2.4.2 Isotropic defect in anisotropic porous medium

<sup>κ</sup>ð Þ<sup>o</sup> <sup>∂</sup>pð Þ<sup>o</sup>

Acoustics of Materials

In these equations, κ

of the pressure:

<sup>p</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup>; <sup>φ</sup>

where

anisotropic sphere.

ð Þi

and (20).

v ð Þi <sup>j</sup> ¼ �κ

80

<sup>∂</sup><sup>r</sup> <sup>r</sup>¼<sup>a</sup> <sup>¼</sup> <sup>κ</sup>

3Aκð Þ<sup>o</sup> 2κð Þ<sup>o</sup> þ κ

> þ Ba<sup>3</sup> r2

ð Þi

ð Þi 3

� � ð Þi 11 ∂pð Þ<sup>i</sup> ∂r þ κ ð Þi 12 ∂pð Þ<sup>i</sup> <sup>r</sup>∂<sup>θ</sup> <sup>þ</sup> <sup>κ</sup>

r cos θ þ

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup>; <sup>φ</sup> Ar cos <sup>θ</sup> <sup>þ</sup> Brsin <sup>θ</sup> cos <sup>φ</sup> <sup>þ</sup> Drsin <sup>θ</sup> sin <sup>φ</sup> <sup>þ</sup>

ð Þi 1

When E<sup>∞</sup> is along the Oz axis and for φ<sup>0</sup> ¼ 0, θ<sup>0</sup> ¼ 0 and κ

of the pressure gradient and of the velocity field inside the defect.

ð Þi 1 2κð Þ<sup>o</sup> þ κ<sup>1</sup>

inclusion when the pressure gradient is along the axis Ox (19).

sin <sup>2</sup>θ<sup>0</sup> sin <sup>2</sup>φ0κ

2κð Þ<sup>e</sup> þκ ð Þi 2

sin <sup>2</sup>θ<sup>0</sup> sin <sup>2</sup>φ0κ

2κð Þ<sup>e</sup> þκ ð Þ<sup>i</sup> ð Þ <sup>2</sup>

the internal fluid velocity is deflected by the anisotropy of the inclusion:

ð Þi 1

> ð Þi 1

<sup>j</sup> <sup>∂</sup>jpð Þ<sup>i</sup> , from which we obtain:

¼

<sup>v</sup>ð Þ<sup>i</sup> ¼ � <sup>3</sup>B<sup>κ</sup>

<sup>κ</sup>ð Þ<sup>o</sup> � <sup>κ</sup> ð Þi 1

2κð Þ<sup>o</sup> þ κ

� � � �

coordinates given in Appendix D. These relations lead to the following expressions

3Bκð Þ<sup>o</sup> 2κð Þ<sup>o</sup> þ κ

sin θ cos φ þ

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

we find the internal and external pressures of isotropic spherical inclusions (19)

The inside fluid velocity results from (94); its components are given by

<sup>i</sup> � <sup>3</sup>D<sup>κ</sup>

This is the generalization to the 3D case of the result obtained for the spherical

The inner product of vð Þ<sup>i</sup> and of the incident field U∞, gives the angle γ whose

∥U∞∥∥vð Þ<sup>i</sup> ∥

ð Þi 2

> ð Þi 2

<sup>s</sup> : (99)

<sup>þ</sup> cos <sup>2</sup>θ0<sup>κ</sup>

<sup>2</sup> <sup>þ</sup> cos <sup>2</sup>θ0<sup>κ</sup>

ð Þi 3 2κð Þ<sup>e</sup> þκ ð Þi 2

ð Þi 3 2κð Þ<sup>e</sup> þκ ð Þ<sup>i</sup> ð Þ <sup>2</sup> 2

<sup>þ</sup> sin <sup>2</sup>θ<sup>0</sup> cos <sup>2</sup>φ0<sup>κ</sup>

2κð Þ<sup>e</sup> þκ ð Þi 1

2κð Þ<sup>e</sup> þκ ð Þ<sup>i</sup> ð Þ <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> sin <sup>2</sup>θ<sup>0</sup> cos <sup>2</sup>φ0<sup>κ</sup>

cos <sup>γ</sup> <sup>¼</sup> <sup>U</sup><sup>∞</sup> � <sup>v</sup>ð Þ<sup>i</sup>

