Open access peer-reviewed chapter

# Tortuosity Perturbations Induced by Defects in Porous Media

By Fatma Graja and Claude Depollier

Submitted: July 24th 2018Reviewed: January 4th 2019Published: April 18th 2019

DOI: 10.5772/intechopen.84158

## Abstract

In this chapter, we describe the effects of defects in a homogeneous saturated porous medium. Defects are modelized by inclusions which disturb the motion of the viscous fluid flowing in the pore space of the medium. The seepage rate of the fluid in the host medium and in the inclusion is given by the Darcy’s law. Disturbances thus produced modify the shape of the stream lines from which we establish the tortuosity induced by the defects and its implications on the acoustic waves propagation in saturated porous media.

### Keywords

• tortuosity
• defects
• porous media
• refractive index

## 1. Introduction

Among the essential physical parameters to describe the microstructure of porous media, tortuosity is one of the most important parameters. For a review, we can refer to the paper of Ghanbarian et al. [1].

Tortuosity was introduced as a correction to the permeability of Kozeny’s model [2] of porous media defined by the Darcy’s law relating the fluidic characteristics and pore space of the medium [3]:

v=kηp,E1

where vis the seepage rate of the fluid, ηthe viscosity coefficient of the fluid, pis the pressure gradient applied to the medium, and kis its permeability. The Kozeny’s model was developed in the framework of straight and parallel streamlines in porous media. Carman has generalized it to neither straight nor parallel streamlines by introducing the hydraulic tortuosity τdefined by:

τ=<λ>L.E2

When a fluid flows through a porous medium from point Ato point Bdistant from L(Euclidean distance) (Figure 1), it follows different paths whose mean length is <λ>, where λis the length of the different paths connecting these two points. In isotropic media, the tortuosity is a scalar number greater than unit (<λ>L), whereas for the low porous media, its values may be greater than 2; they range from 1 to 2 for high porosity media such as fibrous materials and some plastic foams.

The lengthening of the field paths in porous media due to tortuosity does not only occur in the flow of fluids in porous media, but is a more general result. So we meet this concept in processes such as transport phenomena, particles diffusion, electric conductivity, or wave propagation in fluid saturated porous media. Researchers have thus developed many theoretical models adapted to their concerns to introduce the tortuosity, leading to unrelated definitions of this concept. For instance, Saomoto and Katgiri [4] presented numerical simulations to compare hydraulic and electrical tortuosities. Thus, using numerical models of fluid flow and electric conduction in same media, i.e., with the same local solid phase arrangements, the authors show that while electrical tortuosity remains close to the unit whatever the porosity and the shape of the grains, the stream lines of hydraulic flow are much more concentrated in some parts of the medium, leading to a much greater tortuosity.

This example shows that although the physical meaning of this parameter is obvious, in practice, it is not consistent and its treatment is often misleading. The conclusion that emerges from these observations is that tortuosity should not be viewed as an intrinsic parameter of the environment in which the transport process develops, but rather as a property of this process. This partly explains why there are different definitions of tortuosity, each with its own interpretation.

In acoustics of porous media, tortuosity has been introduced to take into account the frequency dependence of viscous and thermal interactions of fluid motion with the walls of pores. In [5], Johnson uses it to renormalize the fluid density ρf. When the viscous skin depth is much larger than the characteristic dimensions of the pore, Lafarge et al. [6] have shown that the density of the fluid is equal to ρfτ0, where τ0is the static tortuosity for a constant flow (ω=0) defined by:

τ0=<v2><v>2,E3

where <.>denotes averaging over the pore fluid volume Vf. Thereafter, in this chapter, we adopt this definition of tortuosity.

Through the definition (3), we see that the tortuosity is given as soon as the permeability of the porous medium is known in each of its points. As it is well known, many factors can affect the fluid flow in porous media, including pore shape, distribution of their radii, and Reynolds number to name a few. It follows that the presence of defects in an initially homogeneous medium (for instance, a local change of an intrinsic parameter) can be an important disturbance of the fluid motion, the result being a modification of the shape of the streamlines.

Taking into account the presence of defects that change the permeability of the porous medium leads to the notion of effective permeability (keff). In general, the keffvalue is not unique but depends on the chosen model for the homogenization of the porous medium. The homogenization process only makes sense for lower scales than the spatial variations of incident excitation, which therefore justifies that mobility is calculated for a low-frequency filtration rate (quasi-static regime). These considerations lead us to be interested only in the instantaneous individual response of defects to external solicitations. Since in our case only media with low levels of defect are considered, it is legitimate to ignore their mutual interactions.

The present chapter is organized as follows. Section 2 describes the mathematical model of the defects and gives the solution of the fluid flow in the presence of homogeneous and layered spherical and ellipsoidal defects. Then, the results are generalized to anisotropic defects. Finally, the hydraulic polarizability is introduced. Section 3 is relative to tortuosity. The expression of effective mobility is given for some particular defects. The induced tortuosity is deduced from the previous results and its effects on the wave propagation are given.

## 2. Defect model

In this chapter, what is called defect is a local change of permeability k. Such a change is due, for instance, to variations in porosity in the microstructure of the medium. In this chapter, a defect is modelized as a porous inclusion Ωcharacterized by its shape and own parameters: intrinsic permeability kiand porosity ϕi. Intrinsic permeability is expressed in darcy: 1D=0.97×1012m2. The porous media we are interested in have permeabilities of the order of 10D. Moreover, it is supposed that the fluid saturating the inclusion Ωis the same (with viscosity coefficient η) as that flowing in the porous medium. Thereafter the mobility of the fluid defined by κ=k/ηis used. This notion combines one property of the porous medium (permeability) with one property of the fluid (viscosity). The inclusion is embedded in a porous medium with porosity ϕoand permeability ko. The saturating fluid is subject to action of a uniform pressure gradient p0. In the sequel, we use indifferently the words defect or inclusion.

