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Investigation of gas transport and sorption in shales

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Investigation of gas transport and sorption in shales

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Content INVESTIGATION OF GAS TRANSPORT AND SORPTION IN SHALES by Ye Lyu 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 (PETROLEUM ENGINEERING) December 2023 Copyright 2023 Ye Lyu ii Dedication To my loving wife, Peiheng Huang, and our precious son, Royce Huang Lyu. To my parents, Cuihong Zhang and Guofang Lyu iii Acknowledgments Commencing this Ph.D. endeavor has proven to be an enlightening voyage, replete with obstacles and triumphs, just as any significant effort accomplishes. This journey would not have been possible without the assistance of many people; to them I am extremely indebted for holding my hand, bolstering my spirit, and igniting my determination. First and foremost, I would like to express my deepest gratitude to my advisor, Professor Kristian Jessen, for his invaluable guidance, support, and mentorship throughout this research journey. His wisdom, knowledge, and commitment to the highest standards have been an inspiration and kept me on track. I am immensely grateful to Professor Theodore Tsotsis for his continuous guidance, feedback, and unlimited support throughout my laboratory work, our publication preparation, my qualifying exam and dissertation, and my future commitments. I am thankful to Professor Doug Hammond for his insightful feedback and support on both my qualifying exam and dissertation. I would also like to thank Professor Iraj Ershaghi and Professor Donald Paul for their teaching and mentoring throughout my education and for serving on my qualifying exam committee. Their perspectives and challenges pushed me to think more deeply and holistically about my research. Additionally, I would like to extend my appreciation to Shokry Bastorous and Tina Silva for their unwavering backing in ensuring that I consistently adhered to the utmost safety iv protocols during my laboratory work. Annie Lee Houang, Heidelita Torres, and Monina Letargo have significantly contributed to the seamless execution of the monthly payment, business reimbursement, and research procurement procedures. Throughout my graduate studies, I also valued the advice and assistance of Andy Chen, Anthony Tritto, Diana Vuong, and Brenda Ornelas. I also wish to acknowledge the financial support and resources provided for my research by the Viterbi School of Engineering, the Mork Family Department of Chemical Engineering and Materials Science, Energi Simulation, the Center for Mechanistic Control of Unconventional Formations (CMC-UF), an Energy Frontier Research Center funded by the U.S. Department of Energy, and the Aramco Houston Research Center. Sincere appreciation is extended to my colleagues and the lifelong friends I have gained throughout the years at USC. A special mention to Cheng Wu, Dr. Syamil Razak, Jodel Cornelio, Junjie Yu, Fangning Zheng, Dr. Lin Sun, Dr. Zhuofan Shi, Dr. Sheng Hu, Jiyue Wu, Linghao Zhao, Saeed Alahmari, Mohammed S. Raslan, and Dr. Pooya Khodaparast, their friendship has been my daily dose of joy, laughter, and inspiration. Through the countless brainstorming sessions, coffee breaks, and late-night discussions, you've been the backbone of my Ph.D. life. My heartfelt appreciation goes to my family: my parents, Cuihong Zhang and Guofang Lyu, for their undying faith in my abilities and for their sacrifices that allowed me to pursue my dreams; and to my beloved Peiheng Huang and our precise son, Royce Huang Lyu, for their patience, understanding, and endless love during the long days and nights of this journey. v Last but not least, I extend my gratitude to all those who, directly or indirectly, contributed to this dissertation and supported me along the way. Your belief in me has made all the difference. vi Table of Contents Dedication .......................................................................................................................... v Acknowledgments............................................................................................................ vi List of Tables.................................................................................................................... ix List of Figures.................................................................................................................... x Abstract........................................................................................................................... xvi CHAPTER 1. Introduction........................................................................................... 1 1.1. Motivation and Challenges (Importance of unconventional resources)......................... 1 1.2. Objective and research outline........................................................................................ 8 1.3. Organization of the dissertation.................................................................................... 10 1.4. References..................................................................................................................... 14 CHAPTER 2. A numerical investigation of surface diffusion and sorption in nanoporous media........................................................................................................... 19 2.1. Introduction................................................................................................................... 19 2.2. Methodology................................................................................................................. 26 2.3. Results and discussion .................................................................................................. 35 2.4. Summary and conclusions............................................................................................ 58 2.5. Appendix....................................................................................................................... 61 2.6. References..................................................................................................................... 66 CHAPTER 3. An integral model of gas diffusion and sorption in tight dualporosity systems .............................................................................................................. 74 3.1. Introduction................................................................................................................... 74 3.2. Formulations and Methods........................................................................................... 77 3.3. Results and Discussion ................................................................................................. 88 vii 3.4. Summary and Conclusions........................................................................................... 97 3.5. Appendix....................................................................................................................... 98 3.6. Nomenclature.............................................................................................................. 104 3.7. References................................................................................................................... 107 CHAPTER 4. Characterization of shales using helium and argon at high pressures ………………………………………………………………………………111 4.1. Introduction................................................................................................................. 111 4.2. Experimental approach ............................................................................................... 114 4.3. Modeling Approach.................................................................................................... 119 4.4. Experimental results and interpretation ...................................................................... 129 4.5. Summary and conclusions.......................................................................................... 142 4.6. References................................................................................................................... 144 CHAPTER 5. Investigation of methane mass transfer and sorption in Marcellus shale under variable net-stress .................................................................. 154 5.1. Introduction................................................................................................................. 154 5.2. Experimental approach ............................................................................................... 159 5.3. Modeling approach ..................................................................................................... 164 5.4. Results......................................................................................................................... 171 5.5. Summary and Discussion ........................................................................................... 184 5.6. Conclusions................................................................................................................. 187 5.7. Nomenclature.............................................................................................................. 187 5.8. Appendix..................................................................................................................... 190 5.9. References................................................................................................................... 193 CHAPTER 6. Analysis of gas storage and transport in shales using pressure pulse decay measurements with He, Kr and CO2 ...................................................... 201 viii 6.1. Introduction................................................................................................................. 201 6.2. Materials and Methods ............................................................................................... 205 6.3. Results......................................................................................................................... 219 6.4. Discussion................................................................................................................... 235 6.5. Conclusions................................................................................................................. 238 6.6. Appendix..................................................................................................................... 239 6.7. References................................................................................................................... 247 CHAPTER 7. Summary of dissertation and recommendations for future work ………………………………………………………………………..257 7.1. Summary of dissertation............................................................................................. 257 7.2. Recommendations for future work ............................................................................. 259 References...................................................................................................................... 262 ix List of Tables Table 2.1. Summary of default parameters including high and low values...................... 49 Table 3.1. Parameters for the generalized Vermeulen approximation.............................. 80 Table 3.2. Evaluated BET sorption model parameters..................................................... 89 Table 3.3. Computation costs comparison for the proposed method vs. fully discretized solution ........................................................................................................... 92 Table 4.1. Langmuir parameters from TGA experiments – Ar ...................................... 136 Table 4.2. Evaluation of pore volume in core holder from He expansion...................... 138 Table 4.3. Model parameters estimated from the He expansion experiment.................. 139 Table 5.1. Porosity measurements for the shale cube sample using He ......................... 171 Table 5.2: Langmuir parameters corresponding to the various adsorbate density models............................................................................................................................. 172 Table 5.3: CH4 in place calculations (scm3 ) at the end of loading experiments A and B............................................................................................................................... 179 Table 5.4: Gas production and recovery for depletion experiments A and B................. 184 Table 6.1. Eagle Ford shale core dimensions and properties (Elkady and Kovscek, 2020b) ............................................................................................................................. 206 Table 6.2. Model parameters for Kr and CO2 adsorption isotherms .............................. 223 Table 6.3. Parameter estimation using the 1 st and 2nd He pulses of pressure decay ....... 225 Table 6.4. Rescaling of (σD) for prediction of additional pulse-decay experiments...... 229 Table E1. Model parameters for CO2 adsorption isotherms with revised density model ..………………………………………………………………………………… 245 x List of Figures Figure 1.1. U.S. energy consumption by fuel (EIA, 2021)................................................. 1 Figure 1.2. Dry shale gas production estimates by play (EIA, 2021)................................. 2 Figure 1.3. Dry natural gas production from shale resources (EIA, 2023)......................... 3 Figure 1.4. Research outline flow chart ............................................................................ 10 Figure 2.1. Schematics of gas transport in micropores (not scaled) ................................. 27 Figure 2.2. Models with initial and boundary conditions for loading and unloading cases.................................................................................................................................. 35 Figure 2.3. Normalized gas amount with different adsorption rate (sorption kinetics) and isotherms .................................................................................................................... 37 Figure 2.4. Normalized gas amount for cases LA and UA (communicating surface boundary and constant gas concentrations) ...................................................................... 40 Figure 2.5. Normalized gas amount for case LB and UB (communicating surface boundary and variable gas concentrations)....................................................................... 43 Figure 2.6. Normalized gas amount for case LC and UC (isolated surface boundary and constant gas concentrations) ...................................................................................... 46 Figure 2.7. Normalized gas amount for case LD and UD (isolated surface boundary and variable gas concentrations)....................................................................................... 48 Figure 2.8. Normalized total gas amount for case LD and UD with the default, high, and low values of the dimensionless groups - , , and ............................................... 51 Figure 2.9. Normalized total gas amount for case LD and UD with the default, high, and low settings for ........................................................................................................ 52 Figure 2.10. Normalized total gas amount for case LD and UD with the default, high, and low settings for ........................................................................................................ 53 Figure 2.11. Estimation of and from artificial loading and unloading data (LA and UA with sorption isotherm and =100, were used to generate the artificial data)................................................................................................................................... 55 Figure 2.12. Estimation of from artificial loading and unloading data (LB and UB with sorption isotherm and =100, were used to generate the artificial data)................................................................................................................................... 56 xi Figure 3.1. Comparison of series solution (Crank, 1975) and generalized Vermeulen approximation – spherical geometry................................................................................. 80 Figure 3.2. Schematics of gas transport in the DP system (not scaled) ............................ 84 Figure 3.3. BET interpretation of CO2 adsorption isotherm on an Eagle Ford shale core at 70 C...................................................................................................................... 89 Figure 3.4. Comparison of PDE (symbol) and ODE (line) solutions to diffusion and sorption in planar geometry from = 0.001 to = 100............................................. 91 Figure 3.5. Sensitivity check with 2, & 2,.......................................................... 94 Figure 3.6. Sensitivity checks with 0.5, & 0.5, .................................................. 94 Figure 3.7. Sensitivity checks with = 0.5 ...................................................................... 95 Figure 3.8. Sensitivity checks with = 0.2 ...................................................................... 95 Figure 3.9. Sensitivity checks for a cylindrical system .................................................... 96 Figure 3.10. Sensitivity checks for a spherical system ..................................................... 96 Figure 4.1. Cube and machined impermeable epoxy (left) and top-view of core/slab assembly forming a cylindrical shape (right).................................................................. 115 Figure 4.2. Schematic of the whole core gas expansion experimental system ............... 118 Figure 4.3. Characteristic time for He and Ar mass transfer in a 1 cm3 shale cube for an average pore diameter of 10 nm................................................................................. 123 Figure 4.4. Illustration of slabbed core (top) and conceptual model (bottom) ............... 125 Figure 4.5. Fractional micropore surface area as a function of fractional micropore volume............................................................................................................................. 128 Figure 4.6. Helium buoyancy tests on the shale cube at 49 °C....................................... 130 Figure 4.7. Argon excess adsorption isotherms for the shale cube at 49 °C................... 131 Figure 4.8. Dynamics of Ar excess sorption at 49˚C – From recorded data (TGA)....... 133 Figure 4.9. Dynamics of Ar excess sorption at 49˚C – Processed data .......................... 133 Figure 4.10. Extrapolation of Ar sorption dynamics (0-10 bar and 10-20 bar shown)... 134 xii Figure 4.11. Ar isotherm (squares) and extrapolated dynamics (asterisk). Solid line from matching isotherm data, and broken line from matching dynamic data ................ 135 Figure 4.12. Estimation of Cs,max, K and bka from Ar dynamics – RMSE = 0.046 mg/g ....................................................................................................... 137 Figure 4.13. Characteristic time for sorption as a function of bulk phase pressure........ 137 Figure 4.14. He (solid) and Ar (dashed) expansion experiments at 49ºC....................... 138 Figure 4.15. He expansion – Parameter estimation – RMSE = 0.024 bar...................... 139 Figure 4.16. Ar expansion – Prediction – RMSE = 0.33 bar.......................................... 141 Figure 5.1. Shale cube and core and epoxy insert assembly (not scaled to size)............ 160 Figure 5.2. Schematic of the high-pressure experimental system for CH4 expansion and depletion experiments .............................................................................................. 162 Figure 5.3. Conceptual model for interpretation of shale core experiments................... 165 Figure 5.4. Characteristic time for mass transfer in a single cylindric pore with a 10 nm diameter using the DGM and BKC models......................................................... 170 Figure 5.5. (a) CH4 excess isotherm data and models using various adsorbate density models; (b) corresponding calculated absolute adsorption isotherms ............................ 173 Figure 5.6. CH4 loading experiments at 49 °C................................................................ 175 Figure 5.7. CH4 loading experiment A - estimation of bka - RMSE = 0.36 bar............. 177 Figure 5.8. Prediction of the CH4 loading experiment B – RMSE = 0.52 bar................ 178 Figure 5.9. Cumulative CH4 production during depletion experiments A and B Accuracy of cumulative production is ±1.5%................................................................. 180 Figure 5.10. Predictions of gas depletion rate data for experiments A and B ................ 181 Figure 5.11. Sorption hysteresis model with δ = 0.050, as estimated from depletion experiment A................................................................................................................... 183 Figure 6.1. Eagle Ford shale sample core #A (left) and a reconstructed 3D CT image of the in-situ Kr mass fraction after ~ 2 PV of nitrogen (N2) injection from the top (right) (Elkady, 2020). ........................................................................................ 207 xiii Figure 6.2. Pressure-pulse decay experimental set up for measuring mass transfer and sorption in the shale core (Modified from Elkady et al., 2020) ............................... 208 Figure 6.3. High-Pressure pulse decay experimental system for measuring mass transfer and sorption in the shale core at reservoir relevant conditions.......................... 209 Figure 6.4. Conceptual model for PPD experiments ...................................................... 210 Figure 6.5. Pressure-pulse decay experiments with He (T = 20 oC)............................... 220 Figure 6.6. Pore volume accessible with He at different pressure stages (T = 70 oC).... 220 Figure 6.7. Gas storage capacity in the shale sample (left - T = 20 oC, right - T = 70 oC)........................................................................................................................ 221 Figure 6.8. Experimental excess adsorption (left) and calculated absolute adsorption (right) – core #A (T = 20 oC)........................................................................................... 222 Figure 6.9. Experimental excess adsorption (left) and calculated absolute adsorption (right) – core #B (T = 70 oC)........................................................................................... 222 Figure 6.10. Characterization of the mass transfer rate for the mesoporous matrix segment using the modified analytical approach: Data shown for the 1st and 2nd He pressure decay................................................................................................................. 224 Figure 6.11. Comparison of calculations and data for the 1st and 2nd He pressure decay experiments after model calibration ..................................................................... 226 Figure 6.12. Experimental observations and model prediction for the 3rd He PPD experiment....................................................................................................................... 230 Figure 6.13. Comparison of model predictions with experimental observations for the 1st (top), 2nd (left-bottom), and 3rd (right-bottom) Kr experiments........................... 232 Figure 6.14. Comparison of model predictions and experimental data for the 1st (left) and 2nd (right) CO2 PPD experiments ......................................................... 233 Figure 6.15. Comparison of data and calculations for the 1st (left - calibration) and 2nd (right - prediction) CO2 PPD experiment ........................................................... 235 Figure A1. Normalized gas amount variations with a range of segment selections……. 62 Figure A2. Shift in critical properties of CH4 as a function of pore size........................ 190 Figure A3. Concentration of CH4 under confinement at different pressures (T=50 ºC).. 191 xiv Figure A4. Sensitivity test of the fracture volume on modeling results for a He PPD experiment using a TPM (full pressure range - left, zoom-in - right)............................. 240 Figure B1. Transition from sorption kinetics to sorption isotherm according to ……...63 Figure B2. Comparison of series solutions (Crank, 1975) with the generalized Vermeulen approximation for planar and cylindrical geometries .................................. 100 Figure B3. Flow regime classification based on the Knudsen number for different pore diameters (1 nm - left, 10 nm - middle, 100 nm - right) at relevant conditions of this work ..................................................................................................................... 241 Figure C1. Boundary variations for the base loading and unloading cases…………….. 63 Figure C2. Comparison of normalized value for (cycles) vs. ∙ (dash line) ......... 101 Figure C3. Knudsen number for CH4 (at 49 ºC) in a single cylindrical pore with varying diameters at relevant experimental conditions .................................................. 192 Figure C4. Effective diffusivity and their ratios among gases determined by DGM and Beskok-Karniadakis-Civan (BKC) in a 10 nm cylindric pore ................................. 243 Figure C5. Effective diffusivity and their ratios among gases determined by DGM and BKC in a 50 nm cylindric pore ................................................................................ 243 Figure D1. Estimation of and from artificial loading and unloading data (LC and UC with =10 and =100, were used to generate the artificial data)…………65 Figure D2. Estimation of and from artificial loading and unloading data (LD and UD with =10 and =100, were used to generate the artificial data) ............... 65 Figure D3. Schematic of workflow of proposed method................................................ 103 Figure D4. Improved results of cases with = 0.2 using proposed workflow............... 104 Figure D5. Workflow summary of this work.................................................................. 244 Figure E1. Excess adsorption (left) and absolute adsorption (right) modeling results…245 Figure F1. The porosity reduction for pressure-pulse decays of Kr (1st pulse – top, 2 nd pulse – bottom left, 3rd pulse - bottom right)………………………………………..246 Figure F2. The porosity reduction for pressure-pulse decays of CO2 (1st pulse – left, 2 nd pulse - right) with = 1.10 g/cm3 ...................................................................... 247 xv Figure F3. The porosity reduction for pressure-pulse decays of CO2, the 1st pulse (left) and the 2nd pulse (right) with ρads= 12.26e-3∙θ2/3 mol/cm3 ............ 247 xvi Abstract Accurate estimation of natural gas recovery from unconventional reservoirs and carbon storage in ultra-tight formations is an essential requirement in the energy industry. Detailed characterization of gas transport and storage in low-permeability and organic-rich shales is associated with an array of challenges due to the complex morphology of the pore space, which represents a broad range of pore sizes, combined with the heterogeneous fabric of the shale matrix. These factors and their interplay during gas transport and sorption complicate a) the analysis of shale samples at the laboratory scale (~ up to a ft. in length) and b) the prediction of natural gas production and carbon sequestration potential at larger scales. This dissertation focuses on the comprehensive characterization of gas transport and storage in nanoporous media, particularly gas shales, across varying scales, which is crucial for optimizing natural gas recovery and carbon storage within the energy industry. The study explores multiple gas transport and sorption mechanisms, including viscous flow, transition flow, Knudsen diffusion, surface diffusion, and sorption, integral to understanding gas loading (injection) and unloading (production) processes. The work commences with a detailed study exploring the interplay of gas transport and sorption. This exploration uses a one-dimensional, fully discretized, unit-dimensionless model, with methane (CH4) as the test gas. A diverse range of boundary conditions, sorption kinetics, and sorption isotherms are employed to analyze the roles of surface diffusion and sorption kinetics in the overall mass transfer during gas loading and unloading. xvii The findings highlight the critical role of sorption kinetics and surface diffusion in these processes. Cases where the influence of surface diffusion on the overall mass transfer becomes less significant are identified, particularly when the bulk phase serves as the only communication channel, a scenario similar to the conditions in shale formations with CH4. Furthermore, the study calls into question the reliance on sorption isotherms for characterizing the diffusivities of bulk and surface phases. Instead, it advocates for separate evaluations of sorption kinetics and bulk phase diffusion coefficients for a more accurate portrayal of surface diffusion. Subsequently, this dissertation introduces a general dual-porosity model of gas transport and sorption in shale for both low (helium – He) and highly (carbon dioxide – CO2) adsorptive gases. This model was validated by comparing results with fully discretized numerical solutions, affirming its ability to accurately estimate gas transport and sorption in shale. This novel integral model for mass transfer and storage in multiporosity shale systems bridges the scale gap between laboratory and field-scale observations and enhances understanding of gas storage, production, and recovery methods across practical applications. To characterize shale at laboratory scale, He and argon (Ar) were used as probe gases to measure sorption behaviors at a small scale (~ 1cm) from the Marcellus shale. A series of high-pressure experiments was conducted with He, Ar, and CH4 on a full-diameter core sample from the same depth and formation to further understand gas transport and sorption behavior under reservoir-relevant conditions. These techniques facilitated the measurement of sorption kinetics/isotherms and mass transfer, providing critical insights xviii into the behavior of these gases in the mesoporous and microporous regions of the shale samples. Ar and CH4 sorption behaviors were evaluated using the Langmuir adsorption model and then combined with the novel integral model, which was specifically modified into a Triple-Porosity Model (TPM). This combined modeling approach proved highly effective in forecasting the behavior of gas loading (pressure-decay) and depletion (production) experiments under reservoir-relevant conditions. The study findings indicate the need for caution when using Ar to estimate the true porosity and permeability of shales and underscore the potential of He and Ar as effective tools for characterizing shales in terms of mass transfer and sorption dynamics across scales. This integrative approach also demonstrated the importance of an accurate representation of sorption hysteresis in predicting shale gas production, revealing that without accounting for this factor, the prediction of CH4 production could be overestimated by 10%. Lastly, pressure pulse-decay (PPD) measurements using He, krypton (Kr), and CO2 were conducted on Eagle Ford shale core samples to develop an efficient workflow for characterizing and modeling transport and sorption at the laboratory scale. The non-sorbing gas He was used to probe the overall porosity, and the TPM was applied to interpret gas transport. A modified analytical approach was introduced to extract effective transport parameters directly from the PPD measurements of He. These parameters, subsequently adjusted for fluid property differences, were applied to the Kr and CO2 PPD experiments. Adsorption isotherms from Kr and CO2 were integrated into the TPM to predict combined gas transport and sorption, showing excellent agreement with the experimental xix observations. This research validates an analytical approach that effectively characterizes mass transfer rates in shales, enabling TPM representation of mass transfer and sorption. Overall, the research combines robust methodologies, experimental studies, and modeling techniques to advance understanding and efficiency in gas storage and recovery from nanoporous media, thereby proposing potential pathways for large-scale CO2 sequestration. 1 CHAPTER 1.Introduction 1.1. Motivation and Challenges (Importance of unconventional resources) 1.1.1. Oil and gas production from shale Over the past two decades, there has been a significant transformation in the energy sector, characterized by the emergence of unconventional hydrocarbon reservoirs, specifically shales. Shale formations are sedimentary rocks that, in the past, were mostly regarded as either source rocks, responsible for supplying oil and/or gas to reservoir rocks, or as cap rocks situated above the reservoirs. The development of advances in technology, e.g., horizontal drilling and multistage hydraulic fracturing, and methodological approaches that made the extraction of oil and gas from these tight rock formations feasible have drastically changed the dynamics of the energy industry, as shown in Figure 1.1. Figure 1.1. U.S. energy consumption by fuel (EIA, 2021) 2 Since 2009, there has been a substantial rise in the production of oil and gas from shale formations. Major geological formations such as the Marcellus, Permian, and Utica basins have become known as leading regions of focus for the exploration and extraction of natural resources, making substantial contributions to the overall energy security of the nation. Intensified activity has been observed in shale plays such as Barnett and Haynesville. As indicated by Figure 1.2, additional important shale sources include the Eagle Ford, Woodford, and Monterey formations. This rise has led to economic growth and reduced dependence on imported fossil fuels. Figure 1.2. Dry shale gas production estimates by play (EIA, 2021) The significant impact of shale may be assessed from the observation that shale gas production in the United States has been steadily increasing and reached at about 27 trillion cubic feet (TCF) in 2022, as show in Figure 1.3, representing about 75% of the country's overall gas production (EIA, 2023). 3 Figure 1.3. Dry natural gas production from shale resources (EIA, 2023) The production of shale commonly has a substantial decrease in rate after the first phases of production. The significant drop in production may be attributed to several factors, such as the extremely low permeabilities and porosity of shale formations, intricate pore structures, closure of fractures, slow interactions between the matrix and fractures, etc. The ultra-low permeabilities means that once the free gas or oil adjacent to the more permeable fracture network has been extracted, the production rate diminishes sharply. The hydrocarbon resources contained inside the matrix require a longer duration and a greater pressure differential in order to move toward the fracture network for production. Additionally, the complex fracture networks created by hydraulic fracturing can extend several hundred feet. Proppant is usually injected with the hydraulic fluid to prevent these artificial fractures from closing. However, these proppants have limited accessibility to serve for the whole fracture network, thus, closure as a result of net stress increase, proppant embedment, fines migration, or geomechanical effects occurs. Furthermore, once 4 the easily accessible hydrocarbon from natural fractures and macro-pores is depleted, further recoveries rely on slower diffusion processes in shale matrix and desorption of gas from organic materials, which also leads to declining production rates. Thus, the combination of ultra-low permeability, reliance on artificial fractures, and the sorption related storage mechanisms in shales results in the steep decline curves observed in the industry. 1.1.2. Potential for carbon sequestration In conjunction with hydrocarbon extraction, there has been a growing emphasis on carbon capture, utilization, and storage (CCUS) due to environmental concerns about global warming and climate change. With their unique geology, shales are currently being recognized not only as energy sources but also as prospective sites for the sequestration of anthropogenic carbon dioxide. The use of the extensive pore space inside these rocks for the purpose of carbon sequestration has the potential to significantly impact the ongoing worldwide efforts to address climate change. 1.1.3. Shale pore structure and mineralogy Shale formations are normally associated with the complex hierarchical structures of pores (Chalmers et al. 2012; Yu et al. 2019; Lyu et al. 2021). The International Union of Pure and Applied Chemistry (IUPAC) has recommended the following classification of pores according to their diameter (Sing, 1985): macropores (> 50 nm), mesopores (2-50 nm), and micropores (< 2 nm). Additionally, the size, shape, and distribution of shale pores all affect the resource's storage capacity and ability to transmit fluids (Lyu et al., 2021). 5 The composition of the shale plays a role in determining the initial gas in place and flow characteristics of the resource. The primary constituents of shales are clay minerals, comprising a substantial proportion of their matrix. The clay minerals, including illite, smectite, chlorite, and kaolinite, are of utmost importance in the assessment of the rock's porosity, permeability, and its interaction with drilling and fracturing fluids. Clay-rich shales are more porous and permeable than silica- or carbonate-rich shales because of the increased pore surface area created by mesopores and micropores when the clay concentration increases (Bustin et al., 2008). These larger quantities of mesopores and micropores lead to a bigger surface area for gas to adsorb, hence boosting gas storage capacity. Additionally, mineralogy of the shale contributes to the formation of microcracks, which give additional paths for gas flow and storage. Due to the brittle nature of quartz and calcite, natural fractures may arise when subjected to in-situ stress. 1.1.4. Gas storage in shale The majority of surface areas in shale are found at the micro- and nanoscale levels. The economic impact of shale gas is significantly dependent on its hydrocarbon generation capacity. Methane (CH4) constitutes the primary constituent of shale gas, accounting for a significant proportion of its composition (88-98 wt%). The storage of gas in shale formations exhibits greater complexity compared to conventional sandstone reservoirs. In addition to the storage of free gas within the pores, sorption is an additional mechanism that enhances the storage capacity in shale formations. The primary mechanism responsible for the sorption of gas molecules onto a solid surface is the "van der Waals" attractive forces (Hutson and Yang, 1997). A multitude of 6 adsorption phenomena have been extensively reported in the existing literature. The IUPAC has classified six adsorption isotherms as Types I through VI (Everett, 1972). The term "isotherm" is used to denote the practice of conducting measurements at varying pressures while maintaining a constant temperature. The prevalent isotherms identified in shale deposits are Type I and Type II. The initial type, referred to as Langmuir behavior, is distinguished by the adsorption of a monolayer of molecules on solid surfaces that are either nonporous or microporous (Everett, 1972). Under low-pressure conditions, the observed trend of this curve often follows a linear pattern. However, as the pressure increases, the curve reaches a point where it levels off and remains relatively constant. The isotherm in question can be characterized using the Langmuir model, as proposed by Langmuir in 1918. Type II adsorption involves the adsorption of adsorbates in many layers on nonporous or macroporous adsorbents. The Brunauer-Emmett-Teller (BET) model, proposed by Brunauer, Emmett, and Teller in 1938, is a commonly employed methodology for the analysis of multilayer adsorption data. 1.1.5. Gas transport in shale During the process of shale gas production, the initial stage involves the extraction of gas from fracture networks and macropores, where it exists in its free gas form. This occurs within a relatively brief timeframe. Subsequently, the extraction process continues with the recovery of free gas from mesopores, which takes place over a longer duration. Finally, gas is extracted from the meso/micropores in its adsorbed state, which happens over an extended length of time. The apparent permeability of shale, a significant petrophysical property, is frequently observed to be significantly lower than that of conventional 7 formations, typically differing by many orders of magnitude. Gas transport mechanisms in shale reservoirs include multiple mechanisms, such as viscous flow, slip/transition flow, and Knudsen diffusion (Schaaf and Chambre, 1961; Javadpour, 2009; Civan, 2010; Alnoaimi and Kovscek, 2019; Lyu et al., 2023a). Permeability of shale is stress sensitive due to natural or hydraulic fractures and microcracks (Heller et al., 2014; Wasaki and Akkutlu, 2015) and finely distributed pore throats within clay minerals (Dewhurst and Siggins, 2006). The effective stress is computed from the difference between the applied confining stress and pore pressure (Terzaghi, 1923). Therefore, it is worthy to study gas transport and storage mechanisms in shale including effective stress effects. The phenomenon of surface diffusion has intrigued the scientific community since the beginning of the 20th century. Fundamentally, this procedure revolves around the displacement of molecules that have been adsorbed onto the solid surface of porous materials, under the impact of several forces such as van der Waals and electrostatic forces. The origin of this phenomenon may be traced back to hopping models (Hill et al., 1956), hydrodynamic models (Gilliland et al., 1958), and Fickian models. Among these models, hopping models are especially relevant for shales due to their tendency for single-layer adsorption. The proposed model posits that molecules in the adsorbed phase exhibit a phenomenon of jumping between neighboring places on surfaces of materials. Foundational understandings and mathematical representations of surface diffusion have been established by key experimental investigations conducted by Hwang and Kammermeyer (1966), as well as theoretical endeavors undertaken by Langmuir and other 8 researchers. However, ongoing discussions continue to exist. There is a divergence of opinions among specialists on the mobility of the adsorbed phase. However, it is widely recognized throughout several disciplines that sorption kinetics, which govern the rate of sorption, play a crucial role in these processes. 1.1.6. Current knowledge gaps As mentioned in Section 1.1.1, with anticipated increases in natural gas consumption in the foreseeable future, producers will increasingly rely on unconventional gas sources, such as shale gas, to fulfill the nation's energy requirements. Yet, a significant proportion of estimated shale reserves remain unrecovered, necessitating further advancements in the integration of interdisciplinary knowledge covering numerous disciplines, including: i. Accurate characterization of petrophysical properties of shale formations at various scales ii. Understanding of fluid communications between fractures and matrix iii. A comprehensive understanding of nanoporous mechanisms, including nano-scale confinement and sorption phenomena iv. Seamless understanding of fluid dynamics and geomechanics across scales v. A better understanding of the enhanced oil/gas recovery techniques 1.2. Objective and research outline The primary of objective of this dissertation is to address the knowledge gaps related to the development of unconventional resources such as gas shale formations. The work presented in this dissertation aims to tackle the following related questions: 9 a. Do we understand the role of surface diffusion in the processes of shale development enough? b. Can we establish an integral model for mass transfer and sorption in shale? c. Can we use small-scale experiments to predict the behavior at a larger scale? d. How can we relate observations from different length-scales in the laboratory? e. Can we characterize shale samples with non-sorbing and sorbing gases separately? f. What percentage of free gas and adsorbed gas can be depleted, primarily at the core scale? g. Can we characterize sorption and mass transfer with inert gases and predict shale gas production? To address these questions, our methodology is to separate them into two main efforts that are then merged together, as illustrated in Figure 1.4. In each effort, we use a variety of numerical models and laboratory methodologies to uncover information at the relevant scales. An integral model has been established from a numerical point of view. For core scale investigations (mm to dm), pressure pulse decay and gas loading and unloading measurements are utilized, while the thermogravimetric analysis (TGA) are used for sorption characterization at cube scale (mm to cm). 10 Figure 1.4. Research outline flow chart 1.3. Organization of the dissertation The dissertation outline is provided as follows:  Chapter 2: A Numerical Investigation of Surface Diffusion and Sorption in Nanopoous Media This chapter investigates the interplay of gas transport and sorption in nanoporous media using a one-dimensional, fully discretized, dimensionless model. Methane (CH4) is used as the base case, and various boundary conditions, sorption kinetics, and sorption isotherms are employed to explore the roles of surface diffusion and sorption kinetics in the overall mass transfer during gas loading and unloading. The findings highlight the importance of sorption kinetics and surface diffusion in gas loading and unloading processes. The diffusion mechanism on the surface may be discarded under 11 certain circumstances, and the reliability of characterizing diffusivities of bulk and surface phases based on sorption isotherms for experimental interpretation is cast into doubt. The study emphasizes the necessity of recognizing the transition of sorption when distinguishing gas transport in bulk and surface phases. Separate evaluations of sorption kinetics and bulk phase diffusion coefficients in experimental practices can yield a more accurate characterization of surface diffusion. This study introduces a novel exploration of diffusion and sorption, deepening our understanding of surface diffusion and sorption kinetics in the storage and production of natural gas from nanoporous media.  