Physicochemical Fluid Dynamics in Porous Media

Mikhail Panfilov, D.Sc., is Professor at the Institute of Mathematics Elie Cartan - University of Lorraine/CNRS, and at the Institute Jean le Rond d'Alembert - Sorbonne University/CNRS. Born and studied in Moscow. He is twice graduated in applied mathematics/mechanics and petroleum engineering. He worked at the Oil & Gas Research Institute of the Academy of Sciences in Moscow. In 2000 he moved to France. He published more than 80 papers in international reviews and two monographs. He is a State Prize Laureate of Russia for Science (1997) and several Excellence Awards of the French Ministry of Higher Education and Research.

• Preface xvIntroduction xvii1 Thermodynamics of Pure Fluids 11.1 Equilibrium of Single-phase Fluids – Equation of State 21.1.1 Admissible Classes of EOS 21.1.2 van derWaals EOS 31.1.3 Soave-Redlish-Kwong EOS 31.1.4 Peng–Robinson EOS 51.1.5 Mixing Rules for Multicomponent Fluids 51.2 Two-phase Equilibrium of Pure Fluids 51.2.1 Pseudo-liquid/Pseudo-gas and True Liquid/Gas 61.2.2 Equilibrium Conditions in Terms of Chemical Potentials 61.2.3 Explicit Relationship for Chemical Potential 71.2.4 Equilibrium Conditions in Terms of Pressure and Volumes 81.2.5 Solvability of the Equilibrium Equation – Maxwell’s Rule 91.2.6 Calculation of Gas–Liquid Coexistence 101.2.7 Logarithmic Representation for Chemical Potential – Fugacity 112 Thermodynamics of Mixtures 132.1 Chemical Potential of an Ideal Gas Mixture 132.1.1 Notations 132.1.2 Definition and Properties of an Ideal Gas Mixture 142.1.3 Entropy and Enthalpy of Ideal Mixing 152.1.4 Chemical Potential of Ideal Gas Mixtures 162.2 Chemical Potential of Nonideal Mixtures 172.2.1 General Model for Chemical Potential of Mixtures 172.2.2 Chemical Potential of Mixtures through Intensive Parameters 192.3 Two-phase Equilibrium Equations for a Multicomponent Mixture 202.3.1 General Form of Two-phase Equilibrium Equations 202.3.2 Equilibrium Equations in the Case of Peng–Robinson EOS 212.3.3 K-values 232.3.4 Calculation of the Phase Composition (“flash”) 242.3.5 Expected Phase Diagrams for Binary Mixtures 242.4 Equilibrium in Dilute Mixtures 262.4.1 Ideal Solution 262.4.2 Chemical Potential for an Ideal Solution 272.4.3 Equilibrium of Ideal Gas and Ideal Solution: Raoult’s Law 272.4.4 Equilibrium of Dilute Solutions: Henry’s Law 282.4.5 K-values for Ideal Mixtures 282.4.6 Calculation of the Phase Composition 293 Chemistry of Mixtures 313.1 Adsorption 313.1.1 Mechanisms of Adsorption 313.1.2 Langmuir’s Model of Adsorption 323.1.3 Types of Adsorption Isotherms 343.1.4 Multicomponent Adsorption 353.2 Chemical Reactions: Mathematical Description 363.2.1 Elementary Stoichiometric System 363.2.2 Reaction Rate 373.2.3 Particle Balance through the Reaction Rate in a Homogeneous Reaction 373.2.4 Particle Balance in a Heterogeneous Reaction 383.2.5 Example 393.3 Chemical Reaction: Kinetics 393.3.1 Kinetic Law of Mass Action: Guldberg–Waage Law 393.3.2 Kinetics of Heterogeneous Reactions 403.3.3 Reaction Constant 413.4 Other Nonconservative Effects with Particles 423.4.1 Degradation of Particles 423.4.2 Trapping of Particles 423.5 Diffusion 423.5.1 Fick’s Law 433.5.2 Properties of the Diffusion Parameter 443.5.3 Calculation of the Diffusion Coefficient in Gases and Liquids 453.5.3.1 Diffusion in Gases 453.5.3.2 Diffusion in Liquids 463.5.4 Characteristic Values of the Diffusion Parameter 463.5.5 About a Misuse of Diffusion Parameters 473.5.5.1 A Misuse of Nondimensionless Concentrations 473.5.5.2 Diffusion as the Effect of Mole Fraction Anomaly but not the