Porous media play a key role in chemical processes, gas and water purification, gas storage and the development of new multifunctional materials. Understanding hydrodynamics in porous media is decisive for enabling a wide range of applications in materials science and chemical engineering. This all-encompassing book offers a timely overview of all flow and transport processes in which chemical or physicochemical phenomena such as dissolution, phase transition, reactions, adsorption, diffusion, capillarity, and surface phenomena are essential. It brings together both theoretical and experimental results and includes important industrial applications.
Physicochemical Fluid Dynamics in Porous Media: Applications in Geoscience and Petroleum Engineering explains the thermodynamics of phase equilibria for multicomponent fluids, physicochemical models of single-phase and immiscible two-phase flow, based on the macroscopic theory of oil displacement by water. It also covers the theory of two-phase flow with partial miscibility and describes partially miscible flows with phase transitions by means of the negative saturation approach. The final chapters are devoted to flow with chemical reactions, based on the example of in-situ leaching of uranium, and flow with bio-chemical reactions in terms of the underground storage of hydrogen.
-Brings together the theoretical and experimental results necessary for the understanding of hydrodynamics in porous media
-Covers important industrial applications such as underground leaching of uranium and underground storage of hydrogen
-Presents a state-of-the-art overview and summarizes the research results usually found only scattered in the literature
Physicochemical Fluid Dynamics in Porous Media: Applications in Geoscience and Petroleum Engineering will appeal to chemical engineers, materials scientists, applied physicists, and mechanical engineers.
Table of Contents
Preface xv
Introduction xvii
1 Thermodynamics of Pure Fluids 1
1.1 Equilibrium of Single-phase Fluids - Equation of State 2
1.1.1 Admissible Classes of EOS 2
1.1.2 van derWaals EOS 3
1.1.3 Soave-Redlish-Kwong EOS 3
1.1.4 Peng-Robinson EOS 5
1.1.5 Mixing Rules for Multicomponent Fluids 5
1.2 Two-phase Equilibrium of Pure Fluids 5
1.2.1 Pseudo-liquid/Pseudo-gas and True Liquid/Gas 6
1.2.2 Equilibrium Conditions in Terms of Chemical Potentials 6
1.2.3 Explicit Relationship for Chemical Potential 7
1.2.4 Equilibrium Conditions in Terms of Pressure and Volumes 8
1.2.5 Solvability of the Equilibrium Equation - Maxwell’s Rule 9
1.2.6 Calculation of Gas-Liquid Coexistence 10
1.2.7 Logarithmic Representation for Chemical Potential - Fugacity 11
2 Thermodynamics of Mixtures 13
2.1 Chemical Potential of an Ideal Gas Mixture 13
2.1.1 Notations 13
2.1.2 Definition and Properties of an Ideal Gas Mixture 14
2.1.3 Entropy and Enthalpy of Ideal Mixing 15
2.1.4 Chemical Potential of Ideal Gas Mixtures 16
2.2 Chemical Potential of Nonideal Mixtures 17
2.2.1 General Model for Chemical Potential of Mixtures 17
2.2.2 Chemical Potential of Mixtures through Intensive Parameters 19
2.3 Two-phase Equilibrium Equations for a Multicomponent Mixture 20
2.3.1 General Form of Two-phase Equilibrium Equations 20
2.3.2 Equilibrium Equations in the Case of Peng-Robinson EOS 21