Chapter 3: An Integral Model of Mass Transfer and Sorption in Tight Dual-porosity Media This chapter aims to provide accurate simulation of mass transfer of shale gas and a feasible method for upscaling during operations by integrating a general dual-porosity model of gas transport and sorption in organic-rich formations. The model is suitable for both low and highly adsorptive gases and extends and validates the applicability of the generalized Vermeulen's approach for cases with varying sorption speeds. The model offers accurate estimation of gas transport and sorption in a dual-porosity system and characterizes it without discretizing corresponding domains. The calculation results closely align with analytical solutions and fine-grid numerical solutions. This study presents a novel attempt to integrate a model of diffusion and sorption, demonstrating an integral characterization model that can lead to a better understanding of gas storage and production, and pathways to enhanced gas recovery, particularly 12 across different scales of practical applications in ultra-tight and organic-rich formations.  Chapter 4: Characterization of Shales Using Helium and Argon at High Pressures This chapter focuses on accurately characterizing the petrophysical and transport properties of gas shales, such as porosity, permeability, diffusivity, and storage capacity, to estimate shale gas in place and recovery during operations. Thermogravimetric analysis (TGA) and gas expansion experiments were used to measure sorption kinetics and mass transfer in shale samples from the same depth/location in the Marcellus shale formation. Thermogravimetric analysis showed that Ar, a non-sorbing and inert gas, adsorbs onto the surfaces of mesoporous and microporous regions of shale samples according to a Langmuir-type behavior. He expansion experiments measured overall porosity and delineated mass transfer across the inherent hierarchy of pore sizes. A triple-porosity model (TPM) was used to interpret He expansion experiments and to extract transport parameters. The study found an excellent agreement between model predictions and experimental data. The experimental observations and interpretation suggest that caution is warrented when using Ar to estimate true porosity and permeability of shales. The chapter also demonstrated that He and Ar probe gases can be effectively used as tools to characterize shales in terms of mass transfer and sorption dynamics across scales.  Chapter 5: Investigation of Methane Mass Transfer and Sorption in Marcellus Shale Under Variable Net-stress 13 This chapter aims to further bridge the gap between laboratory- and field-scale observations with natural gas behaviors in porous media by interpreting different scales experiments at laboratory on Marcellus shale. High-pressure experiments were conducted on a full-diameter core sample, including gas loading and depletion experiments with pure methane (CH4). A novel integral model for mass transfer and storage in multiporosity shale systems was formulated and applied. The model allows for effective investigation of transport and sorption phenomena, demonstrating that sorption hysteresis is crucial for predicting and guiding shale gas production. The study found that without accounting for sorption hysteresis, CH4 production could be overestimated by 10%. The integral, triple-porosity model provides an effective approach for interpreting and predicting gas transport and sorption behavior during loading and production experiments on shale cores under variable net-stress conditions. The chapter validates a characterization approach for improved understanding of shale gas production and defines a potential pathway for translating laboratory-scale experimentation to larger-scale applications.  Chapter 6: Analysis of Gas Storage and Transport in Shale using Pressure Pulse Decay Measurements with He, Kr and CO2 This chapter focuses on characterization of gas transport and storage in lowpermeability and organic-rich shales, addressing challenges due to the complex pore space morphology and the heterogeneous shale matrix. The study uses tandem experiments with inert and adsorbing gases to develop an efficient workflow for characterizing and modeling transport and sorption at the laboratory scale. Pressure 14 pulse-decay (PPD) measurements were conducted on two Eagle Ford shale core samples using He, Kr, and CO2. The triple-porosity model (TPM) was adopted to interpret gas transport in the sample. A modified analytical approach was introduced to extract effective transport parameters directly from PPD measurements with He. The effective transport parameters were then converted for application to Kr and CO2 PPD experiments, accounting for relevant transport modes and differences in fluid properties. Excess adsorption isotherms were extracted from equilibrium pressures of the experiments and integrated into the TPM to predict combined gas transport and sorption for Kr and CO2. The predictions for Kr and CO2 are in excellent agreement with experimental observations, demonstrating the proposed analytical approach provides an effective characterization of mass transfer rates in shales.  Chapter 7: Summary and recommendations for future work This chapter provides a comprehensive overview of the key findings and contributions of the research presented in this dissertation. Additionally, it offers recommendations for future studies for further narrowing the existing knowledge gaps. 15 1.4. References Alnoaimi, K.R., Kovscek, A.R., 2019. Influence of microcracks on flow and storage capacities of gas shales at core scale. Transport Porous Media 127 (1), 53–84. https://doi.org/10.1007/s11242-018-1180-5. Brunauer, S., Emmett, P.H., Teller, E., 1938. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60 (2), 309–319. https://doi.org/10.1021/ja01269a023. Bustin, R.M., Bustin, A.M.M., Cui, X., Ross, D.J.K., Murthy Pathi, V.S., 2008. Impact of shale properties on pore structure and storage characteristics. In: SPE Paper No. 119892- pp for Presentation at the 2008 SPE Shale Gas Production Conference, 16–18 November, Fort Worth, Texas. Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous media. Transport Porous Media 82, 375–384. Clarkson, C., Solano, N., Bustin, R., Bustin, A., Chalmers, G., He, L., Melnichenko, Y., Radliński, A., & Blach, T. (2013). Pore structure characterization of North American shale gas reservoirs using USANS/SANS, gas adsorption, and mercury intrusion. Fuel, 103, 606–616. https://doi.org/10.1016/j.fuel.2012.06.119 Dewhurst, D.N., Siggins, A.F., 2006. Impact of fabric, microcracks and stress field on shale anisotropy. Geophysical Journal International 165, 135–148. Energy Information Adminstration (EIA), 2021. Annual energy outlook 2020 with projections to 2050. U.S. Energy Information Administration Office of Energy Analysis and U.S. Department of Energy. Energy Information Adminstration (EIA), 2023. Annual Energy Outlook 2023 with projections to 2050. Energy Information Administration, Washington. Everett, D. H. (1972). Manual of Symbols and Terminology for Physicochemical Quantities and Units, Appendix II: Definitions, Terminology and Symbols in Colloid and Surface Chemistry. Pure and Applied Chemistry, 31(4), 577–638. https://doi.org/10.1351/pac197231040577 16 Gay, J. M., Suzanne, J., & Coulomb, J. P. (1990). Wetting, surface melting, and freezing of thin films of methane adsorbed on MgO(100). Physical Review B, 41(16), 11346–11351. https://doi.org/10.1103/physrevb.41.11346 Gilliland, E.R., Baddour, R.F., Russell, J.L., 1958. Rates of flow through microporous solids. AICHE J. 4, 90–96. Heller, R., Vermylen, J., Zoback, M., 2014. Experimental investigation of matrix permeability of gas shales. AAPG Bull. 98 (5), 975–995. Hill, T. L., 1956. Surface Diffusion and Thermal Transpiration in Fine Tubes and Pores, 1956. J. Chem. Phys., 25, 730. Hutson, N. D., & Yang, R. T. (1997). Theoretical basis for the Dubinin-Radushkevitch (DR) adsorption isotherm equation. Adsorption, 3, 189-195. Hwang, S., Kammermeyer, K., 1966. Surface diffusion in microporous media, Can. J. Chem. Eng. 44, 82–89. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Petrol. Technol. 48 (8), 16–21. https://doi.org/10.2118/09-08-16- DA. Langmuir, I, 1918. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40 (9), 1361–1403. https://doi.org/10.1021/ja02242a004. Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023a. Investigation of Methane Mass Transfer and Sorption in Marcellus Shale Under Variable Net Stress. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211846. Lyu, Y., Dasani, D., Tsotsis, T.T., Jessen, K., 2021. Characterization of shale using Helium and Argon at high pressures. J. Petrol. Sci. Eng. https://doi.org/10.1016/j. petrol.2021.108952. Rani, S., Padmanabhan, E., & Prusty, B. K. (2019). Review of gas adsorption in shales for enhanced methane recovery and CO2 storage. Journal of Petroleum Science and Engineering, 175, 634–643. https://doi.org/10.1016/j.petrol.2018.12.081 17 Schaaf, S.A., Chambre, P.L., 1961. Flow of Rarefied Gases. Princeton University Press, Princeton, NJ. Sing, K. S., 1985, Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity: Pure and Applied Chemistry, 57, 603- 619. Terzaghi, K., 1923. Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen, Sitzungsber. Akad. Wissen. Wien Math. Naturwiss. Kl., Wien, Abt. 2A, 132, pp. 105–124. Wasaki, A., Akkutlu, I.Y., 2015. Permeability of organic-rich shale. Spe Journal 20 (06), 1384–1396. https://doi.org/10.2118/170830-PA. 18 CHAPTER 2.A numerical investigation of surface diffusion and sorption in nanoporous media1 2.1. Introduction Surface diffusion can play a central role during gas mass transfer in organic nanopores due, in part, to the substantial surface area and the sorption affinity of gases in such materials (Kang et al., 2011; Xiong et al., 2012; Etminan et al., 2014; Fathi and Akkutlu, 2014). The process involves the adsorption of molecules on the pore wall and their subsequent migration along the wall in an adsorbed state. Accordingly, the amounts of gas transported by surface diffusion hinge on both the adsorption process and the mobility of the adsorbed species on the surface. Surface diffusion has been a subject of scientific interest since the 1920s. Volmer and Adhikari (1925) were the first to discover the mobility of physically adsorbed molecules in their study of benzophenone crystals, uncovering that the migration of benzophenone, on the solid surface, was driven by a surface concentration gradient. Later, Wentworth (1944) noted that measured fluxes exceeded predictions for continuum flow, slip flow, or diffusion flow, and posited surface diffusion as an additional mechanism for gas transport. 1 Parts of this chapter have been taken verbatim from our manuscript: Lyu, Y., & Jessen, K. A numerical investigation of surface diffusion and sorption in nanoporous media (will be submitted to Transport in Porous Media). 19 Earlier research conducted by Damkohler (1935) and Wicke (1939) was centered on the sorption kinetics of porous sorbent granules. Despite their assumption of a constant diffusion coefficient and constant pressure in the gaseous adsorbate phase, this remains a relatively complicated system as conditions within the pore space of the sorbent are continuously changing. Given that surface diffusion is typically accompanied by molecular diffusion and Knudsen diffusion, the determination of the surface diffusion coefficient presents a challenge, and the interpretation of experimental data, therefore, often entails considerable uncertainty. Recognizing this complexity, Wicke and his colleagues (Wicke and Kallenbach, 1941; Wicke and Voigt 1947) advocated for maintaining a constant state of diffusion through plugs of porous sorbent, by applying constant conditions on opposite sides of the plug. Carman and Malherbe (1950) later adopted the same methodology. Surface diffusion is attributed to the interplay between diffusing molecules and a solid surface, as well as the interactions between molecules, including van der Waal's forces, electrostatic force, and other direct or indirect interaction forces. Several theories have been developed to describe the surface diffusion of a fluid in porous medium, which can be classified into three categories (Wu et al. 2016): i. Theory derived from hopping models: This model, proposed by Hill et al. (1956), assumes an adsorbed-phase molecule jumps from one adsorption site to a nearby site on the material surface, resulting from an activation of the adsorbed molecule (Higashi et al. 1963; Yang et al. 1973; Chen and Yang 1991). This model is only applicable to monolayer adsorption. 20 ii. Theory derived from hydrodynamic models: The hydrodynamic model assumes that the adsorbed phase surface transport is due to viscous flow in a surface film. This model was initially proposed by Gilliland et al. (1958). Petropoulos (1996) suggested a different hydrodynamic model for mesopores (2-50 nm pore size), followed by extensions of homogeneous hydrodynamic models to heterogeneous surfaces (Kainourgiakis et al., 1996; Kikkinides et al., 1997). This theory is only applicable for multilayer adsorbate diffusion. iii. Theory derived from Fickian models: According to this theory, surface diffusion and free-phase mass transfer in porous media are independent, with the overall flux equal to the sum of the two. Kärger and Ruthven (2016) considered and validated the application of the Fickian model to diffusion in nanoporous media and to hierarchical pore systems. Over recent years, there have been conflicting arguments presented in the literature (Wasaki and Akkutlu 2014). Xiong et al. (2012), for instance, observed that surface diffusion is anticipated to be more significant at low pressure for pores smaller than 5 nm, depending on the value of surface diffusivity. Sigal (2013) suggested that adsorbed-layer diffusional transport could serve as a secondary mechanism for methane (CH4) transport, contingent on substantial values of the diffusion coefficient. Other scholars have proposed that the adsorbed phase might be affected by a gradient in concentrations or chemical potentials and represented this concept through the inclusion of surface diffusion in their flux models (Fathi and Akkutlu 2013; Wu et al. 2015). Fathi and Akkutlu (2013) demonstrated, using the lattice Boltzmann method, that the adsorbed phase can be mobile under reservoir conditions. Wu et al. (2015) revised Hwang's model to account for 21 heterogeneity in the surface energy and isosteric heat of adsorption, assuming ideal gas behavior while incorporating adsorption behaviors into the flux model. Yang et al. (2016) evaluated the surface diffusivity of adsorbed CH4 in the nanopores of organic matter in shale by analyzing the later stage of pressure pulse decay measurements. In contrast, some contributions consider the adsorbed phase to be immobile (Cui et al. 2009; Sakhaee-Pour and Bryant 2012). Many previous efforts on delineating mass transfer in shales, whether based on experimental or numerical methods, have applied sorption isotherms and hence assumed that equilibrium conditions are established fast between the free gas and the surface phase in shales (Do and Do, 2000; Kang et al., 2011; Xiong et al., 2012; Fathi and Akkutlu, 2014; Wu et al., 2015; Song et al., 2018; Qu et al., 2020). The equilibrium state is dictated by the gas (or liquid)-solid interaction and determines the adsorption capacity of the adsorbent. Under circumstances where the adsorption rate constant is relatively high, adsorption equilibrium is approached fast, and sorption kinetics can be approximated accurately by an adsorption isotherm. Numerous scholars have demonstrated that the Langmuir isotherm (Eq. 2.1) is an appropriate approach for describing gas sorption on shale, subject to the assumption of single-layer coverage (Cui et al., 2009; Ambrose et al., 2010). The Langmuir isotherm can be written as (Langmuir, 1918): = , ∙ , = ∙ 1+∙ , (2.1) 22 where (mol/kg) is the adsorbed amount, , (mol/kg) is maximum adsorption capacity, is the fractional coverage of the sorption sites, (bar-1 ) is the equilibrium constant, and (bar) is free-gas pressure. In the context of the surface diffusion of adsorbed gas in shales, theories derived from hopping models may therefore be reasonably applicable. In this work, we consider both sorption kinetics and sorption isotherms, where sorption kinetics are used to represent the rate of fluid adsorption at the gas (or liquid)-solid interface. Fathi and Akkutlu (2012) incorporated sorption kinetics in their upscaling approach to highlight the role of sorption, while the application of sorption kinetics in this work allows us to explore the importance of the dynamic interaction between the free gas and the surface phase. Several hopping models have been proposed over the years. For instance, Hwang and Kammermeyer (1966) proposed and validated an analytic model for surface diffusion in Vycor microporous glass at low pressures based on a hopping model. Chen and Yang (1991) introduced a surface-diffusion model that accounts for adsorbed-gas coverage on the surface. Based on experimental data and an analytical model, Guo et al. (2008) developed an empirical expression for the surface-diffusion coefficient in a CH4/activatedcarbon system. However, the influence of pressure was not considered, and this correlation cannot be used to calculate the surface-diffusion coefficient of an adsorbed gas under highpressure conditions. The experimental results of Carman and Malherbe (1950) and Carman and Raal (1951) demonstrated that surface diffusivity increases with loading until full monolayer coverage 23 is reached. Ash et al. (1963) discovered that the surface diffusivity decreases when the monolayer layer coverage is exceeded and subsequently increases in the region of capillary condensation. Higashi et al. (1963) presented the following correlation to model the concentration dependence of surface diffusivity: = 1 1− , (2.2) where (m2 /s) is the surface diffusion coefficient at the limit of zero coverage: = Ω exp (− ). (2.3) Here, Ω (m2 /s/K0.5) is a constant related to the molecule weight, T (K) is temperature, is a dimensionless constant, (J/mol) is the activation energy for a jump, and R (J/K/mol) is the gas constant. Yang et al. (1973) updated Eq. 2.2 to eliminate the asymptote when the surface coverage approaches unity ( = 1). Later, Chen and Yang (1991) developed a formulation to include blocking of hopping among molecules: = (1−)+ 2 (2−)+[(1−)](1−) 2 2 (1−+ 2 ) 2 , (2.4) (1 − ) = { 0, ≥ 1 1, 0 ≤ ≤ 1 , (2.5) = . (2.6) Here, (1 − ) is the Heaviside step function (dimensionless), (m/s) and (m/s) are forward-velocity and blocking-velocity coefficients of the adsorbed molecule, 24 respectively. When molecules are diffusing on an infinite surface, the blocking coefficient = 0, and Eq. 2.4 simplifies to Eq. 2.2. A central question that we address in this work is if surface diffusion can be quantified accurately from laboratory-scale experiments with shale samples. To this end, we investigate the role of surface diffusion in nanoporous media, with a particular emphasis on micropores (< 2 nm) within the shale matrix. As the contribution of surface diffusion to the overall flux in mesopores is moderate, typically less than 5% (Ren et al., 2015; Sun et al., 2017; Song et al., 2018), these larger pores, which often serve as a source or sink to the micropores, have been excluded from our analysis of the role of surface diffusion. The transport in the gas phase of a nanopore may include viscous flow, slip/transition flow, and Knudsen diffusion (Lyu et al., 2023a). For the analysis of this work, an effective diffusion coefficient that reflects the combined transport behavior in the gas phase is sufficient. Typical gas phase diffusion coefficients in nanopores fall within the range of 10- 9 to 10-8 m2 /s (Medveď and Černý, 2011; Du et al., 2020), while surface diffusion coefficients in organic pores may reach magnitudes of 10-7 to 10-6 m2 /s (Kang et al., 2011; Xiong et al., 2012; Wu et al., 2015). Surface diffusion is also affected by temperature and typically becomes marginal at high temperatures (Kärger et al., 2012) where sorption is, in general, less favorable. In this work, we consider isothermal systems, reflecting relevant subsurface conditions that are often applied in e.g., pulse-decay measurements, for shale samples in the lab. In our previous work (Lyu et al., 2023b), we demonstrated different sorption affinities for different gases in organic-rich materials. We also showed that the adsorbate phase density 25 impacts both the estimation of gas in place (Lyu et al., 2023a) and the mass transfer in shale formations (Lyu et al., 2023b). Building on these findings, the present study focuses on the interplay between sorption behaviors and surface diffusion during gas loading and unloading processes. The organization of the reminder of the manuscript is as follows: We start by presenting the model formulation, including simplifying assumptions, and provide a discussion of relevant initial and boundary conditions. Next, we outline the governing equations in dimensional and dimensionless form and solve them numerically to study the interplay of sorption and surface diffusion during gas transport with CH4 as the probe gas. We then explore the dimensionless parameter space to extend the applicability of the analysis and discuss new insights from the modeling work in the context of the interpretation of experimental observations. Finally, we provide a summary of key findings and a set of conclusions to complete the manuscript. 2.2. Methodology In this work, a homogeneous one-dimensional (1D) model is employed to simulate the transport of gas (CH4) in the micropores of ultra-tight formations (e.g., shales). As illustrated in Figure 2.1, gas transport in micropores includes free gas transport and adsorbed phase diffusion that are closely coupled by sorption dynamics. 26 Figure 2.1. Schematics of gas transport in micropores (not scaled) 2.2.1. Model formulation - Diffusion and sorption in the micropores The process of gas mass transfer in micropores includes contributions from both free and adsorbed phases. Mass conservation in the free gas phase, including transport by an effective diffusional flux and mass exchange between the free gas and the surface phases, can be stated as: = ( ) − (̇ − ̇), (2.7) where is the porosity, (mol/m3 ) is gas concentration, (s) is time, is an effective diffusion coefficient for the free phase that represents relevant transport modes in a nanopore, (m) is the distance along the pore, and ̇ (mol/m3 /s) is molar rate of exchange between free and surface phase per unit volume of sample (subscripts ads and des denote adsorption and desorption, respectively). 27 Similarly, the mass conservation for the surface phase, including transport via surface diffusion and mass exchange with the free gas phase, can be written as: (1 − 0 ) = (1 − 0 ) [ ] + (̇ − ̇), (2.8) where 0 is the initial porosity, (kg/m3 ) is the skeleton density of the porous material, (mol/kg) is adsorbed gas amount, and (m2 /s) is the surface diffusion coefficient. To model the mass exchange between the free gas and the surface phase, the kinetic model of Langmuir is adopted: (̇ − ̇) = (1 − 0 ) ∙ [ ∙ ∙ (, − ) − ∙ ∙ ] = (1 − 0 ) ∙ [(, − ) − ]. (2.9) Eq. (2.9 includes the internal surface area in the pores per unit skeletal volume, (m2 /m3 ), the rate constant for adsorption, (kg/bar/m2 /s), the gas pressure, (bar), the maximum absolute adsorption capacity of the porous material, , (mol/kg), the rate constant for desorption, (kg/m2 /s), and the equilibrium constant for sorption, (bar-1 ) = /. The change in porosity due to adsorption, reflecting the density of the adsorbate phase, (mol/m3 ), can be expressed as: = 0 − (1−0) . (2.10) To generalize and simplify the analysis, we introduce the dimensionless variables: 28 = , = , , = = (), = 0 , = , (2.11) where and are dimensionless gas concentrations for the free and adsorbed phases, respectively, is the dimensionless gas pressure, is the normalized porosity, and is the dimensionless distance along the domain. Furthermore, we introduce characteristic times for sorption ( ), transport in the free phase ( ), sorption ( ), and a dimensionless time : = ∙ , = 2 , = 2 , = . (2.12) Upon substituting of the dimensionless parameters and characteristic times in Eqs. 2.7, 2.8, and 2.10, we arrive at their dimensionless forms: + = ( ) − ∙ ∙ [ (1 − ) − ], (2.13) = ( ) + [ (1 − ) − ], (2.14) = 1 − , (2.15) with = (1−0), 0∙ , = (1−0), 0∙ , = 1 ∙ , = , = = . (2.16) 29 The dimensionless coefficients reported in Eq. 2.16 are associated with specific physical attributes: defines the ratio of gas storage between adsorbed and free phases at the maximum pressure, while defines the maximum reduction of the normalized porosity due to adsorption. A larger value of reflects a lower fractional coverage on the surface and/or high desorption rates. The parameter is the ratio of characteristic times for sorption and transport in the free phase. In our previous work, this ratio was estimated from measurements with argon (=7, see Lyu et al., 2021) and CH4 (=5, see Lyu et al., 2023a). Sorption kinetics represent the transient states when equilibrium has not been reached in the mass exchange between the free and surface phases. At equilibrium, the adsorption and desorption rates balance, and Eq. 2.9 reduces to Eq. 2.1. In the event of rapid gas adsorption (e.g., a large value of ), the adsorbed phase concentration will reach equilibrium fast. Such settings are represented by large values of (refer to Section 2.3.1 for further analysis and discussion). Finally, is the ratio of characteristic times for transport in the free phase and in the surface phase, as dictated by the magnitudes of the relevant diffusivities. While the surface diffusion coefficient is theoretically a function of gas pressure (see Eqs. 2.2 and 2.4), past research has indicated that the dependency is marginal and that it can be omitted (Wu et al., 2015; Javadpour et al., 2021). In this work, the surface diffusion coefficient is assumed to be constant during gas loading and unloading. However, the impact of the magnitude of the surface diffusion coefficient is explored through the dimensionless parameter . 30 2.2.2. Initial and boundary and initial conditions Dimensionless initial conditions ( and ,) and boundary conditions ( for the free phase and , for the adsorbed phase) for the domain are defined as follows: = , , = , , , = , , = , , . (2.17) The initial concentrations in free and surface phases for loading (I.C.1) and unloading (I.C.2) process are: I.C.1: = 0, , = 0, (2.18) representing no initial gas inside the micropores, and I.C.2: = 1, , = 1 1+ , (2.19) setting the maximum free gas concentration (at the maximum pressure) inside the micropores and the associated adsorption on the surface. 31 In settings where gas transport in the larger pores, e.g., mesopores, is significantly faster than in the micropores, a constant boundary condition can be applied for both the free and surface phases. This setting represents gas transport into micropores via high-permeable channels such as natural or induced fractures. We consider two extreme cases for the free and surface phases at the boundary for loading (B.C.1) and unloading (B.C.2) processes: B.C.1 (loading): = 1, , = 1 1+ , (2.20) representing the maximum free gas concentration and sorption isotherms at the boundary. B.C.2 (unloading): = 0, , = 0, (2.21) representing no gas at the boundary. If the gas transport rate in the mesopores is comparable to that in the micropores, a variable gas concentration at the boundary is necessary (e.g., a lag in gas build-up at the entrance to the micropores during loading). To represent this scenario, a dual-porosity (DP) model is employed to provide an appropriate boundary condition for the microporous region. The DP model is based on the existence of two distinct and connected domains within a porous medium, such as mesopores (2-50 nm) and micropores (< 2 nm). Much effort (Zimmerman et al., 1993; Lim and Aziz, 1995; Zhang et al., 2022) has been made on this topic following the introduction of the transfer function by Barenblatt et al. (1960) and Warren & Root (1963), who established the foundation for representing mass transfer between the matrix and fracture segments. Barenblatt et al. (1960) proposed a DP model to describe single-phase fluid flow in porous materials in the form: 32 = − ( − ), (2.22) where (kg/m3 /s) is the mass flow rate per unit rock volume, (m-2 ) is the shape factor, (kg/m3 ) is the fluid density, (m2 ) is the permeability, (Pa·s) is fluid viscosity, (Pa) is gas pressure, and , denote fracture and matrix segments, respectively. We adopt this equation to enable the mesopores to serve as boundaries to the micropores. This equation can be rewritten in molar rate, ̇ (mol/s), using gas concentrations, and (mol/m3 ), and an effective diffusion coefficient, (cm2 /s): ̇ = −( − ). (2.23) With the primary focus on transport in micropores, Eq. 2.23 is strictly applied to represent the boundary state between mesopores and micropores. The free and surface phases at the boundary for gas loading are then given by: = () ∙ ( − ) − 1− , , (2.24) , = [(, − ,) − , ], (2.25) where () (1/s) is the inverse of the characteristic time for transport (1⁄,) in the mesopores. By introducing new dimensionless variables, = ⁄,, = (1−) 0 (1−0) , (2.26) Eq. 2.24 and 2.25 can be re-written as B.C.3 and B.C.4: 33 B.C.3 (loading): = ∙ (1 − ) − , , (2.27) B.C.4 (unloading): = ∙ (0 − ) − , , (2.28) with , = [,(1 − ,) − ,]. (2.29) Generally, should be greater than unity because gas transport in the mesopores is faster than in the micropores, and as increases, the variable boundary gas concentration approaches a constant condition. 2.2.3. Surface continuity at entrance to micropores Two possibilities for surface continuity on the domain are considered in this work: a) communicating surface boundaries or b) isolated surface boundaries. Communicating surface boundaries allow diffusion of the surface phase into (and out of) the domain across the boundary, which is realistic for nanomaterials such as activated carbon and porous silica. Conversely, in some porous media, such as shales with a hierarchical pore structure, the surface phase may not be connected or communicating across the boundary. In that case, the surface can be considered isolated, and no mass enters or exits the domain via the surface. Hence, the adsorbed phase can only move within the domain or interact with the free phase. Therefore, we examine four scenarios for gas loading (LA-LD) and unloading (UA-UD). These include constant boundary gas concentrations with a communicating pore surface (LA and UA), variable boundary gas concentrations with a communicating pore 34 surface (LB and UB), constant boundary gas concentrations with an isolated pore surface (LC and UC), and variable boundary gas concentrations with an isolated pore surface (LD and UD). These scenarios are depicted in Figure 2.3. Figure 2.2. Models with initial and boundary conditions for loading and unloading cases. Left-top: case LA (B.C.1) and LB (B.C.3), right-top: case LC (B.C.1) and LD (B.C.3), Left-bottom: case UA (B.C.2) and UB (B.C.4), right-top: case UC (B.C.2) and UD (B.C.4) 2.3. Results and discussion The governing equations, presented above, were solved using a standard finite difference method, incorporating relevant initial and boundary conditions for gas loading (storage or sequestration) and unloading (production) processes. In the following, we present a detailed analysis of the role of surface diffusion subject to varying sorption rates (kinetics), with a particular emphasis on the effect of key parameters including sorption capacity, rock porosity, skeleton density, and adsorbate density. 35 To provide a uniform representation of the calculation results, we report the normalized gas amount, ̂, that either accumulates (during loading) or remains (during unloading) within the domain (per unit bulk volume): ̂ = () () . (2.30) where (and ) accounts for both free gas and surface phase: = (1 − ∙ ) ∙ 0 + ∙ 0, (2.31) = (1 − + 1 ) ∙ 0 + + 1 ∙ 0. (2.32) The default parameters/properties used here include: the maximum gas pressure = 100 bar, isothermal gas concentrations from the NIST database, as well as rock and sorption information for CH4/shale from Lyu et al. (2023a): = 0.052, , = 0.16 mol/kg, = 0.032 bar-1 , = 6.25 × 10−3 kg/bar/m3 /s, = 26.41 × 103 mol/m3 , = 10, and = 1. 2.3.1. The transition from sorption kinetics to sorption isotherms In an initial set of calculations, we explore the transition from sorption kinetics to sorption isotherms by comparing the normalized gas amounts over time for relevant model parameters. We consider two cases LA (constant gas concentration at inlet and open surface boundary) and UD (variable gas concentration at inlet and closed surface boundary). The free gas and adsorbed phase diffusivities are set constant ( = = 36 100), based on diffusivities reported in prior work (Kang et al., 2011; Xiong et al., 2012; Wu et al., 2015; Du et al., 2019). The transition from sorption kinetics to isotherms was then explored by adjusting the adsorption kinetics, , via the dimensionless parameter = ⁄ . Figure 2.3 illustrates that sorption kinetics transition to isotherm behavior only if the adsorption rate is high (i.e., =1,000 for LA and =100,000 for UD). In other words, it is a reasonable approximation to use isotherms for large values of . Equivalent results for the additional 6 cases are presented in Appendix B. Figure 2.3. Normalized gas amount with different adsorption rate (sorption kinetics) and isotherms; arrows indicate direction for increasing values of In subsequent calculation examples, results are presented for values of 0.1, 10, and 1,000 to explore the impact of sorption kinetics and isotherm-like behavior. It is important to note that values of in gas shales (at relevant subsurface conditions) fall between 1 and 10, as reported in our previous studies (Lyu et al., 2021; Lyu et al., 2023a). 37 2.3.2. Communicating surface boundary and constant gas concentration (LA and UA) Next, we consider settings with an open surface boundary and constant gas concentrations at the boundary during loading and unloading. The ratio of diffusivities ( = ⁄) spans the range from 0 to 100, while sorption behavior transitions from kinetics to isotherms ( ranging from 0.1 to 1,000) for a constant value of . The normalized total gas amount and the amount in the surface phase over the microporous domain are plotted against dimensionless time in Figure 2.4 that demonstrates the intricate interplay between transport in the free and surface phases due to sorption. If transport in the free phase is fast compared to sorption ( = 0.1) during gas loading (LA), Figure 2.4 demonstrates a transition from free-gas dominated transport ( = 0), where the adsorbate concentration lags the free-gas concentration (the gap between the normalized total gas amount and the amount in the surface phase), to a surface dominated transport mode ( = 100 ), where surface concentration outpaces the free gas concentration even as the characteristic time for free-gas transport is less than that for sorption: For = 100, we observe that the total gas in the system is largely equal to the surface concentration until the sorption capacity is satisfied and that an additional increase in total concentration at later times follows closely the behavior of settings with =10. As the characteristic time for sorption is gradually decreased (approaching the isotherm for = 1000), we observe that the concentrations in free and adsorbed phases mirror each other and that an increase in surface diffusivity exclusively shifts the buildup of gas in the pore to earlier times (akin to the diffusion equation when including a retardation factor). 38 During gas unloading (UA), when transport in the free phase is fast compared to sorption ( = 0.1), we observe that in the absence of surface transport ( = 0) that the free gas is depleted before the adsorbate phase can be mobilized through desorption and transport in the free phase. In contrast, as the surface flux dominates transport ( = 100), we observe that the adsorbate phase is largely depleted before the free gas concentration is reduced and that the overall process completes faster. As the characteristic time for sorption is decreased, the timing of variations in surface and free concentrations becomes similar, as we also observe in the loading calculation examples. One key observation from these scenarios is that during gas loading and unloading processes with an open surface boundary and constant gas concentrations at the boundary, surface diffusion operates concurrently with transport in the free phase and can significantly influence the overall gas transient in the micropores if it exhibits a higher diffusivity than free phase diffusion, regardless of the sorption rates. 39 Figure 2.4. Normalized gas amount for cases LA and UA (communicating surface boundary and constant gas concentrations); arrows indicate direction for increasing values of 40 2.3.3. Communicating surface boundary and variable gas concentration (LB and UB) When the gas transport in macropores/mesopores is within an order of magnitude of the transport in micropores, the use of a constant gas concentration as a boundary condition is no longer accurate. To investigate such settings, we assume that the external transport is ten times faster than in the microporous domain ( =10) to let the dimensionless concentrations at the boundary vary in time (see Appendix C for additional details). As in the previous calculations, we keep the surface boundary communicating for transport in this example. The normalized gas amounts are reported as a function of dimensionless time for different sorption rates (=0.1, 10, and 1,000) and surface diffusivities ( =0, 1, 10, and 100) in Figure 2.5. For models LB and UB (the first row) in Figure 2.5, we find that when the sorption rate is slower than the free gas diffusion (=0.1), the surface concentration lags the free gas at the boundary. Consequently, the variation in the adsorbate amount in the micropores lags the free-gas concentration, like in the previous example for LA and UA with free-gasdominated transport ( = 0): the surface phase contributes to the overall mass transfer only after the transport in the free phase has reached or is nearing completion. As a result, the impact of surface diffusion on the overall mass transfer is marginal, as illustrated by the overlapping gas amounts for a broad range of surface diffusivities, even as the surface diffusivity exceeds the free-gas diffusion coefficient by several orders of magnitude (=100,000). This observation reveals that when the rate of gas accumulation or depletion on the surface lags that in the free phase, both at the boundary and within the domain, the 41 transport of the adsorbed phase into and within the domain is constrained by sorption kinetics at the boundary. However, as the sorption rate is gradually increased (approaching the isotherm for = 1000), these lags between the adsorbate and free phases become marginal and negligible. Consequently, we observe that the total gas amount is largely attributed to the adsorbate amount and the buildup of gas in the micropore is shifted to earlier times due to the increasing surface diffusivity as previous cases (LA and UA with = 1000). A central finding of these examples is that the impact of surface diffusion on the gas transport in the micropores, during loading and/or unloading processes, can be marginal when sorption is slower than free phase transport, irrespective of the surface diffusivity. However, as the sorption rate increases, the role of surface diffusion becomes increasingly significant in these processes. Ultimately, when sorption kinetics are closely approximated by an sorption isotherm ( = 1000), it poses a challenge to distinguish surface diffusion from free gas diffusion. 42 Figure 2.5. Normalized gas amount for case LB and UB (communicating surface boundary and variable gas concentrations); arrows indicate direction for increasing values of 43 2.3.4. Isolated surface boundary and constant gas concentration (LC and UC) In the next example, we consider settings with an isolated surface boundary and constant a gas concentration at the inlet. In these situations, the free-gas phase provides the sole channel for transport in and out of the micropore, and transport in the surface phase within the domain is only activated through interactions with the free phase. In other words, the adsorbate phase is constrained by the free gas phase mass transfer and is mobilized only later in the transport process. Therefore, the rate of free gas transport and the rate of sorption dictate the contribution from transport in the surface phase, as shown in Figure 2.6. When the free gas transport is faster than sorption (e.g., = 0.1 in first row of Figure 2.6), the predominant contribution to overall mass transfer, at early times, is attributed to free gas transport, with only a marginal contribution from the adsorbate phase. As the free gas transport reaches completion (filling of the micropores during the loading or the depletion of original free gas during unloading), the adsorbed phase starts building up or starts depleting during loading and unloading processes, respectively. The overlapping curves for the total and surface gas amounts illustrate that surface diffusion plays a very limited role in the overall mass transfer under such conditions. As the characteristic time for sorption decreases ( =10), the adsorbed phase mobilizes earlier and complements the transport in the free gas. Once activated, an increased surface diffusivity (indicated by a larger value of ) results in an increasing contribution from the adsorbate phase. However, the surface contribution is restricted to a limited range and eventually reaches a threshold. Beyond this point, any further increases in the surface 44 diffusivity do not enhance the contributions from the adsorbate phase, as evidenced by the overlapping responses for =10 and =100. When sorption kinetics are closely approximated by an isotherm ( = 1000), the contribution from the adsorbed phase is clearly observed. The enhancement of mass transfer attributed to larger values of surface diffusivity becomes more obvious and shifts the completion of the mass transfer process to earlier times. 45 Figure 2.6. Normalized gas amount for case LC and UC (isolated surface boundary and constant gas concentrations); arrows indicate direction for increasing values of 46 2.3.5. Isolated surface boundary and variable gas concentration (LD and UD) A similar set of calculations were performed with an isolated surface boundary and variable gas concentrations (LD and UD) as shown in Figure 2.7. The results align closely with those of the preceding settings (cases LC and UC). However, we observe a shift towards slower mass transfer, attributable to variations in the gas concentration at the boundary, in contrast to settings with constant concentrations at the boundary. A central observation from both constant (cases LC and UC) and variable (cases LD and UD) gas concentrations at the boundary is that surface diffusion may be ignored when sorption is slower or comparable to the rate of free gas diffusion (see Figure 2.6 and Figure 2.7). In other circumstances where sorption kinetics are approximately an order of magnitude faster than free gas transport (=10), surface diffusion tends to play a minor role in the overall gas transport. When the sorption kinetics are even faster and approaching isotherm behavior, the flux from the surface phase can overwhelm the free gas flux and dominate the mass transfer in the micropores. However, in cases where the interaction between free gas and the adsorbed phase follows isothermal behavior, the contribution from surface diffusion to the overall mass transfer cannot be discerned solely by examining the transient of the total gas amount in the system. This observation and related practical implications are discussed further in section 2.3.7. 47 Figure 2.7. Normalized gas amount for case LD and UD (isolated surface boundary and variable gas concentrations); arrows indicate direction for increasing values of 48 2.3.6. Sensitivity analysis of dimensionless groups/parameters In this section, we explore the impact of various key parameters (e.g., rock properties and sorption capacity, etc.) on the significance of surface diffusion. This investigation is performed by varying the dimensionless groups and parameters, i.e., , , , and . Our selection of parameters and dimensionless groups is informed by the works of Curtis (2002), Weniger et al. (2010), Wang et al. (2016), and Yang & Liu (2020) related to CH4 and CO2 on shales from Brazil, the USA, and China. Table 2.1. Summary of default parameters including high and low values Parameter (unit) Default value Low value High value 0 (-) 0.052 0.02 0.15 (-) 0.052 0.02 0.15 , (mol/kg) 0.16 0 0.57 (bar-1 ) 0.032 0.008 0.14 (mol/cc) 0.026 0.010 0.035 (bar) 100 50 500 (mol/cc) 0.004 0.001 0.019 = (1−0), 0∙ (-) 1.80 0 69.83 = (1−0), 0∙ (-) 0.28 0 1 = 1 ∙ (-) 0.31 0.014 2.5 = , (-) 10 1 100 49 = (1−)0 (1−0) (-) 1 0.5 2 It should be noted here that the default parameters for porous samples and sorption behaviors are reported in Lyu et al., 2023a, and that the dimensionless group has an upper limit of unity dictated by its physical significance (porosity change due to sorption). To explore how the mass transfer contribution from surface diffusion may change for different property settings, various values of , , and were utilized to compute the normalized gas amount. These values are denoted as , , and , where = , , represents the default setting (), the low value (), and the high value (), as reported in Table 2.1. An enhancement of the surface contribution requires greater values of and/or , in general, combined with a greater value of for unloading processes and a smaller value of for loading processes. Therefore, we apply , , and for case LD (loading), while , , and for UD (unloading) to maximize the contribution of surface transport. It is noteworthy that there will be no sorption and surface phase if or is set to be zero, hence, for the purpose of this sensitive check, , , and are defined as the reciprocal of , , and , respectively. Similarly, dimensionless parameter and are altered and compared with the base case. We use LD and UD (loading and unloading with an isolated surface boundary and variable gas concentration at the boundary) as representative examples to demonstrate how surface diffusion contributes differently to the overall mass transfer in micropores with varying values of , , (see Figure 2.8) and with different values of and (see Figure 2.9 and Figure 2.10). Furthermore, we consider settings where the sorption behavior is controlled 50 by kinetics (=10) and compare behaviors without surface diffusion (=0) and with surface diffusion (=100). Figure 2.8. Normalized total gas amount for case LD and UD with the default, high, and low values of the dimensionless groups - , , and Figure 2.8 depicts the normalized gas amount vs. the dimensionless time , distinguished by different color codes representing varying parameter settings: blue for the default settings, gray indicating results with maximized contribution from surface transport (, , for LD and , , for UD) and green for reduced contribution from surface diffusion. An initial observation from contrasting the pairs of curves in different colors is that greater values of and/or combined with a smaller for loading or a greater for unloading impede the mass transfer process. This delay is attributed to the surface requiring a more substantial adsorbed phase, necessitating an extended time durations to reach saturation. Furthermore, by examining the gap between the solid lines (representing behaviors without surface diffusion, =0) and the dashed lines (with surface diffusion, =100), one finds that an increased surface phase promotes the impact and contribution of 51 surface diffusion to the overall mass transfer in such processes. When the capacity for gas sorption is sufficiently low (considering both quantity and volume), the effect of surface diffusion can be minimal/negligible, as illustrated by the equivalency of the solid and dashed green lines in Figure 2.8. Next, we explore how the dimensionless parameter (remind reader what this is) influences the overall mass transfer and the corresponding contribution from surface diffusion. As demonstrated in Figure 2.9, a higher value of , indicating that transport of free gas in upstream macropores/mesopores is faster than that in micropores, accelerates the overall mass transfer within micropores. An increased value of marginally enhances the role of surface diffusion during loading processes (case LD). However, this does not significantly affect the contribution of surface diffusion during unloading (case UD). Figure 2.9. Normalized total gas amount for case LD and UD with the default, high, and low settings for ; arrows indicate direction for increasing values of Figure 2.10, in turn, indicates that a larger value of , representing a greater skeleton to pore volume ratio in the mesopores compared to the micropores, results in a slight decrease 52 in the overall mass transfer within micropores. This is because the equilibrium in the mesoporous region is established later as increases. However, the contribution from surface diffusion remains unaffected by changes in . Figure 2.10. Normalized total gas amount for case LD and UD with the default, high, and low settings for ; arrows indicate direction for increasing values of 2.3.7. Practical implications Laboratory-scale experiments are often used to delineate relevant transport modes during gas production/injection and form the basis for upscaling transport processes to well- or field-scale. One inch or full-diameter core samples are routinely used in experimentation with shales and used in e.g., pulse-decay or production studies. For a given material, critical parameters for gas transport include the diffusivity of the free phase ( ), diffusivity of the surface phase (), and the sorption kinetics (). Both and are often estimated via history matching of the gas uptake or production obtained from laboratory measurements. As demonstrated in the previous sections, sorption behaviors play an important role in the coupling between transport in the free phase and in the surface phase. 