RESERVOIR ANALYSIS USING INTERMEDIATE ...Dr. Khalid Aziz I certify that I have read this thesis and that in my opin-ion it is fully adequate, in scope and in quality, as a dissertation

RESERVOIR ANALYSIS USING INTERMEDIATE

FREQUENCY EXCITATION

a dissertation

submitted to the department of petroleum engineering

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

By

Yan Pan

August 1999

c© Copyright by Yan Pan 1999All Rights Reserved

ii

I certify that I have read this thesis and that in my opin-

ion it is fully adequate, in scope and in quality, as a

dissertation for the degree of Doctor of Philosophy.

Dr. Roland N. Horne(Principal Advisor)




Dr. Khalid Aziz




Dr. Thomas A. Hewett

Approved for the University Committee on Graduate

Studies:

iii

Acknowledgements

I would like to thank my research advisor Professor Roland N. Horne for his constant

support, encouragement and guidance during the course of this study. Professor

Horne is always available to answer my questions, or to help me in finding the right

path to the answers. When I felt depressed because of some unsuccessful strugglings,

Professor Horne kept his belief in my ability of solving problems and encouraged me

to overcome the difficulties eventually. I could not have finished this research project

without Professor Horne’s guidance and encouragement.

Sincere thanks are due to my friends, Jing Wan, Suwat Athichanagorn, Erik Skjetne,

Hongkai Zhao for generously providing valuable ideas and techniques for this research.

I would like to express my special appreciation to my boyfriend, Jorge Landa, for

his love, kindness, patience and constant spiritual and academic support. I always

benefit from his generosity of intelligence and experiences.

I would like to thank my family for their love and support. I am lucky to be able to

stay in a warm and comfortable home with my family while I am pursuing the degree

of PhD. My gratitude is endless to my parents, Zhengpu Pan and Ruyuan Zhang,

and to my brother Lei Pan.

Financial support from the members of the SUPRI-D Research Consortium on In-

novation in Well Testing, the Department of Petroleum Engineering, and Chevron

Scholarship Program is gratefully acknowledged.

iv

Abstract

This study investigated mathematical models to describe the motion of fluids in

porous media, and applied these models to harmonic well testing at intermediate

frequencies. The purpose was to examine the possibility of obtaining more informa-

tion about reservoirs than can be usually derived in conventional well testing (low

frequency excitation) and seismic data processing (high frequency excitation). The

problem of fluid flow in the pores or small channels of a periodic elastic solid matrix

was studied at pore scale, and the homogenization technique was applied to predict

the macroscopic behavior of reservoirs.

The theoretical analysis and the numerical results show that the responses of a porous

medium to harmonic perturbations depend on the parameters of the pore structure,

the properties of fluid and solid and the frequencies of the excitation signals. The

effective parameters, such as dynamic permeability and porosity, are also functions of

perturbation frequencies. From the investigation of the coupling effects of fluid and

solid motions, it seems that the elastic solid vibration has positive impact on fluid

flow under harmonic perturbations of intermediate frequencies, which may provide a

potential new technique for stimulation of oil production.

Based on the homogenization study, five separate characteristic macroscopic model

were identified according to the relation between a length scale parameter and a prop-

erty contrast number. These five models can be used to interpret the corresponding

responses of a reservoir. It is possible to infer the effective parameters of porous

media, such as porosity and fracture density, by analyzing the diphasic macroscopic

v

response to cyclic excitation at various frequencies.

vi

Contents

Acknowledgements iv

Abstract v

1 Introduction 1

1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline of Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Homogenization Techniques 16

2.1 Acoustics of an Empty Porous Medium . . . . . . . . . . . . . . . . . 18

2.1.1 Local Description . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Macroscopic Description . . . . . . . . . . . . . . . . . . . . . 20

2.2 A Prior Estimate for Saturated Porous Media . . . . . . . . . . . . . 23

2.3 Local Description of Saturated Porous Media . . . . . . . . . . . . . . 25

2.4 Acoustics of a Fluid in a Rigid Porous Medium . . . . . . . . . . . . 27

2.5 Diphasic Macroscopic Behavior . . . . . . . . . . . . . . . . . . . . . 29

2.6 Monophasic Elastic Macroscopic Behavior . . . . . . . . . . . . . . . 36

2.7 Monophasic Viscoelastic Macroscopic Behavior . . . . . . . . . . . . . 37

3 Mathematical Model 41

3.1 Pore-Scale Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Motion of Elastic Solid . . . . . . . . . . . . . . . . . . . . . . 43

3.1.2 Fluid Flow in Elastic Channel . . . . . . . . . . . . . . . . . . 44

vii

3.2 Macroscopic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.1 Acoustics of Empty Channels . . . . . . . . . . . . . . . . . . 54

3.2.2 Acoustics of a Fluid in a Rigid Channel . . . . . . . . . . . . . 59

3.2.3 Diphasic Macroscopic Behavior . . . . . . . . . . . . . . . . . 63

3.2.4 Monophasic Elastic Behavior . . . . . . . . . . . . . . . . . . . 70

3.2.5 Monophasic Viscoelastic Behavior . . . . . . . . . . . . . . . . 75

3.3 Relation to Existing Theories . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Telegrapher’s Equation . . . . . . . . . . . . . . . . . . . . . . 76

3.3.3 Biot’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Numerical Methods 82

4.1 Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Finite Element Method for Elasticity . . . . . . . . . . . . . . . . . . 82

4.2.1 Strong Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Variational Form . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.3 Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.4 Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.5 Bossak’s Algorithm for Structural Dynamics . . . . . . . . . . 88

4.2.6 Bilinear Quadrilateral Element . . . . . . . . . . . . . . . . . 90

4.2.7 Element Matrices and Force . . . . . . . . . . . . . . . . . . . 93

4.2.8 Gaussian Quadrature Numerical Integration . . . . . . . . . . 95

4.3 Finite Difference Method for Fluid Flow . . . . . . . . . . . . . . . . 97

4.3.1 Discretized Governing Equations . . . . . . . . . . . . . . . . 97

4.3.2 Treatment of Velocity-Pressure Linkage . . . . . . . . . . . . . 99

4.3.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Analysis of Results 102

5.1 Hagen-Poiseuille Flow in Elastic Channels . . . . . . . . . . . . . . . 102

5.1.1 Effect of Initial Porosity . . . . . . . . . . . . . . . . . . . . . 103

5.1.2 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . 106

5.1.3 Effect of P-Wave Velocity . . . . . . . . . . . . . . . . . . . . 106

viii

5.1.4 Effect of Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . 108

5.1.5 Effect of Density Ratio . . . . . . . . . . . . . . . . . . . . . . 108

5.1.6 Effect of Channel Tortuosity . . . . . . . . . . . . . . . . . . . 109

5.2 Harmonic Motion of Fluid in Rigid Channels . . . . . . . . . . . . . . 112

5.3 Harmonic Motion of Fluid in Elastic Channels . . . . . . . . . . . . . 115

5.3.1 Effects of Frequency on Fluid Motion . . . . . . . . . . . . . . 126

5.3.2 Effect of Initial Porosity on Fluid Motion . . . . . . . . . . . . 126

5.3.3 Resonant Behavior . . . . . . . . . . . . . . . . . . . . . . . . 127

6 Applications 132

6.1 Selection of Macroscopic Models . . . . . . . . . . . . . . . . . . . . . 132

6.2 Inference of Fracture Density From Pulse Decay Test . . . . . . . . . 134

6.3 Inference of Porosity From Harmonic Test . . . . . . . . . . . . . . . 135

6.4 Optimal Frequency Range for Stimulation of Oil Production . . . . . 137

6.5 Benefit to Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.6 Attenuation of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Conclusions 142

Nomenclature 144

References 147

Appendix 156

A Computer Programs 157

A.1 General Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

A.2 Files in makefile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

A.3 Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.3.1 Input Data File . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.3.2 Output Data Files . . . . . . . . . . . . . . . . . . . . . . . . 161

A.4 .gps Files for Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

ix

List of Tables

4.1 Nodal Coordinates in ξ-space . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Integration Point Indices . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Variable, Parameter and Source Term . . . . . . . . . . . . . . . . . . 98

4.4 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1 Natural Frequencies of Solid Matrix . . . . . . . . . . . . . . . . . . . 129

6.1 Selection of Macroscopic Models . . . . . . . . . . . . . . . . . . . . . 133

x

List of Figures

1.1 A Periodic Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Homogenization Procedure . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Periodic Unit Cell of the Porous Media . . . . . . . . . . . . . . . . . 17

3.1 Pore-Scale Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 One-Dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Coordinate System for Macroscopic Behavior Analysis . . . . . . . . . 57

4.1 Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Solid Problem Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Bilinear Quadrilateral Element Domain . . . . . . . . . . . . . . . . . 90

4.4 Block-Centered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Hagen-Poiseuille Flow φc = 0.25 - Example 1 . . . . . . . . . . . . . . 104

5.2 Hagen-Poiseuille Flow φc = 0.60 - Example 2 . . . . . . . . . . . . . . 105

5.3 Effect of Elasticity at Different Initial Porosity . . . . . . . . . . . . . 106

5.4 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Effect of P-Wave Velocity of Solid . . . . . . . . . . . . . . . . . . . . 107

5.6 Effect of Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7 Effect of Density Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.8 Hagen-Poiseuille Flow g(x) = 0.1 sin(2πx/xl) - Example 3 . . . . . . . 110