ð Þi 2 2κð Þ<sup>o</sup> þ κ<sup>2</sup>

<sup>j</sup> � <sup>3</sup>A<sup>κ</sup>

ð Þi 3 2κð Þ<sup>o</sup> þ κ<sup>3</sup>

k: (97)

, (98)

Moreover, from the relations (94) and (95), it is possible to obtain the directions

A ¼ E<sup>∞</sup> cos θ0, B ¼ E<sup>∞</sup> sin θ<sup>0</sup> cos φ0, D ¼ E<sup>∞</sup> sin θ<sup>0</sup> sin φ0: (96)

ð Þi 1

ð Þi 13

nm are the components of the tensor κð Þ<sup>i</sup> in the spherical

rsin θ cos φ þ

Da<sup>3</sup> r2

<sup>κ</sup>ð Þ<sup>o</sup> � <sup>κ</sup> ð Þi 2

2κð Þ<sup>o</sup> þ κ

∂pð Þ<sup>i</sup> rsin θ∂φ � � � � r¼a

3Dκð Þ<sup>o</sup> 2κð Þ<sup>o</sup> þ κ

Aa<sup>3</sup> r2

> ð Þi 2

> > ð Þi <sup>1</sup> ¼ κ ð Þi <sup>2</sup> ¼ κ ð Þi <sup>3</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>i</sup> ,

ð Þi 2

<sup>κ</sup>ð Þ<sup>o</sup> � <sup>κ</sup> ð Þi 3

2κð Þ<sup>o</sup> þ κ

sin θ sin φ,

ð Þi 3

: (93)

rsin θ sin φ,

(94)

(95)

cos θ

Consider an isotropic sphere of radius r whose mobility is κð Þ<sup>i</sup> which is included in an anisotropic host medium with its own mobility κð Þ<sup>o</sup> . The incompressibility of the saturating fluid imposes that the outside pressure is the solution of the equation:

$$\partial\_i \left( \kappa\_{ij}^{(o)} \partial\_j p \right) = \mathbf{0}. \tag{100}$$

In the system of Cartesian coordinate defined by the principal directions of the tensor κð Þ<sup>o</sup> , this equation is written as:

$$\kappa^{\prime(o)}\left(\frac{\kappa^{(o)}\_x}{\kappa^{\prime(o)}}\frac{\partial^2 p^{(o)}}{\partial \kappa^2} + \frac{\kappa^{(o)}\_y}{\kappa^{\prime(o)}}\frac{\partial^2 p^{(o)}}{\partial \mathcal{Y}^2} + \frac{\kappa^{(o)}\_x}{\kappa^{\prime(o)}}\frac{\partial^2 p^{(o)}}{\partial \mathbf{z}^2}\right) = \mathbf{0},\tag{101}$$

where κ ð Þo <sup>j</sup> are the eigenvalues of the mobility tensor and <sup>κ</sup>0ð Þ<sup>o</sup> is an arbitrary scalar such that the ratio κ ð Þo r,j ¼ κ ð Þo <sup>j</sup> <sup>=</sup>κ0ð Þ<sup>o</sup> is a dimensionless quantity. Using the linear transformation of coordinates:

$$\mathbf{x}' = \frac{\mathbf{x}}{\sqrt{\kappa\_{r,x}^{(o)}}}, \quad \mathbf{y}' = \frac{\mathbf{y}}{\sqrt{\kappa\_{r,y}^{(o)}}}, \quad \mathbf{z}' = \frac{\mathbf{z}}{\sqrt{\kappa\_{r,x}^{(o)}}}, \tag{102}$$