53 The complexity of the coupling ranges from relatively simple equilibrium relations (isotherm behavior) to a more complex transient when sorption kinetics control the mass exchange between free and sorbed phases. A key question then arises: What parameters can we determine, with high certainty, from e.g., pulse-decay measurements that explore only the overall gas uptake as a function of time. To this end, we simulate gas loading and unloading based on a set of assumptions and use the observed behavior as artificial data. The artificial data are then interpreted by modeling based on a different set of assumptions (e.g., isotherm vs. kinetics, or surface diffusion vs. no surface diffusion) to examine how accurately we can model and match a given process based on different (and incorrect) assumptions. We start by considering loading and unloading with a constant gas concentration at the boundary and with surface diffusion active across the interface between micro- and mesopores (models LA and UA). We assume that the sorption behavior follows an isotherm and that surface diffusion is two orders of magnitude greater than free-gas diffusion ( = Dsurf/Dfree = 100). The artificial data were then interpreted by using a model that assumes sorption kinetics (δ =10), and the free-gas and surface diffusivities ( and ) were adjusted to match the data. Figure 2.11 compares calculations and (synthetic) data for this example, and we observe a very good agreement between data and model (R2 > 0.999 for loading and unloading): The fitted model parameters include a free-gas diffusivity that is ~ 20 times larger than the value used for generating the data, while the surface diffusivity is ~ 20 times smaller. From an interpretation perspective, the increased value of the free-gas diffusivity (relative to what would be expected from e.g., Knudsen 54 diffusion), should in this case suggest that the underlying model assumptions may be incorrect. This is, however, not always the case, as we demonstrate in the next example. Figure 2.11. Estimation of and from artificial loading and unloading data (LA and UA with sorption isotherm and =100, were used to generate the artificial data) Next, we repeat the exercise for the case where the gas concentration at the boundary is variable (i.e., transport in mesopores is up to approximately an order of magnitude greater than transport in the micropores). In such settings, we have demonstrated that the role of surface diffusion depends strongly on the characteristic time for sorption. Synthetic data were generated using models LB and UB (communicating surface boundary and variable gas concentration at the boundary) with sorption isotherms and ω =100 (fast surface diffusion), as in the example discussed above. Based on this setting, as demonstrated in previous section 2.3.3 (see Figure 2.5), surface diffusion dominates the overall mass transfer. We then interpret the data by assuming that sorption is controlled by kinetics (δ = 10) and that surface diffusion does not contribute to the transport process (ω = 0). Accordingly, we adjust only Dfree to match the data. A comparison of the synthetic data 55 with model calculations (after parameter estimation) is provided in Figure 2.12, which shows a very good agreement between data and model (R2 > 0.999): The estimated value of Dfree is ~ 5 times larger than the default value when surface diffusion is not included in the data interpretation. For these settings, it would be difficult to reject the model assumptions, particularly in the context of natural porous materials where the micropores are often represented by a pore-size distribution (PSD) and a tortuosity factor that complicates a separate estimation of effective free gas diffusivities. Figure 2.12. Estimation of from artificial loading and unloading data (LB and UB with sorption isotherm and =100, were used to generate the artificial data) In these examples (Figure 2.11 and Figure 2.12), we illustrate how the interpretation of experiments where sorption equilibrium is established fast (isotherm), based on sorption kinetics and transport coefficients for surface and free-phase transport, can result in a misleading characterization of the transport process. These observations also hold true if the actual sorption is controlled by kinetics (far from isothermal behaviors), and we assume sorption equilibrium (isotherm) to fit the artificial 56 data: The estimated diffusivities may be substantial errors (for cases LA and UA in Figure 2.11). Moreover, this assumption may artificially assign a significant role to surface diffusion, which in reality may not be important or even inexist (for cases LB and UB in Figure 2.12). Additional examples of the impact of modeling assumptions with isolated surface boundaries (LC/UC and LD/UD) are provided in Appendix D. The observations discussed above suggest that disregarding the dynamic behavior of sorption kinetics might result in significant uncertainties in the estimated transport coefficient (diffusivity) for both surface diffusion and free gas diffusion. Furthermore, although surface diffusion may not be crucial for experimental interpretations in some conditions (specifically in cases LB, UB, LC, and UC) when considering sorption kinetics, there is still a risk of incorrectly estimating the diffusivity associated with the free phase. These findings strongly emphasize the importance of understanding and integrating the characteristic time for sorption into the interpretation of mass transfer measurements. Furthermore, the presented examples demonstrate the challenge of separating surface diffusion from the combined effects of sorption kinetics and diffusion in the free phase when interpreting experimental results. Consequently, we recommend independent investigations of dynamic sorption behaviors, for instance, through gravimetric analysis on smaller samples, to eliminate diffusive mass transfer limitations (see e.g., Lyu et al., 2021), towards a more accurate portrayal of the sorption process. As a step in the characterization procedure, the diffusivity in the free phase can be evaluated using a non-sorbing gas, such as helium. After this evaluation, a translation of helium diffusivity to the gases of interest can be performed based on the average pore size and 57 transport models such as the Dusty-Gas Model (DGM) or the Beskok-Karniadakis-Civan (BKC) model. For additional discussion of translation procedures, readers are referred to our previous studies (Lyu et al., 2021; Lyu et al., 2023a). This integrated approach can contribute to a more robust understanding of gas transport processes and further enhance the reliability and accuracy of diffusion-related research in nanoporous materials. At the laboratory scale, experiments could also take advantage of the distinct temperaturedependent behaviors of diffusivity in the free phase and surface phase. Specifically, the diffusivity of the free-gas phase typically increases monotonically with temperature, whereas the diffusivity of the adsorbed phase is anticipated to initially increase and subsequently decrease as temperature increases. Hence, diffusivity measurements across various temperature conditions can provide a path to differentiate between surface diffusion and free-phase diffusion. 2.4. Summary and conclusions This work aimed to investigate the interplay between free gas transport, surface diffusion, and sorption kinetics during mass transfer of gas in ultra-tight materials, with a specific emphasis on shale gas. The investigation accounted for gas transport in the free phase, adsorbed phase, and sorption with a focus on micropores, given the larger contribution from surface diffusion in smaller pores, as highlighted by previous research. To facilitate our investigation, a 1D model was established and solved numerically in dimensionless form. This approach allowed us to probe the influence of several key parameters. In characterizing the gas transport within the free phase, we utilized an effective diffusivity, whereas surface diffusion was represented by a constant surface diffusivity and a gradient 58 in the adsorbate concentration. Our analysis included both sorption kinetics and sorption isotherms. To simulate gas loading (injection or sequestration) and unloading (production) processes in synthetic and natural porous materials, the combined mass transfer in the free phase and surface phase was simulated based on various boundary conditions. Different boundary conditions were evaluated: communicating and isolated surface boundaries combined with a constant or variable free gas concentration (eight cases in total). Throughout the transient period of mass transfer, the total and adsorbed gas amounts were evaluated for a broad range of characteristic times (diffusion and sorption) and other relevant key parameters. The significance of surface diffusion was analyzed and discussed in each setting. Finally, the practical applications, as related to the interpretation of experimental observations at the laboratory scale, were demonstrated and discussed. Based on the examples and analysis presented above, we arrive at the following set of conclusions: • Sorption kinetics can play an important role in the overall mass transfer process during the loading and unloading of gas from micropores. The rate of adsorption dictates, in part, the potential significance of surface diffusion relative to mass transfer by diffusion in the free gas phase. As the rate of adsorption increases, the combined behavior of sorption and transport can be approximated by using an sorption isotherm rather than a kinetics model. From the calculation examples, we observe that the use of sorption isotherms provides for a good approximation, e.g., when the sorption process is 1000 times faster than that of the free-phase transport ( = ⁄ =1,000) for case LA. 59 • In nanoporous materials where mass transport on the surface can occur across the interface between mesopores and micropores (e.g., activated carbon or microporous silica), the diffusion in the free phase and the surface phase determines, in parallel, the overall mass transfer processes when the gas concentration remains constant at the boundary (see examples for LA and UA). However, when the gas concentration at the boundary is variable (see examples for LB and UB), reflecting a transient in the larger pores, surface diffusion contributes only marginally to the overall mass transfer if the sorption is slower than or equivalent to the free gas diffusion rate. This is true even if the surface diffusivity is several orders of magnitude larger than the diffusivity in the free gas phase. • In systems with isolated surface boundaries (cases LC, LD, UC, and UD), which are representative of natural porous materials (heterogeneous surface chemistry), such as shales with a hierarchical pore structure, surface diffusion is marginal and can be disregarded under two conditions: 1) when the sorption rate is comparable to that of free gas diffusion, and 2) when the surface diffusivity is less than the free phase diffusivity (potentially relevant for high-temperature settings). In other conditions, the contribution from surface diffusion to the total transport in micropores may not be ignored. • Surface diffusion contributes substantially to the overall mass transfer process in situations where sorption is fast (and approximated accurately by an isotherm) across all examples considered in this work. However, in such settings, it becomes challenging to differentiate surface diffusion from free gas diffusion based on observations from e.g., pressure pulse-decay measurements. 60 • The contribution of surface diffusion to overall mass transport also depends on rock properties and sorption parameters, including porosity, skeleton density, sorption capacity, and adsorbate phase density. The impact of the various parameters can be effectively investigated via the dimensionless groups/parameters defined in this study. • In practical applications, the estimation of diffusivities (surface and free gas) can be subject to significant uncertainty without prior knowledge of the sorption behavior (kinetics or isotherm). Therefore, separate and detailed assessments of the dynamic sorption behaviors (using small samples to eliminate transport limitations) and the free-phase diffusivity (using helium as a probe gas) are recommended to promote accurate estimations of the diffusivity for surface transport. In summary, this study highlights the importance of delineating sorption kinetics and surface diffusion, including their interplay, during mass transfer in nanoporous materials. It is recommended to perform separate measurements of sorption behaviors to guide the assumptions we make in modeling and interpretation of experimental observations that reflect the combined contributions of sorption and transport in the free gas and adsorbate phases. 2.5. Appendix Appendix A. Number of segments of 1D homogeneous model The impact of number of cells discretized in the model is test using the simple case LA with =10 and =100. The minimum segments (n=200) are selected to save computational 61 costs but still can achieve reliable results (see left and a zoom-in view on the right in Figure A1). Figure A1. Normalized gas amount variations with a range of segment selections Appendix B. Transition from sorption kinetics to isotherms We also test other cases for the transition from sorption kinetics to sorption isotherms by comparing the corresponding results of normalized gas amount over time. For case LB and UB, the threshold value of is 1,000 while that for other cases are =1e5 to 1e6. 62 Figure B1. Transition from sorption kinetics to sorption isotherm according to Appendix C. Variable boundary conditions The boundary concentrations are solved by the DP model (B.C.3 and B.C.4) with default setting, =10 and =1. Figure C1. Boundary variations for the base loading and unloading cases 63 Appendix D. Practical application analysis on cases with isolated surface boundary In the subsequent examples, we examine outcomes of using an isotherm model to interpret a process controlled by sorption kinetics. We generate synthetic data by assuming sorption kinetics (δ =10) and rapid surface diffusion ( = Dsurf/Dfree = 100). These data are then interpreted employing a model that assumes sorption isotherm, and the surface and free gas diffusivities ( and ) were adjusted to match the data. As discussed in section 2.3.7, here we first consider loading and unloading with a constant gas concentration at the boundary (cases LC and UC). Figure D1 compares the artificial data with computational results for this example, and it illustrates good agreements between the model, based on sorption isotherms and the generated data in the total gas amount (R2 > 0.998). The estimated values of are in comparable ranges to the default value, especially for the loading case (LC). Nevertheless, the surface diffusivity estimates are substantially lower, corroborating the insubstantial role of surface phase transport in the overall mass transfer for these scenarios, particularly for the loading case (LC), where the surface diffusion is estimated to be close to zero under the assumption of sorption isotherm application. In such instances, it becomes challenging to dismiss the assumption of sorption isothermal behavior and the absence of surface diffusion. Similar observations are drawn from the cases with variable gas concentrations at the boundary (cases LD and UD, R2 > 0.999), as depicted in Figure D2: The history-matching analysis yields free-phase diffusivities that are within a plausible range, though the estimates for surface diffusivities are fraught with considerable uncertainty. 64 These examples further demonstrate that the assumption of a sorption isotherm may not always serve as optimal for interpreting mass transfer processes in nanoporous materials. Figure D1. Estimation of and from artificial loading and unloading data (LC and UC with =10 and =100, were used to generate the artificial data) Figure D2. Estimation of and from artificial loading and unloading data (LD and UD with =10 and =100, were used to generate the artificial data) 65 2.6. References Ambrose, R. J., Hartman, R. C., Diaz Campos, M., Akkutlu, I. Y., and Sondergeld, C., 2010. New pore-scale considerations for shale gas in place calculations. SPE Paper 131772 Presented at the SPE Unconventional Gas Conference, 23-25 February, Pittsburgh, Pennsylvania. Ash, R., Barrer, R.M., Pope, C.G., 1963. Flow of adsorbable gases and vapours in a microporous medium. II. Binary mixtures, in: Proc. R. Soc. A Math. Phys. Eng. Sci., The Royal Society, pp. 19–33. Barenblatt G.I., Zheltov, I.P., and Kochina, I.N., 1960. Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks. Journal of Applied Mathematics and Mechanics, 24, 1286-1303. doi: https:// doi.org/10.1016/0021-8928(60)90107-6. Carman, P. C., Malherbe, P. le R., 1950. Proc. Roy. Soc. A, 203, 165. Carman, P. C., Raal, F. A., 1951. Diffusion and Flow of Gases and Vapours through Micropores. III. Surface Diffusion Coefficients and Activation Energies. Proc. Roy. Soc. Lond. A 209 (1096): 38–58. http://dx.doi.org/10.1098/rspa.1951.0186. Chen, Y. D. and Yang, R. T., 1991. Concentration Dependence of Surface Diffusion and Zeolitic Diffusion. AIChE J. 37 (10): 1579–1582. http:// dx.doi.org/10.1002/aic.690371015. Cui, X., Bustin, A. M. M., Bustin, R. M., 2009. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids, 9(3), 208-223. Curtis, J.B., 2002. Fractured shale-gas systems. AAPG bulletin, 86(11), pp.1921-1938. Damkohler, G., 1935. Z. Phys. Chem. A, 174, 222. Do, D. D. and Do, H. D., 2000. Analysis of Dual Diffusion and Non-Linear Adsorption Isotherm with a Time Lag Method. Adsorption 6 (2) 111-123. https://doi.org/10.1023/A:1008959517075 66 Du, F. and Nojabaei, B., 2020. Estimating diffusion coefficients of shale oil, gas, and condensate with nano-confinement effect, Journal of Petroleum Science and Engineering, Volume 193, 107362, ISSN 0920-4105, https://doi.org/10.1016/j.petrol.2020.107362 Etminan, S. R., Javadpour, F., Maini, B. B., et al. 2014. Measurement of Gas Storage Processes in Shale and of the Molecular Diffusion Coefficient in Kerogen. Int. J. Coal Geol. 123 (1 March): 10–19. http://dx.doi.org/10.1016/j.coal.2013.10.007. Fathi, E., and Akkutlu, I. Y., 2012. Multiscale Gas Transport in Shales With Local Kerogen Heterogeneities. SPE J. 17 (04): 1002–1011. https://doi.org/10.2118/146422-PA Fathi, E., and Akkutlu, I. Y., 2013. Lattice Boltzmann Method for Simulation of Shale Gas Transport in Kerogen. SPE J. 18 (2013): 27–37. doi: https://doi.org/10.2118/146821-PA Fathi, E., and Akkutlu, I. Y., 2014. Multi-Component Gas Transport and Adsorption Effects during CO2 Injection and Enhanced Shale Gas Recovery. Int. J. Coal Geol. 123 (1 March): 52–61. http://dx.doi.org/10.1016/j.coal.2013.07.021. Gilliland, E.R., Baddour, R.F., Russell, J.L., 1958. Rates of flow through microporous solids. AICHE J. 4, 90–96. Guo, L., Peng, X., Wu, Z., 2008. Dynamical Characteristics of Methane Adsorption on Monolith Nanometer Activated Carbon. J. Chem. Ind. Eng. (China) 59, 2726−2732. Higashi, K., Ito, H. and Oishi, J., 1963. Surface Diffusion Phenomena in Gaseous Diffusion, I. Surface Diffusion of Pure Gas. J. Atomic Energy Soc. Japan. 5: 846–853. Hill, T. L., 1956. Surface Diffusion and Thermal Transpiration in Fine Tubes and Pores, 1956. J. Chem. Phys., 25, 730. Hwang, S., Kammermeyer, K., 1966. Surface diffusion in microporous media, Can. J. Chem. Eng. 44, 82–89. Kainourgiakis, M. E., Stubos, A. K., Konstantinou, N. D., Kanellopoulos, N. K, Milisic, V., 1996. A Network Model for the Permeability of Condensable Vapours through Mesoporous Media. J. Membrane Sci. 114 (2): 215–225. https://doi.org/10.1016/0376- 7388(95)00322-3. 67 Kang, S. M., Fathi, E., Ambrose, R. J., et al. 2011. Carbon Dioxide Storage Capacity of Organic-Rich Shales. SPE J. 16 (4): 842–855. SPE134583-PA. http://dx.doi.org/10.2118/134583-PA. Kärger, J., Ruthven, D. M. and Theodorou, D. N., 2012. Diffusion in Nanoporous Materials. Wiley, New York. DOI:10.1002/9783527651276. Kärger, J., Ruthven, D.M., 2016. Diffusion in nanoporous materials: fundamental principles, insights and challenges. New Journal of Chemistry, 40(5), pp.4027-4048. Kikkinides, E. S., Tzevelekos, K. P., Stubos, A. K., Kainourgiakis, M. E., Kanellopoulos, N. K., 1997. Application of effective medium approximation for the determination of the permeability of condensable vapours through mesoporous media, Chem. Eng. Sci. 52, 2837–2844. Langmuir, I, 1918. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40 (9), 1361–1403. https://doi.org/10.1021/ja02242a004. Lim, K.T., and Aziz, K., 1995. Matrix-Fracture Transfer Shape Factors for Dual-Porosity Simulators. J Pet Sci & Eng 13 (3): 169–178. doi: https://doi.org/10.1016/0920- 4105(95)00010-F. Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023a. Investigation of Methane Mass Transfer and Sorption in Marcellus Shale Under Variable Net Stress. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211846. Lyu, Y., Dasani, D., Tsotsis, T.T., Jessen, K., 2021. Characterization of shale using Helium and Argon at high pressures, Journal of Petroleum Science and Engineering, doi: https://doi.org/10.1016/j.petrol.2021.108952. Lyu, Y., Elkady, Y., Kovscek, A.R., Jessen, K., 2023b. Analysis of Gas Storage and Transport in Shale using Pressure Pulse Decay Measurements with He, Kr and CO2. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211951. Medveď, I., Černý, R., 2011. Surface diffusion in porous media: A critical review, Microporous and Mesoporous Materials, Volume 142, Issues 2–3, Pages 405-422, ISSN 1387-1811, https://doi.org/10.1016/j.micromeso.2011.01.015. 68 Petropoulos, J.H., 1996. Langmuir 12, 4814–4816. Ren, J., Guo, P., Guo, Z., Wang, Z., 2015. A lattice Boltzmann model for simulating gas flow in kerogen pores. Transport Porous Media 106 (2), 285–301. Sakhaee-Pour, A. and Bryant, S. L., 2012. Gas Permeability of Shale. SPE Res Eval & Eng 15 (4): 401–409. SPE-146944-PA. http://dx.doi.org/10.2118/146944-PA. Sigal, R. F., 2013. The Effects of Gas Adsorption on Storage and Transport of Methane in Organic Shales. Oral presentation of paper SPWLA2013-Y given at the SPWLA 54th Annual Logging Symposium, New Orleans, 22–26 June. Song, W., Yao, J., Ma, J., Sun, H., Li, Y., Yang, Y., Zhang, L., 2018. Numerical simulation of multiphase flow in nanoporous organic matter with application to coal and gas shale systems. Water Resour. Res. 54 (2), 1077–1092. Sun, Z., Li, X., Shi, J., Zhang, T., Sun, F., 2017. Apparent permeability model for real gas transport through shale gas reservoirs considering water distribution characteristic. Int. J. Heat Mass Tran. 115, 1008–1019. Volmer, M. and Adhikari, G. 1925. Z. Phys. 35, 170. Wang, Y., Zhu, Y., Liu, S. and Zhang, R., 2016. Pore characterization and its impact on methane adsorption capacity for organic-rich marine shales. Fuel, 181, pp.227-237. Warren, J.E., and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. doi: https:// doi.org/10.2118/426-PA. Wasaki, A. and Akkutlu, I. Y., 2014. Permeability of Organic-rich Shale. Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, 27–29 October. SPE170830-MS. http://dx.doi.org/10.2118/170830-MS. Wentworth, C. K., 1944. Amer. J. Sci., 242, 478. Weniger, P., Kalkreuth, W., Busch, A. and Krooss, B.M., 2010. High-pressure methane and carbon dioxide sorption on coal and shale samples from the Paraná Basin, Brazil. International Journal of Coal Geology, 84(3-4), pp.190-205. 69 Wicke, E. and Kallenbach, R., 1941. Kolloidzschr. 97, 135. Wicke, E. and Voigt, U., 1947. Angew. Chem. B, 19, 94. Wicke, E., 1939. Kolloidzschr. 86, 167. Wu, K., Li, X., Guo, C., Wang, C., Chen, Z., 2016. A Unified Model for Gas Transfer in Nanopores of Shale-Gas Reservoirs: Coupling Pore Diffusion and Surface Diffusion. SPE J. 21: 1583–1611. doi: https://doi.org/10.2118/2014-1921039-PA. Wu, K., Li, X., Wang, C., Yu, W., Chen, Z., 2015. Model for Surface Diffusion of Adsorbed Gas in Nanopores of Shale Gas Reservoirs. Ind Eng Chem Res 54:3225–36. https:// doi.org/10.1021/ie504030v. Xiong, X., Devegowda, D., Michel, G., et al. 2012. A Fully-Coupled Free and Adsorptive Phase Transport Model for Shale Gas Reservoirs Including Non-Darcy Flow Effects. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8–10 October. SPE-159758-MS. http://dx.doi.org/10.2118/159758-MS. Yang, B., Kang, Y., You, L., Li, X., Chen, Q., 2016. Measurement of the surface diffusion coefficient for adsorbed gas in the fine mesopores and micropores of shale organic matter, Fuel 181, 793–804. Yang, R.T., Fenn, J.B., Haller, G.L., 1973. Modification to the Higashi model for surface diffusion, AIChE J. 19 (1973) 1052–1053. Yang, Y. and Liu, S., 2020. Review of shale gas sorption and its models. Energy & Fuels, 34(12), pp.15502-15524. Zhang, J., Raslan, M., Wu, C., Jessen, K., 2022. "A Generalized Dynamic Transfer Function for Ultra-Tight Dual-Porosity Systems." Paper presented at the SPE Western Regional Meeting, Bakersfield, California, USA, April 2022. doi: https://doi.org/10.2118/209324-MS. Zimmerman, R.W., Chen, G., Hadgu, T. et al, 1993. A Numerical Dual-Porosity Model with Semianalytical Treatment of Fracture/Matrix Flow. Water Res Res 29 (7): 2127–2137. doi: https://doi.org/10.1029/93WR00749. 70 71 CHAPTER 3.An integral model of gas diffusion and sorption in tight dual-porosity systems2 3.1. Introduction The dual-porosity (DP) and triple-porosity (TP) models are widely applied approximations used to represent flow and transport in fractured systems. The concept was initially proposed by Barenblatt et al. (1960) for fissured rocks, and subsequently adopted by the petroleum industry through the work of Warren and Root (1963). They visualized the concept via uniform sets of fractures embedded in a matrix to characterize naturally fractured formations. The popularity stems from a reduced computational cost, and the models have been refined continuously over the years (Kazemi et al., 1976; Blaskovich et al., 1983; Kuchuk and Biryukov, 2014). Within a DP system, fractures and matrix represent distinct characteristics, with the former serving as boundaries to the latter. Shales are often viewed as a multi-porosity system, wherein gas production derives initially from free gas in fracture networks followed by both free gas and adsorbed gas in the shale matrix. We note here that the shale matrix itself is often classified as a dual-porosity system. While gas transports in fractures and pores as free gas via viscous flow and Knudsen diffusion (Javadpour, 2009; Alnoaimi and Kovscek, 2019; Lyu et al., 2023a), a 2 Parts of this chapter have been taken verbatim from our manuscript: Lyu, Y., & Jessen, K.. An integral model of gas diffusion and sorption in tight dual-porosity systems (will be submitted to Transport in Porous Media). 72 clear understanding of the interaction/exchange between fractures and the shale matrix is fundamental to studying shale gas transport and storage. The mass transfer rate between fractures and matrix typically necessitates introduction of a shape factor, σ, representing interface (surface) area per unit of rock volume. Lim and Aziz (1995) derived shape factors for DP systems from analytical solutions to the pressure diffusion problem in various idealized geometries. Zhang et al. (2022) introduced a generalized dynamic transfer function that accurately predicts pressure response in ultra-tight DP formations, independent of geometry, based on the approximate closed-form solution of Vermeulen (Vermeulen, 1953). In addition to free gas mass transfer, the phenomenon of adsorption plays a central role in gas transport and storage within shales (Javadpour, 2009; Akkutlu et al., 2018; Lyu et al., 2023b). Methane (CH4), for instance, is often analyzed using the Langmuir isotherm model (Langmuir, 1918), which posits a relationship between surface coverage and the rates of adsorption/desorption, ultimately leading to equilibrium (Lyu et al., 2023a). The BrunauerEmmett-Teller (BET) isotherm model (Brunauer et al., 1938) extends on the ideas of Langmuir to multilayer adsorption of gases of relevance to e.g., carbon dioxide (CO2). While the ideas of Langmuir provide for modeling of kinetics during adsorption and desorption processes, a similar representation of kinetics is not convenient for multilayer sorption via the BET model. In such settings (e.g., CO2 adsorption on shale), the Linear Driving Force (LDF) model is therefore often used to provide a mechanistic representation of the kinetics during the sorption process. The LDF model assumes that the rate of adsorption or desorption is proportional to the difference between the equilibrium and 73 actual adsorbed amounts and hence combines the isotherm model with a rate constant (Zhang et al., 2011). Helium (He), a non-adsorbing reference gas (Lyu et al., 2021), is commonly used in tandem with other gases to study the sorption potential of a given porous sample. For gases, including CH4 and CO2, which adsorb onto both mineral surfaces and the organic constituents of shale, previous work has demonstrated that the free gas in shale represents only a fraction of the total storage capacity (Nuttall et al., 2005; Vega et al., 2014; Aljamaan et al., 2017; Elkady and Kovscek, 2020). The measurement of excess adsorption provides a path to estimate the absolute adsorbed gas by assuming/assigning a density of the adsorbate phase, as direct laboratory measurement of absolute adsorption is not readily available (Lyu et al., 2021). The estimation of absolute adsorption, which is crucial for assessing the adsorbed phase volume within pore spaces (Lin and Kovscek, 2014), becomes more relevant when pore pressure increases (i.e., sorption increases). We have previously demonstrated the influence of adsorbate density on the estimation of gas-inplace in shale formation (Lyu et al., 2023a), and the related impact on mass transfer due to sorption induced reductions in the pore volume (Lyu et al., 2023b). Furthermore, the complexity of sorption kinetics and the differences between gas adsorption and desorption isotherms (sorption hysteresis) can greatly influence modeling of shale gas extraction processes (Lyu et al., 2023a). Overall, despite the extensive research activities in this areas, accurate representation of gas transport dynamics in shales still poses a challenge, attributed in part to gaps in our understanding of gas transport and storage combined with the intricate architecture of the shale matrix. 74 In our previous work (Lyu et al., 2021; Lyu et al., 2023a; Lyu et al., 2023b), a tripleporosity model (TPM) was developed to interpret and predict laboratory-scale observations based on the assumption that sorption is slower or similar to the transport rate in free gas phase transport rate. In this work, we formulate and validate a volume averaged representation of mass transfer and sorption in the meso/microporous segments of shale to further facilitate the application of DPM and TPM representations of fractured shale cores. Specifically, we utilize analytical solutions for transient pressure diffusion with a constant diffusion coefficient (see e.g., Crank, 1975) and the closed-form approximation of Vermeulen (1953) to estimate shape factors and a Vermeulen parameter, denoted as , for various geometries (see details below). This facilitates the development of an integrated model capable of accurately interpreting gas behaviors in tight DP/TP systems, ranging from inert gases (e.g., He) to strongly adsorptive gases (e.g., CO2). In the subsequent sections, we start by introducing the model formulation, followed by a discussion of how to estimate relevant shape factors via analytical and numerical solutions. The model is then validated with results from finite difference calculations on fully discretized domains. We consider a range of different settings and represent the transition from slow sorption kinetics to fast sorption kinetics that are approximated accurately by a sorption isotherm. Finally, we summarize our findings and provide a relevant set of conclusions. 3.2. Formulations and Methods 3.2.1. Transport in the free phase gas Mass conservation during transport in the gas phase of a single-porosity system can be stated as: 75 + ∇ ∙ = 0, (3.1) where is the porosity of the porous medium, (mol/m3 ) is the free gas concentration, (s) is time, and (mol/m2 /s) is molar flux. The average gas concentration ̅(mol/m3 ) in the pore space of the domain can then be written as: ̅ + 1 ∫ ∇ ∙ = 0. (3.2) where (m3 ) is domain volume. We use bar notation throughout this study to represent average quantities over the domain. In the following sections, we explore how to approximate the volume average of the divergence of the flux (assumed diffusive in nature). 3.2.2. Analytical solutions Analytical solutions for transient diffusion processes with a constant diffusion coefficient (e.g., Eq. 3.2) on unit volume basis are provided for different geometries by Crank (1975). Here, a case of diffusion in a sphere is used to illustrate the approach, while solutions in other geometries are reported in Appendix A. If the fluid concentration (mol/m3 ) is uniformly distributed at t=0, and a constant concentration (mol/m3 ) is imposed at the boundary, the concentration (x,t) within the domain can be calculated from: − − = 1 + 2 ∑ (−1) (− 2 2 / 2 ) ∞ =1 . (3.3) The average concentration in the domain at time can be evaluated from: 76 ̅− − = ∞ = 1 − 6 2 ∑ 1 2 exp(− 2 2 / 2 ) ∞ =1 . (3.4) Here, (mol) is the total amount of gas that has entered the domain at time t, ∞ (mol) represents the amount after infinite time, (m2 /s) is the constant diffusion coefficient, while (m) is the radius of the domain and (m) is the location within the domain. 3.2.3. Generalized Vermeulen approximation Vermeulen et al. (1953) introduced a closed-form approximation to the diffusion problem in spherical geometry: = ̅− − = (1 − (− 2 2 )) 1/2 . (3.5) In this study, we generalize the approach to include planar and cylindrical geometries. Spherical geometry is used for illustration, while results for planar and cylindrical geometries are provided in Appendix B. The generalized form of (Eq. 3.5) is given by Zhang et al. (2022): = [1 − (−)] 1/ , (3.6) where (1/s) is considered the inverse of the characteristic time for diffusion. The two parameters and (m-2 ) can be determined from analytical solutions (or fine-grid numerical solutions) to the diffusion problem for a given geometry (e.g., Crank, 1975). By differentiation of Eq. 3.6 wrt. time and subsequent elimination of time, we arrive at 77 ̅ = ( − ̅)(), (3.7) () = 1 − −̅ ( 1− − ). (3.8) Figure 3.1 compares the series solution (Crank, 1975) to the generalized Vermeulen approximation for spherical geometry (see Appendix B for planar and cylindrical geometries), while Table 3.1 reports the model parameters (a, ) for all three geometries. The dimensionless time is discussed further in Section 3.2.6. Figure 3.1. Comparison of series solution (Crank, 1975) and generalized Vermeulen approximation – spherical geometry Table 3.1. Parameters for the generalized Vermeulen approximation Geometry a Planar 1.57 2.18 78 Cylindrical 1.91 4.99 Spherical 2.07 9.01 Zimmerman et al. (1993) demonstrated for the original form (Eq. 3.5) that the approximation works well for variable boundary conditions. 3.2.4. Sorption formulations 3.2.4.1 Sorption isotherms For gases with a strong affinity to the shale surfaces (e.g., CO2), the BET adsorption model (Brunauer et al., 1938) is often used to represent multilayer sorption. The BET model can be written as: = , ∙ = , ( ) (1− ) [ 1−(+1)( ) +( ) +1 1+(−1)( )−( ) +1 ]. (3.9) Here, (mol/kg) is the absolute sorption that depends on the maximum monolayer adsorption capacity of the shale , (mol/kg) and the fractional surface coverage (that can be greater than one for multilayer sorption) at a given high pressure (psia). (psia) is the saturation pressure at the system temperature (= 1070 psia for CO2, corresponding to a temperature of 343K), is a dimensionless constant, related to the heat of adsorption, and is the maximum number of adsorbed layers. 3.2.4.2 Sorption kinetics In this work, we use the LDF model to describe the transient behavior on the surface as follows: 79 ̇ − ̇ = = (, − ), (3.10) where ̇ (mol/kg/s) and ̇ (mol/kg/s) are rates of adsorption and desorption, respectively, (1/s) is the adsorption rate constant, and , (mol/kg) is the absolution sorption at equilibrium for a given pressure (obtained from an isotherm). 3.2.5. Model formulations For a single porosity system, the interplay of transport and adsorption can be expressed as Free phase: + ̇ − ̇ + ∇ ∙ = 0. (3.11) Adsorbed phase: (1 − ) + ̇ − ̇ = 0. (3.12) Here, denotes the original porosity of the system, (1 − ) represents the skeleton volume and (kg/m3 ) is skeleton density. Once the adsorbed phase density (kg/m3 ) is specified, the change in porosity due to the adsorbate volume is given by: = − (1− ) . (3.13) The overall conservation equation for the domain can then be expressed as: + (1 − ) + ∇ ∙ = 0. (3.14) By integrating over the domain V, Eqs. 3.10, 3.13, and 3.14 can be rewritten as: 80 ̅ = (̅ , − ̅ ), (3.15) ̅= − (1− ) ̅ , (3.16) ̅̅̅ + (1 − ) ̅ + 1 ∫ ∇ ∙ = 0. (3.17) If the characteristic time for diffusion is much smaller than that for sorption, i.e., ≪ , ̅̅̅ ≈ − 1 ∫ ∇ ∙ . (3.18) Next, we assume that ̅̅̅ ≅ ̅̅, which is strictly true if the volume of the adsorbate is ignored (i.e., no sorption or infinite adsorbate density). With an adsorbed phase gradually occupying more volume inside the pores, this approximation becomes less accurate (see validation via numerical calculations in Appendix C). Insert Eq. 3.7 into Eq. 3.18: ̅ ̅ + (1 − ) ̅ = ̅( − ̅)(). (3.19) This formulation can be extended to a DP representation of the shale matrix. We consider the matrix as two distinct and communicating pore scales, including mesopores (m) and micropores (μ). Gas will first diffuse and sorb in the mesopores, followed by the micropores. The total bulk volume (Vt) and the total pore volume (Vp) of the porous medium are divided into a mesoporous and a microporous segment: = , + , , (3.20) 81 = , + ,. (3.21) The porosity of the mesoporous (εm) and microporous (εμ) segments is then defined by: = , , = , , (3.22) and the total porosity of the porous medium is given by: = + . (3.23) Figure 3.2. Schematics of gas transport in the DP system (not scaled) Eq. 3.19 can then be rewritten for the mesopores: ̅ ̅ + (1 − − ) + ̅ , = ̅ ( − ̅)() − ̅(̅ − ̅ )(), (3.24) and for the micropores: 82 ̅ ̅ + (1 − − ) + ̅ , = ̅(̅ − ̅ )(). (3.25) Sorption in mesopores or micropores and the corresponding change in porosity due to the adsorbed phase are as describes as: ̅ , = (̅ ,, − ̅ ,), (3.26) ̅ = − (1− − ) + ̅ , , = . (3.27) Next, we introduce dimensionless variables and characteristic times: ̅ = ̅ , ̅ , = ̅ , , , ̅ , = ̅ = (̅ ) , ̅ , = ̅ (3.28) = 1 , , = ∙ , = , , , = , (3.29) Upon substitution in Eqs. 3.24-3.27, we arrive at ̅ + (−̅) ̅ , ̅ , = ( − ̅)() − ̅ , ̅ , ( 1 − 1) (̅ − ̅ )(), (3.30) ̅ + (−̅) ̅ , ̅ , = (̅ − ̅ )(), (3.31) ̅ , = (̅ ,, − ̅ ,), = , , (3.32) ̅ , = 1 − ̅ , , = , , (3.33) with 83 = (1− − ), , = (1− − ), , (3.34) = (1− − ) (1−), , = (1− − ) (1−), , (3.35) = + . (3.36) Here ( = , ) represents the ratio of maximum storage capacities between the surface and free gas phase, and ( = , ) denotes the maximum volume of the sorbed phase in the pore space for mesopores and micropores, respectively. As one may notice here, = and = as the sorption in this dual-porosity system is assumed to be uniform. Further laboratory efforts are required to separate the sorption behavior in the two domains. By formulating the gas transport and sorption governing equations this way, the process of gas diffusion coupled with sorption can be solved integrating a set of coupled ordinary differential equations (ODE’s). Additionally, the dimensionless groups reduce the complexity of studying how different parameters will affect the validity of the underlying assumptions. 3.2.6. General formulations To validate the proposed method proposed above, we compare the volume averaged solution to implicit finite difference calculations with the governing partial differential equations (Eqs. 3.37-3.40). This method is referred to as PDE for solving direct partial differential equations in this study to differ it from the proposed method. 84 + (1 − − ) + , = 1 ( ) − 1 ( ), (3.37) + (1 − − ) + , = 1 ( ), (3.38) , = (,, − ,), = , (3.39) = − (1− − ) + , , = , (3.40) where = 0, 1, 2 for planar, cylindric, and spherical geometries, respectively. Based on the previously defined dimensionless groups and characteristic length , , Eqs. 3.37-3.40 can be rewritten into dimensionless form as: , + , = 1 , , (, , , ) − (1 − 1 ) 1 , , (, , , ), (3.41) , + , = 1 , , (, , , ), (3.42) , = (,, − ,), = , (3.43) , = 1 − , , = . (3.44) 3.2.7. Initial and boundary conditions for both proposed and general formulations In practical cases, natural gas production and CO2 injection are achieved by a controlled pressure or rate in a well. Thus, a constant boundary condition is applied in this work to 85 mimic a CO2 injection process with constant pressure in the well. Accordingly, the mesoporous region is exposed to constant pressure, , by default = 1. To simplify the modeling work, we assumed a vacuum environment to begin with for the system, i.e., = 0. 3.3. Results and Discussion 3.3.1. Sorption The sorption behavior of gas in this work is represented by the LDF model, where the BET sorption isotherm model provides the equilibrium states. Laboratory measurements of the CO2 adsorption isotherm on an Eagle Ford shale core at 70 C using the pressure-pulse decay technique were used to estimate model parameters for the BET isotherm. As Figure 3.3 illustrates, the BET model provides a reliable interpretation of multilayer adsorption on the shale core. The relevant sorption parameters are summarized in Table 3.2 and are utilized in the following numerical calculations. The characteristic time for sorption (1/) is assumed to be half of that for methane (see Lyu et al., 2023a). 86 Figure 3.3. BET interpretation of CO2 adsorption isotherm on an Eagle Ford shale core at 70 C Table 3.2. Evaluated BET sorption model parameters , (mol/g) 1.53e-3 2.5e-2 2 Based on and the maximum concentration , which corresponds to CO2 at 100 bars, approximately 50% of the total gas inside the system is stored in the adsorbed phase at equilibrium. 3.3.2. Proposed method (ODE) vs. Finite difference method (PDE) The default parameters/properties used in the following calculation examples include: the maximum gas pressure set to 100 bars, isothermal gas concentrations from the NIST database, as well as rock and sorption information provided by Lyu et al. (2023b): = 0.10, = 0.8, = 2.60 × 10−3 kg/m3 . 