### 2.1 Mathematical formulation

When the fluid flows through the porous medium, its motion is perturbed by the defects in the microstructure of the medium. Within the porous medium, the velocity vand the pressure gradient pare related by the Darcy’s law:

vmx=kmxηpmx,E4

where m=iif xΩand m=oif xis in the host medium (xΩ). These equations are subject to the following boundary conditions on ∂Ω:

• continuity of fluid flow

ϕovo.no=ϕivi.ni,E5

• continuity of normal stress component

τijonjo=τijinjiE6

where noand niare unit vectors perpendicular to the interface. The hypothesis that Darcy’s law governs the dynamics of the flow of fluid in a porous excludes inclusions filled with fluid. Indeed, within such inclusions, the movement of the fluid is governed by the Navier-Stokes equations, which are impossible to reconcile with the law of Darcy in the porous medium with the available boundary conditions [7]. Figure 2 represents an oriented inclusion in a fluid in motion.

A porous medium of infinite extension is considered in which a viscous fluid flows at a constant uniform velocity Uunder the action of the pressure gradient along the Oxaxis. We want to determine the local changes of the fluid velocity when defects are present in the medium.

In the following, we give the solutions of Eqs. (4)(6) for some particular defects in such situation. Analytical solutions are possible for homogeneous spherical and ellipsoidal inclusions. We show that the most important characteristic of these inclusions is their hydraulic dipole moment. For layered defects, i.e., when their permeability is a piecewise constant function, we give a matrix-based method to get their dipolar moment.

### 2.2 Isotropic homogeneous defect

In this section, we assume that the background and the defect (embedded inclusion) are homogeneous each with its own parameters: porosity ϕoand ϕiand permeability koand kiwhich have constant values. A static incident pressure with a constant gradient along Oxaxis is applied to this system. What we seek is the pressure perturbation produced by the defect acting as a scatter and the expressions of the resulting seepage rate of the fluid inside and outside the porous inclusion.

#### 2.2.1 Spherical defect

The simplest type of inclusion is the homogeneous spherical one, and we consider a porous sphere of radius r=a, centered at the origin of axes with a constant permeability ko. Using the spherical coordinates (r, θ, φ), Eqs. (4)(6) become

vm=kmηpmwherem=iifr<am=oifr>aE7

with the boundary conditions at r=a:

ϕivrir=a=ϕivrir=a,τrror=a=τrrir=aθ,E8

where

τrr=ϕp+2ηvrr,E9

vrbeing the radial component of the velocity. For an incompressible fluid, Eq. (4) becomes

Δpm=0m=i,o.E10

In spherical coordinates, this equation is written as:

Δp=rr2pr+1sinθpθsinθpθ+1sin2θ2p2φ.E11

The spherical symmetry of the problem (Figure 3) implies that the solution does not depend on φ. It follows that its solution is:

pmrθ=lAlmrl+Blmrl+1Plcosθ,E12

Plxbeing the Legendre polynomial of degree l. The coefficients Almand Blmare related by the boundary conditions (8), those at r=0and when r. Inside the inclusion, the pressure must be finite at r=0. This condition leads to Bli=0for all l. So pibecomes:

pirθ=lAlirlPlcosθ.E13

Outside, far from the inclusion, the pressure is:

po1κoUrcosθ.E14

It follows that the condition

por=aθ=pir=aθθ,E15

which implies that only the term with l=1remains in the sum. Thus, we get:

A1o=1κoU.E16

The expressions of the pressure are then:

pi=A1ircosθifr<a,E17
po=Uκorcosθ+B1or2cosθifr>a,E18

A1iand B1obeen given by the conditions (8). Finally, these expressions are:

pirθ=ϕoϕiUκo3+12koa22+kiko+12kia2rcosθ,r<aE19
porθ=Uκorcosθ+Uκoa3r2kiko12+kiko+12kia2cosθ,r>aE20

For defects with radius r102,103m, the quantities ki/a2and ko/a2are very small compared to unit (105,106) and can be neglected.

The velocity is deduced from the pressure due to Darcy’s law.

• The pressure inside the inclusion (19) describes a constant velocity field. When the porosities of host medium and inclusion are substantially equal, then the fluid velocity is uniform (constant and aligned with the applied pressure gradient) with value

vi=U31+2κoκi.E21

If κi>κo, then vi>U. In this case, the fluid passes preferentially through the inclusion and in the vicinity of the inclusion, the streamlines in the host medium are curved toward the inclusion. We have the inverse conclusion if κi<κo. When κo0, i.e., for low permeability ko, then vi3U. If κi0, then vi0.

Eq. (21) can also be written in the form:

vi=Uκiκo+13κiκoE22

which will be generalized for the ellipsoidal inclusion.

• Outside the inclusion, the pressure is the sum of the applied pressure plus a dipolar contribution due to a induced dipole centered at the origin, the dipolar moment Psphof which is

Psph=4πa3Uκikiko12+kiko.E23

The corresponding density of induced “hydraulic” surface charges is

σsphθ=4πa3Uκikiko12+kikocosθ.E24

In the host medium (r>a), the components of the seepage rate are:

vro=Ucosθ1+2a3r3κiκo12+κiκo,E25
vθo=Usinθ1a3r3κiκo12+κiκo.E26

Figure 4 represents the seepage rate in the porous medium for κi>κo. Figure 4a shows the levels of the amplitude of the velocity, while Figure 4b shows its stream lines.

So the response developed by a defect when submitted to a pressure gradient is an induced dipole P. When dealing with linear phenomena (low filtration speed), this response is proportional to its cause, the proportionality factor depending only on the shape of the defect, its volume, and on the ratio of its mobility to that of the host medium. In our case, the dipole moment is

Psph=αU,E27

where αis a susceptibility which assesses the polarizability, i.e., the capacity of the porous inclusion to induce a dipole Psphunder the action of excitation U/κo. For a spherical inclusion of volume V, the susceptibility αis:

α=3Vκikiko12+kiko.E28

Often, the quantity χ=α/κiVwhich does not depend on the volume of the inclusion is more relevant. Its variations as function of the ratio κ=κi/κoare shown in Figure 5. They range from −3/2 when κi<<κoto 3 when κi>>κo. In the following, it is convenient to put the external pressure in the form:

porθ=Uκorcosθ+Psph4πcosθr2E29

#### 2.2.2 Ellipsoidal defect

In addition to the interest that ellipsoidal inclusion has an exact analytical solution, its study (its study) allows us to understand the effects of the shape of the defects on the fluid motion. Indeed, ellipsoidal surface can be seen as a generic element of a set of volumes comprising the disc, the sphere, and the oblong shape (needle) and so the nonsphericity can be appreciated through the values of its polarizability. The general ellipsoidal inclusion having semiaxes a, b, and caligned with the axes of the Cartesian coordinates system and centered at the origin is described by the following equation:

x2u+a2+y2u+b2+z2u+c2=1,E30

where x, y, and zare the position coordinates of any point on the surface of the ellipsoid. Eq. (30) has three roots ξ, η, and ζwhich define the surfaces coordinates: surfaces with constant ξare ellipsoids, while surfaces with constant ηor ζare hyperboloids. Surfaces of confocal ellipsoids are described by adjusting the scalar u. So, two ellipsoids defined by (30) with u=u1and u=u2are called confocal if their semi axes obey to the conditions

a12a22=b12b22=c12c22E31

where ai, bi, and ciare their respective semiaxes. Some results related to the ellipsoid are given in Appendix A.

Let prbe the pressure with a constant gradient directed along the Oxaxis applied to the porous medium (Figure 6). In absence of defect, its expression is:

pr=p0x=Ex=Ea2+ξa2+ηa2+ζb2a2c2a2E32

where E=U/κo. In this relation, ξ, η, and ζare the ellipsoidal coordinates given in Appendix A. Here, the field p0xcan be viewed as an “incident pressure.”

When the inclusion is embedded in the porous medium, it produces perturbations in the fluid motion giving rise to the “scattered” pressure. When the filtration rate is low, the scattering by the inclusion is a linear phenomenon, leading to the following expression of the scattered pressure:

pscr=p0xFξ.E33

where Fξis a proportional coefficient. For pscrto be a solution of the Laplace equation, it must verify the differential equation:

d2Fdξ2+dFdlnRξξ+a2=0E34

where Rσ=a2+σb2+σc2+σ. Fξis then the sum of the two functions F1ξ+F2ξ, where F1x=Ais a constant and F2xis

F2ξ=ξσ+a2Rσ.E35

Thus, the pressure outside the inclusion is

pr=p0xAB2ξσ+a2Rσ.E36

The two constants Aand Bare determined by the boundary conditions at the surface ξ=0of the inclusion:

1. continuity of fluid flow at ξ=0:

ϕeκe1hξpeξ=ϕiκi1hξpiξ,E37

2. continuity of the stress component τξξin fluid at ξ=0:

ϕepe+2η1hξvξeξ=ϕipi+2η1hξvξiξE38

where vξis the normal component of the seepage flow to the surface ξ=0given by the Darcy’s law:

vλi=κ1hλiλip,λi=ξ,η,ζ.E39

In these relations, the coefficients hλiare the scale factors of ellipsoidal coordinates given in Appendix A. We obtain the values of the pressure such that:

1. within the inclusion:

pix=κoκo+Nxκiκop0x,E40

1. outside the inclusion:

pox=p0x+pscx=p0xabcκiκoκo+Nxκiκop0xF2ξ.E41

Far from the center of the inclusion, ξr2, the scattered pressure can be approximate by:

pscxabcκiκoκo+Nxκiκop0xr2drr5/2abcκiκoκo+Nxκiκox3r3E.E42

The right-hand side of Eq. (42) is the expression of pressure produced by a dipole aligned with the axis Ox. From the expression of the speed within the inclusion:

vi=Uκoκo+Nxκiκo,E43

the dipole moment is then

Pelli,a=VUκiκiκoκo+Nxκiκo.E44

Here, Vis the volume of the ellipsoidal inclusion. The factor Nx,

Nx=abc30σ+a2σ+a2)σ+b2σ+c2,E45

describes how the dipole moment of the inclusion changes with its shape and its orientation in relation with the incident pressure field. The geometric parameters Nx, Ny, and Nzappear for the first time in hydrodynamics [8] to describe the disturbance brought by a solid immersed in an infinite fluid in uniform motion. Their values were computed by Stoner [9] and Osborn [10]. The name “depolarization factors” comes from electromagnetism (see, for example, Landau and Lifchitz [11]).

From Eq. (163), it is possible to find the values of the depolarization factors of some particular inclusions such as:

• spherical inclusion Nx=Ny=Nz=1/3,

• inclusion in disc form (axis Oz) Nx=Ny=0,Nz=1, and

• oblong inclusion Nx=Ny=1/2,Nz=0.

Figure 7 is a plot of depolarization factors of a ellipsoidal inclusion having two equal semiaxes according to their ratio.

The expression of pscxgiven by (42) is not exact since it is a result of the approximation ξr2, i.e., far from the inclusion, where only the dipolar effects are relevant.

In the case of an ellipsoidal inclusion, the polarizability is no longer a scalar but is a tensor. Its eigenvalues are polarizabilities along the axes of the ellipsoid. So, we can write the dipole moment (44) as Pelli,a=αaU, where αais the eigenvalue of the tensor polarizability along its principal direction Oxwhich defines the polarizability along this axis. In Figure 8, we depict the variations of the susceptibility χas function of κi/κofor different sets of the depolarization factors when incident pressure is along Ox.

The pressure outside the inclusion is then:

porθ=Uκorcosθ+Pelli,a4πcosθr2.E46

This relation is similar to (29), differing from it only by the expression of the dipole moment. The fundamental difference between the spherical and ellipsoidal inclusions is that the pressure scattered by the sphere contains only a dipolar field, whereas in strictness, the ellipsoid also scatters high-order multipolar fields. We can then deduce from this remark that the more the shape of the inclusion is distant from that of the sphere, the more the scattered pressure contains high-order multipolar terms.

The result obtained in (42) does not show these terms since the calculation of the integral F2 is an approximate computation when ξ/r1. When we move away from the ellipsoidal inclusion, the multipolar terms of order greater than 2 decrease very quickly, leaving only the contribution of the incident pressure and the dipole term of the scattered pressure.

### 2.3 Inhomogeneous defect

In fact, defects rarely exhibit homogeneous structure. The parameter that characterizes the defect (in our case the permeability) is generally a variable varying according to a law which depends on the way in which the defect develops.

