Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks
Description
Understanding and predicting fluid flow in hydrocarbon shale and other non-conventional reservoir rocks
Oil and natural gas reservoirs found in shale and other tight and ultra-tight porous rocks have become increasingly important sources of energy in both North America and East Asia. As a result, extensive research in recent decades has focused on the mechanisms of fluid transfer within these reservoirs, which have complex pore networks at multiple scales. Continued research into these important energy sources requires detailed knowledge of the emerging theoretical and computational developments in this field.
Following a multidisciplinary approach that combines engineering, geosciences and rock physics, Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks provides both academic and industrial readers with a thorough grounding in this cutting-edge area of rock geology, combining an explanation of the underlying theories and models with practical applications in the field.
Readers will also find:
* An introduction to the digital modeling of rocks
* Detailed treatment of digital rock physics, including decline curve analysis and non-Darcy flow
* Solutions for difficult-to-acquire measurements of key petrophysical characteristics such as shale wettability, effective permeability, stress sensitivity, and sweet spots
Physics of Fluid Flow and Transport in Unconventional Reservoir Rocks is a fundamental resource for academic and industrial researchers in hydrocarbon exploration, fluid flow, and rock physics, as well as professionals in related fields.
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Persons
Behzad Ghanbarian, PhD is Associate Professor of Engineering Geology and Director of the Porous Media Research Lab in the Department of Geology, Kansas State University, Manhattan, Kansas, USA.
Feng Liang, PhD is Production Technology Leader at Aramco Research Center Houston, Aramco Americas, Houston, Texas, USA.
Hui-Hai Liu, PhD is a Senior Petroleum Engineering Consultant at Aramco Research Center Houston, Aramco Americas, Houston, Texas, USA.
Content
List of Contributors xvii
Preface xxi
Introduction 1
1 Unconventional Reservoirs: Advances and Challenges 3
Behzad Ghanbarian, Feng Liang, and Hui-Hai Liu
1.1 Background 3
1.2 Advances 4
1.2.1 Wettability 4
1.2.2 Permeability 5
1.3 Challenges 7
1.3.1 Multiscale Systems 7
1.3.2 Hydrocarbon Production 9
1.3.3 Recovery Factor 9
1.3.4 Unproductive Wells 9
1.4 Concluding Remarks 11
References 11
Part I Pore-Scale Characterizations 15
2 Pore-Scale Simulations and Digital Rock Physics 17
Junjian Wang, Feifei Qin, Jianlin Zhao, Li Chen, Hari Viswanathan, and Qinjun Kang
2.1 Introduction 17
2.2 Physics of Pore-Scale Fluid Flow in Unconventional Rocks 18
2.2.1 Physics of Gas Flow 18
2.2.1.1 Gas Slippage and Knudsen Layer Effect 18
2.2.1.2 Gas Adsorption/Desorption and Surface Diffusion 20
2.2.2 Physics of Water Flow 22
2.2.3 Physics of Condensation 23
2.3 Theory of Pore-Scale Simulation Methods 23
2.3.1 The Isothermal Single-Phase Lattice Boltzmann Method 23
2.3.1.1 Bhatnagar-Gross-Krook (BGK) Collision Operator 24
2.3.1.2 The Multi-Relaxation Time (MRT)-LB Scheme 24
2.3.1.3 The Regularization Procedure 26
2.3.2 Multi-phase Lattice Boltzmann Simulation Method 27
2.3.2.1 Color-Gradient Model 27
2.3.2.2 Shan-Chen Model 28
2.3.3 Capture Fluid Slippage at the Solid Boundary 29
2.3.4 Capture the Knudsen Layer/Effective Viscosity 30
2.3.5 Capture the Adsorption/Desorption and Surface Diffusion Effects 30
2.3.5.1 Modeling of Adsorption in LBM 30
2.3.5.2 Modeling of Surface Diffusion Via LBM 31
2.4 Applications 32
2.4.1 Simulation of Gas Flow in Unconventional Reservoir Rocks 32
2.4.1.1 Gas Slippage 32
2.4.1.2 Gas Adsorption 33
2.4.1.3 Surface Diffusion of Adsorbed Gas 35