87 The total amount of gas that is stored in the shale consists of a free phase and an adsorbed (surface) phase, thereby, concentrations in both phases are compared over time. In the following sections, normalized concentrations (̅ and ̅ , ) calculated by ODE are plotted over dimensionless time for varying ( = , ), representing a range of sorption rates. Based on the identical settings, i.e., parameters and conditions, the benchmark for the results is obtained from the PDE with 50 segments for the mesoporous region and 50 micropores connected to each mesopore, which leads to 2,500 microporous cells overall. The results are then normalized and plotted as shown in Figure 3.4. As is based on the characteristic time for diffusion in the micropores, it is worth noting that the different dimensionless time in subsequent results represents how long the process needs to establish equilibrium relative to free phase diffusion. Figure 3.4 demonstrates the comparisons of both free gas and adsorbed gas accumulation in the DP system over time between solutions derived by PDE and ODE. In the case of substantially faster diffusion than sorption, = 0.001 & = 0.01, the sorbed phase would be barely observable during free gas accumulation. This example could approximate the loading process of a non-sorbing, such as helium. As expected, gas accumulation in the free phase and the adsorbed phase speeds up with faster sorption kinetics, i.e., a greater . Until = 10 & = 100, the acceleration phenomena diminish away as the sorption kinetics is close to equilibrium states, i.e., sorption isotherms. This is consistent with our previous observations (Section 2.3.1). When sorption equilibrium status can be achieved rapidly where the characteristic time for sorption is greatly shorter than that for diffusion 88 (e.g., = 100 & = 1000), the fluid accumulations in the free gas and adsorbed phases are simultaneous. Thus, this case mimics the coupled modeling of gas diffusion with sorption isotherms. Figure 3.4. Comparison of PDE (symbol) and ODE (line) solutions to diffusion and sorption in planar geometry from = 0.001 to = 100 89 As illustrated in Figure 3.4, the proposed method (ODE) can deliver reliable results that are similar to those solved by a fully discretized method (PDE), for cases ranging from slow or non-sorbing gases (left-top) to fast-sorbing gases (right-bottom). However, the proposed method avoids the necessary understanding of explicit geometric properties of the system, such as tortuosity, fracture networks, etc. Moreover, it would substantially minimize the required computation cost for a larger-scale simulation, as tabulated in Table 3.3. It is true that the grid cells utilized in the PDE method may outnumber the required number of segments for scenarios included in this study, which means similar results can be provided by the PDE method using a coarser grid of discretization. However, the grid settings employed in this work aim to represent general cases at a more reservoir-related scale. Table 3.3. Computation costs comparison for the proposed method vs. fully discretized solution Method ODE PDE Computation time (s) 8.6 1724.5 Grid cells 2 50+50*50 Some slight difference in results on micropores between ODE and PDE in the early stages can be noticed in cases when the sorption process is faster than free phase diffusion. This observation may be attributed to the inefficiency of ODE solvers in accurately capturing the transition from free gas to adsorbed gas within one single solver. Therefore, we proposed a workflow to guide this transition in a more explicit way and through iteration; see details in Section 3.3.2.2. 90 3.3.2.1 Sensitivity check In this subsection, the performance sensitivity of the proposed method is evaluated by manipulating the dimensionless groups of parameters, i.e., , and ( = , ). To simplify the discussion and keep a concise writing style, we have focused on two specific cases with less accuracy (namely = 0.1 and = 1). To boost the impact of sorption, a combination of greater and is employed, which is denoted , and ,, respectively. The opposite option would be a combination of less and , as , & , . This exercise modifies these two dimensionless groups by doubling or halving their default values, i.e., , = 2, & , = 2,, , = 0.5, & , = 0.5,. In the final state, it is possible for the adsorbed phase to constitute 70% and the gas phase to contribute 30% of the total gas within the system. As expected, the accuracy of the proposed method was reduced with greater adsorption contributions included in the system (, and ,). On the other hand, when the gas is loaded or injected at a higher pressure, i.e., higher ( ) at the boundary, the reliability of the proposed method would be enhanced, as observed in the cases with , and , . 91 Figure 3.5. Sensitivity check with 2, & 2, Figure 3.6. Sensitivity checks with 0.5, & 0.5, Besides, two alternative values of are employed to investigate different splits of pore volume into mesopores and micropores. When the system is richer in micropores, there is a noticeable divergence between the results obtained from ODE the PDE analysis. In cases when the system is mostly microporous, e.g., = 0.2 , noticeable gaps in the accumulations of free phase can be observed, as shown in Figure 3.8. These errors can be attributed to the distinction between the conditions of the generalized Vermeulen’s approximation and the lower-level porosity system, specifically referring to the presence of micropores in this particular situation. The former approximates fluid diffusion with a 92 constant boundary, while the latter has a boundary that keeps updating during the process. Furthermore, the aforementioned boundary receives updates under the average concentration observed within the mesopores. Therefore, a method that involves an iterative workflow is recommended to improve the performance of such cases (see Appendix D). Figure 3.7. Sensitivity checks with = 0.5 Figure 3.8. Sensitivity checks with = 0.2 Additionally, the reliability of the proposed method for other geometries, i.e., cylindrical and spherical, is also examined, as shown in Figure 3.9 and Figure 3.10. The proposed 93 model (ODE) also accurately delineates the gas transport and storage process as the general formulation model (PDE). Figure 3.9. Sensitivity checks for a cylindrical system Figure 3.10. Sensitivity checks for a spherical system For unknown geometries or heterogeneous systems, the parameters of generalized Vermeulen’s approximation and should be used as adjustable parameters. 94 3.4. Summary and Conclusions In this work, we have introduced a general dual-porosity model that integrates fluid diffusion and sorption kinetics/isotherms. The present model incorporates diffusion in the free gas phase and dynamic sorption in the adsorbed phase, considering a wide variety of ratios between the characteristic rates (time) of transport in free phase and adsorbate phase. The objective of this study is to advance the applicability of the proposed model by testing its effectiveness with different gases, e.g., from fast diffusing but slow or no sorption gas, helium, to slow diffusing but fast or great sorption gas, CO2. The proposed model leverages the generalized Vermeulen approximation for given geometries, which utilizes the average behavior of the system, which allows us to approximate the partial differential equations (PDEs) into ordinary differential equations (ODEs). These ODEs for mesopores and micropores are coupled with sorption kinetics using a collaboration of the Linear Driving Force model (LDF) and the Brunauer-Emmett-Teller (BET). The averaged adsorbed phase concentration over the domain ̅ ,, is generated based on the average gas concentration in free phase, ̅ , , and then implemented into ODEs ( = represents mesopores or micropores, respectively). To validate the proposed model, the generated results by the model are compared to those obtained by solving the governing partial differential equations using the finite difference method with fully discretized domains. The scenario under investigation involves the loading of CO2 into a dual-porosity system, whereby a constant boundary condition. This particular case mimics the process of CO2 injection into an empty and organic-rich reservoir, which can contribute to carbon sequestration in subsurface formations. 95 The presented comparisons demonstrate the consistent and precise performance of the integral model in precisely representing fluid diffusion and sorption in a dual-porosity system, especially when the system is cylindrical. Additionally, it serves the purpose when sorption is relatively slower or similar to diffusion in the free phase ( = 0.001 − 1). While sorption and diffusion are more comparable in terms of characteristic rate or sorption is faster than diffusion, e.g., = 1,10,100, or higher, accumulations in both the free and adsorbed phases obtained by the proposed method differ from those obtained by a fully discretized finite difference method. The present study introduces a novel method that extends the feasibility and reliability of approximation to analytical solutions, while also avoiding the need for discretization of complex matrixes of ultra-tight formations. This novel model not only results in cost savings in computation but also offers a valuable approach for upscaling and solving challenges associate with heterogeneous systems. Based on the findings and analysis reported in this study, we recommend some potential areas for future efforts: 1). The investigation of fluid production (unloading) processes holds significant value and deserves examination; 2). A refined workflow that improves performance should be developed; 3). The establishment of a modified version specifically designed for binary or multicomponent mixtures is also practically necessary. 3.5. Appendix Appendix A. Analytical solutions Planar domain: the concentration within the domain can be calculated by: 96 − − = 1 − 4 ∑ (−1) 2+1 exp {− (2+1) 2 2 4 2 } cos (2+1) 2 ∞ =0 . (3.50) The average concentration in the domain at time can be estimated as: ̅− − = ∞ = 1 − ∑ 8 (2+1) 22 exp {− (2+1) 2 2 4 2 } ∞ =0 , (3.51) where (m) is the radius of the domain and (m) is the location within the domain. Cylindric domain: the concentration within the domain can be calculated by: − − = 1 − 2 ∑ exp(− 2)0() 1() ∞ =1 . (3.52) The average concentration in the domain at time can be estimated as: ̅− − = ∞ = 1 − ∑ 4 2 2 exp{− 2 } ∞ =1 , (3.53) where (m) is the radius of the domain, and the method to find 0 () and 1 () can be found in Crank’s book (Crank, 1975). Appendix B. Matching results for planar and cylindrical geometries 97 Figure B2. Comparison of series solutions (Crank, 1975) with the generalized Vermeulen approximation for planar and cylindrical geometries Appendix C. Validation of assumptions The assumption of ̅̅̅ ≅ ̅̅is validated by numerical calculations, and the results are normalized by dividing for a clearer illustration. 98 Figure C2. Comparison of normalized value for ̅̅̅ (cycles) vs. ̅∙ ̅ (dash line) Appendix D. Proposed method for improvement A refined workflow is introduced here to enhance the performance of the proposed model. First, Eqs. 3.30 and 3.31 can be rearranged as: [1 + (−̅) ̅ , ̅ ̅ , ] ̅ = ( − ̅)() − ̅ , ̅ , (1 − 1 ) (̅ − ̅ )(), (3.45) [1 + (−̅) ̅ , ̅ ̅ , ] ̅ = (̅ − ̅ )(). (3.46) Further introduce dimensionless variables = ̅ , ̅ , = (−̅ ) ̅ , . (3.47) 99 Hence, Eqs. 3.45 and 3.46 can be finalized as: [1 + ] ̅ = ( − ̅)() − ̅ , ̅ , (1 − 1 ) (̅ − ̅ )(), (3.48) [1 + ] ̅ = (̅ − ̅ )(). (3.49) The innovative strategy is to combine analytical solutions and ODE solutions at every step. The workflow is indicated in Figure D1. The basic idea is to take advantage of the analytical solutions (Eq. 3.3), which is capable of providing the gas concentration () profile within the domain. Based upon that, the key parameter ( = ̅ , ̅ ) for the ODE to obtain a reliable average concentration can be updated for each time step. A more detailed explanation is as follows: a. Initial guesses for 0 and 0 b. At each time step, both ODE and Eq. 3.4 are solved, and their results are matched by adjusting the diffusion coefficient c. The concentration profile along the system is provided by solving Eq. 3.3 with the obtained d. Using an ODE solver to compute the adsorption profile based on Eq. 3.44, and then the average ̅ , and ̅ need to be evaluated by integrating the domain V e. The key parameter is calculated and employed in the ODE solver that is built based on Eqs. 3.48 and 3.49, respectively. 100 Figure D3. Schematic of workflow of proposed method With this workflow employed, the results of the cases shown in Figure 3.8 can be improved in performance, as illustrated in Figure D4. The accuracy in simulating the accumulation of free and adsorbed phases in micropores is clearly enhanced (shown in red), which is 101 beneficial from the individual guidance of the analytical solution for the micropores at each time step. Figure D4. Improved results of cases with = 0.2 using proposed workflow 3.6. Nomenclature geometric parameter, dimensionless BET model parameter, dimensionless fluid concentration, mol/m3 initial fluid concentration, mol/m3 boundary fluid concentration, mol/m3 absolute adsorption, mol/kg , absolution sorption at equilibrium, mol/kg , maximum monolayer adsorption capacity, mol/kg diffusion coefficient, m2 /s 102 diffusion coefficient of mesoporous segment ( = ) or microporous segment ( = ), m2 /s porosity, dimensionless original porosity, dimensionless total porosity, dimensionless porosity of mesoporous segment ( = ) or microporous segment ( = ), dimensionless original porosity of mesoporous segment ( = ) or microporous segment ( = ), dimensionless molar flux, mol/m2 domain length, m fluid amount has entered the domain at time t, mol ∞ fluid amount enters the domain after infinite time, mol maximum number of adsorbed layers, dimensionless ̇ rate of adsorption, mol/kg ̇ rate of desorption, mol/kg fluid pressure, bar saturation pressure, bar characteristic sorption rate, 1/s , characteristic radius (length), m radius of a spherical domain, m adsorbed phase density, kg/m3 103 skeleton density, kg/m3 shape factor, m-2 shape factor of mesoporous segment ( = ) or microporous segment ( = ), m-2 time, s dimensionless time, dimensionless characteristic time for diffusion, s characteristic time for sorption, s surface coverage, dimensionless V domain volume, m3 total bulk volume of the system, m3 , bulk volume of mesoporous segment ( = ) or microporous segment ( = ), m3 total pore volume, m3 , pore volume of mesoporous segment ( = ) or microporous segment ( = ), m3 location in domain, m * the use of bar notation to add to any symbols in this study denotes the average behavior of the associated entities 104 3.7. References Akkutlu, I. Y., Efendiev, Y., Vasilyeva, M., & Wang, Y. (2018). Multiscale model reduction for shale gas transport in poroelastic fractured media. Journal of Computational Physics, 353, 356–376. doi: 10.1016/j.jcp.2017.10.023 Aljamaan H, Alnoaimi K, Kovscek AR (2013). In-depth experimental investigation of shale physical and transport properties. SPE Paper 1582146 Presented at the SPE Unconventional Resources Technology Conference, 12–14 August, Denver, Colorado. Alnoaimi, K. R., Kovscek, A. R., 2019. Influence of Microcracks on Flow and Storage Capacities of Gas Shales at Core Scale. Transport in Porous Media, 127(1), 53–84. doi: 10.1007/s11242-018-1180-5 Barenblatt, G.I., Zheltov, I.P., Kochina, I.N., 1960. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. of Applied Mathematics and Mechanics 24 (5):1286–1303. DOI:10.1016/0021-8928(60)90107-6. Blaskovich, F.T., Cain, G.M., Sonier, F., Waldren, D., Webb, S.J., 1983. "A Multicomponent Isothermal System for Efficient Reservoir Simulation." Paper presented at the Middle East Oil Technical Conference and Exhibition, Manama, Bahrain, March 1983. doi: https://doi.org/10.2118/11480-MS Brunauer, S., Emmett, P.H., Teller, E., 1938. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60 (2), 309–319. https://doi.org/10.1021/ja01269a023. Crank, J. 1975. The Mathematics of Diffusion, second ed. Claredon Press, Oxford, pp. 44– 103. Elkady, Y., Lyu, Y., Jessen, K., Kovscek, A.R., 2020. Three-dimensional imaging and quantification of gas storativity in nanoporous media via X-rays computed tomography. Energies 13 (23), 6199. https://doi.org/10.3390/en13236199. Elkady, Y., Kovscek, A.R., 2020. Laboratory visualization of enhanced gas recovery in shale. In: Paper Presented at the SPE Annual Technical Conference and Exhibition, Virtual, October. https://doi.org/10.2118/201707-MS. 105 Javadpour, F., 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). Journal of Canadian Petroleum Technology, 48(08), 16–21. doi: 10.2118/09-08-16-DA Kang, S. M., Fathi, E., Ambrose, R. J., et al. 2011. Carbon Dioxide Storage Capacity of Organic-Rich Shales. SPE J. 16 (4): 842–855. SPE134583-PA. http://dx.doi.org/10.2118/134583-PA. Kazemi, M., & Takbiri-Borujeni, A. (2016). Non-equilibrium molecular dynamics simulation of gas flow in organic nanochannels. Journal of Natural Gas Science and Engineering. Kuchuk, F., Biryukov, D., Fitzpatrick, T., 2014. "Rate Transient and Decline Curve Analyses for Continuously (Dual-Porosity) and Discretely Naturally Fractured Reservoirs." Paper presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands. doi: https://doi.org/10.2118/170698-MS Langmuir, I, 1918. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40 (9), 1361–1403. https://doi.org/10.1021/ja02242a004. Lim, K., Aziz, K., 1995. Matrix-fracture transfer shape factors for dual-porosity simulators. J. Petrol. Sci. Eng. 13 (3–4), 169–178. https://doi.org/10.1016/ 09204105(95)00010-F. Lin, W., Kovscek, A.R. Gas Sorption and the Consequent Volumetric and Permeability Change of Coal I: Experimental. Transp Porous Med 105, 371–389 (2014). https://doi.org/10.1007/s11242-014-0373-9 Lyu, Y., Dasani, D., Tsotsis, T.T., Jessen, K., 2021. Characterization of shale using Helium and Argon at high pressures, Journal of Petroleum Science and Engineering, doi: https://doi.org/10.1016/j.petrol.2021.108952. Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023a. Investigation of Methane Mass Transfer and Sorption in Marcellus Shale Under Variable Net Stress. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211846. Lyu, Y., Elkady, Y., Kovscek, A.R., Jessen, K., 2023b. Analysis of Gas Storage and Transport in Shale using Pressure Pulse Decay Measurements with He, Kr and CO2. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211951. 106 Nuttall, B.C., Eble, C.F., Drahovzal, J.A.D., Bustin, R.M., 2005. Analysis of Devonian Black Shales in Kentucky for Potential Carbon Dioxide Sequestration and Enhanced Natural Gas Production. https://doi.org/10.2172/920185. Warren, J.E., and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. doi: https:// doi.org/10.2118/426-PA. Vega, B., Dutta, A., & Kovscek, A. R., 2014. CT imaging of low-permeability, dualporosity systems using high X-ray contrast gas. Transport in porous media, 101(1), 81-97. Vermeulen, T., 1953. Theory for Irreversible and Constant-Pattern Solid Diffusion. Industrial & Engineering Chemistry, 45(8), 1664–1670. doi: 10.1021/ie50524a025 Zhang, J., Raslan, M., Wu, C., Jessen, K., 2022. A generalized dynamic transfer function for ultra-tight dual-porosity systems. In: Paper presented at the SPE Western Regional Meeting, Bakersfield, California, USA. https://doi.org/10.2118/ 209324MS. April. Zhang, Z., Huang, S., Xian, S., Xi, H., Li, Z., 2011. Adsorption Equilibrium and Kinetics of CO2 on Chromium Terephthalate MIL-101. Energy Fuels, 25, 835−842. Zimmerman, R. W., Chen, G., Hadgu, T., & Bodvarsson, G. S. (1993). A numerical dualporosity model with semianalytical treatment of fracture/matrix flow. Water Resources Research, 29(7), 2127–2137. doi: 10.1029/93wr00749 107 CHAPTER 4.Characterization of shales using helium and argon at high pressures3, 4 4.1. Introduction Porosity and permeability are two of the most important petrophysical properties of shales. An accurate measurement of shale porosity and permeability leads to an improved estimation of the gas storage capacity and provides for a better understanding of the gas flow through various levels of the interconnected pores. Helium (He) is routinely used to characterize shale core samples in terms of their porosity and permeability prior to fluid flow or other relevant studies (e.g., imbibition experiments), as its adsorption amount is considered negligible in shale (Sigal and Qin, 2008; Wang et al., 2016a, 2016b) in comparison with the major NG components such as methane (CH4) and ethane (C2H6). However, He has a kinetic diameter of 0.26 nm, which is smaller than the diameters of CH4 and C2H6, with mean kinetic diameters of 0.38 nm and 0.44 nm, respectively. This may, 3 Parts of this chapter have been taken verbatim from our publication: Lyu, Y., Dasani, D., Tsotsis, T., & Jessen, K. (2021). Characterization of shale using Helium and Argon at high pressures. Journal of Petroleum Science and Engineering, 108952. 4 Parts of this chapter have been taken verbatim from our publication: Lyu, Y., Dasani, D., Tsotsis, T., & Jessen, K. (2023). Investigation of methane mass transfer and sorption in Marcellus shale under variable netstress. Geoenergy Science and Engineering, 211846. 108 then, lead to an over- estimation of shale porosity available to NG (Ross and Bustin, 2007; Bustin et al., 2008; Aljamaan et al., 2013; Gaus et al., 2020). Argon (Ar) has been reported to have a similar sorption potential to that of CH4 (Salem et al., 1998), and a kinetic diameter of 0.34 nm, which is close to that of CH4. It may serve, therefore, as a better-suited candidate to measure the porosity and permeability of shales relevant to NG, while avoiding the risks of working with a flammable gas like CH4 during routine core measurements. Recently, studies have been per- formed to show that shales demonstrate a lower permeability to Ar than to He (Cui et al., 2013; Sinha et al., 2013; Ghanizadeh et al., 2014; Gensterblum et al., 2015). Sorption of Ar has been studied in carbons (Malbrunot et al., 1992; Do and Do, 2003), and has attracted some recent interest from shale gas researchers (Sinha et al., 2013; Jin et al., 2015; Guo et al., 2015; Kazemi and Takbiri-Borujeni, 2016; Zhang et al., 2017). There is still, however, a noticeable lack of literature investigating Ar sorption phenomena at temperature and pressure conditions relevant to NG extraction, and specifically the impact of various shale characteristics including its organic content. Furthermore, most previous experimental works on Ar sorption in shales were conducted on crushed samples rather than whole cores (Klewiah et al., 2020). 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 subsequently over longer periods of time from the adsorbed gas in the meso/micropores. Accordingly, gas transport mechanisms in shale reservoirs includes viscous flow and Knudsen diffusion (Javadpour, 2009; Alnoaimi and Kovscek, 2019) as 109 well as adsorption and desorption (Wang et al., 2016a, 2016b; Akkutlu et al., 2018; Jia et al., 2019). The pore-size distribution (PSD) is a key parameter that controls phase behavior and gas transport in shale formations (Nelson, 2009; Sanaei et al., 2014a). The PSD, in turn, can be influenced by the effective stress (Cho et al., 2013; Song et al., 2016; Peng et al., 2018; Alnoaimi and Kovscek, 2019). Hence, transitional flow between viscous flow and Knudsen diffusion could occur during gas transport as a result of changes in effective stress (Cao et al., 2017; Cui et al., 2020; Song et al., 2020; Zhang et al., 2020). In this work, we have used the TGA technique to measure Ar adsorption isotherms and kinetics in a cubic shale sample (~1 cm3 ) from a specific gas producing depth of the Marcellus formation for a broad range of pressures of up to 110 bar at 49 ºC. In the analysis of the data, we assume that the sorption is controlled by non-linear, Langmuir-type kinetics. We use He as a test gas to measure the buoyancy force on the shale sample at various He densities (i.e., pressures), to calculate the skeletal (i.e., “true” solid) volume of the sample to be used to analyze the Ar adsorption data. Our experiments show a linear correlation between the sample’s apparent weight and the He density, indicative of insignificant He adsorption on the shale sample under these conditions, which justifies its use for measuring the initial sample porosity (and the skeletal volume). We have also used a full-diameter core sample, with a diameter of 8.89 cm (3.5’’) and a length of 18.03 cm (7.1’’), from the same depth of the Marcellus formation to perform gas expansion experiments using both He and Ar. The Ar sorption data generated employing the TGA technique are used to interpret the whole core gas expansion experiments. Combining sorption and mass transfer experiments both at the shale matrix (1 cm3 ) and the 110 full-core levels provides a very effective means for the complete characterization of these materials. A triple-porosity model (TPM) is introduced to combine the sorption dynamics, obtained at the shale-cube scale (which for Ar employs a Langmuir-type dynamic sorption model) with the mass transfer observations from He expansion experiments in order to describe the combined sorption and mass transfer of Ar in the full-diameter core. Furthermore, we demonstrate that sorption phenomena at the cube-scale can be combined with mass transfer observations from experiments with larger samples to predict the combined effect of sorption and mass transfer at larger (full-diameter core) scale. The organization of the rest of the manuscript is as follows: First, we describe the experimental and modeling approaches employed for both the TGA and the gas expansion measurements. We, then, present the experimental results and their interpretation using the relevant modeling approaches. Finally, we conclude with a summary and a discussion of our observations. 4.2. Experimental approach 4.2.1. Sample preparation The shale cube and the full-diameter core samples were both chosen from a depth of 2395.73 m (7,860 ft) from the Marcellus formation in the Appalachian Basin. The samples were boxed and stored in a zip-locked bag to prevent further contamination prior to their use. The shale core sample, with a diameter of 8.89 cm (3.5 in), as received, was slabbed 2.54 cm (1 in) from its circumference. A section of the core 18.03 cm (7.1 in) in length was 111 chosen for the gas expansion study. A cube with a volume of ~1 cm3 was cut and prepared using a mechanical saw from a section of the full core immediately above the 18.03 cm (7.1 in) core sample. In order to be able to load the whole core sample in a cylindrical Viton sleeve, which in turn is mounted inside a core-holder, an impermeable heavy-duty epoxy piece was molded using the Pro-Set M1012 resin/M2010 hardener and machined precisely to replace the missing (originally slabbed) part of the core. Figure 4.1 below shows the cube, the machined epoxy part and the combined core/core slab cylindrical assembly before being loaded in the Viton sleeve of the core-holder. Figure 4.1. Cube and machined impermeable epoxy (left) and top-view of core/slab assembly forming a cylindrical shape (right) The original full-diameter core sample was extracted in the vertical direction (perpendicular to the bedding plane) of the shale formation. The space in between the epoxy piece and the slabbed core is intended to mimic a “macro-fracture” in the vertical direction in the shale formation, perpendicular to the minimum horizontal stress (or parallel to the principal overburden stress). 112 4.2.2. TGA experiments – Shale cube He buoyancy and Ar sorption measurements were performed with the shale cube using a magnetic suspension balance (Rubotherm, Germany) under constant gas flow at a constant temperature of 49 °C, selected to match the prevailing subsurface conditions at the site from where the core was extracted from. The TGA specifications and the associated theory have been described in a prior publication (Wang et al., 2015). The weight and volume of the sample container were first determined following the procedure described by Wang et al. (2015). The cubic shale sample was then loaded into the sample container and heated under the vacuum at 120 °C for 24 hr to remove volatile contaminants, including water, and any residual gas present in the sample. This step was followed by pressurization of the TGA sample chamber with He in a stepwise manner, e.g., from 0 to 5 bar, 5 to 10 bar, 10 to 20 bar, up to a maximum of 110 bar at 49 °C, to perform the He buoyancy tests on the shale cube sample. At each step, from vacuum to 5 bar, for example, He was allowed to flow into the sample chamber at 300 scm3 /min, initially, until the pressure set-point (5 bar) was reached, after which, the flow rate was reduced to 35 scm3 /min. The gas was then allowed to flow until the measured weight of the shale cube stabilized (changes in weight <10 μg, which is the accuracy of the balance). The same protocol was followed when studying Ar in order to generate its isotherm. While the equilibrium TGA measurements are used to generate the sorption isotherm data, the transient weight measurements can be potentially used to extract important information about the sample’s sorption kinetics. As discussed further below, the length-scale of the cube is sufficiently small that mass transfer will occur very fast. Accordingly, at a weight measuring interval of 6 s employed by the 113 TGA instrument, it is not possible to delineate the mass transfer (viscous flow and Knudsen diffusion) from the transient data. 4.2.3. Gas expansion experiments – Full-diameter core In our work, we have used the gas expansion technique (e.g., see Santos and Akkutlu, 2013), to characterize the pore structure of the shale core under a confining stress. This method is in many ways similar to the GRI technique, typically used for the characterization of crushed shale samples (e.g., Luffel et al., 1993; Cui et al., 2009; Tinni et al., 2012). We have performed gas expansion experiments using both He and Ar. The He equilibrium pressure measurements from these gas expansion experiments are used to calculate the core’s accessible porosity. The dynamics of gas expansion provide valuable insight into the mass transfer and sorption characteristics of the full core and allow us to compare the data across length scales with observations from the TGA experiments. A schematic of the gas expansion experimental system used in our study is shown in Figure 4.2 below. To initiate the experiments, the core, including the machined epoxy slab, was wrapped in shrink-tubing, which was then inserted into a Viton sleeve and loaded into the core-holder (C). The core-holder and reference vessel (A) with a volume of 500 cm3 were installed inside an oven to facilitate constant temperature conditions. A confining stress was then applied to the core using a hydraulic pump (B). Initially, with all the valves (V1, V3, V4) closed and the backpressure regulator (BPR) completely open, the reference vessel was charged with He or Ar, depending on the experiment. 114 Figure 4.2. Schematic of the whole core gas expansion experimental system The pressure in the vessel was monitored by a digital pressure gauge (G1). Valve V3 was then opened in order to evacuate the core and the internal volume of the system using the vacuum pump (D). After 3 hrs of evacuation at a pressure of 6.6×10-3 bar (as measured by the gauge G3), the vacuum pump was disconnected from the rest of the system. Valve (V1) was then opened, and He (or Ar) from the reference vessel was allowed to expand into the core. The pressure was monitored electronically using pressure gauge G4. During the expansion process, the pressure first drops rapidly when the gas expands into the empty tubing and fittings of the system (dead volume). The gas then expands into the “macrofracture”, i.e., the space in between the epoxy slab and the core, and finally, the gas permeates into the core along the bedding plane that is perpendicular to the artificially induced macro-fracture. 115 4.3. Modeling Approach 4.3.1. Steady-state TGA experiments - modeling In this work, we use the Langmuir isotherm model to describe the Ar sorption behavior in shale samples. This approach assumes monolayer coverage on the pore walls and provides a consistent approach for interpretation of sorption kinetics (see Section 4.3.2). The Langmuir isotherm model includes two parameters, namely the maximum absolute sorption capacity of the shale Cs,max (mol/kg) and the equilibrium constant K (bar-1 ): = , ∙ 1+∙ = ,, (4.1) where, Cs is the absolute sorption (mol/kg), p is the equilibrium gas pressure (bar) and θ is the fractional coverage (dimensionless). The TGA experiments, however, measure the excess sorption, Cex (mol/kg) of Ar rather than the absolute sorption that is represented by the Langmuir model. In order to utilize the Langmuir model, the measured excess sorption data must be converted into absolute sorption, according to the following relationship: (1 − ) = . (4.2) Here, ρg is the molar density of the bulk-phase gas (mol/m3 ), and ρa is the molar density of the adsorbate phase (mol/m3 ). In this work, we use the data in the NIST database (Lemmon et al., 2005) to calculate the molar density (and viscosity) of the bulk phase. Wang (2016) performed an in-depth comparison between various adsorbate phase density models to 116 calculate the density of CH4, C2H6, and their binary mixtures in the sorbed phase. Following the observations made by Wang (2016), we use the lattice density functional theory (LDFT) approach, originally developed by Ottiger et al. (2008), to estimate the adsorbate phase density ρa. In this approach, the adsorbate density of Ar is expressed as a function of the fractional (lattice) coverage, θ, = (1−)−(1−2) . (4.3) where ρmax represents the maximum molar density (mol/m3 ) and ρc is the critical molar density (mol/m3 ). We set the maximum density equal to the molar density at the normal boiling point. The adsorbate phase density calculated using the LDFT method increases with increasing bulk phase pressure, an observation consistent with the molecular simulation study of Malheiro et al. (2015). By combining Eqs. 4.2 and 4.3, the excess sorption measured via the TGA technique can be converted into absolute adsorption and allow for subsequent estimation of the Langmuir parameters. 4.3.2. Dynamic TGA experiments - modeling Various adsorption and transport models, with corresponding numerical or analytical solutions, have been developed and applied to shale over the past years. Many use some version of the Bi-disperse Pore Model (BPM) model, first introduced by Ruckenstein et al. (1971), and since applied to study mass transfer in rocks, such as shale, by a number of investigators (Ning et al., 1993; Shi and Durucan, 2003; Cui et al., 2009; Ambrose et al., 2010; Guo et al., 2015). 117 In the BPM, the shale matrix is envisioned to consist of spherical regions with average characteristic size, , which are predominantly mesoporous, and contain microporous spherical inclusions with average characteristic size, . Neither nor are known a priori, and can be considered as the average size of spherical inclusions that defines the macroporous pore space. However, the dynamic TGA data does not allow one to accurately probe the dynamics of the macropores, and thus incorporating an additional length scale in the model is not warranted. To further validate the BPM for interpretation of our TGA experiments, we consider a 1 cm3 shale cube and calculate the equivalent radius of a sphere with the same volume, to arrive at a characteristic length Rc = 0.0062 m. We further assume that the shale volume does not include micro-cracks that would facilitate mass transfer. Since micro-cracks are found in Marcellus shales on a regular basis, the time scale for mass transfer, as estimated below, therefore, represents a conservative estimate. To estimate the characteristic time associated with mass transfer into the cube by viscous flow and Knudsen diffusion, we assume a simple pore model of straight cylindrical nonintersecting pores to arrive at = 2 3 √ 8 , = 2 8 . (4.4) Here, rm is the average pore radius of the mesopores (m), and Mw is the molecular weight (kg/mol), Dk is the Knudsen diffusivity (m2 /s), Rg is the gas constant, B is the viscous flow parameter (m2 ). The molar flux, N (mol/m2 /s) can then be written as 118 = −C ∇ − ∇~ − ( + ) ∇, (4.5) and the continuity equation in terms of the gas concentration C (mol/m3 ) - without sorption - for the cube can be written as = ∇ ∙ (M∇), = ( + ). (4.6) The effective diffusivity, M, includes contributions from both Knudsen diffusion and viscous flow, and will vary as a function of concentration (or pressure). The characteristic time for mass transfer, t*, can now be estimated as a function of pressure (or concentration) from t*=Rc 2 /M by assuming an average pore radius. Figure 4.3 reports the characteristic time for mass transfer for He and Ar in an average pore diameter of 10 nm. The average radius of 10 nm was selected following the work of Roychaudhuri et al. (2013) based on BET analysis of several Marcellus shale samples. From Figure 4.3, we observe that the mass transfer of both gases is fast relative to the sampling rate of our TGA equipment, that records the apparent weight every 6 s. Accordingly, we can only study the sorption kinetics of Ar with the selected shale cube size, while mass transfer must be studied at larger scale. 119 Figure 4.3. Characteristic time for He and Ar mass transfer in a 1 cm3 shale cube for an average pore diameter of 10 nm The sorption on the cube sample is then described by the initial value problem, = ((, − ) − 1 ), (4.7) where ρs is the skeletal (true solid) density (kg/m3 ), b is the internal mesopore/micropore surface area per unit skeletal adsorbent volume (m2 /m3 ), ka is the rate constant for adsorption (kg/Pa/m2 /s), K is the sorption equilibrium constant (Pa-1 ), given by K = ka/kd, where kd is the rate constant for desorption (kg/m2 /s), and , is the maximum adsorbed concentration corresponding to a complete monolayer coverage (mol/kg). 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 5 10 15 20 25 30 0 25 50 75 100 Ar/He ratio (-) Characteristic time (sec) Pressure (bar) Helium Argon Argon/Helium ratio 120 4.3.3. Gas expansion experiments – modeling To formulate a model for interpretation of the gas expansion experiments in the fulldiameter core, we revisit the system schematic provided in Figure 4.2. The system includes a 500 cm3 reference volume (Vu) upstream of the core-holder, a dead volume (Vd) in the form of tubing, valves, pressure gauges, etc. upstream and downstream of the core-holder, and pore volume (Vp) inside the core holder. From careful analysis of the system parts, the overall dead volume was evaluated to be ~15 cm3 . Furthermore, 7 cm3 of the dead volume is located upstream of the core holder to arrive at a volume V1 = 507 cm3 (upstream) and a volume V2 = 8 cm3 (downstream). The pore volume is divided into the volume of the macrofracture (i.e., the free space in between the slabbed core and the epoxy piece), Vf, and the total pore volume of the shale matrix, Vmt. The pore volume in the matrix is then further divided into a volume (Vm) consisting of mesopores and any microfractures that may exist in the shale matrix and a micropore volume (Vμ), so that Vmt = Vm+Vμ. Figure 4.4 illustrates the slabbed full-diameter core and the conceptual model that was used to interpret experimental observations. The macrofracture volume (in between the slabbed core and the epoxy piece) is characterized by a width, W, a length, L, and an average aperture, H. If we approximate the macrofracture as the volume between two parallel plates, and that the gas flow is laminar, we can write: = = − 2 12 = − , (4.8) 121 where Q is the volumetric flow rate (m3 /s), v is the superficial velocity (m/s) and kf is the effective fracture permeability (m2 ). The continuity equation for the macrofracture can then be written as: = − () − Г, (4.9) where Cf is the molar concentration of gas in the macrofracture (mole/m3 ), and Гm is the transfer rate (mole/m3 /s) between the macrofracture and the slabbed core (see further discussion below). Figure 4.4. Illustration of slabbed core (top) and conceptual model (bottom) We assume in Eq. 4.9 that the macrofracture volume remains constant during the expansion experiments. If we assume that the upstream and downstream dead volumes V1 and V2 exchange mass exclusively through the macrofracture (in between the epoxy piece and the slabbed core), then we can write the following continuity equations for them: 122 1 1 = −()=0 , (4.10) 2 2 = −()= , (4.11) where A is the cross-sectional area of the macrofracture (A = W∙H = Vf /L). This assumption is supported by the work of Roychaudhuri et al. (2013), which demonstrates that the vertical permeability, perpendicular to the bedding plane, is at least two orders of magnitude less than the horizontal permeability (perpendicular to the macrofracture). Next, we write the continuity equation for the mesopores/microcracks and micropores in the matrix as: + , = Г − Г, (4.12) + , = Г. (4.13) Here, Cm and Cμ are the average molar concentrations of gas (mol/m3 ) in the mesopores and micropores, respectively, and Ns,m and Ns,μ are the total moles of the adsorbate phase in the mesopores/microcracks and micropores, respectively. The right-hand sides of Eqs. 4.12-4.13 represent the mass transfer between the macrofracture and the mesopores/microcracks (Гm), and the corresponding mass transfer between mesopores/microcracks and the micropores (Гμ). These approximate mass transfer functions (mol/m3 /s) are derived from the work of Vermuelen (1953), and are given by = ( − ) ∙ , (4.14) 123 = ( − ) ∙ . (4.15) The key assumption used in Eqs. 4.12-4.15 is that mass transfer rates in both mesoporese/microcracks and micropores are similar or faster than the adsorption rates. Both transfer functions Гj (j=m,μ) include a shape factor σj (m-2 ), and an effective diffusivity Dj (m2 /s), while βj is a function of initial and boundary conditions that accounts for the evolution of gradients within the mesopores/micropores (see Vermeulen, 1953 and Zimmerman et al. 1993). The shape factors represent the interface area (e.g., interface between the macrofracture and mesopores/microcracks in the slabbed core) per volume, and per characteristic length. We note that the product σjDj can be considered as the inverse of the characteristics time for mass transfer. The material balance equations for the adsorbate phase in the mesopores and micropores are written as: , = ((, − ,) − , ), (4.16) , = ((, − ,) − , ), (4.17) where Wc is the core weight (kg) and Aj (j=m,μ) is the surface area, of mesopores and micropores per unit mass (m2 /kg). Aj is evaluated from the total surface area per unit mass (b/ρs) and the fraction of the total surface area associated with mesopores and micropores, respectively. The fraction of the total surface area occupied by either the mesopores or micropores can be determined by BET analysis. Figure 4.5, for example, reports the specific surface area that resides in the micropores (<2 nm) relative to the total specific 124 surface area, as a function of the specific pore volume (cc/g) that resides in the micropores relative to the total specific pore volume. The figure includes BET analysis data from 10 Marcellus shale samples over a vertical span of 30.48 m (100 ft), including the sample depth investigated here, as reported by Xu (2013). Figure 4.5. Fractional micropore surface area as a function of fractional micropore volume While the volume of the macrofracture (Vf) remains approximately constant, the pore volumes associated with the mesopores (Vm) and the micropores (Vμ) will change due to adsorption (and the volume of the adsorbate phase). The change in pore volume, at any point in time, is evaluated from = ,0 − , , (4.18) , = ,,/, , (4.19) y = 1.1682x R² = 0.9902 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 Fractional micropore surface area Fractional micropore volume 125 where Vj (j = m, μ) is the current pore volume, Vj,0 is the pore volume for a non-sorbing gas, Va,j is the volume of the adsorbate phase (m3 ) and ρa,j is the molar density (mol/m3 ) of the adsorbate phase in the mesopores (j=m) and macropores (j=μ). The molar density of the adsorbate phase is evaluated from Eq. 4.3. To close the model formulation, the initial conditions applicable to the conservation equations provided above are t = 0: Cf (x,0)=Cm (0)=Cμ (0)=Cs,m (0)=Cs,μ (0)=C2 = 0, C1 = Cini. (4.20) 4.4. Experimental results and interpretation 4.4.1. TGA steady-state experiments (isotherms) The apparent weight recorded by the TGA represents the mass of the sample (Ms), sample container (Msc) and adsorbate phase (Ma), subject to the buoyancy force corresponding to a given bulk phase pressure (density). In general, we have = + + − ( + + ), (4.21) where Vs is the skeleton volume of the sample (cm3 ), Vsc is the volume of the sample container (cm3 ), and Vads is the volume of the adsorbate phase (cm3 ). Helium buoyancy measurements were performed on the shale cube sample in the TGA at 49 °C, where the gas pressure was increased in a stepwise manner from 0 to 110 bar. Figure 4.6 below reports the apparent weight of the shale cube sample at steady state as a function of He density at each pressure set-point. 126 Figure 4.6. Helium buoyancy tests on the shale cube at 49 °C From the two He buoyancy tests, we observe that the experimental results are repeatable, with R 2 values of ~1. The linear trend of the apparent weight with respect to the (NIST) gas density, indicates that the change in apparent weight of the sample with increasing bulk phase pressure is for the most part due to buoyancy effects. This, in turn, supports the idea that He acts mostly as an inert non-sorbing gas (relative to Ar), at the pressure range and resolution of the microbalance used here. The slope and the intercept from Figure 4.6 accordingly provide the volume (1.95983 cm3 ) and weight (8.56341 g) of the shale sample plus the sample-holder, respectively. A separate buoyancy test was performed with the empty sample container, to find the mass (Msc = 5.02459 g) and volume (Vsc = 0.67501 cm3 ) of the sample container (Wang, 2016). By subtraction, we find the Ms = 3.53882 g, Vs = 1.28482 cm3 , and finally, a sample skeleton density of 2.75433 g/cm3 . 1: y = -1.959928x + 8.563323 R² = 1.000000 2: y = -1.959740x + 8.563490 R² = 0.999976 8.530 8.535 8.540 8.545 8.550 8.555 8.560 8.565 0.000 0.003 0.006 0.009 0.012 0.015 0.018 Apparent Weight (g) Helium NIST density (g/cc) 1: Helium Buoyancy @49C 2: Helium Buoyancy @49C 127 The sample was subsequently regenerated by evacuation (as discussed above) prior to the steady state measurement of Ar isotherms in the range from 0 to 110 bar. From the true mass and volume of sample and sample container, we evaluate the excess adsorption of Ar on the shale cube (by rearranging Eq. 4.21) = − = − − + ( + ), (4.22) Figure 4.7 reports two excess sorption isotherms for Ar measured on the shale cube at 49°C for the pressure range of 0-110 bar Figure 4.7. Argon excess adsorption isotherms for the shale cube at 49 °C The root-mean-square error (RMSE) between the two Ar isotherms was calculated to be 0.042 mg/g, indicating a good experimental reproducibility. The experimental results show that Ar adsorbs onto the shale surfaces with no apparent maximum in excess sorption over 128 the range of the pressures probed in this study. This highlights the importance of accounting for sorption of Ar when using it to characterize shale samples. 4.4.2. Dynamic TGA experiments Figure 4.8 reports the data, as recorded by the TGA, when converted to excess sorption (see Eq. 4.22), for Ar on shale at 49˚C and at pressures in the range of 0-80 bar. We observe the short-lived perturbations of the balance introduced by step changes in the gas pressure. To interpret the sorption kinetics of Ar on the sample, we perform a filtering step to eliminate the perturbations based on a maximum rate of increase in mass (2.5 μg/sec used here). We further require the excess sorption to be continuous with respect to time and correct for sudden shifts in the recorded mass (see e.g., circle in Figure 4.8 at 80 bar) that may signify external perturbations of the system. The result of this pre-processing step is reported in Figure 4.9. 