For the spherical defect, the simplest situation is the radial variation of the permeability. The fundamental difference between homogeneous and inhomogeneous spherical inclusions is that in the latter case, the velocity field loses its uniformity. The determination of the dipole moment requires a different approach from that previously developed. Two cases are considered: (i) the permeability is a piecewise constant function and (ii) the permeability is a continuously varying function.

#### 2.3.1 Layered spherical defect

Consider an inhomogeneous sphere of porous medium embedded in a homogeneous host medium. We assume that the permeability of the sphere depends only on the radius and is a piecewise constant function, i.e., the sphere is a set of nested spherical layers. The permeability of the background medium is ko, that of the outermost layer is k1, and so on to the central sphere whose permeability is ki.

To calculate the perturbation of the incident pressure due to the sphere seen as a scatter, we proceed the following: the pressure field is calculated in each layer of the sphere. The pressure field in the layer number nis related to those in the layer number n+1and n1by the boundary conditions. It is assumed that the defect is a set of Nconcentric spherical layers in which the value of permeability is constant kn. Let κndenotes the ratio kn/ηin the layer number ndelimited by the spheres of radii rnand rn+1such that rn>rn+1. The core of the sphere has index i=N(Figure 9). The determination of pressure and velocity in this type of inclusion consists in solving the Laplace equation Δpn=0in each layer and in connecting the solutions using the boundary conditions: continuity of fluid flow and that of the radial component of the stress.

In the layer number n, the pressure is noted:

pn=Anrcosθ+Bnr2cosθ.E47

The coefficients Aand Bof two consecutive layers are connected by the following conditions in r=rn+1:

ϕnvrnr=rn+1=ϕn+1vrn+1r=rn+1,E48
τrrnr=rn+1=τrrn+1r=rn+1.E49

Transfer matrix:The linearity of the problem makes it possible to write that the pairs of coefficients (An, Bn) and (An+1, Bn+1) are linked by a matrix equation such that:

AnBn=Tn,n+1An+1Bn+1,E50

where Tn,n+1is the transfer matrix between the two consecutive layers nand n+1, the entries of which are:

T11=ϕn+1ϕnkn+1kn+21kn+1kn3+12knrn2,E51
T12=2ϕn+1ϕnrn3kn+1kn+1+2kn+1kn+12kn+1rn23+12knrn2,E52
T21=ϕn+1ϕnrn31kn+1kn3+12knrn2,E53
T22=ϕn+1ϕn1+2kn+1kn+12kn+1rn23+12knrn2.E54

From A0=U/κoand Bi=0, it is possible to get the pressure in each layer of the sphere.

The effects of two consecutive layers are obtained by the product of the transfer matrices of each of these layers. So from the matrix equations:

AnBn=Tn,n+1An+1Bn+1,E55
An+1Bn+1=Tn+1,n+2An+2Bn+2,E56

we get:

AnBn=Tn,n+2An+2Bn+2,E57

where Tn,n+2=Tn,n+1Tn+1,n+2.

Scattering matrix:Another way of linking the coefficients Aand Bof two consecutive layers is the use of the scattering matrix Sn,n+1. The scattering matrix is sometimes more efficient than the transfer matrix for calculating the amplitudes of the waves reflected and transmitted by an object subjected to incident waves. It is such that:

BnAn+1=Sn,n+1AnBn+1.E58

Its entries are:

Sn,n+1=1κn+1+2κnκn+1κnrn+133κn+13κn2κn+1κnrn+13.E59

The scattering matrix of two consecutive layers is given by their Redheffer product. From

BnAn+1=Sn,n+1AnBn+1,E60
Bn+1An+2=Sn+1,n+2An+1Bn+2,E61

we get:

BnAn+2=Sn,n+2AnBn+2,E62

where Sn,n+2=Sn,n+1Sn+1,n+2. In this relation, the right-hand side is the Redheffer star product. For more details about the Redheffer star product, we can refer to [12].

#### 2.3.2 Spherical inclusion with continuously variable permeability

When κis a continuous function of the variable r, the matrix Eq. (50) becomes the system of differential equations:

ddrr3κ2dA/dr/dr+r2κA=0E63
ddrκ2dB/drr3/dr2κr4B=0.E64

#### 2.3.3 Layered ellipsoidal inclusion

The generalization of the radial variation of the permeability of the spherical inclusion to the ellipsoidal requires that the permeability only depends on ξ. This is true in orthogonal directions at its surface. This condition entails that inside the ellipsoid, the strata are limited by confocal ellipsoidal surfaces ξ=ξk, i.e., having the same foci as the surface ξ; hence, their semiaxes are related by the following relations:

a2ak2=b2bk2=c2ck2.E65

Consider a porous inhomogeneous ellipsoidal inclusion having the permeability kiembedded in a background medium of homogeneous mobility κo. We assume that the mobility of the inclusion is stratified, i.e., it is a constant piecewise function and each layer has its own mobility κn. The mobility in the outermost layer is κ1, that of the next layer is κ2, etc. to the central layer whose the mobilty is κi. The layers are limited by the confocal surfaces ξ=ξkwhose semiaxes obey (65) and are numbered from 1 to i=Nfrom the outside to the inside, such that the strata nand n+1have the common boundary ξ=ξn+1(Figure 10). In each of these strata, the pressure is the solution of the Laplace equation given by:

pjr=ExAjBj2ξ(σ+a2RjσE66

where

Rjσ=aj2+σbj2+σcj2+σ.E67

In the layers nand n+1, the solutions of the Laplace equations are:

pnr=ExAnBn2ξ(σ+a2Rnσ,E68
pn+1r=ExAn+1Bn+12ξ(σ+a2Rn+1σE69

with the boundary condition at ξ=ξn+1:

ϕn1hξnpnξ=ϕn+11hξn+1pn+1ξ,E70

and

ϕnpn+2ηvξnhξnξ=ϕn+1pn+1+2ηvξn+1hξn+1ξ.E71

By proceeding in the same way as for the spherical cavity, one finds the transfer matrix and the scattering matrix.

