summary of parameters in the Langmuir model. ............................................... 126
Table B.3. Peng-Robinson EOS model parameters. ............................................................ 128
Table C.1. Summary of model parameters in the LRD approach. ....................................... 129
Table D.1. MPTA model parameters obtained from pure component sorption isotherms. . 131
Table D.2. Summary of parameters in the Langmuir model. ................................................. 132
Table E.1. Summary of parameters in the Langmuir model. ................................................. 133
Table 5.1. RMS errors for single and mixed gases isotherms using the DFT
and the LRD density approaches. ......................................................................................... 150
Table 5.2. Langmuir model parameters for CH
4
and C
2
H
6
sorption
on shale (cube) at 60
o
C. ...................................................................................................... 150
Table S.1. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 40
o
C. ............. 154
Table S.2. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 50
o
C. ............. 155
xi
Table S.3. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 60
o
C. ............. 156
Table S.4. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 40
o
C. ...................................................................................................... 157
Table S.5. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 50
o
C. ...................................................................................................... 158
Table S.6. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 60
o
C. ...................................................................................................... 159
xii
List of Figures:
Figure 1.1. The resource triangle for oil and gas reservoirs [1]. .............................................. 2
Figure 1.2. Map of basins with assessed shale oil and shale gas formations,
as of May 2013 from EIA. ....................................................................................................... 4
Figure 2.1. Schematic diagram of the static experimental set-up. ......................................... 18
Figure 2.2. Static system He leak test. ................................................................................... 20
Figure 2.3. He expansion for volume measurement. ............................................................... 23
Figure 2.4. CH
4
and C
2
H
6
sorption on dry shale sample at 60
o
C. ......................................... 30
Figure 2.5. CH
4
and C
2
H
6
sorption on moist shale sample at 60
o
C. ..................................... 32
Figure 2.6. CH
4
sorption on dry/moist shale sample at 60
o
C. ............................................... 32
Figure 2.7. C
2
H
6
sorption on dry/moist shale sample at 60
o
C. ............................................. 33
Figure 3.1. CH
4
and C
2
H
6
pure component sorption on shale at 40
o
C. ................................. 43
Figure 3.2. CH
4
and C
2
H
6
pure component sorption on shale at 50
o
C. ................................. 43
Figure 3.3. CH
4
and C
2
H
6
pure component sorption on shale at 60
o
C. ................................. 44
Figure 3.4. Reproducibility test of CH
4
sorption on shale at 60
o
C. ...................................... 45
Figure 3.5. CH
4
adsorption/desorption isotherms on shale at 60
o
C. ..................................... 46
Figure 3.6. C
2
H
6
adsorption/desorption isotherms on shale at 60
o
C. .................................... 47
Figure 3.7. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 40
o
C. .................................................................................................................... 49
Figure 3.8. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 50
o
C. .................................................................................................................... 50
xiii
Figure 3.9. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale at 60
o
C. .................................................................................................................... 50
Figure 3.10. Comparison of MPTA calculations and experimental observations
for CH
4
and C
2
H
6
excess sorption on shale at 40
o
C. ............................................................. 56
Figure 3.11. Comparison of MPTA calculations and experimental observations
for CH
4
and C
2
H
6
excess sorption on shale at 50
o
C. ............................................................. 57
Figure 3.12. Comparison of MPTA calculations and experimental observations
for CH
4
and C
2
H
6
excess sorption on shale at 60
o
C. ............................................................. 57
Figure 3.13. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 40
o
C. .... 59
Figure 3.14. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 50
o
C. .... 59
Figure 3.15. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 60
o
C. .... 60
Figure 3.16. Comparison of CH
4
adsorption isotherms on shale at 60
o
C
measured by static and TGA apparatus. ................................................................................. 62
Figure 3.17. Comparison of C
2
H
6
adsorption isotherms on shale at 60
o
C
measured by static and TGA apparatus. ................................................................................. 62
Figure S.1. Experimental CH
4
densities and calculated values by the PR-EOS. ................... 72
Figure S.2. Experimental C
2
H
6
densities and calculated values by the PR-EOS. .................. 73
Figure S.3. Experimental 90%-10% CH
4
-C
2
H
6
densities and calculated values
by the PR-EOS. ...................................................................................................................... 73
Figure S.4. Experimental 93%-7% CH
4
-C
2
H
6
densities and calculated values
by the PR-EOS. ...................................................................................................................... 74
xiv
Figure S.5. Experimental 96%-4% CH
4
-C
2
H
6
densities and calculated values
by the PR-EOS. ...................................................................................................................... 74
Figure 4.1. CH
4
and C
2
H
6
pure component sorption isotherms on shale
at 40, 50, and 60 ° C. ............................................................................................................... 85
Figure 4.2. Sorption dynamic process of C
2
H
6
on shale at 60 °C. ......................................... 85
Figure 4.3. Sorption process of C
2
H
6
on shale at 60 °C: pressure step from 2 to 5 bar. ........ 86
Figure 4.4. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% −10%,93% −7%, 96% −4%)
on shale at 40 ° C. ................................................................................................................... 87
Figure 4.5. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% −10%,93% −7%, 96% −4%)
on shale at 50 ° C. ................................................................................................................... 87
Figure 4.6. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% − 10%,93% − 7%, 96% − 4%)
on shale at 60 ° C. ................................................................................................................... 88
Figure 4.7. Sorption dynamic process of 90%-10% CH
4
−C
2
H
6
on shale at 60 ° C. ............... 89
Figure 4.8. Sorption process of 90%-10% CH
4
−C
2
H
6
on shale at 60 °C:
pressure step from 20 to 30 bar. ............................................................................................. 89
Figure 4.9. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60 °C:
ads
estimated by the DFT method. ....................................................... 95
Figure 4.10. ELM correlation of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale at 60 °C:
ads
estimated by the DFT method. ....................................................... 96
Figure 4.11. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60 °C:
ads
estimated by the LRD method. ....................................................... 96
xv
Figure 4.12. ELM correlation of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale at 60 °C:
ads
estimated by the LRD method. ....................................................... 97
Figure 4.13. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60 °C:
ads
estimated by the MPTA method. .................................................... 97
Figure 4.14. ELM prediction of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale at 60 °C:
ads
estimated by the MPTA method. .................................................... 98
Figure 4.15. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60 °C:
ads
estimated by the VLE method. ....................................................... 98
Figure 4.16. ELM correlation of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale at 60 °C:
ads
estimated by the VLE method. ....................................................... 99
Figure 4.17. Comparison of estimated CH
4
adsorbed phase densities on shale at 60 °C. ... 101
Figure 4.18. Comparison of estimated C
2
H
6
adsorbed phase densities on shale at 60 °C. .. 101
Figure 4.19. Comparison of estimated 90%-10% CH
4
- C
2
H
6
adsorbed phase densities
on shale at 60 °C. ................................................................................................................. 102
Figure 4.20. Comparison of estimated 93%-7% CH
4
−C
2
H
6
adsorbed phase densities
on shale at 60 °C. ................................................................................................................. 102
Figure 4.21. Comparison of estimated 96%-4% CH
4
−C
2
H
6
adsorbed phase densities
on shale at 60 °C. ................................................................................................................. 103
Figure 4.22. Schematic of TGA sorption measurements. .................................................... 105
Figure 4.23. Sorption dynamics of CH
4
sorption on the ground shale sample at 60 °C:
Experimental vs. Modeling results:
ads
estimated by the DFT method. ......................... 111
xvi
Figure 4.24. Sorption dynamics of C
2
H
6
sorption on the ground shale sample at 60 °C:
Experimental vs. Modeling results:
ads
estimated by the DFT method. ......................... 112
Figure 4.25. Sorption dynamics of CH
4
sorption on the ground shale sample at 60 °C:
Experimental vs. Modeling results:
ads
estimated by the LRD method. ......................... 113
Figure 4.26. Sorption dynamics of C
2
H
6
sorption on the ground shale sample at 60 °C:
Experimental vs. Modeling results:
ads
estimated by the LRD method. ......................... 113
Figure 4.27. Sorption dynamics of 90%-10% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the DFT method. .................................................................................... 115
Figure 4.28. Sorption dynamics of 93%-7% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the DFT method. .................................................................................... 116
Figure 4.29. Sorption dynamics of 96%-4% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the DFT method. .................................................................................... 116
Figure 4.30. Sorption dynamics of 90%-10% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the LRD method. ................................................................................... 117
xvii
Figure 4.31. Sorption dynamics of 93%-7% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the LRD method. ................................................................................... 117
Figure 4.32. Sorption dynamics of 96%-4% CH
4
−C
2
H
6
sorption on the
ground shale sample at 60 °C: Experimental vs. Modeling results:
ads
estimated by the LRD method. ................................................................................... 118
Figure 5.1. Picture of the shale sample cube. ...................................................................... 137
Figure 5.2. CH
4
and C
2
H
6
sorption isotherms on shale (cube) at 40, 50 and 60 °C. ........... 139
Figure 5.3. Sorption of CH
4
−C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 40 °C. ...................................................................................................... 140
Figure 5.4. Sorption of CH
4
−C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 50 °C. ...................................................................................................... 140
Figure 5.5. Sorption of CH
4
−C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%)
on shale (cube) at 60 °C. ...................................................................................................... 141
Figure 5.6. Sorption dynamic process of CH
4
on shale (cube) at 60 ° C. ............................. 142
Figure 5.7. Sorption dynamic process of CH
4
on shale (cube) at 60 ° C:
from 10 to 20 bar. ................................................................................................................. 143
Figure 5.8. Sorption dynamic process of C
2
H
6
on shale (cube) at 60 ° C. ............................ 143
Figure 5.9. Sorption dynamic process of C
2
H
6
on shale (cube) at 60 ° C:
from 5 to 10 bar. ................................................................................................................... 144
xviii
Figure 5.10. Sorption dynamic process of 90%-10% CH
4
−C
2
H
6
on shale (cube)
at 60 ° C. ................................................................................................................................ 145
Figure 5.11. Sorption dynamic process of 93%-7% CH
4
−C
2
H
6
on shale (cube) at 60 ° C. .. 146
Figure 5.12. Sorption dynamic process of 96%-4% CH
4
−C
2
H
6
on shale (cube) at 60 ° C. .. 146
Figure 5.13. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale (cube) at 60 °C:
ads
estimated by the DFT method. .......................................... 148
Figure 5.14. ELM correlation of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale (cube) at 60 °C:
ads
estimated by the DFT method. .......................................... 148
Figure 5.15. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale (cube) at 60 °C:
ads
estimated by the LRD method. ......................................... 149
Figure 5.16. ELM correlation of the excess sorption isotherms of CH
4
−C
2
H
6
mixtures
on shale (cube) at 60 °C:
ads
estimated by the LRD method. ......................................... 149
xix
Abstract
In the global context of energy crisis and greenhouse gases reduction, shale gas is emerging as a
clean and abundant unconventional energy resource. The U.S. Energy Information
Administration (EIA) predicts that by 2040, over 50 % of the total US natural gas production
would come from shale gas. Sorption of gases in the complex shale matrix system is the key
mechanism via which shale gas is stored in such formations, and has as a result received
attention in a number of previous studies related to shale gas recovery. Methane (CH
4
) is the
single largest component of shale gas, however, ethane (C
2
H
6
) is typically the second largest
component accounting for more than 15 vol. % in certain cases. The present work mainly
focuses on generating experimental data of pure component and competitive CH
4
-C
2
H
6
sorption
on shale samples under reservoir pressure and temperature conditions. In parallel, modeling
approaches, such as the Multicomponent Potential Theory of Adsorption (MPTA) and a
self-proposed sorption dynamic model, have been undertaken to model the sorption data. The
experimental observations and their interpretation pave a path towards new knowledge of shale
gas recovery by leveraging an improved understanding of sorption and other mass transfer
mechanisms of natural gas mixtures in shale.
In Chapter 1, a brief introduction of shale gas and a literature review of previous and current
research on the shale-gas system are presented. The knowledge gaps are identified and a
systematic research plan is proposed to close these gaps.
In Chapter 2, a static experimental system which employs the manometric method is utilized to
measure gas adsorption on both dry and moist shale samples. Single gas (CH
4
, C
2
H
6
) sorption
xx
isotherms at 60
o
C are obtained from the static experimental measurements. The effect of
moisture on the shale sorption characteristics is studied.
In Chapter 3, measurements of pure component sorption isotherms for CH
4
and C
2
H
6
for
pressures up to 114 bar and 35 bar, respectively, have been performed using a thermogravimetric
method in the temperature range (40-60
o
C), typical of storage formation conditions. Sorption
experiments of binary (CH
4
-C
2
H
6
) gas mixtures containing up to 10% (mole fraction) of C
2
H
6
,
typical of shale-gas compositions, utilizing ground samples, for pressures up to 125 bar under the
aforementioned temperature conditions have also been conducted. In the study, the
Multicomponent Potential Theory of Adsorption (MPTA) approach is utilized to model the
sorption data. The MPTA model is shown capable in representing the pure component sorption
data, and also provides reasonable predictive capability when applied to predict the total sorption
for CH
4
-C
2
H
6
binary mixtures in shale over a range of compositions and temperatures.
In Chapter 4, we study the sorption behavior of CH
4
-C
2
H
6
binary mixture (and its individual
components) in ground shale samples using thermogravimetric analysis (TGA). The sorption
isotherms generated are important to predict the gas storage capacity of the shale samples, while
the study of sorption dynamics/kinetics help us understand the role of desorption during the later
times of gas production. A dynamic Langmuir-type sorption model is proposed, that allows us to
isolate sorption kinetics from diffusive mass transfer. This, in turn, facilitates our modeling and
interpretation of the experimental observations. Work is currently ongoing towards studying the
role of sorption using a whole (cube) shale sample with a volume of ~1 cm
3
in the same TGA
set-up (Chapter 5). The whole sample offers an advantage over the powdered sample due to the
longer diffusion times, and captures the diffusion characteristics of the sample more accurately.
xxi
In Chapter 6, a few remaining tasks with regard to the shale gas study are discussed and future
work which could potentially address these issues is outlined as well.
1
Chapter 1
Introduction
1.1. Unconventional Oil & Gas
Unconventional oil and gas are petroleum products that are extracted from low porosity and low
permeability formations using techniques other than the conventional method. Regardless of how
they are produced or the formation they come from, unconventional oil and gas are essentially
the same as their conventional counterparts. Since the erection of the first modern oil well in
1848, nearly two centuries of oil and gas production around the globe has exhausted most of the
“easy” oil that can be extracted simply by drilling a well (see Fig. 1.1). The world has seen a
peak oil production during the 1970s and then a constant decline in conventional resources
globally. Thanks to the development of new technologies, i.e., directional drilling and hydraulic
fracturing, oil and gas producers are now able to economically extract “unconventional” oil and
to utilize gas resources that were previously impossible to access. Oil companies across the globe
are investing heavily in unconventional oil and gas due to the increasing scarcity of
conventional reserves. The United States has seen a renaissance in petroleum production driven
by these technology improvements. Application of directional drilling and hydraulic fracturing,
for example, has enabled natural gas to be economically produced from shale and other
unconventional formations, and has contributed to the United States becoming the world’s
largest natural gas producer in 2009. Use of the same technologies has also contributed to the rise
in the U.S. oil production over the last few years as well. In 2009, U.S. annual oil production
increased over 2008, the first annual rise since 1991, and has continued to increase each year
2
since then. Between October 2007 and October 2013, U.S. monthly crude oil production rose by
2.7 million barrels per day, with about 92 % of the increase coming from shale and related tight
oil formations in Texas and North Dakota. Other tight oil plays are also being developed, and
have helped raise the prospect of energy independence for North America [1]. Unconventional
oil and natural gas, shale gas, in particular, has been called the future of gas supply in North
America.
Figure 1.1. The resource triangle for oil and gas reservoirs [1].
1.2. Shale Gas in the U.S. and Worldwide
Among all types of unconventional gas resources, shale gas is of particular interest because of its
huge reserves and great potential in the U.S. and worldwide (see Table 1.1 and Fig. 1.2). Shale
3
gas is natural gas that is formed through biogenic or thermogenic mechanisms within shale
formations [2]. It is composed primarily of methane and other hydrocarbons such as ethane,
propane and butane [3]. With an increasingly growing demand for natural gas resources, shale
gas has become one of the most important and promising sources of natural gas in the U.S. over
the past decade. In 2000, shale gas provided only 1 % of the U.S. natural gas production; by
2010 it was over 20 %.
Table 1.1. Estimated worldwide shale gas resources [6].
Region Gas Resource in Fractured Shales (Tcf)
NAM-North America 3,842
LAM-Latin America 2,117
WEU-Western Europe 510
EEU-Eastern Europe 39
FSU-Former Soviet Union 627
MEA-Middle East Asia 2,548
AFR-Africa 274
CPA-Central Pacific 3,528
PAO-Asia and China 2,313
PAS-Other Asia Pacific 314
World 16,112
4
The U.S. government's Energy Information Administration (EIA) predicts that by 2035 46 % of
the U.S. natural gas supply will come from shale gas [4]. The EIA’s annual energy outlook of
2012 projects that the U.S. natural gas production will increase from 21.6 trillion cubic feet (Tcf)
in 2010 to 27.9 Tcf in 2035, a 29 % increase. Almost all of this increase in domestic natural gas
production is due to the projected growth in shale gas production, with an expected growth from
5.0 Tcf in 2010 to 13.6 Tcf in 2035 [5]. With the success of the Barnett shale plays in the 1990’s,
interest has spread to potential gas shales in the rest of the world. A global energy study in 1997
estimated that abundant shale gas resources are distributed in North America, Latin America, and
the Asia Pacific region. Recent estimates suggest that shale gas reserves range from 1,483 to
1,859 Tcf in the U.S., and 500 to 600 Tcf in Canada. In other regions of the world, shale gas
resources have been studied to only a limited extent [6].
Figure 1.2. Map of basins with assessed shale oil and shale gas formations, as of May 2013 from
EIA.
5
1.3. Review of Sorption Studies on Shale
In the context of the reduction of greenhouse gas emissions, gas shales have received renewed
research attention in the past decades because of their emergence as key hydrocarbon reservoirs.
Natural gas is stored in shale in three ways: 1) free gas in the macroscopic cleat, the natural
fracture system of shale; 2) sorbed gas in micropore space within the shale matrix, including
adsorption and absorption; 3) dissolved gas in moisture which is contained in the shale layer.
During shale gas production, gas is first produced from fracture networks and macropores where
it exists as free gas, over a short period of time, followed by the free gas in mesopores, and from
the sorbed gas in the micropores for a longer period of time. As the primary mechanism of gas
storage in shale, sorption phenomena of CH
4
and other hydrocarbons in the micropores and
mesopores are critical to the estimate of gas-in-place (GIP) and of the long-term productivity
from a given shale play.
Research in the laboratory and field scales is currently under way to improve the understanding
of the transfer mechanisms which control the shale gas recovery. Lu et al. [7] measured
adsorption isotherms on various Devonian shales in the Appalachian basin for pressures up to 8
MPa and at temperatures ranging from 25 to 60
o
C. An approximately linear correlation between
adsorption capacities and total organic content (TOC) of shale samples is observed. Strapoc et al.
[8] studied the gas content of New Albany Shale from the eastern Illinois Basin. They discovered
that the production potential of the GIP is dependent on the total organic content and the
micropore volume of the shales. The positive correlation between TOC and sorbed gas was later
recognized by Daniel et al. [9], whose study revealed the effect of shale composition and pore
structure on gas storage potential of shale gas reservoirs. They stated that the organic fraction in
shales is an important measure of CH
4
storage capacity. Also, the relationship between organics
6
and CH
4
sorption is affected by mineral matter. Clay minerals such as illite have micropore
structures capable of sorbing gas. In addition, Daniel et al. [9] found that thermally mature shales
have larger micropore volumes and surface areas per wt% TOC. Hence the ratio of sorbed gas to
TOC is greater in thermally mature shales than immature shales. A similar study of the effect of
organic-matter type and thermal maturity on methane adsorption in shale-gas systems was
carried out by Zhang et al. [10]. According to them, generally, the higher the TOC content, the
greater the gas-sorption capacity. Differences in organic matter type greatly affect gas sorption
rates in organic-rich shales. The value of the Langmuir constant varies directly with kerogen type.
Organic matter and thermal maturation mainly affect the gas sorption capacity of organic rich
shales at lower pressure with gas sorption capacity being higher in shales with higher thermal
maturity. Another important factor which affects shale gas sorption is the moisture content.
Martina et al. [11] studied the effect of moisture on the sorption process of CO
2
on coal. Sorption
experiments with moisturized coal samples proved that an increasing amount of moisture leads
to a decrease in the CO
2
sorption capacity. Also, moisture has a significant impact on sorption
rates. The presence of moisture decreases the sorption rates of CO
2
on the coal samples. Day et al.
[12] studied the effect of moisture on the CO
2
and CH
4
sorption capacity of three bituminous
coals at 55 ° C and at pressures up to 20 MPa. They reported that moist coal had a significantly
lower maximum sorption capacity for both CO
2
and CH
4
than dry coal. However, the extent to
which the capacity was reduced was dependent upon the rank of the coal. Higher rank coals were
less affected by the presence of moisture than low rank coals. All coals exhibited a certain
moisture content beyond which further moisture did not affect the sorption capacity. In a recent
study, Chen et al. [13] investigated the sorption-introduced swelling of organic rich shale. In their
study, an experimental study of shale strain change with gas injection was carried out for two
7
shale samples from China. Both helium, a non-adsorbing gas, and methane, an adsorbing gas,
were used. They found the gas adsorption induced volumetric swelling strain is in the magnitude
of 0.1% at methane pressure of 10 MPa, which is about one magnitude lower than the methane
adsorption-induced swelling for coal. Also, the adsorption-induced swelling strain shows a
Langmuir-like curve with respect to gas pressure and shows a linear relationship with the
methane adsorption amount. A lot of experimental work regarding shales/coals sorption of pure
and multi-component gases for the application of shale gas production and enhanced coalbed
methane recovery is being conducted. For example, Nuttall et al. [14] studied methane and
carbon dioxide adsorption isotherms on New Albany shales and Ohio shales for CO
2
sequestration and enhanced CH
4
production purposes. In a recent study, Weniger et al. [15]
reported the first comprehensive set of high-pressure (up to 20 MPa) sorption data for CH
4
and
CO
2
for coals, carbonaceous shales and petroleum source rocks from the Paraná Basin, Brazil.
They reported an adsorption preference of CO
2
over CH
4
, as indicated by CO
2
/CH
4
sorption
capacity ratios that varied between 1.9 and 6.9 for several coal and shale samples. These studies
provided solid theoretical basis and experimental evidence for the application of methane
recovery from shale/coal formations.
In parallel, significant modeling efforts have been undertaken to date to represent pure and
multicomponent sorption on shales/coals. The extended Langmuir model (ELM) and ideal
adsorbate solution (IAS) theory are usually exclusively used in literature studies due to their
simplicity and low/moderate computational cost [16-19]. Fitzgerald et al. [20] used the
Langmuir/loading ratio correlation (LRC) and the Zhou–Gasem–Robinson (ZGR)
two-dimensional equation of state (2-D EOS) to model methane, nitrogen, CO
2
and their binary
and ternary mixtures on a wet Tiffany coal sample. They discovered that both LRC and ZGR 2-D
8
EOS are capable of representing the total adsorption for the pure, binary and ternary systems
within their expected experimental uncertainties. Pongtorn et al. [21] applied the simplified local
density (SLD) model to describe the adsorption data of methane, nitrogen, and CO
2
on a New
Albany shale, and found that the model was able to represent these data within the expected
experimental uncertainties. Ottiger et al. [22] used a lattice density functional theory (DFT)
model based on the Ono-Kondo equations to describe pure methane, nitrogen, CO
2
, and their
binary and ternary mixture adsorption on a dry coal. However, it has been proven that neither of
these models is capable to accurately describe the multicomponent sorption phenomenon that is
relevant to shale gas recovery processes. It is a challenge to model sorption in materials like
shale with heterogeneous nature and hierarchical pore size distribution.