Number of Moles 473.5.6 Stefan–Maxwell Equations for Diffusion Fluxes 484 Reactive Transport with a Single Reaction 514.1 Equations of Multicomponent Single-Phase Transport 514.1.1 Material Balance of Each Component 514.1.2 Closure Relationships 524.1.2.1 Chemical Terms 524.1.2.2 Total Flow Velocity – Darcy’s Law 534.1.2.3 Diffusion Flux – Fick’s Law 534.1.3 Transport Equation 534.1.4 Transport Equation for Dilute Solutions 554.1.5 Example of Transport Equation for a Binary Mixture 554.1.6 Separation of Flow and Transport 564.2 Elementary Fundamental Solutions of 1D Transport Problems 564.2.1 Convective Transport – TravelingWaves 574.2.2 Transport with Diffusion 584.2.3 Length of the Diffusion Zone 594.2.4 Peclet Number 594.2.5 Transport with Linear Adsorption – Delay Effect 604.2.6 Transport with Nonlinear Adsorption: Diffusive TravelingWaves 604.2.7 Origin of Diffusive TravelingWaves 624.2.8 Transport with a Simplest Reaction (or Degradation/Trapping) 624.2.9 Macrokinetic Effect: Reactive Acceleration of the Transport 634.3 Reactive Transport in Underground Storage of CO2 644.3.1 Problem Formulation and Solution 654.3.2 Evolution of CO2 Concentration 664.3.3 Evolution of the Concentration of Solid Reactant 674.3.4 Evolution of the Concentration of the Reaction Product 674.3.5 Mass of Carbon Transformed to Solid 685 Reactive Transport with Multiple Reactions (Application to In Situ Leaching) 71ISL Technology 715.1 Coarse Monoreaction Model of ISL 735.1.1 Formulation of the Problem 735.1.2 Analytical Solution 745.2 MultireactionModel of ISL 755.2.1 Main Chemical Reactions in the Leaching Zone 755.2.2 Transport Equations 775.2.3 Kinetics of Gypsum Precipitation 785.2.4 Definite Form of the MathematicalModel 795.3 Method of Splitting Hydrodynamics and Chemistry 805.3.1 Principle of the Method 805.3.2 Model Problem of In Situ Leaching 815.3.3 Analytical Asymptotic Expansion: Zero-Order Terms 825.3.4 First-Order Terms 835.3.5 Solution in Definite Form 845.3.6 CaseWithout Gypsum Deposition 845.3.7 Analysis of the Process: Comparison with Numerical Data 855.3.8 Experimental Results: Comparison withTheory 865.3.9 Recovery Factor 886 Surface and Capillary Phenomena 916.1 Properties of an Interface 916.1.1 Curvature of a Surface 916.1.2 Signed Curvature 926.1.3 Surface Tension 946.1.4 Tangential Elasticity of an Interface 956.2 Capillary Pressure and Interface Curvature 966.2.1 Laplace’s Capillary Pressure 966.2.2 Young–Laplace Equation for Static Interface 976.2.3 Soap Films and Minimal Surfaces 996.2.4 Catenoid as a Minimal Surface of Revolution 1016.2.5 Plateau’s Configurations for Intercrossed Soap Films 1026.3 Wetting 1036.3.1 Fluid–Solid Interaction: Complete and PartialWetting 1036.3.2 Necessary Condition of Young for PartialWetting 1046.3.3 Hysteresis of the Contact Angle 1066.3.4 CompleteWetting – Impossibility of Meniscus Existence 1066.3.5 Shape of Liquid Drops on Solid Surface 1076.3.6 Surfactants – Significance ofWetting for Oil Recovery 1096.4 Capillary Phenomena in a Pore 1106.4.1 Capillary Pressure in a Pore 1106.4.2 Capillary Rise 1126.4.3 CapillaryMovement – Spontaneous Imbibition 1136.4.4 Menisci in Nonuniform Pores – Principle of Pore Occupancy 1146.4.5 Capillary Trapping – Principle of Phase Immobilization 1156.4.6 Effective Capillary Pressure 1166.5 Augmented Meniscus and Disjoining Pressure 1186.5.1 Multiscale Structure of Meniscus 1186.5.2 Disjoining Pressure in Liquid Films 1196.5.3 Augmented Young–Laplace Equation 1207 MeniscusMovement in a Single Pore 1237.1 Asymptotic Model for Meniscus near the Triple Line 1237.1.1 Paradox of the