Transfer matrix:The transfer matrix of the ellipsoidal inclusion is:

Tn,n+1=1κnκn+Nx,n+1κn+1κnNx,n+11Nx,n+1an+1bn+1bn+1κn+1κnan+1bn+1cn+1κn+1κnκn+1+Nx,n+1κnκn.E72

Scattering matrix:The scattering matrix of the ellipsoidal inclusion is:

Sn,n+1=1κn+Nx,n+1κn+1κnκn+1κnVn+1κn+1κnNx,n+11Nx,n+1κn+1κnVn+11E73

where Vn=anbncn.

Dipole moment:When we are only interested in the scattered far field, the inclusion can be replaced by an equivalent dipole. When ris large in front of the lengths of the axes of the ellipsoid (ra,b,c), ξis of the order of r2:

ξr2E74

and the dipolar term is

Bo2ξσ+a2RσBo2r2σ5/2=Bo3r3E75

The component Pelli,aof dipole moment along the direction of the fluid flow is obtained by identification of the terms in r2in relations (46) and (75), namely:

Pelli,a=4π3UBo.E76

#### 2.3.4 Inclusion with continuously variable permeability

For ellipsoidal inclusion, we assume that mobility depends only on the variable ξ. From (33), the problem comes down to finding of the differential equation of the function Fξ. Eq. 33 is then:

dκξξ+a2RξdF+ξRξ2FξξRξ2E=0.E77

It is easy to verify that when κis constant, we find the case of the homogeneous inclusion, and that if we put a=b=c, then we find the result of spherical inclusion.

### 2.4 Anisotropic defects

Often the defects occurring in porous media are anisotropic, i.e., some of their physical parameters like permeability are no longer scalar quantities but are tensors. For an anisotropic porous medium, assuming the Einstein convention, the Darcy’s law is

vi=kijηjpwherejp=pxj.E78

The permeability is then defined by nine components kij, i.e., it has different values in different directions of the space. Liakopoulos [13] had shown that the permeability is a symmetric tensor of second rank. This leads to great simplifications for the study of such porous media. If in isotropic media the fluid velocity is aligned with the hydraulic gradient, in anisotropic media, this is true only along the principal directions of the tensor. It is therefore not surprising that the flow movement of the fluid is seriously disturbed by this type of defects.

In a 3D space, the permeability tensor has three principal directions perpendicular to each other and for which the permeability corresponds to the tensor eigenvalues. In the coordinates system defined by these directions, the permeability tensor is diagonal:

k=k1000k2000k3.E79

By its definition, mobility inherits properties of symmetry of permeability and is therefore a symmetric tensor such that κij=kij/η.

The principal directions of the permeability tensors of the host medium and of the inclusion define coordinate systems which generally do not coincide. The system linked to the host environment is called the primary system, while that of the inclusion is called the secondary system.

In this part, the study of the effects of an anisotropic spherical inclusion in a porous medium explores three different configurations:

1. the host medium is isotropic and the defect is anisotropic,

2. the host medium is anisotropic and the defect is isotropic, and

3. the host medium and the defect are anisotropic.

#### 2.4.1 Anisotropic defect in isotropic medium

In this configuration, the primary system is such that the incident pressure gradient is along the Oz axis and the secondary coordinate system is defined by the principal directions of the permeability tensor. Then, let (θ0, φ0) and (θ, φ) be the angular directions of the incident pressure gradient poand of the observation vector OM=rin the primary system. We note βas the angle between these two directions.

The external pressure verifies the classical Laplace equation, while the internal one is solution of the following equation:

κ1ipix2+κ2ipiy2+κ3ipiz2=0.E80

This one is transformed into a Laplace equation as follows: at first, we dimensionalize the mobilities κjiby introducing the scalar quantity κi. Eq. (80) then becomes

κiκr,1ipix2+κr,2ipiy2+κr,3ipiz2=0,E81

where κr,ji=κji/κi. Using the linear transformation:

xi=κr,iixi,E82

Eq. (80) takes the form:

κi2pi2x+2pi2y+2pi2z=0.E83

The solutions piand po, respectively, are:

pirθφ=m,nAm,nirnPnmcosθcos+m,nBm,nirnPnmcosθsin,E84
porθφ=m,nAm,norn+Bm,norn+1Pnmcosθcos+m,nCm,norn+Dm,norn+1Pnmcosθsin.E85

In (84), only the finite terms at r=0appear.

The amplitudes Am,no, Bm,no, Cm,no, and Dm,no, are determined by the boundary conditions at r=aand when r.

When r, pois:

poErcosβ,withE=Uκo,E86

where βis the angle between vector Eand the direction of the observer OM=r. If (θ, φ) resp. (θ0, φ0) are the angular coordinates of the observer (resp. of the incident pressure gradient), then

cosβ=sinθsinθ0cosφφ0+cosθcosθ0.E87

The expressions (86) and (87) show that, in the expansion (85), only nonzero terms are those for which n=1. Taking into account the relations P10cosθ=P1cosθ=cosθand P11cosθ=sinθ, we obtain:

porθφ=A0,1orcosθ+A1,1orsinθcosφ+C1,1orsinθsinφ+m,nBm,norn+1Pnmcosθcos+m,nDm,norn+1Pnmcosθsin.E88

By identification with (86) with help of (87), we find:

A0,1o=Ecosθ0,E89
A1,1o=Esinθ0cosφ0,E90
C1,1o=Esinθ0sinφ0.E91

The boundary conditions at r=a(continuity of the stress component τrrand conservation of the fluid flow through the inclusion surface) are:

por=a=pir=aE92
κoporr=a=κ11ipir+κ12ipirθ+κ13ipirsinθφr=a.E93

In these equations, κnmiare the components of the tensor κiin the spherical coordinates given in Appendix D. These relations lead to the following expressions of the pressure:

pirθφ=3Aκo2κo+κ3ircosθ+3Bκo2κo+κ1irsinθcosφ+3Dκo2κo+κ2irsinθsinφ,E94
porθφ=Arcosθ+Brsinθcosφ+Drsinθsinφ+Aa3r2κoκ3i2κo+κ3icosθ+Ba3r2κoκ1i2κo+κ1isinθcosφ+Da3r2κoκ2i2κo+κ2isinθsinφ,E95

where

A=Ecosθ0,B=Esinθ0cosφ0,D=Esinθ0sinφ0.E96

The first three terms of the right-hand side of (95) are due to the pressure gradient applied to the porous medium. The last three terms are the pressure induced by the hydraulic dipoles directed along the principal directions of the anisotropic sphere.