1.4. Knowledge Gaps and Objectives of This Study
Similarly to most previous studies that are related to shale gas recovery, the emphasis in this
work is on the sorption of gases in the complex shale matrix system. As the primary mechanism
of gas storage in shale, sorption of methane (CH
4
) and other hydrocarbons in the micropores and
mesopores is critical to the estimate of the gas-in-place (GIP) and the long-term production from
a given shale play. Since ethane (C
2
H
6
)
is typically the second largest component, accounting for
more than 15 vol. % in certain cases [23], knowledge of CH
4
-C
2
H
6
binary sorption on shale is of
fundamental significance and plays a central role in understanding the physics that control fluid
storage, transport and subsequent shale gas production. Despite the fact that pure and
multicomponent gas sorption on shale has received research attention, there is still a lack of
knowledge regarding CH
4
-C
2
H
6
competitive sorption on shale at in situ pressure and temperature
9
conditions. Some knowledge gaps that need to be addressed in the short-term are: (i) additional
reliable sorption isotherm and dynamic data of CH
4
and C
2
H
6
in shales need to be generated; (ii)
there is currently lack of understanding of the mass transfer mechanisms in gas-shale systems,
i.e., whether diffusion, sorption and free gas flow (or a combination thereof) prevail inside the
shale matrix and within the shale fractures; (iii) there is currently lack of reliable models to
simulate the mass transfer processes in gas-shale systems and for retrieving useful shale
characteristics and dynamic properties. In order to close these knowledge gaps in the context of
shale-gas systems, we have carried out the following systematic research program:
First, a manometric method (Static apparatus) is employed to measure single component sorption
of CH
4
and C
2
H
6
on dry/moist shale rock samples. For pure component sorption, important
sorption information such as CH
4
and C
2
H
6
sorption equilibrium constants and maximum
sorption capacities can be extracted from the Langmuir model equation by a non-linear
regression method. The comparison of sorption isotherms on dry and moist samples allows us to
study the effect of moisture on shale rock sorption characteristics.
Second, Thermogravimetric Analysis (TGA) experiments are carried out using an advanced
magnetic suspension balance which allow us to measure pure and multicomponent excess
adsorption directly. Sorption isotherms of pure CH
4
and C
2
H
6
and their binary mixtures with
various compositions are measured at 40, 50 and 60
o
C. The Multicomponent Potential Theory of
Adsorption (MPTA) approach [24-27] is utilized to model the sorption data. The MPTA model is
shown capable of representing the pure component sorption data, and also provides adequate
predictive capability when applied to predict the total sorption for CH
4
-C
2
H
6
binary mixtures
over a range of compositions and temperatures.
10
Third, an experimental and modeling study of adsorption/desorption dynamics of CH
4
-C
2
H
6
mixture (and its individual components) in ground and whole shale rock samples is conducted
using thermogravimetric analysis (TGA). The sorption dynamic data (adsorbed amount vs. time)
contains valuable information regarding the dynamics of the diffusion-sorption processes taking
place inside the shale sample. In our initial modeling work focusing on the interpretation of the
sorption dynamic experiments, a Langmuir-type of sorption dynamic model is proposed, which
allows us to isolate sorption kinetics from diffusive and convective mass transfer. This, in turn,
facilitates our modeling and interpretation of the experimental observations.
In summary, the main objective of the research is to close the knowledge gaps in terms of shale
gas sorption and transport properties, and to improve the fundamental understanding of shale gas
mass transfer mechanisms. Based on the systematic studies of shale-gas systems, as outlined
above, we plan to improve the interpretation of production data from shale-gas wells by
leveraging an improved understanding of sorption dynamics and mass transfer of natural gas
mixtures in shale. The experimental and modeling efforts of this study will provide a better
understanding towards new knowledge in the context of shale gas production operations.
1.5. References
[1] Ratner M, Tiemann M. An overview of unconventional oil and natural gas: resources and
federal actions. Congressional Research Service. 2014 Nov 21;21.
[2] U.S. Energy Information Administration (EIA). “What is shale gas and why is it important?”
http://www.eia.gov/energy_in_brief/about_shale_gas.cfm. July 9, 2012.
[3] Naturalgas.org. "Composition of natural gas".
11
http://www.naturalgas.org/overview/background.asp. July 14, 2012.
[4] Stevens P. The ‘shale gas revolution’: Developments and changes. Chatham House. 2012
Aug:2-3.
[5] Outlook AE. US Energy Information Administration (EIA). Department of Energy (DoE).
2012.
[6] National Petroleum Council (NPC) Global Oil & Gas Study. “Unconventional gas subgroup
of the technology task group of the NPC committee on global oil and gas”. July 18, 2007.
[7] Lu XC, Li FC, Watson AT. Adsorption measurements in Devonian shales. Fuel. 1995 Apr
30;74(4):599-603.
[8] Strapoc D, Mastalerz M, Schimmelmann A, Drobniak A, Hasenmueller NR. Geochemical
constraints on the origin and volume of gas in the New Albany Shale (Devonian–Mississippian),
eastern Illinois Basin. AAPG bulletin. 2010;94(11):1713-40.
[9] Ross DJ, Bustin RM. The importance of shale composition and pore structure upon gas
storage potential of shale gas reservoirs. Marine and Petroleum Geology. 2009 Jun
30;26(6):916-27.
[10] Zhang T, Ellis GS, Ruppel SC, Milliken K, Yang R. Effect of organic-matter type and
thermal maturity on methane adsorption in shale-gas systems. Organic Geochemistry. 2012 Jun
30;47:120-31.
[11] Švábová M, Weishauptová Z, Přibyl O. The effect of moisture on the sorption process of
CO
2
on coal. Fuel. 2012 Feb 29;92(1):187-96.
12
[12] Day S, Sakurovs R, Weir S. Supercritical gas sorption on moist coals. International Journal
of Coal Geology. 2008 May 7;74(3):203-14.
[13] Chen T, Feng XT, Pan Z. Experimental study of swelling of organic rich shale in methane.
International Journal of Coal Geology. 2015 Oct 1;150:64-73.
[14] Nuttall BC, Eble CF, Drahovzal JA, Bustin RM. Analysis of Devonian black shales in
Kentucky for potential carbon dioxide sequestration and enhanced natural gas production.
Kentucky Geological Survey Report DE-FC26-02NT41442. 2005 Dec 30.
[15] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide
sorption on coal and shale samples from the Paraná Basin, Brazil. International Journal of Coal
Geology. 2010 Dec 1;84(3):190-205.
[16] Smith DH, Bromhal G, Sams WN, Jikich S, Ertekin T. Simulating Carbon Dioxide
Sequestration/ECBM Production in Coal Seams: Effects of Permeability Anisotropies and Other
Coal Properties. SPE Reservoir Evaluation & Engineering. 2005 Apr 1;8(02):156-63.
[17] Seto CJ, Jessen K, Orr FM. A multicomponent, two-phase-flow model for CO
2
storage and
enhanced coalbed-methane recovery. SPE Journal. 2009 Mar 1;14(01):30-40.
[18] Manik J. Compositional modeling of enhanced coalbed methane recovery. 1999.
[19] Jessen K, Lin W, Kovscek AR. Multicomponent sorption modeling in ECBM displacement
calculations. In SPE Annual Technical Conference and Exhibition 2007 Jan 1. Society of
Petroleum Engineers.
13
[20] Fitzgerald JE, Pan Z, Sudibandriyo M, Robinson Jr RL, Gasem KA, Reeves S. Adsorption
of methane, nitrogen, carbon dioxide and their mixtures on wet Tiffany coal. Fuel. 2005 Dec
31;84(18):2351-63.
[21] Chareonsuppanimit P, Mohammad SA, Robinson RL, Gasem KA. High-pressure adsorption
of gases on shales: Measurements and modeling. International Journal of Coal Geology. 2012
Jun 1;95:34-46.
[22] Ottiger S, Pini R, Storti G, Mazzotti M. Measuring and modeling the competitive adsorption
of CO
2
, CH
4
, and N
2
on a dry coal. Langmuir. 2008 Aug 5;24(17):9531-40.
[23] Bulba KA, Krouskop PE. Compositional variety complicates processing plans for US shale
gas. Oil & Gas Journal. 2009;107(10):50-5.
[24] Polanyi M. Section III.—theories of the adsorption of gases. A general survey and some
additional remarks. Introductory paper to section III. Transactions of the Faraday Society.
1932;28:316-33.
[25] Shapiro AA, Stenby EH. Potential theory of multicomponent adsorption. Journal of colloid
and interface science. 1998 May 15;201(2):146-57.
[26] Monsalvo MA, Shapiro AA. Prediction of adsorption from liquid mixtures in microporous
media by the potential theory. Fluid Phase Equilibria. 2007 Dec 1;261(1):292-9.
[27] Monsalvo MA, Shapiro AA. Study of high-pressure adsorption from supercritical fluids by
the potential theory. Fluid Phase Equilibria. 2009 Sep 15;283(1):56-64.
14
Chapter 2
Static Experiments of CH
4
and C
2
H
6
Sorption on Dry and Moist Shale
2.1. Introduction
Among all relevant techniques that are used to measure gas adsorption on coals/shales, the
manometric method is the one that is mostly employed [1-7]. The static experimental system
employed in this research uses the manometric technique for measuring CH
4
and C
2
H
6
adsorption on the shale sample. Adsorption experiments on both dry and water-saturated samples
were carried out to study the effect of moisture on gas sorption on shale. The static experimental
approach involves successive filling of a calibrated reference cell with the working gas, i.e., CH
4
or C
2
H
6
, and expanding it into a calibrated sample cell containing the shale sample of known
mass and grain volume. Details of this experimental approach were described by Krooss et al. [8],
Busch et al. [9], Siemons et al. [10], Gensterblum et al. [17].
Krooss et al. [8] studied CH
4
and CO
2
sorption on dry and moisture-equilibrated Pennsylvanian
coals. Adsorption isotherms of the two gases were measured up to pressure of 20 MPa, at 40, 60
and 80
o
C using the manometric method. They observed that both CH
4
and CO
2
sorption
capacities for moisture-equilibrated coals are lower than those for the dry samples. A strong
bimodal character (excess sorption reaches a minimum at certain pressure range, but increases
with pressure beyond this range) of CO
2
excess sorption isotherms on moist coals was identified
and interpreted as the result of a swelling effect caused by supercritical CO
2
and further enhanced
15
by water. Busch et al. [9] studied sorption kinetics of CH
4
and CO
2
on a Pennsylvanian coal in
the dry and moisture-equilibrated states using a manometric set-up. It was found that for moist
coals, sorption rates for both gases were reduced by a factor of more than 2 with respect to dry
coals. In addition, the sorption rate was found to be positively correlated with temperature. The
experiments were conducted on 6 different grain size fractions. Generally, adsorption rates
decreased with increasing grain size for all experimental conditions. Busch et al. [11-12] reported
that preferential adsorption of CH
4
was observed clearly and reproducibly for certain coal
samples in the low-pressure range, and the preferential desorption of CO
2
was observed also in
their study. They claim that, although the preferential adsorption of CO
2
on coals appears to be
the common case at high pressures, CH
4
may be preferentially adsorbed by certain coals
particularly in the low-pressure range. More importantly, for coals exhibiting preferential CH
4
sorption, a preferential desorption of CO
2
can be observed even at high pressures. Weniger et al.
[13] reported the first comprehensive set of high-pressure sorption data for CH
4
and CO
2
on
coals, carbonaceous shales and petroleum source rocks from the Paraná Basin, Brazil. The
manometric technique was used in their study to measure CH
4
and CO
2
sorption isotherms on
coals/shales at different temperatures and pressure up to 20 MPa. Zhang et al. [14] found in their
study important “cause and effect” relationships between organic matter type, thermal maturity
and gas sorption. Differences in organic matter type greatly affect gas sorption rates in organic
rich shales while organic matter thermal maturation affects the gas sorption capacity on organic
rich shales at lower pressure. Goodman et al. [15-16] carried out a series of inter-laboratory
comparison of CO
2
isotherms measured on both dry and moisture-equilibrated Argonne premium
coal samples. Three types of equipment used by laboratories, i.e., manometric, volumetric and
gravimetric systems, were used to measure the CO
2
sorption isotherms. The overall agreement
16
among laboratories was very good for high-rank coals and pressure up to 8 MPa, but the mid-
and low-rank coals and sorption isotherms at CO
2
pressures above 8 MPa showed large
variations. Gensterblum et al. [17-18] also conducted an inter-laboratory study of high-pressure
sorption isotherms of CO
2
on coals using the manometric and the gravimetric methods up to
16 MPa among four European research laboratories. The study shows that CO
2
sorption in the
supercritical range can be determined accurately with both gravimetric and manometric
equipment but requires thorough optimization of instrumentation and measuring as well as
proper sample preparation procedures.
2.2. Experimental
2.2.1. Sample Preparation
The shale sample used in this study is from the Marcellus formation in the Appalachian Basin.
The sample was extracted from a depth of 7,802.5 ft. Prior to its use in the sorption study, the
sample was stored in a zip-lock bag as received to avoid further oxidation and water
uptake/release. To reduce mass transfer effects and facilitate faster sorption of gases on the shale,
the sample was ground and sieved. 15.6 g of ground sample particles with diameters in the range
of 1-1.18 mm (US Mesh 16-18) were collected and subsequently used in the static experiments.
Prior to the initiation of these experiments, the sample was evacuated at 120
o
C for 24 hrs. At the
end of each sorption experiment, the sample was again regenerated under vacuum at 120
o
C for
24 hrs. Such a procedure assured that all water/gas that may potentially remain adsorbed at the
end of an experiment is desorbed prior to the initiation of next experiment. For the sorption
experiments with the moisture-equilibrated samples, deionized water was used to fully saturate
17
the shale sample (at a given humidity). To accomplish this, the shale sample was placed in an
environment where the relative humidity (RH) was kept constant at 100 % at 60
o
C for 24 hrs.
Over time, there has been discussion about the effect of sample size over sorption characteristics.
Weniger et al. [13] stated that grain size of coal/shale controls gas diffusion and sorption kinetics:
gas can diffuse faster to the adsorption sites in smaller sample particles so that the equilibrium
time will be shorter compared to larger particles. Cloke et al. [19] reported that grinding of coal
and size fractionation (sieving) may cause compositional change in sub-samples of various
particle sizes: maceral composition will differ from the smallest fraction (inertinite enriched) to
larger fractions (vitrinite enriched). Spears and Booth [20] observed that mineral matter tends to
be enriched in the smaller size fractions. Busch et al. [9] found these compositional changes can
also result in differences in sorption capacity. In our study, to avoid any compositional
fractionation, a small size interval of the sample particle, i.e. 1-1.18 mm, was chosen for the
static sorption experiments.
2.2.2. Experimental Set-up
A lab-scale static experimental set-up was built and tested to ensure accuracy and reducibility of
the measurement of sorption isotherms. Fig. 2.1 shows the schematic diagram of the static
apparatus.
18
Figure 2.1. Schematic diagram of the static experimental set-up.
The static experimental system consists of two cells (Swagelok stainless steel miniature
cylinders), which are identified as the sample cell and the reference cell, respectively. The
nominal volumes of both cells are 10 cm
3
(see Sec 2.2.4. for the He expansion experiment which
accurately determines these volumes). Both the sample cell and the reference cell are located in a
GC (Varian 3400) oven to allow for strict temperature control (temperature fluctuation of less
than 0.1 K around the set point as stated by the GC manufacturer). Two ultra-precision digital
pressure gauges P
s
and P
r
(3D Instruments accu-cal plus digital test gauge with accuracy of 0.04 %
GC Oven
V1
He
CH
4
/C
2
H
6
V2
V3
Sample
Cell
Reference
Cell
Pressure
Gauge
Pressure
Gauge
V4
V5
Vacuum
Pump
V6
19
of reading plus 0.01 % of 3000 psig full scale) were applied to monitor the pressures inside the
sample cell and the reference cell, respectively. For water saturation purpose, two water tanks
(Swagelok stainless steel miniature cylinders) were filled with deionized water and installed
sideways towards the reference and sample cells to introduce water to the adsorbate and
adsorbent, respectively. A vacuum pump (Alcatel 2012A) is connected at the end for evacuating
the system.
2.2.3. System Leak Test
Since the manometric method relies on the pressure readings before and after each sorption step,
it is crucial to ensure that the system is free from any leakage. Failure to comply with this
requirement will result in over-estimate of sorption isotherms and will subsequently cause
significant experimental uncertainty. To assess the system’s leak rate, a leak test using an inert
gas (He in our case) was conducted prior to the sorption measurements. Since the static sorption
experiments will not exceed the maximum pressure of 100 bar at 60
o
C, the leak test was
performed at 120 bar at 60
o
C. The reference cell and the sample cell were connected during the
leak test. Fig. 2.2 shows the pressure reading (P
r
) during the 25-hr leak test.
20
Figure 2.2. Static system He leak test.
It can be seen that the system pressure has a tendency to decline over the period of the leak test.
Calculation shows that the leak rate is less than 0.015 bar/hr under the testing conditions, i.e.,
120 bar and 60
o
C. An estimate of system acceptable leak rate shows the percentage error of
excess adsorption is less than 5 % if the leak rate < 0.05 bar/hr at 100 bar. So, the effect of the
observed leakage can be considered insignificant and acceptable.
2.2.4. Volume Measurement of the Reference Cell and the Sample Cell
As the first step prior to the initiation of the sorption experiments, we have to measure the
volumes of the reference cell and the sample cell. He expansion with the aid of known-volume
quartz was implemented in this measurement. Clear fused quartz with definitive density (2.203
g/cm
3
) manufactured by National Scientific Company was used. The volume was calculated by
the quartz weight divided by the density. The experimental protocol for the volume measurement
is as follows:
120
120.1
120.2
120.3
120.4
120.5
0 5 10 15 20 25 30
Pressure (bar)
Time (hr)
21
1. Close the valves that connect the reference cell and sample cell to the water tanks.
2. Make sure that the valve (V3) between V
R
and V
S
is opened. Vacuum V
R
and V
S
for 10
min until the pressure gauge reads -1.01 bar and stabilizes.
3. Close V3 and V6. Charge V
R
with He up to a certain pressure (around 20 bar in our
experiment) and wait until the pressure reading becomes stable. Denote this initial
pressure as p
1
.
4. Open V3 to let He expand from the reference cell into the sample cell. Wait until the
pressure equilibrates between two cells. Denote the new pressure as p
2
.
5. Add a certain amount of quartz of accurately known-volume V
Ki
into the sample cell.
6. Repeat Step 2 – 5 above until the sample cell is filled up with quartz.
Table 2.1 below shows the experimental data obtained from the He expansion measurement.
Table 2.1. He expansion pressure data.
Volume of quartz
Ki
V (cm
3
)
Reference cell pressure
p
1
(bar, abs)
Equilibrium pressure
p
2
(bar, abs)
(p
1
-p
2
)/p
2
0.0000
18.03 9.83 0.8342
1.2879 17.61 10.10 0.7436
2.6276 17.65 10.70 0.6495
4.3406 17.63 11.49 0.5344
5.4158 17.63 12.10 0.4570
22
According to Boyle’s Law:
Ki S R R
V V V p V p
2 1
or
R
Ki S
V
V V
p
p p
2
2 1
(i=1, 2, … ,) (2.1)
where:
R
V --- reference volume including the reference cell, fittings, and the dead volume of
the pressure gauge P
r
;
S
V --- volume of the sample cell plus dead volume of fittings and the pressure gauge P
s
;
Ki
V --- volume of the non-absorbable quartz.
If we plot the ratio of (p
1
-p
2
)/p
2
as a function of
Ki
V , as shown in Fig. 2.3, since
R
S
Ki
R R
Ki S
V
V
V
V V
V V
p
p p
1
2
2 1
(2.2)
The slope of the above line is (-1/V
R
) and the intercept is (V
S
/V
R
). We can hence get the values for
both V
R
and V
S
from the He expansion experiment.
23
Figure 2.3. He expansion for volume measurement.
The accuracy of the above volume measurement, as reflected by the 95 % confidence interval,
was calculated as follows:
V
R
= 14.41 ± 0.14 cc
V
S
= 12.01 ± 0.03 cc
The experimental uncertainty of the volume measurement via the helium expansion method is
± 1 %. That is acceptable in the shale sorption measurements because the ± 1 % uncertainty in the
volume measurement results in only ± 2 % uncertainty in our sorption experiments, which is
considered satisfactory.
y = -0.0694x + 0.8334
R² = 0.9999
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5 6
(p
1
-p
2
)/p
2
Volume of quartz (cm
3
)
24
2.2.5. Sorption Measurement Procedures for CH
4
and C
2
H
6
Sorption on
Dry/Moist Shale Sample
2.2.5.1. Procedure for Gas Sorption on Dry Sample
The procedure for measuring sorption on dry samples is as follows:
1. Load as much ground shale sample into the sample cell as possible. Leak-test the system
using He to ensure a He leakage rate of less than 0.05 bar/hr at 100 bar (the leakage rate
is determined to ensure that the experimental error caused by the leakage does not exceed
5 %).
2. Shut valves V4 and V5. Evacuate the sample at a pressure level of less than 30 mbar at
120
o
C for 24 hrs. Then close valves V3 and V6.
3. Set the GC oven to 60
o
C (the accuracy of the GC oven is 0.1 K).
4. Repeat steps 3 – 4 from the volume measurement procedure described in Sec. 2.2.4. to
get the skeletal volume of shale sample. Evacuate the system again at 60
o
C for 5 hrs.
Then close valves V3 and V6.
5. Open valve V2 to let CH
4
charge the reference cell to 10 bar (total pressure). Wait until
the pressure and temperature equilibrate in the reference cell. (The steady state is reached
if the reference cell pressure does not change by more than the preset acceptable leakage
rate over a period of 3-5 hrs).
6. Open valve V3 to allow the gas to enter the sample cell. Wait until the adsorption reaches
equilibrium (for definition of equilibrium see Step 5 above).
25
7. Close valve V3. Repeat 5 – 6 while increasing the total pressure in the reference cell from
10 to 20, 30, 40 bar… until the final pressure level, i.e. 100 bar, is reached.
2.2.5.2. Procedure for Gas Sorption on Moist Sample
The procedure for measuring sorption on moist samples is as follows:
1. Leak-test the system again using He to ensure a He leakage rate of less than 0.05 bar/hr at
100 bar and to make sure that no new leaks have developed during the dry sample
adsorption tests.
2. Shut valves V4 and V5. Evacuate the sample at a pressure level of less than 30 mbar at
120° C for 24 hrs. This treatment was shown to completely desorb the residual gas from
the sample. Close valves V3 and V6.
3. Add deionized water in both side cells (water tanks) that connect with the reference cell
and the sample cell.
4. Set the GC oven to 60
o
C (the accuracy of the GC oven is 0.1 K).
5. Open valve V2 to let CH
4
charge the reference cell to 10 bar (total pressure). Wait until
the pressure and temperature equilibrate in the reference cell (The equilibrium state is
reached if the reference cell pressure does not change by more than the preset acceptable
leakage rate over a period of 3-5 hrs).