2.3.3 K-values 23
2.3.4 Calculation of the Phase Composition (“flash”) 24
2.3.5 Expected Phase Diagrams for Binary Mixtures 24
2.4 Equilibrium in Dilute Mixtures 26
2.4.1 Ideal Solution 26
2.4.2 Chemical Potential for an Ideal Solution 27
2.4.3 Equilibrium of Ideal Gas and Ideal Solution: Raoult’s Law 27
2.4.4 Equilibrium of Dilute Solutions: Henry’s Law 28
2.4.5 K-values for Ideal Mixtures 28
2.4.6 Calculation of the Phase Composition 29
3 Chemistry of Mixtures 31
3.1 Adsorption 31
3.1.1 Mechanisms of Adsorption 31
3.1.2 Langmuir’s Model of Adsorption 32
3.1.3 Types of Adsorption Isotherms 34
3.1.4 Multicomponent Adsorption 35
3.2 Chemical Reactions: Mathematical Description 36
3.2.1 Elementary Stoichiometric System 36
3.2.2 Reaction Rate 37
3.2.3 Particle Balance through the Reaction Rate in a Homogeneous Reaction 37
3.2.4 Particle Balance in a Heterogeneous Reaction 38
3.2.5 Example 39
3.3 Chemical Reaction: Kinetics 39
3.3.1 Kinetic Law of Mass Action: Guldberg-Waage Law 39
3.3.2 Kinetics of Heterogeneous Reactions 40
3.3.3 Reaction Constant 41
3.4 Other Nonconservative Effects with Particles 42
3.4.1 Degradation of Particles 42
3.4.2 Trapping of Particles 42
3.5 Diffusion 42
3.5.1 Fick’s Law 43
3.5.2 Properties of the Diffusion Parameter 44
3.5.3 Calculation of the Diffusion Coefficient in Gases and Liquids 45
3.5.3.1 Diffusion in Gases 45
3.5.3.2 Diffusion in Liquids 46
3.5.4 Characteristic Values of the Diffusion Parameter 46
3.5.5 About a Misuse of Diffusion Parameters 47
3.5.5.1 A Misuse of Nondimensionless Concentrations 47
3.5.5.2 Diffusion as the Effect of Mole Fraction Anomaly but not the Number of Moles 47
3.5.6 Stefan-Maxwell Equations for Diffusion Fluxes 48
4 Reactive Transport with a Single Reaction 51
4.1 Equations of Multicomponent Single-Phase Transport 51
4.1.1 Material Balance of Each Component 51
4.1.2 Closure Relationships 52
4.1.2.1 Chemical Terms 52
4.1.2.2 Total Flow Velocity - Darcy’s Law 53
4.1.2.3 Diffusion Flux - Fick’s Law 53
4.1.3 Transport Equation 53
4.1.4 Transport Equation for Dilute Solutions 55
4.1.5 Example of Transport Equation for a Binary Mixture 55
4.1.6 Separation of Flow and Transport 56
4.2 Elementary Fundamental Solutions of 1D Transport Problems 56
4.2.1 Convective Transport - TravelingWaves 57
4.2.2 Transport with Diffusion 58
4.2.3 Length of the Diffusion Zone 59
4.2.4 Peclet Number 59
4.2.5 Transport with Linear Adsorption - Delay Effect 60
4.2.6 Transport with Nonlinear Adsorption: Diffusive TravelingWaves 60
4.2.7 Origin of Diffusive TravelingWaves 62
4.2.8 Transport with a Simplest Reaction (or Degradation/Trapping) 62
4.2.9 Macrokinetic Effect: Reactive Acceleration of the Transport 63
4.3 Reactive Transport in Underground Storage of CO2 64
4.3.1 Problem Formulation and Solution 65
4.3.2 Evolution of CO2 Concentration 66
4.3.3 Evolution of the Concentration of Solid Reactant 67
4.3.4 Evolution of the Concentration of the Reaction Product 67
4.3.5 Mass of Carbon Transformed to Solid 68
5 Reactive Transport with Multiple Reactions (Application to In Situ Leaching) 71
ISL Technology 71
5.1 Coarse Monoreaction Model of ISL 73
5.1.1 Formulation of the Problem 73
5.1.2 Analytical Solution 74
5.2 MultireactionModel of ISL 75
5.2.1 Main Chemical Reactions in the Leaching Zone 75