5.9 Effect of Channel Tortuosity . . . . . . . . . . . . . . . . . . . . . . . 111

5.10 Response of a Fluid in a Rigid Channel to Harmonic Perturbation . . 113

5.11 Phase Map of Fluid Motion in a Rigid Channel . . . . . . . . . . . . 113

xi

5.12 Effects of Perturbation Signal Frequency - Model II . . . . . . . . . . 114

5.13 Harmonic Motion of Fluid in Elastic Channels - Example 4 . . . . . . 116

5.14 Normalized Maps of Solid Displacements - Example 4 . . . . . . . . . 117


5.16 Phase Maps - Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . 120








5.24 Effects of Perturbation Frequency on Fluid Motion in Elastic Channels 126

5.25 Effect of Initial Porosity on Fluid Motion in Elastic Channels . . . . . 127

5.26 Effects of Perturbation Frequency on Fluid and Solid Motions . . . . 130

5.27 Effects of Perturbation Frequency on Fluid and Solid Motions . . . . 131

6.1 Setup of Pulse Decay Test . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 Pressure Transient Responses in Pulse Decay Test . . . . . . . . . . . 136

6.3 Relation Between Vibration Frequency and Fracture Density . . . . . 137

6.4 Optimal Frequencies for Stimulation . . . . . . . . . . . . . . . . . . 138

xii

Chapter 1

Introduction

1.1 Statement of the Problem

In petroleum engineering and hydrology, well testing is used to investigate the prop-

erties (permeability, porosity, etc.) of reservoirs. Normally the transient change of

pressure in the porous medium during conventional well testing is described by the

diffusion equation. This equation neglects the inertial effects on the fluid flow which

in some cases are important. On the other hand, seismic data used in geophysics pro-

vide other types of information about the rocks, such as compression and shear wave

velocities, reflection and transmission coefficients, dry and saturated elastic moduli,

etc.. The underlying mechanism is the acoustic propagation of high frequency signals,

which can be described by the wave equation. In fact, the motion of fluids and solids

in porous media is a complicated nonlinear process which involves the two-phase cou-

pling of fluid and solid, which may be approximated as a diffusion process or a wave

propagation only under certain conditions. If a new test could be designed to measure

signals propagating at an intermediate frequency higher than the pressure pulse yet

lower than the high frequency seismic signals, it may be possible to determine addi-

tional reservoir information beyond that usually derived from well testing or seismic

data processing.

The effects of fluid and solid two-phase coupling can be observed in many natural

1

CHAPTER 1. INTRODUCTION 2

phenomena. One of the early observations of periodic tidal effects on fluid accumula-

tions in porous strata in the earth can be dated back more than one hundred years.

This phenomenon is a result of the relative displacement of the particles composing

a celestial body caused by the gravitational attraction between the sun, moon and

the earth. In petroleum reservoirs, observations accumulated during the last 40 years

suggest that seismic waves generated from earthquakes and cultural noise may alter

water and oil production. It has also been observed in some laboratory measure-

ments and field applications that imposing harmonic signals into cores or reservoirs

sometimes may induce higher fluid flow rate. The mechanism of these phenomena is

still unclear and the observations themselves are not unambiguous. This study will

investigate this problem and seek a theoretical explanation of such a phenomenon. If

true, this mechanism may provide a potential new technique for enhanced oil recovery.

Fluid flow in the reservoir is a two-scale problem. The small scale involves fluid

flow in the pores and small channels. The larger scale macroscopic behavior of the

reservoir can be observed in well testing and seismic data processing. The macro-

scopic response to harmonic signals depends on the pore structure, the fluid and solid

properties and the signal frequency. It is not appropriate to apply Darcy’s law to

both scales without first analyzing the individual problem. For each particular case,

a corresponding macroscopic model has to be chosen to describe the large scale be-

havior of the reservoir according to the scale of two characteristic parameters: one

permeability related length-scale ratio and one property contrast number. Special

attention has to be paid to a finely heterogeneous medium to determine whether an

equivalent macroscopic description exists. These mathematical treatments not only

clarify on which assumptions a macroscopic law is based, but also allow development

of techniques that can be applied to similar and related problems. The homogeniza-

tion approach provides better understanding of the influence of the structure of the

porous media and of the concepts of effective permeability and other macroscopic

properties. Therefore this study uses the homogenization procedure to study the

two-scale problem and apply the results to reservoir characterization.


1.2 Literature Review

The modification of Darcy’s law to include momentum balance for steady single phase

flow through porous media has been discussed for many years. There have been sev-

eral different approaches to derive more accurate equations to describe the motion of

fluids in porous media. Forchheimer (1901) suggested adding terms of higher order

in the velocity. Klinkenberg (1941) demonstrated that the permeability coefficient in

Darcy’s law depends on the absolute pressure or, alternatively, on the density field.

Oroveanu and Pascal (1959) noted that the time derivative of the momentum density

must be included in the equations of motion, their differential equation for pressure

is commonly known as the Telegrapher’s equation. Slattery (1963) presented an ex-

tension of Darcy’s law to unsteady state flow in anisotropic media. Slattery (1966)

discussed local volume-averaging of the equations of continuity and of momentum

balance. Foster, McMillen and Odeh (1967) proposed complete equations of average

linear momentum balance for a single-phase fluid in an incompressible homogeneous,

porous medium, and demonstrated that for transient flow of compressible Newto-

nian fluid, the space-time description of the pressure is determined with the lowest

approximation by the Telegrapher’s equation. Foster, McMillen and Wallick (1968)

performed an experimental study with a perfect gas in a porous medium, and derived

an approximate nonlinear description of the space-time behavior of a small propa-

gating pressure pulse. Odeh and McMillen (1972) did theoretical research on the

diffusion equation and performed experimental work on pulse propagation in linear

cores saturated with air. Due to the technical limitation at that time, these ideas

were not explored in any detail, and no reservoir testing procedure was proposed.

The theory of acoustic propagation in fluid-saturated porous media was developed

by Biot (1956, 1962). Since then the model has been extended and applied widely in

geophysics. Bonnet (1987) reviewed some extensions of Biot equations and derived

the complete basic singular solution in the frequency domain for dynamic poroelastic-

ity problems in analogy with thermoelasticity. Depollier, Allard and Lauriks (1988)

compared the equations of continuity of the models traditionally used to predict the


acoustical properties of sound-absorbing materials to simplified expressions obtained

from the Biot model, and pointed out the inconsistencies in traditional models. Nor-

ris (1992) explored the analogy between the equations of static poroelasticity and

the equations of thermoelasticity including entropy, and discussed the method of de-

termining the effective parameters in an inhomogeneous poroelastic medium using

known results from the literature on the effective thermal expansion coefficient and

the effective heat capacity of a disordered thermoelastic continuum. Zimmerman and

Stern (1994) derived the pressure-solid displacement form of the harmonic equations

of motion for a poroelastic solid from the form of equations originally proposed by

Biot. The analytical solutions for several basic problems were presented. Dai, Vafidis

and Kanasewich (1995) considered Biot’s hyperbolic system and developed a particle

velocity-stress, finite-difference method for the simulation of wave propagation in two-

dimensional heterogeneous poroelastic media. Abousleiman, Bai and Roegiers (1995)

studied the effects of fluid flow coupled with displacements in and around pressurized

tunnels in fractured formations using Biot’s theory. Chen, Harstad and Teufel (1996)

followed Biot’s theory and extended the conventional fluid-flow dual-porosity for-

mulations to a coupled fluid-flow-geomechanics model. Interpolation of pore volume

compressibilities and the associated effective stress laws were identified to be the most

critical coupling considerations. Abousleiman et al. (1996) addressed the phenomena

of mechanical creep and deformation in rock formations coupled with the hydraulic

effects of fluid flow. The theory was based on Biot’s poroelasticity, generalized to

encompass viscoelastic effects through the correspondence principle. Atkinson and

Craster (1996) studied a porous elastic material with an isotropic elastic response,

but highly anisotropic permeability. It was shown that for a crack with impermeable

faces and oriented to be parallel to the fluid flow direction, the highly anisotropic

results were good approximations to those found for a material with isotropic per-

meability. Chen (1996) presented a purely poroelastodynamic Boundary Element

Method formulation in the Laplace domain, and solved the transient response of a

long cylindrical cavity in a poroelastic medium under different boundary conditions

and the transient response of a fluid-saturated porous half-space to a sudden strip

load on the surface. De Campos and Neto (1996) implemented the direct Boundary


Element Method to analyze the poroelastic effects on two linear quasistatic elasticity

problems, namely the infinite plate with a circular hole and the thick walled cylinder.

Cui et al (1996) investigated the classical Mandel’s problem, which demonstrated

the existence of a nonmonotonic pressure response for a saturated porous medium

subject to constant external loading, with an extension to transversely isotropic case,

and explored the problem of an inclined borehole using the finite element equations

for incrementally nonlinear anisotropic poroelasticity. Ochs, Chen and Teufel (1997)

examined the flow-induced rock-stress responses to the transient fluid-pressure for

a well with a stationary vertical fracture within the framework of Biot’s theory of

poroelasticity.