6. CH
4
saturation: Open valve V4 to let the water vapor saturate the CH
4
in the reference
cell. Wait until the pressure and temperature equilibrate in the reference cell (for
definition of equilibrium see Step 5 above). Then close valve V4.
26
7. Sample saturation: Open valve V5 to let water vapor saturate the sample in the sample
cell. Wait until the pressure and temperature equilibrate in the sample cell, in our case
more than 24 hrs, in order to make sure the 100 % relative humidity is achieved in the
sample cell and that the shale sample was fully saturated. Then close valve V5.
(Note: The CH
4
saturation and sample saturation can be done simultaneously. The CH
4
saturation needs to be repeated every time dry CH
4
is introduced into the system. The
sample saturation only needs to be done once.)
8. Open valve V3 to allow the gas to enter the sample cell. Wait until the adsorption reaches
equilibrium (for definition of equilibrium see Step 5 above).
9. Close valve V3. Repeat 5 – 8 while increasing the total pressure in the reference cell from
10 to 20, 30, 40 bar… until the final pressure level, i.e. 100 bar, is reached.
10. After the above steps are finished, take out the regenerated shale sample from sample cell
and weigh the sample.
2.2.6. Analysis Approach for Gas Sorption in the Static Apparatus
The excess sorption of single gas sorption, i.e., CH
4
or C
2
H
6
,
for the individual step is
calculated via the real gas law, i.e. Eq. 2.3. The accumulated amount of sorbed gas is
subsequently computed by Eq. 2.4.
si
si
sf
sf
V
rf
rf
ri
ri
R
s iso g
eas
i
Z
p
Z
p
V
Z
p
Z
p
V
m T R
n
1
(2.3)
eas
i
eas eas eas eas
i
n n n n n ...
3 2 1
(2.4)
27
where:
g
R --- molar gas constant, mol/ (mol.K);
iso
T --- isothermal temperature, K (controlled by a thermostat with an accuracy of ± 0.1 K);
s
m --- mass of the sample, gr (measured by a SARTORIUS analytical scale with an accuracy of ±
0.00001 g);
R
V --- volume of the reference cell, cm
3
(calculated through a method of helium expansion with
the aid of known-volumes of inert quartz with an accuracy of ± 1.0 %);
V
V --- void volume in the sample cell, cm
3
(
solid sample S V
m V V ) (
solid
is the solid density of
the shale sample and was calculated through helium expansion measurement in the static
apparatus);
p --- pressures in the cells, bar (ri denotes the reference initial, rf denotes the reference final, si
denotes the sample initial, and sf the sample final. The error in pressure readings is the pressure
gauge accuracy, i.e., 0.04 % of reading plus 0.01 % of full scale of 3000 psig);
Z --- compressibility factors at the relevant conditions, dimensionless (estimated based on the
NIST Chemistry WebBook);
eas
i
n --- excess sorption from the i
th
step, mg/g;
eas
i
n --- sum of the excess sorption from all of the previous i steps, mg/g.
The static experimental method measures the excess sorption, which is defined physically as the
total amount of gas adsorbed minus the mass of gas of the same volume at the bulk density
28
conditions of the experiment. For adsorbing gases of high density, i.e., at high pressures, the
“excess adsorption” is quantity most often measured in the laboratory. However, in the context of
thermodynamic properties calculations and theoretical models and molecular simulations, the
absolute sorption arises naturally. Therefore, the conversion of the excess to the absolute sorption
is a crucial step toward the understanding and analysis of the sorption data. A detailed discussion
pertaining to the relationship between the excess sorption and the absolute sorption can be found
in Chapter 4.
2.3. Results
2.3.1. CH
4
and C
2
H
6
Sorption on Dry Sample
As noted in Sec. 2.2.1., upon the completion of one sorption experiment, and prior to the
initiation of another, the shale sample is evacuated for 24 hrs at 120
o
C. This has been shown to
not only evaporate water completely, but also to restore the surface of the sample to its original
state via desorption of any residual gases that may remain adsorbed [22]. CH
4
and C
2
H
6
pure
component sorption isotherms for pressures up to 90 bar and 35 bar at 60
o
C, respectively, were
obtained by the experimental procedure (Sec. 2.5.2.1.) discussed above, and are reported in Fig.
2.4 and Table 2.2.
29
Table 2.2. CH
4
and C
2
H
6
sorption data on dry shale at 60
o
C.
CH
4
C
2
H
6
Pressure (bar) m
eas
(mg/g) Pressure (bar) m
eas
(mg/g)
6.35 0.420923 4.03 1.724965
14.39 0.741094 7.12 2.240572
24.45 0.994902 13.75 2.867448
34.54 1.185820 19.10 3.200240
43.87 1.313271 23.55 3.400034
54.69 1.431973 29.99 3.487980
63.97 1.526536 36.55 3.387485
74.08 1.609413
83.78 1.671738
90.03 1.700048
30
Figure 2.4. CH
4
and C
2
H
6
sorption on dry shale sample at 60
o
C.
The error bars were calculated using the propagation-of-error theory. Due to the fact that the
accumulative sorption was calculated by summing the excess sorption from each step, the
maximum error arises at the maximum pressure for both CH
4
and C
2
H
6
isotherms. From Fig. 2.4
we observe that the CH
4
sorption isotherm has increased continuously with pressure and
appeared to approach an asymptote. However, the same is not observed for C
2
H
6
excess sorption,
which shows a maximum sorption at ~30 bar followed by a decline. A maximum excess sorption
has frequently been reported for CO
2
sorption on coals/shales, particularly near its critical
density [17-18]. Theoretically, once sorption reaches saturation, i.e., C
2
H
6
molecules fill all the
adsorption sites in the shale matrix, the excess sorption will start to decline, and the decline rate
is a constant if the adsorbed-phase density and volume are assumed to be constant.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100
Excess sorption (mg/g)
Pressure (bar)
CH4 C2H6
31
2.3.2. CH
4
and C
2
H
6
Sorption on Moist Sample
CH
4
and C
2
H
6
sorption measurements on the moist sample were performed by following the
procedure described in Sec. 2.5.2.2. Fig. 2.5 presents the experimental results with experimental
data listed in Table 2.3.
Table 2.3. CH
4
and C
2
H
6
sorption data on moist shale at 60
o
C.
CH
4
C
2
H
6
Pressure (bar) m
eas
(mg/g) Pressure (bar) m
eas
(mg/g)
5.34 0.254504 5.77 1.247451
13.66 0.488653 9.96 2.060077
23.38 0.679265 15.17 2.52998
33.39 0.871688 20.7 2.77139
43.11 1.057656 25.87 2.838756
52.83 1.195621 31.45 2.840918
62.75 1.312097 36.95 2.586969
72.27 1.398591
81.74 1.469774
88.83 1.520803
32
Figure 2.5. CH
4
and C
2
H
6
sorption on moist shale sample at 60
o
C.
To better illustrate the effect of moisture content on shale sorption characteristics, comparisons
of gas sorption on dry and moist samples for CH
4
and C
2
H
6
are reported in Figs. 2.6 to 2.7,
respectively.
Figure 2.6. CH
4
sorption on dry/moist shale sample at 60
o
C.
0
0.5
1
1.5
2
2.5
3
3.5
0 20 40 60 80 100
Excess sorption (mg/g)
Pressure (bar)
CH4 C2H6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 20 40 60 80 100
Excess sorption (mg/g)
Pressure (bar)
dry moist
33
Figure 2.7. C
2
H
6
sorption on dry/moist shale sample at 60
o
C.
It can be seen from Figs. 2.6 to 2.7 that excess sorption capacity on the moist sample decreases
by ~20 % more than on the dry sample for both CH
4
and C
2
H
6
cases. It is generally found that
increasing moisture content results in a reduction of the shale/coal sorption capacity. Krooss et al.
[8] measured high-pressure CH
4
and CO
2
adsorption isotherms on several dry and
moisture-equilibrated Dutch coals. For CH
4
, they found the sorption capacity of the moist coals
to be as much as 25 % lower than for the dry coal. Clarkson and Bustin [2] studied several
Canadian coals and found that both methane and CO
2
sorption capacity (in pure gases) was
reduced by approximately 25–30 % for moist coals relative to the dry material. Joubert et al. [21]
suggested that CH
4
sorption capacity decreases with increasing moisture content up to a certain
value of moisture content that was characteristic for each coal. Moisture present in excess of the
critical value had no further effect on methane sorption capacity. Clearly, the results of our CH
4
and C
2
H
6
experiments on the dry/moist sample support the observation that the presence of water
would reduce the sample sorption capacity. One possible explanation is that the accessible
0
0.5
1
1.5
2
2.5
3
3.5
4
0 10 20 30 40
Excess sorption (mg/g)
Pressure (bar)
dry moist
34
micropore volume in a moist sample is much less than in a dry sample, due to either a reduction
of pore size caused by water adsorption or due to swelling of the sample [8].
2.4. Conclusion
CH
4
and C
2
H
6
sorption isotherms for a ground shale sample at 60
o
C were measured in a static
experimental apparatus. We observe that CH
4
sorption isotherm increase continuously over the
pressure range, while C
2
H
6
excess sorption shows a maximum sorption at ~30 bar followed by a
decline. The ratio of C
2
H
6
/CH
4
excess sorption is quite large, varying between 1.5 and 2.5 (on a
molar basis) over the common pressure range investigated. The impact of moisture on the
sorption isotherms was studied by performing sorption experiments on water-equilibrated shale
samples. It is found that the excess sorption capacity on the moist sample decreases by ~20 %
relative to the dry sample for both CH
4
and C
2
H
6
, which implies competitive sorption between
water and gas molecules in the shale matrix, or possibly, swelling of the shale sample due to
water uptake.
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35
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2
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[20] Spears DA, Booth CA. The composition of size-fractionated pulverised coal and the trace
element associations. Fuel. 2002 Mar 31;81(5):683-90.
[21] Joubert JI, Grein CT, Bienstock D. Sorption of methane in moist coal. Fuel. 1973 Jul
31;52(3):181-5.
[22] Wang Y, Tsotsis TT, Jessen K. Competitive Sorption of Methane/Ethane Mixtures on Shale:
Measurements and Modeling. Industrial & Engineering Chemistry Research. 2015 Nov
20;54(48):12187-95.
38
Chapter 3
Competitive Sorption of Methane/Ethane
Mixtures on Shale on a TGA Apparatus
3.1. Introduction
1
Gas shales have emerged as key hydrocarbon reservoirs in the past couple of decades or so, and
have received renewed research attention in recent years in the context of reducing greenhouse
gas emissions. The U.S. government's Energy Information Administration (EIA) predicts that by
2035 46% of the U.S. natural gas supply will come from shale gas [1]. Sorption of gases in the
complex shale matrix system is the key mechanism via which shale-gas is stored in such
formations, and has as a result received attention in a number of previous studies related to shale
gas recovery [2-5]. However, shale-gas (consisting primarily of methane in its mixture with
various other low MW hydrocarbons) sorption data on shale under realistic reservoir pressure
and temperature conditions are still lacking, and generating such data is the key motivation
behind this work.
It is, in general, a challenge to model sorption in natural materials, such as shale, due to their
heterogeneous nature and hierarchical pore structures (manifested, typically, by multi-modal pore
size distributions), which makes the representation of competitive sorption phenomena in these
systems quite difficult. Past modeling efforts, for example, have attempted to describe
1
The material in this Chapter is part of a paper published in Industrial & Engineering Chemistry Research: Yu Wang, Theodore
T. Tsotsis, and Kristian Jessen. Competitive Sorption of Methane/Ethane Mixtures on Shale: Measurements and Modeling.
39
multicomponent gas sorption in shales from pure component isotherm data via the use of the
extended Langmuir model (ELM), and also the Ideal Adsorbate Solution (IAS) theory, the main
motivation behind such efforts being the simplicity and low/moderate computational cost of
these commonly utilized methods [6-9]. However, it has been demonstrated that neither of these
models is capable of accurately describing the multicomponent sorption behavior that is relevant
to shale-gas recovery processes [10-12]. Here, we test, instead, the accuracy of the
Multicomponent Potential Theory of Adsorption (MPTA) approach for the calculation of gas
sorption on shale. The choice of the MPTA method, which originated from the potential theory
concept suggested by Polanyi [13], is because it has been utilized before to model the sorption of
gases and liquids in microporous materials with promising results [14-16].
3.2. Experimental
3.2.1. Sample Preparation
The shale sample used in this study is from the Marcellus formation in the Appalachian Basin. It
was extracted from a depth of 7,802.5 ft. Prior to its use in the present sorption study, the sample
was stored in a zip-lock bag as received to avoid further oxidation and water uptake. To reduce
internal mass transfer limitation effects and to facilitate faster sorption of gases on the shale, the
sample was ground and sieved. Sample particles with diameters in the range of 1-1.18 mm (US
Mesh 16-18) were then collected and subsequently used in the sorption experiments. Prior to the
initiation of these experiments, the sample was then evacuated at 120
o
C for 24 hrs. At the end of
each sorption experiment, the sample was again regenerated under vacuum at 120
o
C for 24 hrs.
Such a procedure assures that all gases that may potentially remain adsorbed at the end of an
40
experiment get desorbed prior to the initiation of the next experiment. This, then, guarantees
reproducibility among replicate runs, as discussed further in Sec. 3.3.1.
3.2.2. Experimental Approach
The thermogravimetric analysis (TGA) technique is used for the measurement of the sorption
data. The heart of the TGA set-up is a Magnetic Suspension Balance (Rubotherm, Germany),
which is capable of measuring weight changes down to 1 μg. During an experiment, the weight
change of the sample due to sorption is transmitted from the sorption chamber to the analytical
balance in a contactless manner via the magnetic suspension mechanism. A sorption experiment
consists of several steps: First, one must measure the weight of the empty sample container,
sc
M
and its volume
sc
V at the temperature of the experiment. This is accomplished by pressurizing
the sorption chamber with the sample container alone (without the sample in place) in a
step-wise manner using a flowing inert gas (Helium) and recording its apparent weight
app sc
M
,
at various pressures.
app sc
M
,
relates to
sc
M and
sc
V according to the following relationship
(the He density
He
in Eq. 3.1 is measured also in situ in the TGA set-up by using a reference
stainless steel insert of known volume and weight):
sc He sc app sc
V M M
,
. (3.1)
By plotting
app sc
M
,
vs. the He density
He
, one can calculate both the
sc
M (as the intercept)
and the
sc
V (as the slope). The above experimental step is then repeated after the shale sample is
placed into the sorption chamber to calculate the combined weight of the sample and the sample
container, (
sc s
M M ), and their total volume (
sc s
V V ). In a third step, after evacuating once
41
more the sample chamber at the eventual temperature of the experiment for 5 hrs, the sorption
chamber is filled with the flowing gas to be studied, i.e., CH
4
, C
2
H
6
or one of their mixtures at an
initial pre-determined pressure and the apparent weight is monitored until it stabilizes. The
chamber pressure is then raised in a step-wise manner consistent with steps 1 and 2 above. The
TGA again measures the apparent weight T M
b
m
, at the desired conditions (T, p):
a
sc s
b
m
a
sc s
b
m
V V V m M M T M , , (3.2)
where
a
m is the mass sorbed on the sample,
a
V is its volume, and
b
m
is the mass density of
the bulk fluid (CH
4
, C
2
H
6
or their mixtures) as measured in situ using the aforementioned
method for measuring the He density. The experimental density values are quite accurate, as Figs.
S.1 to S.5 in the Supplemental Materials Section show that compare these values with modeling
via the Peng-Robinson (PR) equation of state. By rearranging Eq. 3.2, one can directly obtain the
measurable quantity, i.e., the excess sorbed mass
eas
m by the following Eq. 3.3:
sc s
b
m sc s
b
m
a b
m
a b
m
eas
V V M M T M V m T m , , . (3.3)
TGA (but also volumetric and chromatographic sorption) measurements cannot distinguish
between amounts that are adsorbed and absorbed. For a solid material, an indication that
absorption may be taking place is sample swelling. However, for heterogeneous samples like
shales consisting of both an inorganic backbone and potential organic inclusions it is not possible
to accurately measure such swelling effects.
42
3.3. Results
3.3.1. CH
4
and C
2
H
6
Pure Component Isotherms
CH
4
and C
2
H
6
pure component sorption were measured on the shale sample at three different
temperatures, namely, 40
o
C, 50
o
C and 60
o
C. For each temperature, the weight and volume of
the shale sample were measured. During the experiments, we observed that the shale sample
undergoes a slight thermal expansion and volume change in response to the change
in temperature. Specifically, the measured volume of the sample used in our TGA experiments,
which weighs 2.2880 g, is 0.8414 cc, 0.8421cc and 0.8439 cc, at the three different temperatures
of 40
o
C, 50
o
C and 60
o
C, respectively. The impact of this volume change is rather small, but it
is also taken into account in the calculations of excess adsorption via Eqs. 3.2 and 3.3. CH
4
and
C
2
H
6
pure component isotherms for pressures up to 114 bar and 35 bar, respectively, were
obtained by the experimental procedure discussed above, and are reported in Figs. 3.1- 3.3 (and
Tables S.1-S.3, in the Supplemental Section).
43
0 20 40 60 80 100 120
0
1
2
3
4
5
CH4
C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.1. CH
4
and C
2
H
6
pure component sorption on shale at 40
o
C.
0 20 40 60 80 100 120
0
1
2
3
4
5
CH4
C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.2. CH
4
and C
2
H
6
pure component sorption on shale at 50
o
C.
44
0 20 40 60 80 100 120
0
1
2
3
4
CH4
C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.3. CH
4
and C
2
H
6
pure component sorption on shale at 60
o
C.
In the range of experimental pressures utilized (up to ~114 bar) as can be seen from Figs. 3.1- 3.3
(and also Tables S.1-S.3, in the Supplemental Section) the CH
4
sorption isotherm has approached
an asymptotic behavior. However, the same is not true for C
2
H
6
excess sorption, which for the
pressure range utilized (up to ~35 bar) increases monotonically over the entire range of pressures.
Several other studies on methane sorption on coal and shale samples have also reported similar
behavior of maximum CH
4
loadings attained ~100 bar [17-23]. Prior studies, for example, have
found that this maximum loading behavior happens when the gas reaches its supercritical fluid
state and beyond [24, 25]. For CH
4
, T
c
= -82.59 ° C and p
c
= 45.99 bar, while for C
2
H
6
, T
c
=
32.17 ° C and p
c
= 48.72 bar, so our experimental observations are consistent with those of the
prior studies. At higher pressures, assuming that adsorption is the only mechanism for sorption,
one expects the excess sorbed amount after reaching a maximum value to start declining as the
pressure keeps on rising. Clear from Figs. 3.1-3.3 are the notably different sorption affinities of
45
the shale towards CH
4
and C
2
H
6
, with C
2
H
6
displaying the stronger affinity: The ratio of
C
2
H
6
/CH
4
excess sorption is quite large, varying between 1.5 and 2.5 (on a molar basis) over the
common pressure range investigated.
As noted in Sec. 3.2.1., upon the completion of one sorption experiment and prior to the
initiation of another the shale sample is evacuated for 24 hrs at 120
o
C. This has been shown to
restore the surface of the sample to its original state via the desorption of any residual gases that
may remain adsorbed. The regeneration approach that we have used has been found effective in
assuring experimental reproducibility among the various runs. For example, Fig. 3.4 shows the
results of two consecutive methane sorption runs at 60
o
C. Between the runs the shale sample
was subjected to the aforementioned regeneration step. The data in Fig. 3.4 indicate very good
reproducibility between the two consecutive runs.
0 20 40 60 80 100 120
0
1
2
Run 1
Run 2
Excess sorption (mg/g)
Pressure (bar)
Figure 3.4. Reproducibility test of CH
4
sorption on shale at 60
o
C.
46
3.3.2. CH
4
and C
2
H
6
Pure Component Adsorption/Desorption Hysteresis
Though adsorption isotherm data are of fundamental importance, from an engineering
perspective desorption phenomena are of significance as well, as they dominate during the
shale-gas recovery process. As part of this study, therefore, we have investigated the desorption
characteristics of CH
4
and C
2
H
6
on the shale sample. Figs. 3.5 and 3.6, for example, show CH
4
and C
2
H
6
adsorption/desorption isotherms at 60
o
C as measured with the TGA set-up.
0 20 40 60 80 100 120
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
adsorption
desorption
Excess sorption (mg/g)
Pressure (bar)
Figure 3.5. CH
4
adsorption/desorption isotherms on shale at 60
o
C.
47
0 5 10 15 20 25 30 35 40
0
1
2
3
4
adsorption
desorption
Excess sorption (mg/g)
Pressure (bar)
Figure 3.6. C
2
H
6
adsorption/desorption isotherms on shale at 60
o
C.
Two interesting features are observed in Figs. 3.5 and 3.6: First, moderate hysteresis behavior is
observed between the loading (adsorption) and unloading (desorption) curves. Second, the
desorption isotherm does not return to the origin at p = 0 bar. Though in recent years hysteresis
phenomena have been observed with microporous materials as well, they are typically associated
with mesoporous systems (i.e., those containing pores in the range between 2 and 50 nm) with
complex 3-D pore structures. It has been reported that sorption hysteresis in such materials is
associated with liquid-gas phase transitions under porous confinement. It is thought, that during
adsorption the pores are filled by the liquid in the order of increasing pore radii, while on
desorption the dimension of the pore necks (narrowest parts of the pore structure) controls the
emptying of the pores. This explains why adsorbed gas molecules get “stuck” in these
mesoporous materials and consequently create the adsorption hysteresis. The shale sample
studied here has a hierarchical pore structure (as the data in Table 3.1 indicate) characterized by
48
microporous, mesoporous, and macroporous regions, but the largest fraction of the pore space
(>76 %) resides in the mesopore range.
Interestingly enough, as Figs. 3.5 and 3.6 indicate, a fraction of the adsorbed CH
4
and C
2
H
6
resists coming out of the shale sample even when it is evacuated (p = 0 bar) for > 12 hrs. This
may signify a part of the shale sample (perhaps an organic inclusion) where both these gases are
held more tightly. Heating under vacuum at 120
o
C for 24 hrs results in the desorption of these
residual amounts and restores the sample to its original state (see Fig. 3.4).
Table 3.1. Distribution of micropore, mesopore, and macropore volumes in the shale sample as
measured by BET.
Cumulative micropore
(<2 nm)
volume (cc/g)
Cumulative
mesopore (2-50 nm)
volume (cc/g)
Cumulative
macropore (50-500 nm)
volume(cc/g)
Total pore
volume
(cc/g)
0.0081 0.0255 0.0010 0.0346
3.3.3. Competitive Adsorption of CH
4
-C
2
H
6
Binary Mixtures
Sorption isotherms were also measured for binary mixtures of CH
4
-C
2
H
6
on the shale sample at
the same temperatures (i.e., 40
o
C, 50
o
C and 60
o
C) as those employed for the single gas
experiments. Three different CH
4
-C
2
H
6
binary gas compositions (90%-10%, 93%-7%, and
96%-4% mole fraction – certified pre-mixed gases purchased from Matheson) were used in these
experiments, and the isotherms were measured by varying, in a step-wise manner, the total
pressure of the mixture. Estimates of the rate of uptake indicate it to be < 0.2 % of the gas flow
49
rate through the TGA chamber, meaning that the mixture composition stays virtually constant
throughout the sorption experiment. Figs. 3.7-3.9 (and Tables S.4-S.6 in Supplemental Materials
Section) report the total (CH
4
+ C
2
H
6
) excess sorption data measured for these mixtures.
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.7. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 40
o
C.
50
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.8. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 50
o
C.
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 3.9. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 60
o
C.
