Homogenization and Porous Media

For several decades developments in porous media have taken place in almost independent areas. In civilengineering, many papers were publisheddealing with the foundations offlow and transport through porous media. The method used in most cases is called averaging, and the notion ofa representative elementary vol- ume(REV)playsanimportantrole. Inchemicalengineering,papersonconceptual models were written on the theory ofmixtures. Intheoretical physics and stochas- tic analysis, percolation theory has emerged, providing probabilistic models for systems where theconnectedness propertiesofsomecomponentdominatethebe- havior. In mathematics, atheoryhasbeendevelopedcalled homogenizationwhich deals with partial differential equations having rapidly oscillating coefficients. Early work in these and related areas was - among others - done by the fol- lowing scientists: Maxwell [Max81] and Rayleigh [Ray92] studied the effective conductivity of media with small concentrations of randomly and periodically, respectively, arranged inclusions. Einstein [Ein06] investigated the effective vis- cosityofsuspensions with hard spherical particles in compressible viscous fluids. Marchenko and Khrouslov [MK64] looked at the asymptotic nature of homog- enization; they introduced a general approach of averaging based on asymptotic tools which can handle a variety ofdifferent physical problems. Unfortunately, up to now, little efforthas been made to bridge the gap between these different fields of research. Consequently, many results were and are dis- covered independently, and scientists are almost unable to understand each other because the respective languages have been developing in different directions.

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.