# Nonstationary Flows of Viscous Fluids through Porous Elastic Media. Homogenization Method

**Author(s):**Bielski W.

**Volume:**388

**Series:**A-29

Porous media found in the natural environment represent a good example of media with microstructure. Their pores contain different phases of liquids and gases filling them as well as the interfaces between the phases. This issue has been frequently discussed by soil physicists and chemists, chemical and petro engineers, seismologists and other researchers interested in geosciences.

CONTENTS

Introduction, ...5

1. Macroscopic equations for nonstationary flow of Stokesian fluid through porous elastic reservoirs, ...9

1.1 Notations and basic relations, ...10

1.2 Equations of microperiodic porous media, ...12

1.3 Derivation of effective relations. Homogenization, ...15

1.4 Justification of the asymptotic analysis by the two-scale convergence, ...21

1.5 Passage to the stationary case, ...25

1.6 Comments on related papers, ...27

1.7 An example of one-dimensional flow, ...32

1.8 Concluding remarks, ...33

2. Nonstationary flow of two immiscible fluids, ...35

2.1 Overview of previous results on two-phase flow through porous media, ...36

2.2 Notations and basic relations, ...39

2.3 Basic equations for the motion of porous medium with biphasic fluid, ...40

2.4 Results of homogenization, ...42

2.5 Macroscopic constitutive relations, ...43

2.6 Mechanics of porous media with biphasic liquid, ...44

2.7 Flow of biphasic fluid in porous media, ...45

2.8 Consolidation equation and Darcy's law nonlocal in time, ...49

2.9 Justification of the asymptotic analysis by the two-scale convergence, ...50

2.10 Passage to the stationary case, ....53

2.11 An example of the flow of two-immiscible fluids, local problems, ...55

2.12 Final remarks, ...58

3. Porous hierarchical structures: double-scale porosity, ...59

3.1 Microscopic model: periodic distribution of micropores, ...60

3.2 Multi-scale convergence and applications to hierarchical structure, ...62

3.3 Macroscopic model, ...63

3.4 Final remarks, ...65

4. Stochastic approach: random distribution of pores, random properties of elastic reservoir, ...67

4.1 Random porous media and stochastic two-scale convergence in the mean, ...68

4.2 Basic equation to nonstationary flow of viscous fluid through porous media with random microstructure, ...69

4.3 Effective model, ...71

4.4 Ergodicity of dynamical system and comments, ...73

4.5 Stochastic multi-scale convergence in the mean, ...75

4.6 Application to reiterated stochastic homogenization of stationary diffusion equation, ...80

4.7 Final remarks, ...82

5. Stationary diffusion in random porous media composed of nonhomogeneous material, ...83

5.1 The statement of the problem, ...84

5.2 Main result, ...86

6. Final comments and remarks, ...91

Appendix A. Asymptotic expansion, ...95

Appendix B. Two scale convergence, ...97

Appendix C. Equations with expanded terms, ...101

Appendix D. Stochastic homogenization, ...107

Acknowledgements, ...118

Bibliography, ...119

Streszczenie, ...139