51
From Figs. 3.7-3.9, we observe an increase in the total excess sorption amount as the
concentration of C
2
H
6
in the binary mixture increases. This is consistent with the single-gas
sorption data that indicate the preferential sorption of C
2
H
6
over CH
4
in the shale sample. In
addition, we observe a maximum in the total excess loading at ~100 bar beyond which the total
excess sorption amount starts to decrease, while this was not observed from the CH
4
or the C
2
H
6
pure component sorption isotherms. We attribute the maximum excess loading observed for
binary mixtures to interactions between the two species in the sorbed phase, and defer additional
discussion of this behavior to the modeling section below.
3.4. Modeling
3.4.1. Multicomponent Potential Theory of Adsorption (MPTA)
Based on the original potential theory of adsorption (PTA), originally suggested by Polanyi
[13],
Shapiro and Stenby [14] introduced the multicomponent potential theory of adsorption (MPTA).
In this theory, the adsorbate is considered to be a separate phase subjected to an external
attractive potential field z
i
exerted by the adsorbent itself [26]. For the i
th
component in the
adsorbed phase, the isothermal equilibrium state is reached when the sorption potential on that
component z
i
exerted by the adsorbent at any position z within the adsorbed phase equals
the difference between its chemical potential in the bulk phase y p
y ig
, and the chemical
potential in the adsorbed phase ) ( ), ( z x z p
i
at location z:
y p z z x z p
y ig i i
, ) ( ), ( nc i ,..., 1 . (3.4)
52
In Eq. 3.4 above, ) (z p is the pressure in the sorbed phase,
y
p is the pressure in the bulk
phase, x and y are the mole fractions of component i in the sorbed phase and bulk phase,
respectively. Eq. 3.4 can be rewritten in the form of fugacities:
RT
z
y p f z x z p f
i
y ig i
, ln ) ( ), ( ln . (3.5)
In order to determine the distribution of pressures ) (z p and mole fractions ) (z x in the sorbed
phase, in addition to an appropriate equation of state (EOS) that describes the bulk and sorbed
phases, one has to also assume a representation of the sorption potentials z
i
. Traditionally, in
MPTA the same EOS is used to describe both phases. Though this does not represent an intrinsic
limitation for the theory since, in principle, one could use a different EOS to describe the
adsorbed phase, in practice no such experimentally-validated EOS exist today.
Shapiro and Stenby
[14] assumed in their original paper the following generalized
Dubinin–Astakhov (DA) potential to describe sorption in porous media:
1
0
0
ln
z
z
z
i i
, (3.6)
where
0
z is the total pore volume of the adsorbent,
i 0
is the characteristic potential for
component i , and is the so-called Dubinin exponent [27, 28]. The DA potential has been
shown adequate to describe microporous solids with a narrow pore-size distribution (PSD) [29].
As the PSD becomes broader, the DA equation may no longer be applicable, however.
In the MPTA approach,
0
z ,
i 0
and are treated as adjustable parameters and can be
regressed from modeling the relevant pure component sorption isotherms. A robust procedure for
53
solving the MPTA equations is utilized here, that was first developed by Shojaei and Jessen
[26]
and for which further details are provided in the Appendix. In the spirit of the original papers by
Dubinin and coworkers
[30-32]
and also Shapiro and coworkers [14-16, 27, 28] the model
parameters are taken here to be temperature-independent. For example, Dubinin and coworkers
[30-32], in their studies of the adsorption of vapors on various solids, reported that for a given
adsorbate–adsorbent pair, when plotting the quantity (
0
z z ) vs. , they obtained a nearly
invariant curve for different temperatures. Generally good success has been obtained in fitting
experimental data with temperature-invariant DA parameters by Shapiro and coworkers [14-16,
27, 28] as well.
The excess sorption of component i , i.e., the difference between the actual amount sorbed and
the amount that would exist under bulk-phase conditions in a volume of the pore space equal to
that of the adsorbed phase is represented by the following integral:
dz y z z x
z
y i i i
0
0
. (3.7)
where
0
z is the total porous volume, is the molar density of the sorbed phase, which can be
estimated by an equation of state. Here the PR-EOS was employed to describe both the bulk
phase and the adsorbed phase. As noted above, the PR-EOS has been shown to accurately
describe the experimental bulk densities, see Figs. S.1 - S.5 in the Supplemental Section and
further discussion to follow.
3.4.2. Modeling of CH
4
and C
2
H
6
Pure Component Isotherms
Using the original form of the MPTA model [14], we extracted relevant model parameters, i.e.,
the characteristic energies, and the Dubinin exponent by matching the model to the pure
54
component sorption isotherms. In the interest of integrating the experimental pore volume
information from the shale characterization (Table 3.1) into the MPTA model, two different DA
potential functions were utilized, one corresponding to the microporous region covering the pore
volume from 0 to 0.0081 cc/gr, and another to the meso- to macro-porous regions covering the
pore volume from 0.0081 to 0.0346 cc/gr. Specifically, we keep the functional form of the DA
potential to be the same in both regions but allow the use of different characteristic parameters in
each region. This is done by introducing the following potential relationships
0
1
0
0
ln b
z
b
z
i i i
,
0
0 b z (3.8)
1
0
0
ln
z
z
z
i i
,
0 0
z z b , (3.9)
where
0
b represents the pore volume of the microporous region, and
0
z is the total pore
volume. Eq. 3.8 ensures continuity in the chemical potential in the transition from the
microporous to the mesoporous region. The new composite potential function is used with fixed
values of
0
b and
0
z set to 0.0081 and 0.0346 cc/gr, respectively, as measured experimentally
for this shale sample. In this form, a total of 6 parameters must be estimated simultaneously from
the CH
4
, C
2
H
6
pure-component isotherms. They include the energy parameters for each species
in each region (a total of four parameters) and the Dubinin exponent for each region, taken to be
independent of the species (a total of two additional parameters). We assume that in each porous
region, the characteristic potential
0
should vary among different components, because in the
DA it is interpreted as a measure of the affinity between the component and the adsorbent.
However, we keep the Dubinin exponent ( β) the same for all species in each porous region, since
55
the DA model is thought to provide a measure of the heterogeneity of the porous regions, and to
thus be independent of the components (however, we allow it to vary among the pore regions, as
they are likely to have different degrees of heterogeneity).
As noted above, we have used the PR-EOS
[33] in our calculations. The relevant PR-EOS model
parameters are listed in Table 3.2, with the binary interaction coefficient between CH
4
and C
2
H
6
is set to zero. As discussed previously, and also shown in Figs. S.1-S.5 in the Supplemental
Materials Section, the PR-EOS calculations are in excellent agreement with both our single
component and also binary mixture experimental density data. The six MPTA model parameters
were fitted to the experimental pure component isotherms (54 points), and Table 3.3 reports the
final MPTA parameters. Figs. 3.10-3.12 present the MPTA model representation of the CH
4
and
C
2
H
6
pure component isotherms corresponding to the parameters listed in Table 3.3.
Table 3.3. MPTA model parameters obtained from pure component sorption isotherms at 40
o
C,
50
o
C and 60
o
C.
MPTA CH
4
C
2
H
6
682.5 684.4
11.2 120.4
1.284
0.314
gr cc b
0
0.0081
gr cc z
0
0.0346
K R
i
0
K R
i
0
56
Table 3.2. Peng-Robinson EOS model parameters.
Critical temperature
c
T (K)
Critical pressure
c
p (bar)
Acentric factor
(dimensionless)
V olume shift
i
S (dimensionless)
CH
4
190.56 45.99 0.011 -0.03463
C
2
H
6
305.32 48.72 0.099 -0.34350
0 20 40 60 80 100 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
CH4 Expt.
C2H6 Expt.
CH4 MPTA
C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.10. Comparison of MPTA calculations and experimental observations for CH
4
and C
2
H
6
excess sorption on shale at 40
o
C.
57
0 20 40 60 80 100 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
CH4 Expt.
C2H6 Expt.
CH4 MPTA
C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.11. Comparison of MPTA calculations and experimental observations for CH
4
and C
2
H
6
excess sorption on shale at 50
o
C.
0 20 40 60 80 100 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
CH4 Expt.
C2H6 Expt.
CH4 MPTA
C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.12. Comparison of MPTA calculations and experimental observations for CH
4
and C
2
H
6
excess sorption on shale at 60
o
C.
58
We observe from Figs. 3.10-3.12 that the MPTA model is reasonably accurate in representing the
CH
4
, C
2
H
6
pure component sorption isotherms, with low/moderate Root-Mean-Square (RMS)
errors (representing the standard deviation of the differences between predicted values and
observed values) as listed in Table 3.4. Interestingly, the model predicts very similar amounts of
CH
4
and C
2
H
6
sorbed in the microporous regions (note the very similar
i 0
values in Table 3.3),
with the significant differences in excess sorption between the two species resulting from
differences in the amounts sorbed in the mesoporous regions.
Table 3.4. RMS errors for CH
4
and C
2
H
6
isotherms at 40
o
C, 50
o
C and 60
o
C.
RMS error
(mg/g)
CH
4
C
2
H
6
40
o
C 0.1147 0.1194
50
o
C 0.0326 0.0819
60
o
C 0.0758 0.1109
3.4.3. Prediction of CH
4
-C
2
H
6
Adsorption Behavior
Based on the MPTA model parameters obtained from the pure component sorption isotherms at
various temperatures, we applied the MPTA model, in a predictive mode, to calculate the binary
sorption of CH
4
-C
2
H
6
mixtures on shale at relevant conditions. In this work, we have studied
binary mixtures with 90%, 93% and 96% of CH
4
by mole. Figs. 3.13-3.15 compare the model
predictions for overall CH
4
-C
2
H
6
binary excess sorption with experimental observations.
59
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
3.0
90%-10% CH4-C2H6 Expt.
93%-7% CH4-C2H6 Expt.
96%-4% CH4-C2H6 Expt.
90%-10% CH4-C2H6 MPTA
93%-7% CH4-C2H6 MPTA
96%-4% CH4-C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.13. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 40
o
C.
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
3.0
90%-10% CH4-C2H6 Expt.
93%-7% CH4-C2H6 Expt.
96%-4% CH4-C2H6 Expt.
90%-10% CH4-C2H6 MPTA
93%-7% CH4-C2H6 MPTA
96%-4% CH4-C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.14. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 50
o
C.
60
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
3.0
90%-10% CH4-C2H6 Expt.
93%-7% CH4-C2H6 Expt.
96%-4% CH4-C2H6 Expt.
90%-10% CH4-C2H6 MPTA
93%-7% CH4-C2H6 MPTA
96%-4% CH4-C2H6 MPTA
Excess sorption (mg/g)
Pressure (bar)
Figure 3.15. Measured and calculated total CH
4
-C
2
H
6
excess sorption on shale at 60
o
C.
From Figs. 3.13-3.15, we observe that the total excess sorption of CH
4
-C
2
H
6
mixtures are
predicted fairly accurately by the MPTA model, with the RMS errors reported in Table 3.5.
Table 3.5. RMS errors for CH
4
-C
2
H
6
binary mixtures at 40
o
C, 50
o
C and 60
o
C.
RMS error
(mg/g)
90%-10%
CH
4
-C
2
H
6
93%-7%
CH
4
-C
2
H
6
96%-4%
CH
4
-C
2
H
6
40
o
C 0.2746 0.2566 0.1445
50
o
C 0.2825 0.2153 0.1529
60
o
C 0.2601 0.2289 0.1289
61
Due to the interactions and competitions between the two sorbed species, the CH
4
-C
2
H
6
binary
sorption demonstrates a maximum in excess loading for all cases, and the MPTA model is able to
represent this feature. We note that the MPTA model tends to over-estimate the excess sorption,
the difference between theory and experiments widening with an increasing mole fraction of
C
2
H
6
in the mixture. This implies that the interaction and competition between the two sorbed
species become increasingly complicated with more evenly distributed composition of the bulk
phase.
Another interesting experimental observation is the slight but noticeable deviation between the
sorption isotherms measured by the static apparatus and the TGA system. Figs. 3.16 and 3.17
show the comparison of CH
4
and C
2
H
6
adsorption isotherms on the shale sample at 60
o
C
measured by the two different methods. Possible reasons for the discrepancy between the
experimental results obtained from the static apparatus and the TGA system are: 1) the shale
sample is highly heterogeneous. Grinding and sieving may cause compositional change in the
sub-samples. The shale sample used in the TGA experiments (2.2880 g) was sampled from the
static experiments (17.0 g). It is possible that there might be slight differences in the sample
composition; 2) the experimental uncertainties associated with the static measurements are
relatively large (as shown by the error bars in Fig. 2.4 in Chapter 2), which could contribute to
the overall disagreement as well.
62
Figure 3.16. Comparison of CH
4
adsorption isotherms on shale at 60
o
C measured by static and
TGA apparatus.
Figure 3.17. Comparison of C
2
H
6
adsorption isotherms on shale at 60
o
C measured by static and
TGA apparatus.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 20 40 60 80 100 120
Excess adsorption (mg/g)
Pressure (bar)
Static
TGA
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 10 20 30 40
Excess sorption (mg/g)
Pressure (bar)
Static
TGA
63
3.5. Discussion and Conclusion
In the previous sections we have reported new experimental observations of excess sorption of
CH
4
and C
2
H
6
and their binary mixtures for a range of pressures and temperatures. At these
experimental conditions, the pure component isotherms do not show any extrema in the excess
sorption as commonly observed for CO
2
sorption in coal. In contrast, the sorption isotherms for
the binary CH
4
/C
2
H
6
mixtures exhibit extrema in the total excess loading, predominantly at high
C
2
H
6
concentrations and at low temperatures.
Sorption hysteresis is observed for both the pure component isotherms, indicating a multi-modal
pore size distribution. BET measurements (reported in Table 3.1) further support this
interpretation by demonstrating that the sample’s pore volume consists of micropores, mesopores
and macropores, with the mesopores commonly thought responsible for adsorption/desorption
hysteresis. The sorption hysteresis is observed to be more significant for CH
4
than for C
2
H
6
. This
suggests that the common industry practice of using the loading curve to evaluate gas in place
and to evaluate production dynamics may not be appropriate.
Mineralogical analysis of these samples [34] reveals that the shale sample is composed largely of
clay and quartz (> 80 wt.% ), but also contains ~3 % of total organic content (TOC). The
non-zero unloading at p = 0 bar is also consistent with the concept of a heterogeneous sample
containing both organic and inorganic components with gases being held perhaps more tightly in
the organic inclusions.
The multi-porosity and inherent heterogeneous nature of shale present a challenge for the
successful application of MPTA (but also all other continuum, semi empirical-type theories)
to model sorption phenomena. The MPTA model, in its original form, was first proposed for
64
sorption calculations in microporous materials with narrow PSD, such as activated carbons and
zeolites. The pore size distribution of a shale sample, however, is more widely dispersed, with
micropores, mesopores and macropores all contributing to the overall porosity and adsorption in
the sample. It is notable, therefore, that the model still manages to provide an adequate fit of the
experimental mixture data based solely on parameters that were estimated from monotonic pure
component isotherms, and that is even capable of identifying the observed maximum in the total
excess loading.
3.6. References
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Aug:2-3.
[2] Chareonsuppanimit P, Mohammad SA, Robinson RL, Gasem KA. High-pressure adsorption
of gases on shales: Measurements and modeling. International Journal of Coal Geology. 2012
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[3] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide
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[4] Lu XC, Li FC, Watson AT. Adsorption measurements in Devonian shales. Fuel. 1995 Apr
30;74(4):599-603.
[5] Ross DJ, Bustin RM. The importance of shale composition and pore structure upon gas
storage potential of shale gas reservoirs. Marine and Petroleum Geology. 2009 Jun
30;26(6):916-27.
65
[6] Smith DH, Bromhal G, Sams WN, Jikich S, Ertekin T. Simulating Carbon Dioxide
Sequestration/ECBM Production in Coal Seams: Effects of Permeability Anisotropies and Other
Coal Properties. SPE Reservoir Evaluation & Engineering. 2005 Apr 1;8(02):156-63.
[7] Seto CJ, Jessen K, Orr FM. A multicomponent, two-phase-flow model for CO
2
storage and
enhanced coalbed-methane recovery. SPE Journal. 2009 Mar 1;14(01):30-40.
[8] Manik J. Compositional modeling of enhanced coalbed methane recovery. 1999.
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calculations. InSPE Annual Technical Conference and Exhibition 2007 Jan 1. Society of
Petroleum Engineers.
[10] Rexer TF, Benham MJ, Aplin AC, Thomas KM. Methane adsorption on shale under
simulated geological temperature and pressure conditions. Energy & Fuels. 2013 May
28;27(6):3099-109.
[11] Ambrose RJ, Hartman RC, Akkutlu IY . Multi-component sorbed phase considerations for
shale gas-in-place calculations. InSPE Production and Operations Symposium 2011 Jan 1.
Society of Petroleum Engineers.
[12] Hartman RC, Ambrose RJ, Akkutlu IY , Clarkson CR. Shale gas-in-place calculations part
II-multicomponent gas adsorption effects. InNorth American Unconventional Gas Conference
and Exhibition 2011 Jan 1. Society of Petroleum Engineers.
[13] Polanyi M. Section III.—theories of the adsorption of gases. A general survey and some
additional remarks. Introductory paper to section III. Transactions of the Faraday Society.
1932;28:316-33.
66
[14] Shapiro AA, Stenby EH. Potential theory of multicomponent adsorption. Journal of colloid
and interface science. 1998 May 15;201(2):146-57.
[15] Monsalvo MA, Shapiro AA. Prediction of adsorption from liquid mixtures in microporous
media by the potential theory. Fluid Phase Equilibria. 2007 Dec 1;261(1):292-9.
[16] Monsalvo MA, Shapiro AA. Study of high-pressure adsorption from supercritical fluids by
the potential theory. Fluid Phase Equilibria. 2009 Sep 15;283(1):56-64.
[17] Fitzgerald JE, Pan Z, Sudibandriyo M, Robinson Jr RL, Gasem KA, Reeves S. Adsorption
of methane, nitrogen, carbon dioxide and their mixtures on wet Tiffany coal. Fuel. 2005 Dec
31;84(18):2351-63.
[18] Pini R, Ottiger S, Burlini L, Storti G, Mazzotti M. Sorption of carbon dioxide, methane and
nitrogen in dry coals at high pressure and moderate temperature. International Journal of
Greenhouse Gas Control. 2010 Jan 31;4(1):90-101.
[19] Pini R, Ottiger S, Storti G, Mazzotti M. Pure and competitive adsorption of CO
2
, CH
4
and
N
2
on coal for ECBM. Energy Procedia. 2009 Feb 28;1(1):1705-10.
[20] Busch A, Gensterblum Y , Krooss BM, Siemons N. Investigation of high-pressure selective
adsorption/desorption behaviour of CO
2
and CH
4
on coals: an experimental study. International
Journal of Coal Geology. 2006 Feb 3;66(1):53-68.
[21] Krooss BM, Van Bergen F, Gensterblum Y , Siemons N, Pagnier HJ, David P. High-pressure
methane and carbon dioxide adsorption on dry and moisture-equilibrated Pennsylvanian coals.
International Journal of Coal Geology. 2002 Jul 31;51(2):69-92.
67
[22] Busch A, Gensterblum Y , Krooss BM. Methane and CO
2
sorption and desorption
measurements on dry Argonne premium coals: pure components and mixtures. International
Journal of Coal Geology. 2003 Aug 31;55(2):205-24.
[23] Zhang T, Ellis GS, Ruppel SC, Milliken K, Yang R. Effect of organic-matter type and
thermal maturity on methane adsorption in shale-gas systems. Organic Geochemistry. 2012 Jun
30;47:120-31.
[24] Li D, Liu Q, Weniger P, Gensterblum Y , Busch A, Krooss BM. High-pressure sorption
isotherms and sorption kinetics of CH
4
and CO
2
on coals. Fuel. 2010 Mar 31;89(3):569-80.
[25] Fitzgerald JE, Robinson RL, Gasem KA. Modeling high-pressure adsorption of gas
mixtures on activated carbon and coal using a simplified local-density model. Langmuir. 2006
Nov 7;22(23):9610-8.
[26] Shojaei H, Jessen K. Application of potential theory to modeling of ECBM recovery. InSPE
Western North American Region Meeting 2011 Jan 1. Society of Petroleum Engineers.
[27] Monsalvo MA, Shapiro AA. Modeling adsorption of binary and ternary mixtures on
microporous media. Fluid phase equilibria. 2007 Jun 15;254(1):91-100.
[28] Monsalvo MA, Shapiro AA. Prediction of adsorption from liquid mixtures in microporous
media by the potential theory. Fluid Phase Equilibria. 2007 Dec 1;261(1):292-9.
[29] Burevski D. The application of the Dubinin-Astakhov equation to the characterization of
microporous carbons. Colloid and Polymer Science. 1982 Jun 1;260(6):623-7.
68
[30] Dubinin MM, Astakhov V A. Development of ideas of volume filling of micropores during
adsorption of gases and vapours by microporous adsorbents. 1. Carbonaceous adsorbents.
Izvestiya Akademii Nauk Sssr-Seriya Khimicheskaya. 1971 Jan 1(1):5.
[31] Dubinin MM, Astakhov V A. Development of ideas of volume filling of micropores during
adsorption of gases and vapours by microporous adsorbents. 2. General fundamentals of theory
of gas and vapour adsorption on zeolites. Izvestiya Akademii Nauk SSSR-Seriya Khimicheskaya.
1971 Jan 1(1):11.
[32] Astakhov V A, Dubinin MM. Development of Ideas of V olume Filling of Micropores during
Adsorption of Gases and Vapours by Microporous Adsorbents. 3. Zeolites with Large V oids and
Considerable Number of Adsorption Centers. Izvestiya Akademii Nauk Sssr-Seriya
Khimicheskaya. 1971 Jan 1(1):17.
[33] Mathias PM, Copeman TW. Extension of the Peng-Robinson equation of state to complex
mixtures: evaluation of the various forms of the local composition concept. Fluid Phase
Equilibria. 1983 Dec 31;13:91-108.
[34] Xu J. The study of mass transfer in gas shales and the optimization of hydraulic stimulation
processes via additives. University of Southern California; 2013.
[35] Michelsen ML. Saturation point calculations. Fluid phase equilibria. 1985 Dec
31;23(2):181-92.
69
3.7. Appendix
Numerical Approach for the MPTA Model
Evaluation of Eq. (3.7) requires numerical integration, and hence repeated solution of the
equilibrium problem stated by Eq. (3.5) at the nodes of any selected integration rule. Given the
similarity of the equilibrium problem stated by Eq. (3.5) to a dew-point calculation where the
solution is given by an incipient phase (here the adsorbed phase) and the associated equilibrium
pressure (in the adsorbed phase), we adapt the approach by Michelsen
[35] for calculation of
saturation points.
At any given node ( z ) of the integration rule, we start by rewriting Eq. (3.5) in terms of the
fugacity coefficients,
RT
z
p y p x
i
y
y
i i
x
i i
ln ln ln ln ln ln nc i ,..., 1 , (S.1)
In Eq. (S.1), we know the temperature (T ), the bulk phase composition ( y ) and the pressure of
the bulk phase (
y
p ). From these, we wish to calculate the composition ( x ) and pressure ( p ) of
the adsorbate. We introduce the equilibrium K-value
i
x
i i i i
p x y K ln ln ln ln ln nc i ,..., 1 , (S.2)
with
RT
z
p
i
y
y
i i
ln ln . (S.3)
The requirement for equilibrium can now be stated as
70
0 1
1
nc
i i
i
K
y
f , (S.4)
and we can proceed by solving Eq. (S.4) for the equilibrium pressure ( p ) and the incipient
phase composition ( x ) following the ideal solution based method proposed by Michelsen
[35].