5.2.2 Transport Equations 77
5.2.3 Kinetics of Gypsum Precipitation 78
5.2.4 Definite Form of the MathematicalModel 79
5.3 Method of Splitting Hydrodynamics and Chemistry 80
5.3.1 Principle of the Method 80
5.3.2 Model Problem of In Situ Leaching 81
5.3.3 Analytical Asymptotic Expansion: Zero-Order Terms 82
5.3.4 First-Order Terms 83
5.3.5 Solution in Definite Form 84
5.3.6 CaseWithout Gypsum Deposition 84
5.3.7 Analysis of the Process: Comparison with Numerical Data 85
5.3.8 Experimental Results: Comparison withTheory 86
5.3.9 Recovery Factor 88
6 Surface and Capillary Phenomena 91
6.1 Properties of an Interface 91
6.1.1 Curvature of a Surface 91
6.1.2 Signed Curvature 92
6.1.3 Surface Tension 94
6.1.4 Tangential Elasticity of an Interface 95
6.2 Capillary Pressure and Interface Curvature 96
6.2.1 Laplace’s Capillary Pressure 96
6.2.2 Young-Laplace Equation for Static Interface 97
6.2.3 Soap Films and Minimal Surfaces 99
6.2.4 Catenoid as a Minimal Surface of Revolution 101
6.2.5 Plateau’s Configurations for Intercrossed Soap Films 102
6.3 Wetting 103
6.3.1 Fluid-Solid Interaction: Complete and PartialWetting 103
6.3.2 Necessary Condition of Young for PartialWetting 104
6.3.3 Hysteresis of the Contact Angle 106
6.3.4 CompleteWetting - Impossibility of Meniscus Existence 106
6.3.5 Shape of Liquid Drops on Solid Surface 107
6.3.6 Surfactants - Significance ofWetting for Oil Recovery 109
6.4 Capillary Phenomena in a Pore 110
6.4.1 Capillary Pressure in a Pore 110
6.4.2 Capillary Rise 112
6.4.3 CapillaryMovement - Spontaneous Imbibition 113
6.4.4 Menisci in Nonuniform Pores - Principle of Pore Occupancy 114
6.4.5 Capillary Trapping - Principle of Phase Immobilization 115
6.4.6 Effective Capillary Pressure 116
6.5 Augmented Meniscus and Disjoining Pressure 118
6.5.1 Multiscale Structure of Meniscus 118
6.5.2 Disjoining Pressure in Liquid Films 119
6.5.3 Augmented Young-Laplace Equation 120
7 MeniscusMovement in a Single Pore 123
7.1 Asymptotic Model for Meniscus near the Triple Line 123
7.1.1 Paradox of the Triple Line 123
7.1.2 Flow Model in the Intermediate Zone (Lubrication Approximation) 124
7.1.3 Tanner’s Differential Equation for Meniscus 125
7.1.4 Shape of the Meniscus in the Intermediate Zone 127
7.1.5 Particular Case of Small 𝜃: Cox-Voinov Law 128
7.1.6 Scenarios of Meniscus Spreading 128
7.2 Movement of the Augmented Meniscus 130
7.2.1 Lubrication Approximation for Augmented Meniscus 130
7.2.2 Adiabatic Precursor Films 132
7.2.3 Diffusive Film 132
7.3 Method of Diffuse Interface 133
7.3.1 Principle Idea of the Method 133
7.3.2 Capillary Force 134
7.3.3 Free Energy and Chemical Potential 135
7.3.4 Reduction to Cahn-Hilliard Equation 137
8 Stochastic Properties of Phase Cluster in Pore Networks 139
8.1 Connectivity of Phase Clusters 139
8.1.1 Connectivity as a Measure of Mobility 139
8.1.2 Triple Structure of Phase Cluster 140
8.1.3 Network Models of Porous Media 140
8.1.4 Effective Coordination Number 142
8.1.5 Coordination Number and Medium Porosity 143
8.2 Markov Branching Model for Phase Cluster 144
8.2.1 Phase Cluster as a Branching Process 144
8.2.2 Definition of a Branching Process 145
8.2.3 Method of Generating Functions 147
8.2.4 Probability of Creating a Finite Phase Cluster 148
