Homogenization and Porous Media (Interdisciplinary Applied Mathematics)

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9781461273394: Homogenization and Porous Media (Interdisciplinary Applied Mathematics)

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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.

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This book discusses methods and results from the theory of homogenization and their applications to flow and transport in porous media. 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, non-Newtonian phenomena, two phase flow, miscible displacement, 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 main underlying ideas and concepts of modern mathematical theory, including the most recently obtained results and applications. Mathematicians, soil physicists, geo-hydrologists, chemical engineers, researchers working in an oil reservoir simulation and the environmental sciences, will find this book of particular interest.

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