| List of Symbols |
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xi | |
| Introduction |
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xv | |
| Chapter 1 Fluids, Porous Media and REV: Basic Concepts |
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1 | (20) |
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1.1 Geologic porous media: basic concepts |
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1 | (5) |
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1 | (1) |
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2 | (1) |
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1.1.3 Geologic porous media: examples |
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3 | (3) |
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1.2 Porous media: basic concepts, porosity and specific area |
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6 | (3) |
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7 | (2) |
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1.3 Single-phase flow and Darcy's law: basic concepts |
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9 | (1) |
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1.3.1 Darcy's flux-gradient law |
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9 | (1) |
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1.4 The Darcy-Buckingham law and the Richards equation: basic concepts of unsaturated flow |
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10 | (1) |
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1.4.1 Remarks on unsaturated water flow |
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11 | (1) |
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1.5 Capillarity and two-phase flow systems at different scales: basic concepts |
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11 | (3) |
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11 | (1) |
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1.5.2 Capillarity pressure jump at different scales |
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12 | (2) |
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1.5.3 Moving from one scale to another: upscaling |
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14 | (1) |
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1.6 A basic approach to pore scale two-phase flow |
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14 | (1) |
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1.7 A basic approach for continuum scale description of two-phase flow in porous media: the Darcy-Muskat model |
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15 | (1) |
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1.7.1 The Buckey-Leverett model |
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16 | (1) |
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1.8 Other issues: capillarity vs. gravity and viscosity, heterogeneity and upscaling |
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16 | (5) |
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1.8.1 Capillarity plus gravity and viscous dissipation |
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17 | (1) |
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1.8.2 Scales and the representative elementary volume |
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17 | (1) |
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1.8.3 Objectives at various scales of analysis |
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17 | (1) |
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1.8.4 Upscaling: first and second upscaling problems |
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18 | (3) |
| Chapter 2 Two-Phase Physics: Surface Tension, Interfaces, Capillary Liquid/Vapor Equilibria |
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21 | (56) |
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2.1 Summary and objectives |
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21 | (1) |
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2.2 Physics of capillarity and surface tension at equilibrium |
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22 | (21) |
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2.2.1 Observations and practical applications of surface tension, capillary forces and contact angles |
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23 | (8) |
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2.2.2 Interfacial tension: from molecular scale to fluid scale |
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31 | (3) |
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2.2.3 Laplace-Young pressure jump law (capillary pressure) |
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34 | (4) |
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2.2.4 Solid/liquid contact angle theta at equilibrium (Young) |
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38 | (2) |
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2.2.5 Measurements of interfacial tension |
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40 | (2) |
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2.2.6 Immiscibility versus miscibility at fluid interfaces (examples) |
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42 | (1) |
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2.3 Dimensionless groups (characteristic forces, length scales, timescales) |
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43 | (8) |
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2.3.1 Introduction: three forces driving multiphase systems |
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43 | (1) |
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2.3.2 Reynolds and Reynolds-Darcy number, viscous dissipation |
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44 | (2) |
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2.3.3 Capillary forces, surface tension and capillary number Ca |
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46 | (3) |
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2.3.4 Gravitational buoyancy forces and the Bond number Bo |
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49 | (1) |
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2.3.5 Dimensionless contrast ratios (viscosity and density contrasts) |
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49 | (1) |
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2.3.6 Recap of dimensionless groups for a two-phase system |
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50 | (1) |
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2.4 Thermodynamics, Gibbs energy, pressure, suction |
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51 | (16) |
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2.4.1 Interpretation of large suctions, bonding forces and Gibbs energy |
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51 | (7) |
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2.4.2 Thermodynamical systems (isolated or not) |