Investigation of the diffusion model and other related models by using field data and

experimental measurements is important. Some field testing and laboratory work

using pulse pressure or harmonic signals have been done to estimate the permeabili-

ties and other properties of the porous media. It appears that harmonic testing may

be superior to conventional well testing in some cases. Crosnier, Fras and Jouanna

(1985) applied harmonic techniques to a physical laboratory model simulating frac-

tured media, and proposed a mathematical method to determine the parameters, such

as the number of sets of cracks and the different thickness of each set, from the spec-

tral signatures. The comparison between theory and laboratory tests showed a good

fit in the medium range of frequencies. Saeedi and Standen (1987) designed a layer

pulse test and implemented and analyzed it in a pinnacle reef in Alberta. Charlaix

et al. (1988a, 1991) did a series of experimental studies on hydrodynamic dispersion

in networks of capillaries. Charlaix et al. (1988b, 1992) used harmonic techniques in

experiments to study the dynamics of fluid flow in capillaries. The dynamic perme-

ability was measured at a frequency range of 0.1 Hz to 1 kHz. Gilicz (1991) applied

the pulse-decay technique for radial cores. Kamath, Boyer and Nakagawa (1992)

developed analytical methods and an experimental framework for obtaining and in-

terpreting the time dependence of pressure response to an initial disturbance, and

demonstrated that the effective permeability calculated from an interference pressure

transient test could be a function of the direction of the pressure disturbance, and


could differ significantly from the effective steady state value. Rigord, Caristan and

Hulin (1993) presented an experimental study and a model of the diffusion of sinu-

soidal pressure waves through porous media, and showed that measurements of the

hydraulic admittance in the sine wave mode allow us to probe the structure of porous

samples with an adjustable investigation depth depending on the frequency. Rigord,

Charlaix and Petit (1996) measured the dynamic viscosity and shear modulus on

suspension of non-Brownian hard spheres subjected to an oscillating flow in a tube

at the frequency range of 0.1 Hz to 80 Hz. Rashidi et al. (1996) studied chemical

flow and transport in an experimental porous medium. The microscopic medium

geometry, velocity and concentration field, dispersive solute fluxes and reasonable es-

timates of a representative elementary volume (REV) for the porous medium were

obtained. Chin and Proett (1997) invented the reservoir description tester (RDT)

and provided a method to determine geologic formation properties using phase shift

periodic pressure pulse testing. The spherical transient flow model used to evaluate

the test results was further discussed by Skinner et al. (1997) and Proett and Chin

(1998). Most of these laboratory studies were based on traditional theories of fluid

flow in porous media. Few experiments were performed to investigate approaches

other than the diffusion process.

Harmonic techniques have also been used in numerical studies. Ni (1988) analyzed the

one-dimensional nonlinear oscillations of an ideal gas in a pipe due to a periodically

varying pressure perturbation. Ganiev, Petrov and Ukrainskii (1990) examined the

steady-state oscillations of a simple model of a well face region, which is a straight

circular cylindrical channel filled with fluid. Oscillations were caused by harmonic

pressure perturbation at the end of the channel connected with the well face. The

model was based on Biot’s theory. The results showed that the principal resonant

frequency only depends weakly on parameters of the porous solid and strongly on

channel length. Rosa and Horne (1991) formulated a mathematical model for the

pressure behavior due to sinusoidal flow rates in reservoirs with continuously varying

radial permeability distributions. The solution was obtained by the application of


a regular perturbation analysis and a correlation between the radius of cyclic influ-

ence and the frequency of a sinusoidal flow rate. Only pressure diffusion behavior

was considered. Siginer and Valenzuela-Rendón (1995) investigated the unsteady

nonviscometric motion of a memory integral fluid of order three in a rigid, circular,

straight tube. The parametric studies of the longitudinal oscillatory velocity field

and the change in mass transport were presented for a range of liquids and driving

conditions. The results showed that elasticity of the fluid tends to increase the lon-

gitudinal steady velocity if driven by longitudinal waves, but elasticity of the fluid

tends to decrease the enhancement in the case of transversal waves. Bernabé (1997)

calculated the dynamic permeability of heterogeneous networks of cracks, tubes and

spheres by simulating the harmonic flow of an interstitial fluid for a wide range of

frequencies. The frequency dependence of the real and imaginary parts of the per-

meability showed a transition from viscous macroscopic flow at low frequencies to

inertial flow at high frequencies. Buschmann (1997) developed a two-stage numerical

method to represent harmonic response functions from creep and stress-relaxation

tests for linear viscoelastic materials.

The analysis of reservoir responses to earth tides is one of the early studies of fluid

and solid two-phase coupling effects on fluid flow in porous media. Arditty and

Ramey (1978) derived an expression for the pressure induced at the borehole by a

periodic tidal stress. Hemala and Balnaves (1986) provided an overview of analyt-

ical interpretation of tidal effects from a petroleum engineering point of view and

proposed the application of the phase shift of ocean tide effects to estimate reservoir

heterogeneities. Dean et al. (1991) introduced a method to monitor compaction and

compressibility changes in offshore chalk reservoirs by measuring formation pressure

variations caused by the sea tide. Pinilla et al. (1997) presented a model to describe

the oceanic tidal effects on an infinite reservoir by coupling geomechanic principles

with fluid flow equations in a deformable porous media. At the microscopic scale or

pore scale, the two-phase coupling effects are often studied by analyzing the fluid flow

in deformable tubes or small channels. Padmanabhan and Pedley (1987) analyzed the


steady streaming generated in an infinite elliptical tube containing a viscous, incom-

pressible fluid when the boundary oscillates in such a way that the area and ellipticity

of the cross-section vary with time but remain independent of the longitudinal coor-

dinate. Ganiev et al. (1986) analyzed the linear stability of viscous incompressible

flow in a circular viscoelastic tube and demonstrated that the Poiseuille flow in an

elastic tube with respect to infinitesimal axisymmetric perturbations can be unstable.

Ganiev et al. (1988a, 1988b) further considered disturbances with different azimuthal

wave numbers and small arbitrary three-dimensional perturbations and analyzed the

flow stability over a broad interval of values of the elasticity parameter. Ganiev et

al. (1989) studied the wave mechanism of the fluid motion acceleration in capillar-

ies and porous media and observed that small-scale pulsation of the pressure and

velocity may lead to the appearance of unidirectional flows with velocities substan-

tially exceeding the filtration velocities. Ganiev and Ukrainskii (1992) investigated

the bifurcation problem near the neutral curves constructed for Poiseuille flow in a

compliant pipe. The bifurcated self-excited oscillation modes were determined, and

the effects of Reynolds number, the compliance and internal viscosity of the pipe ma-

terial was analyzed. These studies assumed that the motions of the tube walls follow

the couple-stress theory of thin shells. In biomechanics, a similar phenomenon has

been studied by analyzing blood flow in elastic arteries. Hughes and Lubliner (1973)

developed a theory of one-dimensional flow through distensible tubes with perme-

able walls. Reuderink et al. (1989) evaluated the accuracy of nonlinear and linear

one-dimensional models in describing pulse wave propagation in a uniform cylindrical

viscoelastic tube at 1 Hz. The calculations were compared with experimental results.

Dutta et al. (1992) simulated numerically the oscillatory and pulsatile flows of New-

tonian fluids in straight elastic tubes, and indicated that the flow field and associated

wall shear stress are extremely sensitive to the phase angle between oscillatory pres-

sure and flow waves. Wang and Tarbell (1995) further considered the nonlinear effects

on the amplitude of the wall shear rate, on the amplitude of the pressure gradient,

and on the mean velocity profile, and developed a perturbation solution. There are

also studies on how the flow of body fluid influences the mechanical properties of

bones. Lim and Hong (1994) obtained a uniaxial poroelastic model to investigate the


mechanics of trabecular bone. Pore pressure and total stress in response to the given

strain input were predicted to vary as a function of intrinsic properties of trabecular

bone, such as permeability and Skempton’s coefficient. These results suggest that the

body fluid in the bone may play a significant role in enhancing the the mechanical

properties of trabecular bone. Mak et al. (1997) developed a finite element model to

study the contributions from various hierarchical flow channels in bone. Cortical bone

was modeled as a fully hydrated biphasic poroelastic material with a superposing net-

work of one-dimensional channels radiating from the Haversian canals. The model

was subjected to stress fields simulating uniform compression and pure bending. The

effects of the interfacial permeability and the solid content within the channels on the

drag forces in the channels were assessed. In solid mechanics, the effects of fluid mo-

tion to solid structure have been studied by analyzing the vibrations of fluid-saturated

poroelastic structures. Li et al. (1996, 1997) solved the problems of transverse vibra-

tions of a poroelastic beam with axial fluid diffusion and of a poroelastic plate with

fluid diffusion in the in-plane directions based on Biot’s theory and Darcy’s law. The

results showed that the resonant frequencies of the fluid-saturated structures increase

as the influence of fluid increases, and the amplitude response is significantly reduced

as the fluid influence increases. In petroleum engineering, the effect of elastic solid

motion on the permeability of saturated rock has been confirmed in numerous labo-

ratory experiments. Many observations in the field also show that seismic waves may

alter water and oil production. Beresnev (1994) reviewed the methods and results of

elastic-wave stimulation of oil production. Nycal Royalty Corporation (1997) investi-

gated the application of intermediate frequency waves for stimulation of immobile oil

stocks. The mechanism of this phenomenon needs further testing and understanding.

Homogenization techniques have been applied to acoustic studies concerning poroelas-

ticity, the mechanics of porous elastic solids with fluid-filled pores. These approaches

start with the detailed microstructure of the pores, the linearized equations of elastic-

ity, and the linearized equations of fluid dynamics. Burridge and Keller (1981) derived

equations which govern the linear macroscopic mechanical behavior of a porous elas-

tic solid saturated with a compressible viscous fluid. Auriault, Borne and Chambon


(1985) obtained a formulation similar to Biot’s results, and presented some proper-

ties of the generalized Darcy coefficient and an experimental checking. Kazi-Aoual,

Bonnet and Jouanna (1988) applied homogenization techniques to study Green’s func-

tions in an infinite, transversely isotropic, saturated, poroelastic medium for harmonic

displacements. Zobnin, Kudryavtsev and Parton (1988) developed an equation de-

scribing the motion of a viscous fluid in a rigid porous medium of periodic structure.