2.4.2 Simulation of Water Flow in Unconventional Reservoir Rocks 35
2.4.3 Simulation of Immiscible Two-Phase Flow 39
2.4.4 Simulation of Vapor Condensation 43
2.4.4.1 Model Validations 44
2.4.4.2 Vapor Condensation in Two Adjacent Nano-Pores 44
2.5 Conclusion 48
References 49
3 Digital Rock Modeling: A Review 53
Yuqi Wu and Pejman Tahmasebi
3.1 Introduction 53
3.2 Single-Scale Modeling of Digital Rocks 54
3.2.1 Experimental Techniques 54
3.2.1.1 Imaging Technique of Serial Sectioning 54
3.2.1.2 Laser Scanning Confocal Microscopy 54
3.2.1.3 X-Ray Computed Tomography Scanning 55
3.2.2 Computational Methods 55
3.2.2.1 Simulated Annealing 56
3.2.2.2 Markov Chain Monte Carlo 56
3.2.2.3 Sequential Indicator Simulation 56
3.2.2.4 Multiple-Point Statistics 57
3.2.2.5 Machine Learning 58
3.2.2.6 Process-Based Modeling 58
3.3 Multiscale Modeling of Digital Rocks 59
3.3.1 Multiscale Imaging Techniques 60
3.3.2 Computational Methods 60
3.3.2.1 Image Superposition 60
3.3.2.2 Pore-Network Integration 61
3.3.2.3 Image Resolution Enhancement 63
3.3.2.4 Object-Based Reconstruction 63
3.4 Conclusions and Future Perspectives 65
Acknowledgments 66
References 66
4 Scale Dependence of Permeability and Formation Factor: A Simple Scaling Law 77
Behzad Ghanbarian and Misagh Esmaeilpour
4.1 Introduction 77
4.2 Theory 78
4.2.1 Funnel Defect Approach 78
4.2.2 Application to Porous Media 79
4.3 Pore-network Simulations 80
4.4 Results and Discussion 81
4.5 Limitations 86
4.6 Conclusion 86
Acknowledgment 86
References 87
Part II Core-Scale Heterogeneity 89
5 Modeling Gas Permeability in Unconventional Reservoir Rocks 91
Behzad Ghanbarian, Feng Liang, and Hui-Hai Liu
5.1 Introduction 91
5.1.1 Theoretical Models 91
5.1.2 Pore-Network Models 92
5.1.3 Gas Transport Mechanisms 93
5.1.4 Objectives 93
5.2 Effective-Medium Theory 93
5.3 Single-Phase Gas Permeability 95
5.3.1 Gas Permeability in a Cylindrical Tube 95
5.3.2 Pore Pressure-Dependent Gas Permeability in Tight Rocks 96
5.3.3 Comparison with Experiments 96
5.3.4 Comparison with Pore-Network Simulations 98
5.3.5 Comparaison with Lattice-Boltzmann Simulations 99
5.4 Gas Relative Permeability 100
5.4.1 Hydraulic Flow in a Cylindrical Pore 100
5.4.2 Molecular Flow in a Cylindrical Pore 101
5.4.3 Total Gas Flow in a Cylindrical Pore 101
5.4.4 Gas Relative Permeability in Tight Rocks 101
5.4.5 Comparison with Experiments 102
5.4.6 Comparison with Pore-Network Simulations 107
5.5 Conclusions 108
Acknowledgment 109
References 109
6 NMR and Its Applications in Tight Unconventional Reservoir Rocks 113
Jin-Hong Chen, Mohammed Boudjatit, and Stacey M. Althaus
6.1 Introduction 113
6.2 Basic NMR Physics 113
6.2.1 Nuclear Spin 114
6.2.2 Nuclear Zeeman Splitting and NMR 114
6.2.3 Nuclear Magnetization 115
6.2.4 Bloch Equations and NMR Relaxation 116
6.2.5 Simple NMR Experiments: Free Induction Decay and CPMG Echoes 117
6.2.6 NMR Relaxation of a Pure Fluid in a Rock Pore 118
6.2.7 Measured NMR CPMG Echoes in a Formation Rock 119
6.2.8 Inversion 119
6.2.8.1 Regularized Linear Least Squares 120
6.2.8.2 Constrains of the Resulted NMR Spectrum in Inversion 120
6.2.9 Data from NMR Measurement 121
6.3 NMR Logging for Unconventional Source Rock Reservoirs 121
6.3.1 Brief Introduction of Unconventional Source Rocks 121
6.3.2 NMR Measurement of Source Rocks 122
6.3.2.1 NMR Log of a Source Rock Reservoir 122
6.3.3 Pore Size Distribution in a Shale Gas Reservoir 124
6.4 NMR Measurement of Long Whole Core 125
6.4.1 Issues of NMR Instrument for Long Sample 125
6.4.2 HSR-NMR of Long Core 126
6.4.3 Application Example 128
6.5 NMR Measurement on Drill Cuttings 130
6.5.1 Measurement Method 131
6.5.1.1 Preparation of Drill Cuttings 131
6.5.1.2 Measurements 131
6.5.2 Results 132
6.6 Conclusions 133
References 135