The iterative procedure is as follows: Given an initial estimate of p and x , we can calculate
the K-values from Eq. (S.2) at iteration level k
i
k k x
i
k
i
p K ln ln ln , nc i ,..., 1 . (S.5)
Next, we evaluate the trial function, i.e., Eq. (S.4) and the derivative w.r.t. p
nc
i
k
i
i k
K
y
f
1
1 , (S.6)
p p K
y
dp
df
f
k x
i
k
nc
i
k
i
i
k
ln 1
1
. (S.7)
The pressure is then updated to the next level by a Newton correction
f
f
p p
k
k k
1
, (S.8)
and the mole fractions of the adsorbed phase are finally updated by direct substitution
k
i
i k
i
K
y
x
1
. (S.9)
Eqs. (S.5) to (S.9) are then repeated until convergence. The overall algorithm for calculation of
excess sorption is as follows:
71
1. Select a number of interior points, z , within the integration interval [0
0
z ], as dictated by
the selected integration rule.
2. Start at the upper integration limit and obtain initial estimates for p and x in the adsorbed
phase. For z =
0
z ,
y
p is a good initial estimate for p , and the extended Langmuir
approach provides a good initial estimate of x .
3. For subsequent (smaller) values of z , the converged values of p and x from previous
integration node provide good initial estimates. The initial estimate of p can be refined by
using the derivative of Eq. (S.4) w.r.t. z from the previously converged level
dp
df
dz
df
dz
dp
. (S.10)
4. Evaluate the surface excess for each component using Eq. (3.7) and integration rule.
72
3.8. Supplementary Data
A. Bulk Phase Density Data and Calculations using the PR-EOS
0 20 40 60 80 100 120
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
40
o
C TGA
40
o
C PR EOS
50
o
C TGA
50
o
C PR EOS
60
o
C TGA
60
o
C PR EOS
Density (g/cm
3
)
Pressure (bar)
Figure S.1. Experimental CH
4
densities and calculated values by the PR-EOS.
73
0 10 20 30 40
0.00
0.01
0.02
0.03
0.04
0.05
0.06
40
o
C TGA
40
o
C PR EOS
50
o
C TGA
50
o
C PR EOS
60
o
C TGA
60
o
C PR EOS
Density (g/cm
3
)
Pressure (bar)
Figure S.2. Experimental C
2
H
6
densities and calculated values by the PR-EOS.
0 20 40 60 80 100 120
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
40
o
C TGA
40
o
C PR EOS
50
o
C TGA
50
o
C PR EOS
60
o
C TGA
60
o
C PR EOS
Density (g/cm
3
)
Pressure (bar)
Figure S.3. Experimental 90%-10% CH
4
-C
2
H
6
densities and calculated values by the PR-EOS.
74
0 20 40 60 80 100 120 140
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
40
o
C TGA
40
o
C PR EOS
50
o
C TGA
50
o
C PR EOS
60
o
C TGA
60
o
C PR EOS
Density (g/cm
3
)
Pressure (bar)
Figure S.4. Experimental 93%-7% CH
4
-C
2
H
6
densities and calculated values by the PR-EOS.
0 20 40 60 80 100 120 140
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
40
o
C TGA
40
o
C PR EOS
50
o
C TGA
50
o
C PR EOS
60
o
C TGA
60
o
C PR EOS
Density (g/cm
3
)
Pressure (bar)
Figure S.5. Experimental 96%-4% CH
4
-C
2
H
6
densities and calculated values by the PR-EOS.
75
B. Sorption Data
Table S.1. CH
4
and C
2
H
6
pure component sorption data on shale at 40
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
0.0 0.000000 0.0 0.000000
5.0 0.787136 2.0 1.896739
10.0 1.028196 5.0 2.696548
20.0 1.299004 8.0 3.202626
29.9 1.496764 12.0 3.679534
40.0 1.630212 15.9 4.060011
60.0 1.811959 19.9 4.314307
79.3 1.912745 25.0 4.571672
100.0 1.964381 30.0 4.753208
112.0 1.972641 34.9 4.842111
76
Table S.2. CH
4
and C
2
H
6
pure component sorption data on shale at 50
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
0.0 0.000000 0.0 0.000000
5.0 0.604154 2.0 1.663590
10.1 0.826611 5.0 2.395841
20.0 1.107061 7.9 2.929381
29.9 1.319961 12.0 3.370582
39.9 1.454585 16.0 3.732761
59.8 1.658571 20.0 3.981616
80.0 1.747551 24.9 4.234912
100.0 1.785428 30.0 4.425110
113.7 1.790652 34.9 4.558140
77
Table S.3. CH
4
and C
2
H
6
pure component sorption data on shale at 60
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
0.0 0.000000 0.0 0.000000
5.1 0.395034 1.0 0.884247
10.0 0.621168 2.0 1.308857
19.9 0.917190 5.0 2.055943
29.9 1.132800 8.0 2.510221
40.0 1.281893 12.0 2.935124
59.9 1.481999 16.0 3.277300
79.8 1.600072 20.0 3.526033
99.7 1.663468 24.9 3.770927
113.9 1.692039 35.6 4.126262
78
Table S.4. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 40
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
0.0 0.000000 0.0 0.000000 0.0 0.000000
5.0 0.999994 5.0 0.791409 5.0 0.728690
10.0 1.341076 10.0 1.100994 10.0 1.046426
20.0 1.695882 20.0 1.506929 20.0 1.406008
30.0 1.921900 29.9 1.728177 30.0 1.628997
39.9 2.086311 39.9 1.902473 40.0 1.782557
60.1 2.276040 60.0 2.066002 59.9 1.956562
80.2 2.308244 79.3 2.124567 79.7 2.033089
99.9 2.278735 100.0 2.129467 100.1 2.029928
114.7 2.226834 124.8 2.066679 124.7 1.963429
79
Table S.5. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 50
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p
(bar)
eas
m
(mg/g)
p
(bar)
eas
m
(mg/g)
0.0 0.000000 0.0 0.000000 0.0 0.000000
5.0 0.767901 5.1 0.705612 5.1 0.658156
10.0 1.094441 10.0 1.009578 10.0 0.920039
20.0 1.487462 19.9 1.391041 19.9 1.252520
30.2 1.733733 30.0 1.620130 30.0 1.485792
39.9 1.903225 40.0 1.775578 40.1 1.633879
60.1 2.096772 59.9 1.958856 59.9 1.807177
80.2 2.165913 79.9 2.033903 79.6 1.889033
99.9 2.159869 99.6 2.064248 99.6 1.920153
114.9 2.121849 125.1 2.007951 125.3 1.878029
80
Table S.6. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale at 60
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
0.0 0.000000 0.0 0.000000 0.0 0.000000
5.0 0.646543 5.0 0.528495 4.9 0.497076
10.0 0.969728 10.0 0.824503 10.1 0.798392
20.0 1.337430 19.9 1.212261 20.0 1.148174
30.0 1.579109 30.0 1.451898 30.1 1.382094
40.0 1.744338 40.0 1.613266 40.1 1.521842
59.7 1.962677 59.8 1.806861 60.0 1.726034
80.1 2.052123 79.9 1.909414 79.8 1.829699
100.2 2.073626 100.0 1.951769 99.7 1.853641
115.0 2.066996 125.1 1.929907 124.8 1.832309
81
Chapter 4
Sorption Kinetics and Mass Transfer in
Powdered Shale Sample
4.1. Introduction
In the context of increasing shale gas production in the U.S. and around the world, sorption of
CH
4
, C
2
H
6
and their mixtures on shale has become a subject undergoing intense research
investigation. Sorption of gases in the complex shale matrix system is the key mechanism via
which shale gas is stored in such formations, and has as a result received a lot of attention in a
number of previous studies related to shale gas recovery [1-5]. An overwhelming majority of
these studies focus on the measurement/modeling of sorption isotherms of gases on shale [6-9],
which is justifiable because the sorption isotherms carry intrinsic value in terms of estimating the
sorption capacity. However, in addition to the sorption isotherms, it is also of great importance to
improve the understanding of pure and mixed gases sorption dynamics on shale. In this study, we
focus our attention on the measurement/modeling of CH
4
, C
2
H
6
pure components and binary
mixtures sorption isotherms/dynamics on shale.
In sorption studies where excess sorption is measured experimentally, one has to face the issue of
converting the excess sorption into the absolute sorption, which is thought to be more descriptive
and physically accurate. The knowledge of the adsorbed phase density, i.e.,
ads
, is required in
this conversion. However, under the existing technical conditions, one can only estimate
ads
82
because it is not readily accessible for measurement. This chapter summarizes a number of
methods that people commonly use to estimate
ads
, and compares them for their application to
CH
4
-C
2
H
6
-shale systems.
A Langmuir-type of sorption dynamic model is proposed, which allows us to readily investigate
sorption kinetics coupled to diffusive and convective mass transfer.
4.2. Experimental
4.2.1. Sample Preparation
In the laboratory-scale research related to gas sorption and mass transfer in shales, there are
typically two sample preparation approaches, i.e., those using ground samples and those using
whole rock/shale samples. It is known that the grain size (and distribution) of a sample controls
gas diffusion and sorption uptake rates. This is because gas can diffuse faster to the adsorption
sites in smaller particles so that the equilibrium time will be shorter compared to larger particles
[7]. In our study of sorption kinetics of shale in this chapter, a ground sample was used to reduce
internal mass transfer limitation effects and to facilitate faster sorption of gases on the shale. The
shale sample used in this study is from the Marcellus formation in the Appalachian basin. It was
extracted from a depth of 7802.5 ft. Prior to its use in the present study, the sample was stored in
a zip-lock bag as received to avoid further oxidation and water uptake. The shale sample was
ground and sieved. Sample particles with diameters in the range of 1-1.18 mm (US Mesh 16-18)
were then collected and subsequently used in the sorption experiments. Prior to the initiation of
these experiments, the sample was evacuated at 120
o
C for 24 hrs. At the end of each sorption
experiment, the sample was regenerated under vacuum at 120
o
C for 24 hrs. Such a procedure
83
assures that all gases that may potentially remain adsorbed at the end of an experiment are
desorbed prior to the initiation of the next experiment.
4.2.2. Experimental Approach
The thermogravimetric analysis (TGA) technique is used for the measurement of sorption
dynamic data (pure components and mixtures). The heart of the TGA set-up is a magnetic
suspension balance (Rubotherm, Germany), which is capable of measuring weight changes down
to 1 μg. During a sorption experiment, the weight change of the sample due to sorption is
transmitted from the sorption chamber to the analytical balance in a contactless manner via the
magnetic suspension mechanism. Prior to the sorption experiment, the measurements of the
weight of the sample container
sc
M and its volume
sc
V , the weight of the sample
s
M and its
skeletal volume
s
V have to be carried out. Further details of the measurement technique can be
found elsewhere [5]. The sorption experiments were carried out by increasing the gas-phase
pressure inside the sorption chamber in a step-wise manner, e.g., from 10 to 20, to 30, to 40 bar,
etc. for CH
4
. However, it was found that an abrupt overshoot/undershoot could occur
immediately after a pressure step, which was caused by the zero-point fluctuation introduced by
the turbulent flow created by the abrupt pressure change. To eliminate the effect of a zero-point
fluctuation, a blank run with a nonabsorbent, i.e., quartz, was performed. In the blank run with
quartz, the same procedure was followed with each gas used in the sorption experiment on the
shale sample. The zero-point fluctuation was then recorded and subtracted from the shale
sorption dynamic data to eliminate the overshoot/undershoot. A detailed description about the
zero-point correction can be found in the Supplementary Material in this Chapter.
84
4.3. Experimental Results
4.3.1. Sorption Isotherms and Dynamics of Pure Components
The pure component sorption isotherms were measured on the shale sample at three different
temperatures, namely, 40 ° C, 50 ° C and 60 ° C. The thermogravimetric measurements provide
both the sorption isotherms and the sorption dynamics. Fig. 4.1 (refer also to Figs. 3.1 to 3.3 in
Chapter 3) reports the pure-component isotherms for CH
4
and C
2
H
6
at 40 ° C, 50 ° C and 60 ° C, as
obtained from the TGA measurements. From Fig. 4.1, we observe the notably different sorption
affinities of the shale toward CH
4
and C
2
H
6,
with C
2
H
6
displaying the stronger affinity: The ratio
of C
2
H
6
/CH
4
excess sorption is quite large, varying between 1.5 and 2.5 (on a molar basis) over
the common pressure range investigated. Fig. 4.2 reports the C
2
H
6
sorption dynamics from 2 to
25 bar at 60 ° C while the dynamics of a single pressure step (from 2 to 5 bar) is reported in Fig.
4.3. Prior to the recording of sorption dynamic data, sufficient time (over 3 hrs) was allowed at 2
bar to ensure the sample reaches sorption equilibrium. It can be seen that the bulk phase pressure
is increased in a stepwise manner, and stabilizes relatively fast (within 2-3 min). In comparison,
the excess sorption process happens in a rather slow manner, and reaches equilibrium state in
about 3 hrs. This implies that the sorption is the rate-limiting step among all mass transfer
mechanisms of gases in shale, which explains why desorption is thought to dominate the
long-term dynamics of shale-gas production.
85
Figure 4.1. CH
4
and C
2
H
6
pure component sorption isotherms on shale at 40, 50, and 60 ° C
(Isotherm data can be found in the Supplementary Data section in Chapter 3).
Figure 4.2. Sorption dynamic process of C
2
H
6
on shale at 60
o
C.
0 20 40 60 80 100 120
0
2
4
6
CH4 @ 40C
CH4 @ 50C
CH4 @ 60C
C2H6 @ 40C
C2H6 @ 50C
C2H6 @ 60C
Excess sorption (mg/g)
Pressure (bar)
0
5
10
15
20
25
30
0
0.5
1
1.5
2
2.5
3
3.5
4
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Pres.
86
Figure 4.3. Sorption process of C
2
H
6
on shale at 60
o
C: pressure step from 2 to 5 bar.
4.3.2. Sorption Isotherms and Dynamics of Binary Mixtures
Sorption isotherms were also measured for binary mixtures of CH
4
−C
2
H
6
(90% −10%, 93% −7%,
96% −4% mole fraction - certified premixed gases purchased from Matheson) on the shale
sample at the same temperatures (i.e., 40, 50, and 60 ° C) as those employed for the single-gas
experiments. Figs. 4.4-4.6 (refer also to Figs. 3.7 to 3.9 in Chapter 3) report the sorption
isotherms of the three binary mixtures on the shale sample at various temperatures.
0
1
2
3
4
5
6
7
8
1.2
1.4
1.6
1.8
2
2.2
0 50 100 150 200
Excess sorption (mg/g)
Time (min)
Exc. Pres.
87
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 4.4. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% −10%, 93% −7%, 96% −4%) on shale
at 40 ° C (Isotherm data can be found in the Supplementary Data section in Chapter 3).
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 4.5. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% −10%,93% −7%, 96% −4%) on shale
at 50 ° C.
88
0 20 40 60 80 100 120 140
0.0
0.5
1.0
1.5
2.0
2.5
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
Excess sorption (mg/g)
Pressure (bar)
Figure 4.6. Sorption isotherms of CH
4
−C
2
H
6
mixtures (90% −10%,93% −7%, 96% −4%) on shale
at 60 ° C.
In Figs. 4.4-4.6, we observe an increase in the total excess sorption amount as the concentration
of C
2
H
6
in the binary mixture increases. This is consistent with the single-gas sorption data
which indicate the preferential sorption of C
2
H
6
over CH
4
in the shale sample. Along with
isotherms, the sorption dynamics were also obtained during the mixed gases sorption process.
Fig. 4.7 shows the sorption dynamics for the 90% −10% CH
4
−C
2
H
6
sorption at 60
o
C. Fig. 4.8
provides an enlarged diagram for a single pressure step (from 30 to 40 bar), to show details. The
sorption dynamics behavior for the mixtures is very similar to that for the pure components. The
excess sorption approaches an asymptotic behavior during the sorption process. The sorption
dynamics are a prolonged process compared to the sudden change in the bulk-phase pressure.
This, again, features the significant role that sorption plays in the shale-gas system.
89
Figure 4.7. Sorption dynamic process of 90%-10% CH
4
-C
2
H
6
on shale at 60
o
C.
Figure 4.8. Sorption process of 90%-10% CH
4
-C
2
H
6
on shale at 60
o
C: pressure step from 20 to
30 bar.
0
10
20
30
40
50
60
70
80
90
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Pres.
10
15
20
25
30
35
1.3
1.4
1.5
1.6
1.7
1.8
0 50 100 150 200
Excess sorption (mg/g)
Time (min)
Exc. Pres.
90
4.4. Adsorbed Phase Density Estimates
4.4.1. Introduction of Various Methods to Estimate Adsorbed Phase Density
In the sorption studies where excess sorption is measured experimentally, one faces the issue of
converting the excess sorption into absolute sorption because it arises naturally when modeling
approaches, such as the Langmuir model, are used to interpret the experimental observations. In
the thermogravimetric method used here, the only experimentally measurable quantity is the
excess sorption, namely, the amount adsorbed in excess of what would be present if the
adsorbed-phase volume was replaced with the bulk-phase gas, i.e., Eq. 4.1
gas ads ads
exc
V n (4.1)
where
ads
V is the volume of the adsorbed phase;
ads
and
gas
are the adsorbed phase
density and gas phase density, respectively. The absolute sorption, by definition, can be
represented by Eq. 4.2.
ads ads
abs
V n (4.2)
Combining Eqs 4.1 and 4.2 yields
ads
gas exc abs
n n
1 (4.3)
In Eq. 4.3, the gas phase density,
gas
, can be either calculated by an equation of state (EOS), or
measured directly by the TGA apparatus [5]. However, the adsorbed phase density,
ads
, can
only be estimated because it is not readily accessible for measurement. Historically, people have
tried a number of approaches to estimate the adsorbed phase density. These methods can be
91
classified into two categories: One category assumes constant adsorbed phase density, while the
other assumes the adsorbed phase density varies in accordance with the bulk phase conditions.
In the constant adsorbed phase density assumption, people generally use the saturated liquid
density at the triple point (TP) or the liquid density at the atmospheric pressure boiling point (BP)
to approximate the adsorbed phase density of the pure substances [10-11]. These methods are
generally simple to apply but lack a theoretical framework. Another constant adsorbed phase
density method uses a graphical estimate from the excess adsorption isotherms [19]. As Eq. 4.3
indicates, the adsorbed phase density is equal to the bulk phase density when the excess
adsorption is equal to zero. This method assumes that after the adsorption reaches saturation, i.e.,
ads ads
V becomes a constant at high pressures, the excess adsorption beyond that point will show
a linear decrease with increasing bulk phase density
gas
. The extrapolation of this linear
correlation yields an x-axis intercept, where
gas ads
. The graphical estimation approach has
an advantage of sound theoretical framework because it is based on the definition of excess
adsorption. However, the use of this technique requires sufficient data in the linear
(high-pressure) region beyond the maximum in the excess adsorption isotherms, which is not
often available for most cases. Fitzgerald et al. [12] demonstrate that the adsorbed phase density
should approach the reciprocal co-volume (1/b) at higher pressures in their simplified local
density/Peng–Robinson model (SLD-PR).
All of the methods that assume constant adsorbed phase density might be undermined if the
adsorption is viewed as a process whereby the adsorbate molecules gradually occupy the empty
sorption sites. Mazzotti et al. [13] proposed a one-to-one mapping correlation between (lattice
occupancies) and (adsorbate densities) in order to describe the experimentally measured
92
excess adsorption data through the lattice density functional theory (DFT) model. The mapping
function g is presented in Eq. 4.4
2 1 1
max
max
c
c
g (4.4)
where
max
and
c
are the molar maximum density of the gas in the pore and its critical
density, respectively. In our implementation of Eq. 4.4,
max
is estimated from the
close-packing of equal spheres of adsorbate molecules whose diameters are assumed equal to the
collision diameter of the species. The Mazzotti’s method (subsequently referred to as the DFT
method) gives a rising estimate of adsorbate layer densities with increasing extent of lattice
occupancy (further details can be found in Sec. B in the Supplementary Materials).
Srinivasan et al. [14] proposed a method for the estimation of the adsorbed phase volume as
shown in Eq. 4.5
1
*
1
g
a
v
A v , (4.5)
where BT A A
*
a linear function of temperature; v is the specific volume, with the
subscripts a and g denoting adsorbed and bulk gas phases, respectively. Thus, the excess
adsorption for a species can be calculated as follows (assuming it follows a Langmuir isotherm):
1
*
max
1
1
1
g
g A
A abs exc
v
v
A
p K
p K
n n . (4.6)
Srinivasan’s approach (subsequently referred to as the DFT method) results in a decreasing trend
of the adsorbed phase density with increasing bulk-phase pressure (further details can be found
93
in Sec. C in the Supplementary Materials). This approach has been used for both pure and
multicomponent gas adsorption on activated carbon [14-15].
Another way of estimating the adsorbed phase density is through the use of the multicomponent
potential theory of adsorption (MPTA) [16]. In this approach the adsorbate layer is considered to
be a segregated phase in the potential field emitted by the adsorbent. For component i in the
adsorbed phase, the isothermal equilibrium state is reached when the sorption potential on that
component z
i
exerted by the adsorbent at any position z within the adsorbed phase equals
the difference between its chemical potential in the bulk phase y p
y ig
, and the chemical
potential in the adsorbed phase ) ( ), ( z x z p
i
at location z:
y p z z x z p
y ig i i
, ) ( ), ( nc i ,..., 1 (4.7)
In Eq. 4.7 above, is the pressure in the sorbed phase, is the pressure in the bulk
phase, x and are the mole fractions of component in the sorbed phase and bulk phase,
respectively. Eq. 4.7 can be rewritten in the form of fugacities:
(4.8)
In Eq. 4.8, the bulk and sorbed phases are represented by an appropriate EOS. z
i
is often
described by the Dubinin–Astakhov (DA) potential:
i
z
z
z
i i
1
0
0
ln
(4.9)
) (z p
y
p
y i
RT
z
y p f z x z p f
i
y ig i
, ln ) ( ), ( ln
94
where
0
z is the total pore volume of the adsorbent,
i 0
is the characteristic potential for
component i , and
i
is the so-called Dubinin exponent. The excess sorption of component i
is represented by the following integral:
dz y z z x
z
y i i i
0
0
(4.10)
where is the molar density of the sorbed phase. The MPTA model can be used to represent
the excess sorption. Along with the model representation, the adsorbed phase density is obtained
and later used in the conversion from excess to absolute sorption. The MPTA method gives an
estimate of increasing adsorbed phase density as the bulk phase pressure increases (further
details can be found in Sec. D in the Supplementary Materials).
In this study, we also propose a new method for estimating the adsorbed phase density. We
assume that the bulk gas phase and the liquid-like adsorbed phase coexist in a vapor-liquid
equilibrium (denoted by VLE subsequently). In this assumption, the adsorbed phase density
equals to the liquid density of the pure substance at its saturation temperature. Accordingly, the
adsorbed phase density varies with the gas phase pressure along the saturation pressure curve.
This assumption provides for a variable adsorbed phase density depending on the gas phase
pressure, and is, in our opinion, more reasonable than the assumption of a constant adsorbed
phase density (further details can be found in Sec. E in the Supplementary Materials).
