Mathematical Models for Poroelastic Flows
The book is devoted to rigorous derivation of macroscopic mathematical models as a homogenization of exact mathematical models at the microscopic level. The idea is quite natural: one first must describe the joint motion of the elastic skeleton and the fl
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Anvarbek Meirmanov
Mathematical Models for Poroelastic Flows
Atlantis Studies in Differential Equations Volume 1
Series editor Michel Chipot, Zürich, Switzerland
For further volumes: www.atlantis-press.com
Aims and Scope of the Series The ‘‘Atlantis Studies in Differential Equations’’ publishes monographs in the area of differential equations, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. For more information on this series and our other book series, please visit our website www.atlantis-press.com/publications/books AMSTERDAM – PARIS – BEIJING ATLANTIS PRESS Atlantis Press 29, avenue Laumière 75019 Paris, France
Anvarbek Meirmanov
Mathematical Models for Poroelastic Flows
Anvarbek Meirmanov Mathematics and Cybernetics Kazakh-British Technical University Almaty Kazakhstan
ISSN 2214-6253 ISBN 978-94-6239-014-0 DOI 10.2991/978-94-6239-015-7
ISSN 2214-6261 (electronic) ISBN 978-94-6239-015-7 (eBook)
Library of Congress Control Number: 2013949487 Atlantis Press and the authors 2014 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
To my wife Liudmila and my daughters Anastasia and Aliya
Preface
This book is devoted to the rigorous mathematical modeling of physical processes in underground continuous media, namely, the correct description of porous elastic solids with fluid-filled pores. Recently this subject has been attracting increased attention for many reasons: the recovery of oil and gas, liquid waste disposal into the ground, seismic phenomena, acoustic wave propagation in the water-saturated porous bottom of the ocean, diffusion-convection in porous media, etc. Such continuous media are called heterogeneous continuous media. That is, those continuous media which consist of two or more different components (phases) and in any sufficiently small amount of a continuum there are different phases. The minimum size of this volume is different for different heterogeneous media, but usually it is in the range from several microns to several tens of microns. There are two different approaches to the description of heterogeneous media. The first approach, which we call the phenomenological approach, is based upon the notion of a continuous medium as a kind of conglomerate, where at each point all phases of such a medium are present. In this approach, the main difficulty is the physical modeling: the choice of axioms that define the dependence of the stress tensor on the basic characteristics of motion and thermodynamic relations. The second approach is based on precise physical modeling with further simplification of the mathematical model using the methods of mathematical analysis. As a rule, the differential equ
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