8.2.5 Length of the Phase Cluster 149
8.2.6 Probability of an Infinite Phase Cluster 150
8.2.7 Length-Radius Ratio Υ: Fitting with Experimental Data 151
8.2.8 Cluster of Mobile Phase 153
8.2.9 Saturation of the Mobile Cluster 154
8.3 Stochastic Markov Model for Relative Permeability 155
8.3.1 Geometrical Model of a Porous Medium 155
8.3.2 Probability of Realizations 156
8.3.3 Definition of Effective Permeability 156
8.3.4 Recurrent Relationship for Space-Averaged Permeability 157
8.3.5 Method of Generating Functions 158
8.3.6 Recurrent Relationship for the Generating Function 159
8.3.7 Stinchcombe’s Integral Equation for Function F(x) 160
8.3.8 Case of Binary Distribution of Permeabilities 161
8.3.9 Large Coordination Number 162
9 Macroscale Theory of Immiscible Two-Phase Flow 165
9.1 General Equations of Two-Phase Immiscible Flow 165
9.1.1 Mass and Momentum Conservation 165
9.1.2 Fractional Flow and Total Velocity 167
9.1.3 Reduction to the Model of KinematicWaves 167
9.2 Canonical Theory of Two-Phase Displacement 168
9.2.1 1D Model of KinematicWaves (the Buckley-Leverett Model) 168
9.2.2 Principle of Maximum 169
9.2.3 Nonexistence of Continuous Solutions 170
9.2.4 Hugoniot-Rankine Conditions at a Shock 171
9.2.5 Entropy Conditions at a Shock 172
9.2.6 Entropy Condition for Particular Cases 174
9.2.7 Solution Pathway 175
9.2.8 Piston-Like Shocks 176
9.3 Oil Recovery 177
9.3.1 Recovery Factor and Average Saturation 177
9.3.2 Breakthrough Recovery 178
9.3.3 Another Method of Deriving the Relationship for the Recovery Factor 179
9.3.4 Graphical Determination of Breakthrough Recovery 179
9.3.5 Physical Structure of Solution. Structure of Nondisplaced Oil 180
9.3.6 Efficiency of Displacement 181
9.4 Displacement with Gravity 182
9.4.1 1D-model of KinematicWaves with Gravity 182
9.4.2 Additional Condition at Shocks: Continuity w.r.t. Initial Data 183
9.4.3 Descending Flow 185
9.4.4 Ascending Flow 186
9.5 Stability of Displacement 187
9.5.1 Saffman-Taylor and Raleigh-Taylor Instability and Fingering 187
9.5.2 Stability Criterion 188
9.6 Displacement by Immiscible Slugs 189
9.6.1 Setting of the Problem 190
9.6.2 Solution of the Problem 191
9.6.3 Solution for the Back Part 192
9.6.4 Matching Two Solutions 192
9.6.5 Three Stages of the Evolution in Time 192
9.7 Segregation and Immiscible Gas Rising 196
9.7.1 Canonical 1D Model 196
9.7.2 Description of Gas Rising 197
9.7.3 First Stage of the Evolution: Division of the Forward Bubble Boundary 198
9.7.4 Second Stage: Movement of the Back Boundary 199
9.7.5 Third Stage: Monotonic Elongation of the Bubble 200
10 NonlinearWaves in Miscible Two-phase Flow (Application to Enhanced Oil Recovery) 203
Expected Scenarios of Miscible Gas-Liquid Displacement 203
10.1 Equations of Two-Phase Miscible Flow 205
10.1.1 General System of Equations 205
10.1.2 Formulation through the Total Velocity and Fractional Flow 206
10.1.3 Ideal Mixtures; Volume Fractions 207
10.1.4 Conversion to the Model of KinematicWaves 208
10.1.5 Particular Case of a Binary Mixture 209
10.1.5.1 Conclusion 209
10.2 Characterization of Species Dissolution by Phase Diagrams 209
10.2.1 Thermodynamic Variance and Gibbs’ Phase Rule 209
Example 210
10.2.2 Ternary Phase Diagrams 211
10.2.3 Tie Lines 213
10.2.4 Tie-Line Parametrization of Phase Diagrams (Parameter 𝛼) 214
