Homogenization and porous media by Ulrich HornungHomogenization and porous media by Ulrich Hornung

Homogenization and porous media

EditorUlrich Hornung

Hardcover | November 26, 1996

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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.
Title:Homogenization and porous mediaFormat:HardcoverDimensions:295 pagesPublished:November 26, 1996Language:English

The following ISBNs are associated with this title:

ISBN - 10:0387947868

ISBN - 13:9780387947860

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Table of Contents

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.

From Our Editors

The methods and results of the theory of homogenization and their applications to flow and transport in porous media are discussed in this book. It offers a systematic and rigorous treatment of upscaling procedures related to physical modeling for porous media on micro-, meso-, and macro-scales. The chapters are devoted to percolation, Newtonian and non-Newtonian phenomena, two-phase flow, miscible displacement, and thermal and elastic effects. Detailed studies of micro-structure systems and computational results for dual-porosity models are presented. This book will be of interest to readers who want to learn the principal underlying ideas and concepts of modern mathematical theory, including the most recently obtained results and applications. Mathematicians, soil physicists, geo-hydrologists, chemical engineers, and researchers working in oil reservoir simulation or in the environmental sciences will find this book of particular interest.