Homogenization and porous media.

*(English)*Zbl 0872.35002
Interdisciplinary Applied Mathematics. 6. New York, NY: Springer. xvi, 275 p. (1997).

[The articles of this volume will not be indexed individually.]

This book contains ten articles devoted to the most actual themes of the mathematical theory of flow through a porous medium; articles are written from the point of view of homogenization ideology. The authors are leading specialists in the subject. The book is dedicated to the memory of Serguei M. Kozlov. The contents of the book are as follows.

1. Introduction (U. Hornung). The basic problems are formulated. The method of two-scale asymptotic expansion is illustrated with the help of classical examples. 2. Percolation models for porous media (K. M. Golden). The presented theory is useful in a number of applications. For two- or three-phase flow there are regimes for which a percolation theoretical approach seems to be more appropriate than the standard Darcy-type description. 3. One-phase Newtonian flow (G. Allaire and A. Mikelić). The most of strong homogenization results have been obtained for flow in a saturated medium. In this chapter, the derivation of Darcy’s and Brinkman’s law is described in detail; the influence of inertia and time-dependent effects are studied. The last section is devoted to the very recent results on the interface condition between free and porous medium flow. 4. Non-Newtonian flow (A. Mikelić). This chapter deals with the flow of a one-phase quasi-Newtonian fluid. The Ostwald-de Waele, Carreau-Yasuda, and Bingham models are analyzed. Due to the nonlinearity of the governing equations advanced tools from nonlinear functional analysis are used. 5. Two-phase flow (A. Bourgeat). Homogenization of two-phase flow from the porescale to the messoscale and also from the measascale to the macroscale is investigated. In addition the flow in media with random heterogeneities is considered. 6. Miscible displacement (U. Hornung). This chapter deals with the process of transporting solutes in fluids. The process is important in soil pollution and remediation techniques; it is closely related to absorption and adsorption and to chemical reactions. 7. Thermal flow (H. Ene). The transport of heat in a moving one-phase fluid (convection) is considered. The cases of low and high values of the Rayleigh number are investigated. 8. Poroelastic media (J.-L. Auriault). The acoustic macroscopic behavior of a saturated porous medium is investigated. A quasistatic flow of the saturating fluid and an elastic matrix of the porous medium is assumed. 9. Microstructure models of porous media (R. E. Showalter). This chapter gives a survey on the application of models that contain two scales. As an example a single-phase flow in various types of fissured media is considered. 10. Computational aspects of dual-porosity models (T. Arbogast). Dual porosity models are compared with mesoscopic models, i.e. with models that explicitly account for two-phase flow in fissures and blocks of fractured medium. It is shown that dual-porosity models have far better performance as far as computer time is concerned.

According to the editor, the authors of this book have tried to write their contributions so that the text is understood by engineers with only a basic knowledge of the PDE-theory. For the more mathematically inclined reader, the important methods, tools, and results of a general nature from homogenization theory are collected in Appendix A (G. Allaire). Appendix B contains symbols, notations and definitions most often used in the book.

This book contains ten articles devoted to the most actual themes of the mathematical theory of flow through a porous medium; articles are written from the point of view of homogenization ideology. The authors are leading specialists in the subject. The book is dedicated to the memory of Serguei M. Kozlov. The contents of the book are as follows.

1. Introduction (U. Hornung). The basic problems are formulated. The method of two-scale asymptotic expansion is illustrated with the help of classical examples. 2. Percolation models for porous media (K. M. Golden). The presented theory is useful in a number of applications. For two- or three-phase flow there are regimes for which a percolation theoretical approach seems to be more appropriate than the standard Darcy-type description. 3. One-phase Newtonian flow (G. Allaire and A. Mikelić). The most of strong homogenization results have been obtained for flow in a saturated medium. In this chapter, the derivation of Darcy’s and Brinkman’s law is described in detail; the influence of inertia and time-dependent effects are studied. The last section is devoted to the very recent results on the interface condition between free and porous medium flow. 4. Non-Newtonian flow (A. Mikelić). This chapter deals with the flow of a one-phase quasi-Newtonian fluid. The Ostwald-de Waele, Carreau-Yasuda, and Bingham models are analyzed. Due to the nonlinearity of the governing equations advanced tools from nonlinear functional analysis are used. 5. Two-phase flow (A. Bourgeat). Homogenization of two-phase flow from the porescale to the messoscale and also from the measascale to the macroscale is investigated. In addition the flow in media with random heterogeneities is considered. 6. Miscible displacement (U. Hornung). This chapter deals with the process of transporting solutes in fluids. The process is important in soil pollution and remediation techniques; it is closely related to absorption and adsorption and to chemical reactions. 7. Thermal flow (H. Ene). The transport of heat in a moving one-phase fluid (convection) is considered. The cases of low and high values of the Rayleigh number are investigated. 8. Poroelastic media (J.-L. Auriault). The acoustic macroscopic behavior of a saturated porous medium is investigated. A quasistatic flow of the saturating fluid and an elastic matrix of the porous medium is assumed. 9. Microstructure models of porous media (R. E. Showalter). This chapter gives a survey on the application of models that contain two scales. As an example a single-phase flow in various types of fissured media is considered. 10. Computational aspects of dual-porosity models (T. Arbogast). Dual porosity models are compared with mesoscopic models, i.e. with models that explicitly account for two-phase flow in fissures and blocks of fractured medium. It is shown that dual-porosity models have far better performance as far as computer time is concerned.

According to the editor, the authors of this book have tried to write their contributions so that the text is understood by engineers with only a basic knowledge of the PDE-theory. For the more mathematically inclined reader, the important methods, tools, and results of a general nature from homogenization theory are collected in Appendix A (G. Allaire). Appendix B contains symbols, notations and definitions most often used in the book.

Reviewer: I.Aganović (Zagreb)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

76S05 | Flows in porous media; filtration; seepage |