The procedure for averaging linearized hydrodynamic equations with small viscosity

coefficients was used. Auriault (1991) proved that the homogenization process is the

appropriate method to determine whether an equivalent macroscopic description ex-

ists for a finely heterogeneous medium submitted to some excitation. Auriault and

Boutin (1992, 1993, 1994) investigated the macroscopic quasistatic description of a

deformable porous medium with a double porosity constituted by pores and fractures.

It appeared that the macroscopic description is sensitive to the ratios between the

characteristic lengths of the pores, the fractures and the macroscopic medium. Ra-

soloarijaona and Auriault (1994) studied the nonlinear seepage of a fluid through a

porous medium by using the homogenization theory for very small Reynolds numbers

and performed an experimental and a numerical checking of the results. Norris and

Grinfeld (1995) derived explicit motion equations for a medium consisting of alter-

nating solid and fluid layers. The wave solutions were discussed and compared with

other studies. Auriault and Lewandowska (1996) developed a macroscopic model for

pollutant transport in a heterogeneous medium, including diffusion, advection and

adsorption, and discussed the issue of the existence of nonhomogenizable problems.

Lee et al. (1996) computed numerically the permeability and dispersivities of solute

and heat for a periodic porous medium with geometry consisting of a cubic array of

uniform Wigner-Seitz shape grains. Lenoach (1998) studied wave propagation in a

random poroelastic medium and derived an upscaled one-dimensional representation

of a reservoir with a heterogeneous fluid distribution consisting of several pore fluids

with very different constitutive parameters. Some researchers studied in detail the

microscopic problems for periodic structures. Saeger et al. (1995) solved numerically

the Stokes equation system and Ohm’s law for fluid in periodic bicontinuous porous

media of simple cubic, body-centered cubic and face-centered cubic symmetry. The


dependence of Darcy permeability and conductivity on porosity was investigated.

Skjetne (1995) presented analytical and numerical studies on high-velocity flow in

spatially periodic media. The results showed that the most pronounced inertial ef-

fect is the formation of narrow straight flow tubes impacting with the pore walls at

obstructions. Lee and Mei (1997) derived nonlinear effective poroelastic equations

at the macroscale applying the method of homogenization. The matrix displacement

corresponding to the global strain was assumed to be comparable to the granular

size. The constitutive coefficients of the nonlinear terms were analyzed. The result

suggests that the linear effective equations may be adequate even for practical prob-

lems involving moderate deformation or loading, as long as the microscale geometry

is isotropic in the statistical sense.

The mathematical models introduced previously provide some possible approaches

to improve the description of fluid flow in porous media. The idea of applying har-

monic signals in well testing or as a possible new method for enhanced oil recovery

appears promising, but needs yet to be fully understood and further to be imple-

mented in reservoir characterization and production control. This study focuses on

exploring the theoretical models generated from the homogenization process in har-

monic testing to provide accurate description and characterization of oil, gas and

groundwater reservoirs.

1.3 Outline of Approach

This study applies homogenization techniques to investigate the two-scale problems

in reservoirs due to harmonic perturbations. The objectives are:

• To study the sensitivity of pore structure parameters and the properties of fluidand solid to harmonic signal frequencies. This provides a method to choose

appropriate frequency ranges for diagnosis of different properties.

• To investigate the mechanism of acceleration of fluid flow due to harmonic per-turbation in pore geometry. If we know the compatible porous media conditions


and the effective frequency range, a harmonic stimulation could be performed

to increase production in reservoirs.

• To analyze different macroscopic behavior of porous media corresponding tocertain parameter scales. It is important to apply the appropriate macroscopic

model to interpret the reservoir responses in well testing, seismic data processing

and other kinds of measurements.

• To obtain effective macroscopic parameters for the purposes of reservoir charac-terization and large scale simulation, which may lead to the prediction of future

production.

A porous medium with periodic pore structure (Figure 1.1) is investigated in this

study. The ratio of the period to the overall size of the porous medium, denoted by

�, is the small parameter of asymptotic analysis in the procedure because the pore

size is usually much smaller than the characteristic length of the reservoir. The wave

length λ of the excitation signal is associated with the characteristic length L at the

macroscopic scale.

��

��

��

��

λ ~ L

L

l

~ l / Lε

P

S

Figure 1.1: A Periodic Porous Medium

This approach follows the procedure of homogenization (Figure 1.2). The input of

the system includes the unit cell geometry, two length scales, the excitation signal

frequency, the pressure gradient and the basic properties of the fluid and the solid in

the microscopic structure. The output will be the macroscopic behavior of the porous

medium which includes the distributions of solid displacement, fluid relative velocity


and pore pressure.

The principal steps are as follows:

1. For a given microscopic geometry, solve the periodic unit cell problem under

harmonic perturbation to obtain the solid displacement field, the fluid velocity

field and the pressure distribution in the unit cell.

2. Perform volume averaging in the unit cell to calculate all the effective coefficients

that may appear in the macroscopic equations.

3. Study the sensitivity of pore scale parameters and the effective macroscopic

coefficients to the excitation frequencies.

4. According to the specific physical problems, evaluate the scale parameter � based

on the periodic unit cell geometry and the macroscopic scale, and calculate the

value of the contrast number C = µω/a, where µ is fluid viscosity, ω is signal

frequency, and a is the scale of pore pressure and solid stress.

5. By asymptotic analysis, determine whether an equivalent macroscopic descrip-

tion exists for the relation between � and C evaluated in the previous step. If

an intrinsic description is not possible, using an average theorem will lead to

a large-scale description depending directly upon the external boundary condi-

tions of the medium.

6. In homogenizable situations, the macroscopic equations independent of external

boundary conditions can be obtained. There are different macroscopic models

corresponding to the scale of the contrast number.

7. Solve the macroscopic equations to describe the large-scale behavior of the

porous medium.

The review of literature reveals some important and interesting areas that were poorly

developed and hence are targets of this study. These areas include:


, ,

K(ω)

fluid and solid propertiesperturbation: ω

Evaluation of

dp,

l, Lpore geometry,

Description ofMacroscopic Behavior

Model II

StudySensitivity

ModelMacroscopicSelection of

YesCoefficients

EffectiveCalculation of

Exists ?Model

MacroscopicIntrinsic

ProblemMacroscopic

ParticularSolving

NoAveragingVolume

Asymptotic Analysis

Cell ProblemSolving Unit

C, εScale Parameters

Figure 1.2: Homogenization Procedure


• At the microscopic scale, the fluid flow in channels of smooth trajectory (notnecessarily straight) and of smooth width variation. The channel walls (the

solid part) are elastic and are of random thickness.

• Study of sensitivity of porous medium properties to harmonic signal frequencies.

• The mechanism of alternation of fluid flow rate due to harmonic perturbation.

• Diagnosis of effective parameters from the porous medium responses to har-monic excitation.

Chapter 2

Homogenization Techniques

The homogenization theory was first developed in mathematics to deal with partial

differential equations with rapidly oscillating coefficients. Many papers have been

devoted to the derivation of Darcy’s law by homogenization using asymptotic ex-

pansions (Sanchez-Palencia 1980, Lions 1981). In recent years, because of the rapid

mathematical development in this area, many basic phenomena are now far better

understood. Several nonstandard models were developed including Darcy’s law with

memory and nonlinear Darcy’s law. A survey on the method of homogenization ap-

plied to problems in porous media was presented in the book edited by Hornung

(1997). This chapter focuses on the derivation of some linear macroscopic laws for

poroelastic porous media following the steps contributed by Auriault in Hornung’s

book, but in much more detail. The mathematical convergence proofs can be found

in the references, and will not be reproduced here. The purpose of this chapter is to

introduce the general ideas of homogenization to people not very familiar with the

techniques. Our nonlinear approach will be discussed in Chapter 3.

Assumptions

To derive the macroscopic quasistatic behavior, classically, the inertial terms are

neglected. The material of the porous matrix is assumed elastic and the saturating

fluid is assumed viscous, Newtonian and incompressible. The local pore structure

16

CHAPTER 2. HOMOGENIZATION TECHNIQUES 17

is assumed periodic with a period Y of characteristic length l (Figure 1.1). The

characteristic length L at the macroscopic scale is assumed to be associated with the

excitation wave length λ: L = λ/2π. The two characteristic lengths introduce two

dimensionless space variables:

y =X

l

x =X

L

X is the physical space variable. We denote the solid part of the period S, the pores

P , and the interface Γ (Figure 2.1). An important small parameter of scale separation

is defined as:

� =l

L� 1

and

x = �y

Different types of macroscopic behavior of the saturated porous medium are studied

in relation to the value of a dimensionless number as a function of the small parameter

� in the following sections.

ΓS

P

Figure 2.1: Periodic Unit Cell of the Porous Media


2.1 Acoustics of an Empty Porous Medium

For a porous medium filled with a very compressible and low viscosity fluid, the fluid

can be ignored. This represents the most basic problem to be used in all the following

investigations.

2.1.1 Local Description

The medium satisfies the Navier equation in the solid part.