Triple Line 1237.1.2 Flow Model in the Intermediate Zone (Lubrication Approximation) 1247.1.3 Tanner’s Differential Equation for Meniscus 1257.1.4 Shape of the Meniscus in the Intermediate Zone 1277.1.5 Particular Case of Small 𝜃: Cox–Voinov Law 1287.1.6 Scenarios of Meniscus Spreading 1287.2 Movement of the Augmented Meniscus 1307.2.1 Lubrication Approximation for Augmented Meniscus 1307.2.2 Adiabatic Precursor Films 1327.2.3 Diffusive Film 1327.3 Method of Diffuse Interface 1337.3.1 Principle Idea of the Method 1337.3.2 Capillary Force 1347.3.3 Free Energy and Chemical Potential 1357.3.4 Reduction to Cahn–Hilliard Equation 1378 Stochastic Properties of Phase Cluster in Pore Networks 1398.1 Connectivity of Phase Clusters 1398.1.1 Connectivity as a Measure of Mobility 1398.1.2 Triple Structure of Phase Cluster 1408.1.3 Network Models of Porous Media 1408.1.4 Effective Coordination Number 1428.1.5 Coordination Number and Medium Porosity 1438.2 Markov Branching Model for Phase Cluster 1448.2.1 Phase Cluster as a Branching Process 1448.2.2 Definition of a Branching Process 1458.2.3 Method of Generating Functions 1478.2.4 Probability of Creating a Finite Phase Cluster 1488.2.5 Length of the Phase Cluster 1498.2.6 Probability of an Infinite Phase Cluster 1508.2.7 Length-Radius Ratio Υ: Fitting with Experimental Data 1518.2.8 Cluster of Mobile Phase 1538.2.9 Saturation of the Mobile Cluster 1548.3 Stochastic Markov Model for Relative Permeability 1558.3.1 Geometrical Model of a Porous Medium 1558.3.2 Probability of Realizations 1568.3.3 Definition of Effective Permeability 1568.3.4 Recurrent Relationship for Space-Averaged Permeability 1578.3.5 Method of Generating Functions 1588.3.6 Recurrent Relationship for the Generating Function 1598.3.7 Stinchcombe’s Integral Equation for Function F(x) 1608.3.8 Case of Binary Distribution of Permeabilities 1618.3.9 Large Coordination Number 1629 Macroscale Theory of Immiscible Two-Phase Flow 1659.1 General Equations of Two-Phase Immiscible Flow 1659.1.1 Mass and Momentum Conservation 1659.1.2 Fractional Flow and Total Velocity 1679.1.3 Reduction to the Model of KinematicWaves 1679.2 Canonical Theory of Two-Phase Displacement 1689.2.1 1D Model of KinematicWaves (the Buckley–Leverett Model) 1689.2.2 Principle of Maximum 1699.2.3 Nonexistence of Continuous Solutions 1709.2.4 Hugoniot–Rankine Conditions at a Shock 1719.2.5 Entropy Conditions at a Shock 1729.2.6 Entropy Condition for Particular Cases 1749.2.7 Solution Pathway 1759.2.8 Piston-Like Shocks 1769.3 Oil Recovery 1779.3.1 Recovery Factor and Average Saturation 1779.3.2 Breakthrough Recovery 1789.3.3 Another Method of Deriving the Relationship for the Recovery Factor 1799.3.4 Graphical Determination of Breakthrough Recovery 1799.3.5 Physical Structure of Solution. Structure of Nondisplaced Oil 1809.3.6 Efficiency of Displacement 1819.4 Displacement with Gravity 1829.4.1 1D-model of KinematicWaves with Gravity 1829.4.2 Additional Condition at Shocks: Continuity w.r.t. Initial Data 1839.4.3 Descending Flow 1859.4.4 Ascending Flow 1869.5 Stability of Displacement 1879.5.1 Saffman–Taylor and Raleigh–Taylor Instability and Fingering 1879.5.2 Stability Criterion 1889.6 Displacement by Immiscible Slugs 1899.6.1 Setting of the Problem 1909.6.2 Solution of the Problem 1919.6.3 Solution for the Back Part 1929.6.4 Matching Two Solutions 1929.6.5 Three Stages of the Evolution in Time 1929.7 Segregation and Immiscible Gas Rising 1969.7.1 Canonical 1D Model 1969.7.2 Description of Gas Rising 1979.7.3 First Stage of the Evolution: Division of the Forward Bubble Boundary 