When Eis along the Ozaxis and for φ0=0,θ0=0and κ1i=κ2i=κ3i=κi, we find the internal and external pressures of isotropic spherical inclusions (19) and (20).

Moreover, from the relations (94) and (95), it is possible to obtain the directions of the pressure gradient and of the velocity field inside the defect.

The inside fluid velocity results from (94); its components are given by vji=κjijpi, from which we obtain:

vi=3Bκ1i2κo+κ1i3Dκ2i2κo+κ2j3Aκ3i2κo+κ3k.E97

This is the generalization to the 3D case of the result obtained for the spherical inclusion when the pressure gradient is along the axis Ox (19).

The inner product of viand of the incident field U, gives the angle γwhose the internal fluid velocity is deflected by the anisotropy of the inclusion:

cosγ=UviUvi,E98
=sin2θ0sin2φ0κ1i2κe+κ2i+sin2θ0cos2φ0κ2i2κe+κ1i+cos2θ0κ3i2κe+κ2isin2θ0sin2φ0κ1i2κe+κ2i2+sin2θ0cos2φ0κ2i2κe+κ1i2+cos2θ0κ3i2κe+κ2i2.E99

#### 2.4.2 Isotropic defect in anisotropic porous medium

Consider an isotropic sphere of radius rwhose mobility is κiwhich is included in an anisotropic host medium with its own mobility κo. The incompressibility of the saturating fluid imposes that the outside pressure is the solution of the equation:

iκijojp=0.E100

In the system of Cartesian coordinate defined by the principal directions of the tensor κo, this equation is written as:

κoκxoκo2pox2+κyoκo2poy2+κzoκo2poz2=0,E101

where κjoare the eigenvalues of the mobility tensor and κois an arbitrary scalar such that the ratio κr,jo=κjo/κois a dimensionless quantity. Using the linear transformation of coordinates:

x=xκr,xo,y=yκr,yo,z=zκr,zo,E102

Eq. (101) becomes a Laplace equation. Correspondingly, the sphere is transformed into an ellipsoid with the semiaxes ax=r/κr,xo, ay=r/κr,yo, and az=r/κr,zo. Since the principal directions of the inside permeability κicoincide with the axes of the ellipsoid, for each direction j, we find, for each of the components of the pressure gradient, the result of the ellipsoidal inclusion (40). The internal pressure gradient is then:

jpi=κoκo+Njκi/κrjoκojpo,E103

or

jpi=κjoκjo+Njκiκjojpo.E104

In this equation, the depolarization factor Njis

Nj=axayaz2Odss+aj2s+ax2s+ay2s+az2forj=x,y,z.E105

Thus, the anisotropy induced in the sphere by the change of variables appears through the depolarization factor Nj.

#### 2.4.3 Anisotropic defect in anisotropic porous medium

We assume now that the host medium and the defect have their own anisotropic microstructure with the mobilities tensors κijoand κije. The velocity of the fluid flowing in each part of the porous medium is given by equations:

vio=κijojpo,vii=κijijpi.E106

Without restricting the generality of the problem, the first relation of (106) can be written as:

vio=κiopoxi,E107

where κjo, j=1,2,3, are the eigenvalues of the tensor κoand vioand poxiare the components of the velocity and of the pressure gradient along the principal directions of this tensor.

The incompressibility of the fluid implies the condition:

ivio=0,E108

or

κ1o2pox12+κ2o2pox22+κ3o2pox32=0.E109

To transform this equation into a Laplace equation, we proceed as before by using the change of variables

xi=κr,ioxi.E110

Then, the external environment becomes an isotropic medium and the outside pressure is the solution of the Laplace equation:

pox12+pox22+2pox32=0.E111

The new xivariables constitute a new coordinate system. The host medium is transformed into an isotropic medium, while the inclusion medium becomes anisotropic. In the new coordinate system, the pressure gradient is transformed according to:

p=κr,jop,E112

while the components of the position vector become:

ri=κr,io1/2ri.E113

Using the Darcy’s law and (112), the incompressibility of the fluid inside the inclusion

iκijijpi=0E114

implies the new equation:

iκio1/2κijiκjo1/2jpi=0.E115

In the new coordinates system, the mobility in the inclusion defined by the equation:

iκijiipji=0E116

is such that:

κiji'=κr,io1/2κijiκr,jo1/2.E117

In the coordinates xi, the semiaxes of the inclusion can be calculated from the equation of the ellipsoidal inclusion surface written in matrix form as RtAR, where Ris the position vector of a point of this surface (Rt=xyz) and Ais the diagonal matrix whose entries are the lengths of the half-axes:

A=a1000a2000a3.E118

Then, the linear transformation (113) changes Ainto A:

A=a1000a2000a3,E119

with

ai2==κr,io1/2ai2κr,io1/2.E120

Thus, the operation that transforms the anisotropic host medium into an isotropic one transforms the ellipsoidal inclusion with the semiaxes (a1, a2, a3) into another one with the new semiaxes (a1, a2, a3) and the new mobility κijigiven respectively by (120) and.

We recover the previous case where the outer medium is isotropic and the inner medium is anisotropic. So, in accordance with (104):

jpi=κoκo+Njκi/κrjoκojpo,E121

where the depolarization factors of the new inclusion are given by:

Ni=detA20ai+σdetA2+σI.E122

### 2.5 Hydraulic polarisability

As mentioned above, the reaction of a saturated porous inclusion subject to a pressure gradient is to induce a hydraulic dipole whose dipole moment is P. This dipole results from the appearance of pressure discontinuities at the inclusion-host interface. They have different signs depending on whether the flow is incoming or outgoing, but have the same absolute value. They are the hydraulic analogues of electrostatic charges induced by an electric field in a dielectric medium. The resulting hydraulic polarization is only nonzero if the contrast between the mobility of the host environment and that of inclusion is itself nonzero.