129 Figure 4.8. Dynamics of Ar excess sorption at 49˚C – From recorded data (TGA) Figure 4.9. Dynamics of Ar excess sorption at 49˚C – Processed data 130 To validate the pre-processing step, we extrapolate the Ar excess sorption data, for each pressure step, to t = ∞ to estimate the isotherm points. This is done by fitting the exponential function: = ∙ (1 + exp (− ∙ )), (4.23) to the last 50% of the data points for each pressure step. Figure 4.10 demonstrates this approach for 2 pressure steps (0-10 bar and 10-20 bar) in terms of normalized time (over a pressure step). Figure 4.11 compares the extrapolated isotherm points from this approach with the steady state isotherms reported in Figure 4.7. We observe from Figure 4.11 that the extrapolated values are in good agreement with the steady state measurements, indicating that the preprocessing step discussed above is valid. Figure 4.10. Extrapolation of Ar sorption dynamics (0-10 bar and 10-20 bar shown) 131 Figure 4.11. Ar isotherm (squares) and extrapolated dynamics (asterisk). Solid line from matching isotherm data, and broken line from matching dynamic data We then combine Eq. 4.7 with Eq. 4.22 to estimate the sorption kinetics of Ar on shale. Two different approaches were investigated for estimating the adsorption kinetic constant: First, we use the isotherm data to estimate the maximum sorption capacity (Cs,max) and the equilibrium constant (K). The values are reported in Table 4.1 and represents the solid line in Figure 4.11. In this approach, only one parameter, bka, is estimated from the data reported in Figure 4.9. We note that the adsorbate density, used to convert excess to absolute sorption and, in particular, the dependency of the adsorbate density on the fractional coverage (pressure and time), is closely linked to the observed sorption dynamics (Figure 4.9). Accordingly, the use of Cs,max and K from equilibrium data, may not be optimal for correlating sorption dynamics. From this approach, we arrive at a RMSE of 0.096 mg/g. In the second approach, we estimate Cs,max, K and bka simultaneously from the 132 dynamics data to arrive at a RMSE of 0.046 mg/g. The parameters from this approach are also listed in Table 4.1 and represented by a broken line in Figure 4.11. Table 4.1. Langmuir parameters from TGA experiments – Ar Cs,max K bka RMSE Cs,max and K from: (10-3 mole/g) (1/bar) (g/bar/cm3 /s) (mg/g) Steady state data 0.34215 1.1490e-02 6.52708e-06 0.096 Dynamic data 0.45946 8.0337e-03 8.63795e-06 0.046 From Figure 4.11, we observe that both approaches for estimating the adsorption rate constant provide a reasonable agreement with the isotherm data, and we proceed by using the sorption rate and equilibrium constants from the second approach: Figure 4.12 provides a comparison of the dynamic data with model calculations after parameter estimation. From the interpretation of the sorption kinetics, one can evaluate the characteristic time for adsorption (tsorp). This is done by rewriting Eq. 4.7 in dimensionless form and results in: = . (4.24) As would be expected, the characteristic time for adsorption is a function of the bulk-phase pressure, with slow sorption at lower pressures and higher rates at higher pressures. Figure 4.13 reports tsorp as a function of bulk phase pressure, and we observe that the sorption process is substantially slower than the mass transfer in a 1cm3 cube, as reported in Figure 4.3. 133 Figure 4.12. Estimation of Cs,max, K and bka from Ar dynamics – RMSE = 0.046 mg/g Figure 4.13. Characteristic time for sorption as a function of bulk phase pressure 1.E+03 1.E+04 1.E+05 1.E+06 0 20 40 60 80 100 Characteristic time for sorption (sec) Pressure (bar) 134 4.4.3. Gas expansion experiments Following the experimental protocol presented in Section 4.2.3, gas expansion experiments were performed with He and Ar as probe gases. The variation in pressure during each of the experiments is reported in Figure 4.14. Figure 4.14. He (solid) and Ar (dashed) expansion experiments at 49 ºC We use the initial and final pressures from the He expansion experiment to evaluate the volumes associated with core holder (Vf +Vm), as depicted in Figure 4.4, and the relevant pressures and volumes are reported in Table 4.2. Table 4.2. Evaluation of pore volume in core holder from He expansion pini (bar) pend (bar) Vref (cm3 ) Vd (cm3 ) Vcore (cm3 ) Vf+Vp (cm3 ) 63.5±0.02 57.5±0.02 507.0±0.5 8.0±0.5 855.3±8 44.4±0.5 135 Eqs. 4.8-4.20 were implemented and solved in Matlab using an implicit finite volume formulation to simulate the expansion experiments: We used 50 sub-volumes to represent the core in our 1-D triple-porosity interpretation. The model includes four parameters (for non-sorbing gases) that must be estimated from the He experiment, including kf, σDm, σDμ and α = Vm/Vm,t. We note that kf defines the aperture H (as well as the cross-sectional area A = H∙W and Vf =A∙L) of the macrofracture, while α is the fraction of the total matrix pore volume (Vm,t) that resides in the mesopores (Vm). The parameters were estimated by a nonlinear least squares approach (Levenberg–Marquardt algorithm used here) to arrive at the values reported in Table 4.3 below. Figure 4.15 compares the experimental observations with model calculations, after parameter estimation, and we observe a good agreement with a RMSE of 0.024 bar. Figure 4.15. He expansion – Parameter estimation – RMSE = 0.024 bar Table 4.3. Model parameters estimated from the He expansion experiment 136 kf Vm/Vm,t cm2 1/sec 1/sec - Results 1.50e-9 1.35e-3 91.39e-6 0.79 95% CI [1.42, 1.58]e-9 [0.96, 1.73]e-3 [76.70, 106.10]e-6 [0.77, 0.82] From the estimated model parameters, we find a fracture volume (Vf) of 0.02 cm3 (H = 1.34e-4 cm), a mesoporous volume of 35.1 cm3 and a microporous volume of 9.3 cm3 . The total pore volume in the shale core corresponds to a porosity of 5.2% that is consistent with measurements reported for samples from the same Marcellus well (Roychaudhuri et al., 2013). The model parameters from Table 4.3 were then combined with the Ar sorption parameters reported in Table 4.1 to predict the behavior of the Ar expansion experiment. To model the sorption behavior of Ar in the core, we rescale the estimated adsorption capacity Cs,max from the cube, based on the difference in porosities between the core and the cube, to account for the variations in the adsorbent surface area accessible in the core. The observed differences in porosity may be caused by several factors, including: 1) pore closure and/or deformation due to confinement, and 2) any additional pore volume that was connected/created during the cube sample preparation. In this work, we rescale the adsorption capacity of the core (shale matrix) from the cube based on the specific pore volume to reflect differences in access to the internal structures between the two samples. The impact of net stress on the matrix porosity of the core is assumed to be insignificant based on previous studies on samples also from Marcellus shale (Elsaig, 2016; Goral et al., 2020). Accordingly, the core sorption capacity was calculated as: 137 Cs,max,core=Cs,max,cube∙ pore volume per core weight pore volume per cube weight =0.46 mol/kg× 44.4 cc/2024 gram 0.20 cc/3.54 gram =0.18 mol/kg. To apply the transport parameters, listed in Table 4.3 from the He expansion experiment, to predict the Ar expansion experiment, we need to rescale the transport parameter to reflect differences in the Knudsen diffusivity and fluid viscosity between He and Ar. If transport was controlled exclusively by Knudsen diffusion, the scaling factor would simply be the square root of the ratio of molecular weights (see Eq. 4.4), and σDj should be multiplied by a factor of 0.32. However, since viscous flow and Knudsen diffusion both contribute to the mass transfer in the mesopores over the range of conditions experienced during the expansion experiments, we use the average ratio of characteristic times from Figure 4.3 (assuming an average pore diameter of 10 nm) to arrive at a factor of 0.41. A comparison of the experimental data with the model predictions is provided in Figure 4.16, where we observe a good agreement with a RMSE of 0.33 bar. Figure 4.16. Ar expansion – Prediction – RMSE = 0.33 bar 138 4.5. Summary and conclusions In this work, we have characterized shale samples from the Marcellus formation at centimeter and decimeter length-scales using He and Ar as test gases. A full-diameter core was studied in terms of its matrix/fracture systems. A shale cube (~1 cm3 ) from the same depth in the Marcellus formation was also studied, and we hypothesized that the cube is representative of the matrix region of the full-diameter core. We measured the sorption behavior of Ar in terms of isotherms and kinetics for the matrix region of the shale, by performing TGA analysis on the shale cube. We then investigated the matrix/fracture interaction by performing gas expansion experiments, using He and Ar with the fulldiameter core. Based on time-scale arguments, a simple Langmuir-type model was introduced to interpret sorption isotherms and dynamics for the shale cube. A tripleporosity model (TPM) was introduced to interpret the mass transfer and sorption of Ar in the full-diameter core based on sorption analysis on the cube and a He expansion test on the core. Our TGA experimental results, using He as test gas, demonstrate a linear dependency between the weight of the shale cube in equilibrium with the bulk phase pressure, and the bulk phase density. This indicates that He does not adsorb onto the shale surfaces (in any significant amount), and that we can use these measurements to evaluate the sample mass and skeleton volume for subsequent use in studying Ar excess sorption on the shale cube. The excess sorption measurements on the shale cube demonstrate that Ar adsorbs substantially on the shale. Furthermore, no apparent sorption maximum is observed in the range of pressures probed in this study (0-110 bar). We applied the Langmuir model, 139 combined with the LDFT model for adsorbate density to interpret the Ar excess sorption isotherm and to evaluate the maximum sorption capacity and equilibrium constant. These parameters were subsequently used to evaluate the adsorption rate constant for Ar from a separate dynamic TGA experiment. The dynamic sorption behavior of Ar was demonstrated to be in good agreement with the Langmuir model (assuming monolayer coverage) with a RMSE = 0.096 mg/g. However, we observe an even better agreement between the Langmuir-type model and the dynamic data, when the isotherm and adsorption rate constants are estimated simultaneously (RMSE = 0.046 mg/g). The observed difference illustrates the importance of the transient in the adsorbate density during dynamic sorption experiments that has received little (if any) attention in the literature to date. Helium expansion was then performed on the full-diameter core to characterize the porosity and mass transfer in the relevant portions of the pore space, including the macrofracture (interface between epoxy insert and core) as well as in the mesopores/microcracks and micropores. We used a TPM to interpret the He behavior in the core and estimated the permeability of the macrofrature (1.5x10-9 cm2 or ~ 150 millidarcy) and the split between mesoporosity and microporosity (79% of the pore volume in mesoporous region). Furthermore, we estimate the characteristic times for mass transfer between macrofracture/mesopores (~700 s) and mesopores/micropores (~11,000 s). We note that the characteristic times, reported here, are evaluated from the product of the shape factor (σ) and an effective diffusivity (D). The shape factor represents the surface area per volume and characteristic length for mass transfer. Furthermore, the characteristic time for mass transfer in the micropores is linked to the characteristic time for mass transfer in the 140 mesopores, based on the assumed hierarchical structure of the shale. The experimental observations presented in this work, however, do not allow for further separation the σD product into additional information regarding length scales or effective diffusivities. A central contribution of the presented experimental work, modeling and interpretations presented in this manuscript, pertains to the combination of sorption behaviors from a sample, at the cm scale, with mass transfer observations at larger scale (length and volume): We demonstrate that sorption measurements, in terms of isotherms and kinetics and smaller scale, allow for reasonably accurate predictions of coupled sorption and mass transfer at larger scale. Finally, the use of transfer functions to delineate mass transfer between macrofractures and mesoporous and microporous segments of the matrix, provides a feasible path for upscaling of sorption behavior, observed at the laboratory scale, to larger scale. 141 4.6. References Akkutlu, I. Y., Efendiev, Y., Vasilyeva, M., & Wang, Y. (2018). Multiscale model reduction for shale gas transport in poroelastic fractured media. Journal of Computational Physics, 353, 356–376. doi: 10.1016/j.jcp.2017.10.023 Aljamaan H, Alnoaimi K, Kovscek AR (2013). In-depth experimental investigation of shale physical and transport properties. SPE Paper 1582146 Presented at the SPE Unconventional Resources Technology Conference, 12–14 August, Denver, Colorado. Alnoaimi, K. R., & Kovscek, A. R. (2019). Influence of Microcracks on Flow and Storage Capacities of Gas Shales at Core Scale. Transport in Porous Media, 127(1), 53–84. doi: 10.1007/s11242-018-1180-5 Ambrose, R. J., Hartman, R. C., Diaz Campos, M., Akkutlu, I. Y., & Sondergeld, C. (2010, January). New pore-scale considerations for shale gas in place calculations. SPE Paper 131772 Presented at the SPE Unconventional Gas Conference, 23-25 February, Pittsburgh, Pennsylvania. Ambrose, R. J., Hartman, R. C., Diaz-Campos, M., Akkutlu, I. Y., & Sondergeld, C. H. (2012). Shale Gas-in-Place Calculations Part I: New Pore-Scale Considerations. SPE Journal, 17(01), 219–229. doi: 10.2118/131772-PA Bustin RM, Bustin AMM, Cui X, Ross DJK, Murthy Pathi VS (2008) Impact of shale properties on pore structure and storage characteristics, SPE Paper No. 119892-pp for Presentation at the 2008 SPE Shale Gas Production Conference, 16–18 November, Fort Worth, Texas. Cao, C., Zhao, Q., Gao, C., Sun, J., Xu, J., Zhang, P., & Zhang, L. (2017). Discrete fracture model with multi-field coupling transport for shale gas reservoirs. Journal of Petroleum Science and Engineering, 158, 107–119. doi: 10.1016/j.petrol.2017.08.033 Cho, Y., Ozkan, E., & Apaydin, O. G. (2013, May 1). Pressure-Dependent Natural-Fracture Permeability in Shale and Its Effect on Shale-Gas Well Production. Society of Petroleum Engineers. doi:10.2118/159801-PA Cui A, Wust R, Nassichuk B, Glover K, Brezovski R, Twemlow C. (2013) A nearly complete characterization of permeability to hydrocarbon gas and liquid for unconventional reservoirs: A challenge to conventional thinking. SPE paper No. 168730 142 presented at the SPE Unconventional Resources Technology Conference, 12–14 August, Denver, CO. Cui, G., Xia-Ting, F., Pan, Z., Chen, T., Liu, J., Elsworth, D., … Wang, C. (2020). Impact of shale matrix mechanical interactions on gas transport during production. Journal of Petroleum Science and Engineering, 184, 106524. doi: 10.1016/j.petrol.2019.106524 Cui, X., Bustin, A. M. M., & Bustin, R. M. (2009). Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids, 9(3), 208-223. Do, D. D., & Do, H. D. (2003). Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density. Carbon, 41(9), 1777-1791. EIA (2020). Annual Energy Outlook 2020 with projections to 2050. Energy Information Administration, Washington. Elsaig, M., 2016. "Characterizations of the Marcellus Shale petrophysical properties". Graduate Theses, Dissertations, and Problem Reports. 5541. https://researchrepository.wvu.edu/etd/5541. Gaus, G., Kalmykov, A., Krooss, B. M., & Fink, R. (2020). Experimental Investigation of the Dependence of Accessible Porosity and Methane Sorption Capacity of Carbonaceous Shales on Particle Size. Geofluids, 2020, 1–13. doi: 10.1155/2020/2382153 Gensterblum, Y., Ghanizadeh, A., Cuss, R. J., Amann-Hildenbrand, A., Krooss, B. M., Clarkson, C. R., … Zoback, M. D. (2015). Gas transport and storage capacity in shale gas reservoirs – A review. Part A: Transport processes. Journal of Unconventional Oil and Gas Resources, 12, 87–122. doi: 10.1016/j.juogr.2015.08.001 Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., & Krooss, B. M. (2014). Experimental study of fluid transport processes in the matrix system of the European organic-rich shales: I. Scandinavian Alum Shale. Marine and Petroleum Geology, 51, 79-99. Goral, J., Panja, P., Deo, M., Andrew M., Linden S., Schwarz JO., Wiegmann A., 2020. Confinement Effect on Porosity and Permeability of Shales. Sci Rep 10, 49. https://doi.org/10.1038/s41598-019-56885-y. 143 Guo, C., Wei, M., & Liu, H. (2015). Modeling of Gas Production from Shale Reservoirs Considering Multiple Transport Mechanisms. PloS one, 10(12), e0143649. Javadpour, F. (2009). Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). Journal of Canadian Petroleum Technology, 48(08), 16–21. doi: 10.2118/09-08-16-DA Jia, B., Tsau, J.-S., & Barati, R. (2019). A review of the current progress of CO2 injection EOR and carbon storage in shale oil reservoirs. Fuel, 236, 404–427. doi: 10.1016/j.fuel.2018.08.103 Jin, G., Pérez, H. G., Agrawal, G., Khodja, M. R., Ali, A. Z., Hussaini, S. R., & Jangda, Z. Z. (2015). The impact of gas adsorption and composition on unconventional shale permeability measurement. SPE Paper No. 172744 presented at the SPE Middle East Oil & Gas Show and Conference, 8-11 March, Manama, Bahrain. Kazemi, M., & Takbiri-Borujeni, A. (2016). Non-equilibrium molecular dynamics simulation of gas flow in organic nanochannels. Journal of Natural Gas Science and Engineering. Klewiah, I., Berawala, D. S., Walker, H. C. A., Andersen, P. Ø., & Nadeau, P. H. (2020). Review of experimental sorption studies of CO2 and CH4 in shales. Journal of Natural Gas Science and Engineering, 73, 103045. doi: 10.1016/j.jngse.2019.103045 Lemmon, E. W., McLinden, M. O., & Friend, D. G. (2005). Thermophysical properties of fluid systems. NIST chemistry webbook, NIST standard reference database, 69. Lim, K., & Aziz, K. (1995). Matrix-fracture transfer shape factors for dual-porosity simulators. Journal of Petroleum Science and Engineering, 13(3-4), 169–178. doi: 10.1016/0920-4105(95)00010-F Luffel, D. L., Hopkins, C. W., & Schettler Jr, P. D. (1993). Matrix permeability measurement of gas productive shales. SPE Paper No. 26633 presented at the SPE Annual Technical Conference and Exhibition, 3-6 October, Houston, Texas. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R., & Bose, T. K. (1992). Adsorption measurements of argon, neon, krypton, nitrogen, and methane on activated carbon up to 650 MPa. Langmuir, 8(2), 577-580. 144 Malheiro, C., Mendiboure, B., Plantier, F., Guatarbes, B., & Miqueu, C. (2015). An accurate model for the filling pressure of carbon slit-like micropores. Fluid Phase Equilibria, 404, 118-123. Nelson, P. H. (2009). Pore-throat sizes in sandstones, tight sandstones, and shales. AAPG Bulletin, 93(3), 329–340. doi: 10.1306/10240808059 Ning, X., Fan, J., Holditch, S. A., & Lee, W. J. (1993). The measurement of matrix and fracture properties in naturally fractured cores. SPE Paper No. 25898 presented at the Low Permeability Reservoirs Symposium, 26-28 April, Denver, Colorado. Ottiger, S., Pini, R., Storti, G., & Mazzotti, M. (2008). Measuring and modeling the competitive adsorption of CO2, CH4, and N2 on a dry coal. Langmuir,24(17), 9531-9540. Peng, Y., Liu, J., Pan, Z., Qu, H., & Connell, L. (2018). Evolution of shale apparent permeability under variable boundary conditions. Fuel, 215, 46–56. doi: 10.1016/j.fuel.2017.11.024 Ross, D. J., & Bustin, R. M. (2007). Impact of mass balance calculations on adsorption capacities in microporous shale gas reservoirs. Fuel, 86(17), 2696-2706. Roychaudhuri, B., Tsotsis, TT. and Jessen, K (2013), An experimental investigation of spontaneous imbibition in gas shales, Journal of Petroleum Science and Engineering 111, 87-97. Ruckenstein, E., Vaidyanathan, A. S., & Youngquist, G. R. (1971). Sorption by solids with bidisperse pore structures. Chemical Engineering Science,26(9), 1305-1318. Salem, M. M. K., Braeuer, P., Szombathely, M. V., Heuchel, M., Harting, P., Quitzsch, K., & Jaroniec, M. (1998). Thermodynamics of high-pressure adsorption of argon, nitrogen, and methane on microporous adsorbents. Langmuir, 14(12), 3376-3389. Sanaei, A., Jamili, A., & Callard, J., 2014a. Effect of Pore Size Distribution and Connectivity on Phase Behavior and Gas Condensate Production From Unconventional Resources. SPE Unconventional Resources Conference. doi: 10.2118/168970-MS Santos, J. M., & Akkutlu, I. Y., 2013. Laboratory Measurement of Sorption Isotherm under Confining Stress with Pore-Volume Effects. SPE Journal,18(05), 924-931. 145 Sigal, R. F., & Qin, B. (2008, June 1). Examination of the Importance of Self Diffusion In the Transportation of Gas In Shale Gas Reservoirs. Society of Petrophysicists and WellLog Analysts. Sinha, S., Braun, E.M., Determan, M.D., Passey, Q.R., Leonardi, S.A., Boros, J.A., Wood, A.C., Zirkle, T., Kudva, R.A. (2013). Steady-state permeability measurements on intact shale samples at reservoir conditions – effect of stress, temperature, pressure, and type of gas. , SPE No. 164263 presented at Society of Petroleum Engineers (SPE) Middle East Oil and Gas Show and Exhibition, 10-13 March, Manama, Bahrain. Shi, J. Q., & Durucan, S. (2003). A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel, 82(10), 1219-1229. Song W., Yao J., Li Y., Sun H., Zhang L., Yang Y., Zhao J., & Sui H., 2016. Apparent Gas Permeability in Organic-rich Shale Reservoir. Fuel. Vol 181, 973–984. Song, W., Yao, J., Wang, D., Li, Y., Sun, H., & Yang, Y., 2020. Dynamic pore network modelling of real gas transport in shale nanopore structure. Journal of Petroleum Science and Engineering, 184, 106506. doi: 10.1016/j.petrol.2019.106506 Tinni, A., Fathi, E., Agarwal, R., Sondergeld, C. H., Akkutlu, I. Y., & Rai, C. S. (2012). Shale permeability measurements on plugs and crushed samples. SPE No. 162235 presented at SPE Canadian Unconventional Resources Conference, 10 October-1 November, Calgary, Alberta, Canada. Vermeulen, T. (1953). Theory for Irreversible and Constant-Pattern Solid Diffusion. Industrial & Engineering Chemistry, 45(8), 1664–1670. doi: 10.1021/ie50524a025 Wang, J., Yang, Z., Dong, M., Gong, H., Sang, Q., Li, Y., 2016a. Experimental and numerical investigation of dynamic gas adsorption/desorption–diffusion process in shale. Energy Fuels 30 (12), 10080–10091. https://doi.org/10.1021/acs. energyfuels.6b01447. Wang, J., Wang, B.E., Li, Y., Yang, Z., Gong, H., Dong, M., 2016b. Measurement of dynamic adsorption–diffusion process of methane in shale. Fuel 172, 37–48. https:// doi.org/10.1016/j.fuel.2015.12.069. 146 Wang, Y., Tsotsis, T. T., & Jessen, K. (2015). Competitive Sorption of Methane/Ethane Mixtures on Shale: Measurements and Modeling. Industrial & Engineering Chemistry Research, 54(48), 12187-12195. Wang, Y. (2016). Investigation of Gas Sorption and Mass Transfer in Gas Shales (Doctoral dissertation). University of Southern California, Los Angeles. Xu, J. (2013). The study of mass transfer in gas shales and the optimization of hydraulic stimulation processes via additives. (Doctoral dissertation). University of Southern California. Zhang, L., Xiong, Y., Li, Y., Wei, M., Jiang, W., Lei, R., & Wu, Z. (2017). DFT modeling of CO2 and Ar low-pressure adsorption for accurate nanopore structure characterization in organic-rich shales. Fuel, 204, 1–11. doi: 10.1016/j.fuel.2017.05.046 Zhang, L., Liang, H., Zhao, Y., Xie, J., Peng, X., & Li, Q. (2020). Gas transport characteristics in shale matrix based on multiple mechanisms. Chemical Engineering Journal, 386, 124002. doi: 10.1016/j.cej.2019.124002 Zimmerman, R. W., Chen, G., Hadgu, T., & Bodvarsson, G. S. (1993). A numerical dualporosity model with semianalytical treatment of fracture/matrix flow. Water Resources Research, 29(7), 2127–2137. doi: 10.1029/93wr00749 147 CHAPTER 5.Investigation of methane mass transfer and sorption in Marcellus shale under variable net-stress5 5.1. Introduction Gas production from shales is promoted by horizontal drilling and multi-stage hydraulic fracturing. Induced fractures, but also natural (micro) fractures that shales often contain, represent high conductivity pathways in shales, which are typically characterized by ultralow matrix permeability. To accurately estimate and predict gas recovery from shales, gas transport and storage mechanisms in both the fractures and matrix must be well understood. One goal of this work is to facilitate the interpretation of laboratory-scale experiments under relevant conditions to bridge the scale-gap between laboratory-scale and field-scale observations. The Appalachian Basin has a long history of shale gas production and contributes substantially to the overall NG output of the United States. Monthly dry NG production from shale formations in the Appalachian Basin, which stretches over Pennsylvania, West Virginia, and Ohio, has been increasing dramatically since 2008 and has recently achieved new highs in terms of volume. Reservoirs in the Appalachian Basin include two shale 5 Parts of this chapter have been taken verbatim from our publication: Lyu, Y., Dasani, D., Tsotsis, T., & Jessen, K. (2023). Investigation of methane mass transfer and sorption in Marcellus shale under variable netstress. Geoenergy Science and Engineering, 211846. 148 formations, the Marcellus and the Utica, which accounted for 30% of total dry NG production in the United States in the first half of 2021 (EIA, 2021). The Marcellus Middle Devonian-age organic-rich formation, generally known as Marcellus shale, is the most productive NG-producing formation in the Appalachian basin, extending in the subsurface through New York State in the north to northeastern Kentucky and Tennessee in the south (EIA, 2017). The formation’s entire size is estimated to be approximately 95,000 square miles, with depths ranging from 4,000 to 8,000 feet, according to the United States Geological Survey (USGS). This more than 350-millionyear-old deposit is thought to hold about 97 trillion cubic feet (tcf) of NG (de Witt et al., 1993; USGS, 2019), enough to meet the energy demands of US customers for hundreds of years to come. Because of their low porosity, poor permeability, and high organic and clay content, these shales are distinctive in nature (Roychaudhuri et al., 2013), and only in the past few decades have substantial attempts been made to recover NG from the Marcellus shale for commercial purposes (Boyer et al., 2006; Harper, 2008). During shale gas production, free gas is first produced from fracture networks and macropores (natural or induced), followed by the free gas in mesopores and the adsorbed gas in the meso/micropores. Gas transport in these pore subsystems of the shale occurs at different characteristic times and length scales. These transport processes take place simultaneously, however, and it is complicated to characterize the transition between separate flow regimes (Ghanizadeh et al., 2014). One of the main obstacles to a detailed representation of fluid flow and transport in shales relates to the omnipresent heterogeneity in natural geological media (Zachara et al., 2016; Loschko et al., 2019; Zhang et al., 2021). 149 The computational cost associated with performance predictions can be substantial, particularly if high-resolution models are used to, e.g., account for spatial variability in shale properties. Consequently, effective prediction of flow and transport behaviors at large scale, based on knowledge of parameter variations at smaller scale, is of basic and practical relevance (Soltanian et al., 2015), and can facilitate the use of low-cost computational methods via "upscaling" (Rubin, 2003): the translation of processes in a heterogeneous domain to a homogeneous domain via effective properties/processes. To investigate the flow/transport behavior in complex porous media, such as shales, it is necessary to identify the relevant processes. These can be classified by the Knudsen number, Kn, defined as the ratio between the characteristic size (Dp) of a given pore and the mean free path of the molecules (Schaaf and Chambre, 1961; Civan, 2010): • Viscous flow (Darcy flow): Kn ≤ 0.001 - Fluid flow is pressure-driven, and the fluid velocity is proportional to the pressure gradient. This mechanism is dominant in fractures and macropores (Dp > 50 nm). • Slip/Transition flow (Klinkenberg phenomenon): 0.001 < Kn < 10 - Fluid flow is in transition between viscous flow and Knudsen diffusion regimes, and this occurs when intermolecular collisions and collisions between gas molecules and pore walls are equally prevalent. Transitional flow is often associated with mesopores (2 nm < Dp < 50 nm). • Knudsen diffusion (free molecular flow): Kn ≥ 10 - The mean free path of gas molecules is close to the pore diameters, which means collisions between gas molecules and pore walls dominate mass transfer. The driving force is the gas 150 concentration gradient. This mode of transport can be dominant in meso- and micro-pores (Dp < 50 nm), depending on the pore pressure. • Surface diffusion (adsorbed gas): This type of mass transfer describes molecules moving along the pore surfaces in an adsorbed layer and is driven by the concentration gradient of the adsorbed phase. This mode of transport is also active in micropores (Dp < 2 nm). Based on the above classification, researchers have developed models that attempt to delineate the gas transport mechanisms via an apparent permeability (Chen et al., 2015; Sheng et al., 2018). Examples include: • The Dusty-Gas Model (DGM) (Mason and Malinauskas, 1983; Freeman et al., 2011; Lyu et al., 2021): The total flux for a pure-component gas is represented as a combination of viscous flow and Knudsen diffusion. • The Javadpour model (Javadpour, 2009; Akkutlu and Fathi, 2012; Wasaki and Akkutlu, 2015): Coupled-flow equations include viscous flow, slippage effect, Knudsen diffusion, and surface diffusion, and the combined transport is related to the intrinsic permeability, the Knudsen diffusion coefficient, and a slip factor based on pore size(s). • The Beskok, Karniadakis, and Civan model (BKC): Beskok and Karniadakis (1999) proposed a second-order correlation for apparent permeability to include gas slippage effects based on the Knudsen number; Civan (2010) extended this model and introduced an expression for the rarefaction coefficient. 151 A key attribute of shale that controls phase behavior and gas transport is the pore-size distribution (PSD) (Nelson, 2009; Sanaei et al., 2014b), which can be influenced by effective stress (Cho et al., 2013; Song et al., 2016; Peng et al., 2018). Hence, as the effective stress changes during shale gas production, transitional flow between viscous flow and Knudsen diffusion can occur (Cao et al., 2017; Song et al., 2020). To further complicate matters, several researchers have studied gas sorption hysteresis on coals or shales (e.g., Jessen et al., 2008; Zhao et al., 2017) and demonstrated that a clear lag between adsorption and desorption timescales and amounts can exist. Furthermore, gas densification effects due to fluid confinement may also affect mass transfer and storage in shales (Zarragoicoechea and Kuz, 2002; Sanaei et al., 2014b). However, for the experimental conditions of this work, the impact is marginal (see Appendix A). Gas transport and storage mechanisms in shales have been extensively studied over the past few years, including experimental work (e.g., Ghanizadeh et al., 2014; Alnoaimi and Kovscek, 2019; Elkady et al., 2020) and modeling work (e.g., Javadpour, 2009; Civan, 2010; Freeman et al., 2011; Akkutlu et al., 2018; Lyu et al., 2021) on shale samples from different formations. However, it is still challenging to comprehensively delineate gas transport in porous media, such as shales, with a hierarchical pore structure and a broad PSD within each layer of the hierarchy, especially at reservoir scale. Therefore, in this work, we propose a method that takes advantage of a volume-averaging approach to incorporate all relevant mechanisms into characteristic times for mass transfer in mesoporous and microporous subdomains. The characteristic times for a porous medium can be calibrated with non-sorbing gases, e.g., helium (He), and can subsequently be rescaled for use with other gases. 152 In our previous work (Lyu et al., 2021), we characterized several shale samples at different length scales from the same depth of the Marcellus formation with He and argon (Ar). The sorption behavior of Ar in small cube shale samples and the mass transfer behavior of He and Ar in a full-diameter shale core were studied. We demonstrated that understanding gas sorption behavior at a small length scale allows for reasonably accurate predictions of coupled sorption and mass transfer phenomena at a larger scale using a triple-porosity model (TPM) with appropriate transfer functions (Vermeulen, 1953; Zimmerman et al., 1993; Zhang et al., 2022). Here, we extend our previous work (Lyu et al., 2021) to interpret and predict gas loading and production behavior for a full-diameter shale core. We use the same full-diameter core from the Marcellus formation as in our previous work (Lyu et al., 2021) and report on a series of measurements with methane (CH4), which is the major component of shale gas. Companion measurements of CH4 adsorption isotherms on a small cube sample from the same depth of the Marcellus formation were performed. Extending on our previous work, we combine gas loading experiments with separate sorption measurements to predict the subsequent gas production behavior. 5.2. Experimental approach 5.2.1. Sample preparation The full-diameter core and the shale cube sample were extracted from the same depth of the Marcellus formation in the Appalachian Basin. The samples contain about 30 vol.% of clay and 7.5 wt.% of TOC, with an average pore diameter of ~10 nm (for further details, see Roychaudhuri et al., 2013). Details for the sample preparation of the cube (~1 cc) can 153 be found in our group’s prior work (Wang, 2016). The shale core sample was extracted vertically (perpendicular to the bedding plane) from the subsurface, with a diameter of 8.89 cm (3.5 in) and a length of 18.03 cm (7.1 in). The core was slabbed after extraction, and an impermeable heavy-duty epoxy piece, molded using the Pro-set M1012 resin/M2010 hardener and carefully machined and trimmed to reduce surface roughness, was used to complete the cylindrical shape of the slabbed shale core. The gap between the epoxy piece and the main core body is intended to mimic an induced macro-fracture parallel to the overburden stress, as illustrated in Figure 5.1 below. Additional information regarding sample preparation is also available in Section 4.2.1 or Lyu et al. (2021). Figure 5.1. Shale cube and core and epoxy insert assembly (not scaled to size) 5.2.2. Porosity and sorption isotherm experiments – Shale cube We measured the porosity of the sample cube by a series of He expansion experiments at room temperature (25 °C) using a porosimeter. The excess sorption isotherm for pure CH4 on the shale cube was previously measured by employing a high-precision magnetic suspension balance (TGA) at 50 °C (see Wang, 2016 for details). The steady-state TGA measurements generated the excess sorption isotherm for CH4 in a stepwise manner, from 0 to 98 bar. 154 5.2.3. Gas expansion and depletion experiments – Full-diameter core We used a high-pressure experimental system that can accommodate full-diameter cores up to 10.16 cm (4 in) in diameter and 30.48 cm (12 in) in length and operate at pressures up to ~400 bar. A schematic of the experimental setup utilized is shown in Figure 5.2 below, and includes the following components: A – gas cylinder, V1, V2, V3, V4, V5, V6, V7 – needle valves, G1, G2, G3, G4, G5 – pressure gauges, B – reference vessel, C – coreholder, D – hydraulic pump, E – vacuum pump, BPR – back-pressure regulator, F – mass flow meter (MFM) (0-1000 scc/min), G – MFM (0-100 scc/min), H – MFM (0-10 scc/min). The reference vessel B, core-holder C, and pressure gauges G2 and G4 were installed inside a Despatch LBB 2-27 oven to maintain constant temperature conditions during the experiments. To keep the system’s dead volume to a minimum, stainless-steel tubing with an outer diameter (OD) of 1/16” and an inner diameter (ID) of 0.043” was selected. In a similar manner, the tee-connections (HiP fittings) and valves were selected based on their small interior volumes (0.02-0.06 cm3 ). The entire internal volume of the system was calculated to be around 15 cm3 and was taken into consideration throughout our subsequent data processing and interpretation. The confining pressure was applied in the radial direction, perpendicular to the core axis (uniaxial core-holder) by the hydraulic pump D. 155 Figure 5.2. Schematic of the high-pressure experimental system for CH4 expansion and depletion experiments The core studied in this work was previously characterized using He and Ar (Lyu et al., 2021). Following the same protocol for the gas expansion experiments (loading), CH4 (with an ultra-high purity of 99.97%) was loaded into the reference vessel B at a high pressure (~87 bar for each experiment) and monitored with the pressure gauge G2 at a resolution of 6.89×10-3 bar. Meanwhile, valve V5 was kept open to evacuate the core and the internal volume of the system via the vacuum pump E. After evacuation of the core and the downstream portion of the system (from V2 to V6) for 9+ days at a pressure of 6.6×10-3 bar (G4) and a temperature of 49 C, most of the removable moisture in the core and the rest of the system was evacuated, and V5 was closed. More importantly, the exact same protocol was used throughout all loading experiments, to maintain a consistent residual moisture content in the core. After introducing a confining pressure via hydraulic pump D, valve V2 was opened to allow gas (CH4) to expand into the evacuated core in the core- 156 holder C at a constant confining pressure. Gas initially fills the macro-fracture (the space in between the epoxy and the core – see Figure 5.1) and subsequently transports into the shale matrix. The mesoporous region and any micro-cracks present in the matrix will be filled first, followed by the micropores. As reported by Xu (2013), the specific surface (internal) area of the shale sample was 33.52 m2 /g, which corresponds to a total internal surface area of 7.4×104 m2 for the full-diameter core. Even considering the potential surface roughness of the epoxy, macro-fracture, and any void volumes in the system, the total surface area belonging to these regions would be several orders of magnitude smaller than that of the shale matrix. Therefore, we assume that sorption can only take place in the shale matrix. During the expansion experiment, we continuously monitored the pressure in the reference vessel B via the pre-calibrated pressure gauge G2 (3D Instruments 76514- 35B55 Digital Test Gauge, Vac To 340 bar (5000 Psi), ±0.05% FS accuracy) and recorded the pressure decay over time. Once the pressure in the reference vessel B had stabilized (a pressure drop of less than 6.89×10-3 bar over a 5 hr period, resolution of G2), we concluded that a steady state had been achieved and that the core was ready for a depletion/production experiment to commence. The gas stored inside the core was then released during the depletion experiment (unloading) by controlling the pressure downstream of the core using the BPR with only valve V6 open. The produced gas flow rates were recorded by the three precalibrated MFM’s (Aalborg, Mass Flow Meter GFM, ±1% FS accuracy): F, G, and H. 157 5.3. Modeling approach 5.3.1. Sorption isotherm modeling – Shale cube Our previous work has demonstrated that the Langmuir isotherm model accurately describes the Ar sorption behavior in the shale cube sample (Lyu et al. 2021) and we, therefore, use the same approach here to represent the CH4 sorption isotherm at 50 °C: The Langmuir model describes the absolute adsorption isotherms Cs (mol/kg) as a function of bulk-phase pressure p (bar) according to Eq. 5.1: = , = , ∙ 1+∙ , (5.1) where Cs,max (mol/kg) and K (bar-1 ) represent the maximum absolute sorption capacity of the shale and the equilibrium constant, respectively, while θ is the fractional coverage (dimensionless). To utilize the Langmuir model, the measured excess sorption data Cs ex (mol/kg) must be converted into absolute sorption by the following relationship: (1 − ) = . (5.2) This conversion requires knowledge of ρ g , the molar density of the bulk-phase gas (mol/m3 ), and ρ a , the molar density of the adsorbed-phase (mol/m3 ). The bulk CH4 density (and viscosity) is readily available from the NIST webbook (Lemmon et al., 2005). The adsorbate density, in contrast, is not known, and we proceed here by testing different adsorbed phase density models (see more details below). By combining Eqs. 5.1 and 5.2 158 and matching the excess sorption data, the Langmuir adsorption parameters, Cs,max (mol/kg) and K (bar-1 ), can be estimated for CH4 for each adsorbate density model. 5.3.2. Modeling of gas loading and production – Shale core To interpret gas loading and to predict the gas production experiments with the fulldiameter core, we utilized the TPM that was introduced and described in detail in our previous work on shale characterization with He and Ar (Lyu et al., 2021). Following the schematic provided in Figure 5.3, the overall pore volume (Vt) of the shale core is divided into the volume of the macro-fracture (i.e., the space in between the epoxy insert and the shale core), Vf (cm3 ), and the total pore volume of the shale matrix, Vp (cm3 ). The matrix pore volume consists of the mesopore volume, Vm (cm3 ), which includes the mesopores and any microcracks in the shale matrix, and the micropore volume Vμ (cm3 ). We assume that the upstream (reference + tubing) and downstream (tubing + BPR) volumes, Vref and Vd respectively, only exchange mass through the macro-fracture (Vf ). Figure 5.3. Conceptual model for interpretation of shale core experiments Gas flow in the macro-fracture (between core and epoxy) is described by viscous flow between parallel plates, corresponding to an effective fracture permeability kf (m2 ). As for 159 the mass transfer in the shale matrix, by volume-averaging the matrix under the assumption that gas must transport from the main fracture to the mesopores before reaching the micropores, the mass exchange rate between the main fracture and the mesopores, and the mass exchange rate between the mesopores (m) and micropores (μ) are described by transfer functions. The mass transfer functions are derived from the work of Vermeulen (1953) and include: a) the product of a shape factor σj (m -2 ) (j=m,μ) and the effective diffusivity Dj (m 2 /s) (j=m,μ), which can be considered as the inverse of the characteristic time for mass transfer in the relevant shale domain; b) the average concentrations of gas in the macro-fracture, mesopores, and micropores; and c) a correction factor that accounts for gradients within each volume-averaged domain (additional details are provided in 4.3.3). The key assumption of the mass transfer functions (following the work of Vermeulen, 1953 and Zimmerman et al., 1993) is that the mass transfer rates in mesopores/microcracks and micropores are similar to or faster than the adsorption rates. This assumption has been validated for Ar in a previous study (Lyu et al., 2021), and considering the similarity in sorption and transport behaviors observed for Ar and CH4 (Salem et al., 1998; Dasani, 2017), we consider the assumption to be equally valid for CH4. We note that the characteristic times for mass transfer, associated with the meso- and micro-porous subdomains of the matrix, provide an average/approximate representation of the relevant physical processes as discussed above. Accordingly, for non-sorbing gases (e.g., He), the model includes 4 parameters: the effective permeability of the macro-fracture (kf ), the characteristic time for mass transfer in mesopores (1/σD) m and micropores (1/σD) μ , and the fraction of the matrix pore volume that resides in the mesopores (α=Vm/Vp ). As discussed further below, these parameters can be extracted from pulse-decay experiments with He. 160 For adsorbing gases, we use a dynamic Langmuir model: = ((, − ) − 1 ), (5.3) where Cs is the absolute sorption (mol/kg), ρ s is the skeletal (solid) density (kg/m3 ), b is the internal mesopore/micropore surface area per unit skeletal volume (m2 /m3 ), and ka is the rate constant for adsorption (kg/bar/m2 /s). With Cs,max and K available from the sorption isotherm (see Eq. 5.1), we introduce the characteristic time for adsorption, ts , as follows: = , (5.4) where pmax is the maximum pressure (bar) of interest. He expansion/loading experiments were performed and interpreted previously (Lyu et al., 2021) to calibrate the model parameters, including the effective fracture permeability, kf , characteristic time for mass transfer in mesopores/microcracks and micropores, (1/σD) m and (1/σD) μ , and the fraction of the total pore volume that resides in the mesopores/microcracks, α=Vm/Vp . To interpret and predict CH4 loading and depletion experiments, however, the mass transfer rates in the shale matrix, (σD)m and (σD)μ , must be translated from He to CH4. σm and σμ , were considered unchanged because the experiments were following the same protocol, and we assume that there was no change in the matrix between experiments. 161 To translate the effective diffusivities Dj (j=m, μ) between He and CH4, we investigate the ratio of diffusivities for a single cylindrical pore with an average pore size that is representative of the shale matrix: the gas transport in the mesopores falls into the slip/transition flow regime at relevant conditions, while Knudsen diffusion will dominate transport in the micropores (see Appendix B). Mass transfer between the mesopores and micropores is assumed to only happen in the free phase, considering the complex pore structure and geometry in the shale matrix. Hence, surface diffusion can be ignored not only due to its modest contribution (less than 5%) to the overall flux in the mesopores (Song et al., 2018; Liu et al., 2021), but also because the surface/gas interaction, including adsorption, desorption, and surface diffusion, is controlled by the characteristic time of sorption, ts . Based on the investigation for Ar in our previous work (Lyu et al., 2021), the characteristic times for adsorption and diffusion are comparable, which means the adsorption rate is the bottleneck for the mass transfer on the internal surfaces. It is therefore reasonable to ignore the surface diffusion in the micropores, in this work, according to the similarity of Ar and CH4, and this assumption is further supported by the modeling results (see Section 5.4). The translation of the transport in the mesopores, was investigated by computing the effective diffusivity Dm from two models, including 1) BKC: = (1 + ) ∙ , (5.5) = ∙ , = 2 8 , (5.6) 162 = (1 + 1.358 1+0.17 −0.4348) [1 + 4 1+ ], (5.7) and 2) DGM: Dm= p μ B (1+C z dz dC)+Dk , B= rp 2 8 , Dk= 2rp 3 √ 8RT πMw . (5.8) Here μ (bar·s) is the gas viscosity, kapp and kint (m2 ) are the apparent and the intrinsic permeability, respectively. C (mol/m3 ) is the gas concentration and z is the gas compressibility factor. f c is the BKC correction factor. B (m2 ) is the viscous flow parameter and Dk (m2 /s) is the Knudsen diffusion coefficient. R (J/mol/K) is the gas constant, T (K) is the temperature, and Mw (kg/mol) is the molecular weight. The Knudsen number, Kn, was calculated at relevant conditions and are reported in Appendix B. We use an average pore diameter (Dp=2rP) of 10 nm for the mesopores, based on previous BET analysis of several relevant Marcellus shale samples (Roychaudhuri et al., 2013). The characteristic time for mass transfer, t * , in the mesopores can then be estimated as a function of pressure from tm * =Lc 2 /Dm , where Lc (m) is the characteristic length of the sample: For a 1 cm3 shale cube, the characteristic length Lc is ~ 0.0062 m (from a sphere with equivalent volume). The characteristic time for mass transfer of He and CH4 and their ratio at relevant experimental conditions are reported as a function of pressure in Figure 5.4. 