4.4.2. Langmuir Adsorption Isotherms Calculated from the Various Density
Models
In the previous work [5], we have measured the excess sorption for both pure and mixed gases
(CH
4
and C
2
H
6
) on a shale sample with the TGA apparatus. In the model that we propose here to
95
describe the sorption dynamics, one of the assumptions is that the sorption of CH
4
and C
2
H
6
molecules on the shale surface form a monolayer, such that the sorption rate can be described by
the Langmuir type equations. Accordingly, and prior to the modeling of sorption dynamics, we
have to extract the Langmuir model parameters by fitting the pure component sorption isotherms.
Once the Langmuir parameters are obtained, the extended Langmuir model (ELM) can then be
used to predict the binary sorption isotherms. Figs. 4.9-4.16 provide a comparison of
experimental and simulated excess sorption isotherms for both pure and binary mixtures based
on the various adsorbed phase density estimating methods discussed in Sec. 4.4.1 (further details
about these simulations can be found in the Supplementary materials at the end of this Chapter).
Figure 4.9. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60
o
C:
ads
estimated by the DFT method.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
96
Figure 4.10. ELM correlation of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale at
60
o
C:
ads
estimated by the DFT method.
Figure 4.11. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60
o
C:
ads
estimated by the LRD method.
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120 140
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
97
Figure 4.12. ELM correlation of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale at
60
o
C:
ads
estimated by the LRD method.
Figure 4.13. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60
o
C:
ads
estimated by the MPTA method.
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120 140
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
98
Figure 4.14. ELM prediction of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale at
60
o
C:
ads
estimated by the MPTA method.
Figure 4.15. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale at 60
o
C:
ads
estimated by the VLE method.
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120 140
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
99
Figure 4.16. ELM correlation of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale at
60
o
C:
ads
estimated by the VLE method.
The root-mean square (RMS) errors (representing the standard deviation of the differences
between predicted values and observed values) are listed in Table 4.1 for both single and mixed
gases.
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120 140
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
100
Table 4.1. RMS errors for single and mixed gases isotherms using various density approaches.
RMS error (mg/g)
CH
4
C
2
H
6
90:10
CH
4
/C
2
H
6
93:7
CH
4
/C
2
H
6
96:4
CH
4
/C
2
H
6
DFT 0.0433 0.1385 0.1028 0.0626 0.0986
LRD 0.0272 0.1259 0.1168 0.0947 0.0726
MPTA 0.0497 0.5319 0.0646 0.0503 0.0436
VLE 0.0942 0.1443 0.3135 0.2568 0.1643
We observe from Figs. 4.9 to 4.12 and Table 4.1 that both the DFT and the LRD approaches are
reasonably accurate in representing the CH
4
and C
2
H
6
pure component sorption isotherms, with
low RMS errors. On the basis of the Langmuir model parameters obtained from the pure
component sorption isotherms (Table 4.2), we applied a revised extended Langmuir model plus a
newly proposed mixing rule (further details can be found in the Supplementary Material in this
chapter) to predict the binary sorption of CH
4
-C
2
H
6
mixtures with various compositions. It is
found that both the DFT and the LRD approaches are doing a decent job in reaching a good
quantitative agreement with the experimental observations. The MPTA approach (Figs. 4.13 to
4.14 and Table 4.1), although it gives better predictions for the binary mixtures, is less accurate
when to represent the C
2
H
6
sorption isotherms. The VLE approach (Figs. 4.15 to 4.16 and Table
4.1), although it reasonably represents the pure component sorption isotherms, provides the
worst predictions for the binary mixtures. To illustrate the differences among various density
101
estimating methods, a comparison of the estimated adsorbate phase densities for the various
approaches is presented in Figs. 4.17 to 4.21.
Figure 4.17. Comparison of estimated CH
4
adsorbed phase densities on shale at 60
o
C.
Figure 4.18. Comparison of estimated C
2
H
6
adsorbed phase densities on shale at 60
o
C.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Adsorbed phase densities (g/cm3)
Pressure (bar)
DFT LRD MPTA VLE
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Adsorbed phase densities (g/cm3)
Pressure (bar)
DFT LRD MPTA VLE
102
Figure 4.19. Comparison of estimated 90%-10% CH
4
-C
2
H
6
adsorbed phase densities on shale at
60
o
C.
Figure 4.20. Comparison of estimated 93%-7% CH
4
-C
2
H
6
adsorbed phase densities on shale at
60
o
C.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 20 40 60 80 100 120
Adsorbed phase densities (g/cm3)
Pressure (bar)
DFT LRD MPTA VLE
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 20 40 60 80 100 120 140
Adsorbed phase densities (g/cm3)
Pressure (bar)
DFT LRD MPTA VLE
103
Figure 4.21. Comparison of estimated 96%-4% CH
4
-C
2
H
6
adsorbed phase densities on shale at
60
o
C.
The Langmuir model parameters obtained by various approaches are reported in Table 4.2.
Table 4.2. Langmuir model parameters for CH
4
and C
2
H
6
sorption on shale at 60
o
C:
ads
estimated by various estimating methods.
abs
n
max
(mmol/g)
A
K (bar
-1
)
DFT
CH
4
0.1938 0.0251
C
2
H
6
0.1653 0.1559
LRD
CH
4
0.1494 0.0327
C
2
H
6
0.1538 0.1702
MPTA
CH
4
0.1925 0.0275
C
2
H
6
0.1505 0.1116
VLE
CH
4
0.3795 0.0082
C
2
H
6
0.1649 0.1495
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 20 40 60 80 100 120 140
Adsorbed phase densities (g/cm3)
Pressure (bar)
DFT LRD MPTA VLE
104
Interestingly, all the other methods except the VLE generate approximately the same
abs
n
max
for
CH
4
and C
2
H
6
. The VLE method predicts that the ratio between the maximum sorption capacity
of CH
4
and C
2
H
6
is very close to 2:1 (Table 4.2), which suggests that CH
4
and C
2
H
6
molecules
occupy different surface area: a C
2
H
6
molecule may occupy as twice the surface area as a CH
4
molecule does. This is reasonable, considering that the ratio between C
2
H
6
/CH
4
collision
diameter is around 1.3:1 (CH
4
: 4.1 Å; C
2
H
6
: 5.3 Å). A comprehensive comparison in terms of
single and mixed gases sorption isotherms representation, among various adsorbed phase density
estimating methods, leads us to believe the DFT and the LRD approaches provide better
description of the adsorbed phase density for both single and mixed gases. In addition, the DFT
and the LRD approaches represent two types of adsorbed phase density estimating methods. We
observe from Figs. 4.17 to 4.21 that the adsorbed phase density estimated by the DFT approach
increases with increasing bulk phase pressure, whereas the adsorbed phase density estimated by
the LRD approach decreases over the bulk phase pressure. Mosher et al. [20] reported in their
molecular simulation that adsorption layer density of CH
4
grew with rising bulk phase pressure
in coal and gas shale systems. Malheiro et al. [21] predicted the upward trend of adsorbed phase
density as well in their model of nitrogen and argon sorption in carbon slit-like micropores.
However, Srinivasan et al. [14] argued in their publication that the LRD approach has been
found to be applicable to phase change between liquid and vapour phases along the coexistence
curve [22]. El-sharkawy et al. [15] supported Srinivasan’s idea by applying the LRD approach to
the adsorption of a 3-gas mixture onto activated carton. Thus, in the subsequent sorption
dynamic modeling section, these two approaches will be implemented.
105
4.5. Sorption Dynamic Model Derivation
The simple yet effective Langmuir model has been used widely in sorption studies. Previous
work [7, 17-18] has shown that it is sufficiently accurate to represent pure component sorption
isotherms on shale. In our present work, the Langmuir model is selected not only because of its
low computational complexity, but also because the useful parameters, such as the sorption
equilibrium constant and the maximum sorption capacity, can be extracted and subsequently
used in the modeling of sorption dynamics. In the derivation of the dynamic model the sample
particles are assumed to be spherical. Fig. 4.22 shows the schematic diagram of the TGA
sorption experiment.
p(t)
CH
4
Figure 4.22. Schematic of TGA sorption measurements.
Specifically, before the start of a given experiment (t≤0+), the adsorbent is assumed to be in
equilibrium with the working gas (CH
4
, C
2
H
6
or their mixture) at pressure
0
p . When the
experiment begins (t=0
+
), the bulk phase pressure changes with time, and it can be represented
by a function, i.e., t p . Within the sample particle porous volume, we assume, for a start, that
two different transport mechanisms prevail: viscous flow and Knudsen diffusion. Within the
106
sample particle porous volume, the flux J (mol.m
-2
.s
-1
) of the binary gas mixture is described
by the following equation:
r
p
C
B
r
C
B D J
M
t
0
1
(4.11)
where,
2
1
J
J
J (
i
J molar flux of component i inside the sample particle, mol.m
-2
.s
-1
);
2
1
C
C
C (
i
C molar density of component i in the sample pore space, mol/m
3
);
sample porosity, dimensionless;
t
sample tortuosity, dimensionless;
M
D
1
Knudsen
diffusivity of component 1 (CH
4
) in the shale matrix, m
2
/s; r radial coordinate in the sample
particle, m;
0
B viscous flow parameter (assuming that the pore volume is represented by a
bundle of parallel capillaries with average pore diameter
p
D , then 32
2
0 p
D B ), m
2
;
viscosity of CH
4
-C
2
H
6
binary mixture, kg.m
-1
.s
-1
; p total pressure of the bulk phase gas, Pa.
Assuming that the following equation applies for the CH
4
-C
2
H
6
mixture in the sample pore space
2 1
C C T ZR p
g
(4.12)
where Z is the mixture gas compressibility factor, which can be calculated by the PR-EOS
which we have shown well describes the CH
4
-C
2
H
6
mixtures at different compositions and
temperatures. The matrix B in Eq. 4.11 is given by
107
2
2
1
2
1 1
1
2
2 1
1
x
WM
WM
x
x x
WM
WM
x x
B
(4.13)
with
M
D
D
2
12
,
2 1
1
1
C C
C
x
,
2 1
2
2
C C
C
x
,
1
WM molecular weight of component 1 g/mol;
2
WM molecular weight of component 2, g/mol;
12
D binary diffusion coefficient of CH
4
-C
2
H
6
mixture in the shale matrix, m
2
/s;
M
D
2
Knudsen diffusivity of component 2 (C
2
H
6
) in the shale
matrix, m
2
/s. The mass balance equation for the sample particle is
J r
r r t
C
t
C
s
2
2
0
1
1
(4.14)
where
0 0 1 2
(1 ) ( ) /
bulk
CC
with
being the molar density (mol/m
3
) of the
adsorbed phase of CH
4
-C
2
H
6
mixture. Substitute Eqs. 4.11-4.13 into Eq. 4.14,
r
p B
Z
p
r
Z p
B
D
r T R t
C
t
Z p
T R
t t
M
g
s
g
0 1
0
1
1
1
(4.15)
2 ,
1 ,
C
C
C (
i
C
,
adsorbed phase concentration of component i per unit weight of the
adsorbent, mol/kg), and can be represented by the following equation:
2 ,
2 ,
2 ,
2 , 1 ,
max
2 2 ,
1 ,
1 ,
1 ,
2 , 1 ,
max
1 1 ,
C
K
k
a aC C C p k
C
K
k
aC C C p k
b
t
C
A
a
s a
A
a
s a
s
(4.16)
108
where
2 ,
1 ,
s
s
C
C
a
ratio of adsorption sites occupied by a single C
2
H
6
molecule to that occupied
by a single CH
4
molecule; b internal surface area per unit volume of adsorbent, m
2
/m
3
;
i a
k
,
rate constant for adsorption of component i, kg/(Pa.m
2
.s);
i d i a i A
k k K
, , ,
sorption equilibrium
constant of component i, bar
-1
;
max
s
C
the maximum moles of adsorption sites per unit weight of
the sample (the maximum of
1 , s
C
and
2 , s
C
). The initial and boundary conditions for the Eqs.
4.15-4.16 are as follows,
; 0 t
2 0
1 0
y p
y p
p (4.17)
; 0 r
0
0
r
p
(4.18)
;
0
r r
2
1
y t p
y t p
p
(4.19)
; 0 t
2 0 A , 2 1 0 1 ,
2 0 2 ,
2 ,
2 0 A , 2 1 0 1 ,
1 0 1 ,
1 ,
K 1
K 1
y p y p K
y P K
C
y p y p K
y p K
C
C
A
A
s
A
A
s
(4.20)
where
i
y mole fraction of species i in the bulk phase gas;
0
r average radius of the sample
particle, m. The unknown parameters in the above equations are
i a
bk
,
,
t
M
D
1
and
t
B
0
. In the
ground sample, due to the small particle diameter, the diffusion and viscous flow characteristic
times are sufficiently short that neither of these mass transfer mechanisms can be effectively
109
captured in these TGA experiments. In contrast, the sorption process has a much larger
characteristic time, as reflected by the longer sorption equilibrium time observed in Figs. 4.3 and
4.8. Accordingly, only Eq. 4.16 is solved and only
i a
bk
,
is extracted from dynamics of ground
samples. The TGA balance always measures the sample weight (the adsorbent + amount
adsorbed) minus the buoyancy force as shown below
exc
s
b
m s sc
b
m sc
b
m
m V M V M T M , , (4.21)
where T M
b
m
, is the apparent weight (of the sample + sample-holder) that the balance
measures,
sc
M the true weight of the sample-holder and
sc
V its skeletal volume,
s
M the true
weight of the sample and
s
V its skeletal volume,
b
m
the mass density of the fluid phase.
abs
m
and
exc
m are the total absolute and excess adsorption, correspondingly, given by the following
equations:
i
i i s
abs
MW C M m
,
, (4.22)
m
b
m
abs exc
m m
1 , (4.23)
where
m
the mass density of the adsorbed phase. The model parameters
i a
bk
,
are extracted
by minimizing the squares of the deviation between the calculated excess sorption dynamics and
experimentally obtained results for the pure components. Then we applied the proposed model,
in a predictive mode, to calculate the binary sorption dynamics of CH
4
−C
2
H
6
mixtures on shale
at relevant conditions and compare them with the experimental observations.
110
4.6. Modeling Results - Sorption Dynamics of Pure Components and Binary
Mixtures
4.6.1. CH
4
and C
2
H
6
Pure Component Sorption Dynamics
Two adsorbed phase density estimating methods – DFT and LRD were utilized in the modeling
of sorption dynamics. Using the DFT method, we extracted relevant model parameters, i.e., the
maximum sorption amount and the equilibrium sorption constant by matching the Langmuir
model to the pure component sorption isotherms for CH
4
and C
2
H
6
, respectively (further details
can be found in Sec. B in the Supplementary Material). These parameters were subsequently
used in the single-gas formulation of the sorption dynamic model that was described in Sec. 4.5.
During the modeling of pure CH
4
and C
2
H
6
sorption dynamics, all the experimental data from six
pressure steps were used simultaneously in the parameterization process. The goal is to minimize
the squares of deviation between the simulated dynamic excess sorption results and the
experimental observations. The results from dynamic sorption experiments and modeling of the
soprtion processes for CH
4
and C
2
H
6
are presented in Figs. 4.23 and 4.24. The extracted model
parameters are summarized in Table 4.3.
111
Figure 4.23. Sorption dynamics of CH
4
sorption on the ground shale sample at 60
o
C:
Experimental vs. Modeling results:
ads
estimated by the DFT method.
Table 4.3. Summary of the model parameters for CH
4
and C
2
H
6
sorption dynamics:
ads
estimated by the DFT method.
a
bk (kg/(Pa.m
3
.s))
CH
4
2.99× 10
-6
C
2
H
6
1.08× 10
-5
0
20
40
60
80
100
120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
112
Figure 4.24. Sorption dynamics of C
2
H
6
sorption on the ground shale sample at 60
o
C:
Experimental vs. Modeling results:
ads
estimated by the DFT method.
Similarly, the modeling results of the sorption dynamics for CH
4
and C
2
H
6
using the LRD
method are presented in Figs. 4.25 and 4.26. The extracted model parameters are summarized in
Table 4.4.
0
5
10
15
20
25
30
0
0.5
1
1.5
2
2.5
3
3.5
4
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
113
Figure 4.25. Sorption dynamics of CH
4
sorption on the ground shale sample at 60
o
C:
Experimental vs. Modeling results:
ads
estimated by the LRD method.
Figure 4.26. Sorption dynamics of C
2
H
6
sorption on the ground shale sample at 60
o
C:
Experimental vs. Modeling results:
ads
estimated by the LRD method.
0
20
40
60
80
100
120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
0
5
10
15
20
25
30
0
0.5
1
1.5
2
2.5
3
3.5
4
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
114
Table 4.4. Summary of the model parameters for CH
4
and C
2
H
6
sorption dynamics:
ads
estimated by the LRD method.
a
bk (kg/(Pa.m
3
.s))
CH
4
3.01× 10
-6
C
2
H
6
1.00× 10
-5
The model parameters
i a
bk
,
for the pure component extracted from the DFT and the LRD
approaches are very close (within 10% difference). This is consistent with the low and adjacent
RMS errors reported in Table 4.1. In addition, both approaches report the C
2
H
6
adsorption rate
constant is approximately 3 times larger than the CH
4
adsorption rate constant. This, again,
implies the different sorption affinities of the shale toward CH
4
and C
2
H
6,
with C
2
H
6
displaying
the stronger affinity.
4.6.2. CH
4
-C
2
H
6
Binary Mixture Sorption Dynamics
Based on the model parameters obtained from fitting the pure component sorption isotherms
(
i s
C
,
and
i A
K
,
) and sorption dynamics (
i a
bk
,
), we applied the sorption dynamic model, in a
predictive mode, to calculate the binary sorption process of CH
4
-C
2
H
6
mixtures on shale at
relevant conditions. In this work, we have studied binary mixtures with 90%, 93% and 96% of
CH
4
by mole. Figs. 4.27-4.29 compare the model predictions of CH
4
-C
2
H
6
binary excess
sorption dynamics with experimental observations using the DFT method. Figs. 4.30-4.32
115
compare the model predictions of CH
4
-C
2
H
6
binary excess sorption dynamics with experimental
observations using the LRD method.
Figure 4.27. Sorption dynamics of 90%-10% CH
4
-C
2
H
6
sorption on the ground shale sample at
60
o
C: Experimental vs. Modeling results:
ads
estimated by the DFT method.
0
10
20
30
40
50
60
70
80
90
100
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
116
Figure 4.28. Sorption dynamics of 93%-7% CH
4
-C
2
H
6
sorption on the ground shale sample at 60
o
C: Experimental vs. Modeling results:
ads
estimated by the DFT method.
Figure 4.29. Sorption dynamics of 96%-4% CH
4
-C
2
H
6
sorption on the ground shale sample at 60
o
C: Experimental vs. Modeling results:
ads
estimated by the DFT method.
0
10
20
30
40
50
60
70
80
90
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
0
10
20
30
40
50
60
70
80
90
100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
117
Figure 4.30. Sorption dynamics of 90%-10% CH
4
-C
2
H
6
sorption on the ground shale sample at
60
o
C: Experimental vs. Modeling results:
ads
estimated by the LRD method.
Figure 4.31. Sorption dynamics of 93%-7% CH
4
-C
2
H
6
sorption on the ground shale sample at 60
o
C: Experimental vs. Modeling results:
ads
estimated by the LRD method.
0
10
20
30
40
50
60
70
80
90
100
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
0
10
20
30
40
50
60
70
80
90
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
118
Figure 4.32. Sorption dynamics of 96%-4% CH
4
-C
2
H
6
sorption on the ground shale sample at 60
o
C: Experimental vs. Modeling results:
ads
estimated by the LRD method.
It can be seen from Figs. 4.27 to 4.32 that for both methods, while the general trend is captured
by the modeling based on the model equations reported in Sec. 4.5, the calculated results
increasingly depart from the experimental observations as C
2
H
6
mole fraction increases from 4%
to 10%. This departure is, in part, due to inaccurate modeling of the sorption isotherms of the
binary system via the extended Langmuir isotherm (further details can be found in Sec. 4.4.2 and
Sec. B and C in the Supplementary Materials), and is the topic of ongoing investigations. In
addition, the widening difference between theory and experiments implies that, with increasing
C
2
H
6
mole
percent, the interaction and competition between the two sorbed species become
increasingly complicated.
0
10
20
30
40
50
60
70
80
90
100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 200 400 600 800 1000 1200
Excess sorption (mg/g)
Time (min)
Exc. Simu. Pres.
119
4.7. Conclusion and Discussion
This study is an extension beyond the measurement of sorption isotherms of pure CH
4
, C
2
H
6
and
their binary mixtures on shale as discussed in Chapter 3. This chapter mainly focuses on the
investigation of sorption dynamics of pure CH
4
, C
2
H
6
and their mixtures on the shale sample. A
Langmuir-type sorption dynamic model is proposed to describe the sorption process of pure
component, i.e., CH
4
, C
2
H
6
on shale. Corresponding parameters were extracted by matching the
pure component excess sorption dynamics. It is found the model was sufficiently accurate to
represent the sorption process for the single gases. Based on the model parameters obtained from
fitting CH
4
and C
2
H
6
sorption dynamics, the model is used in a predictive mode, to calculate the
sorption dynamics for CH
4
-C
2
H
6
mixtures. Departure from the experimental observations was
observed for all 3 mixtures, with a widening deviation as the C
2
H
6
mole fraction increases in the
bulk phase. Taking into account the fact that the interaction and competition between the two
sorbed species become increasingly complicated as C
2
H
6
mole fraction increases, the model is
still successful in terms of predicting the sorption isotherms and capturing the general trend of
the sorption dynamics for the mixtures.
This study summarizes a number of methods that are commonly used to estimate the adsorbed
phase density. Each method has been examined in terms of its ability to reproduce the single and
mixed gases sorption isotherms. The DFT approach, which predicts upward trend of the
adsorbed phase density with increasing bulk phase pressure, generates a maximum number of
moles of CH
4
that can be adsorbed per unit weight of the sample 17% larger than that of C
2
H
6
.
Considering the effective size and shape of the molecules, it is reasonable to predict that more
CH
4
molecules are able to be adsorbed than C
2
H
6
on the same surface. For the mixtures, the
revised mixing rule calculates the interaction coefficient 9513 . 0
12
w , which is reasonable if
120
one considers the potential volume change of the adsorbed phase when two different types of
molecules are competing for sorption sites. The LRD approach, which employs a fairly simple
assumption that the sum of the densities of the adsorbed phase and the bulk phase is a constant at
constant temperature, predicts the opposite: 3% more C
2
H
6
molecules are adsorbed than CH
4
molecules on the same surface. In addition, the interaction coefficient 4447 . 0
12
w also seems
unrealistic. These observations imply that the downward trend in the adsorbed density as
predicted by the LRD approach may not be realistic and the presumption of this method may also
be defective. The MPTA approach, despite the similar trend in the adsorbed phase density as the
DFT approach, appears to have a difficulty in representing both the pure component and binary
mixture isotherms without using a binary interaction parameter to describe the mixture adsorbed
phase. The VLE approach predicts that CH
4
molecules are able to access a larger number of
adsorption sites (2:1 compared to C
2
H
6
). This is still acceptable considering the different size and
shape of the two adsorbates. However, the VLE approach requires an interaction coefficient
6982 . 0
12
w , which is borderline unrealistic in terms of the corresponding volume change of
the adsorbate phase. Since an improved representation of the adsorbed phase density is of crucial
importance in the sorption study of shale-gas systems, it is the topic of ongoing investigations.
The dynamic sorption model that we propose in this study provides a better understanding of the
role of adsorption/desorption at play during shale gas production. As part of our efforts to
delineate the complex transport phenomena during shale gas production, work is currently
underway to study the role of sorption and other mass transfer mechanisms using whole shale
(cube) samples – See next Chapter. Working with whole samples offers an advantage over the
powdered sample due to longer diffusion and viscous flow characteristic times, and will allow us
to more accurately capture these mass transfer characteristics of the shales.