7 Tight Rock Permeability Measurement in Laboratory: Some Recent Progress 139
Hui-Hai Liu, Jilin Zhang, and Mohammed Boudjatit
7.1 Introduction 139
7.2 Commonly Used Laboratory Methods 140
7.2.1 Steady-State Flow Method 140
7.2.2 Pressure Pulse-Decay Method 141
7.2.3 Gas Research Institute Method 143
7.3 Simultaneous Measurement of Fracture and Matrix Permeabilities from Fractured Core Samples 144
7.3.1 Estimation of Fracture and Matrix Permeability from PPD Data for Two Flow Regimes 144
7.3.2 Mathematical Model 146
7.3.3 Method Validation and Discussion 148
7.4 Direct Measurement of Permeability-Pore Pressure Function 150
7.4.1 Knudsen Diffusion, Slippage Flow, and Effective Gas Permeability 150
7.4.2 Methodology for Directly Measuring Permeability-Pore Pressure Function 152
7.4.3 Experiments 155
7.5 Summary and Conclusions 159
References 159
8 Stress-Dependent Matrix Permeability in Unconventional Reservoir Rocks 163
Athma R. Bhandari, Peter B. Flemings, and Sebastian Ramiro-Ramirez
8.1 Introduction 163
8.2 Sample Descriptions 164
8.3 Permeability Test Program 165
8.4 Permeability Behavior with Confining Stress Cycling 166
8.5 Matrix Permeability Behavior 170
8.6 Concluding Remarks 172
Acknowledgments 174
References 174
9 Assessment of Shale Wettability from Spontaneous Imbibition Experiments 177
Zhiye Gao and Qinhong Hu
9.1 Introduction 177
9.2 Spontaneous Imbibition Theory 178
9.3 Samples and Analytical Methods 179
9.3.1 SI Experiments 179
9.3.2 Barnett Shale from United States 180
9.3.3 Silurian Longmaxi Formation and Triassic Yanchang Formation Shales from China 180
9.3.4 Jurassic Ziliujing Formation Shale from China 182
9.4 Results and Discussion 183
9.4.1 Complicated Wettability of Barnett Shale Inferred Qualitatively from SI Experiments 183
9.4.1.1 Wettability of Barnett Shale 184
9.4.1.2 Properties of Barnett Samples and Their Correlation to Wettability 186
9.4.1.3 Low Pore Connectivity to Water of Barnett Samples 187
9.4.2 More Oil-Wet Longmaxi Formation Shale and More Water-Wet Yanchang Formation Shale 188
9.4.2.1 TOC and Mineralogy 188
9.4.2.2 Pore Structure Difference Between Longmaxi and Yanchang Samples 188
9.4.2.3 Water and Oil Imbibition Experiments 191
9.4.2.4 Wettability of Longmaxi and Yanchang Shale Samples Deduced from SI Experiments 197
9.4.3 Complicated Wettability of Ziliujing Formation Shale 197
9.4.3.1 TOC and Mineralogy 197
9.4.3.2 Pore Structure 197
9.4.3.3 Water and Oil Imbibition Experiments 200
9.4.3.4 Wettability of Ziliujing Formation Shale Indicated from SI Experiments and its Correlation to Shale Pore Structure and Composition 201
9.4.4 Shale Wettability Evolution Model 201
9.5 Conclusions 204
Acknowledgments 204
References 204
10 Permeability Enhancement in Shale Induced by Desorption 209
Brandon Schwartz and Derek Elsworth
10.1 Introduction 209
10.1.1 Shale Mineralogical Characteristics 209
10.1.2 Flow Network 210
10.1.2.1 Bedding-Parallel Flow Network 211
10.1.2.2 Bedding-Perpendicular Flow Paths 212
10.2 Adsorption in Shales 214
10.2.1 Langmuir Theory 214
10.2.2 Competing Strains in Permeability Evolution 215
10.2.2.1 Poro-Sorptive Strain 215
10.2.2.2 Thermal-Sorptive Strain 218
10.3 Permeability Models for Sorptive Media 218
10.3.1 Strain Based Models 219
10.4 Competing Processes during Permeability Evolution 220
10.4.1 Resolving Competing Strains 220
10.4.2 Solving for Sorption-Induced Permeability Evolution 221
10.5 Desorption Processes Yielding Permeability Enhancement 223
10.5.1 Pressure Depletion 223
10.5.2 Lowering Partial Pressure 224
10.5.3 Sorptive Gas Injection 225
10.5.4 Desorption with Increased Temperature 225
10.6 Permeability Enhancement Due to Nitrogen Flooding 225
10.7 Discussion 226
10.8 Conclusion 228
References 229
11 Multiscale Experimental Study on Interactions Between Imbibed Stimulation Fluids and Tight Carbonate Source Rocks 235