10.2.5 Saturation of Gas 216
10.2.6 Phase Diagrams for Constant K-Values 216
10.2.7 Phase Diagrams for Linear Repartition Function: 𝛽 = −𝛾𝛼 219
10.3 Canonical Model of Miscible EOR 221
10.3.1 Problem Setting 221
10.3.2 Fractional Flow of a Chemical Component 222
10.4 Shocks 224
10.4.1 Hugoniot-Rankine and Entropy Conditions at a Shock. Admissible Shocks 225
10.4.2 Mechanical Shock (C-shock) and Its Graphical Image 226
10.4.3 Chemical Shock (C𝛼-shock) and Its Graphical Image 227
10.4.4 Shocks of Phase Transition 228
10.4.5 Weakly Chemical Shock 230
10.4.6 Three Methods of Changing the Phase Composition 231
10.4.7 Solution Pathway 231
10.5 Oil Displacement by Dry Gas 232
10.5.1 Description of Fluids and Initial Data 232
10.5.2 Algorithm of Selecting the Pathway 233
10.5.3 Behavior of Liquid and Gas Composition 235
10.5.4 Behavior of Liquid Saturation 236
10.5.5 Physical Behavior of the Process 237
10.5.6 EOR Efficiency 239
10.6 Oil Displacement byWet Gas 239
10.6.1 Formulation of the Problem and the Pathway 239
10.6.2 Solution to the Problem. Physical Explanation 240
10.6.3 Comparison with Immiscible Gas Injection 242
10.6.4 Injection of Overcritical Gas 243
10.6.5 Injection of Overcritical Gas in Undersaturated Single-Phase Oil 245
10.7 Gas Recycling in Gas-Condensate Reservoirs 246
10.7.1 Techniques of Enhanced Condensate Recovery 246
10.7.2 Case I: Dry Gas Recycling: Mathematical Formulation 247
10.7.3 Solution to the Problem of Dry Gas Recycling 247
10.7.4 Case II: Injection of Enriched Gas 249
10.7.4.1 Conclusion 251
10.8 Chemical Flooding 251
10.8.1 Conservation Equations 251
10.8.2 Reduction to the Model of KinematicWaves 252
10.8.3 Diagrams of Fractional Flow ofWater F(s, c) 253
10.8.4 Shocks and Hugoniot-Rankine Conditions 253
10.8.5 Solution of the Riemann Problem 255
10.8.6 Impact of the Adsorption 256
11 Counter Waves in Miscible Two-phase Flow with Gravity (Application to CO2 &H2 Storage) 257
Introducing Notes 257
11.1 Two-component Two-phase Flow in Gravity Field 258
11.1.1 Formulation 259
11.1.2 Solution before Reaching the Barrier 261
11.1.3 ReverseWave Reflected from Barrier 261
11.1.4 Calculation of the Concentrations at the Shocks 263
11.1.5 Rate of Gas Rising and Bubble Growth under the Barriers 264
11.1.6 Comparison with Immiscible Two-phase Flow 264
11.2 Three-component Flow in Gravity Field 265
11.2.1 Problem Setting 265
11.2.2 Solution of the Riemann Problem 266
11.2.3 Propagation of the Reverse Wave under the Barrier 268
12 Flow with Variable Number of Phases: Method of Negative Saturations 271
12.1 Method NegSat for Two-phase Fluids 271
12.1.1 Interface of Phase Transition and Nonequilibrium States 271
12.1.2 Essence of the Method Negsat 273
12.1.3 Principle of Equivalence 275
12.1.4 Proof of the Equivalence Principle 276
12.1.5 Density and Viscosity of Fictitious Phases 277
12.1.6 Extended Saturation - Detection of the Number of Phases 277
12.1.7 Equivalence Principle for Flow with Gravity 279
12.1.8 Equivalence Principle for Flow with Gravity and Diffusion 279
12.1.9 Principle of Equivalence for Ideal Mixing 281
12.1.10 Physical and Mathematical Consistency of the Equivalent Fluids 282
12.2 Hyperbolic-parabolic Transition 282
12.2.1 Phenomenon of Hyperbolic-parabolic Transition (HP Transition) 282