∑j

∂σsij∂Xj

= ρs∂2usi∂t2

in S (2.1)

and the stress-strain relation is:

σsij = aijklekl(us) in S (2.2)

On the solid surface Γ, the normal stress is zero due to the pore emptiness.

σsijνj = 0 on Γ (2.3)

νj is the unit normal vector of the surface pointing outside of the solid.

The stress tensor can be resolved into normal components σii and shear components

σij . For shear components, the first suffix denotes the direction of the stress, and the

second suffix denotes the normal to the plane on which the component is acting. For

normal stresses, only one suffix is needed since the direction of component is the same

as that of the normal to the surface. The elastic tensor a represents the stiffness of

the material. It is a periodic function of y, which satisfies:

aijkl = ajikl = aijlk = aklij (2.4)

and the ellipticity condition:

aijkleijekl ≥ γeijeij , γ > 0. (2.5)


The strain tensor is related to the solid displacement by:

eij(u) =1

2(∂ui∂Xj

+∂uj∂Xi

) (2.6)

Equation 2.1 introduces a dimensionless number P :

P =|ρs ∂2usi∂t2 ||∂σsij∂Xj

| (2.7)

We consider a wave of pulsation ω with a wave length at the order of the macroscopic

scale L, and we use the pore scale l as the characteristic length to make the equations

dimensionless. We assume that all the components of elastic tensor a are of the same

order of magnitude a, and the strain tensor e is small as the order of O(l/l) = O(1),

then,

|∂σsij∂Xj

| ∼ al

(2.8)

|ρs∂2usi∂t2

| ∼ ρω2l (2.9)

We notice that the propagation velocity of a compression wave (P-wave) can be

written as:

vp =

√a

ρ∼ ωL = ω λ

2π(2.10)

then,

Pl = O(ρω2l2

a) = O(

l2

L2) = O(�2) (2.11)

We define the dimensionless variables as:

yj =Xjl, u∗si =

usil, σ∗sij =

σsija, ω∗ =

ωL√a/ρ

, ρ∗s =ρsρ

(2.12)

then,

∂σsij∂Xj

=∂σ∗sij∂yj

(a

l)

ρs∂2usi∂t2

= −ω2ρsusi = −ω∗2ρ∗su∗si(al

L2)


Finally, the dimensionless form of the local description is derived. The superscript *

is omitted without confusion. The subscript y shows operators with respect to the

space variable y.

∑j

∂σsij∂yj

= −�2ω2ρsusi in S (2.13)

σsij = aijkleykl(us) in S (2.14)

σsijνj = 0 on Γ (2.15)

2.1.2 Macroscopic Description

We consider two-scale problems and we write the unknown us in the form:

usi(x, y) = u(0)i (x, y) + �u

(1)i + �

2u(2)i (x, y) + · · · (2.16)

with x = �y. We substitute this expansion into Equations 2.13 - 2.15, and use the

following differentiation rule:

d

dy=

∂

∂y+ �

∂

∂x(2.17)

We look at the left hand side and the right hand side of Equation 2.13 respectively.

d

dyj(∂uk∂yl

) = (∂

∂yj+ �

∂

∂xj)[∂u

(0)k

∂yl+ �(

∂u(1)k

∂yl+∂u

(0)k

∂xl) + �2(

∂u(2)k

∂yl+∂u

(1)k

∂xl) + · · ·]

=∂2u

(0)k

∂yj∂yl+ �[

∂

∂yj(∂u

(1)k

∂yl+∂u

(0)k

∂xl) +

∂2u(0)k

∂xj∂yl]

+ �2[∂

∂yj(∂u

(2)k

∂yl+∂u

(1)k

∂xl) +

∂

∂xj(∂u

(1)k

∂yl+∂u

(0)k

∂xl)] +O(�3)

dekldyj

=∂e

(0)ykl

∂yj+ �[

∂

∂yj(e

(1)ykl + e

(0)xkl) +

∂e(0)ykl

∂xj]

+ �2[∂

∂yj(e

(2)ykl + e

(1)xkl) +

∂

∂xj(e

(1)ykl + e

(0)xkl)] +O(�

3)

then,

L.H.S. =∂

∂yj(aijkle

(0)ykl) + �{

∂

∂yj[aijkl(e

(1)ykl + e

(0)xkl)] +

∂

∂xj(aijkle

(0)ykl)}


+ �2{ ∂∂yj

[aijkl(e(2)ykl + e

(1)xkl)] +

∂

∂xj[aijkl(e

(1)ykl + e

(0)xkl)]}+O(�3) (2.18)

R.H.S. = −�2ρsω2∞∑n=0

(�nu(n)i ) = −ρsω2

∞∑n=0

(�n+2u(n)i ) (2.19)

The boundary-value problem for the solid displacement at the order of �0 becomes:

∑j

∂

∂yj[aijkleykl(u

(0))] = 0 in S (2.20)

aijkleykl(u(0))νj = 0 on Γ (2.21)

Because a and u(0) are Y periodic in y, the solution is an arbitrary function of x,

independent of y.

u(0) = u(0)(x) (2.22)

The boundary-value problem for the solid displacement at the order of �1 becomes:

∑j

∂

∂yj[aijkl(eykl(u

(1)) + exkl(u(0))] = 0 in S (2.23)

aijkl[eykl(u(1)) + exkl(u

(0))]νj = 0 on Γ (2.24)

Since u(1) is Y periodic in y, it can be expressed as a linear vectorial function of

exkl(u(0)) to an arbitrary vector ū

(1)i (x) independent of y.

u(1)i = ξ

khi (y)exkh(u

(0)) + ū(1)i (x) (2.25)

where the third order tensor ξlmi (y) is taken of zero average to insure the uniqueness.∫S

ξ dY = 0 (2.26)

Finally, the boundary-value problems for u(2) is obtained from the order of �2.

∑j

(∂σ

(1)sij

∂yj+∂σ

(0)sij

∂xj) = −ρsω2u(0)i in S (2.27)

σ(1)sijνj = 0 on Γ (2.28)


where σ(0)s and σ(1)s are the zero- and first-order terms respectively in the asymptotic

expansion of σs.

σ(0)sij = aijkh(eykh(u

(1)) + exkh(u(0))) (2.29)


(2)) + exkh(u(1))) (2.30)

Equation 2.27 is a balance equation for the periodic quantity σ(1)s with the source

terms −∂σsij/∂xj and −ρsω2u(0)i . The compatibility condition for the existence ofu(2) can be obtained by integrating Equation 2.27 in S and using the boundary

condition 2.28 and the periodicity of σ(1).

∑j

∂〈σ(0)sij 〉s∂xj

= −ρeffs ω2u(0)i (2.31)

and ρeffs = 〈ρs〉s

The volume average is defined by:

〈•〉s = |Y |−1∫S• dY (2.32)

From Equation 2.25, we can get:

eylm(u(1)) = eylm(ξ

kh)exkh(u(0))

σ(0)sij = aijkhexkh(u

(0)) + aijlmeylm(ξkh)exkh(u

(0)) (2.33)

To obtain the macroscopic description, define the total stress by:

σT =

σs in S0 in P (2.34)

and 〈σTij〉s = 〈σ(0)ij 〉s = aeffijkhexkh(u(0)) (2.35)with aeffijkh = 〈aijkh + aijlmeylm(ξkh)〉s (2.36)

then,

∑j

∂〈σTij〉s∂xj

= −ρeffs ω2usi (2.37)


where usi = u(0)si .

Finally, by returning Equation 2.37 to dimensional quantities, the macroscopic de-

scription (xj = Xj/L, u ∼ L) at the first order of approximation is obtained. Wewill refer to this as Model I.

∑j

∂〈σTij〉s∂Xj

= ρeffs∂2usi∂t2

+Or(�) (2.38)

and 〈σTij〉s = aeffijkhekh(us) +Or(�) (2.39)It can be shown that aeff is an elastic tensor. Therefore, the set of Equations 2.38

and 2.39 is a classical Navier description. The symbolOr(�) shows that the description

is an approximation at the relative order �.

2.2 A Prior Estimate for Saturated Porous Media

To anticipate the different macroscopic description of the acoustics of saturated porous

media, consider the volume balance of such media at the macroscopic scale assuming

that the material constituting the porous matrix is incompressible. Then,

∂θ

∂t= −∂vri

∂Xi(2.40)

where vr is the fluid velocity relative to the matrix and θ is the volume change

percentage of the matrix. This introduces the Strouhal number:

Sh =|∂θ∂t|

|∂vri∂Xi

| (2.41)

Obviously Sh = O(1) corresponds to diphasic behavior with distinguishable macro-

scopic displacements of the matrix and the fluid. For Sh ≤ O(�) the matrix dis-placement is negligible compared to the fluid displacement. When Sh ≥ O(�−1) thefluid has a relative small velocity compared to the matrix velocity. The macroscopic

behavior is monophasic.

Xj ∼ L, u ∼ L, ∂u∂t

∼ ωL (2.42)

|∂θ∂t| ∼ 1

L3∂u3

∂t∼ uω

L, |∂vri

∂Xi| ∼ vr

L(2.43)


then,

Sh = O(uω

vr) (2.44)

This expression of the Strouhal number can be generalized as follows. The displace-

ment in the solid is related to the solid stress by:

u = O(σL

a) (2.45)

The relative fluid velocity is governed by a Darcy-like relation,

vri = −kij ∂p∂Xj

(2.46)

The filtration tensor is of the order:

k = O(l2

µ) (2.47)

and µ is the fluid viscosity. From relations 2.46 and 2.47,

vr = O(l2p

µL) (2.48)

Then the Strouhal number becomes:

Sh = O(L2

l2σ

p

µω

a) (2.49)

Assuming that the stress in the matrix is at the same order as the fluid pressure at

the macroscopic scale, the Strouhal number is simplified to:

Sh = O(�−2C), where C =µω

a. (2.50)

C is the property contrast number. Actually it is the ratio of the characteristic

diffusion velocity in the fluid and the wave propagation velocity in the solid.