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58 | (3) |
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2.4.3 Gibbs free energy, heat, work |
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61 | (6) |
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2.5 Kelvin's liquid/vapor relation (suction vs. air humidity) |
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67 | (10) |
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2.5.1 Introduction to Kelvin's law (applications in flow modeling) |
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67 | (1) |
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2.5.2 Qualitative discussion of Kelvin's law (liquid/vapor relations) |
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68 | (1) |
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2.5.3 Thermodynamical variables (pressure, air humidity, etc) |
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69 | (1) |
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2.5.4 Perfect gases (dry air and water vapor) |
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70 | (1) |
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2.5.5 Kelvin's law: relative air humidity vs. capillary pressure |
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71 | (4) |
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2.5.6 Extended discussion on liquid/vapor thermodynamics (review) |
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75 | (2) |
| Chapter 3 Capillary Equilibria in Pores, Tubes and Joints |
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77 | (74) |
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3.1 Introduction and summary |
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77 | (1) |
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3.2 Capillary equilibrium in a single tube or planar joint of constant diameter or aperture |
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78 | (13) |
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3.2.1 Introduction: problem formulation and notations |
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78 | (1) |
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3.2.2 Capillary tube: pressure jump (Laplace-Young) |
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79 | (4) |
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3.2.3 Capillary tube: water height (Jurin) |
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83 | (1) |
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3.2.4 Capillary tube: extensions and examples (other fluids, etc.) |
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84 | (2) |
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3.2.5 Example of water/air equilibrium in a capillary tube: calculation of water height for a tube of diameter 100 μm |
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86 | (1) |
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3.2.6 Planar joint: introduction - planar geometry of the meniscus |
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87 | (2) |
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3.2.7 Planar joint: pressure jump across the water/air meniscus |
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89 | (1) |
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3.2.8 Planar joint: equilibrium height of meniscus (capillary rise) |
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90 | (1) |
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3.2.9 Example: parameter values for water and "light oil" in a joint |
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90 | (1) |
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3.3 Capillary equilibria in variable tubes and joints (a(x)) |
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91 | (7) |
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3.3.1 Introduction, description of the problem, and hypotheses |
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91 | (2) |
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3.3.2 Non-existence of two-phase equilibria, depending on initial state |
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93 | (1) |
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3.3.3 Geometric correction for variable tubes/joints: wetting angle thetaφ(x) in a fixed frame |
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94 | (4) |
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3.4 Capillary equilibrium in a random set of tubes: calculation of water retention curve θψ |
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98 | (22) |
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3.4.1 Introduction and summary |
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98 | (1) |
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3.4.2 Capillary water/air equilibrium in a random set of "pores"; moisture retention curve theta(pc) for uniformly distributed radii |
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99 | (10) |
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3.4.3 Capillary water/air equilibrium and moisture retention curve θ(pc) for Pareto distributed radii with exponent ω = 2 |
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109 | (6) |
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3.4.4 Limitations of the Boolean model of random tubes |
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115 | (4) |
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3.4.5 Soil water retention curves in hydro-agriculture (overview) |
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119 | (1) |
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3.5 Capillary equilibrium of soap films: minimal area surfaces and Euler-Lagrange equations |
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120 | (10) |
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3.5.1 Introduction and summary |
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120 | (1) |
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3.5.2 Soap film surface (preliminary formulation) |
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121 | (1) |
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3.5.3 Euler-Lagrange equations for minimizing integrals |
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122 | (3) |
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3.5.4 Euler-Lagrange equation minimizing the area |
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125 | (5) |
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3.6 Case study of soap film equilibrium between two circular rings: minimal area surface (catenoid) |
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130 | (18) |
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3.6.1 Presentation of the case study: soap film between two rings |
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130 | (1) |
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3.6.2 Formulation: minimal area surface between two coaxial circles |