Feng Liang, Hui-Hai Liu, and Jilin Zhang
11.1 Introduction 235
11.2 Fluid Uptake Pathways 236
11.2.1 Experimental Methods 236
11.2.1.1 Materials 236
11.2.1.2 Experimental Procedure 237
11.2.2 Results and Discussion 237
11.2.2.1 Surface Characterization 237
11.2.2.2 Spontaneous Imbibition Tests 239
11.3 Mechanical Property Change After Fluid Exposure 240
11.3.1 Experimental Methods 242
11.3.1.1 Materials 242
11.3.1.2 Experimental Procedure 242
11.3.2 Results and Discussion 243
11.3.2.1 UCS and Brazilian Test on Cylindrical Core Plugs 243
11.3.2.2 Microindentation Test 243
11.4 Morphology and Minerology Changes After Fluid Exposure 245
11.4.1 Experimental Methods 247
11.4.1.1 Materials 247
11.4.1.2 Experimental Procedure 248
11.4.2 Results and Discussion 248
11.4.2.1 SEM and EDS Mapping of Thin-Section Surface before Fluid Treatment 248
11.4.2.2 SEM and EDS Mapping of Thin-Section Surface after Fluid Treatment 251
11.4.2.3 Quantification of Dissolved Ions in the Treatment Fluids 256
11.5 Flow Property Change After Fluid Exposure 257
11.5.1 Experimental Methods 258
11.5.1.1 Materials 258
11.5.1.2 Experimental Procedure 258
11.5.2 Results and Discussion 258
11.5.2.1 Changes in Flow Characteristics 258
11.6 Conclusions 259
References 261
Part III Large-Scale Petrophysics 265
12 Effective Permeability in Fractured Reservoirs: Percolation-Based Effective-Medium Theory 267
Behzad Ghanbarian
12.1 Introduction 267
12.1.1 Percolation Theory 267
12.1.2 Effective-Medium Theory 268
12.2 Objectives 269
12.3 Percolation-Based Effective-Medium Theory 269
12.4 Comparison with Simulations 270
12.4.1 Chen et al. (2019) 270
12.4.1.1 Two-Dimensional Simulations 271
12.4.1.2 Three-Dimensional Simulations 273
12.4.2 New Three-Dimensional Simulations 274
12.5 Conclusion 275
Acknowledgment 277
References 277
13 Modeling of Fluid Flow in Complex Fracture Networks for Shale Reservoirs 281
Hongbing Xie, Xiaona Cui, Wei Yu, Chuxi Liu, Jijun Miao, and Kamy Sepehrnoori
13.1 Shale Reservoirs with Complex Fracture Networks 281
13.2 Complex Fracture Reservoir Simulation 281
13.3 Embedded Discrete Fracture Model 283
13.4 EDFM Verification 286
13.5 Well Performance Study - Base Case 290
13.6 Effect of Natural Fracture Connectivity on Well Performance 294
13.6.1 Effect of Natural Fracture Azimuth 294
13.6.2 Effect of Number of Natural Fractures 295
13.6.3 Effect of Natural Fracture Length 298
13.6.4 Effect of Number of Sets of Natural Fractures 301
13.6.5 Effect of Natural Fracture Dip Angle 305
13.7 Effect of Natural Fracture Conductivity on Well Performance 306
13.8 Conclusions 311
References 312
14 A Closed-Form Relationship for Production Rate in Stress-Sensitive Unconventional Reservoirs 315
Hui-Hai Liu, Huangye Chen, and Yanhui Han
14.1 Introduction 315
14.2 Production Rate as a Function of Time in the Linear Flow Regime Under the Constant Pressure Drawdown Condition 317
14.3 An Approximate Relationship Between Parameter A and Stress-Dependent Permeability 318
14.4 Evaluation of the Relationship Between Parameter A and Stress-Dependent Permeability 321
14.5 Equivalent State Approximation for the Variable Pressure Drawdown Conditions 327
14.6 Discussions 328
14.7 Concluding Remarks 329
Nomenclature 329
Subscript 330
Appendix 14.A Derivation of Eq. (14.22) with Integration by Parts 330
References 331
15 Sweet Spot Identification in Unconventional Shale Reservoirs 333
Rabah Mesdour, Mustafa Basri, Cenk Temizel, and Nayif Jama
15.1 Introduction 333
15.2 Reservoir Characterization 334
15.3 Sweet Spot Identification 334
15.3.1 The Method Based on Organic, Rock and Mechanical Qualities 335
15.3.2 Methods Based on Geological and Engineering Sweet Spots 337
15.3.3 Methods Based on Other Quality Indicators 340
15.3.4 Methods Based on Data Mining and Machine Learning 343
15.4 Discussion 345