C =µω

a=

(µ/ρω)ω2

a/ρ=

(δω)2

v2p=v2Dv2p

(2.51)

where δ =√

µρω

is the viscous skin length. This characteristic length represents the

diffusion of the wave imposed on the fluid by an oscillating impermeable boundary

wall.

From the physical definition of the Strouhal number, the following cases will be in-

vestigated.


• Model II, C = O(�3): acoustics of a fluid in a rigid porous matrix

• Model III, C = O(�2): diphasic macroscopic behavior

• Model IV, C = O(�): monophasic elastic macroscopic behavior

• Model V, C = O(1): monophasic viscoelastic macroscopic behavior

As the fluid viscosity or the perturbation frequency increases, the scale of C increases,

and the macroscopic behavior of a saturated porous medium changes from fluid mo-

tion dominated to elastic solid motion dominated.

2.3 Local Description of Saturated Porous Media

When the matrix is not rigid (Models III, IV and V), the solid part satisfies the Navier

equation and the the fluid part satisfies the Navier-Stokes equation. The interface

between the fluid and solid satisfies the continuity equations of the normal stress and

the displacement.

∑j

∂σsij∂Xj

= ρs∂2usi∂t2

in S (2.52)

σsij = aijklekl(us) in S (2.53)∑j

∂σpij∂Xj

= ρp(∂vi∂t

+∑j

(vj∂

∂Xj)vi) in P (2.54)

σpij = 2µDij − pIij in P (2.55)∑i

∂vi∂Xi

= 0 in P (2.56)

usi = upi on Γ (2.57)

σsijνj = σpijνj on Γ (2.58)

For compressible Newtonian fluid, the tensor Dij is defined as:

2µD = µ(∇v + (∇v)T ) + η(∇ · v)I (2.59)∇ · (2µD) = µ∇2v + (η + µ)∇(∇ · v) (2.60)


where µ is the fluid shear viscosity coefficient and η is the fluid bulk viscosity coeffi-

cient. For incompressible fluid (∇ · v = 0), the tensor becomes:

Dij =1

2(∂vi∂Xj

+∂vj∂Xi

) (2.61)

Here we consider the fluid and solid movements at constant pulsation. To neglect

the nonlinear terms in Equation 2.54, we assume that the amplitude of the pertur-

bation is sufficiently small. For simplicity, we restrict the problems to the following

assumptions:

• The displacements and densities are of the same order of magnitude in the solidand the fluid parts respectively.

|us| ∼ |up|, ρs ∼ ρp (2.62)

• At the macroscopic scale (estimated with the characteristic length L), The pres-sure in the fluid and the stress in the solid are of the same order of magnitude.

Σ = |σsp|, ΣL = au

pL= O(1), and Σl =

au

pl= O(�−1) (2.63)

As shown in Section 2.1.1, the acoustic behavior of the matrix at the pore scale

(X ∼ l) is governed by the ratio:

Pl =|ρsω2usi||∂σsij∂Xj

| = O(ρω2l2

a) = O(�2) (2.64)

Therefore, for the fluid at the pore scale,

Tl =|ρp ∂vi∂t || ∂p∂Xi| = O(

ρω2u

p/l) = O(

ρω2ul

au/L) = O(�) (2.65)

From the above estimates, there is only one free dimensionless number that concerns

the acoustic behavior of the fluid.

Ql =|µ ∂2vi

∂Xj∂Xj|

| ∂p∂Xi| = O(

µωu

pl) = O(�−1

µω

a) (2.66)

The value of Ql is directly related to the property contrast number C = µω/a. Let

Ql = O(�−r), then following the discussion in Section 2.2 leads to the evaluation of

Ql for macroscopic Models III, IV and V.


• Model III, r = −1: diphasic macroscopic behavior

• Model IV, r = 0: monophasic macroscopic elastic behavior

• Model V, r = 1: monophasic macroscopic viscoelastic behaviorDefine the following dimensionless variables:

yj =Xjl, ω∗ = ωL√

a/ρ, (2.67)

ρ∗s =ρsρ, u∗si =

usil, σ∗sij =

σsija

ρ∗p =ρpρ, v∗i =

vi

�√a/ρ, σ∗pij =

σpija�

p∗ =p

a�, µ∗ = µ

�−rl√aρ , k∗ =

k

µ

�−r√aρ

l

Finally, the dimensionless equations for pore scale local description of a saturated

porous medium are obtained (* is omitted).

∑j

∂σsij∂yj

= −�2ρsω2usi in S (2.68)

σsij = aijkleykl(us) in S (2.69)∑j

∂σpij∂yj

= �ρpiωvi in P (2.70)

σpij = 2µ�−rDyij(v)− pIij in P (2.71)∑

i

∂vi∂yi

= 0 in P (2.72)

usi = upi on Γ (2.73)

(σsij − �σpij)νj = 0 on Γ (2.74)

2.4 Acoustics of a Fluid in a Rigid Porous Medium

When the matrix can be considered rigid (Model II, r = −1), the fluid is in motionrelative to the matrix. The dimensionless equations of fluid motion are those of Model

III, with the interface continuity condition 2.73 reduced to the adherence condition.

∑j

∂σpij∂yj

= �ρpiωvi in P (2.75)


σpij = 2µ�Dyij(v)− pIij in P (2.76)∑i

∂vi∂yi

= 0 in P (2.77)

vi = 0 on Γ (2.78)

Introduce the asymptotic expansions for p and v in the form:

p(x, y) = p(0)(x, y) + �p(1)(x, y) + �2p(2)(x, y) + · · · (2.79)vi(x, y) = v

(0)i (x, y) + �v

(1)i (x, y) + �

2v(2)i (x, y) + · · · (2.80)

with x = �y. The p(i)’s and the v(i)’s are Y -periodic in y. From Equations 2.75

and 2.76, we obtain:

∑j

{µ�[ ∂∂yj

(∂vi∂yj

+∂vj∂yi

)]} − ∂p∂yi

= �ρpiωvi (2.81)

Noticing that d/dy = ∂/∂y + �∂/∂x, Equation 2.81 becomes:

∑j

{ µ�[ ∂∂yj

(∂v

(0)i

∂yj+∂v

(0)j

∂yi) + · · ·]} − [∂p

(0)

∂yi+ �(

∂p(1)

∂yi+∂p(0)

∂xi) + · · ·]

= ρpiω�(v(0)i + �v

(1)i + · · ·) (2.82)

From Equation 2.82 at the order of �0, we obtain:

∂p(0)

∂yi= 0 (2.83)

then p(0) = p(0)(x) (2.84)

Equation 2.82 at the order of �1 and Equations 2.77 - 2.78 at the order of �0 are:

∑j

µ∂2v

(0)i

∂y2j− ∂p

(0)

∂xi− ∂p

(1)

∂yi= ρpiωv

(0)i in P (2.85)

∑i

∂v(0)i

∂yi= 0 in P (2.86)

v(0)i = 0 on Γ (2.87)

Because of the linearity of Equations 2.85 - 2.87, v(0) can be written in the form:

v(0)i = −kij

∂p(0)

∂xj(2.88)


where the tensor kij(x, y, ω) depends on ω and is complex-valued. Finally from Equa-

tions 2.77 and 2.78 at the order of �1, we obtain:

∑i

(∂v

(1)i

∂yi+∂v

(0)i

∂xi) = 0 in P (2.89)

v(1)i = 0 on Γ (2.90)

Performing a volume average on P with respect to the variable y on Equation 2.89

and applying the divergence theorem and the boundary condition 2.90, we can obtain

the compatibility condition.

∑i

∂〈v(0)i 〉p∂xi

= 0 (2.91)

〈v(0)i 〉p = −keffij∂p(0)

∂xj(2.92)

and keffij = 〈kij〉p (2.93)

where 〈•〉p is the volume average on P .

