Homogenization and Porous Media

This book offers a systematic, rigorous treatment of upscaling procedures related to physical modeling for porous media on micro-, meso- and macro-scales, including detailed studies of micro-structure systems and computational results for dual-porosity models.

Contenu

1 Introduction.- 1.1 Basic Idea.- 1.2 First Examples.- 1.2.1 One-Dimensional Diffusion.- 1.2.2 Resistor Networks.- 1.2.3 Layered Media.- 1.3 Diffusion in Periodic Media.- 1.3.1 Formal Asymptotic Expansion.- 1.3.2 Application to Layered Media.- 1.3.3 Estimates of the Effective Conductivity Tensor.- 1.3.4 Media with Obstacles and Diffusion in Perforated Domains.- 1.4 Formal Derivation of Darcy's Law.- 1.5 Formal Derivation of a Distributed Microstructure Model.- 1.6 Remarks on Networks of Resistors, Capillary Tubes, and Cracks.- 1.6.1 Monte-Carlo Simulations.- 2 Percolation Models for Porous Media.- 2.1 Fundamentals of Percolation Theory.- 2.2 Exponent Inequalities for Random Flow and Resistor Networks.- 2.3 Critical Path Analysis in Highly Disordered Porous and Conducting Media.- 3 One-Phase Newtonian Flow.- 3.1 Derivation of Darcy's Law.- 3.1.1 Presentation of the Results.- 3.1.2 Proof of the Homogenization Theorem.- 3.1.3 A Priori Estimate of the Pressure in a Porous Medium.- 3.2 Inertia Effects.- 3.2.1 Darcy's Law with Memory.- 3.2.2 Nonlinear Darcy's Law.- 3.3 Derivation of Brinkman's Law.- 3.3.1 Setting of the Problem.- 3.3.2 Principal Results.- 3.4 Double Permeability.- 3.5 On the Transmission Conditions at the Contact Interface between a Porous Medium and a Free Fluid.- 3.5.1 Statement of the Problem and Existing Results from Physics.- 3.5.2 Statement of the Mathematical Results and Comparison with the Literature.- 4 Non-Newtonian Flow.- 4.1 Introduction.- 4.2 Equations Governing Creeping Flow of a Quasi-Newtonian Fluid.- 4.3 Description of a Periodic s-Geometry, Construction of the Restriction Operator, and Review of the Results of Two-Scale Convergence in Lq-Spaces.- 4.4 Statement of the Principal Results.- 4.5 Inertia Effects for Non-Newtonian Flows through Porous Media.- 4.6 Proof of the Uniqueness Theorems.- 4.7 Uniform A Priori Estimates.- 4.8 Proof of Theorem A.- 4.9 Proof of Theorem B.- 4.10 Conclusion.- 5 Two-Phase Flow.- 5.1 Derivation of the Generalized Nonlinear Darcy Law.- 5.1.1 Introduction.- 5.1.2 Obtaining Macroscopic Laws by Volume Averaging.- 5.1.3 Obtaining Macroscopic Laws by Homogenization.- 5.2 Upscaling Two-Phase Flow Characteristics in a Heterogeneous Reservoir with Capillary Forces (Finite Peclet Number).- 5.2.1 Introduction.- 5.2.2 Definition of the Homogenization Problem.- 5.2.3 General Homogenization Result.- 5.2.4 Some Special Cases.- 5.2.5 Randomly Heterogeneous Porous Media.- 5.3 Upscaling Two-Phase Flow Characteristics in a Heterogeneous Core, Neglecting Capillary Effects (Infinite Peclet Number).- 5.3.1 Introduction.- 5.3.2 Homogenization of the Buckley-Leverett System.- 5.3.3 Propagation of Nonlinear Oscillations.- 5.4 The Double-Porosity Model of Immiscible Two-Phase Flow.- 5.4.1 Introduction.- 5.4.2 The Microscopic Model.- 5.4.3 Compactness and Convergence Results.- 6 Miscible Displacement.- 6.1 Introduction.- 6.2 Upscaling from the Micro-to the Mesoscale.- 6.2.1 Diffusion, Convection, and Reaction.- 6.2.2 Adsorption.- 6.2.3 Chromatography.- 6.2.4 Semipermeable Membranes.- 6.3 Upscaling from the Meso-to the Macroscale.- 6.3.1 Mobile and Immobile Water.- 6.3.2 Fractured Media.- 6.4 Discussion.- 7 Thermal Flow.- 7.1 Introduction.- 7.2 Basic Equations.- 7.3 Natural Convection in a Bounded Domain.- 7.4 Natural Convection in a Horizontal Porous Layer.- 7.5 Mixed Convection in a Horizontal Porous Layer.- 7.6 Thermal Boundary Layer Approximation.- 7.7 Conclusion.- 8 Poroelastic Media.- 8.1 Acoustics of an Empty Porous Medium.- 8.1.1 Local Description and Estimates.- 8.1.2 Macroscopic Description.- 8.2 A Priori Estimates for a Saturated Porous Medium.- 8.3 Local Description of a Saturated Porous Medium.- 8.4 Acoustics of a Fluid in a Rigid Porous Medium.- 8.5 Diphasic Macroscopic Behavior.- 8.6 Monophasic Elastic Macroscopic Behavior.- 8.7 Monophasic Viscoelastic Macroscopic Behavior.- 8.8 Acoustics of Double-Porosity Media.- 8.8.1 A Priori Estimate.- 8.8.2 Double-Porosity Macroscopic Models.- 8.9 Conclusion.- 9 Microstructure Models of Porous Media.- 9.1 Introduction.- 9.2 Parallel Flow Models.- 9.2.1 Totally Fissured Media.- 9.2.2 Partially Fissured Media.- 9.3 Distributed Microstructure Models.- 9.3.1 Totally Fissured Media.- 9.3.2 Partially Fissured Media.- 9.4 A Variational Formulation.- 9.5 Remarks.- 9.5.1 Homogenization.- 9.5.2 Further Remarks.- 10 Computational Aspects of Dual-Porosity Models.- 10.1 Single-Phase Flow.- 10.1.1 The Mesoscopic and Dual-Porosity Models.- 10.1.2 Numerical Solution.- 10.2 Two-Phase Flow.- 10.2.1 The Mesoscopic and Dual-Porosity Models.- 10.2.2 Numerical Solution.- 10.3 Some Computational Results.- 10.3.1 Single-Phase.- 10.3.2 Two-Phase.- A Mathematical Approaches and Methods.- A.1.1 F-Convergence.- A.1.2 G-Convergence.- A.1.3 H-Convergence.- A.2 The Energy Method.- A.2.1 Setting of a Model Problem.- A.2.2 Proof of the Results.- A.3 Two-Scale Convergence.- A.3.1 A Brief Presentation.- A.3.2 Statement of the Principal Results.- A.3.3 Application to a Model Problem.- A.4 Iterated Homogenization.- B Mathematical Symbols and Definitions.- B.1 List of Symbols.- B.2 Function Spaces.- B.2.1 Macroscopic Function Spaces.- B.2.2 Micro-and Mesoscopic Function Spaces.- B.2.3 Two-Scale Function Spaces.- B.2.4 Time-Dependent Function Spaces.- C References.