P=ΩκirκoκipirdV.E123

For spherical or ellipsoidal inclusions and for low filtration rates, we have seen that the internal pressure gradient is proportional to the incident one. For this type of inclusions, the dipole moment is written as:

P=αvo,E124

where the value of the susceptibility αmeasures the ability of the inclusion to induce a dipole under the action of a pressure gradient. αcan be seen as the “hydraulic polarisability” of the defect. For a spherical defect of volume V, we have:

P=αUE125

with

α=3Vκikiko12+kiko.E126

For an ellipsoidal inclusion, hydraulic polarisability is not a scalar since the response of the inclusion is a function of the direction of pressure incidence, but a second rank tensor whose eigenvalues are the susceptibilities along the three axes of the ellipsoid:

αi=Vκiκiκoκo+Niκiκo,i=x,y,z.E127

Figure 11 represents the surface hydraulic charges induced by the hydraulic polarization on a spherical inclusion and on ellipsoidal inclusions with different orientations with respect to the direction of the incident flux. The red areas represent the surface “hydraulic charge” density σpol. Its expression depends on the direction of the incident pressure gradient and is written as the sum of the contributions of the dipoles along the three axes of the ellipsoid.

When the permeability of the inclusion is stratified, the dipole moment is given by the dipolar term (Bo) of the external pressure field obtained by the transfer matrix method or by the scattering matrix method.

## 3. Tortuosity induced by defects

In this section, we determine the hydraulic effects of defects on the permeability of porous media. As mentioned above, the shape of the defects is one of the most important factors for the modification of the current lines of the seepage rate in the whole porous medium and thus contributes to its acoustic properties.

### 3.1 Homogenization: generalities

Experiments show that a nonhomogeneous medium subject to excitation behaves in the same way as its different components, but with different parameter values. The homogenization of an inhomogeneous porous medium consists in replacing it with an effective homogeneous medium with the permeability keff. This operation is only possible at a fairly large observation scale. Determining the value of the effective permeability from the mobility values of the structure components and their relative positions is not a simple averaging operation. The calculation of the global mobility of a mixture of porous inclusions immersed in a homogeneous medium is a topic widely addressed in many research fields such as hydrology, oil recovery, chemical industry, etc. As a consequence, a considerable number of works deal with this problem based on various methods: renormalization theory, variational methods, T-Matrix method, field theory methods, nonperturbative approach based on Feynman path integral. To quote some of authors, we can refer to the works of Prakash and Raja-Sekhar [14], King [15, 16], Drummond and Horgan [17], Dzhabrailov and Meilanov [18], Teodorovich [19, 20], Stepanyants and Teodorovich [21], and Hristopulos and Christakos [22].

In the case of media subject to a variable field action, homogenization requires defining a length below which it is no longer relevant. For example, for a periodic field, acting on a medium whose average distance between inhomogeneities is a, this length is the wavelength λif λ/a1. In our case, effective mobility being essentially a low-frequency concept, this remark justifies that the effective mobility should then be calculated from a steady filtration velocity.

### 3.2 Effective mobility

Darcy’s law is often used as the definition of the mobility of a porous medium, and the easiest way to introduce the effective mobility κeffis to use it as follows:

<v>=κeff<E>,E128

where E=pand <>is the averaging operation. The mean values of the filtration rate and the pressure gradient are given by:

<v>=f<κiEi>+1f<κoEo>,E129
<E>=f<Ei>+1f<Eo>,E130

where fis the volumic fraction of the defect. Putting Ei=AEowe show that:

κeff=fAκi+1fκofA+1f.E131

For spherical defects, A=3κo/(2κo+κi, Eq. (131) leads to the result:

κeff=κof3κi2κo+κi+1ff3κo2κo+κi+1f.E132

When f0, then κeffκo, and when f1, then κeffκi. Finally when f<<1, then

κeffκo+3fκoκiκoκi+2κo.E133

For anisotropic inclusion, mobility is a second rank tensor defined by the relationship:

<vi>=κeff,ij<Ej>.E134

As a result, for each main direction, we have:

<vj>=κeff,j<Ej>,j=x,y,z,E135

where κeff,jare the eigenvalues of κeff. Taking into account that Eji=AjEjo, Eq. (135) shows that:

κeff,j=fAjκi+1fκofAj+1f,E136

with

Aj=3κo2κo+κji,j=1,2,3.E137

When the environment has several defects, the calculation of keffis more complicated because their mutual influence must be taken into account. The excitation pressure gradient Eedefined from the filtration rate is introduced by the equation:

κoEe=κoE+LP.E138

In this relationship, Lis an operator that takes into account the shape of the defect and its orientation with respect to the fluid flow, and Pis due to the induced polarization in the inclusion. Pdefined by (123) is related to the dipole moment induced by the interaction between the fluid moving in the porous medium and the defect. When the medium contains nidentical defects per unit volume, P=np, pbeing the dipole moment of each defect, and since pis proportional to the applied field (p=αEe), we have P=Ee. For an ellipsoidal defect, Lis reduced to depolarization factors, i.e., L=Nk, k=x,y,z, which takes into account the direction of fluid flow. In this case, the excitation field is:

Ee=E1Nk/κo,E139

κeff=κo+1Nk/κo,E140

leading, for spherical defects, to expression:

κeff=κo+1/3κo.E141

It is then possible to calculate the effective mobility of a set of ellipsoidal inclusions in different geometries (Figure 12):

• the ellipsoids are aligned with the direction xof the fluid flow:

κeff=κo+nαx1nαxNx/κo.E142

• the ellipsoids are randomly oriented:

κeff=κo+1/3i=x,y,znαi1i=x,y,znαiNi/κo.E143

### 3.3 Induced tortuosity

We restrict ourselves here to the calculation of the tortuosity induced by homogeneous spherical inclusions (Figure 13). Since the dipole moment is the essential element for this calculation element for this calculation, it is easy to generalize the results obtained with other types of inclusions: inhomogeneous spherical, ellipsoidal, etc.

The tortuosity induced by the presence of defects τdis defined by:

τd=<vo2><vo>2E144

where vois the perturbation of filtration rate due to the defects. When the ratio k/a2where ais the characteristic size of the defects is small relative to the unity, it is legitimate to neglect the volume of the defects for the calculation of <v2>, whereas it is taken into account for that of <v>.