〈•〉p = |Y |−1∫P• dY (2.94)

Equation 2.92 represents a monochromatic seepage law with a complex-valued and

ω-dependent filtration tensor keff . Returning to physical quantities, Model II is

expressed by:

∑i

∂〈vi〉p∂xi

= Or(�) (2.95)

〈vi〉p = −keffij∂p

∂xj+Or(�) (2.96)

and keffij = 〈kij〉p (2.97)

2.5 Diphasic Macroscopic Behavior

The diphasic macroscopic behavior is described by Equations 2.68 - 2.74 with r = −1.We search for the solutions of the unknowns us, v and p in the asymtotic expansion


form:

usi(x, y) = u(0)i (x, y) + �u

(1)i (x, y) + �

2u(2)i (x, y) + · · · (2.98)

vi(x, y) = v(0)i (x, y) + �v

(1)i (x, y) + �

2v(2)i (x, y) + · · · (2.99)

p(x, y) = p(0)(x, y) + �p(1)(x, y) + �2p(2)(x, y) + · · · (2.100)with x = �y. u(i),v(i) and p(i) are Y -periodic in y. Introducing expansions 2.98 - 2.100

into Equations 2.68 - 2.74, and applying the relation d/dy = ∂/∂y+�∂/∂x, we obtain:

∑j

{ ∂∂yj

(aijkle(0)ykl) + �[

∂

∂yj(aijkl(e

(1)ykl + e

(0)xkl)) +

∂

∂xj(aijkle

(0)ykl)]

+ �2[∂

∂yj(aijkl(e

(2)ykl + e

(1)xkl)) +

∂

∂xj(aijkl(e

(1)ykl + e

(0)xkl))] + · · ·}

= −ρsω2�2(u(0)i + �u(1)i + · · ·) in S (2.101)∑j

{ µ�−r[ ∂∂yj

(∂v

(0)i

∂yj+∂v

(0)j

∂yi)

+ �(∂

∂yj((∂v

(1)i

∂yj+∂v

(1)j

∂yi) + (

∂v(0)i

∂xj+∂v

(0)j

∂xi)) +

∂

∂xj(∂v

(0)i

∂yj+∂v

(0)j

∂yi))

+ �2(∂

∂yj((∂v

(2)i

∂yj+∂v

(2)j

∂yi) + (

∂v(1)i

∂xj+∂v

(1)j

∂xi))

+∂

∂xj((∂v

(1)i

∂yj+∂v

(1)j

∂yi) + (

∂v(0)i

∂xj+∂v

(0)j

∂xi))) + · · ·] }

− [∂p(0)

∂yi+ �(

∂p(1)

∂yi+∂p(0)

∂xi) + · · ·]

= ρpiω�(v(0)i + �v

(1)i + · · ·) in P (2.102)∑

i

[∂v

(0)i

∂yi+ �(

∂v(1)i

∂yi+∂v

(0)i

∂xi) + · · ·] = 0 in P (2.103)

iω ( u(0)si + �u

(1)si + · · ·) = v(0)i + �v(1)i + · · · on Γ (2.104)

{aijkle(0)ykl + �[aijkl(e(1)ykl + e(0)xkl)] + �2[aijkl(e(2)ykl + e(1)xkl)] + · · ·}νj

= {µ�1−r[(∂v(0)i

∂yj+∂v

(0)j

∂yi) + �((

∂v(1)i

∂yj+∂v

(1)j

∂yi) + (

∂v(0)i

∂xj+∂v

(0)j

∂xi))

+ �2((∂v

(2)i

∂yj+∂v

(2)j

∂yi) + (

∂v(1)i

∂xj+∂v

(1)j

∂xi)) + · · ·]

− �(p(0) + �p(1) + · · ·)}νi on Γ (2.105)


Equations 2.101 - 2.105 are used to solve the boundary-value problems for Models

III, IV and V with different values of r. Here we consider the case when r = −1corresponding to the diphasic behavior of the matrix and the fluid.

From the equations at the order of �0, we can obtain u(0) and p(0) similarly to those

in Sections 2.1 and 2.4 respectively.

u(0) = u(0)(x) (2.106)

p(0) = p(0)(x) (2.107)

From Equations 2.101 and 2.105 at the order of �1, the boundary-value problem for

u(1) is obtained.

∑j

∂∂yj

[aijkh(eykh(u(1)) + exkh(u

(0)))] = 0 in S (2.108)

aijkh(eykh(u(1)) + exkh(u

(0)))νj = −p(0)νi on Γ (2.109)where u(1) is Y periodic in y. When p(0) = 0, this set of equations is identical to

Equations 2.23 - 2.24 for empty pores. Because of the linearity, u(1) can be expressed

in the form:

u(1)i = ξ

lmi exlm(u

(0))− ηip(0) + ū(1)i (x) (2.110)where ξ(y) is a third order tensor introduced in Section 2.1 and η(y) is a vector. Both

are zero averaged. ∫S

ξ dY = 0,∫S

η dY = 0 (2.111)

From Equation 2.110 we can derive that:

eylm(u(1)) = eylm(ξ

kh)exkh(u(0))− eylm(η)p(0)


(1)) + exkh(u(0))) (2.112)

= (aijkh + aijlmeylm(ξkh))exkh(u

(0))− aijkheykh(η)p(0)

The problem for p(1) and v(0) can be obtained from Equation 2.102 at the order of �1

and Equations 2.103 and 2.104 at the order of �0.

∑j

µ∂2v

(0)i

∂y2j− ∂p

(0)

∂xi− ∂p

(1)

∂yi= ρpiωv

(0)i in P (2.113)


∑i

∂v(0)i

∂yi= 0 in P (2.114)

v(0)i = iωu

(0)i on Γ (2.115)

v(0)i and p

(1) are Y -periodic with respect to y. To solve this problem, set wi =

v(0)i − u̇(0)i , then,

∑j

µ∂2wi∂y2j

− ∂p(0)

∂xi− ∂p

(1)

∂yi= ρpiωwi − ρpω2u(0)i in P (2.116)

∑i

∂wi∂yi

= 0 in P (2.117)

wi = 0 on Γ (2.118)

wi and p(1) are Y -periodic in y. This is the linear boundary-value problem in Sec-

tion 2.4 with ∂p(0)

∂xireplaced by ∂p

(0)

∂xi− ρpω2u(0)i . Therefore wi can be expressed as:

wi = v(0)i − iωu(0)i = −kij(

∂p(0)

∂xj− ρpω2u(0)i ) (2.119)

where k is defined in Section 2.4 as the complex-valued permeability tensor. Inte-

grating Equation 2.119 over P , we obtain:

∫Pv

(0)i dY − iωu(0)i |YP | = −

∫PkijdY · (∂p

(0)

∂xj− ρpω2u(0)i ) (2.120)

Further averaging Equation 2.120 in the whole domain (|Y | = |YP |+ |YS|), we obtain:

〈v(0)i 〉p − φiωu(0)i = −keffij (∂p(0)

∂xj− ρpω2u(0)i ) (2.121)

where 〈v(0)i 〉p = |Y |−1∫Pv

(0)i dY (2.122)

keffij = 〈kij〉p = |Y |−1∫PkijdY (2.123)

and φ = |Yp|/|Y | is the porosity.

Two compatibility conditions are required for the existence of solutions of this prob-

lem. One compatibility condition is obtained from Equations 2.101, 2.105 at the order


of �2 and Equation 2.102 at the order of �1.

∑j

(∂σ

(2)sij

∂yj+∂σ

(1)sij

∂xj) = −ρsω2u(0)i in S (2.124)

∑j

∂σ(1)pij

∂yj− ∂p

(0)

∂xi= ρpiωv

(0)i in P (2.125)

σ(2)sijνj = σ

(1)pijνj on Γ (2.126)

where


(2)) + exkh(u(1))) (2.127)


(1)) + exkh(u(0))) (2.128)

σ(1)pij = µ(

∂v(0)i

∂yj+∂v

(0)j

∂yi)− p(1)Iij (2.129)

The total stress can be defined in the whole domain as:

σT =

σs in Sσp in P (2.130)

Integrating Equation 2.124 over S, Equation 2.125 over P , and applying the diver-

gence theorem together with the periodicity condition, we obtain:∮Γsσ(2)s · dΓ +∇x · (

∫Sσ(1)s dY ) = −ω2u(0)

∫SρsdY (2.131)∮

Γpσ(1)p · dΓ−∇xp(0)|YP | = iωρp

∫P

v(0)dY (2.132)

Noticing that dΓs = −dΓp and applying the interface boundary condition 2.126, weobtain:

∇x · (∫Sσ(1)s dY )−∇xp(0)|YP | = −ω2u(0)

∫SρsdY + iωρp

∫P

v(0)dY (2.133)

Define the volume average in the whole domain as:

〈•〉s = |Y |−1∫S•dY and 〈•〉p = |Y |−1

∫P•dY (2.134)

Then we derive:

∇x · 〈σ(1)s 〉s − φ∇xp(0) = −〈ρs〉sω2u(0) + iωρp〈v(0)〉p (2.135)


where φ = |Yp|/|Y | is the porosity. Taking into account the solid stress-strain rela-tion 2.112, Equation 2.135 can be rewritten as:

∑j

∂〈σT (0)ij 〉∂xj

= −ρeffs ω2u(0)i + ρpiω〈v(0)i 〉p (2.136)

〈σT (0)ij 〉 = aeffijkhexkh(u(0))− αeffij p(0) (2.137)

where the effective coefficients are defined by:

ρeffs = 〈ρs〉s (2.138)aeffijkh = 〈aijkh + aijlmeylm(ξkh)〉s (2.139)αeffij = φIij + 〈aijkheykh(η)〉s = αeffji (2.140)

When the medium is isotropic, it can be verified that the value of the effective tensor

αij has the following relation.

φ < α < 1. (2.141)

The second compatibility condition can be obtained from the incompressibility, Equa-

tion 2.103 at the order of �1.

∑i

(∂v

(1)i

∂yi+∂v

(0)i

∂xi) = 0 (2.142)

Integrating Equation 2.142 over P and applying the divergence theorem with the

periodicity and the interface continuity condition 2.104, we obtain:

∇x ·∫P

v(0)dY = −∮Γp

v(1) · dΓ = iω∮Γs

u(1) · dΓ = iω∫S∇y · u(1)dY (2.143)

Performing a volume averaging on Equation 2.143 in the whole domain Y , we have,

∇x · 〈v(0)〉p = iω〈∇y · u(1)〉s (2.144)

From Equation 2.110 we know that

∇y · u(1) = (∇y · ξ) · ex(u(0))− (∇y · η)p(0) (2.145)


Then Equation 2.144 becomes:

∇x · 〈v(0)〉p = iω〈∇y · ξ〉s · ex(u(0))− iω〈∇y · η〉sp(0) (2.146)

Adding the term (−φiω∇x · u(0)) at both sides of Equation 2.146 and noticing that

∇x · u = I · (∇xu + (∇xu)T )/2 = I · ex(u) (2.147)

We derive the second compatibility equation.