Eq. (101) becomes a Laplace equation. Correspondingly, the sphere is transformed into an ellipsoid with the semiaxes ax ¼ r= ffiffiffiffiffiffiffi κ ð Þo r,x q , ay ¼ r= ffiffiffiffiffiffiffi κ ð Þo r, y q , and az ¼ r= ffiffiffiffiffiffiffi κ ð Þo r, z q . Since the principal directions of the inside permeability κð Þ<sup>i</sup> coincide 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:

$$\partial\_{\mathbf{j}}p^{(i)} = \frac{\kappa^{\prime(o)}}{\kappa^{\prime(o)} + N\_{\mathbf{j}}\left(\kappa^{(i)}/\kappa\_{\mathbf{j}}^{(o)} - \kappa^{\prime(o)}\right)}\partial\_{\mathbf{j}}p^{(o)},\tag{103}$$

or

$$
\partial\_j p^{(i)} = \frac{\kappa\_j^{(o)}}{\kappa\_j^{(o)} + N\_j \left(\kappa^{(i)} - \kappa\_j^{(o)}\right)} \partial\_i p^{(o)}.\tag{104}
$$

In this equation, the depolarization factor Nj is

$$N\_j = \frac{a\_x a\_y a\_x}{2} \int\_0^\infty \frac{ds}{\left(\mathfrak{s} + a\_j^2\right) \sqrt{\left(\mathfrak{s} + a\_x^2\right) \left(\mathfrak{s} + a\_y^2\right) \left(\mathfrak{s} + a\_x^2\right)}}\quad \text{for}\quad j = x, y, z. \tag{105}$$

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 κ ð Þo ij and κ ð Þe ij . The velocity of the fluid flowing in each part of the porous medium is given by equations:

$$\boldsymbol{\nu}\_{i}^{(o)} = -\kappa\_{ij}^{(o)} \nabla\_{j} \boldsymbol{p}^{(o)}, \qquad \boldsymbol{\nu}\_{i}^{(i)} = -\kappa\_{ij}^{(i)} \nabla\_{j} \boldsymbol{p}^{(i)}.\tag{106}$$

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

$$v\_i^{(o)} = -\kappa\_i^{(o)} \frac{\partial p^{(o)}}{\partial \mathbf{x}\_i},\tag{107}$$

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

ij <sup>∇</sup>jpð Þ<sup>i</sup> � � <sup>¼</sup> <sup>0</sup> (114)

� � <sup>¼</sup> <sup>0</sup> (116)

, the semiaxes of the inclusion can be calculated from the

1

1

ð Þi 0

¼ 0: (115)

: (117)

CA: (118)

CA, (119)

: (120)

ij given respectively by (120) and.

� � <sup>∂</sup>jpð Þ<sup>o</sup> , (121)

:

AR, where

∇0 j pð Þ<sup>i</sup>

∇<sup>i</sup> κ ð Þi

> κ ð Þi ij κ ð Þo j � ��1=<sup>2</sup>

∇0 <sup>i</sup> κ ð Þi 0 ij ∇<sup>0</sup> i pð Þi j

ð Þo r,i � ��1=<sup>2</sup>

equation of the ellipsoidal inclusion surface written in matrix form as R<sup>t</sup>

0

B@

a0

0

B@

ð Þo r,i � ��1=<sup>2</sup>

3) and the new mobility κ

<sup>∂</sup>jpð Þ<sup>i</sup> <sup>¼</sup> <sup>κ</sup>0ð Þ<sup>o</sup> <sup>κ</sup>0ð Þ<sup>o</sup> <sup>þ</sup> <sup>N</sup><sup>0</sup>

A ¼

Then, the linear transformation (113) changes A into A<sup>0</sup>

a02 <sup>i</sup> ¼¼ κ

medium is anisotropic. So, in accordance with (104):