1989.7.4 Second Stage: Movement of the Back Boundary 1999.7.5 Third Stage: Monotonic Elongation of the Bubble 20010 NonlinearWaves in Miscible Two-phase Flow (Application to Enhanced Oil Recovery) 203Expected Scenarios of Miscible Gas–Liquid Displacement 20310.1 Equations of Two-Phase Miscible Flow 20510.1.1 General System of Equations 20510.1.2 Formulation through the Total Velocity and Fractional Flow 20610.1.3 Ideal Mixtures; Volume Fractions 20710.1.4 Conversion to the Model of KinematicWaves 20810.1.5 Particular Case of a Binary Mixture 20910.1.5.1 Conclusion 20910.2 Characterization of Species Dissolution by Phase Diagrams 20910.2.1 Thermodynamic Variance and Gibbs’ Phase Rule 209Example 21010.2.2 Ternary Phase Diagrams 21110.2.3 Tie Lines 21310.2.4 Tie-Line Parametrization of Phase Diagrams (Parameter 𝛼) 21410.2.5 Saturation of Gas 21610.2.6 Phase Diagrams for Constant K-Values 21610.2.7 Phase Diagrams for Linear Repartition Function: 𝛽 = −𝛾𝛼 21910.3 Canonical Model of Miscible EOR 22110.3.1 Problem Setting 22110.3.2 Fractional Flow of a Chemical Component 22210.4 Shocks 22410.4.1 Hugoniot–Rankine and Entropy Conditions at a Shock. Admissible Shocks 22510.4.2 Mechanical Shock (C-shock) and Its Graphical Image 22610.4.3 Chemical Shock (C𝛼-shock) and Its Graphical Image 22710.4.4 Shocks of Phase Transition 22810.4.5 Weakly Chemical Shock 23010.4.6 Three Methods of Changing the Phase Composition 23110.4.7 Solution Pathway 23110.5 Oil Displacement by Dry Gas 23210.5.1 Description of Fluids and Initial Data 23210.5.2 Algorithm of Selecting the Pathway 23310.5.3 Behavior of Liquid and Gas Composition 23510.5.4 Behavior of Liquid Saturation 23610.5.5 Physical Behavior of the Process 23710.5.6 EOR Efficiency 23910.6 Oil Displacement byWet Gas 23910.6.1 Formulation of the Problem and the Pathway 23910.6.2 Solution to the Problem. Physical Explanation 24010.6.3 Comparison with Immiscible Gas Injection 24210.6.4 Injection of Overcritical Gas 24310.6.5 Injection of Overcritical Gas in Undersaturated Single-Phase Oil 24510.7 Gas Recycling in Gas-Condensate Reservoirs 24610.7.1 Techniques of Enhanced Condensate Recovery 24610.7.2 Case I: Dry Gas Recycling: Mathematical Formulation 24710.7.3 Solution to the Problem of Dry Gas Recycling 24710.7.4 Case II: Injection of Enriched Gas 24910.7.4.1 Conclusion 25110.8 Chemical Flooding 25110.8.1 Conservation Equations 25110.8.2 Reduction to the Model of KinematicWaves 25210.8.3 Diagrams of Fractional Flow ofWater F(s, c) 25310.8.4 Shocks and Hugoniot–Rankine Conditions 25310.8.5 Solution of the Riemann Problem 25510.8.6 Impact of the Adsorption 25611 Counter Waves in Miscible Two-phase Flow with Gravity (Application to CO2 &H2 Storage) 257Introducing Notes 25711.1 Two-component Two-phase Flow in Gravity Field 25811.1.1 Formulation 25911.1.2 Solution before Reaching the Barrier 26111.1.3 ReverseWave Reflected from Barrier 26111.1.4 Calculation of the Concentrations at the Shocks 26311.1.5 Rate of Gas Rising and Bubble Growth under the Barriers 26411.1.6 Comparison with Immiscible Two-phase Flow 26411.2 Three-component Flow in Gravity Field 26511.2.1 Problem Setting 26511.2.2 Solution of the Riemann Problem 26611.2.3 Propagation of the Reverse Wave under the Barrier 26812 Flow with Variable Number of Phases: Method of Negative Saturations 27112.1 Method NegSat for Two-phase Fluids 27112.1.1 Interface of Phase Transition and Nonequilibrium States 27112.1.2 Essence of the Method Negsat 27312.1.3 Principle of Equivalence 27512.1.4 Proof of the Equivalence Principle 27612.1.5 Density and Viscosity of Fictitious Phases 