With the pressure scattered field by the inclusions being limited to the dipolar terms, the expression of <vo2>is then:

<vo2>=<vro2>+<vθo2>E145

where

vro=κopor,andvθo=κoporθ.E146

For ellipsoidal inclusion, the external pressure is:

porθ=Uκorcosθ+Pd4πcosθr2E147
=Uκorαr2cosθE148

where Pdand αare, respectively, the dipol moment and the polarisability of the inclusion. By keeping only the terms greater than or equal to r2, one obtains:

vo2Uκo212αr32cos2θsin2θE149

The average value <vo2>is calculated by integration on the volume between two spheres of radius a(characteristic size of the defect) andRsufficiently large so that the dipolar effects are negligible. For a spherical inclusion, it results:

<vo2>=U2+U21a3R3κiκo12+κiκo2.E150

<vo>2is calculated from the definition of effective mobility:

<v>=κeff<E>E151

where κeffand <E>are given by (131) and (130). When f<<1, we have

κeffκo+fAκiκoand<E>Uκo1+fA.E152

From these two relations, we obtain the expression of the induced tortuosity:

τd=1+1a3R3κiκo12+κiκo212f3κiκo2+κiκo.E153

Results of numerical simulations:The results of a numerical simulation for κi/κo=10and κi/κo=100are shown in Figure 14. The tortuosity value is calculated on square domains around the inclusion (Figure 13). Inside the inclusion, τb=1. As xincreases, the tortuosity increases to reach its maximum value at x=1.7for κi/κo=10and x=1.6when κi/κo=10. For larger values of x, it decreases toward 1 since, far from inclusion, the field lines again become parallel to the direction of the incident pressure gradient. This result confirms the behavior of the field lines of Figure 4b.

## 4. Conclusion

In this chapter, we studied the effect of defects on the circulation of the fluid saturating a porous medium. We have shown that the modification of the stream lines of the filtration velocities leads to a modification of the value of the tortuosity and thus on the local velocity of the waves susceptible to propagate in such media. The induced tortuosity was calculated from the pressure field scattered by the inclusions. The model used is based on the Darcy’s law. in addition to being general, its major interest is to lead to a very practical mathematical expression of tortuosity

## Acknowledgments

C. Depollier is supported by Russian Science Foundation grant number 14-49-00079.

The ellipsoidal coordinates (ξ, η, ζ) are the solutions of the cubic equation:

x2a2+u+y2b2+u+z2c2+u=1.E154

They are connected to the Cartesian coordinates (x, y, z) by the relations:

x2=a2+ξa2+ηa2+ζb2a2c2a2,E155
y2=b2+ξb2+ηb2+ζa2b2c2b2,E156
z2=c2+ξc2+ηc2+ζa2c2b2c2,E157

subject to the conditions ξ<c2<η<b2<ζ<a2.

The scalar factors are the vector norms:

hqi=rqioùqi=ξ,η,ζ.E158

Their values are:

hξ=12ηξζξa2ξb2ξc2ξ,E159
hη=12ξηζηa2ηb2ηc2η,E160
hζ=12ηζξζa2ζb2ζc2ζ.E161

The depolarization factors are important quantities for the expression of solutions of the Laplace equation. They take into account the form of the domain in which this solution is sought and its orientation in relation to the excitation field. Their expression is:

Nk=abc30σ+qk2σ+a2σ+b2σ+c2E162

where k=x(resp. y, z) et qk=a, (resp. b, c) and satisfy the relation:

Nx+Ny+Nz=1.E163

Consider the rectangular coordinate systems (x, y, z) and (x, y,z). We are looking for the relations between the spherical coordinates (r, θ, φ) and (r, θ, φ) associated with each of them. From

x=rsinθcosφx=rsinθcosφ,E164
y=rsinθsinφy=rsinθsinφ,E165
z=rcosθz=rsinθ,E166

one deduces

r2=x2+y2+z2=x2κ1+y2κ2+z2κ3=r2sin2θcos2φκ1+sin2θsin2φκ2+cos2θκ3

or

r=rΔoùΔ=sin2θcos2φκ1+sin2θsin2φκ2+cos2θκ3.E167

From (164) and (165), one has:

x=xκ1rsinθcosφ=rκ1sinθcosφy=yκ2rsinθsinφ=rκ2sinθsinφ.

By eliminating φ, one finds:

sinθ=sinθδΔavecδ=cos2φκ1+sin2φκ2.E168

In the same way, from (166), one can establish

cosθ=1Δcosθκ3.E169

Similar relationships between angles φand φare deduced from (164) and (165):

sinφ=1δsinφκ2,E170
cosφ=1δcosφκ1.E171

Let κbe a tensor of rank 2. We denote by κr, its expression in the system of rectangular coordinates defined by its principal directions, and κs, its expression in the corresponding spherical coordinates system. So we have

κr=κ1000κ2000κ3κs=κ11κ12κ13κ21κ22κ23κ31κ32κ33E172

with

κ11=κ1sin2θcos2φ+κ2sin2θsin2φ+κ3cos2θ,E173
κ12=κ1cosθsinθcos2φ+κ2cosθsinθsin2φκ3cosθsinθ,E174
κ13=κ2κ1sinθcosφsinφ,E175
κ22=κ1cos2θcos2φ+κ2cos2θsin2φ+κ3sin2θ,E176
κ23=κ2κ1cosθcosφsinφ,E177
κ33=κ1sin2φ+κ2cos2φ,E178
κ21=κ12,E179
κ31=κ13,E180
κ32=κ23.E181

Or, alternatively in the matrix form:

k=κ1I+κ2κ1A+κ3κ1B,E182

where Iis the unit matrix 3×3and Aand Bare given by:

A=sin2θsin2φcosθsinθsin2φsinθcosφsinφcosθsinθsin2φcos2θsin2φcosθcosφsinφsinθcosφsinφcosθcosφsinφcos2θ,E183
B=cos2θcosθsinθ0cosθsinθsin2θ0000.E184

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Fatma Graja and Claude Depollier (April 18th 2019). Tortuosity Perturbations Induced by Defects in Porous Media, Acoustics of Materials, Zine El Abiddine Fellah and Erick Ogam, IntechOpen, DOI: 10.5772/intechopen.84158. Available from:

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