∑l

∂

∂xl(〈v(0)l 〉p − φiωu(0)l ) = −γeffij iωexij(u(0))− βeff iωp(0) (2.148)

where γeff is a effective tensor and βeff is a positive scalar effective coefficient.

γeffij = φIij − 〈∑l

∂ξijl∂yl

〉s (2.149)

βeff = 〈∑l

∂ηl∂yl

〉s (2.150)

The compatibility relations 2.136 and 2.148 represent the macroscopic behavior of the

acoustics of a saturated porous medium at the first order approximation. Returning

to physical quantities, the macroscopic Model III is expressed as:

∑j

∂〈σTij〉∂Xj

= ρeffs∂2usi∂t2

+ ρp∂〈vi〉p∂t

+Or(�) (2.151)

〈σTij〉 = aeffijkheXkh(us)− αeffij p+Or(�) (2.152)∑l

∂

∂Xl(〈vl〉p − φ∂usl

∂t) = −γeffij

∂eXij(us)

∂t− βeff ∂p

∂t+Or(�) (2.153)

〈vi〉p − φ∂usi∂t

= −keffij (∂p

∂Xj+ ρp

∂2usi∂t2

) +Or(�) (2.154)

where aeff ,αeff ,γeff and βeff are elastic effective coefficients. The permeability

keff is complex-valued and ω-dependent. The model introduces two macroscopic

displacement fields. It can be shown that this model has a structure similar to Biot’s

theory.


2.6 Monophasic Elastic Macroscopic Behavior

When the fluid viscosity is bigger or the perturbation frequency is higher, and the

contrast number C = µω/a is of order � (r = 0), the porous medium is dominated by

monophasic behavior. The fluid stress tensor becomes:

σpij = 2µDyij(v)− pIij (2.155)

From Equation 2.101 and the interface condition 2.105 at the order of �0, we can still

obtain that

u(0) = u(0)(x) (2.156)

The fluid boundary-value problem for v(0) is obtained from Equations 2.102, 2.103

and 2.104 at the order of �0.

∑j

µ∂2v

(0)i

∂y2j− ∂p

(0)

∂yi= 0 in P (2.157)

∑i

∂v(0)i

∂yi= 0 in P (2.158)

v(0)i − iωu(0)i = 0 on Γ (2.159)

Obviously, the solution is:

v(0)i = iωu

(0)i (x) (2.160)

p(0) = p(0)(x) (2.161)

Similar to Section 2.5, u(1)i is obtained from Equation 2.101 and the interface condi-

tion 2.105 at the order of �1.

u(1)i = ξ

lmi exlm(u

(0))− ηip(0) + ū(1)i (x) (2.162)

The compatibility relation for the total momentum balance is also unchanged.

∑j


= −ρeffs ω2u(0)i + ρpiω〈v(0)i 〉p (2.163)

= −ρeffs ω2u(0)i − φρpω2u(0)i〈σT (0)ij 〉 = aeffijkhexkh(u(0))− αeffij p(0) (2.164)


Using v(0)i = iωu

(0)i and integrating the compressibility equation 2.103 at the order of

�1 over P , the second compatibility relation can be obtained.

φ∇x · v(0) = φiω∇x · u(0) = iω〈∇y · u(1)〉s (2.165)φI · ex(u(0)) = 〈∇y · ξ〉s · ex(u(0))− 〈∇y · η〉sp(0) (2.166)

Then,

γeffij exij(u(0)) + βeffp(0) = 0 (2.167)

where the effective coefficients γeff and βeff are the same as defined in Equa-

tions 2.149 and 2.150. Eliminating p(0) between Equations 2.164 and 2.167 leads

to the first order macroscopic description.

∑j


= −(ρeffs + φρp)ω2u(0)i (2.168)

〈σT (0)ij 〉 = a∗effijkh exkh(u(0)) (2.169)

where a∗eff is an elastic tensor defined by:

a∗effijkh = aeffijkh + α

effij γ

effkh (β

eff)−1 (2.170)

Returning to the physical quantities, the macroscopic Model IV can be expressed as:

∑j

∂〈σTij〉∂Xj

= (ρeffs + φρp)∂2usi∂t2

+Or(�) (2.171)

〈σTij〉 = a∗effijkh eXkh(us) +Or(�) (2.172)

At the macroscopic scale, the porous medium behaves like a monophasic elastic

medium with the elastic tensor a∗eff and the density (ρeffs + φρp).

2.7 Monophasic Viscoelastic Macroscopic Behav-

ior

For certain high viscosity fluid, the contrast number C = µω/a = O(1) with r = 1,

there is no contrast between the mechanical properties of the constituents in the pore


system. The fluid stress tensor then can be written as:

σpij = 2µ�−1Dyij(v)− pIij (2.173)

In this case, it is convenient to define the total displacement in the porous medium

by:

u =

us in Sup in P (2.174)

The first boundary-value problem for u(0) is obtained from Equation 2.102 at the

order of �−1 and Equations 2.101, 2.103, 2.104 and 2.105 at the order of �0.∑j

∂

∂yj(aijkleykl(u

(0))) = 0 in S (2.175)

∑j

µiω∂2u

(0)i

∂y2j= 0 in P (2.176)

∑i

∂u(0)i

∂yi= 0 in P (2.177)

u(0)si = u

(0)pi on Γ (2.178)

aijkleykl(u(0)))νj − 2µiωeyij(u(0))νj = 0 on Γ (2.179)

where u(0) is Y -periodic with respect to y. The solution can be expressed as:

u(0) = u(0)(x) (2.180)

The boundary-value problem for u(1) and p(0) is obtained from Equation 2.102 at the

order of �0 and Equations 2.101, 2.103, 2.104 and 2.105 at the order of �1.∑j

∂

∂yj[aijkl(eykl(u

(1)) + exkl(u(0)))] = 0 in S (2.181)

∑j

µiω∂2u

(1)i

∂y2j− ∂p

(0)

∂yi= 0 in P (2.182)

∑i

(∂u

(1)i

∂yi+∂u

(0)i

∂xi) = 0 in P (2.183)

u(1)si = u

(1)pi on Γ (2.184)

aijkl[eykl(u(1)) + exkl(u

(0))]νj

= [2µiω(eyij(u(1)) + exij(u

(0)))− p(0)Iij]νj on Γ (2.185)


From the linearity of the equations, the solutions are derived in the following form.

u(1)i = χ

lmi exlm(u

(0)) + ū(1)i (x) (2.186)

p(0) = ζlmexlm(u(0)) (2.187)

Here ū(1)i (x) is an arbitrary vector independent of y. The tensors χ(y, ω) and ζ(y, ω)

are complex-valued and depend on y and ω. Due to the stress continuity on Γ at the

order of �0, there is no additive constant in the expression of p(0).

The only compatibility relation is obtained from Equation 2.102 at the order of �1

and Equations 2.101 and 2.105 at the order of �2.

∑j

(∂σ

(2)sij

∂yj+∂σ

(1)sij

∂xj) = −ρsω2u(0)i in S (2.188)

∑j

(∂σ

(1)pij

∂yj+∂σ

(0)pij

∂xj) = −ρpω2u(0)i in P (2.189)

σ(2)sijνj = σ

(1)pijνj on Γ (2.190)

where the solid stress tensors are the same as defined in Equations 2.127 and 2.128,

and the zero and first order terms in the asymptotic expansion of the fluid stress

tensors are:

σ(1)pij = 2µiω[eyij(u

(2)) + exij(u(1))]− p(1)Iij (2.191)

σ(0)pij = 2µiω[eyij(u

(1)) + exij(u(0))]− p(0)Iij (2.192)

By integrating Equation 2.188 over S and Equation 2.189 P respectively, and applying

the divergence theorem with the stress continuity condition 2.190 at the interface, we

obtain:

∇x · 〈σ(1)s 〉s +∇x · 〈σ(0)p 〉p = −〈ρs〉sω2u(0) − φρpω2u(0) (2.193)Substituting the solutions of u

(1)i and p

(0) into stress-strain relations 2.128 and 2.192,

and then into Equation 2.193, the compatibility relation can be obtained as:

∑j


= −(ρeffs + φρp)ω2u(0)i (2.194)

〈σT (0)ij 〉 = a∗∗effijkh exkh(u(0)) (2.195)


where the effective coefficient is given by:

a∗∗effijkh = 〈aijkh + aijlmeylm(χkh)〉s + 〈2µiω(eyij(χkh) + IijIkh)− ζijIkh〉p (2.196)

Returning to physical quantities, the monochromatic macroscopic model is expressed

in the form:

∑j

∂〈σTij〉∂Xj

= −(ρeffs + φρp)ω2ui +Or(�) (2.197)

〈σTij〉 = a∗∗effijkh eXkh(u) +Or(�) (2.198)

The fourth-order tensor a∗∗eff is complex-valued and depends on ω. The Model V

describes a monophasic viscoelastic medium at constant pulsation.

RESERVOIR ANALYSIS USING INTERMEDIATE ...Dr. Khalid Aziz I certify that I have read this thesis and that in my opin-ion it is fully adequate, in scope and in quality, as a dissertation

Documents