1, a<sup>0</sup> 2, a<sup>0</sup> A<sup>0</sup> ¼

κ ð Þi ij κ ð Þo r,j � ��1=<sup>2</sup>

<sup>R</sup> is the position vector of a point of this surface (R<sup>t</sup> <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ) and <sup>A</sup> is the diagonal

a<sup>1</sup> 0 0 0 a<sup>2</sup> 0 0 0 a<sup>3</sup>

<sup>1</sup> 0 0 0 a<sup>0</sup>

<sup>2</sup> 0 0 0 a<sup>0</sup>

> a2 <sup>i</sup> κ ð Þo r,i � ��1=<sup>2</sup>

Thus, the operation that transforms the anisotropic hostmedium into anisotropic one transforms the ellipsoidal inclusion with the semiaxes (a1, a2, a3) into another one with

We recover the previous case where the outer medium is isotropic and the inner

<sup>j</sup> κð Þ<sup>i</sup> =κ

ð Þo rj � <sup>κ</sup>0ð Þ<sup>o</sup>

3

κ ð Þi 0 ij ¼ κ

matrix whose entries are the lengths of the half-axes:

i

� �

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

inclusion

equation:

with

83

the new semiaxes (a<sup>0</sup>

is such that:

In the coordinates x<sup>0</sup>

implies the new equation:

∇0 <sup>i</sup> κ ð Þo i � ��1=<sup>2</sup>

Tortuosity Perturbations Induced by Defects in Porous Media

DOI: http://dx.doi.org/10.5772/intechopen.84158

where κ ð Þo <sup>j</sup> , <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3, are the eigenvalues of the tensor <sup>κ</sup>ð Þ<sup>o</sup> and <sup>v</sup> ð Þo <sup>i</sup> and <sup>∂</sup>pð Þ<sup>o</sup> <sup>∂</sup>xi are 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:

$$\nabla\_i \boldsymbol{\nu}\_i^{(o)} = \mathbf{0},\tag{108}$$

or

$$
\kappa\_1^{(o)} \frac{\partial^2 \mathfrak{p}^{(o)}}{\partial \kappa\_1^2} + \kappa\_2^{(o)} \frac{\partial^2 \mathfrak{p}^{(o)}}{\partial \kappa\_2^2} + \kappa\_3^{(o)} \frac{\partial^2 \mathfrak{p}^{(o)}}{\partial \kappa\_3^2} = 0. \tag{109}
$$

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

$$\frac{\partial}{\partial \mathbf{x}\_i'} = \sqrt{\mathbf{x}\_{r,i}^{(o)}} \frac{\partial}{\partial \mathbf{x}\_i} \,. \tag{110}$$

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

$$\frac{\partial p^{(o)} }{\partial \mathbf{x}\_1'^2} + \frac{\partial p^{(o)} }{\partial \mathbf{x}\_2'^2} + \frac{\partial^2 p^{(o)} }{\partial \mathbf{x}\_3'^2} = \mathbf{0}. \tag{111}$$

The new x<sup>0</sup> <sup>i</sup> variables 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:

$$
\nabla' p = \sqrt{\kappa\_{r,j}^{(o)}} \nabla p,\tag{112}
$$

while the components of the position vector become:

$$r\_i' = \left(\kappa\_{r,i}^{(o)}\right)^{-1/2} r\_i. \tag{113}$$

Tortuosity Perturbations Induced by Defects in Porous Media DOI: http://dx.doi.org/10.5772/intechopen.84158

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

$$\nabla\_i \left( \kappa\_{ij}^{(i)} \nabla\_j p^{(i)} \right) = \mathbf{0} \tag{114}$$

implies the new equation:

$$\nabla\_i' \left[ \left( \kappa\_i^{(o)} \right)^{-1/2} \kappa\_{ij}^{(i)} \left( \kappa\_j^{(o)} \right)^{-1/2} \nabla\_j' p^{(i)} \right] = \mathbf{0}. \tag{115}$$

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

$$\nabla'\_i \left( \kappa^{(i)'}\_{\vec{\eta}} \nabla'\_i p^{(i)}\_{\vec{\jmath}} \right) = \mathbf{0} \tag{116}$$

is such that:

2.4.3 Anisotropic defect in anisotropic porous medium

flowing in each part of the porous medium is given by equations:

ð Þo

v ð Þo <sup>i</sup> ¼ �κ

The incompressibility of the fluid implies the condition:

microstructure with the mobilities tensors κ

v ð Þo <sup>i</sup> ¼ �κ

κ ð Þo 1 ∂2 pð Þ<sup>o</sup> ∂x<sup>2</sup> 1 þ κ ð Þo 2 ∂2 pð Þ<sup>o</sup> ∂x<sup>2</sup> 2 þ κ ð Þo 3 ∂2 pð Þ<sup>o</sup> ∂x<sup>2</sup> 3

pressure is the solution of the Laplace equation:

be written as:

Acoustics of Materials

where κ

or

ð Þo

directions of this tensor.

using the change of variables

The new x<sup>0</sup>

according to:

82

We assume now that the host medium and the defect have their own anisotropic

Without restricting the generality of the problem, the first relation of (106) can

ð Þo i

∂pð Þ<sup>o</sup> ∂xi

ij <sup>∇</sup>jpð Þ<sup>o</sup> , vð Þ<sup>i</sup>

<sup>j</sup> , <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3, are the eigenvalues of the tensor <sup>κ</sup>ð Þ<sup>o</sup> and <sup>v</sup>

the components of the velocity and of the pressure gradient along the principal

∇iv ð Þo

To transform this equation into a Laplace equation, we proceed as before by

ffiffiffiffiffiffiffi κ ð Þo r,i q ∂

Then, the external environment becomes an isotropic medium and the outside

transformed into an isotropic medium, while the inclusion medium becomes aniso-

ffiffiffiffiffiffiffi κ ð Þo r,j q

tropic. In the new coordinate system, the pressure gradient is transformed

∇0 p ¼

r 0 <sup>i</sup> ¼ κ ð Þo r,i � ��1=<sup>2</sup>

while the components of the position vector become:

∂xi

<sup>i</sup> variables constitute a new coordinate system. The host medium is

∂ ∂x0 i ¼

∂pð Þ<sup>o</sup> ∂x0<sup>2</sup> 1 þ ∂pð Þ<sup>o</sup> ∂x0<sup>2</sup> 2 þ ∂2 pð Þ<sup>o</sup> ∂x0<sup>2</sup> 3

ð Þo ij and κ

ð Þe

ð Þi

<sup>i</sup> ¼ �κ

ij . The velocity of the fluid

ij ∇jpð Þ<sup>i</sup> : (106)

, (107)

<sup>i</sup> ¼ 0, (108)

ð Þo <sup>i</sup> and <sup>∂</sup>pð Þ<sup>o</sup>

¼ 0: (109)

: (110)

¼ 0: (111)

∇p, (112)

ri: (113)

<sup>∂</sup>xi are

$$
\kappa\_{ij}^{(i)'} = \left(\kappa\_{r,i}^{(o)}\right)^{-1/2} \kappa\_{ij}^{(i)} \left(\kappa\_{r,j}^{(o)}\right)^{-1/2}.\tag{117}
$$

In the coordinates x<sup>0</sup> i , the semiaxes of the inclusion can be calculated from the equation of the ellipsoidal inclusion surface written in matrix form as R<sup>t</sup> AR, where <sup>R</sup> is the position vector of a point of this surface (R<sup>t</sup> <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ) and <sup>A</sup> is the diagonal matrix whose entries are the lengths of the half-axes:

$$A = \begin{pmatrix} a\_1 & 0 & 0 \\ 0 & a\_2 & 0 \\ 0 & 0 & a\_3 \end{pmatrix}. \tag{118}$$