121
4.8. References
[1] Bulba KA, Krouskop PE. Compositional variety complicates processing plans for US shale gas.
Oil & Gas Journal. 2009;107(10):50-5.
[2] Chareonsuppanimit P, Mohammad SA, Robinson RL, Gasem KA. High-pressure adsorption of
gases on shales: Measurements and modeling. International Journal of Coal Geology. 2012 Jun
1;95:34-46.
[3] Gasparik M, Ghanizadeh A, Bertier P, Gensterblum Y, Bouw S, Krooss BM. High-pressure
methane sorption isotherms of black shales from the Netherlands. Energy & fuels. 2012 Jul
20;26(8):4995-5004.
[4] Yuan W, Pan Z, Li X, Yang Y, Zhao C, Connell LD, Li S, He J. Experimental study and
modelling of methane adsorption and diffusion in shale. Fuel. 2014 Jan 30;117:509-19.
[5] Wang Y, Tsotsis TT, Jessen K. Competitive Sorption of Methane/Ethane Mixtures on Shale:
Measurements and Modeling. Industrial & Engineering Chemistry Research. 2015 Nov
20;54(48):12187-95.
[6] Lu XC, Li FC, Watson AT. Adsorption measurements in Devonian shales. Fuel. 1995 Apr
30;74(4):599-603.
[7] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide
sorption on coal and shale samples from the Paraná Basin, Brazil. International Journal of Coal
Geology. 2010 Dec 1;84(3):190-205.
[8] Chareonsuppanimit P, Mohammad SA, Robinson RL, Gasem KA. High-pressure adsorption of
gases on shales: Measurements and modeling. International Journal of Coal Geology. 2012 Jun
1;95:34-46.
[9] Zhang T, Ellis GS, Ruppel SC, Milliken K, Yang R. Effect of organic-matter type and thermal
122
maturity on methane adsorption in shale-gas systems. Organic Geochemistry. 2012 Jun
30;47:120-31.
[10] Arri LE, Yee D, Morgan WD, Jeansonne MW. Modeling coalbed methane production with
binary gas sorption. In SPE Rocky Mountain Regional Meeting 1992 Jan 1. Society of Petroleum
Engineers.
[11] Sudibandriyo M, Pan Z, Fitzgerald JE, Robinson RL, Gasem KA. Adsorption of methane,
nitrogen, carbon dioxide, and their binary mixtures on dry activated carbon at 318.2 K and
pressures up to 13.6 MPa. Langmuir. 2003 Jun 24;19(13):5323-31.
[12] Fitzgerald JE, Sudibandriyo M, Pan Z, Robinson RL, Gasem KA. Modeling the adsorption of
pure gases on coals with the SLD model. Carbon. 2003 Dec 31;41(12):2203-16.
[13] Ottiger S, Pini R, Storti G, Mazzotti M. Measuring and modeling the competitive adsorption
of CO2, CH4, and N2 on a dry coal. Langmuir. 2008 Aug 5;24(17):9531-40.
[14] Srinivasan K, Saha BB, Ng KC, Dutta P, Prasad M. A method for the calculation of the
adsorbed phase volume and pseudo-saturation pressure from adsorption isotherm data on activated
carbon. Physical Chemistry Chemical Physics. 2011;13(27):12559-70.
[15] El-sharkawy MM, Askalany AA, Harby K, Ahmed MS. Adsorption isotherms and kinetics of
a mixture of Pentafluoroethane, 1, 1, 1, 2-Tetrafluoroethane and Difluoromethane (HFC-407C)
onto granular activated carbon. Applied Thermal Engineering. 2016 Jan 25;93:988-94.
[16] Shojaei H, Jessen K. Application of potential theory to modeling of ECBM recovery. InSPE
Western North American Region Meeting 2011 Jan 1. Society of Petroleum Engineers.
[17] Busch A, Gensterblum Y, Krooss BM. Methane and CO
2
sorption and desorption
measurements on dry Argonne premium coals: pure components and mixtures. International
Journal of Coal Geology. 2003 Aug 31;55(2):205-24.
123
[18] Li D, Liu Q, Weniger P, Gensterblum Y, Busch A, Krooss BM. High-pressure sorption
isotherms and sorption kinetics of CH4 and CO2 on coals. Fuel. 2010 Mar 31;89(3):569-80.
[19] Pini R, Ottiger S, Burlini L, Storti G, Mazzotti M. Sorption of carbon dioxide, methane and
nitrogen in dry coals at high pressure and moderate temperature. International Journal of
Greenhouse Gas Control. 2010 Jan 31;4(1):90-101.
[20] Mosher K, He J, Liu Y, Rupp E, Wilcox J. Molecular simulation of methane adsorption in
micro-and mesoporous carbons with applications to coal and gas shale systems. International
Journal of Coal Geology. 2013 Apr 1;109:36-44.
[21] Malheiro C, Mendiboure B, Plantier F, Guatarbes B, Miqueu C. An accurate model for the
filling pressure of carbon slit-like micropores. Fluid Phase Equilibria. 2015 Oct 25;404:118-23.
[22] Guggenheim EA. The principle of corresponding states. The Journal of Chemical Physics.
1945 Jul 1;13(7):253-61.
4.9. Supplemental Materials
A. Sorption Dynamic Data Correction
The heart of the TGA apparatus used in this study is the German made Rubotherm magnetic
suspension balance (MSB), whose key technical specifications are listed in Table A.1.
124
Table A.1. Specifications of the Rubotherm magnetic suspension balance.
Pressure range Vacuum to 350 bar
Temperature range Up to 350
o
C
Maximum load 10 g
Resolution 0.01 mg to 1 ~µ g
Reproducibility (standard deviation)
± 0.02 mg to ± 2 µ g
Relative error <= 0.002% of measured value
The MSB operation cycle consists of zero point, measuring point 1 and measuring point 2.
During the process of a sorption measurement, the MSB zero point might drift because of even
the slightest disturbance introduced from the measurement or surrounding environment, and will
need to be corrected periodically. The TGA zero point correction assumes that the zero point
drift happens in a linear manner, so that the drift away from the zero point has a constant slope
within the time period investigated. This zero point correction, as we discovered from the
sorption measurement observation, manifests itself as an overshoot or undershoot. This behavior
is likely caused by a zero-point fluctuation introduced by the short-lived turbulent flow created
by the pressure change because we found it usually occurs immediately after the bulk phase
pressure is changed. To eliminate the effect of the zero-point fluctuation, a blank run with an
inert substance (here quartz) was performed. The procedure for the blank run, CH
4
for example,
is shown below.
125
1. Load quartz (ground) with the same volume, V
q
,
as the shale sample into the TGA apparatus.
The weight of the quartz, W
q
, can be measured by the TGA.
2. Sorption experiment on quartz with CH
4
is carried out by following exactly the same
procedure as that in the sorption experiment on shale.
For the quartz sample, given that the CH
4
pressure, thus the bulk phase density, is recorded, we
know the true (apparent) weight that the MSB should record
q CH q
V W t W
4
(A.1)
Then, we assume that the zero point change during the sorption process is described by t D .
Then the uncorrected weight that the balance will record in the blank run is
t D t W t W
b
0
(A.2)
Substitute Eq. A.1 into Eq. A.2, and after some algebraic manipulations, we obtain
q CH q b
V W t W t D
4
0
(A.3)
Then the sorption experiment with CH
4
is repeated on the shale sample. Extreme care needs to be
taken to ensure that the same procedure was followed exactly as in the blank run. Then the
corrected weight in the CH
4
sorption experiment on the shale sample will be
t D t W t W
b c
(A.4)
where t W
b
is the uncorrected weight that the balance records in the CH
4
sorption on the shale
sample ; t D is obtained from Eq. A.3. Note that the above derivation is based on the
assumption that CH
4
doesn't absorb on the quartz sample. This presumption was later proved
valid by CH
4
sorption experiment on quartz, which confirmed that quartz was indeed
non-adsorbent.
126
B. DFT Method
The model coefficients that appear in Eq. 4.4 are summarized in Table B.1.
Table B.1. Summary of parameters in the DFT mapping function.
c
(mol/cm
3
)
max
(mol/cm
3
)
CH
4
0.01014 0.02338
C
2
H
6
0.00687 0.01567
The Langmuir model parameters extracted from the DFT approach for estimating the pure
component adsorbed phase density are summarized in Table B.2.
Table B.2. Summary of parameters in the Langmuir model.
abs
n
max
(mmol/g)
A
K (bar
-1
)
CH
4
0.1938 0.0251
C
2
H
6
0.1653 0.1559
For mixtures, the adsorbed phase densities are estimated from the following mixing rule,
2 ,
2
1 ,
1
,
12
a a mix a
x x w
(B.1)
where
mix a,
the adsorbed phase density for the mixtures corresponding to the total bulk phase
pressure;
1 , a
the adsorbed phase density of component 1, i.e., CH
4
, corresponding to its partial
127
pressure in the bulk phase;
2 , a
the adsorbed phase density of component 2, i.e., C
2
H
6
,
corresponding to its partial pressure in the bulk phase;
1
x the mole fraction of component 1 in
the adsorbed phase;
2
x the mole fraction of component 2 in the adsorbed phase. They are
calculated by taking into account that CH
4
and C
2
H
6
molecules occupy different sorption areas.
a p K p K
p K
x
A A
A
2 2 , 1 1 ,
1 1 ,
1
(B.2)
a p K p K
a p K
x
A A
A
2 2 , 1 1 ,
2 2 ,
2
(B.3)
where
1 , A
K and
2 , A
K are the sorption equilibrium constants of CH
4
and C
2
H
6
, respectively;
1
p and
2
p are the partial pressures of CH
4
and C
2
H
6
in the bulk phase, respectively;
abs
abs
n
n
a
2 max,
1 max,
is the ratio of adsorption sites occupied by a single C
2
H
6
molecule and that occupied
by a single CH
4
molecule. Here in this case, the value of a equals 1.172. The coefficient
12
w
can be parameterized by minimizing the following objective function,
NP
n mix a
mix g abs abs exc
t
n MW n MW n
1 ,
,
2 2 1 1 exp
1
(B.4)
where NP represents the total number of experimental data points for the binary mixtures
adsorption.
abs
n
1
and
abs
n
2
are the absolute adsorption for CH
4
and C
2
H
6
, respectively, which
are estimated by the following revised extended Langmuir model (ELM),
abs
A A
A abs
n
p K p K
p K
n
max
2 2 , 1 1 ,
1 1 ,
1
1
(B.5)
128
a
n
p K p K
p K
n
abs
A A
A abs max
2 2 , 1 1 ,
2 2 ,
2
1
(B.6)
where
abs
n
max
(0.1938 mmol/g) is the maximum moles of adsorption sites per unit weight of the
sample. The objective function is to minimize the deviation of squares of differences between the
experimental observations and the revised ELM predictions. The binary interaction coefficient
12
w is obtained,
0.9513 =
12
w .
In the calculation of the bulk phase gas density,
gas
, throughout this work, unless noted
otherwise, the relevant PR-EOS model parameters are listed in Table B.3, with the binary
interaction coefficient between CH
4
and C
2
H
6
set to zero.
Table B.3. Peng-Robinson EOS model parameters.
Critical temperature
c
T (K)
Critical pressure
c
p (bar)
Acentric factor
(dimensionless)
Volume shift
i
S (dimensionless)
CH
4
190.56 45.99 0.011 -0.03463
C
2
H
6
305.32 48.72 0.099 -0.34350
C. LRD Method
In the LRD method of estimating the adsorbed phase density, the model parameters that appear
in Eq. 4.6 are summarized in Table C.1.
129
Table C.1. Summary of model parameters in the LRD approach.
*
A (g/cm
3
)
abs
n
max
(mmol/g)
A
K (bar
-1
)
CH
4
0.7066 0.1494 0.0327
C
2
H
6
0.9951 0.1538 0.1702
For mixtures, the adsorbed phase densities are estimated from the following mixing rule,
2 ,
2
1 ,
1
,
12
a a mix a
x x w
(C.1)
where
mix a,
the adsorbed phase density for the mixtures corresponding to the total bulk phase
pressure;
1 , a
the adsorbed phase density of component 1, i.e., CH
4
, corresponding to its partial
pressure in the bulk phase;
2 , a
the adsorbed phase density of component 2, i.e., C
2
H
6
,
corresponding to its partial pressure in the bulk phase;
1
x the mole fraction of component 1 in
the adsorbed phase;
2
x the mole fraction of component 2 in the adsorbed phase. Since the
adsorbed phase mole fractions are unknown, they have to be estimated. In this case, the revised
extended Langmuir model (ELM) has to reflect the observation that the maximum sorption
amounts are not exactly the same for CH
4
and C
2
H
6
: CH
4
and C
2
H
6
molecules occupy different
sorption areas. The revised version of the ELM is showing below,
a n
p K p K
p K
n
abs
A A
A abs
max
2 2 , 1 1 ,
1 1 ,
1
1
(C.2)
abs
A A
A abs
n
p K p K
p K
n
max
2 2 , 1 1 ,
2 2 ,
2
1
(C.3)
130
where
1 , A
K and
2 , A
K are the sorption equilibrium constants of CH
4
and C
2
H
6
, respectively;
1
p and
2
p are the partial pressures of CH
4
and C
2
H
6
, respectively;
abs
n
max
(0.1538 mmol/g) is
the maximum number of mole of adsorption sites per unit weight of the sample;
abs
abs
n
n
a
2 max,
1 max,
is
the ratio of adsorption sites occupied by a single C
2
H
6
molecule and that occupied by a single
CH
4
molecule. Here in this case, the value of a equals 0.971;
abs
n
1
and
abs
n
2
are the number
of mole of CH
4
and C
2
H
6
molecules adsorbed on the surface per unit weight of the sample when
they compete with each other. Using the revised version of ELM, the adsorbed phase mole
composition can then be estimated as follows,
abs abs
abs
n n
n
x
2 1
1
1
(C.4)
abs abs
abs
n n
n
x
2 1
2
2
(C.5)
Substitute Eqs. C.2 and C.3 into Eqs. C.4 and C.5,
2 2 , 1 1 ,
1 1 ,
1
p K p aK
p aK
x
A A
A
(C.6)
2 2 , 1 1 ,
2 2 ,
2
p K p aK
p K
x
A A
A
(C.7)
Eqs. C.6 and C.7 provide us with the adsorbed phase mole fractions (
1
x and
2
x ). Then the
adsorbed phase density for the mixtures can be estimated by Eq. C.1. Eq. C.1 is revised based on
the ideal mixing rule which assumes volume additivity. The newly added coefficient
12
w can be
parameterized by minimizing the following objective function,
131
NP
n mix a
mix g abs abs exc
t
n MW n MW n
1 ,
,
2 2 1 1 exp
1
(C.8)
where NP represents the total number of experimental points for the binary mixtures. The
objective function is to minimize the deviation between the experimental excess sorption and
ELM predicted results,
4447 . 0
12
w
D. Multicomponent Potential Theory of Adsorption (MPTA) Method
The five-parameter MPTA model was used to fit the experimental pure component isotherms at
60
o
C. Table D.1 reports the model parameters.
Table D.1. MPTA model parameters obtained from pure component sorption isotherms.
MPTA CH
4
C
2
H
6
K R
i 0
595.6 854.3
i
1.178 1.443
gr cc z
0
0.0100
In the MPTA approach, alongside the model fit of the experimental excess sorption isotherms for
the pure components and binary mixtures, the molar density of the adsorbed phase is calculated
by the PR-EOS (model parameters are listed in Table B.3.). For the pure components, taking
advantage of the adsorbed phase density attained from MPTA model fit of the excess sorption
isotherms, one can convert the excess sorption into the absolute sorption, and subsequently fit the
132
absolute sorption isotherms using the Langmuir model. The Langmuir model parameters
extracted from the MPTA approach are summarized in Table D.2. For the mixtures, the total
excess sorption (CH4 + C2H6) is calculated by the extended Langmuir model based on the
parameters listed in Table D.2 and the adsorbed phase density attained from MPTA model fit of
the mixture excess sorption isotherms. In Chapter 3, we have shown that the MPTA is capable of
representing the sorption isotherms, but the main focus in this chapter is the description of the
sorption dynamics, which is done by the Langmuir-type of sorption dynamic model. Still, the
MPTA model can provide the estimate of the adsorbed phase density, which is essential for the
sorption dynamic model.
Table D.2. Summary of parameters in the Langmuir model.
abs
n
max
(mmol/g)
A
K (bar
-1
)
CH
4
0.1925 0.0275
C
2
H
6
0.1505 0.1116
E. Vapor-Liquid Equilibrium (VLE) Method
The Langmuir model parameters extracted from the pure component sorption isotherms using the
VLE approach are summarized in Table E.1.
133
Table E.1. Summary of parameters in the Langmuir model.
abs
n
max
(mmol/g)
A
K (bar
-1
)
CH
4
0.3795 0.0082
C
2
H
6
0.1649 0.1495
For mixtures, the revised extended Langmuir model (ELM) which reflects the observation that
C
2
H
6
molecule occupies about twice as much sorption area as CH
4
molecule was used. The
revised version of the ELM is showing below:
abs
A A
A abs
n
p K p K
p K
n
max
2 2 , 1 1 ,
1 1 ,
1
1
(E.1)
a
n
p K p K
p K
n
abs
A A
A abs max
2 2 , 1 1 ,
2 2 ,
2
1
(E.2)
where
1 , A
K and
2 , A
K are the sorption equilibrium constants of CH
4
and C
2
H
6
, respectively;
1
p and
2
p are the partial pressures of CH
4
and C
2
H
6
, respectively;
abs
n
max
(0.3795 mmol/g) is
the maximum number of mole of adsorption sites per unit weight of the sample; a is the ratio
of number of adsorption sites occupied by a single C
2
H
6
molecule and a single CH
4
molecule.
Here in our case, the value of a equals 2.3;
abs
n
1
and
abs
n
2
are the number of mole of CH
4
and C
2
H
6
molecules adsorbed on the surface per unit weight of the sample. Using the revised
version of the ELM, the adsorbed phase composition can then be estimated as follows,
abs abs
abs
n n
n
x
2 1
1
1
(E.3)
134
abs abs
abs
n n
n
x
2 1
2
2
(E.4)
Substitute Eqs. E.1 and E.2 into Eqs. E.3 and E.4,
a P K P K
P K
x
A A
A
2 2 , 1 1 ,
1 1 ,
1
(E.5)
a P K P K
a P K
x
A A
A
2 2 , 1 1 ,
2 2 ,
2
(E.6)
Eqs. E.5 and E.6 provide us with the adsorbed phase mole fractions (
1
x and
2
x ). The
individual species adsorbed phase densities (
1 , a
and
2 , a
) can be interpolated relative to their
corresponding partial pressures. Once obtaining these values, the mixture adsorbed phase
densities of the mixture are estimated by Eq. E.7.
2 ,
2
1 ,
1
,
12
a a mix a
x x w
(E.7)
The newly added
12
w can be parameterized by minimizing the following objective function,
NP
n mix a
mix g abs abs exc
t
n MW n MW n
1 ,
,
2 2 1 1 exp
1
(E.8)
where NP represents the total number of experimental data points for the binary mixtures
adsorption. The objective function is to minimize the deviation between the experimental excess
sorption results and the revised ELM predictions. The binary interaction coefficient
12
w is
obtained,
0.6982 =
12
w .
135
Chapter 5
Sorption Kinetics and Mass Transfer in a
Cubic Shale Sample
5.1. Introduction
Shale gas, which contains primarily methane (CH
4
) and smaller amounts of other hydrocarbons,
such as ethane (C
2
H
6
) and propane (C
3
H
8
), exists as sorbed gas and free gas. In particular, these
gases are stored in the sorbed state in the micro and mesopores of the shale rocks, and as free gas
in the fracture networks. Though convective and diffusive transport accounts for the short-term
behavior in gas production, desorption is thought to dominate the long-term dynamics of shale
gas generation [1].
The key objective of this chapter is to investigate adsorption/desorption as well as diffusive and
convective transport phenomena of methane/hydrocarbon mixtures in shale-gas rocks. In
particular, we focus our attention on the binary CH
4
/C
2
H
6
mixture, since C
2
H
6
is typically the
second largest component of shale gas, and is thought to compete for the same adsorption sites in
gas shale rocks as CH
4
. We study the adsorption/desorption behavior of this mixture (and its
individual components) in whole shale rock samples using thermogravimetric analysis (TGA).
The sorption isotherms are important to predict the gas storage capacity of the shale samples,
while the study of adsorption/desorption dynamics/kinetics help us understand the role of
desorption during the later times of gas production.
136
In the laboratory-scale research focusing on gas sorption and other mass transfer mechanisms in
shale, there are typically two sample preparation approaches, involving either ground samples or
whole rock samples. Weniger et al. [2] stated that the grain size of coal/shale controls gas
diffusion and sorption kinetics: gas can diffuse faster to the adsorption sites in smaller sample
particles so that the equilibrium time will be shorter compared to for larger particles. Cloke et al.
[3] reported that grinding of coal and size fractionation (sieving) may cause compositional
change in sub-samples of various particle sizes: maceral composition will differ from the
smallest fraction (inertinite enriched) to larger fractions (vitrinite enriched). Spears and Booth [4]
observed that mineral matter tends to be enriched in the smaller size fractions. Busch et al. [5]
found these compositional changes can also result in differences in sorption capacity. In our
present study (Chapter 4) of sorption kinetics and mass transfer in shale, ground sample with a
small size range, i.e. 1-1.18 mm, was chosen for the sorption experiments to reduce internal mass
transfer limitation effects and to facilitate faster sorption of gases on the shale sample. However,
as discussed in Sec. 4.5 in Chapter 4, in the ground sample experiments, due to small size,
diffusion and viscous flow characteristics cannot be effectively captured by the TGA
experiments. As a part of our ongoing effort to study the sorption and mass transfer mechanisms
in the shale-gas system at different scales, an investigation of sorption kinetics and mass transfer
in a whole shale cube (~ 1 cm
3
) was undertaken. The upscaling of sample size would facilitate
deeper understanding of sorption, diffusion and other mass transfer mechanisms associated with
shale gas production.
137
5.2. Experimental
5.2.1. Sample Preparation
The shale sample used in this study is from the Marcellus formation in the Appalachian basin. It
was extracted from a depth of 7860.0 ft (note, that this sample is different from the ground
sample used in Chapter 4). Prior to its use in the sorption study, the sample was stored in a
zip-lock bag as received to avoid further oxidation and water uptake. The shale sample was cut
into a ~1cm
3
cube with sides of approximately same length by a mechanical saw (Fig. 5.1.)
Figure 5.1. Picture of the shale sample cube.
Prior to the initiation of the sorption experiments, the sample was evacuated at 120
o
C for 24 hrs.
At the end of each sorption experiment, the sample was regenerated under vacuum at 120
o
C for
24 hrs. Such a procedure assures that all gases that may potentially remain adsorbed at the end of
an experiment are desorbed prior to the initiation of the next experiment [6].
5.2.2. Experimental Approach
The thermogravimetric analysis (TGA) technique was used for the measurement of the CH
4
and
C
2
H
6
sorption isotherms/dynamics. The heart of the TGA set-up is a magnetic suspension
balance (Rubotherm, Germany), which is capable of measuring weight changes down to 1 μg.