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131 | (2) |
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3.6.3 Expressing Euler-Lagrange for the generating curve Y(x) |
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133 | (2) |
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3.6.4 Solution of Euler-Lagrange equations: catenoid surface between two coaxial circles of different diameters |
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135 | (4) |
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3.6.5 A special solution of the Euler-Lagrange equations: the catenoid surface between two identical coaxial rings |
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139 | (3) |
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3.6.6 Parametric study and conclusions (existence/unicity of the soap film depending on ring geometry) |
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142 | (6) |
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3.7 Additional topic: the equilibrium depth of a bubble |
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148 | (3) |
| Chapter 4 Pore-Scale Capillary Flows (Tubes, Joints) |
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151 | (90) |
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4.1 Introduction and summary: pore-scale flow in capillary tubes and planar joints (steady and transient) |
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151 | (2) |
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4.1.1 Introduction and summary |
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151 | (1) |
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4.1.2 Case of steady flow systems (single phase and two phase) |
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152 | (1) |
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4.1.3 Remark on the quasi-static nature of the water retention curve |
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152 | (1) |
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4.1.4 Case of transient flow problems |
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152 | (1) |
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4.1.5 Numerical experiment (2D visco-capillary invasion) |
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153 | (1) |
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4.2 Single-phase steady flow in tubes: Poiseuille, Darcy, Kozeny-Carman permeability |
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153 | (34) |
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4.2.1 Overview: Stokes, Poiseuille, Specific Area, Darcy, Kozeny permeability |
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153 | (1) |
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4.2.2 Specific area concept |
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154 | (2) |
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4.2.3 Poiseuille flow in a cylindrical tube or a planar joint |
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156 | (8) |
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4.2.4 Kozeny-Carman permeability for single-phase flow (from Poiseuille to Darcy) |
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164 | (23) |
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4.3 Unsaturated and two-phase steady flow in sets of planar joints: equivalent mesoscale quantities (porosity θ permeability k, capillary length λcap) |
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187 | (24) |
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4.3.1 Summary and overview |
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187 | (5) |
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4.3.2 Upscaling unsaturated flow through a set of joints (equivalent permeability, porosity, and capillary length) |
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192 | (2) |
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4.3.3 Upscaling two-phase flow in smooth or rough statistical joints: water retention 0(pc); conductivity curves {Kw(pc), KNw(pc)} |
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194 | (11) |
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4.3.4 Unsaturated or two-phase constitutive curves from statistical pore-scale models (discussion, review) |
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205 | (6) |
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4.4 Transient two-phase visco-capillary dynamics: interface motion X(t) in axially uniform or variable tubes/joints |
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211 | (21) |
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4.4.1 Introduction, objectives, and literature review |
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211 | (2) |
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4.4.2 Eulerian/Lagrangian equations for transient two-phase flow: axial interface displacement in tubes and joints |
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213 | (7) |
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4.4.3 Quasi-analytical results on transient dynamics of immiscible fluids: axial displacement in variably constricted tubes and joints |
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220 | (10) |
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4.4.4 Geometrical correction on interface dynamics X(t) in the case of very rough, highly variable tubes or joints (remarks) |
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230 | (1) |
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4.4.5 Interface dynamics X(t) in tubes, pores, joints (prospects) |
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231 | (1) |
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4.5 Two-dimensional two-phase dynamics: transient drainage in a planar joint with randomly variable aperture field a(x,y) |
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232 | (5) |
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4.5.1 Introduction and summary |
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232 | (1) |
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4.5.2 The 2D "rough fracture" and its random aperture field a(x,y) |
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232 | (1) |
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4.5.3 The 2D synthetic drainage experiment (two-phase flow) |
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233 | (4) |
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4.6 Other transient capillary phenomena in fluid dynamics: waves, bubbles, etc. (brief indications) |
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237 | (4) |
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237 | (1) |
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4.6.2 Rayleigh-Plateau instability |
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238 | (1) |
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4.6.3 Bubble dynamics and cavitation |
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239 | (1) |