15.5 Conclusion 346
References 347
Index 351
1
Unconventional Reservoirs: Advances and Challenges
Behzad Ghanbarian1, Feng Liang2, and Hui-Hai Liu2
1 Porous Media Research Lab, Department of Geology, Kansas State University, Manhattan, KS, USA
2 Aramco Americas: Aramco Research Center-Houston, Houston, TX, USA
1.1 Background
Energy is one of the most important components in the world. Primary sources of energy take various forms, such as fossil energy, nuclear energy, and renewable energy sources. Fossil energy resources (e.g. coal, oil, and natural gas) were formed when plants and animals died and were buried underground. The quality of hydrocarbon accordingly depends on organic content as well as temperature and pressure conditions. Although there are limited reserves of fossil energy resources and despite recent advances in renewable energy, global economy still depends on fossil fuels to a great extent (Figure 1.1). Statistics reported by the British Petroleum (BP) company based on data from 1994 to 2019 show that the world primary energy consumption growth in 2019 slowed to 1.3%. This is less than half the growth rate i.e. 2.8% in 2018. Three-quarters of the energy consumption increase was driven by natural gas and renewable resources in 2019.
Based on analyses reported by the BP company, oil has contributed to the share of global primary energy more than others since 1994 (Figure 1.2) with 33.1% contribution in 2019. After oil, coal and natural gas are the second and third largest contributors. Although coal lost its share to account for nearly 27%, the contribution of natural gas increased to 24% in 2019. The share of renewable resources rose to record highs of 5% in 2019, and they overtook nuclear energy with about 4% contribution. Figure 1.2 shows the share of hydroelectricity has been nearly constant and about 6%.
Unconventional reservoirs, including oil and gas shales and tight sandstones, are distributed around the world (Figure 1.3) with an estimated endowment of several thousand trillion cubic feet (Kim et al. 2000). Since shale reservoirs have been successfully explored and produced in the United States (Figure 1.3), they recently became one of the major contributors to energy supplies.
There exist three general types of unconventional reservoirs, i.e. (i) organic-rich source rocks, (ii) tight oil reservoirs, and (iii) hybrid plays in which production occurs from source rocks and conventional reservoirs (Zoback and Kohli 2019). These types of unconventional reservoirs are different in geologic formations and, therefore, should be optimally exploited using different and appropriate approaches.
Despite numerous practical applications in oil/gas exploration and production as well as recent progress, we are still far from fully understanding all mechanisms of flow and transport in shales and tight sandstones across scales, particularly from pore to reservoir. In the following, we briefly address recent advances in unconventional reservoirs and discuss current challenges in oil and gas exploration and production.
Figure 1.1 World total energy consumption between 1994 and 2019.
Source: BP Statistical Review of World Energy (2020)/BP International Limited.
1.2 Advances
Since 2005, the beginning of shale gas revolution in the United States, unconventional oil and gas resources as well as their developments and productions have received a remarkable amount of attention around the world (Zoback and Kohli 2019). Despite various challenges that still exist, the petroleum engineering community made tremendous progress, particularly in the past decade. In what follows, we briefly address several notable achievements. For further details and comprehensive recent advances, see e.g. Barati and Alhubail (2020), Rezaee (2021), and Moghanloo (2022).