138
During a sorption experiment, the weight change of the sample is transmitted from the sorption
chamber to the analytical balance in a contactless manner via the magnetic suspension
mechanism. The data recording time interval was set to 6 sec. Prior to the sorption experiment,
measurements of the weight of the sample container
sc
M and its volume
sc
V along with the
weight of the sample
s
M and its skeletal volume
s
V were performed by conducting Helium
sorption experiments. Details of the measurement technique can be found elsewhere [6]. The
sorption experiments were performed by varying the gas phase pressure in the sorption chamber
in a step-wise manner, e.g., from 10 to 20, 20 to 30 bar, etc. However, it was found that abrupt
overshoot/undershoot could occur immediately after the pressure alteration, due to a zero-point
fluctuation introduced by the short-lived turbulent flow created by the pressure change. To
eliminate the effect of the zero-point fluctuation, a blank run with an inert substance (here
quartz) was performed. The blank run was performed by following the same procedure as for the
sorption experiments on the shale sample. The zero-point fluctuation was then recorded and
subtracted from the sorption dynamic data from shale experiments to eliminate the
overshoot/undershoot (details can be found in the supplementary material in Chapter 4).
5.3. Experimental Results
5.3.1. CH
4
and C
2
H
6
Pure Component Sorption Isotherms
Prior to the sorption dynamic experiments, CH
4
and C
2
H
6
sorption isotherms on the sample cube
were measured in the TGA apparatus at 40, 50 and 60
o
C. Fig. 5.2 presents the experimental
observations with actual isotherms data reported in Tables S.1 to S.3 in the Supplementary
Materials.
139
Figure 5.2. CH
4
and C
2
H
6
sorption isotherms on shale (cube) at 40, 50 and 60
o
C.
The ratio of C
2
H
6
/CH
4
excess sorption between 1.5 and 2.5 (on a molar basis) over the common
pressure range investigated, indicating clear sorption affinity of C
2
H
6
over CH
4
. These results are
consistent with the experimental observations of the ground shale sample that were reported in
Chapter 4.
5.3.2. CH
4
-C
2
H
6
Binary Mixture Sorption Isotherms
Sorption isotherms were also measured for binary mixtures of CH
4
-C
2
H
6
on the cubic shale
sample at the same temperatures (i.e., 40
o
C, 50
o
C and 60
o
C) as those employed for the single
gas experiments. Three different CH
4
-C
2
H
6
binary gas compositions (90%-10%, 93%-7%, and
96%-4% mole fraction – certified pre-mixed gases purchased from Matheson) were used in these
experiments, and the isotherms are reported in Figs. 5.3 to 5.5, and Tables S.4 to S.6 in the
Supplementary Materials.
0
1
2
3
4
5
6
7
8
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 @ 40C
CH4 @ 50C
CH4 @ 60C
C2H6 @ 40C
C2H6 @ 50C
C2H6 @ 60C
140
Figure 5.3. Sorption of CH4-C2H6 mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 40
o
C.
Figure 5.4. Sorption of CH4-C2H6 mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 50
o
C.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
141
Figure 5.5. Sorption of CH4-C2H6 mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 60
o
C.
From Figs. 5.3-5.5, we observe an increase in the total excess sorption amount as the
concentration of C
2
H
6
in the binary mixture increases. This is consistent with the single-gas
sorption data that indicate the preferential sorption of C
2
H
6
over CH
4
in the shale sample (Fig.
5.2). Interestingly, we do not observe the maximum in the total excess loading at ∼100 bar as
that appears in the CH4-C2H6 competitive sorption in Chapter 4. Also, we notice that the sorption
capacity of the shale sample used in this study is much larger than that used in Chapter 4, even
though they are extracted from the same site, with depths that are less than 60 ft apart. This,
again, highlights the complexity of the shale-gas system and the differences among samples with
different geological ages.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
90%-10% CH4-C2H6
93%-7% CH4-C2H6
96%-4% CH4-C2H6
142
5.3.3. CH
4
and C
2
H
6
Pure Component Sorption Dynamics
CH
4
and C
2
H
6
sorption experiments were carried out through a step-wise pressurization, as
described in Sec. 5.2.2. The pure component sorption dynamics were obtained along with the
sorption isotherms on the cubic shale sample at 60 ° C. Fig. 5.6 shows CH4 sorption dynamics at
60 ° C while the dynamics of a single pressure step (from 10 to 20 bar) is reported in Fig. 5.7. Fig.
5.8 shows C
2
H
6
sorption dynamics at 60 ° C while the dynamics of a single pressure step (from 5
to 10 bar) is reported in Fig. 5.9. It is worth noting that the corrected weight represents the
summation of weight of the sample container, shale sample and excess sorption after taking the
buoyancy effect into account.
Figure 5.6. Sorption dynamic process of CH4 on shale (cube) at 60
o
C.
0
20
40
60
80
100
120
8.4
8.42
8.44
8.46
8.48
8.5
8.52
8.54
8.56
0 500 1000 1500 2000 2500 3000
Corr. weight (g)
Time (min)
Corr. Weight Pressure
143
Figure 5.7. Sorption dynamic process of CH4 on shale (cube) at 60
o
C: from 10 to 20 bar.
Figure 5.8. Sorption dynamic process of C
2
H
6
on shale (cube) at 60
o
C.
0
5
10
15
20
25
8.53
8.532
8.534
8.536
8.538
8.54
8.542
8.544
8.546
0 100 200 300 400 500
Corr. weight (g)
Time (min)
Corr. weight Pressure
0
5
10
15
20
25
30
35
40
8.46
8.47
8.48
8.49
8.5
8.51
8.52
8.53
8.54
8.55
8.56
0 600 1200 1800
Corr. weight (g)
Time (min)
Corr. weight Pressure
144
Figure 5.9. Sorption dynamic process of C
2
H
6
on shale (cube) at 60
o
C: from 5 to 10 bar.
Similar to the sorption dynamics in the ground shale sample, sorption takes a substantial time to
reach equilibrium compared to the sudden change in the bulk phase pressure in the whole sample.
An interesting observation is that it takes longer for the whole sample to reach sorption
equilibrium than the ground sample. For example, as one compares the sorption process of C
2
H
6
on the ground sample (Fig. 4.3 in Chapter 4) and on the whole sample (Fig. 5.9), at t = 30 min
C
2
H
6
has reached 98.5% of the maximum equilibrium sorption amount on the ground sample,
whereas it has only reached 91.5% of the maximum equilibrium sorption amount on the whole
sample. This is consistent with the grain size effect on the sorption kinetics stated by Weniger et
al. [2]. The weight measured by the TGA first decreases rapidly due to the increased buoyancy
effect caused by the increase of the bulk phase pressure (density). Equilibrium is reached later in
a much slower manner. This, again, implies that movement into the shale matrix is the rate
limiting step among all mass transfer mechanisms of gases in shale, thus dominates the shale gas
generation in the later times of a production well’s lifespan.
0
2
4
6
8
10
12
8.538
8.54
8.542
8.544
8.546
8.548
8.55
8.552
0 50 100 150 200 250
Corr. weight (g)
Time (min)
Corr. weight Pressure
145
5.3.4. CH
4
-C
2
H
6
Binary Mixture Sorption Dynamics
Sorption dynamics were also measured for CH
4
-C
2
H
6
binary mixtures (90%-10%, 93%-7%,
and 96%-4% CH
4
-C
2
H
6
, mole fraction) on the cubic shale sample at 60
o
C. Figs. 5.10-5.12
present the experimental results of the sorption dynamic process for the mixtures of various
compositions.
Figure 5.10. Sorption dynamic process of 90%-10% CH
4
-C
2
H
6
on shale (cube) at 60
o
C.
0
20
40
60
80
100
120
8.4
8.42
8.44
8.46
8.48
8.5
8.52
8.54
8.56
0 500 1000 1500 2000
Corr. weight (g)
Time (min)
Corr. weight Pressure
146
Figure 5.11. Sorption dynamic process of 93%-7% CH
4
-C
2
H
6
on shale (cube) at 60
o
C.
Figure 5.12. Sorption dynamic process of 96%-4% CH
4
-C
2
H
6
on shale (cube) at 60
o
C.
The sorption dynamic data in the whole shale sample was recorded much frequently than that in
the ground sample, thus providing us with more information, especially during the initial phase
0
20
40
60
80
100
120
8.4
8.42
8.44
8.46
8.48
8.5
8.52
8.54
8.56
0 500 1000 1500 2000
Corr. weight (g)
Time (min)
Corr. weight Pressure
0
20
40
60
80
100
120
8.42
8.44
8.46
8.48
8.5
8.52
8.54
8.56
0 500 1000 1500 2000
Corr. weight (g)
Time (min)
Corr. weight Pressure
147
of sorption process. Another advantage of using whole sample is to capture the diffusion and
viscous flow characteristics of gases in shale, which was not possible using the ground sample.
5.4. Langmuir Adsorption Isotherms Calculated from the DFT and the LRD
Approaches
In Chapter 4, we have shown that the DFT [7] and the LRD [8] approaches are both capable of
representing the pure component sorption isotherms, and correlating the CH
4
-C
2
H
6
binary
mixture sorption isotherms based on the extracted Langmuir model parameters for the pure
components and a revised mixing rule. As a result, we choose to apply these two approaches to
estimate the adsorbed phase density for single and mixed gases sorption on the whole cubic shale
sample. Figs. 5.13 to 5.16 provide a comparison of experimental and simulated excess sorption
isotherms for both pure and binary mixtures based on these two approaches.
148
Figure 5.13. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale
(cube) at 60
o
C:
ads
estimated by the DFT method.
Figure 5.14. ELM correlation of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale
(cube) at 60
o
C:
ads
estimated by the DFT method.
0
1
2
3
4
5
6
7
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
149
Figure 5.15. Langmuir model fit of the excess sorption isotherms of CH
4
and C
2
H
6
on shale
(cube) at 60
o
C:
ads
estimated by the LRD method.
Figure 5.16. ELM correlation of the excess sorption isotherms of CH
4
-C
2
H
6
mixtures on shale
(cube) at 60
o
C:
ads
estimated by the LRD method.
0
1
2
3
4
5
6
7
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
CH4 Expt.
C2H6 Expt.
CH4 Simu.
C2H6 Simu.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120
Excess sorption (mg/g)
Pressure (bar)
90%-10% C1-C2 Expt.
93%-7% C1-C2 Expt.
96%-4% C1-C2 Expt.
90%-10% C1-C2 Simu.
93%-7% C1-C2 Simu.
96%-4% C1-C2 Simu.
150
The root-mean square (RMS) errors are listed in Table 5.1 for both single and mixed gases for
these two approaches.
Table 5.1. RMS errors for single and mixed gases isotherms using the DFT and the LRD density
approaches.
RMS error (mg/g)
CH
4
C
2
H
6
90:10
CH
4
/C
2
H
6
93:7
CH
4
/C
2
H
6
96:4
CH
4
/C
2
H
6
DFT 0.0822 0.1027 0.1547 0.0952 0.1373
LRD 0.0582 0.0656 0.1226 0.0607 0.1059
We observe from Figs. 5.13 to 5.16 that both the DFT and the LRD approaches are reasonably
accurate in representing the pure component and binary mixture sorption isotherms. Table 5.1
indicates the LRD approach is slightly better than the DFT approach with lower RMS errors. The
Langmuir model parameters obtained by these two approaches are reported in Table 5.2.
Table 5.2. Langmuir model parameters for CH
4
and C
2
H
6
sorption on shale (cube) at 60
o
C.
abs
n
max
(mmol/g)
A
K (bar
-1
)
DFT
CH
4
0.3649 0.0213
C
2
H
6
0.3130 0.0856
LRD
CH
4
0.2747 0.0277
C
2
H
6
0.2736 0.1007
151
It is worth noting the interaction coefficients (
12
w ) that appear in the revised mixing rule (further
details can be found in the Supplementary Materials in Chapter 4) are 1.6914 and 1.5733,
respectively, for the DFT and the LRD approaches for estimating the adsorbed phase density for
the mixtures. This contradicts the observation that the coefficients are less than unity in Chapter
4. This, again, features the complicated interactions between the two species in the sorbed phase
in the shale matrix.
5.5. Conclusion and Discussion
It this chapter, we have measured the sorption isotherms and the sorption dynamics for pure CH
4
,
C
2
H
6
and their binary mixtures at various temperatures in a whole cubic shale sample. Work is
currently underway to study the roles of sorption and other mass transfer mechanisms at play
during shale gas production using a sorption dynamic model that we proposed. As part of our
ongoing efforts to delineate the complex transport phenomena during shale gas production, we
have used a powdered sample to measure the sorption isotherms and dynamics for CH
4
, C
2
H
6
,
and CH
4
-C
2
H
6
binary mixtures using the TGA technique in Chapter 4. However, the small
sample size makes it impossible to extract any diffusion and viscous flow characteristics. This
chapter, however, employs a whole cubic shale sample. Working with whole sample offers an
advantage over the powdered sample due to longer diffusion and viscous flow times, and will
allow us to capture these mass transfer characteristics along with sorption properties of the shales
more accurately.
152
5.6. References
[1] Yang T, Li X, Zhang D. Quantitative dynamic analysis of gas desorption contribution to
production in shale gas reservoirs. Journal of Unconventional Oil and Gas Resources. 2015 Mar
31;9:18-30.
[2] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide
sorption on coal and shale samples from the Paraná Basin, Brazil. International Journal of Coal
Geology. 2010 Dec 1;84(3):190-205.
[3] Cloke M, Lester E, Belghazi A. Characterisation of the properties of size fractions from ten
world coals and their chars produced in a drop-tube furnace. Fuel. 2002 Mar 31;81(5):699-708.
[4] Spears DA, Booth CA. The composition of size-fractionated pulverised coal and the trace
element associations. Fuel. 2002 Mar 31;81(5):683-90.
[5] Busch A, Gensterblum Y , Krooss BM, Littke R. Methane and carbon dioxide
adsorption–diffusion experiments on coal: upscaling and modeling. International Journal of Coal
Geology. 2004 Dec 3;60(2):151-68.
[6] Wang Y , Tsotsis TT, Jessen K. Competitive Sorption of Methane/Ethane Mixtures on Shale:
Measurements and Modeling. Industrial & Engineering Chemistry Research. 2015 Nov
20;54(48):12187-95.
[7] Ottiger S, Pini R, Storti G, Mazzotti M. Measuring and modeling the competitive adsorption
of CO2, CH4, and N2 on a dry coal. Langmuir. 2008 Aug 5;24(17):9531-40.
153
[8] Srinivasan K, Saha BB, Ng KC, Dutta P, Prasad M. A method for the calculation of the
adsorbed phase volume and pseudo-saturation pressure from adsorption isotherm data on
activated carbon. Physical Chemistry Chemical Physics. 2011;13(27):12559-70.
154
5.7. Supplementary Data
Table S.1. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 40
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
5 1.026145 2 3.285041
9.9 1.446735 4.9 4.10082
20.1 1.913198 10 5.115861
30 2.259645 14.9 5.735903
40.1 2.532052 20.1 6.201112
50 2.722036 25.1 6.589337
70 3.027831 30 6.900577
89.9 3.214371 35 7.177019
155
Table S.2. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 50
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
5 0.855234 2.9 2.864747
10 1.245876 5 3.611665
19.9 1.751815 9.9 4.638572
29.6 2.088337 15 5.327568
40.1 2.347694 20 5.765904
50.1 2.541255 25 6.128089
70.1 2.838665 29.9 6.430291
90 3.049514 35 6.68761
98 3.104488
156
Table S.3. CH
4
and C
2
H
6
pure component sorption data on shale (cube) at 60
o
C.
CH
4
C
2
H
6
p (bar)
eas
m (mg/g)
p (bar)
eas
m (mg/g)
5 0.62964 5 2.819644
9.9 1.023879 9.9 4.077302
19.9 1.542906 15 4.792152
29.9 1.900665 20 5.279827
39.8 2.170556 24.9 5.636891
49.6 2.380494 29.9 5.938394
69.8 2.687432 35.1 6.182652
90 2.890141
103.2 2.992716
157
Table S.4. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 40
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
5 1.489252 5 1.312222 5 1.134454
10 2.116528 10 1.913846 10 1.643101
20 2.868154 19.7 2.635087 19.9 2.306001
30.1 3.297908 30.1 3.065116 30.1 2.745251
40 3.620102 40.1 3.404168 40 3.083679
50 3.890485 50 3.601005 50 3.374908
70 4.174714 70 3.912811 70 3.636082
90 4.367695 90 4.100352 89.9 3.831009
99.9 4.461012 99.9 4.146921 99.9 3.922796
158
Table S.5. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 50
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
5.1 1.199568 5 1.05579 5 0.874308
10 1.819808 10 1.732477 10 1.445912
19.9 2.574975 19.9 2.399749 20.1 2.17015
30 3.084055 30 2.848812 30 2.566864
39.9 3.361782 39.9 3.194361 40 2.953544
49.9 3.59344 49.9 3.36515 50 3.202384
69.9 3.893601 69.9 3.699225 70 3.484344
89.8 4.072696 89.8 3.903012 90 3.702347
99.9 4.162263 99.9 3.98511 99.9 3.759914
159
Table S.6. Sorption of CH
4
-C
2
H
6
mixtures (90%-10%, 93%-7%, 96%-4%) on shale (cube) at 60
o
C.
90%-10% CH
4
-C
2
H
6
93%-7% CH
4
-C
2
H
6
96%-4% CH
4
-C
2
H
6
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
p (bar)
eas
m
(mg/g)
5 1.026181 5 0.781928 5 0.71933
10 1.66573 10 1.426889 10 1.291684
19.9 2.357713 19.9 2.1182 19.9 2.068275
29.6 2.862698 29.8 2.552977 29.7 2.429936
40.1 3.198737 40.1 2.85958 40.1 2.704499
50.1 3.418942 50 3.171084 50 2.905375
70.1 3.734572 70 3.456879 70 3.201073
89.9 3.92551 89.9 3.670441 89.9 3.406097
100 3.995863 100 3.741408 99.9 3.474614
160
Chapter 6
Future Work
Sorption studies are of key importance to most research work associated with shale-gas systems.
Among all the methods that are capable of measuring gas sorption on shales, the only
experimentally measurable quantity is the excess sorption. However, in the context of
thermodynamic property calculations using theoretical models and molecular simulations, the
absolute sorption arises naturally. Therefore, the conversion of the excess sorption to the absolute
sorption is a crucial step towards the understanding and interpretation of the sorption data. As
discussed in Chapters 4, the estimate of the adsorbed phase density has a significant impact on
the absolute sorption calculation, which further affects the modeling results. It is found that
neither the constant density assumption nor the varying density hypothesis are capable of
accurately predicting the mixed gases sorption, when based solely on the pure component
sorption characteristics. Researchers have proposed a number of approaches to either
approximate or fit the adsorbed phase density, but still, it is has proven difficult to provide an
approximation that is both physically reasonable and theoretically sound. So, a proper method
which is capable of representing the adsorbed phase density accurately is the key for accurate
interpretation of sorption studies related to shale-gas system.
The study of sorption dynamics and other mass transfer mechanisms of gas in shale plays a
significant role in understanding the fluid transport and subsequent gas recovery. In this work, a
Langmuir type sorption dynamic model was proposed to describe the sorption process of pure
and mixed gases on shale. Our findings demonstrate that the sorption model is sufficient to
represent the pure component sorption process, but the calculated results depart from the
161
experimental observations for the mixtures. This departure reveals the limitations of the
Langmuir model’s capacity and/or suitability in modeling the gas-shale system. The description
of gas sorption on the complex shale matrix characterized by a heterogeneous pore structure
requires more sophisticated modeling approaches beyond the Langmuir model. An appropriate
model should have the ability to account for the shale pore-size-distribution and interactions
between gas molecules of same and different species on the surfaces of the shale. It is
recommended that the development and experimental validation of such model should be the
focus of future investigations.
Abstract (if available)
Abstract
In the global context of energy crisis and greenhouse gases reduction, shale gas is emerging as a clean and abundant unconventional energy resource. The U.S. Energy Information Administration (EIA) predicts that by 2040, over 50 % of the total US natural gas production would come from shale gas. Sorption of gases in the complex shale matrix system is the key mechanism via which shale gas is stored in such formations, and has as a result received attention in a number of previous studies related to shale gas recovery. Methane (CH₄) is the single largest component of shale gas, however, ethane (C₂H₆) is typically the second largest component accounting for more than 15 vol. % in certain cases. The present work mainly focuses on generating experimental data of pure component and competitive CH₄-C₂H₆ sorption on shale samples under reservoir pressure and temperature conditions. In parallel, modeling approaches, such as the Multicomponent Potential Theory of Adsorption (MPTA) and a self-proposed sorption dynamic model, have been undertaken to model the sorption data. The experimental observations and their interpretation pave a path towards new knowledge of shale gas recovery by leveraging an improved understanding of sorption and other mass transfer mechanisms of natural gas mixtures in shale. ❧ In Chapter 1, a brief introduction of shale gas and a literature review of previous and current research on the shale-gas system are presented. The knowledge gaps are identified and a systematic research plan is proposed to close these gaps. ❧ In Chapter 2, a static experimental system which employs the manometric method is utilized to measure gas adsorption on both dry and moist shale samples. Single gas (CH₄, C₂H₆) sorption isotherms at 60 ℃ are obtained from the static experimental measurements. The effect of moisture on the shale sorption characteristics is studied. ❧ In Chapter 3, measurements of pure component sorption isotherms for CH₄ and C₂H₆ for pressures up to 114 bar and 35 bar, respectively, have been performed using a thermogravimetric method in the temperature range (40-60 ℃), typical of storage formation conditions. Sorption experiments of binary (CH₄-C₂H₆) gas mixtures containing up to 10% (mole fraction) of C₂H₆, typical of shale-gas compositions, utilizing ground samples, for pressures up to 125 bar under the aforementioned temperature conditions have also been conducted. In the study, the Multicomponent Potential Theory of Adsorption (MPTA) approach is utilized to model the sorption data. The MPTA model is shown capable in representing the pure component sorption data, and also provides reasonable predictive capability when applied to predict the total sorption for CH₄-C₂H₆ binary mixtures in shale over a range of compositions and temperatures. ❧ In Chapter 4, we study the sorption behavior of CH₄-C₂H₆ binary mixture (and its individual components) in ground shale samples using thermogravimetric analysis (TGA). The sorption isotherms generated are important to predict the gas storage capacity of the shale samples, while the study of sorption dynamics/kinetics help us understand the role of desorption during the later times of gas production. A dynamic Langmuir-type sorption model is proposed, that allows us to isolate sorption kinetics from diffusive mass transfer. This, in turn, facilitates our modeling and interpretation of the experimental observations. Work is currently ongoing towards studying the role of sorption using a whole (cube) shale sample with a volume of ~1 cm³ in the same TGA set-up (Chapter 5). The whole sample offers an advantage over the powdered sample due to the longer diffusion times, and captures the diffusion characteristics of the sample more accurately. ❧ In Chapter 6, a few remaining tasks with regard to the shale gas study are discussed and future work which could potentially address these issues is outlined as well.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Wang, Yu
(author)
Core Title
Investigation of gas sorption and mass transfer in gas shales
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Chemical Engineering
Publication Date
09/28/2016
Defense Date
04/20/2016
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
dynamics,ethane,isotherms,methane,OAI-PMH Harvest,shale,sorption,TGA
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Jessen, Kristian (
committee chair
), Hammond, Douglas (
committee member
), Tsotsis, Theodore (
committee member
)
Creator Email
uscwang54@gmail.com,wang54@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-308414
Unique identifier
UC11281183
Identifier
etd-WangYu-4829.pdf (filename),usctheses-c40-308414 (legacy record id)
Legacy Identifier
etd-WangYu-4829.pdf
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308414
Document Type
Dissertation
Format
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Wang, Yu
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
dynamics
ethane
isotherms
methane
shale
sorption
TGA