# Mathematical Models for Poroelastic Flows

• For each underground physical process the reader finds a set of mathematical models depending on the dimensionless criteria of the given process and describing the process with different degrees of exactness

• The reader may apply the suggested approach to solve many other important problems in filtration and acoustics and obtain macroscopic mathematical models, which are asymptotically exact

• First volume in a new series

Book

Part of the Atlantis Studies in Differential Equations book series (ASDE, volume 1)

1. Front Matter
Pages i-xxxviii
2. Anvarbek Meirmanov
Pages 1-65
3. Anvarbek Meirmanov
Pages 67-87
4. Anvarbek Meirmanov
Pages 89-134
5. Anvarbek Meirmanov
Pages 135-167
6. Anvarbek Meirmanov
Pages 169-239
7. Anvarbek Meirmanov
Pages 241-263
8. Anvarbek Meirmanov
Pages 265-283
9. Anvarbek Meirmanov
Pages 285-316
10. Anvarbek Meirmanov
Pages 317-325
11. Anvarbek Meirmanov
Pages 327-365
12. Anvarbek Meirmanov
Pages 367-381
13. Back Matter
Pages 383-449

### Introduction

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 fluid in pores at the microscopic level by means of classical continuum mechanics, and then use homogenization to find appropriate approximation models (homogenized equations). The Navier-Stokes equations still hold at this scale of the pore size in the order of 5 – 15 microns. Thus, as we have mentioned above, the macroscopic mathematical models obtained are still within the limits of physical applicability. These mathematical models describe different physical processes of liquid filtration and acoustics in poroelastic media, such as isothermal or non-isothermal filtration, hydraulic shock, isothermal or non-isothermal acoustics, diffusion-convection, filtration and acoustics in composite media or in porous fractured reservoirs. Our research is based upon the Nguetseng two-scale convergent method.

### Keywords

Lame´s system Stokes system homogenization mathematical models poroelasticity

#### Authors and affiliations

1. 1.Mathematics and CyberneticsKazakh-British Technical UniversityAlmatyKazakhstan

### Bibliographic information

• Book Title Mathematical Models for Poroelastic Flows
• Authors Anvarbek Meirmanov
• Series Title Atlantis Studies in Differential Equations
• Series Abbreviated Title Atlantis Studies in Differential Equations
• DOI https://doi.org/10.2991/978-94-6239-015-7
• Copyright Information Atlantis Press and the authors 2014
• Publisher Name Atlantis Press, Paris
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-94-6239-014-0
• eBook ISBN 978-94-6239-015-7
• Series ISSN 2214-6253
• Series E-ISSN 2214-6261
• Edition Number 1
• Number of Pages XXXVIII, 449
• Number of Illustrations 22 b/w illustrations, 3 illustrations in colour
• Topics
• Buy this book on publisher's site
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