163 Figure 5.4. Characteristic time for mass transfer in a single cylindric pore with a 10 nm diameter using the DGM and BKC models This allows us to rescale the transport parameters (σD)m from the He experiments to the calculations for CH4 by considering the ratio of characteristic times. We note that the characteristic time based on BKC is not monotonic and that differences in the ratio of characteristic times between DGM and BKC are modest (~10%). In this work, we proceed with the results from the DGM. As discussed above, the translation of the characteristic time for mass transfer in the microporous fraction of the matrix is performed by scaling the characteristic time with the ratio of molecular weights of CH4 and He as dictated by Knudsen diffusivity. This, in turn, assumes that the characteristic time for adsorption is comparable to (or slower than) the characteristic time for transport in the free phase, as demonstrated for Ar by Lyu et al. (2021). 164 5.4. Results 5.4.1. Sorption isotherms – Shale cube The porosity of the shale cube was measured in a series of four expansion experiments using a porosimeter at room temperature with a non-sorbing gas, He. The temperature difference is ignored since each porosity test was promptly performed after 48 hours of evacuation of the cube at 50 °C. The results are tabulated in Table 5.1. The average porosity of this cube was 8.9%. Table 5.1 includes the 95% confidence levels, considering the uncertainty associated with the pressure transducer (Mensor DPG 2400, ±0.03 psig), temperature variations during the gas expansion (±1 °C), and measurement errors for the bulk volume of the shale cube (±0.005 cm3 ). Table 5.1. Porosity measurements for the shale cube sample using He Experiment No. I II III IV average Porosity (%) 9.6±1. 77 8.3±1.78 9.8±1.76 8.1±1.76 8.9±1.77 In Figure 5.5(a), the excess adsorption data, measured via the TGA system by Wang (2016), are shown as black diamonds. The calibration process for the setup, including He buoyancy tests (see e.g., Lyu et al., 2021), results in highly accurate measurements of the apparent sample mass (±5 µg) and the associated excess adsorption (±0.001 mol/kg). Error bars are therefore not included in Figure 5.5. The calculated excess adsorption isotherms, reported in the same figure, were obtained by using the Langmuir isotherm (Eq. 5.1) with parameters (K and Cs,max) extracted from the experimental data. This approach requires an adsorbate phase density model to correlate absolute and excess adsorption via Eq. 5.2. 165 Here, we consider five different models for the adsorbate density ρ a : 1) ρmax , the normal boiling point density; 2) ρ crit, the density at the critical point; 3) ρmean , the average of ρmax and ρ crit ; 4) lattice density functional theory (LDFT) of Ottiger et al. (2008) that interpolates (nonlinearly) between ρmax and ρ crit based on the fractional coverage θ; 5) ρmax θ, a linear function of the fractional coverage. The model parameters Cs,max (mol/kg) and the equilibrium constant K (bar-1 ), corresponding to each density model, are reported in Table 5.2. Table 5.2: Langmuir parameters corresponding to the various adsorbate density models Model: ρmax ρ crit ρmean LDFT ρ max∙θ Cs,max (mol/kg) 0.29 0.52 0.33 0.32 0.31 K (bar-1 ) 0.032 0.014 0.027 0.031 0.032 We observe from Figure 5.5(a) that all but the ρ crit adsorbate density model represent the experimental excess adsorption data reasonably well. However, for the four density models that substantially agree with the experimental data, there are variations between the corresponding calculated absolute adsorption isotherms. The ρmax model predicts the smallest values of absolute adsorption, while the LDFT model predicts the largest values. The difference between the two models is <10% over the whole range of pressures studied, and for the subsequent modeling work, the ρmax density model was selected due to its simplicity. 166 Figure 5.5. (a) CH4 excess isotherm data and models using various adsorbate density models; (b) corresponding calculated absolute adsorption isotherms 5.4.2. Gas loading experiments – Shale core Following the experimental approach outlined above, two CH4 gas loading experiments (exp. A and exp. B) were performed. Figure 5.6 reports the results from these two loading experiments, starting with an initial pressure in the reference cell at ~ 87 bar at 49 °C and a constant confining pressure of 139 bar. The net stress is defined as the difference between the total stress (confining pressure) and the pore pressure σ=σc -p (bar). The net stress will thus decrease from 139 bar to around 67 bar as a result of the increase in pore pressure during the loading process. From our previous work (Lyu et al., 2021), with relevant pressures and volumes reported in the preceding chapter (Table 4.2), the total void volume within the core (Vf+Vp ) was estimated, from the initial and final pressures of He using gas compressibility data from the NIST database (Lemmon et al., 2005). We arrive at a total pore volume of 44.4 cm3 , from which the porosity of the core was calculated to be ϕ core = 44.4⁄855.3 =5.2% (at a net stress of ~80 bar). The uncertainty on the pressure (a) (b) 167 readings represents the fluctuations of the gauge (±0.02 bar), rather than the full-scale (FS) uncertainty (±0.17 bar), given the moderate range of pressures observed during the loading experiment. Following the same approach, the total quantity of CH4 loaded into the core was calculated from the initial and equilibrium pressures from the two experiments. We note that the uncertainties outlined above for the He expansion tests are also in play for the CH4 experiments (see Table 4.2). To fit this quantity of CH4 into the core, the corresponding core total pore volume (Vf+Vp ) should be 125.1 cm3 and 119.8 cm3 , respectively. The larger values for Vf+Vp , relative to 44.4 cm3 inferred from He loading experiments, indicate that a large amount of CH4 gas is stored in the shale core as an adsorbed phase. The difference of 5.3 cm3 in Vf+Vp observed between exp. A (125.1 cm3 ) and exp. B (119.8 cm3 ) can be attributed to insufficient evacuation prior to exp. B (23 days for exp. A vs. 9 days for exp. B). This reduction in pore volume may also be due to the residual adsorbed phase and is consistent with the observed sorption hysteresis behavior discussed in more detail below. 168 Figure 5.6. CH4 loading experiments at 49 °C - Error bars of ± 0.02 bar are not included The mass transfer parameters, including kf , (σD)m, (σD)μ , and Vm/Vp , as obtained from the analysis of a previous He loading experiment, including 95% confidence limits, are reported in the previous chapter (Table 4.3). To interpret and predict the CH4 behavior in the shale core, the mass transfer parameters, (σD)m and (σD)μ , must be rescaled to reflect the difference between the transport of He and CH4. Since slip/transition flow in the mesopores falls in between the viscous flow and Knudsen diffusion regimes, the ratio of the characteristic times for mass transfer for CH4 and He at the average pressure (36 bar) was selected to estimate the rescaling factor of 1/1.6 = 0.63 - (see Figure 5.4). Gas transport in the micropores (Dp<2 nm) is assumed to be dominated by Knudsen diffusion, and as discussed above, the ratio of the molecular weights of CH4 and He was thus used to calculate the rescaling factor of √4/16 = 0.5. 169 As the preceding chapter, to model the sorption behavior of CH4 in the core, we rescale the estimated adsorption capacity Cs,max from the cube, based on the difference in porosities between the core and the cube, to account for the variations in the adsorbent surface area accessible in the core (see details in Section 4.4). Accordingly, the core sorption capacity was calculated as: Cs,max,core=Cs,max,cube∙ pore volume per core weight pore volume per cube weight =0.29 mol/kg× 44.4 cc/2024 gram 0.16 cc/3.92 gram =0.16 mol/kg. We note that the cube isotherm was measured at 50 °C, while the core tests were performed at 49 °C (due to a drift in the thermostat of the oven). However, according to the CH4 isotherms reported at various temperatures by Wang et al. (2015), this minor difference in temperature will only have a marginal impact on the sorption capacity. With the Langmuir parameters Cs,max and K for CH4 available, only one additional model parameter, bka , is needed to describe the CH4 loading (dynamic) behavior. We, therefore, estimate bka by matching the model to one of the CH4 loading experiments using a nonlinear least squares approach (Levenberg-Marquardt algorithm). We selected loading exp. A for this purpose due to the more complete evacuation prior to the experiment and report the result in Figure 5.7 for the fitted value of bka of 6.25×10-6 (g/bar/cm3 /s), corresponding to a root-mean-square error (RMSE) of 0.36 bar. 170 Figure 5.7. CH4 loading experiment A - estimation of bka - RMSE = 0.36 bar Since the sorption behavior in the shale core is described based on sorption measurements on the shale cube, the impact of any kerogen swelling due to adsorption should be included in the shale cube data. However, potential kerogen shrinkage/swelling due to net-stress changes is not accounted for in the cube experiments, which were carried out in the TGA setup, and is therefore not considered in the analysis presented here. An investigation of potential kerogen shrinkage/swelling in shale samples under different net-stress conditions is deferred, instead, to future studies. With all model parameters available, we then predicted the behavior of loading exp. B, see Figure 5.8, which provides a comparison of experimental observations and modeling results. 171 Figure 5.8. Prediction of the CH4 loading experiment B – RMSE = 0.52 bar For consistency, we initialized the model with a residual gas in place (both in the reference cell and the core) equal to 329 scm3 (20°C and 1.01325 bar). The deviation between the data and the modeling results in the early stage (within the first hour of loading) was likely due to a short fluctuation in the temperature control (4 C maximum) of the oven when its door was opened to adjust valves (because the pressure gauge and the reference vessel are close to the oven door, this temperature fluctuation could have potentially shifted the gas pressure by about 1.5 bar, as observed in loading exp. B). The presence of residual gas was reported in Figure 5.6 and is assumed to represent an adsorbed phase from the previous loading test (exp. A) that could not be extracted during the 9 days of evacuation before loading exp. B was initiated. This observation also indicates that sorption hysteresis may have a significant impact on production behavior (see further discussion below). At the end of each loading experiment, the amount of gas accumulated in the core (not all the gas in 172 the system due to the presence of dead volumes) was estimated from the modeling results and is tabulated in Table 5.3. We note that approximately 2/3 of the total gas in the core is stored as adsorbed gas. Table 5.3: CH4 in place calculations (scm3 ) at the end of loading experiments A and B exp. macrofracture mesopores/microcracks micropores whole core Free (scm3 ) Free (scm3 ) Adsorbed (scm3 ) Total (scm3 ) Free (scm3 ) Adsorbed (scm3 ) Total (scm3 ) Total (scm3 ) A 1.35 1977 4159 6136 509 1089 1598 7735 B 1.36 1990 4167 6157 507 1087 1594 7753 5.4.3. Gas production experiments – Shale core Following each CH4 loading experiment (exp. A and exp. B), a depletion/production experiment was performed. Depletion exp. A was performed with a single-step pressure drawdown, while depletion exp. B was performed with a four-stage pressure drawdown (constant pressure steps). Both depletion experiments were completed when core pore pressure reached atmospheric conditions. For exp. B, the outlet pressure (BPR) was reduced in four steps of ~ 17 bar at a time. For each step, unloading was assumed to have been completed when the gas flow rate exiting the core was lower than 0.1 scm3 /min (the lower limit of an accurate measurement of the MFM in the experimental setup). Figure 5.9 reports the cumulative gas production for the two depletion experiments. The MFMs employed in this study were calibrated using a bubble flow meter, resulting in an accuracy 173 of ±1%. An additional ±0.5% of error may be introduced by the temperature variations in the laboratory (~0.2%/°C sensitivity), as the MFMs were not located inside the oven. Consequently, the cumulative production data is associated with an uncertainty of ±1.5%, and we present the data (e.g., Figure 5.9) without including error bars. Figure 5.9. Cumulative CH4 production during depletion experiments A and B. Accuracy of cumulative production is ±1.5% Based on the model parameters obtained from the loading experiments, initial predictions for the depletion experiments were performed, and Figure 5.10 compares the calculated and experimental gas production rates for the two experiments. For both depletion experiments, model predictions are in good agreement with the experimental observations at the early stage of the production process. However, at later times, when the gas pressure approaches atmospheric conditions and gas desorption dominates the production, the model overestimates the gas production. This behavior is indicative of hysteresis behavior 174 between adsorption and desorption processes (at least at the time scale over which these depletion experiments were performed). Figure 5.10. Predictions of gas depletion rate data for experiments A and B In a recent study by our group, investigating Ar sorption on a ~1 cm3 Marcellus shale cube, a similar behavior was observed: steady-state adsorption and desorption experiments were conducted in a stepwise manner, whereby the bulk gas pressure was increased from vacuum conditions to ~ 110 bar and then gradually reduced back to vacuum. The sample weight at the end of the desorption process, after more than 23 hours of evacuation, was larger than the initial weight, prior to the start of the experiments. The residual gas was measured to be 0.0134 moles of Ar adsorbed per kg of shale. Since Ar has a sorption potential and a kinetic diameter that are similar to those of CH4 (Salem et al., 1998), it is likely that a similar hysteresis will also be observed for CH4. 175 To account for hysteresis during the depletion experiments, we introduce a simple hysteresis model that shifts the isotherm according to ′ = + ( − ), (5.9) where p ' (bar) is the desorption pressure used in Eq. 5.3 to calculate desorption rates during a depletion process, p (bar) is the absolute pressure. pmax (bar) is the maximum desorption pressure, i.e., the pressure where the desorption process starts. δ is a positive dimensionless empirical parameter that reflects the shift between adsorption and desorption isotherms: If δ is set to zero, the sorption is fully reversible; if δ is greater than zero, the model represents sorption hysteresis; δ has an upper limit of 1 that represents the maximum hysteresis, where no adsorbed gas will desorb during production; negative values of δ are not physically meaningful as the desorption isotherm would fall below the adsorption isotherm. We then estimated the value of δ by matching the model to the experimental observations from depletion exp. A and arrived at a value for δ = 0.050. A comparison of the final modeling for both experiments, including hysteresis, with the observed behavior is also provided in Figure 5.10: Application of the simple hysteresis model provides improved agreement between experimental data and the model. A comparison of adsorption and desorption isotherms from the hysteresis model is provided in Figure 5.11. We observe a difference of about 0.02 (mol/kg) in the sorption at low pressures, which is in reasonable agreement with the observation for Ar on the shale cube. 176 Figure 5.11. Sorption hysteresis model with δ = 0.050, as estimated from depletion experiment A The original gas in place (OGIP) (void volume plus core pore volume) for the two depletion experiments was calculated to be 8,741 scm3 and 8,767 scm3, respectively, based on data from the corresponding loading experiments and a mass balance. Accordingly, any potential errors in these calculations arise solely from the uncertainties associated with the pressure gauge (±0.02 bar) and upstream volume (±0.5 cm3). This translates into uncertainties of ±31 scm3 for exp. A and ±33 scm3 for exp. B. At the end of each depletion experiment, the gas recovery factor (RF) was calculated from the cumulative gas production. The observed cumulative production and related gas recovery are reported and compared to calculations with and without sorption hysteresis in Table 5.4. Based on the calculations performed, including the hysteresis model, we can further break down the cumulative production to investigate the production from the original adsorbed gas. For depletion experiment A, 77.9% (3242 scm3 ) and 65.1% (709 scm3 ) of the adsorbed 177 gas in the mesopores and micropores are produced. For the depletion experiment B, 77.8% (3244 scm3 ) and 68.1% (or 740 scm3 ) of the adsorbed gas is produced from the mesopores/microcracks and micropores, respectively. Table 5.4: Gas production and recovery for depletion experiments A and B. Dep. exp. Experiment data Calculation results Without hysteresis With hysteresis Prod. (scm3 ) RF (%) Prod. (scm3 ) RF (%) Prod. (scm3 ) RF (%) B (stage 1) 1240±19 14.1±0.3 1436 16.4 1412 16.1 B (stage 2) 1678±25 19.1±0.4 1568 17.9 1513 17.5 B (stage 3) 1709±26 19.5±0.4 1843 21.0 1728 19.7 B (stage 4) 2523±39 28.8±0.5 3068 35.0 2717 31.0 B (total) 7150±108 81.6±1.5 7915 90.3 7370 84.1 A (total) 7334±110 83.9±1.6 7876 90.1 7336 83.9 5.5. Summary and Discussion In this work, we have studied CH4 mass transfer and sorption behavior in a full-diameter shale core from the Marcellus formation under variable net-stress conditions by performing gas loading and depletion experiments. Two sets of CH4 loading/depletion experiments were conducted at 49 °C, for which the gas pressure and production rate were monitored continuously. The two loading experiments were interpreted based on a triple-porosity model (TPM) that has been introduced and validated in our previous work (Lyu et al., 178 2021), combined with CH4 sorption data from sorption isotherm experiments (Wang, 2016) with a small shale cube (from the same depth as the core). The experimental CH4 excess sorption data were interpreted using different adsorbate density models on the assumption that absolute adsorption follows a Langmuir isotherm. Four out of five of these models provide good agreement with the excess adsorption data. Differences, however, exist between the corresponding calculated absolute adsorption isotherms. Additional investigations are therefore much needed to determine the appropriate adsorbate phase density model, since different model choices will result in different estimates of the absolute sorption capacity and will have an impact on the estimation of gas-in-place at reservoir scale. We interpreted the CH4 loading experiments (exp. A and exp. B) on the full-diameter core using the TPM, with mass transfer parameters for CH4 rescaled from previous He experiments: The permeability of the macro-fracture (~1.5 × 10-9 cm2 or ~ 150 millidarcy) and the ratio between mesopore and micropore volumes (~79% of the pore volume exists in the mesoporous range) were taken directly from the He study. The characteristic times for mass transfer in mesopores and micropores were rescaled based on the ratio of characteristic times between He and CH4. We then estimate the characteristic times for CH4 mass transfer between macro-fracture/mesopores (~1,200 s) and mesopores/micropores (~22,000 s), while the estimated characteristic time for CH4 sorption (~ 5,600 s) falls in between the time scales for mass transfer in the meso/microporous regions of the shale matrix, and thus justifies the use of the mass transfer functions in the TPM. 179 The first depletion experiment (exp. A) was performed in one step, with gas being produced continuously until the pore pressure reached atmospheric conditions. The second depletion experiment (exp. B) was performed in a stagewise manner, in four equal pressure steps, via the use of a back-pressure regulator (BPR). Exp. A shows a slightly higher recovery (+2%) of the original gas in place (OGIP), potentially resulting from experimental errors (refer to Table 5.4). It is important to note that any error associated with the upstream volume would affect both experiments in the same way. Therefore, the difference in the recovery factor (RF) between Exp. A and Exp. B would be attributed to the uncertainties in the mass flow meters (MFMs) and the pressure gauge. These two factors, in combination, introduce approximately ±1.3% of error in the experiments and can hence account for the ~2% difference observed in the RF. In both experiments, approximately 20% of the OGIP remained in the core (as residual gas) at the end of the production period. We attribute the residual gas to CH4 sorption hysteresis that introduces a shift between desorption and adsorption for the same bulk phase gas pressure. As a result, some of the adsorbed gas was not produced, even at low pressure. A similar behavior has been observed in separate Ar sorption measurements on a shale cube (~1 cc). We model this behavior using a simple sorption hysteresis model by introducing a ‘lag’ in the bulk pressure during the gas desorption process. We subsequently estimated this desorption ‘lag’ from one of the two depletion experiments and predicted the data from the other depletion experiment, with good agreement. 180 5.6. Conclusions As a central contribution of this work, we demonstrate that sorption behavior, studied with a small-scale (cube) shale sample, combined with the study of mass transfer behavior at a larger scale (full-diameter core), provides for reasonably accurate predictions of the coupled phenomena at the larger scale. It is also important to note, that the sorption behavior measured at zero net-stress allows for the prediction of combined sorption and mass transfer behavior in shales under variable net-stress conditions. This indicates that gas reserves in shale formations can be accurately estimated based on lab-scale sorption isotherm measurements, provided that the adsorbate density is known. We demonstrated that adsorbed CH4 is likely to be trapped as residual gas during the production/depletion process, and that this behavior can be explained and modeled by sorption hysteresis. Finally, a triple-porosity model (TPM) was further validated as a feasible method for translating sorption isotherms across length scales and for predicting coupled mass transfer and sorption in shale without the need for a detailed discretization of the shale matrix. 5.7. Nomenclature B viscous flow parameter, m2 ϕ core porosity of the core, dimensionless b internal mesopore/micropore surface area per unit skeletal volume, m2 /m3 R gas constant, J/mol/K C gas concentration, mol/m3 rP characteristic radius of pores, nm 181 Cs absolute adsorption, mol/kg ρ a molar density of the adsorbed phase, mol/m3 Cs ex excess adsorption, mol/kg ρ crit molar density of the gas phase at the critical point, mol/m3 Cs,max maximum absolute adsorption capacity, mol/kg ρ g molar density of the bulkphase gas, mol/m3 Cs,max,core maximum absolute adsorption capacity on the shale core, mol/kg ρmax molar density of the adsorbed phase gas (normal boiling point density), mol/m3 Cs,max,cube maximum absolute adsorption capacity on the shale cube, mol/kg ρmean the average of ρmax and ρ crit, mol/m3 Dk Knudsen diffusion coefficient, m2 /s ρ s density of the shale skeleton, kg/m3 Dm effective diffusivity of mesopores, m2 /s T temperature, K Dμ effective diffusivity of micropores, m2 /s t * characteristic time for mass transfer, s Dp characteristic diameter of pores, nm ts characteristic time for adsorption, s δ empirical parameter of sorption hysteresis, dimensionless θ fractional coverage, dimensionless f c BKC correction factor, dimensionless σ net stress on the shale sample, bar K adsorption equilibrium constant, bar-1 σc confining pressure on the shale sample, bar ka rate constant for adsorption, kg/bar/m2 /s σm shape factor for mesopores, m-2 182 kapp apparent permeability, m2 σμ shape factor for micropores, m-2 kf effective fracture permeability, m2 (σD)m mass transfer rates in the mesopores, 1/s kint intrinsic permeability, m2 (σD)μ mass transfer rates in the micropores, 1/s Kn Knudsen number, dimensionless Vcore bulk volume of the shale core, cm3 Lc characteristic length, m Vd downstream volume, cm3 Mw molecular weight, kg/mol Vf volume of the macro fracture(s), cm3 μ gas viscosity, bar·s Vm mesopore volume, cm3 p pressure of the bulk-phase gas, bar Vμ micropore volume, cm3 p ' desorption bulk-phase pressure, bar Vp total pore volume of the shale matrix, cm3 pend gas pressure in the end of an experiment, bar Vref upstream volume, cm3 pini initial gas pressure, bar Vt overall pore volume of the shale core, cm3 pmax maximum pressure, bar z gas compressibility factor, dimensionless 183 5.8. Appendix Appendix A - Impact of confinement on Methane density The apparent critical properties of methane (CH4) can be affected by confinement as a function of the ratio of the molecule to the pore size (Zarragoicoechea and Kuz, 2002; Sanaei et al., 2014b). Figure A2 demonstrates the estimated variations in critical pressure and temperature for CH4 as a function of pore size. Figure A2. Shift in critical properties of CH4 as a function of pore size However, the impact on the fluid density (using the Peng-Robinson equation of state) is marginal for the experimental conditions of this work (see Figure A3). 184 Figure A3. Concentration of CH4 under confinement at different pressures (T=50 ºC) Appendix B – Knudsen numbers and transport regimes At relevant experimental conditions, the Knudsen number, Kn, can be calculated for a single cylindrical pore by: = ℎ , (5.10) where lh (m) is the characteristic length of the pore, which generally is considered as the hydraulic diameter Dh=2Rh , while the BKC model uses the hydraulic radius Rh . λ (m) is the mean free path of molecules, given by (Loeb, 1934): = √ 2/ = √ 2∙ , (5.11) 185 where ρ (kg/m3 ) is the gas density and C (mol/m3 ) is the gas concentration. R (J/mol/K) is the gas constant, T (K) is the temperature, and Mw (kg/mol) is the molecular weight. Then the Knudsen number, Kn=λ⁄Dp , for CH4 transport in cylindrical pores of different diameters can be evaluated as shown in Figure C3. Figure C3. Knudsen number for CH4 (at 49 ºC) in a single cylindrical pore with varying diameters at relevant experimental conditions We observe that Kn is greater than one for the micropores (e.g., see the profile with Dp=1 nm) and between 0.01 and 100 for the mesopores (e.g., see the profiles with Dp=10 and 50 nm) at the relevant experimental temperature and pressure conditions covered in this work. Based on the classification of flow conditions according to Schaaf and Chambre (1961) and Civan (2010), the gas transport in this work is mainly controlled by Knudsen diffusion in the micropores, while it falls into the slip/transition flow regime for the mesopores. 186 The molar flux of the gas transport in the shale matrix can then be written as: = − = −, (5.12) = = , (5.13) where, ka is the apparent permeability (including contributions from relevant transport mechanisms). Further, Eq. 5.12 can be re-written and arranged as: F=-C ka μ ∇(C∙zRT)=-C ka μ zRT∙∇C-C ka μ CRT∙∇z=-C∙zRT ka μ ∇C [1+ C z dz dC ], (5.14) and the effective diffusivity can be computed as: = ∙ [1 + ]. (5.15) 187 5.9. References Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y., 2018. Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys. 353, 356–376. https://doi.org/10.1016/j.jcp.2017.10.023. Akkutlu, I.Y., Fathi, E., 2012. Multiscale gas transport in shales with local kerogen heterogeneities, SPE J. 17 (04) 1002–1011. Alnoaimi, K.R., Kovscek, A.R., 2019. Influence of microcracks on flow and storage capacities of gas shales at core scale. Transport Porous Media 127 (1), 53–84. https://doi.org/10.1007/s11242-018-1180-5. Beskok, A., Karniadakis, G., 1999. Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Therm Eng 3, 43–77, DOI: 10.1080/ 108939599199864. Boyer, C., Kieschnick, J., Suarez-Riviera, R., Lewis, R.E., Waters, G., 2006. Producing Gas from its Source, Oilfield Review. Cao, C., Zhao, Q., Gao, C., Sun, J., Xu, J., Zhang, P., Zhang, L., 2017. Discrete fracture model with multi-field coupling transport for shale gas reservoirs. J. Petrol. Sci. Eng. 158, 107–119. https://doi.org/10.1016/j.petrol.2017.08.033. Chen, L., Zhang, L., Kang, Q., Viswanathan H.S., Yao, J., Tao W., 2015. Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity. Sci Rep 5, 8089. https://doi.org/10.1038/srep08089 Cho, Y., Ozkan, E., Apaydin, O. G., 2013, May 1. Pressure-Dependent Natural Fracture Permeability in Shale and its Effect on Shale-Gas Well Production. Society of Petroleum Engineers. https://doi.org/10.2118/159801-PA. Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82, 375–384 Dasani, D. (2017). Investigation of Mass Transfer in Fractured Shales (Doctoral dissertation). University of Southern California, Los Angeles. 188 de Witt, Wallace, Jr., Roen, J.B., Wallace, L.G., 1993, Stratigraphy of Devonian black shales and associated rocks in the Appalachian basin, in Roen, J.B., and Kepferle, R.C., eds., Petroleum geology of the Devonian and Mississippian black shale of eastern North America: U.S. Geological Survey Bulletin 1909–B, p. B1–B57. EIA, 2021. Annual Energy Outlook 2020 with Projections to 2050. Energy Information Administration, Washington. Elkady, Y., Lyu, Y., Jessen, K., Kovscek, AR., Three-Dimensional Imaging and Quantification of Gas Storativity in Nanoporous Media via X-rays Computed Tomography. Energies. 2020; 13(23):6199. https://doi.org/10.3390/en13236199. Freeman, C., Moridis, G., Blasingame, T., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90, 253–268. Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., Krooss, B.M., 2014. Experimental study of fluid transport processes in the matrix system of the European organic-rich shales: I. Scandinavian Alum Shale. Mar. Petrol. Geol. 51, 79–99. Harper, J.A., 2008. The Marcellus Shale — an old “new” gas reservoir, in Pennsylvania Geology: Pennsylvania Department of Conservation and Natural Resources, v. 38, no. 1, 20 p. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Ca n. Petrol. Technol. 48 (8), 16–21. https://doi.org/10.2118/09-08-16- DA. Jessen, K., Tang, G.Q., Kovscek, A.R., 2008. Laboratory and simulation investigation of enhanced coalbed methane recovery by gas injection. Transp. Porous Media 73 (2),141– 159. Zarragoicoechea, G.J., Kuz, Z.A., 2002. Van der Waals equation of state for a fluid in a nanopore. Physical Review, Issue E 65, 021110(1-4). Lemmon, E.W., McLinden, M.O., Friend, D.G., 2005. In: Thermophysical Properties of Fluid Systems. NIST Chemistry Webbook, vol. 69. NIST standard reference database. 189 Liu, L., Wang, Y., Aryana, S.A., 2021. Insights into scale translation of methane transport in nanopores, Journal of Natural Gas Science and Engineering, Volume 96, 104220, ISSN 1875-5100, https://doi.org/10.1016/j.jngse.2021.104220. Loeb, L.B., 1934. The kinetic theory of gases, 2nd ed. McGraw-Hill Co. Inc., New York. Loschko M, W¨ohling T, Rudolph DL, Cirpka OA. An electron-balance based approach to predict the decreasing denitrification potential of an aquifer. Groundwater 2019;57(6):925– 39. Lyu, Y., Dasani, D., Tsotsis, T.T., Jessen, K., Characterization of shale using Helium and Argon at high pressures, Journal of Petroleum Science and Engineering (2021), doi: https://doi.org/10.1016/j.petrol.2021.108952. Mason, E.A., Malinauskas, A.P., 1983. Gas Transport in Porous Media: The Dusty-Gas Model. Nelson, P.H., 2009. Pore-throat sizes in sandstones, tight sandstones, and shales. AAPG (Am. Assoc. Pet. Geol.) Bull. 93 (3), 329–340. https://doi.org/10.1306/10240808059. Ottiger, S., Pini, R., Storti, G., Mazzotti, M., 2008. Measuring and modeling the competitive adsorption of CO2, CH4, and N2 on a dry coal. Langmuir 24 (17), 9531–9540. Peng, Y., Liu, J., Pan, Z., Qu, H., Connell, L., 2018. Evolution of shale apparent permeability under variable boundary conditions. Fuel 215, 46–56. https://doi.org/10.1016/j.fuel.2017.11.024. Roychaudhuri, B., Tsotsis, T.T., Jessen, K., 2013. An experimental investigation of spontaneous imbibition in gas shales. J. Petrol. Sci. Eng. 111, 87–97. Rubin Y., 2003. Applied stochastic hydrogeology. New York: Oxford University Press. Salem, M.M.K., Braeuer, P., Szombathely, M.V., Heuchel, M., Harting, P., Quitzsch, K., Jaroniec, M., 1998. Thermodynamics of high-pressure adsorption of argon, nitrogen, and methane on microporous adsorbents. Langmuir 14 (12), 3376–3389. Sanaei, A., Jamili, A., Callard, J., Mathur, A., 2014b. Production modeling in the Eagle Ford gas condensate window: integrating new relationships between core permeability, 190 pore size, and confined PVT properties. In: SPE Western North American and Rocky Mountain Joint Regional Meeting Held in Denver, Colorado, USA, 16-18 April 2014. Schaaf, S.A., Chambre, P.L., 1961. Flow of Rarefied Gases. Princeton University Press, Princeton, NJ. Sheng, G., Javadpour, F., Su,Y., 2018. Effect of microscale compressibility on apparent porosity and permeability in shale gas reservoirs, International Journal of Heat and Mass Transfer, Volume 120, Pages 56-65, ISSN 0017-9310, https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.014. Soltanian M.R., Ritzi R.W., Huang C., Dai Z., 2015. Relating reactive solute transport to hierarchical and multiscale sedimentary architecture in a Lagrangian-based transport model: 1. Time-dependent effective retardation factor. Water Resources 2015;51(3):1586– 600. Song, W., Yao, J., Li, Y., Sun, H., Zhang, L., Yang, Y., Zhao, J., Sui, H., 2016. Apparent gas permeability in organic-rich shale reservoir. Fuel 181, 973–984. Song, W., Yao, J., Wang, D., Li, Y., Sun, H., Yang, Y., 2020. Dynamic pore network modelling of real gas transport in shale nanopore structure. J. Petrol. Sci. Eng. 184, 106506. https://doi.org/10.1016/j.petrol.2019.106506. Song, W., Yao, B., Yao, J., Li, Y., Sun, H., Yang, Y., Zhang, L., 2018. Methane surface diffusion capacity in carbon-based capillary with application to organic-rich shale gas reservoir, Chemical Engineering Journal, Volume 352, Pages 644-654, ISSN 1385-8947, https://doi.org/10.1016/j.cej.2018.07.050. U.S. Geological Survey, 2019, New Estimates for Marcellus and Point Pleasant-Utica Shales, accessed October 3, 2019, at URL https://www.usgs.gov/news/national-newsrelease/usgs-estimates-214-trillion-cubic-feet-natural-gas-appalachian-basin Vermeulen, T., 1953. Theory for irreversible and constant-pattern solid diffusion. Ind. Eng. Chem. 45 (8), 1664–1670. https://doi.org/10.1021/ie50524a025. Wang, Y., Tsotsis, T.T., and Jessen, K., 2015. Competitive Sorption of Methane/Ethane Mixtures on Shale: Measurements and Modeling. Industrial & Engineering Chemistry Research 54 (48), 12187-12195 DOI: 10.1021/acs.iecr.5b02850 191 Wang, Y., 2016. Investigation of Gas Sorption and Mass Transfer in Gas Shales (Doctoral dissertation). University of Southern California, Los Angeles. Wasaki, A., Akkutlu, I.Y., 2015. Permeability of organic-rich shale, SPE J. 20 (06) 1384– 1396. Xu, J., 2013. The Study of Mass Transfer in Gas Shales and the Optimization of Hydraulic Stimulation Processes via Additives (Doctoral dissertation). University of Southern California, Los Angeles. Zachara, J.M., Chen, X., Murray, C., Hammond, G., 2016. River stage influences on uranium transport in a hydrologically dynamic groundwater-surface water transition zone. Water Resour. Res. 52, 1568–1590. https://doi.org/10.1002/2015WR018009. Zarragoicoechea, G.J., Kuz, Z.A., 2002. Van der Waals equation of state for a fluid in a nanopore. Phys. Rev. Issue E 65 (1–4), 021110. Zhang, X., Ma, F., Yin, S., Wallace, C.D., Soltanian, M.R., Dai, Z., Ritzi, R. W., Ma, Z., Zhan, C., Lü, X., 2021. Application of upscaling methods for fluid flow and mass transport in multi-scale heterogeneous media: A critical review. Applied energy, 303, 117603. doi: 10.1016/j.apenergy.2021.117603 Zhang, J., Raslan, M., Wu, C., Jessen, K., “A Generalized Dynamic Transfer Function for Ultra-Tight Dual-Porosity Systems." Paper presented at the SPE Western Regional Meeting, Bakersfield, California, USA, April 2022. doi: https://doi.org/10.2118/209324MS Zhao, H., Lai, Z., Firoozabadi, A., 2017. Sorption Hysteresis of Light Hydrocarbons and Carbon Dioxide in Shale and Kerogen. Sci Rep 7, 16209. https://doi.org/10.1038/s41598- 017-13123-7. Zimmerman, R.W., 1991. Compressibility of Sandstones, Developments in Petroleum Science 29, Elsevier Science Publishers B.V., Amsterdam, 173p.. 192 CHAPTER 6.Analysis of gas storage and transport in shales using pressure pulse decay measurements with He, Kr and CO2 6 6.1. Introduction The primary objective of this study is to address the challenges related to the prediction of gas transport and storage in shale, at the laboratory-scale, by combining experiments with inert and adsorbing gases. One of the key petrophysical properties of shale is the apparent permeability, which is often several orders of magnitude smaller than that of conventional rocks. Gas transport in shale includes viscous flow, slip flow, transitional flow, and Knudsen diffusion in the free gas phase (Javadpour, 2009; Alnoaimi and Kovscek, 2019; Chen et al., 2019; Lyu et al., 2023). Numerous experimental studies, with shale samples from various formations, have investigated the apparent gas permeability in shale using different techniques. Brace et al. (1968) introduced the pressure-pulse decay (PPD) method for estimating permeability in tight rocks, where gas transports from an upstream reference cell, through a core sample to a downstream reference cell. The sample permeability can then be estimated from the observed pressure variation in up/downstream reference cells, gas properties, and core dimensions. Compared to the traditional steady-state approach, this method significantly 6 Parts of this chapter have been taken verbatim from our publication: Lyu, Y., Elkady, Y., Kovscek, A.R., & Jessen, K. (2023). Analysisof gas storage and transport in Eagle Ford shale using pressure pulse decay measurements with He, Kr and CO2. Geoenergy Science and Engineering, 211951. 193 reduces the time required for laboratory measurement of permeability in tight rocks. Apparent permeability of shale samples can be stress-sensitive, mainly due to natural fractures and micro-cracks (Heller et al., 2014; Wasaki and Akkutlu, 2015). This effect can also be investigated using the PPD technique by varying and controlling the net stress applied to the sample. However, flaws associated with the original approach by Brace et al. (1968) have been identified: These include the sensitivity of the selected time interval (Yamada and Jones, 1980) and the pore volume of the sample (Amaefule et al., 1986) used in the analysis. Dicker and Smits (1988) addressed the flaws and used upstream and downstream reference cells of equal volume for samples with pore volumes that are (slightly) smaller than both reference cells. Cui et al. (2009) presented a formulation to account for the adsorption of gas in shale, an aspect not considered in the original work of Brace. Further advances in the evaluation of apparent gas permeability in shale, using PPD methods, include a bidirectional pressure-pulse test by Feng et al. (2017), simultaneous determination of permeability, porosity, and adsorption capacity by Wang et al. (2019), and clear evidence that the matrix permeability can be determined using the data collected during later times, as suggested by Peng et al. (2019) and Zhang et al. (2020). Other experimental efforts include steady-state and non-steady state flow measurements to study the fluid permeability (helium, argon, methane, and water) in shale samples from the Scandinavian Alum Shale by Ghanizadeh et al. (2014). Computed Tomography (CT), PPD techniques, Scanning Electron Microscopy (SEM), and micro-CT were utilized to investigate the effects of carbon dioxide on fluid transport in Utica and Eagle Ford shales by Elkady & Kovscek (2020a) and Elkady et al. (2020). 194 Besides experimental efforts, apparent permeability in tight rocks has been investigated by numerical studies. Javadpour (2009) introduced a formulation, based on Maxwell’s theory, for a single nanotube, to address deviations in gas transport behaviors between shales and conventional formations, with emphasis on the role of slip flow and Knudsen diffusion. Following the work of Beskok and Karniadakis (1999), Civan (2010) developed a model for the apparent permeability based on the Knudsen number (Schaaf and Chambre, 1961; Roy et al., 2003). Freeman et al. (2011) applied the dusty-gas model (DGM), which combines viscous flow and Knudsen diffusion (Mason and Malinauskas, 1983; Ho and Webb, 2006; Sakhaee-Pour and Bryant, 2012), in the TOUGH+ simulator to explore gas transport in samples with micro- and nano-scale pores. Alnoaimi and Kovscek (2019) determined gas storage capacity and Klinkenberg-corrected permeability simultaneously by interpreting PPD experiments using a dual continuum model. Li et al. (2022) incorporated the Klinkenberg effect and effective stress to formulate multiphysics, multiscale, and multiporosity shale gas transport models. Jia and Xian (2022) integrated PPD measurements in naturally fractured shale using a 3D embedded discrete fracture model, while Lyu et al. (2021 and 2023) utilized a triple-porosity model (TPM) with transfer functions (based on Vermeulen, 1953; Zimmerman et al., 1993; Lim and Aziz, 1995; Zhang et al., 2022) to characterize shale samples from the Marcellus formation at different length scales. Another important aspect of gas transport and storage in shale relates to the role of adsorption (Javadpour, 2009; Akkutlu et al., 2018; Lyu et al., 2021). The Langmuir model (Langmuir, 1918) is commonly employed to determine asymptotic adsorption behaviors as observed for, e.g., methane, assuming that surface coverage correlates with the 195 adsorption/desorption rates and that equilibrium is reached between adsorption and desorption processes. Other formulations, e.g., the Brunauer-Emmett-Teller (BET) model (Brunauer et al., 1938) extend the Langmuir theory to describe multilayer adsorption of highly adsorbing gases, such as carbon dioxide. Helium (He), as a non-sorbing reference gas (Lyu et al., 2021), allows for the determination of the sample sorption capacity for other gases. Sorbing gases, such as methane and carbon dioxide, adsorb onto the mineral surfaces and the organic content of a shale sample at varying magnitudes, resulting in excess storage compared to that of non-sorbing gases (Nuttall et al., 2005; Vega et al., 2014; Aljamaan et al., 2017; Elkady and Kovscek, 2020b). Excess adsorption data are utilized to determine the actual (absolute) adsorbed gas amounts by utilizing the ratio of phase densities (free and adsorbed gas) since absolute adsorption cannot be measured directly in the laboratory (Lyu et al., 2021). Absolute adsorption accounts for the adsorbed phase volume in the pore space (Lin and Kovscek, 2014), which can become significant as the pore pressure increases. Moreover, sorption kinetics and hysteresis, between loading and unloading curves, can play important roles in shale gas production (Lyu et al., 2023). Despite the numerous efforts, the fundamental mechanisms of gas transport in shales remain a topic of debate due to the incomplete understanding of gas storage conditions and the complex pore structure within the shale. While significant advances in the applications of the PPD technique have been presented, the influence of fractures on permeability evaluation in unconventional samples remains a challenge. Accordingly, the development of techniques that can quantify and identify the contributions from cracks and/or from the matrix (Peng et al. 2019) is still an active area of research. In addition, limitations still exist for upscaling PPD solutions to larger scales, including variations in gas properties and 196 integration of sorption effects, as no method has been developed thus far that can fully account for the effects of gas adsorption on transport in shales (Han et al., 2020). This work investigates the gas transport and storage processes/capacity in Eagle Ford shale cores using the PPD technique with inert (He) and adsorbing gases (Kr and CO2). For one of the core, a modified analytical approach is introduced to determine the mass transfer rates in the shale matrix using He, that accounts for variations in gas properties (viscosity and compressibility) during the PPD experiment. The mass transfer coefficients can be rescaled to specific pressure conditions and for application with different gases, such as krypton (Kr) and carbon dioxide (CO2), in a TPM representation of the shale. A TPM is applied to estimate the remaining parameters for interpreting a series of He PPD experiments, at different pressures, and then applied to predict PPD experiments for Kr and CO2, with adsorption isotherms extracted from the equilibrium conditions of the relevant PPD experiments. In the following sections, the experimental and modeling approaches used for performing and interpreting the PPD measurements are presented (Section 6.2). Section 6.3 presents the experimental results and numerical interpretations. Section 6.4 provides a discussion of the results, while Section 6.5 details the relevant conclusions. 6.2. Materials and Methods 6.2.1. Materials A summary of relevant core parameters is provided in Table 6.1. Pressure pulse decay measurements were performed on shale cores (#A and #B) extracted the Eagle Ford 197 formation (Figure 6.1). Although two calcite-filled fractures are visible on the core (Figure 6.1, left), they do not appear to contribute to gas transport. A CT image (Figure 6.1, right) from a N2/Kr displacement experiment, however, indicates the existence of highly permeable pathways (Elkady, 2020) in the sample. The core was dried under vacuum at 50 °C for a week, sleeved, and inserted in a biaxial core holder. Table 6.1. Eagle Ford shale core dimensions and properties (Elkady and Kovscek, 2020b) Core #A Core #B Length (cm) 4.24 8.33 Diameter (cm) 2.54 2.52 Volume (cm3 ) 20.7 41.5 Dry Weight (g) 49.8 91.9 Bulk Density (g/cm3 ) 2.41 2.21 Cored Depth (ft) 11,184 - 198 Figure 6.1. Eagle Ford shale sample core #A (left) and a reconstructed 3D CT image of the in-situ Kr mass fraction after ~ 2 PV of nitrogen (N2) injection from the top (right) (Elkady, 2020). 