1.2.1 Wettability
Characterizing the contact angle of fluids (e.g. water, oil, and gas) and its spatial variability within unconventional reservoirs and under in situ conditions are essential not only to understand the trapping phenomenon and enhance oil and gas recovery but also to improve greenhouse gas (e.g. carbon dioxide and hydrogen) sequestration underground. In the literature, various methods, such as contact angle measurements (Iglauer et al. 2015; Roshan et al. 2016), spontaneous imbibition (Liu et al. 2019; Siddiqui et al. 2019), and nuclear magnetic resonance (Odusina et al. 2011; Su et al. 2018) were proposed to determine wettability in unconventional reservoir rocks. Recently, Arif et al. (2021) collected published data on shale contact angle measurements and developed a repository. They concluded that the oil-brine mixture in shales behaved in terms of wettability over a wide range from water-wet to strongly oil-wet. Although the CO2-brine mixture typically showed weakly water-wet to CO2-wet behavior, the CH4-brine mixture in shales was weakly water-wet. Arif et al. (2021) also investigated what causes high variabilities in shale wettability and found that the main factors were pressure, temperature, thermal maturity, total organic content, and mineralogy of shales.
Figure 1.2 Shares of global primary energy between 1994 and 2019 (BP Statistical Review of World Energy 2020/BP International Limited).
Although our knowledge of shale wettability has improved, further investigations are still needed to study the solid-fluid and fluid-fluid contact angles under realistic reservoir conditions more comprehensively. This would help enhance oil and gas recovery and exploit unconventional reservoirs even more successfully.
1.2.2 Permeability
Liquid and gas transports in shales and tight porous rocks were widely studied, particularly at the pore and core levels. The literature on gas permeability and its modeling is indeed vast and extensive (Javadpour et al. 2021; Liu 2017; Tahmasebi et al. 2020; Zhang et al. 2019). Numerous models were developed to address gas flow in nanostructures of shales by taking the effect of different transport mechanisms, such as slip flow, Knudsen diffusion, surface diffusion, and sorption into account. For example, Beskok and Karniadakis (1999) incorporated the effect of slip flow and modified the Poiseuille equation to describe gas flow in a cylindrical tube. Civan (2010) later applied the Beskok-Karniadakis model to scale up gas permeability in a network of pores. Using concepts from first-order slip flow and Knudsen diffusion, Javadpour (2009) developed another model for gas transport in a nano-scale cylindrical tube.
Figure 1.3 World shale gas resources (top) and shale gas and oil plays in the United States (bottom).
Source: Both maps are from US Energy Information Administration.
Recently, in addition to slip flow and Knudsen diffusion, more complex mechanisms such as sorption and surface diffusion were incorporated to model gas flow in shales. For instance, Wu et al. (2016) proposed a unified theoretical model by coupling various mechanisms including slip flow, Knudsen diffusion, sorption, and surface diffusion. Those authors stated that ". surface diffusion is an important transport mechanism, and its contribution cannot be negligible and even dominates in nanopores with less than 2 nm in diameter." Jia et al. (2018) found that surface diffusion might increase gas flow significantly at low pore pressures (e.g. < 2000 psi) depending on the value of surface diffusivity.
In the literature, most theoretic models developed to study gas permeability are based on the bundle of tortuous capillary tubes approach. However, it is an oversimplifying idealization in which a porous medium with actual interconnected pores is replaced with non-interconnected tortuous tubes of equal length (Purcell 1949). An important inconsistency between the bundle of tubes model and real porous media was noted by Fatt (1956). Fatt's main objection to the parallel-tubes model was that it has no connections between the individual tubes, in contrast to the network models that he developed. An additional problem is that, in a bundle-of-tubes model, individual pores span the entire sample or problem domain, regardless of its size.
Instead of the bundle of tortuous tubes approach, one may theoretically model gas transport in shales using more appropriate upscaling techniques e.g. percolation theory, critical path analysis (CPA), and effective-medium approximation (EMA) from statistical physics (Ghanbarian et al. 2020; Hunt et al. 2014; Sahimi 2011). Such approaches take into account the effect of pore connectivity. For example, Zhang and Scherer (2012) applied the CPA approach to estimate Klinkenberg-corrected permeability in shales from mercury intrusion capillary pressure curve and formation factor. They also used the Kozeny-Carman model, compared their estimations with experiments, and found that measured permeability values matched with CPA estimations more accurately. In another study, Ghanbarian and Javadpour (2017) assumed that slip flow and Knudsen diffusion are dominant...