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4.6.4 Liquid/vapor phase changes, boiling, bubbles in porous media |
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239 | (2) |
| Chapter 5 Darcy-Scale Capillary Flows in Heterogeneous or Statistical Continua (Richards and Muskat) |
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241 | (102) |
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5.1 Introduction, objectives and applications |
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241 | (11) |
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5.1.1 Introduction and summary |
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241 | (1) |
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5.1.2 Flow regimes and potential applications |
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242 | (1) |
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5.1.3 Hierarchy of scales and related issues (discontinuities) |
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243 | (1) |
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5.1.4 Material discontinuities in Darcy-scale flows |
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244 | (8) |
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5.2 Concepts: porous media, Darcy scale and REV (revisited) |
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252 | (3) |
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5.3 Single-phase Darcy-scale continuum flow equations (Navier-Stokes, Poiseuille, Darcy) |
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255 | (19) |
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5.3.1 Introduction and summary |
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255 | (1) |
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5.3.2 Darcy's law: from Navier-Stokes to Darcy in a nutshell |
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256 | (8) |
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5.3.3 Darcy's law for isotropic media (scalar permeability, single phase flow) |
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264 | (2) |
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5.3.4 Darcy's law for anisotropic media with tensorial or directional permeability (single-phase flow) |
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266 | (5) |
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5.3.5 Darcy's law from single-phase "Poiseuille flow" in fractures |
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271 | (3) |
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5.4 Richards equation for unsaturated water flow with fixed air pressure in the porous medium |
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274 | (35) |
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5.4.1 Introduction and summary (unsaturated flow) |
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274 | (1) |
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5.4.2 Darcy-Richards unsaturated flow equations |
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274 | (6) |
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5.4.3 Constitutive relationships theta(h), K(theta), K(h), C(h), D(theta), U(theta) |
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280 | (5) |
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5.4.4 Unsaturated curve models (theta(ψ), K(ψ)): overview |
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285 | (1) |
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5.4.5 Van Genuchten/Mualem (VGM) constitutive model for unsaturated moisture and conductivity curves (8(w), K(w)) |
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286 | (7) |
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5.4.6 Gardner's exponential K(ψ) conductivity curve and extensions |
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293 | (8) |
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5.4.7 Nonlinear relations {K(ψ,x), theta(ψ,x)} for heterogeneous media |
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301 | (1) |
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5.4.8 Matching different nonlinear models for {theta(ψ),K(ψ)}: exponential versus Van Genuchten/Mualem (parameter analyses) |
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302 | (7) |
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5.5 Philip's theory of infiltration - vertical unsaturated flow |
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309 | (15) |
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5.5.1 Introduction: literature and background on infiltration problems |
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309 | (2) |
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5.5.2 Philip's theta-based unsaturated flow equation for theta(z,t) |
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311 | (1) |
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5.5.3 Philip's analytical solution: sorptivity and gravitational term; infiltration rate i(t) and volume I(t); moisture profiles theta(z,t) |
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312 | (3) |
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5.5.4 Philip's analytical solution versus numerical infiltration experiments (comparisons and identification of soil parameters "A" and "5") |
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315 | (5) |
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5.5.5 Ponding time under a fixed rainfall rate, from Philip's quasi-analytical solution i(t) with both gravitational and capillary terms |
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320 | (2) |
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5.5.6 Recapitulation, discussion, conclusions |
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322 | (2) |
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5.6 Darcy-Muskat equations for immiscible two-phase flow |
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324 | (19) |
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5.6.1 Introduction and summary (two-phase flow) |
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324 | (2) |
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5.6.2 Mixed formulation of Darcy-Muskat PDEs governing two-phase flow |
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326 | (8) |
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5.6.3 Nonlinear characteristic curves of porous media for two-phase flow |
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334 | (3) |
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5.6.4 Other two-phase quantities derived from the Darcy-Muskat equations |
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337 | (6) |
| Conclusion to Volume 1 and Outline of Volume 2 |
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343 | (2) |
| References |
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345 | (22) |
| Index |
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367 | |