6.2.2. Experimental approach - Pressure-pulse decay measurements The pulse decay setup, depicted in Figure 6.2, was used to investigate the shale sample at room temperature. The core was initially evacuated via Valve 4 with Valve 1 closed. Valve 3 remained open while Valve 2 and Valve 4 were closed after the evacuation. The upstream reservoir (V1) was then charged by opening Valve 1 to the He supply. Valve 1 was then closed, and the upstream pressure was allowed to stabilize. To start an experiment, Valve 2 was subsequently opened, allowing the He to flow through the core towards the downstream reservoir (V2). Pre-calibrated pressure transducers, rated for 2000 psia (138 bar), were used to record the pressures upstream and downstream of the core. 199 Figure 6.2. Pressure-pulse decay experimental set up for measuring mass transfer and sorption in the shale core (Modified from Elkady et al., 2020) Another high-pressure pulse decay system, as shown in Figure 6.3, was employed to examine sample core at high temperature. This setup was updated based on the system introduced in CHAPTER 4 and CHAPTER 5 (Figure 5.2) by adding an accumulator C inside the oven and an Isco pump D outside. Loaded gas in the reference vessel B can be pressurized by the Isco pump and a piston inside the accumulator up to about 345 bar. 200 Figure 6.3. High-Pressure pulse decay experimental system for measuring mass transfer and sorption in the shale core at reservoir relevant conditions Experiments were concluded when the system pressure stabilized at the final (equilibrium) level: Upstream and downstream reservoir pressures were equal and remained unchanged for 6 to 12 hours. An initial pressure recording period, of a few seconds, was excluded from the analysis to eliminate the impact of adiabatic temperature fluctuations caused by the abrupt changes in the upstream pressure (Brace et al., 1968). The confining pressure was initially set to 500 psia (34 bar) above the average of the upstream and downstream reservoirs pressures. The confining pressure was adjusted throughout a pulse decay experiment to maintain a net stress of 500 psia (34 bar). In subsequent PPD experiments, additional gas was charged to the upstream reservoir (V1), while keeping the equilibrium pressure from the previous test in the reminder of the system (core + V2). 201 6.2.3. Modeling approach 6.2.3.1 Triple-porosity model In this work, the TPM presented by (Lyu et al., 2021) was modified, to analyze gas transport and storage. The overall pore volume, Vt (cm3 ), of the shale core was divided into the volume of natural fractures, Vf (cm3 ), and the overall pore volume of the shale matrix, Vp (cm3 ). Natural fractures, present in the shale core, serve as flow pathways connecting the upstream and downstream reservoirs. Although these fractures may have unknown geometries and tortuosities, they are integrated and represented as an effective fracture system, as depicted in Figure 6.4. Figure 6.4. Conceptual model for PPD experiments The total matrix pore volume Vp includes a mesopore volume, Vm (cm3 ), consisting of mesopores and any microcracks in the shale matrix, as well as a micropore volume Vμ (cm3 ). It is assumed that the effective fracture controls and dominates transport between the up- and down-stream reservoir volumes, V1 (cm3 ) and V2 (cm3 ). The formulation of the TPM considers the fracture system as a 1D object that can exchange mass with the reference cells and the mesoporous fraction of the matrix. The mesoporous 202 fraction of the matrix can, in turn, exchange mass with the microporous fraction of the matrix. With these assumptions, the conservation equations for the fracture and matrix subsystems can be written as: Fracture system: + , + = −, (6.1) Mesoporous matrix: + , = − , (6.2) Microporous matrix: + , = . (6.3) The mass exchange rates between the mesopores and micropores, are described by transfer functions ( Γ ) that represents the volume-averaged response of the relevant matrix segments: = ( − ), (6.4) = ( − ). (6.5) In this formulation, Cf , Cm and Cμ (mol/cm3 ) represent the molar concentrations of free gas in the fracture, mesopores and micropores, respectively. Ff (mol/cm2 /s) is the gas flux in the fracture system. Γm (mol/cm3 /s) represents the transfer rate between the fracture and shale matrix (mesopores) and Γμ (mol/cm3 /s) is the transfer rate between the mesopores and the micropores. Ns,f, Ns,m and Ns,μ (mol) are the adsorbed amounts in the fracture, mesopores and micropores, respectively. Both transfer functions Γj (j=m,μ) include the product of the shape factor σj , and the effective diffusivity Dj . β j is a dimensionless 203 function of initial and boundary conditions that accounts for time-dependent gradients inside the meso/microporous segments (see Lyu et al. (2021) for additional details). Gas transport in the fracture system is defined by an effective group of parameters that includes the apparent permeability kf (cm2 ), the cross-sectional area, Af (cm2 ), and the characteristic length Lf (cm) of the fracture system. The flux Ff (mol/cm2 /s) in the fracture system was evaluated from: = − ( ), (6.6) The apparent gas permeability in the fracture system may include contributions from Darcy flow (viscous flow) and slip flow, and kf was therefore evaluated from the Dusty-GasModel (DGM) = + (1 − ), (6.7) with Cf (mol/cm3 ), μf (psia·s), and pf (psia) denoting the gas concentration, viscosity, and pressure in the fracture, respectively. B (cm2 ) is the viscous flow parameter, Dk (cm2 /s) is the Knudsen diffusivity, and Z is the gas compressibility factor. It should be noted that Vf implicitly defines the effective cross-sectional area of the fracture system as Af=Vf⁄Lf . In this work, Vf represents a small fraction of the overall porosity of the shale core only. A preliminary sensitivity test was performed to investigate the impact of the fracture volume on the PPD process. With other parameters fixed, the fracture volume has a marginal impact on the pressure decay behavior when Vf < 10% the total pore volume (see Appendix 204 A). To reduce the complexity of the modeling, the volume of the effective fracture system was assigned a value of 1% of the total core porosity, resulting in two parameters (BAf /Lf and DkAf /Lf ) defining the gas transport in the fracture system. In the shale matrix, the mass transfer rate (inverse of the characteristic time for mass transfer), is represented by the product of a shape factor σj (cm-2 ) and the effective diffusivity Dj (cm2 /s), where j=m, μ denote the mesoporous and microporous segments, respectively. The product, σjDj , includes the tortuosity of the matrix segments. The porosity, tortuosity, and shape factors, σj , are assumed to remain constant as the net stress is maintained constant throughout the experiments. The mass transfer rates can be estimated directly from the pressure decay data from the He experiments, using a modified analytical approach (see additional discussion below) and subsequently translated to higher pressures and/or other gases. The translation requires knowledge of how the effective diffusivity varies with pressure for different gases. Slip- and transitional-flow are expected to dominate gas transport in the mesopores, as demonstrated in Appendix B based on relevant Knudsen numbers. The effective diffusivity in the mesopores can be represented by the DGM, which combines viscous flow and Knudsen diffusion. The molar flux, Fm (mol/cm2 /s), can be reformulated as a diffusiontype equation with an effective diffusivity, Dm (cm2 /s), that depends on the fluid pressure: = − − = − [ (1 + ) +] = −. (6.8) 205 Another popular model, developed by Beskok and Karniadakis (1999) and modified by Civan (2010), can also be utilized to represent the apparent permeability in tight porous media. However, the explicit calculation/approximation of gas transport coefficients in the shale matrix was not the objective of this work. Instead, the correlations of transport coefficients between different gases and pressures were investigated. For that purpose, the two models (BKC and DGM) can be applied and result in similar correlation of the effective diffusivity used in Eq. 6.8 (see Appendix C). In the microporous region, where pore dimensions are comparable to the size of the molecules, Knudsen diffusion is expected to be the dominant mode of gas transport (see Appendix B). Accordingly, the effective diffusivity in the micropores is assumed to be independent of the pressure and was rescaled based on the molecular weight of the relevant gases. The molar flux (mol/cm2 /s) of gas in the shale matrix (j=m,μ) can now be expressed as: = − , = , . (6.9) Because the fractures in the core are more permeable than the shale matrix, and only account for a very small volume relative to the pore volume in the matrix, the pressuredecay rate at later times is mainly controlled by the matrix. Accordingly, the matrix permeability can be determined using the pressure-decay data collected during the later times (Peng et al., 2019). After initiation of a PPD experiment, the upstream pressure decays exponentially in time according to the analytical approach of Dicker and Smits (1988): 206 1 − 2 = ∆ ∙ − , (6.10) ( 1−2 ∆ ) = () = −, (6.11) where P1 and P2 (psia) denote the pressures in the upstream and downstream reservoirs with volumes V1 (cm3 ) and V2 (cm3 ), respectively. ∆P (psia) represents the initial pressure difference between up- and downstream reservoirs, PD is the dimensionless pressure difference between P1 and P2 (scaled with ∆P), and α (s-1 ) is the exponential decay factor. The effective permeability k (cm2 ) of the core can be estimated from α (Dicker and Smits, 1988) as: = ( 1 1 + 1 2 ), (6.12) where μ (psia·s) is the gas viscosity, cg (psia-1 ) is the isothermal compressibility of the gas, L (cm) is the length of the sample, and A (cm2 ) is the cross-sectional area of the sample. Eq. 6.12 can be re-written for the shale matrix (e.g., mesopores) as: = ( ∙/ ) ( 1 1 + 1 2 ) = ( 1 + 2 ), (6.13) = ( ) , = 1 2 . (6.14) The slope, αm, for transport in the mesopores as obtained from the plot of ln(PD) versus time, now represents the product of the shape factor , the effective diffusivity Dm, and a volume factor (Vm⁄V1 +Vm⁄V2 ). Thus, the mass transfer rate in the mesoporous fraction 207 of the matrix (σmDm) (1/s) can be directly estimated from the pressure decay data if the volume of the mesoporous segment is known. Ideally, a similar approach can be applied to estimate the characteristic time for transport (reciprocal of the mass transfer rate) in the microporous regions of the matrix. Preliminary tests, however, revealed that this method is highly sensitive to the accuracy of the measurements, when the pressure difference approaches the accuracy of the pressure transducers at later times. Consequently, the characteristic time for mass transfer in the micropores was estimated, in this work, from the He PPD data as detailed further below. 6.2.3.2 Adsorption in the shale sample The excess sorption can be calculated from the PPD experiments with sorbing gasesusing the He measurements as reference: = − , (6.15) where nex (mol) denotes the excess adsorption, ntotal (mol) is the amount of gas (free and sorbed) in the sample at the end of a given pulse experiment, Ceq (mol/cm3 ) represents the gas concentration at the equilibrium pressure, and VHe (cm3 ) is the pore volume (free gas volume) calculated from the He PPD measurements. The absolute adsorption for Kr and CO2 can then be evaluated from: = 1−⁄ . (6.16) 208 nabs (mol) is the absolute adsorption, ρ ads (mol/ cm3 ) is the molar density of the adsorbed phase, and ρ gas (mol/cm3 ) is the density of the free gas phase (available, e.g., from the NIST database - Lemmon et al., 2005). It is noteworthy that the density of the adsorbed phase, which cannot be measured directly, is likely to have a considerable influence on the estimation of gas storage capacity in shale (Lyu et al., 2022). The Langmuir adsorption isotherm was used to interpret the absolute adsorption for Kr, and the isotherm was directly applied in the analysis of the pressure decay experiments. This assumes that the characteristic time for adsorption is minimal, and consequently, that sorption kinetics can be ignored (Langmuir, 1918) = , ∙ , = 1+ . (6.17) Cs (mol/kg) is the absolute adsorption, that depends on the maximum monolayer adsorption capacity of the shale Cs,max (mol/kg) and the fractional surface coverage θ. θ can be computed from the equilibrium constant K (psia-1 ) and the bulk phase pressure p (psia). The Langmuir isotherms is appropriate for modeling monolayer sorption (e.g., N2, CH4 and Kr at low/moderate pressures or high temperatures), where θ is less than one. For gases with a strong affinity to the shale surfaces (e.g., CO2), the BET adsorption model (Brunauer et al., 1938) can be used to represent multilayer sorption. The BET model can be written as: = , ∙ = , ( ) (1− ) [ 1−(+1)( ) +( ) +1 1+(−1)( )−( ) +1 ], (6.18) 209 where po (psia) represents the saturation pressure at the experimental temperature (po = 833 psia in this work), c is a dimensionless constant associated with the heat of adsorption, and n is the maximum number of adsorbed layers. To implement the adsorption isotherms in the TPM, the amounts of adsorbed gas can be calculated as: , = , ( = , , ), (6.19) and the impact on the pore volume due to adsorption (adsorbate volume) can be evaluated from: = ,0 − , = ,0 − , /, ( = , , ). (6.20) In Eqs. 6.19-6.20, Wc (kg) denotes the core weight, Aj (j=f,m,μ) is the surface area of fractures, mesopores, and micropores per unit mass and At is the total surface area per unit mass. The fraction of the total surface area assigned to the fracture, mesopores, or micropores, Aj⁄At , can be estimated from the ratio between the specific pore volume and the total pore volume, as demonstrated by Lyu et al. (2021). Vj (cm3 ) represents the current pore volume, Vj,0 (cm3 ) is the pore volume from He experiments, and Vads,j (cm3 ) is the adsorbate volume based on the molar density ρ ads,j (mol/cm3 ) of the adsorbate. 210 6.3. Results A total of eight pulse-decay experiments, including three with He, three with Kr, and two with CO2, were performed as part of this work. In the following sections, the experimental observations are presented and interpreted towards a detailed characterization of transport and sorption in fractured shale. 6.3.1. Porosity and adsorption isotherms Figure 6.5 reports the observed pressure variations for both the upstream (blue) and downstream (black) reservoirs for a series of three He pulse-decay experiments. Each pulse of He provides an estimate of the pore volume, and hence an estimate of the porosity (as He is assumed to be a non-sorbing gas). The He porosity used in this study represents the average of the porosity values obtained from three pulses. As demonstrated by Elkady and Kovscek (2020b), the average He porosity for the core #A is 9.8%, corresponding to a total pore volume of Vt=2.02 cm3 . The total pore volume defines the fracture volume (Vf ) to represent 0.02 cm3 from the assumption of Vf=Vt /100 as discussed above. Similarly, pore volume for core #B is evaluated 70 oC, as shown in Figure 6.6, and it increases with He pressure until the limitation. For adsorbing gases (Kr/CO2), an apparent porosity or storage capacity can be calculated at the end of each pressure pulse. The storage capacity, represented in standard cubic feet (scf) of gas per ton of shale (1 scf/ton = 1.196⋅10-3 mol/kg), at each equilibrium pore pressure was calculated for Kr and CO2 and is reported in Figure 6.7. 211 Figure 6.5. Pressure-pulse decay experiments with He (T = 20 oC) Figure 6.6. Pore volume accessible with He at different pressure stages (T = 70 oC) 212 Figure 6.7. Gas storage capacity in the shale sample (left - T = 20 oC, right - T = 70 oC) It is important to note that the storage capacity (ntotal/mass of sample – see Eq. 6.15 includes contributions from free and adsorbed gas. A comparison of the storage capacities of CO2 and Kr (surrogate for CH4) indicates a clear net storage potential from shale gas operations over the pressure range investigated in this work. The excess adsorption can be expressed in terms of absolute adsorption and the ratio of densities between adsorbed and free gas (see Eq. 6.16. Parameters for the absolute adsorption isotherms (Langmuir and BET - Eqs. 6.17-6.18 can accordingly be estimated from the excess adsorption data, based on an assumption of the adsorbate phase density. The densities of Kr and CO2 at their normal boiling points were initially selected for converting excess to absolute adsorption. The corresponding model parameters for the Langmuir (Kr) and the BET (CO2) isotherms are reported in Table 6.2, while Figure 6.8 (right) reports the calculated absolute adsorption for the two gases. Figure 6.9 indicates CO2 excess (left) and absolute adsorption (right) in core #B at 70 oC. Based on the absolute 213 adsorption amount and the total gas storage amount (Figure 6.7), gas is mainly stored as adsorbed phase at high pressure (67% of the total at 4500 psi). Figure 6.8. Experimental excess adsorption (left) and calculated absolute adsorption (right) – core #A (T = 20 oC) Figure 6.9. Experimental excess adsorption (left) and calculated absolute adsorption (right) – core #B (T = 70 oC) 214 It should be noted that the adsorption isotherms represent an average behavior of the whole sample, and therefore, the maximum number of layers (n) in the BET model is not related to a single pore diameter but rather to the average of the pore-size distribution. Table 6.2. Model parameters for Kr and CO2 adsorption isotherms ρ ads (mol/cm3 ) Core # Cs,max (mol/kg) K (psia-1 ) c (-) n (-) Kr 28.80e-3 89.66e-3 2.60e-3 CO2 25.02e-3 A 109.47e-3 6.88 7 B 1.53 0.025 2 6.3.2. Mass transfer in the matrix In the following sections, experimental measurements on core #A will be comprehensively analyzed. Following the approach of Dickers and Smit (1988), as discussed earlier, data from two He pulses are plotted in terms of ln ( P1 -P2 ∆P ) versus time in Figure 6.10. The slope of data, in this form, were utilized to estimate the mass transfer rates for the mesoporous segment of the matrix, following the modified interpretation outlined in Eqs. 6.13-6.14. Each slope in Figure 6.10 corresponds to a specific pressure range, and once the volume of the mesopores is known, the mass transfer rates for the mesopores can be estimated from () = ( 1 + 2 ⁄ ), (6.21) 215 where subscript m refers to the meoporous segment of the matrix. The determination of the mesopore volume (Vm) is discussed in detail below. Figure 6.10. Characterization of the mass transfer rate for the mesoporous matrix segment using the modified analytical approach: Data shown for the 1st and 2nd He pressure decay 6.3.3. Interpretation of pressure decay experiments Next, the TPM is applied to interpret the PPD experiments. The TPM formulation utilizes several parameters, including the effective permeability of the fracture system (kfAf /Lf ), that can be represented by two parameters (BAf /Lf and DkAf /Lf ), the mass transfer rates in the mesopores and micropores (σmDm and σμDμ ), and the distribution of the pore volume within the shale matrix (Vm⁄Vp with Vp=Vt -Vf ). The workflow for obtaining the relevant model parameters is detailed in Appendix D. Initially, the model parameters were 216 calibrated using two (of three) He pressure decay experiments: The model parameters were estimated by matching the 1st and 2nd He experiments simultaneously, using a non-linear regression approach. Given that the transport parameters are dependent on pressure (i.e., kf depends on pressure, as shown in Eq. 6.7, and σmDm represents an average characteristic time over the relevant pressure range in each pulse), simultaneous regression of two pulses allows for the delineation of the pressure dependence of the transport coefficients. Following the model calibration, the 3 rd He pulse experiment was used to validate the estimated model parameters. The estimated model parameters, including 95% confidence intervals (CI), are reported in Table 6.3. Table 6.3. Parameter estimation using the 1 st and 2nd He pulses of pressure decay BAf /Lf (10-15 cm3 ) DkAf /Lf (10-4 cm3 /s) (σD)μ (10-6 1/s) Vm⁄Vp (%) Results 8.02 5.95 27.31 72.58 95% CI [8.00,8.04] [5.94,5.97] [27.31, 27.31] [72.49, 72.65] Figure 6.11 compares the modeling results and experimental data after regression: A good agreement is observed, with root mean square error (RMSE) values of less than 0.1% of the pressure range. Using the estimated volume split (Vm⁄Vp ) and the slope obtained from the modified analytical approach (Figure 6.10), the mass transfer rates in the mesopores are calculated 217 to (σD)m=574.72×10-6 (1/s) and (σD)m=825.36×10-6 (1/s) for the 1st and 2nd He pressure decay experiments, respectively. Figure 6.11. Comparison of calculations and data for the 1st and 2nd He pressure decay experiments after model calibration To predict the behavior for other pressure decay experiments (He, Kr, and CO2), the transport coefficients must be rescaled to account for variations in gas pressure and fluid properties. Since viscous flow parameter, BAf /Lf , only depends on the fracture geometry (fracture aperture etc.), while DkAf /Lf depends on the fracture geometry and molecular weight of the gas (see Eq. 6.23, the rescaling of fracture transport parameters is relatively simple. However, to rescale the mesoporous transport parameter (σD)m, it is necessary to estimate the ratio of effective diffusivities (Dm) at relevant pressure ranges and fluid properties. The characteristic time for mass transfer t * (s) is estimated following the approach of Lyu et al. (2021). ∗ = 1/() = 2⁄, (6.22) 218 where Lc represents the characteristic length and Dm is the effective diffusivity. To evaluate Dm, as a function of pressure and molecular weight, for scaling-purposes, a single cylindrical pore with an effective radius rm (cm) was considered. The viscous flow parameter and the Knudsen diffusivity can be evaluated as functions of rm from: = 2 8 , = 2 3 √ 8 , (6.23) where Mw (kg/mol) is the molecular weight, Rg (J/mol/K) is the gas constant, and T (K) is the temperature. Dm can then be calculated as a function of pressure for relevant gases using a given value of rm: = (1 + ) + . (6.24) It is assumed that the shape factor remains constant between experiments (at constant net stress), such that the ratio of the products ()m, between two experiments, only depends on Dm. However, in contrast to previous work, the average pore size or the pore size distribution of the shale matrix is unknown for the current shale sample. Therefore, an effective pore size must be estimated based on the ratio of characteristic times (or Dm) for the two He experiments reported in Table 6.3: 1 ∗ /2 ∗ = ,2/,1 . (6.25) To evaluate Dm (as shown in Eq. 6.24) and estimate t * (as shown in Eq. 6.25), an average pressure that accurately represents the pulse decay process is required. To this end, the average of the maximum pressure in the fracture, pf,max, and the minimum pressure in the 219 micropores, p2,ini (initial downstream pressure of each pulse) is used. pf,max can be estimated by assuming a rapid expansion from upstream to downstream using the material balance: 1, 1, 1 + 2, 2, (2 + )~ , , (1 + 2 + ). (6.26) By combining Eqs. 6.24-6.26, with gas properties from NIST, an effective pore radius of 84 nm was determined. As the actual geometry of the shale matrix is far more complex than that of a single pore (approximation), the estimated value of the effective pore radius exceeds the usual pore size range of a mesopore (2–50 nm). This in turn suggests that the characteristic time estimated from the He expansion experiments includes contributions from a range of pores that are not fully aligned with the standard pore-size classification. Based on the effective pore radius, the characteristic time for mass transfer in the mesoporous region can be calculated using Eq. 6.24 for pulses at different pressures and for different gases. In the microporous region, it is assumed that Knudsen diffusivity dominates gas transport. Consequently, the scaling factor between gases (independent of pressure) is determined by the square root of the ratio of molecular weights. Thus, the mass transfer rates, denoted as (σD )µ, were rescaled from He to Kr and CO2 by factors of approximately 0.22 (√4.0026⁄83.798) and 0.30 (√4.0026⁄44.009), respectively. The mass transfer rates for the mesopores and micropores, corresponding to the range of experimental pressures, were evaluated and are reported in Table 6.4. To validate the 220 rescaling approach, the 3rd He PPD experiment was predicted using parameters from Table 6.3 and Table 6.4. The calculated pressure response is compared with the experimental observations in Figure 6.12, and a good agreement between the model and the experiment is observed (RMSE of ~ 0.05% over the relevant pressure range). This validation supports the proposed method for rescaling transport parameters for application at different pressure conditions. Table 6.4. Rescaling of (σD) for prediction of additional pulse-decay experiments Pulse Source (σD) m (10-6 1/s) (σD) μ (10-6 1/s) 1 st He Estimated 574.72 27.31 2 nd He Estimated 825.36 27.31 3 rd He Rescaled 1117.80 27.31 1 st Kr Rescaled 241.24 6.01 2 nd Kr Rescaled 477.24 6.01 3 rd Kr Rescaled 733.46 6.01 1 st CO2 Rescaled 268.69 8.23 2 nd CO2 Rescaled 402.43 8.23 221 Figure 6.12. Experimental observations and model prediction for the 3rd He PPD experiment 6.3.4. Pressure decay predictions for Kr and CO2 To predict the gas transport and storage capacity for adsorbing gases, i.e., Kr and CO2, the observed adsorption behaviors were integrated into the TPM. In this work, gas adsorption is represented directly from the adsorption isotherms, as discussed above, by assuming that free and adsorbed gas coexist in equilibrium. This approach contrasts with previous work (Lyu et al., 2022), where sorption kinetics were included in the modeling. For the present sample, separate measurements of adsorption (including dynamics) were not available. The equilibrium assumption is further supported by the lower temperature of these experiments compared to the previous work (20 oC versus 50 oC). 222 Adsorption of gases affects both the storage capacity and the gas transport due to the link between adsorbed-phase volume, porosity, and apparent permeability (Javadpour, 2009; Wasaki and Akkutlu, 2015; Wu and Zhang, 2016; Elkady and Kovscek, 2020b; Memon et al., 2020; Afagwu et al., 2021; Gao et al., 2023). The apparent permeability of a fracture depends strongly on the fracture geometry. For a planar geometry, B is proportional to the square of the fracture aperture (hf 2 ), while Dk is a function of hf . In this work, the impact of adsorbed gas on the fracture permeability and the porosity reduction in the fracture due to sorption, is represented through Eq. 6.20. The adsorbed phase volume along the fracture is assumed to contribute uniformly to the porosity reduction (Eq. 6.20), providing for a simplified representation of the impact on the apparent permeability: = 0 ∙ ( ,0 ) 2 , = ,0 ∙ ( ,0 ) , (6.27) where B, Dk , and Vf represent the current values, and B0 , Dk,0, and Vf,0 are the reference values obtained from the He experiments (no adsorption). The parameter γ depends on the geometry of the fracture, e.g., γ=1 for planar fracture geometry as assumed in this work. With adsorption isotherm parameters from Table 6.2 and transport properties from Table 6.4 incorporated into the TPM, the pressure responses from the Kr experiments were initially predicted and are compared to the experimental observations in Figure 6.13. A strong agreement between experiments and model predictions is observed, with small RMSE values over the relevant pressure-decay ranges. These results further validate the 223 technique of rescaling the mass transfer rates of He for different gases and different pressure conditions. For the Kr experiments, the impact of adsorption on the porosity and apparent permeability of the fracture (and matrix) is predicted to be marginal, corresponding to a maximum reduction of the fracture volume of 4% (see Figure F1). The selected adsorbate density model for Kr (density at the normal boiling point) appears to provide a good approximation, partially due to the limited amount of adsorption at the relevant conditions (see Figure 6.8). Figure 6.13. Comparison of model predictions with experimental observations for the 1st (top), 2nd (left-bottom), and 3rd (right-bottom) Kr experiments 224 For adsorbing gases with a strong affinity to shale, e.g., CO2, the PPD behavior is expected to be sensitive to the representation of the adsorbed phase density. The default adsorbate density model employed here (the density at the normal boiling point) was initially combined with the translated transport properties (Table 6.4) to predict the two CO2 PPD experiments. A comparison of the model predictions with the experimental observations is provided in Figure 6.14. Figure 6.14. Comparison of model predictions and experimental data for the 1st (left) and 2 nd (right) CO2 PPD experiments The model response is observed to be faster than what is observed from the experiments, suggesting that the transport within the fracture system and/or the fracture aperture was overestimated. This, in turn, indicates that either a) the selected density model of the adsorbed CO2 is not adequate for representing the porosity (and permeability) reduction in the fracture (the predicted porosity reduction is reported in Appendix E), or b) swelling of the matrix occurs in tandem with the porosity reduction due to the volume of the adsorbate (Kamali-Asl et al., 2022). If matrix swelling occurs due to the adsorption of CO2 (Tesson 225 and Firoozabadi, 2019; Yu et al., 2021), additional data, such as strain measurements, are required to separate this impact from the role of the adsorbate phase density. In the absence of strain data, an empirical adsorbate density model that depends on the surface coverage, θ, during a multilayer adsorption process for CO2 was explored: = ∙ 2/3 . (6.28) This model predicts an increase in the adsorbate density/volume as the fractional coverage increases, and Eqs. 6.19-6.20 reduce to: , = , 1/3 /, = , , . (6.29) In this case, the adsorbate density for a monolayer and for full coverage of the adsorbent surface (θ=1) is related to M (mol/cm3 ). M is an unknown parameter that was estimated from simultaneous minimization of errors between the model and 1) the excess sorption data, and 2) the 1st PPD experiment with CO2. An estimated value of M=12.26 x10-3 mol/cm3 provides for the best agreement (see Figure 6.15 – left, and Appendix E). 226 Figure 6.15. Comparison of data and calculations for the 1st (left - calibration) and 2nd (right - prediction) CO2 PPD experiment The revised adsorbate density model was then used to predict the behavior of the 2nd CO2 PPD experiment, and a good agreement was observed (see Figure 6.15 - right). In the revised model, the impact of adsorption, porosity reduction (> 20% in the fracture), and related permeability reduction for the fracture are amplified due to the larger adsorbate volumes (see Appendix F - Figure F3). It should be noted that, although a continuous density model is reasonable, variations in the adsorbate density with pressure require further investigation, particularly for CO2 adsorption on shale. 6.4. Discussion In this work, gas transport and storage in cores from the Eagle Ford shale were investigated using He, Kr, and CO2. Pressure-pulse decay measurements were utilized to study sample porosity, fracture permeability, as well as mass transfer and sorption in the matrix. Excess adsorption isotherms for Kr and CO2 were evaluated based on the average porosity of the shale core, which was extracted from He PPD measurements, as He considered a non- 227 sorbing gas. The excess isotherms were then converted to absolute adsorption by using the Langmuir adsorption model for Kr and the BET model for CO2, respectively, and applying an assumed adsorbate density model (density of liquid at the normal boiling point). An effective fracture system, representing 1% of the total pore volume, was utilized to represent natural fractures/cracks as the main flow conduits in the core. Gas transport in the fracture system was characterized by an apparent permeability kf , cross-sectional area Af , and length Lf of the fracture system. Compared to previous work, e.g., Wang et al. (2019), Li et al. (2022), and Jia and Xian (2022), this effective representation includes both the fracture tortuosity and gas transport coefficient. The apparent permeability was further expanded to represent viscous flow (BAf /Lf ) and Knudsen diffusion (DkAf /Lf ) according to the Dusty-Gas-Model (DGM). To characterize the product of the reciprocal of the characteristic time for mass transfer (σmDm) in the mesopores, and the pore volume of the mesopores relative to up- and downstream volumes, Vm(1⁄V1 + 1⁄V2 ), a modified analytical approach for the interpretation of PPD experiments was introduced. The proposed approach effectively improves upon previous methods (Wang et al., 2019; Li et al., 2022; Jia and Xian, 2022) by incorporating gas properties and material heterogeneity, and avoiding errors caused by sorption (Cui et al., 2009; Wang et al., 2019; Han et al., 2020). Furthermore, the proposed method provides a path for characterizing the effective pore size within a shale matrix. A triple-porosity model (TPM) was applied to estimate the effective permeability of the fracture system, the mass transfer rate for the micropores, and the mesopore volume, by 228 simultaneously matching two He PPD experiments. The estimated parameters allowed for accurate prediction of a 3rd He PPD experiment, to validate the characterization of the gas transport properties and their representation for the shale core. Excess sorption isotherms were extracted from Kr and CO2 PPD measurements and converted to absolute sorption using the Langmuir model for Kr and the BET model for CO2, respectively. The conversion from excess to absolute sorption was performed based on an adsorbate density model (normal boiling point density of liquid). Transport parameters extracted from the He PPD experiments were then translated and combined with the isotherm model to predict the behavior of PPD experiments with Kr. The proposed workflow allows for accurate prediction of the Kr PPD experiments, without the need for additional parameters estimation. It was observed that Kr has a moderate affinity to the shale, relative to that of CO2, and that the representation of porosity variation due to the adsorbate phase only has a marginal impact on the model predictions. A similar approach was employed to achieve a reasonable agreement between model predictions and experimental observations from CO2 PPD measurements. A central difference between Kr and CO2 behaviors was attributed to the stronger sorption affinity of CO2, as observed from the excess sorption data. The model predictions for the CO2 experiments overestimate the apparent permeability of the fracture system, corresponding to an underestimation of the sorption-induced porosity (and permeability) reduction. A reduction in the apparent permeability can be a result of a) matrix swelling (strain) in response to the adsorption in the shale, or b) a build-up of the adsorbate phase volume in the fracture system. The latter explanation was investigated by introducing an alternate 229 representation of the adsorbate density that predicts in a larger porosity reduction due to adsorption. The alternate adsorbate density model requires an additional parameter to be estimated by simultaneously matching the excess isotherm data and one PPD experiment. Following model calibration, the modified model was demonstrated to accurately predicted a second CO2 PPD experiment. However, it is not possible to differentiate between sorption-induced strain and sorption-induced porosity reduction based on the experimental data presented in this work. 6.5. Conclusions In this study, an analytical approach for interpretation of PPD measurements with He was modified and used as an efficient tool to delineate the gas transport coefficients in the shale matrix. Furthermore, a method for translating the mass transfer rates to other gases and pressure conditions was introduced and used in a TPM to efficiently interpret and predict the PPD behavior of He and sorbing gases (Kr and CO2). The TPM approach provides a practical approach for modeling gas transport and storage in shale cores with natural fractures/cracks without the need for a comprehensive discretization of the shale matrix. Application of sorption isotherms was demonstrated to provide for reasonably accurate interpretation of gas transport and sorption in shale at the pressure and temperature conditions studied in this work. Moreover, the study indicates that the density of the adsorbed phase, particularly for a strongly adsorbing gas such as CO2, can significantly impact mass transfer and storage behaviors in shale. The presented experimental observations and analysis provide a better understanding of the mechanisms that control gas recovery and carbon storage in complex unconventional formations. 230 Based on the results and analysis presented here, recommendations for future research activities include: 1) Direct measurements of sorption kinetics and isotherms for CO2 in shale core samples, using techniques such as thermogravimetric analysis (TGA), to validate adsorbate density models for CO2 adsorption, particularly at higher pressures; 2) Development of experimental techniques for isolating the impact of adsorbate density and matrix swelling during PPD measurements is needed to provide for a more complete interpretation of the experimental observations. Overall, these recommendations may contribute to a more detailed characterization of gas transport and storage in shale, facilitate process optimization during gas production operations, and aid evaluation of carbon sequestration potentials in unconventional formations. 6.6. Appendix Appendix A – Sensitivity of Vf on pulse decay calculations In this sensitivity analysis, the volume of the effective fracture was the only variable altered (ranging from 0.1% to 10% of Vt) in the triple-porosity model (TPM), while other parameters were manually assigned and remained fixed. The calculation results demonstrate that the fracture volume has a modest impact on the pressure decay in the core. 231 Figure A4. Sensitivity test of the fracture volume on modeling results for a He PPD experiment using a TPM (full pressure range - left, zoom-in - right) Appendix B – Classification of flow regimes from the Knudsen number The flow regime during gas transport can be classified and determined based on the Knudsen number (Schaaf and Chambre, 1961; Roy et al., 2003): Viscous flow (Kn≤0.001), slip flow (0.001<Kn<0.1), transition flow (0.1<Kn<10), Knudsen flow (Kn≥10). Subject to the gas properties and relevant temperature and pressure conditions in this work, the Knudsen number can be evaluated for gas transport through cylindrical pores of different sizes. 232 Figure B3. Flow regime classification based on the Knudsen number for different pore diameters (1 nm - left, 10 nm - middle, 100 nm - right) at relevant conditions of this work Appendix C – Representation of effective diffusivity in mesopores Beskok and Karniadakis (1999) developed a second-order correction to the apparent permeability (ka) based on the Knudsen number due to gas slippage in tight porous media: = , = (1 + ()) [1 + 4 1−], (6.30) where ki is the intrinsic permeability, b is the slip coefficient (b = -1 for slip flow), α(Kn) is the rarefaction coefficient, as discussed by Civan (2010): () = 1.358 1+0.170−0.4348. (6.31) It should be noted that some contributions, including Beskok and Karniadakis (1999) use the hydraulic radius to calculate the Knudsen number, rather than the hydraulic diameter. Then the effective diffusivity can be computed as: = (1 + ) ∙ , (6.32) 233 where μ (psia·s) is the gas viscosity, ka and kint (cm2 ) are the apparent and the intrinsic permeability, respectively. C (mol/cm3 ) is the gas concentration and Z is the gas compressibility factor. f c is the BKC correction factor. The effective diffusivity in tight porous media can also be described by the DGM (Mason and Malinauskas, 1983), which combines viscous flow and Knudsen diffusion in the effective diffusive flux: = (1 + ) + , (6.33) where B (cm2 ) is the viscous flow parameter and Dk (cm2 /s) is the Knudsen diffusion coefficient. For a cylindrical pore with radius rp, B and Dk can be evaluated from = 2 8 , = 2 3 √ 8 , (6.34) R (J/mol/K) is the gas constant, T (K) is the temperature, and Mw (kg/mol) is the molecular weight. These two models for describing gas transport in tight porous media, the BKC correction and the DGM, are compared below in terms of their predictions for effective diffusivity through capillary tubes. The results show that although the effective diffusivity could differ by as much as 25% for He in a small diameter tubes, the ratio used to rescale the diffusivity between different gases is relatively similar, with a maximum difference of ~ 7%. 234 Figure C4. Effective diffusivity and their ratios among gases determined by DGM and Beskok-Karniadakis-Civan (BKC) in a 10 nm cylindric pore Figure C5. Effective diffusivity and their ratios among gases determined by DGM and BKC in a 50 nm cylindric pore 235 Appendix D – Workflow for estimating model parameters Figure D5. Workflow summary of this work 236 Appendix E – Revised density model and BET isotherm Excess adsorption isotherms of CO2 as evaluated from the BET model with the revised adsorbate density model. Figure E1. Excess adsorption (left) and absolute adsorption (right) modeling results Table E1. Model parameters for CO2 adsorption isotherms with revised density model ρ ads (mol/cm3 ) Cs,max (mol/kg) c (-) n (-) 12.26e-3∙ θ 2/3 121.57e-3 5.39 6 237 Appendix F – Porosity reduction due to gas adsorption Calculated porosity reductions for sorbing gases, Kr and CO2, are reported below: Figure F1. The porosity reduction for pressure-pulse decays of Kr (1st pulse – top, 2nd pulse – bottom left, 3rd pulse - bottom right) 238 Figure F2. The porosity reduction for pressure-pulse decays of CO2 (1st pulse – left, 2nd pulse - right) with = 1.10 g/cm3 Figure F3. The porosity reduction for pressure-pulse decays of CO2, the 1st pulse (left) and the 2nd pulse (right) with ρ ads = 12.26e-3∙θ 2/3 mol/cm3 239 6.7. References Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y., 2018. Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys. 353, 356–376. https://doi.org/10.1016/j.jcp.2017.10.023. Afagwu, C., Alafnan, S., Mahmoud, M.A., Patil, S., 2021. Permeability model for shale and ultra-tight gas formations: Critical insights into the impact of dynamic adsorption, Energy Reports, Volume 7, Pages 3302-3316, ISSN 2352-4847, https://doi.org/10.1016/j.egyr.2021.05.060. Aljamaan, H. C. M. Ross, and A. R. Kovscek, 2017. Multiscale Imaging of Gas Adsorption in Shales. Society of Petroleum Engineers Journal, 22(6), 1760-1777. https://doi.org/10.2118/185054-PA Alnoaimi, K.R., Kovscek, A.R., 2019. Influence of microcracks on flow and storage capacities of gas shales at core scale. Transport Porous Media 127 (1), 53–84. https://doi.org/10.1007/s11242-018-1180-5 Amaefule, J. O., Wolfe, K., Walls, J. D., Ajufo, A. O., Peterson, E., 1986. Laboratory determination of effective liquid permeability in low-quality reservoir rocks by the pulse decay technique. Society of Petroleum Engineers. doi: doi:10.2118/15149-MS. Beskok, A., Karniadakis, G., 1999. Report: a model for flows in channels, pipes, and ducts at micro and nano scales. MIicroscale Therm Eng 3, 43–77, DOI: 10.1080/ 108939599199864. Brace, W. F., Walsh, J. B., Frangos, W. T., 1968. Permeability of granite under high pressure. Journal of Geophysical Research (1896-1977), 73(6):2225–2236. Brunauer, S., Emmett, P. H., Teller, E., 1938. Adsorption of gases in multimolecular layers. Journal of the American Chemical Society, 60(2):309–319. doi: 10.1021/ja01269a023. Chen, M, Kang, Y, Zhang, T, Li, X, Lin, C., 2019. Shale gas transport behavior considering dynamic changes in effective flow channels. Energy Sci Eng; 7: 2059– 2076. https://doi.org/10.1002/ese3.411. 240 Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82, 375–384. Cui, X., Bustin, A. M. M., Bustin, R. M., 2009. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids, 9(3), 208-223. Dicker, A. I., Smits, R. M., 1988. A practical approach for determining permeability from laboratory pressure-pulse decay measurements. Society of Petroleum Engineers. doi: doi:10.2118/17578-MS. Elkady, Y., 2020. Multiscale Investigation of Fluid Transport and Enhanced Recovery in Shale (Doctoral dissertation). Stanford University. Elkady, Y., Kovscek, A. R., 2020a. Multiscale study of the impact of CO2 on fluid transport and carbonate dissolution in Utica and Eagle Ford shale. Petroleum Science and Engineering. J. Petrol. Sci. Eng., 107867. Elkady, Y., Kovscek, A. R., 2020b. "Laboratory Visualization of Enhanced Gas Recovery in Shale." Paper presented at the SPE Annual Technical Conference and Exhibition, Virtual, October. doi: https://doi.org/10.2118/201707-MS Elkady, Y., Lyu, Y., Jessen, K., Kovscek, AR., 2020. Three-Dimensional Imaging and Quantification of Gas Storativity in Nanoporous Media via X-rays Computed Tomography. Energies; 13(23):6199. https://doi.org/10.3390/en13236199. Feng, R., Pandey, R., 2017. Investigation of Various Pressure Transient Techniques on Permeability Measurement of Unconventional Gas Reservoirs. Transp Porous Med 120, 495–514. https://doi.org/10.1007/s11242-017-0936-7. Freeman, C., Moridis, G., Blasingame, T., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90, 253–268. Gao, Q., Liu, J., Leong, Y. K., Elsworth, D., 2023. A Review of Swelling Effect on Shale Permeability: Assessments and Perspectives. Energy & Fuels, 37 (5), 3488-3500. DOI: 10.1021/acs.energyfuels.2c04005 Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., & Krooss, B. M., 2014. Experimental study of fluid transport processes in the matrix system of the 241 European organic-rich shales: I. Scandinavian Alum Shale. Marine and Petroleum Geology, 51, 79-99. Han, G., Liu, X., Huang, J., 2020. Theoretical Comparison of Test Performance of Different Pulse Decay Methods for Unconventional Cores. Energies. 13(17):4557. https://doi.org/10.3390/en13174557 Heller, R., Vermylen, J., Zoback, M., 2014. Experimental Investigation of Matrix Permeability of Gas Shales, AAPG Bull., 98 (5), 975–995. Ho, C. K., Webb, S. W., 2006. Gas transport in porous media. (Springer,Dordrecht, the Netherlands). Javadpour, F., 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). Journal of Canadian Petroleum Technology, 48(08), 16–21. doi: 10.2118/09-08-16-DA. Jia, B., Xian, C., 2022. Permeability measurement of the fracture-matrix system with 3D embedded discrete fracture model, Petroleum Science, Volume 19, Issue 4, Pages 1757- 1765, ISSN 1995-8226. https://doi.org/10.1016/j.petsci.2022.01.010. Kamali-Asl, A., Kovscek, A. R., Zoback, M. D., 2022. Long-term permeability evolution of shale seal rocks with argon and scCO2, Journal of Natural Gas Science and Engineering, Volume 104, 104642, ISSN 1875-5100, https://doi.org/10.1016/j.jngse.2022.104642. Langmuir, I, 1918. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40 (9), 1361–1403. https://doi.org/ 10.1021/ja02242a004. Lemmon, E.W., McLinden, M.O.. Friend, D.G., 2005. In: Thermophysical Properties of Fluid Systems. NIST Chemistry Webbook, vol. 69. NIST standard reference database. Li, Z., Teng, Y., Fan, M., Ripepi, N., Chen, C., 2022. "A Novel Multiphysics Multiscale Multiporosity Shale Gas Transport Model for Geomechanics/Flow Coupling in Steady and Transient States." SPE J. 27: 452–464. doi: https://doi.org/10.2118/206726-PA Lim, K., Aziz, K., 1995. Matrix-fracture transfer shape factors for dual-porosity simulators. J. Petrol. Sci. Eng. 13 (3–4), 169–178. https://doi.org/10.1016/09204105(95)00010-F. 242 Lin, W., Kovscek, A. R., 2014. Gas sorption and the consequent volumetric and permeability change of coal I: Experimental. Transport in Porous Media, 105:371 – 389. ISSN 1573-1634. doi: 10.1007/s11242-014-0373-9. Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2021. Characterization of shale using Helium and Argon at high pressures, Journal of Petroleum Science and Engineering, doi: https://doi.org/10.1016/j.petrol.2021.108952 Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023. Investigation of methane mass transfer and sorption in Marcellus shale under variable net-stress. Geoenergy Science and Engineering 227. https://doi.org/10.1016/j.geoen.2023.211846 Mason, E.A., Malinauskas, A.P., 1983. Gas Transport in Porous Media: The Dusty-gas Model, Elsevier Science Ltd., New York. Memon, A., Li, A., Jacqueline, N., Kashif, M., Ma, M., 2020. Study of gas sorption, stress effects and analysis of effective porosity and permeability for shale gas reservoirs, Journal of Petroleum Science and Engineering, Volume 193, 107370, ISSN 0920-4105, https://doi.org/10.1016/j.petrol.2020.107370. Peng, S., Ren, B., and Meng M., 2019. "Quantifying the Influence of Fractures for MoreAccurate Laboratory Measurement of Shale Matrix Permeability Using a Modified GasExpansion Method." SPE Res Eval & Eng 22: 1293–1304. doi: https://doi.org/10.2118/195570-PA Roy, S., Raju, R., Chuang, H.F., Cruden, B.A., and Meyyappan, M. 2003. Modeling gas flow through microchannels and nanopores. Journal of Applied Physics, 93(8), pp. 4870– 4879. Sakhaee-Pour, A., Bryant, S. L., 2012. Gas permeability of shale. SPE Reserv Eval Eng 15, 401–409, DOI: 10.2118/146944-PA. Schaaf, S.A. and Chambre, P.L. 1961. Flow of Rarefied Gases. Princeton University Press, Princeton, NJ. Tesson, S., Firoozabadi, A., 2019. Deformation and swelling of kerogen matrix in light hydrocarbons and carbon dioxide, J. Phys. Chem. C 123, 29173–29183, https://doi.org/10.1021/acs.jpcc.9b04592. 243 Vermeulen, T., 1953. Theory for Irreversible and Constant-Pattern Solid Diffusion. Industrial & Engineering Chemistry, 45(8), 1664–1670. doi: 10.1021/ie50524a025. Vega, B., Dutta, A., Kovscek. A. R., 2014. CT imaging of low-permeability, dual-porosity systems using high X-ray contrast gas. Transport in Porous Media, 101(1):81–97. ISSN 1573-1634. doi: 10.1007/s11242-013-0232-0. Wang, C., Pan, L., Zhao, Y., Zhang, Y., Shen, W., 2019. Analysis of the Pressure-Pulse Propagation in Rock: A New Approach to Simultaneously Determine Permeability, Porosity, and Adsorption Capacity. Rock Mech Rock Eng 52, 4301–4317. https://doi.org/10.1007/s00603-019-01874-w Wasaki, A., Akkutlu, I.Y., 2015. Permeability of organic-rich shale. SPE J. 20 (6) https://10.2118/170830-PA, 1,384-1,396. Wu, T., Zhang, D., 2016. Impact of Adsorption on Gas Transport in Nanopores. Sci Rep 6, 23629. https://doi.org/10.1038/srep23629 Yamada, S. E. Jones, A. H., 1980. A review of a pulse technique for permeability measurements. Yu, X., Li, J., Chen, Z., Wu, K., Zhang, L., Yang, S., 2021. Determination of CH4, C2H6 and CO2 adsorption in shale kerogens coupling sorption-induced swelling. Chem Eng J. 410:127690. https://doi.org/10.1016/j.cej.2020.127690. Zhang, J., Raslan, M., Wu, C., Jessen, K., "A Generalized Dynamic Transfer Function for Ultra-Tight Dual-Porosity Systems." Paper presented at the SPE Western Regional Meeting, Bakersfield, California, USA, April 2022. doi: https://doi.org/10.2118/209324- MS Zhang, J.J., Liu, H.H., Boudjatit, M., 2020. Matrix permeability measurement from fractured unconventional source-rock samples: Method and application, Journal of Contaminant Hydrology, Volume 233, 103663, ISSN 0169-7722, https://doi.org/10.1016/j.jconhyd.2020.103663 Zimmerman, R. W., Chen, G., Hadgu, T., Bodvarsson, G. S., 1993. A numerical dualporosity model with semianalytical treatment of fracture/matrix flow. Water Resources Research, 29(7), 2127–2137. doi: 10.1029/93wr00749 244 245 CHAPTER 7. Summary of dissertation and recommendations for future work 7.1. Summary of dissertation In this dissertation, a range of current challenges related to the development of unconventional resources has been investigated, with the primary objective of advancing the foundational understanding of the mass transfer and sorption mechanisms in shale formations and, subsequently, enhance future utilization of natural gas resources. To achieve that, some knowledge gaps have been addressed and bridged by extensive experimental characterizations and numerical investigations of gas transport and sorption in shale, to facilitate future exploration and development of shale formations. The investigation began (CHAPTER 2) by exploring the complex interplay between surface diffusion and sorption kinetics during mass transfer, specifically in the microporous segments of ultra-tight organic-rich formations like shale. The investigation considers gas transport in the free and adsorbed phases alongside sorption, with a particular emphasis on micropores. By developing and solving a detailed 1D model, the study revealed the importance of sorption kinetics and surface diffusion in these tight formations. However, under some circumstances, surface diffusion contributes negligibly to the overall transport and sorption behaviors. In practical exercise, surface diffusion is often coupled with sorption kinetics. The study demonstrates the benefits of separate measurements of dynamic sorption behavior to reduce uncertainties and enhance accuracy in evaluating gas diffusivity in nanoporous materials. 246 In CHAPTER 3, a novel dual-porosity model with new transfer functions was introduced to describe the processes of gas diffusion and dynamic sorption, accounting for the variability in characteristic rates. The study highlights the model's fidelity to accommodate a wide range of gases, thereby offering a strong foundation for the analysis of experiments with the very diffusive and non-sorbing helium (He) to the less diffusive but highly sorptive carbon dioxide (CO2). Significantly, the accuracy of the model was demonstrated via validation with a fully finite difference method. CHAPTER 4 focuses specifically on shale samples obtained from the Marcellus formation. The study employed rigorous experimental methods, utilizing He and argon (Ar) as test gases, in order to gain a comprehensive understanding of sorption kinetics and the interaction between the matrix and fracture systems within the shale. Sorption behavior of Ar in terms of isotherms and kinetics for the matrix region of the shale was determined, by performing thermogravimetric analysis (TGA) on a small shale cube. The utilization of a triple-porosity model (TPM) facilitated the description of the mass transport and sorption dynamics within a full-diameter core. Moreover, the research highlighted the importance of the adsorbate density in dynamic sorption tests, which has been disregarded in prior investigations. CHAPTER 5 extends the study to evaluate the mass transfer and sorption behavior of methane (CH4) in the full-diameter core. The TPM is employed to interpret experimental measurements, demonstrating that sorption isotherm measurements conducted at the laboratory scale, even in the absence of net stress, may effectively forecast sorption and mass transfer dynamics across different stress settings. Additionally, the findings of the 247 study imply that CH4 deposits could potentially persist as residual gas because of sorption hysteresis. This highlights the necessity for the development of models that take into account this dynamic phenomenon. In CHAPTER 6, the scope of this work was further broadened by examining the mass transfer and storage of gases in the Eagle Ford shale formation, specifically focusing on the utilization of He, krypton (Kr), and CO2 as probe gases. The porosity, permeability, and sorption properties of a core sample were determined using pressure-pulse decay measurements, which were further interpreted by a modified analytical approach and the TPM. The study emphasized the importance of the density of the adsorbed phase, specifically for CO2, when evaluating the impact of sorption on mass transport and storage in shale formations. In summary, this dissertation provides a comprehensive study of mass transfer and sorption processes within shale formations. Through the utilization of integral models, detailed experiments, and analytical insights, these studies deliver enhanced knowledge for optimizing the efficiency of hydrocarbon production and potential CO2 sequestration. 7.2. Recommendations for future work To further advance the research reported in this dissertation and enhance our comprehension of fluid transport and sorption in shale formations, the following recommendations for future efforts are proposed: i. In order to further validate and develop the integral model of gas diffusion and sorption in shale, it is important to examine the production (unloading) processes, 248 which would provide useful insights. Additionally, it is necessary to establish a refined workflow that can effectively enhance the performance of the model. It is essential to develop a modified version of the model for binary or multi-component mixtures, which may involve multi-phase flow and multi-component sorption kinetics. ii. From a laboratory-scale characterization prespective, flow-through measurements at steady state conducted on sample cores would offer better characterization for the permeability of the fractures or other higher-permeable channels across the sample core. Moreover, gas loading/unloading or pressure pulse decay experiments at different net-stress conditions can facilitate a deeper understanding of how the stress state influences porosity, mass transfer, and sorption in the shale matrix. At the smaller scale, additional adsorption and desorption experiments by TGA with natural gases and CO2 at reservoir relevant temperature and pressure conditions are needed to further understand the sorption hysteresis behavior and the adsorbate density. Development of experimental techniques for isolating the impact of adsorbate density and matrix swelling during PPD measurements is also needed to provide for a more complete interpretation of the experimental observations. iii. Additional projects to investigate the capacity of shale formations for carbon sequestration and hydrogen (H2) storage could involve conducting pressure pulse decay measurements using CO2 and H2 under varying levels of water saturation within the sample or at different compositional ratios. These investigations would serve the purpose of addressing two fundamental inquiries that hold significant importance: a) what are the impacts of CO2 and H2 adsorption on their transport 249 and storage processes? and b) can CO2 serve as an effective cushion gas for H2 storage operations in shale formations. iv. By integrating laboratory characterization and the integral models into reservoir simulators like the MATLAB Reservoir Simulation Toolbox (MRST), the upscaling to the reservoir settings could potentially be further promoted and used to interpret or forecast well or reservoir behaviors with enhanced efficiency. 250 References Afagwu, C., Alafnan, S., Mahmoud, M.A., Patil, S., 2021. Permeability model for shale and ultra-tight gas formations: Critical insights into the impact of dynamic adsorption, Energy Reports, Volume 7, Pages 3302-3316, ISSN 2352-4847, https://doi.org/10.1016/j.egyr.2021.05.060. Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y., 2018. Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys. 353, 356–376. https://doi.org/10.1016/j.jcp.2017.10.023. Akkutlu, I.Y., Fathi, E., 2012. Multiscale gas transport in shales with local kerogen heterogeneities, SPE J. 17 (04) 1002–1011. Aljamaan H, Alnoaimi K, Kovscek AR (2013). In-depth experimental investigation of shale physical and transport properties. SPE Paper 1582146 Presented at the SPE Unconventional Resources Technology Conference, 12–14 August, Denver, Colorado. Aljamaan, H. C. M. Ross, and A. R. Kovscek, 2017. Multiscale Imaging of Gas Adsorption in Shales. Society of Petroleum Engineers Journal, 22(6), 1760-1777. https://doi.org/10.2118/185054-PA Alnoaimi, K. R., Kovscek, A. R., 2019. Influence of Microcracks on Flow and Storage Capacities of Gas Shales at Core Scale. Transport in Porous Media, 127(1), 53–84. https://doi.org/10.1007/s11242-018-1180-5 Amaefule, J. O., Wolfe, K., Walls, J. D., Ajufo, A. O., Peterson, E., 1986. Laboratory determination of effective liquid permeability in low-quality reservoir rocks by the pulse decay technique. Society of Petroleum Engineers. doi: doi:10.2118/15149-MS. Ambrose, R. J., Hartman, R. C., Diaz Campos, M., Akkutlu, I. Y., and Sondergeld, C., 2010. New pore-scale considerations for shale gas in place calculations. SPE Paper 131772 Presented at the SPE Unconventional Gas Conference, 23-25 February, Pittsburgh, Pennsylvania. Ambrose, R. J., Hartman, R. C., Diaz-Campos, M., Akkutlu, I. Y., & Sondergeld, C. H. (2012). Shale Gas-in-Place Calculations Part I: New Pore-Scale Considerations. SPE Journal, 17(01), 219–229. doi: 10.2118/131772-PA 251 Ash, R., Barrer, R.M., Pope, C.G., 1963. Flow of adsorbable gases and vapours in a microporous medium. II. Binary mixtures, in: Proc. R. Soc. A Math. Phys. Eng. Sci., The Royal Society, pp. 19–33. Barenblatt G.I., Zheltov, I.P., and Kochina, I.N., 1960. Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks. Journal of Applied Mathematics and Mechanics, 24, 1286-1303. doi: https://doi.org/10.1016/0021-8928(60)90107-6. Beskok, A., Karniadakis, G., 1999. Report: a model for flows in channels, pipes, and ducts at micro and nano scales. MIicroscale Therm Eng 3, 43–77, DOI: 10.1080/ 108939599199864. Blaskovich, F.T., Cain, G.M., Sonier, F., Waldren, D., Webb, S.J., 1983. "A Multicomponent Isothermal System for Efficient Reservoir Simulation." Paper presented at the Middle East Oil Technical Conference and Exhibition, Manama, Bahrain, March 1983. doi: https://doi.org/10.2118/11480-MS Boyer, C., Kieschnick, J., Suarez-Riviera, R., Lewis, R.E., Waters, G., 2006. Producing Gas from its Source, Oilfield Review. Brace, W. F., Walsh, J. B., Frangos, W. T., 1968. Permeability of granite under high pressure. Journal of Geophysical Research (1896-1977), 73(6):2225–2236. Brunauer, S., Emmett, P.H., Teller, E., 1938. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60 (2), 309–319. https://doi.org/10.1021/ja01269a023. Bustin, R.M., Bustin, A.M.M., Cui, X., Ross, D.J.K., Murthy Pathi, V.S., 2008. Impact of shale properties on pore structure and storage characteristics. In: SPE Paper No. 119892- pp for Presentation at the 2008 SPE Shale Gas Production Conference, 16–18 November, Fort Worth, Texas. Cao, C., Zhao, Q., Gao, C., Sun, J., Xu, J., Zhang, P., & Zhang, L., 2017. Discrete fracture model with multi-field coupling transport for shale gas reservoirs. Journal of Petroleum Science and Engineering, 158, 107–119. doi: 10.1016/j.petrol.2017.08.033 Carman, P. C., Malherbe, P. le R., 1950. Proc. Roy. Soc. A, 203, 165. 252 Carman, P. C., Raal, F. A., 1951. Diffusion and Flow of Gases and Vapours through Micropores. III. Surface Diffusion Coefficients and Activation Energies. Proc. Roy. Soc. Lond. A 209 (1096): 38–58. http://dx.doi.org/10.1098/rspa.1951.0186. Chen, L., Zhang, L., Kang, Q., Viswanathan H.S., Yao, J., Tao W., 2015. Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity. Sci Rep 5, 8089. https://doi.org/10.1038/srep08089 Chen, M, Kang, Y, Zhang, T, Li, X, Lin, C., 2019. Shale gas transport behavior considering dynamic changes in effective flow channels. Energy Sci Eng; 7: 2059– 2076. https://doi.org/10.1002/ese3.411. Chen, Y. D. and Yang, R. T., 1991. Concentration Dependence of Surface Diffusion and Zeolitic Diffusion. AIChE J. 37 (10): 1579–1582. http:// dx.doi.org/10.1002/aic.690371015. Cho, Y., Ozkan, E., & Apaydin, O. G. (2013, May 1). Pressure-Dependent Natural-Fracture Permeability in Shale and Its Effect on Shale-Gas Well Production. Society of Petroleum Engineers. doi:10.2118/159801-PA Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous media. Transport Porous Media 82, 375–384. Clarkson, C., Solano, N., Bustin, R., Bustin, A., Chalmers, G., He, L., Melnichenko, Y., Radliński, A., & Blach, T. (2013). Pore structure characterization of North American shale gas reservoirs using USANS/SANS, gas adsorption, and mercury intrusion. Fuel, 103, 606–616. https://doi.org/10.1016/j.fuel.2012.06.119 Crank, J. 1975. The Mathematics of Diffusion, second ed. Claredon Press, Oxford, pp. 44– 103. Cui A, Wust R, Nassichuk B, Glover K, Brezovski R, Twemlow C., 2013. A nearly complete characterization of permeability to hydrocarbon gas and liquid for unconventional reservoirs: A challenge to conventional thinking. SPE paper No. 168730 presented at the SPE Unconventional Resources Technology Conference, 12–14 August, Denver, CO. 253 Cui, G., Xia-Ting, F., Pan, Z., Chen, T., Liu, J., Elsworth, D., … Wang, C., 2020. Impact of shale matrix mechanical interactions on gas transport during production. Journal of Petroleum Science and Engineering, 184, 106524. doi: 10.1016/j.petrol.2019.106524 Cui, X., Bustin, A. M. M., Bustin, R. M., 2009. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids, 9(3), 208-223. Curtis, J.B., 2002. Fractured shale-gas systems. AAPG bulletin, 86(11), pp.1921-1938. Damkohler, G., 1935. Z. Phys. Chem. A, 174, 222. Damkohler, G., 1935. Z. Phys. Chem. A, 174, 222. Dasani, D., 2017. Investigation of Mass Transfer in Fractured Shales (Doctoral dissertation). University of Southern California, Los Angeles. de Witt, Wallace, Jr., Roen, J.B., Wallace, L.G., 1993, Stratigraphy of Devonian black shales and associated rocks in the Appalachian basin, in Roen, J.B., and Kepferle, R.C., eds., Petroleum geology of the Devonian and Mississippian black shale of eastern North America: U.S. Geological Survey Bulletin 1909–B, p. B1–B57. Dewhurst, D.N., Siggins, A.F., 2006. Impact of fabric, microcracks and stress field on shale anisotropy. Geophysical Journal International 165, 135–148. Dicker, A. I., Smits, R. M., 1988. A practical approach for determining permeability from laboratory pressure-pulse decay measurements. Society of Petroleum Engineers. doi: doi:10.2118/17578-MS. Do, D. D. and Do, H. D., 2000. Analysis of Dual Diffusion and Non-Linear Adsorption Isotherm with a Time Lag Method. Adsorption 6 (2) 111-123. https://doi.org/10.1023/A:1008959517075 Do, D. D., & Do, H. D., 2003. Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density. Carbon, 41(9), 1777-1791. 254 Du, F. and Nojabaei, B., 2020. Estimating diffusion coefficients of shale oil, gas, and condensate with nano-confinement effect, Journal of Petroleum Science and Engineering, Volume 193, 107362, ISSN 0920-4105, https://doi.org/10.1016/j.petrol.2020.107362 EIA, 2020. Annual Energy Outlook 2020 with projections to 2050. Energy Information Administration, Washington. Elkady, Y., 2020. Multiscale Investigation of Fluid Transport and Enhanced Recovery in Shale (Doctoral dissertation). Stanford University. Elkady, Y., Kovscek, A. R., 2020a. Multiscale study of the impact of CO2 on fluid transport and carbonate dissolution in Utica and Eagle Ford shale. Petroleum Science and Engineering. J. Petrol. Sci. Eng., 107867. Elkady, Y., Kovscek, A. R., 2020b. "Laboratory Visualization of Enhanced Gas Recovery in Shale." Paper presented at the SPE Annual Technical Conference and Exhibition, Virtual, October. doi: https://doi.org/10.2118/201707-MS Elkady, Y., Lyu, Y., Jessen, K., Kovscek, AR., 2020. Three-Dimensional Imaging and Quantification of Gas Storativity in Nanoporous Media via X-rays Computed Tomography. Energies; 13(23):6199. https://doi.org/10.3390/en13236199. Elsaig, M., 2016. "Characterizations of the Marcellus Shale petrophysical properties". Graduate Theses, Dissertations, and Problem Reports. 5541. https://researchrepository.wvu.edu/etd/5541. Energy Information Adminstration (EIA), 2021. Annual energy outlook 2020 with projections to 2050. U.S. Energy Information Administration Office of Energy Analysis and U.S. Department of Energy. Energy Information Adminstration (EIA), 2023. Annual Energy Outlook 2023 with projections to 2050. Energy Information Administration, Washington. Etminan, S. R., Javadpour, F., Maini, B. B., et al. 2014. Measurement of Gas Storage Processes in Shale and of the Molecular Diffusion Coefficient in Kerogen. Int. J. Coal Geol. 123 (1 March): 10–19. http://dx.doi.org/10.1016/j.coal.2013.10.007. Everett, D. H., 1972. Manual of Symbols and Terminology for Physicochemical Quantities and Units, Appendix II: Definitions, Terminology and Symbols in Colloid and Surface 255 Chemistry. Pure and Applied Chemistry, 31(4), 577–638. https://doi.org/10.1351/pac197231040577 Fathi, E., and Akkutlu, I. Y., 2012. Multiscale Gas Transport in Shales With Local Kerogen Heterogeneities. SPE J. 17 (04): 1002–1011. https://doi.org/10.2118/146422-PA Fathi, E., and Akkutlu, I. Y., 2013. Lattice Boltzmann Method for Simulation of Shale Gas Transport in Kerogen. SPE J. 18 (2013): 27–37. doi: https://doi.org/10.2118/146821-PA Fathi, E., and Akkutlu, I. Y., 2014. Multi-Component Gas Transport and Adsorption Effects during CO2 Injection and Enhanced Shale Gas Recovery. Int. J. Coal Geol. 123 (1 March): 52–61. http://dx.doi.org/10.1016/j.coal.2013.07.021. Feng, R., Pandey, R., 2017. Investigation of Various Pressure Transient Techniques on Permeability Measurement of Unconventional Gas Reservoirs. Transp Porous Med 120, 495–514. https://doi.org/10.1007/s11242-017-0936-7. Freeman, C., Moridis, G., Blasingame, T., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90, 253–268. Gao, Q., Liu, J., Leong, Y. K., Elsworth, D., 2023. A Review of Swelling Effect on Shale Permeability: Assessments and Perspectives. Energy & Fuels, 37 (5), 3488-3500. DOI: 10.1021/acs.energyfuels.2c04005 Gaus, G., Kalmykov, A., Krooss, B. M., & Fink, R., 2020. Experimental Investigation of the Dependence of Accessible Porosity and Methane Sorption Capacity of Carbonaceous Shales on Particle Size. Geofluids, 2020, 1–13. doi: 10.1155/2020/2382153 Gay, J. M., Suzanne, J., & Coulomb, J. P., 1990. Wetting, surface melting, and freezing of thin films of methane adsorbed on MgO(100). Physical Review B, 41(16), 11346–11351. https://doi.org/10.1103/physrevb.41.11346 Gensterblum, Y., Ghanizadeh, A., Cuss, R. J., Amann-Hildenbrand, A., Krooss, B. M., Clarkson, C. R., … Zoback, M. D., 2015. Gas transport and storage capacity in shale gas reservoirs – A review. Part A: Transport processes. Journal of Unconventional Oil and Gas Resources, 12, 87–122. doi: 10.1016/j.juogr.2015.08.001 Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., & Krooss, B. M., 2014. Experimental study of fluid transport processes in the matrix system of the 256 European organic-rich shales: I. Scandinavian Alum Shale. Marine and Petroleum Geology, 51, 79-99. Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., & Krooss, B. M., 2014. Experimental study of fluid transport processes in the matrix system of the European organic-rich shales: I. Scandinavian Alum Shale. Marine and Petroleum Geology, 51, 79-99. Gilliland, E.R., Baddour, R.F., Russell, J.L., 1958. Rates of flow through microporous solids. AICHE J. 4, 90–96. Goral, J., Panja, P., Deo, M., Andrew M., Linden S., Schwarz JO., Wiegmann A., 2020. Confinement Effect on Porosity and Permeability of Shales. Sci Rep 10, 49. https://doi.org/10.1038/s41598-019-56885-y. Guo, C., Wei, M., & Liu, H., 2015. Modeling of Gas Production from Shale Reservoirs Considering Multiple Transport Mechanisms. PloS one, 10(12), e0143649. Guo, L., Peng, X., Wu, Z., 2008. Dynamical Characteristics of Methane Adsorption on Monolith Nanometer Activated Carbon. J. Chem. Ind. Eng. (China) 59, 2726−2732. Han, G., Liu, X., Huang, J., 2020. Theoretical Comparison of Test Performance of Different Pulse Decay Methods for Unconventional Cores. Energies. 13(17):4557. https://doi.org/10.3390/en13174557 Harper, J.A., 2008. The Marcellus Shale — an old “new” gas reservoir, in Pennsylvania Geology: Pennsylvania Department of Conservation and Natural Resources, v. 38, no. 1, 20 p. Heller, R., Vermylen, J., Zoback, M., 2014. Experimental investigation of matrix permeability of gas shales. AAPG Bull. 98 (5), 975–995. Higashi, K., Ito, H. and Oishi, J., 1963. Surface Diffusion Phenomena in Gaseous Diffusion, I. Surface Diffusion of Pure Gas. J. Atomic Energy Soc. Japan. 5: 846–853. Hill, T. L., 1956. Surface Diffusion and Thermal Transpiration in Fine Tubes and Pores, 1956. J. Chem. Phys., 25, 730. 257 Ho, C. K., Webb, S. W., 2006. Gas transport in porous media. (Springer,Dordrecht, the Netherlands). Hutson, N. D., & Yang, R. T., 1997. Theoretical basis for the Dubinin-Radushkevitch (DR) adsorption isotherm equation. Adsorption, 3, 189-195. Hwang, S., Kammermeyer, K., 1966. Surface diffusion in microporous media, Can. J. Chem. Eng. 44, 82–89. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Petrol. Technol. 48 (8), 16–21. https://doi.org/10.2118/09-08-16-DA. Jessen, K., Tang, G.Q., Kovscek, A.R., 2008. Laboratory and simulation investigation of enhanced coalbed methane recovery by gas injection. Transp. Porous Media 73 (2),141– 159. Jia, B., Tsau, J.-S., & Barati, R., 2019. A review of the current progress of CO2 injection EOR and carbon storage in shale oil reservoirs. Fuel, 236, 404–427. doi: 10.1016/j.fuel.2018.08.103 Jia, B., Xian, C., 2022. Permeability measurement of the fracture-matrix system with 3D embedded discrete fracture model, Petroleum Science, Volume 19, Issue 4, Pages 1757- 1765, ISSN 1995-8226. https://doi.org/10.1016/j.petsci.2022.01.010. Jin, G., Pérez, H. G., Agrawal, G., Khodja, M. R., Ali, A. Z., Hussaini, S. R., & Jangda, Z. Z., 2015. The impact of gas adsorption and composition on unconventional shale permeability measurement. SPE Paper No. 172744 presented at the SPE Middle East Oil & Gas Show and Conference, 8-11 March, Manama, Bahrain. Kainourgiakis, M. E., Stubos, A. K., Konstantinou, N. D., Kanellopoulos, N. K, Milisic, V., 1996. A Network Model for the Permeability of Condensable Vapours through Mesoporous Media. J. Membrane Sci. 114 (2): 215–225. https://doi.org/10.1016/0376- 7388(95)00322-3. Kamali-Asl, A., Kovscek, A. R., Zoback, M. D., 2022. Long-term permeability evolution of shale seal rocks with argon and scCO2, Journal of Natural Gas Science and Engineering, Volume 104, 104642, ISSN 1875-5100, https://doi.org/10.1016/j.jngse.2022.104642. 258 Kang, S. M., Fathi, E., Ambrose, R. J., et al. 2011. Carbon Dioxide Storage Capacity of Organic-Rich Shales. SPE J. 16 (4): 842–855. SPE134583-PA. http://dx.doi.org/10.2118/134583-PA. Kärger, J., Ruthven, D. M., Theodorou, D. N., 2012. Diffusion in Nanoporous Materials. Wiley, New York. DOI:10.1002/9783527651276. Kärger, J., Ruthven, D.M., 2016. Diffusion in nanoporous materials: fundamental principles, insights and challenges. New Journal of Chemistry, 40(5), pp.4027-4048. Kazemi, M., & Takbiri-Borujeni, A., 2016. Non-equilibrium molecular dynamics simulation of gas flow in organic nanochannels. Journal of Natural Gas Science and Engineering. Kikkinides, E. S., Tzevelekos, K. P., Stubos, A. K., Kainourgiakis, M. E., Kanellopoulos, N. K., 1997. Application of effective medium approximation for the determination of the permeability of condensable vapours through mesoporous media, Chem. Eng. Sci. 52, 2837–2844. Klewiah, I., Berawala, D. S., Walker, H. C. A., Andersen, P. Ø., & Nadeau, P. H., 2020. Review of experimental sorption studies of CO2 and CH4 in shales. Journal of Natural Gas Science and Engineering, 73, 103045. doi: 10.1016/j.jngse.2019.103045 Kuchuk, F., Biryukov, D., Fitzpatrick, T., 2014. "Rate Transient and Decline Curve Analyses for Continuously (Dual-Porosity) and Discretely Naturally Fractured Reservoirs." Paper presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands. doi: https://doi.org/10.2118/170698-MS Langmuir, I, 1918. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40 (9), 1361–1403. https://doi.org/ 10.1021/ja02242a004. Lemmon, E. W., McLinden, M. O., & Friend, D. G., 2005. Thermophysical properties of fluid systems. NIST chemistry webbook, NIST standard reference database, 69. Lemmon, E.W., McLinden, M.O.. Friend, D.G., 2005. In: Thermophysical Properties of Fluid Systems. NIST Chemistry Webbook, vol. 69. NIST standard reference database. 259 Li, Z., Teng, Y., Fan, M., Ripepi, N., Chen, C., 2022. "A Novel Multiphysics Multiscale Multiporosity Shale Gas Transport Model for Geomechanics/Flow Coupling in Steady and Transient States." SPE J. 27: 452–464. doi: https://doi.org/10.2118/206726-PA Lim, K., Aziz, K., 1995. Matrix-fracture transfer shape factors for dual-porosity simulators. J. Petrol. Sci. Eng. 13 (3–4), 169–178. https://doi.org/10.1016/09204105(95)00010-F. Lin, W., Kovscek, A. R., 2014. Gas sorption and the consequent volumetric and permeability change of coal I: Experimental. Transport in Porous Media, 105:371 – 389. ISSN 1573-1634. doi: 10.1007/s11242-014-0373-9. Liu, L., Wang, Y., Aryana, S.A., 2021. Insights into scale translation of methane transport in nanopores, Journal of Natural Gas Science and Engineering, Volume 96, 104220, ISSN 1875-5100, https://doi.org/10.1016/j.jngse.2021.104220. Loeb, L.B., 1934. The kinetic theory of gases, 2nd ed. McGraw-Hill Co. Inc., New York. Loschko M, W¨ohling T, Rudolph DL, Cirpka OA. An electron-balance based approach to predict the decreasing denitrification potential of an aquifer. Groundwater 2019;57(6):925– 39. Luffel, D. L., Hopkins, C. W., & Schettler Jr, P. D., 1993. Matrix permeability measurement of gas productive shales. SPE Paper No. 26633 presented at the SPE Annual Technical Conference and Exhibition, 3-6 October, Houston, Texas. Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2021. Characterization of shale using Helium and Argon at high pressures, Journal of Petroleum Science and Engineering, doi: https://doi.org/10.1016/j.petrol.2021.108952 Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023. Investigation of methane mass transfer and sorption in Marcellus shale under variable net-stress. Geoenergy Science and Engineering 227. https://doi.org/10.1016/j.geoen.2023.211846 Lyu, Y., Dasani, D., Tsotsis, T., Jessen, K., 2023a. Investigation of Methane Mass Transfer and Sorption in Marcellus Shale Under Variable Net Stress. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211846. 260 Lyu, Y., Elkady, Y., Kovscek, A.R., Jessen, K., 2023b. Analysis of Gas Storage and Transport in Shale using Pressure Pulse Decay Measurements with He, Kr and CO2. Geoenergy Science and Engineering, doi: https://doi.org/10.1016/j.geoen.2023.211951. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R., & Bose, T. K. (1992). Adsorption measurements of argon, neon, krypton, nitrogen, and methane on activated carbon up to 650 MPa. Langmuir, 8(2), 577-580. Malheiro, C., Mendiboure, B., Plantier, F., Guatarbes, B., & Miqueu, C. (2015). An accurate model for the filling pressure of carbon slit-like micropores. Fluid Phase Equilibria, 404, 118-123. Mason, E.A., Malinauskas, A.P., 1983. Gas Transport in Porous Media: The Dusty-gas Model, Elsevier Science Ltd., New York. Medveď, I., Černý, R., 2011. Surface diffusion in porous media: A critical review, Microporous and Mesoporous Materials, Volume 142, Issues 2–3, Pages 405-422, ISSN 1387-1811, https://doi.org/10.1016/j.micromeso.2011.01.015. Memon, A., Li, A., Jacqueline, N., Kashif, M., Ma, M., 2020. Study of gas sorption, stress effects and analysis of effective porosity and permeability for shale gas reservoirs, Journal of Petroleum Science and Engineering, Volume 193, 107370, ISSN 0920-4105, https://doi.org/10.1016/j.petrol.2020.107370. Nelson, P. H., 2009. Pore-throat sizes in sandstones, tight sandstones, and shales. AAPG Bulletin, 93(3), 329–340. doi: 10.1306/10240808059 Ning, X., Fan, J., Holditch, S. A., & Lee, W. J., 1993. The measurement of matrix and fracture properties in naturally fractured cores. SPE Paper No. 25898 presented at the Low Permeability Reservoirs Symposium, 26-28 April, Denver, Colorado. Nuttall, B.C., Eble, C.F., Drahovzal, J.A.D., Bustin, R.M., 2005. Analysis of Devonian Black Shales in Kentucky for Potential Carbon Dioxide Sequestration and Enhanced Natural Gas Production. https://doi.org/10.2172/920185. Ottiger, S., Pini, R., Storti, G., & Mazzotti, M., 2008. Measuring and modeling the competitive adsorption of CO2, CH4, and N2 on a dry coal. Langmuir,24(17), 9531-9540. 261 Peng, S., Ren, B., and Meng M., 2019. "Quantifying the Influence of Fractures for MoreAccurate Laboratory Measurement of Shale Matrix Permeability Using a Modified GasExpansion Method." SPE Res Eval & Eng 22: 1293–1304. doi: https://doi.org/10.2118/195570-PA Peng, Y., Liu, J., Pan, Z., Qu, H., Connell, L., 2018. Evolution of shale apparent permeability under variable boundary conditions. Fuel 215, 46–56. https://doi.org/10.1016/j.fuel.2017.11.024. Petropoulos, J.H., 1996. Langmuir 12, 4814–4816. Rani, S., Padmanabhan, E., & Prusty, B. K. (2019). Review of gas adsorption in shales for enhanced methane recovery and CO2 storage. Journal of Petroleum Science and Engineering, 175, 634–643. https://doi.org/10.1016/j.petrol.2018.12.081 Ren, J., Guo, P., Guo, Z., Wang, Z., 2015. A lattice Boltzmann model for simulating gas flow in kerogen pores. Transport Porous Media 106 (2), 285–301. Ross, D. J., & Bustin, R. M. (2007). Impact of mass balance calculations on adsorption capacities in microporous shale gas reservoirs. Fuel, 86(17), 2696-2706. Roy, S., Raju, R., Chuang, H.F., Cruden, B.A., and Meyyappan, M. 2003. Modeling gas flow through microchannels and nanopores. Journal of Applied Physics, 93(8), pp. 4870– 4879. Roychaudhuri, B., Tsotsis, T.T., Jessen, K., 2013. An experimental investigation of spontaneous imbibition in gas shales. J. Petrol. Sci. Eng. 111, 87–97. Rubin Y., 2003. Applied stochastic hydrogeology. New York: Oxford University Press. Ruckenstein, E., Vaidyanathan, A. S., & Youngquist, G. R. (1971). Sorption by solids with bidisperse pore structures. Chemical Engineering Science,26(9), 1305-1318. Sakhaee-Pour, A. and Bryant, S. L., 2012. Gas Permeability of Shale. SPE Res Eval & Eng 15 (4): 401–409. SPE-146944-PA. http://dx.doi.org/10.2118/146944-PA. Sakhaee-Pour, A., Bryant, S. L., 2012. Gas permeability of shale. SPE Reserv Eval Eng 15, 401–409, DOI: 10.2118/146944-PA. 262 Salem, M. M. K., Braeuer, P., Szombathely, M. V., Heuchel, M., Harting, P., Quitzsch, K., Jaroniec, M., 1998. Thermodynamics of high-pressure adsorption of argon, nitrogen, and methane on microporous adsorbents. Langmuir, 14(12), 3376-3389. Sanaei, A., Jamili, A., & Callard, J., 2014a. Effect of Pore Size Distribution and Connectivity on Phase Behavior and Gas Condensate Production From Unconventional Resources. SPE Unconventional Resources Conference. doi: 10.2118/168970-MS Sanaei, A., Jamili, A., Callard, J., Mathur, A., 2014b. Production modeling in the Eagle Ford gas condensate window: integrating new relationships between core permeability, pore size, and confined PVT properties. In: SPE Western North American and Rocky Mountain Joint Regional Meeting Held in Denver, Colorado, USA, 16-18 April 2014. Santos, J. M., & Akkutlu, I. Y., 2013. Laboratory Measurement of Sorption Isotherm under Confining Stress with Pore-Volume Effects. SPE Journal,18(05), 924-931. Schaaf, S.A. and Chambre, P.L. 1961. Flow of Rarefied Gases. Princeton University Press, Princeton, NJ. Sheng, G., Javadpour, F., Su,Y., 2018. Effect of microscale compressibility on apparent porosity and permeability in shale gas reservoirs, International Journal of Heat and Mass Transfer, Volume 120, Pages 56-65, ISSN 0017-9310, https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.014. Shi, J. Q., & Durucan, S., 2003. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel, 82(10), 1219-1229. Sigal, R. F., & Qin, B. (2008, June 1). Examination of the Importance of Self Diffusion In the Transportation of Gas In Shale Gas Reservoirs. Society of Petrophysicists and WellLog Analysts. Sigal, R. F., 2013. The Effects of Gas Adsorption on Storage and Transport of Methane in Organic Shales. Oral presentation of paper SPWLA2013-Y given at the SPWLA 54th Annual Logging Symposium, New Orleans, 22–26 June. Sing, K. S., 1985, Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity: Pure and Applied Chemistry, 57, 603- 619. 263 Sinha, S., Braun, E.M., Determan, M.D., Passey, Q.R., Leonardi, S.A., Boros, J.A., Wood, A.C., Zirkle, T., Kudva, R.A. (2013). Steady-state permeability measurements on intact shale samples at reservoir conditions – effect of stress, temperature, pressure, and type of gas. , SPE No. 164263 presented at Society of Petroleum Engineers (SPE) Middle East Oil and Gas Show and Exhibition, 10-13 March, Manama, Bahrain. Soltanian M.R., Ritzi R.W., Huang C., Dai Z., 2015. Relating reactive solute transport to hierarchical and multiscale sedimentary architecture in a Lagrangian-based transport model: 1. Time-dependent effective retardation factor. Water Resources 2015;51(3):1586– 600. Song W., Yao J., Li Y., Sun H., Zhang L., Yang Y., Zhao J., & Sui H., 2016. Apparent Gas Permeability in Organic-rich Shale Reservoir. Fuel. Vol 181, 973–984. Song, W., Yao, B., Yao, J., Li, Y., Sun, H., Yang, Y., Zhang, L., 2018a. Methane surface diffusion capacity in carbon-based capillary with application to organic-rich shale gas reservoir, Chemical Engineering Journal, Volume 352, Pages 644-654, ISSN 1385-8947, https://doi.org/10.1016/j.cej.2018.07.050. Song, W., Yao, J., Ma, J., Sun, H., Li, Y., Yang, Y., Zhang, L., 2018b. Numerical simulation of multiphase flow in nanoporous organic matter with application to coal and gas shale systems. Water Resour. Res. 54 (2), 1077–1092. Song, W., Yao, J., Wang, D., Li, Y., Sun, H., & Yang, Y., 2020. Dynamic pore network modelling of real gas transport in shale nanopore structure. Journal of Petroleum Science and Engineering, 184, 106506. doi: 10.1016/j.petrol.2019.106506 Sun, Z., Li, X., Shi, J., Zhang, T., Sun, F., 2017. Apparent permeability model for real gas transport through shale gas reservoirs considering water distribution characteristic. Int. J. Heat Mass Tran. 115, 1008–1019. Terzaghi, K., 1923. Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen, Sitzungsber. Akad. Wissen. Wien Math. Naturwiss. Kl., Wien, Abt. 2A, 132, pp. 105–124. Tesson, S., Firoozabadi, A., 2019. Deformation and swelling of kerogen matrix in light hydrocarbons and carbon dioxide, J. Phys. Chem. C 123, 29173–29183, https://doi.org/10.1021/acs.jpcc.9b04592. 264 Tinni, A., Fathi, E., Agarwal, R., Sondergeld, C. H., Akkutlu, I. Y., & Rai, C. S. (2012). Shale permeability measurements on plugs and crushed samples. SPE No. 162235 presented at SPE Canadian Unconventional Resources Conference, 10 October-1 November, Calgary, Alberta, Canada. U.S. Geological Survey, 2019, New Estimates for Marcellus and Point Pleasant-Utica Shales, accessed October 3, 2019, at URL https://www.usgs.gov/news/national-newsrelease/usgs-estimates-214-trillion-cubic-feet-natural-gas-appalachian-basin Vega, B., Dutta, A., Kovscek. A. R., 2014. CT imaging of low-permeability, dual-porosity systems using high X-ray contrast gas. Transport in Porous Media, 101(1):81–97. ISSN 1573-1634. doi: 10.1007/s11242-013-0232-0. Vermeulen, T., 1953. Theory for Irreversible and Constant-Pattern Solid Diffusion. Industrial & Engineering Chemistry, 45(8), 1664–1670. doi: 10.1021/ie50524a025 Volmer, M. and Adhikari, G. 1925. Z. Phys. 35, 170. Wang, C., Pan, L., Zhao, Y., Zhang, Y., Shen, W., 2019. Analysis of the Pressure-Pulse Propagation in Rock: A New Approach to Simultaneously Determine Permeability, Porosity, and Adsorption Capacity. Rock Mech Rock Eng 52, 4301–4317. https://doi.org/10.1007/s00603-019-01874-w Wang, J., Yang, Z., Dong, M., Gong, H., Sang, Q., Li, Y., 2016a. Experimental and numerical investigation of dynamic gas adsorption/desorption–diffusion process in shale. Energy Fuels 30 (12), 10080–10091. https://doi.org/10.1021/acs. energyfuels.6b01447. Wang, J., Wang, B.E., Li, Y., Yang, Z., Gong, H., Dong, M., 2016b. Measurement of dynamic adsorption–diffusion process of methane in shale. Fuel 172, 37–48. https:// doi.org/10.1016/j.fuel.2015.12.069. Wang, Y. (2016). Investigation of Gas Sorption and Mass Transfer in Gas Shales (Doctoral dissertation). University of Southern California, Los Angeles. Wang, Y., Tsotsis, T. T., & Jessen, K., 2015. Competitive Sorption of Methane/Ethane Mixtures on Shale: Measurements and Modeling. Industrial & Engineering Chemistry Research, 54(48), 12187-12195. DOI: 10.1021/acs.iecr.5b02850 265 Wang, Y., Zhu, Y., Liu, S. and Zhang, R., 2016. Pore characterization and its impact on methane adsorption capacity for organic-rich marine shales. Fuel, 181, pp.227-237. Warren, J.E., and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. doi: https:// doi.org/10.2118/426-PA. Wasaki, A. and Akkutlu, I. Y., 2014. Permeability of Organic-rich Shale. Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, 27–29 October. SPE170830-MS. http://dx.doi.org/10.2118/170830-MS. Wasaki, A., Akkutlu, I.Y., 2015. Permeability of organic-rich shale. Spe Journal 20 (06), 1384–1396. https://doi.org/10.2118/170830-PA. Weniger, P., Kalkreuth, W., Busch, A. and Krooss, B.M., 2010. High-pressure methane and carbon dioxide sorption on coal and shale samples from the Paraná Basin, Brazil. International Journal of Coal Geology, 84(3-4), pp.190-205. Wentworth, C. K., 1944. Amer. J. Sci., 242, 478. Wicke, E. and Kallenbach, R., 1941. Kolloidzschr. 97, 135. Wicke, E. and Voigt, U., 1947. Angew. Chem. B, 19, 94. Wicke, E., 1939. Kolloidzschr. 86, 167. Wu, K., Li, X., Guo, C., Wang, C., Chen, Z., 2016. A Unified Model for Gas Transfer in Nanopores of Shale-Gas Reservoirs: Coupling Pore Diffusion and Surface Diffusion. SPE J. 21: 1583–1611. doi: https://doi.org/10.2118/2014-1921039-PA. Wu, K., Li, X., Wang, C., Yu, W., Chen, Z., 2015. Model for Surface Diffusion of Adsorbed Gas in Nanopores of Shale Gas Reservoirs. Ind Eng Chem Res 54:3225–36. https:// doi.org/10.1021/ie504030v. Wu, T., Zhang, D., 2016. Impact of Adsorption on Gas Transport in Nanopores. Sci Rep 6, 23629. https://doi.org/10.1038/srep23629 266 Xiong, X., Devegowda, D., Michel, G., et al. 2012. A Fully-Coupled Free and Adsorptive Phase Transport Model for Shale Gas Reservoirs Including Non-Darcy Flow Effects. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8–10 October. SPE-159758-MS. http://dx.doi.org/10.2118/159758-MS. Xu, J. (2013). The study of mass transfer in gas shales and the optimization of hydraulic stimulation processes via additives. (Doctoral dissertation). University of Southern California. Yamada, S. E. Jones, A. H., 1980. A review of a pulse technique for permeability measurements. Yang, B., Kang, Y., You, L., Li, X., Chen, Q., 2016. Measurement of the surface diffusion coefficient for adsorbed gas in the fine mesopores and micropores of shale organic matter, Fuel 181, 793–804. Yang, R.T., Fenn, J.B., Haller, G.L., 1973. Modification to the Higashi model for surface diffusion, AIChE J. 19 (1973) 1052–1053. Yang, Y. and Liu, S., 2020. Review of shale gas sorption and its models. Energy & Fuels, 34(12), pp.15502-15524. Yu, X., Li, J., Chen, Z., Wu, K., Zhang, L., Yang, S., 2021. Determination of CH4, C2H6 and CO2 adsorption in shale kerogens coupling sorption-induced swelling. Chem Eng J. 410:127690. https://doi.org/10.1016/j.cej.2020.127690. Zachara, J.M., Chen, X., Murray, C., Hammond, G., 2016. River stage influences on uranium transport in a hydrologically dynamic groundwater-surface water transition zone. Water Resour. Res. 52, 1568–1590. https://doi.org/10.1002/2015WR018009. Zarragoicoechea, G.J., Kuz, Z.A., 2002. Van der Waals equation of state for a fluid in a nanopore. Physical Review, Issue E 65, 021110(1-4). Zhang, J., Raslan, M., Wu, C., Jessen, K., 2022. "A Generalized Dynamic Transfer Function for Ultra-Tight Dual-Porosity Systems." Paper presented at the SPE Western Regional Meeting, Bakersfield, California, USA, April 2022. doi: https://doi.org/10.2118/209324-MS. 267 Zhang, J.J., Liu, H.H., Boudjatit, M., 2020. Matrix permeability measurement from fractured unconventional source-rock samples: Method and application, Journal of Contaminant Hydrology, Volume 233, 103663, ISSN 0169-7722, https://doi.org/10.1016/j.jconhyd.2020.103663 Zhang, L., Liang, H., Zhao, Y., Xie, J., Peng, X., & Li, Q., 2020. Gas transport characteristics in shale matrix based on multiple mechanisms. Chemical Engineering Journal, 386, 124002. doi: 10.1016/j.cej.2019.124002 Zhang, L., Xiong, Y., Li, Y., Wei, M., Jiang, W., Lei, R., & Wu, Z., 2017. DFT modeling of CO2 and Ar low-pressure adsorption for accurate nanopore structure characterization in organic-rich shales. Fuel, 204, 1–11. doi: 10.1016/j.fuel.2017.05.046 Zhang, X., Ma, F., Yin, S., Wallace, C.D., Soltanian, M.R., Dai, Z., Ritzi, R. W., Ma, Z., Zhan, C., Lü, X., 2021. Application of upscaling methods for fluid flow and mass transport in multi-scale heterogeneous media: A critical review. Applied energy, 303, 117603. doi: 10.1016/j.apenergy.2021.117603 Zhang, Z., Huang, S., Xian, S., Xi, H., Li, Z., 2011. Adsorption Equilibrium and Kinetics of CO2 on Chromium Terephthalate MIL-101. Energy Fuels, 25, 835−842. Zhao, H., Lai, Z., Firoozabadi, A., 2017. Sorption Hysteresis of Light Hydrocarbons and Carbon Dioxide in Shale and Kerogen. Sci Rep 7, 16209. https://doi.org/10.1038/s41598- 017-13123-7. Zimmerman, R.W., 1991. Compressibility of Sandstones, Developments in Petroleum Science 29, Elsevier Science Publishers B.V., Amsterdam, 173p.. Zimmerman, R.W., Chen, G., Hadgu, T. et al, 1993. A Numerical Dual-Porosity Model with Semianalytical Treatment of Fracture/Matrix Flow. Water Res Res 29 (7): 2127–2137. doi: https://doi.org/10.1029/93WR00749.

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Creator Lyu, Ye (author)

Core Title Investigation of gas transport and sorption in shales

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Petroleum Engineering

Degree Conferral Date 2023-12

Publication Date 12/05/2023

Defense Date 10/30/2023

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag carbon storage,characterization,gas diffusion,gas sorption,mass transfer,shale gas

Format theses (aat)

Language English

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Advisor Jessen, Kristian (committee chair), Hammond, Doug (committee member), Tsotsis, Theo (committee member)

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