27712.1.6 Extended Saturation – Detection of the Number of Phases 27712.1.7 Equivalence Principle for Flow with Gravity 27912.1.8 Equivalence Principle for Flow with Gravity and Diffusion 27912.1.9 Principle of Equivalence for Ideal Mixing 28112.1.10 Physical and Mathematical Consistency of the Equivalent Fluids 28212.2 Hyperbolic-parabolic Transition 28212.2.1 Phenomenon of Hyperbolic-parabolic Transition (HP Transition) 28212.2.2 Derivation of the Model (12.23) 28412.2.3 Purely Hyperbolic Case 28412.2.4 Case of Hyperbolic-parabolic Transition 28512.2.5 Generalization of Hugoniot–Rankine Conditions for a Shock of HP-transition 28712.2.6 Regularization by the Capillarity 28812.2.7 Reduction to VOF or Level-set Method for Immiscible Fluids 29013 Biochemical Fluid Dynamics of Porous Media 29113.1 Microbiological Chemistry 29113.1.1 Forms of Existence of Microorganisms 29113.1.2 Bacterial Metabolism 29213.1.3 Bacterial Movement 29313.1.4 Chemotaxis 29413.1.5 Population Dynamics 29513.1.6 Kinetics of Population Growth and Decay: Experiment 29513.1.6.1 Population Decay 29513.1.6.2 Population Growth 29613.1.7 Kinetics of Population Growth: MathematicalModels 29713.1.8 Coupling between Nutrient Consumption and Bacterial Growth 29813.1.9 Experimental Data on Bacterial Kinetics 30013.2 Bioreactive Waves in Microbiological Enhanced Oil Recovery 30013.2.1 The Essence of the Process 30013.2.2 Metabolic Process 30213.2.3 Assumptions 30313.2.4 Mass Balance Equations 30313.2.5 Description of the Impact of the Surfactant 30413.2.6 Reduction to the Model of KinematicWaves 30413.2.7 1D MEOR Problem 30513.2.8 Solution and Analysis of the MEOR Problem 30513.3 NonlinearWaves in Microbiological Underground Methanation Reactors 30813.3.1 Underground Methanation and Hydrogen Storage 30813.3.2 Biochemical Processes in an Underground Methanation Reactor 30913.3.3 Composition of the Injected Gas 31113.3.4 MathematicalModel of Underground Methanation 31113.3.5 KinematicWave Model 31313.3.6 Asymptotic Model for Biochemical Equilibrium 31413.3.7 Particular Case of Biochemical Equilibrium 31513.3.8 Solution of the Riemann Problem 31513.3.9 Comparison with the CaseWithout Bacteria. Impact of Bacteria 31713.4 Self-organization in Biochemical Dynamical Systems (Application to Underground Methanation) 31813.4.1 Integral Material Balance in the Underground Reactor 31813.4.2 Reduction to a Dynamical System 31913.4.3 Singular Point Analysis – Oscillatory Regimes 32013.4.4 Existence of a Limit Cycle – Auto-oscillations 32113.4.5 Phase Portrait of Auto-oscillations 32313.5 Self-organization in Reaction–Diffusion Systems 32513.5.1 Equations of Underground Methanation with Diffusion 32513.5.2 Turing’s Instability 32713.5.3 Limit Space OscillatoryWaves at 𝜀 = 0 32813.5.4 Three Types of Limit Patterns at Large Times 32913.5.5 Exact Analytical Solution of Problem (13.52). Estimation of Parameters 33013.5.6 Limit Two-scale Spatial Oscillatory Patterns at 𝜀 > 0 33113.5.7 Two-scale Asymptotic Expansion of Problem (13.59) 33313.5.7.1 Two-scale Formulation 33313.5.7.2 Two-scale Expansion 33413.5.7.3 Zero-order Terms c0 and n0 33413.5.7.4 First-order Term n1 33513.5.7.5 Second-order Term c2 33613.5.8 2D Two-scale Spatial Patterns 336A Chemical Potential of a Pure Component from the Homogeneity of Gibbs Energy 339B Chemical Potential for Cubic EOS 341C Chemical Potential of Mixtures from the Homogeneity of Gibbs Energy 343D Calculation of the Integral in (2.25a) 347E Hugoniot–Rankine Conditions 349F Numerical Code (Matlab) for Calculating Phase Diagrams of a Pure Fluid 351Bibliography 355Index 363