12.2.2 Derivation of the Model (12.23) 284
12.2.3 Purely Hyperbolic Case 284
12.2.4 Case of Hyperbolic-parabolic Transition 285
12.2.5 Generalization of Hugoniot-Rankine Conditions for a Shock of HP-transition 287
12.2.6 Regularization by the Capillarity 288
12.2.7 Reduction to VOF or Level-set Method for Immiscible Fluids 290
13 Biochemical Fluid Dynamics of Porous Media 291
13.1 Microbiological Chemistry 291
13.1.1 Forms of Existence of Microorganisms 291
13.1.2 Bacterial Metabolism 292
13.1.3 Bacterial Movement 293
13.1.4 Chemotaxis 294
13.1.5 Population Dynamics 295
13.1.6 Kinetics of Population Growth and Decay: Experiment 295
13.1.6.1 Population Decay 295
13.1.6.2 Population Growth 296
13.1.7 Kinetics of Population Growth: MathematicalModels 297
13.1.8 Coupling between Nutrient Consumption and Bacterial Growth 298
13.1.9 Experimental Data on Bacterial Kinetics 300
13.2 Bioreactive Waves in Microbiological Enhanced Oil Recovery 300
13.2.1 The Essence of the Process 300
13.2.2 Metabolic Process 302
13.2.3 Assumptions 303
13.2.4 Mass Balance Equations 303
13.2.5 Description of the Impact of the Surfactant 304
13.2.6 Reduction to the Model of KinematicWaves 304
13.2.7 1D MEOR Problem 305
13.2.8 Solution and Analysis of the MEOR Problem 305
13.3 NonlinearWaves in Microbiological Underground Methanation Reactors 308
13.3.1 Underground Methanation and Hydrogen Storage 308
13.3.2 Biochemical Processes in an Underground Methanation Reactor 309
13.3.3 Composition of the Injected Gas 311
13.3.4 MathematicalModel of Underground Methanation 311
13.3.5 KinematicWave Model 313
13.3.6 Asymptotic Model for Biochemical Equilibrium 314
13.3.7 Particular Case of Biochemical Equilibrium 315
13.3.8 Solution of the Riemann Problem 315
13.3.9 Comparison with the CaseWithout Bacteria. Impact of Bacteria 317
13.4 Self-organization in Biochemical Dynamical Systems (Application to Underground Methanation) 318
13.4.1 Integral Material Balance in the Underground Reactor 318
13.4.2 Reduction to a Dynamical System 319
13.4.3 Singular Point Analysis - Oscillatory Regimes 320
13.4.4 Existence of a Limit Cycle - Auto-oscillations 321
13.4.5 Phase Portrait of Auto-oscillations 323
13.5 Self-organization in Reaction-Diffusion Systems 325
13.5.1 Equations of Underground Methanation with Diffusion 325
13.5.2 Turing’s Instability 327
13.5.3 Limit Space OscillatoryWaves at 𝜀 = 0 328
13.5.4 Three Types of Limit Patterns at Large Times 329
13.5.5 Exact Analytical Solution of Problem (13.52). Estimation of Parameters 330
13.5.6 Limit Two-scale Spatial Oscillatory Patterns at 𝜀 > 0 331
13.5.7 Two-scale Asymptotic Expansion of Problem (13.59) 333
13.5.7.1 Two-scale Formulation 333
13.5.7.2 Two-scale Expansion 334
13.5.7.3 Zero-order Terms c0 and n0 334
13.5.7.4 First-order Term n1 335
13.5.7.5 Second-order Term c2 336
13.5.8 2D Two-scale Spatial Patterns 336
A Chemical Potential of a Pure Component from the Homogeneity of Gibbs Energy 339
B Chemical Potential for Cubic EOS 341
C Chemical Potential of Mixtures from the Homogeneity of Gibbs Energy 343
D Calculation of the Integral in (2.25a) 347
E Hugoniot-Rankine Conditions 349
F Numerical Code (Matlab) for Calculating Phase Diagrams of a Pure Fluid 351
Bibliography 355
Index 363