Then, the linear transformation (113) changes A into A<sup>0</sup> :

$$A' = \begin{pmatrix} a\_1' & 0 & 0 \\ 0 & a\_2' & 0 \\ 0 & 0 & a\_3' \end{pmatrix},\tag{119}$$

with

$$a\_i'^2 = = \left(\kappa\_{r,i}^{(o)}\right)^{-1/2} a\_i^2 \left(\kappa\_{r,i}^{(o)}\right)^{-1/2}.\tag{120}$$

Thus, the operation that transforms the anisotropic hostmedium into anisotropic one transforms the ellipsoidal inclusion with the semiaxes (a1, a2, a3) into another one with the new semiaxes (a<sup>0</sup> 1, a<sup>0</sup> 2, a<sup>0</sup> 3) and the new mobility κ ð Þi 0 ij given 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):

$$\partial\_{\mathbf{j}}p^{(i)} = \frac{\kappa^{\prime(o)}}{\kappa^{\prime(o)} + N\_{\mathbf{j}}^{\prime} \left(\kappa^{(i)} / \kappa\_{\mathbf{j}}^{(o)} - \kappa^{\prime(o)}\right)} \partial\_{\mathbf{j}}p^{(o)},\tag{121}$$

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

$$N\_i' = \frac{\det A'}{2} \Big|\_{0}^{\infty} \frac{d\sigma}{(a\_i' + \sigma)\sqrt{\det\left(A'^2 + \sigma I\right)}} \cdot \tag{122}$$

<sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>V</sup>κð Þ<sup>i</sup> <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup>

contributions of the dipoles along the three axes of the ellipsoid.

whole porous medium and thus contributes to its acoustic properties.

Dzhabrailov and Meilanov [18], Teodorovich [19, 20], Stepanyants and

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 λ if λ=a ≫ 1. In our case, effective mobility being essentially a low-frequency concept, this remark justifies that the 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 κeff is to use it as follows:

Teodorovich [21], and Hristopulos and Christakos [22].

should then be calculated from a steady filtration velocity.

3.2 Effective mobility

85

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],

matrix method or by the scattering matrix method.

Tortuosity Perturbations Induced by Defects in Porous Media

DOI: http://dx.doi.org/10.5772/intechopen.84158

3. Tortuosity induced by defects

3.1 Homogenization: generalities

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

When the permeability of the inclusion is stratified, the dipole moment is given by the dipolar term (Bð Þ<sup>o</sup> ) of the external pressure field obtained by the transfer

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

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Ni <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ, i <sup>¼</sup> x, y, z: (127)

#### 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.

$$\mathcal{P} = \int\_{\Omega} \left( \kappa^{(i)}(\mathbf{r}) - \kappa^{(o)} \right) \kappa^{(i)} \nabla p^{(i)}(\mathbf{r}) dV. \tag{123}$$

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:

$$\mathcal{P} = a\mathbf{v}^{(o)},\tag{124}$$

where the value of the susceptibility α measures the ability of the inclusion to induce a dipole under the action of a pressure gradient. α can be seen as the "hydraulic polarisability" of the defect. For a spherical defect of volume V, we have:

$$\mathcal{P} = a(-\mathbf{U}\_{\infty}) \tag{125}$$

with

$$\alpha = \Im V \kappa^{(i)} \left( \frac{\frac{k^{(i)}}{k^{(o)}} - \mathbf{1}}{2 + \frac{k^{(i)}}{k^{(o)}}} \right). \tag{126}$$

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:

#### Figure 11.

Sketch of polarization surface "charge" density σpol for a spherical inclination and an ellipsoidal inclusion with different orientations with respect to the incident flow.

Tortuosity Perturbations Induced by Defects in Porous Media DOI: http://dx.doi.org/10.5772/intechopen.84158

$$a\_i = V\kappa^{(i)} \frac{\kappa^{(i)} - \kappa^{(o)}}{\kappa^{(o)} + N\_i(\kappa^{(i)} - \kappa^{(o)})}, \quad i = \text{x, y, z}. \tag{127}$$

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 (Bð Þ<sup>o</sup> ) of the external pressure field obtained by the transfer matrix method or by the scattering matrix method.
