3. Tortuosity induced by defects

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

a0

<sup>κ</sup>ð Þ<sup>i</sup> ð Þ� <sup>r</sup> <sup>κ</sup>ð Þ<sup>o</sup> � �

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

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:

> kð Þ<sup>i</sup> <sup>k</sup>ð Þ<sup>o</sup> � <sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>k</sup>ð Þ<sup>i</sup> kð Þ<sup>o</sup>

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:

Sketch of polarization surface "charge" density σpol for a spherical inclination and an ellipsoidal inclusion with

1

0 @

<sup>α</sup> <sup>¼</sup> <sup>3</sup>Vκð Þ<sup>i</sup>

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

dσ

<sup>i</sup> <sup>þ</sup> <sup>σ</sup> � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det A0<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>I</sup>

<sup>r</sup> � � : (122)

<sup>κ</sup>ð Þ<sup>i</sup> <sup>∇</sup>pð Þ<sup>i</sup> ð Þ<sup>r</sup> dV: (123)

<sup>P</sup> <sup>¼</sup> <sup>α</sup>vð Þ<sup>o</sup> , (124)

P ¼ αð Þ �U<sup>∞</sup> (125)

A: (126)

ð<sup>∞</sup> 0

N0

2.5 Hydraulic polarisability

Acoustics of Materials

<sup>i</sup> <sup>¼</sup> detA<sup>0</sup> 2

of the host environment and that of inclusion is itself nonzero.

P ¼ ð Ω

of inclusions, the dipole moment is written as:

with

Figure 11.

84

different orientations with respect to the incident flow.

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 λ if λ=a ≫ 1. 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 κeff is to use it as follows:

$$<\mathbf{v}> = \boldsymbol{\kappa}\_{\text{eff}} < \mathbf{E}>,\tag{128}$$

<sup>κ</sup>ð Þ<sup>o</sup> <sup>E</sup><sup>e</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>E</sup> <sup>þ</sup> <sup>L</sup> � <sup>P</sup>: (138)

. For an ellipsoidal defect, L is reduced to depo-

<sup>1</sup> � <sup>n</sup>αNk=κð Þ<sup>o</sup> , (139)

<sup>1</sup> � <sup>n</sup>αNk=κð Þ<sup>o</sup> , (140)

<sup>1</sup> � <sup>n</sup>α=3κð Þ<sup>o</sup> : (141)

<sup>1</sup> � <sup>n</sup>αxNx=κð Þ<sup>o</sup> : (142)

In this relationship, L is an operator that takes into account the shape of the defect and its orientation with respect to the fluid flow, and P is due to the induced polarization in the inclusion. P defined 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 n identical defects per unit volume, P ¼ np, p being the dipole moment of each defect, and since p is proportional to the applied

larization factors, i.e., L ¼ Nk, k ¼ x, y, z, which takes into account the direction of

nα

nα

nα<sup>x</sup>

It is then possible to calculate the effective mobility of a set of ellipsoidal inclu-

Ee <sup>¼</sup> <sup>E</sup>

<sup>κ</sup>eff <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup>

<sup>κ</sup>eff <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup>

• the ellipsoids are aligned with the direction x of the fluid flow:

<sup>κ</sup>eff <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup>

field (<sup>p</sup> <sup>¼</sup> <sup>α</sup>E<sup>e</sup>

Figure 12.

87

), we have <sup>P</sup> <sup>¼</sup> <sup>n</sup>αE<sup>e</sup>

Tortuosity Perturbations Induced by Defects in Porous Media

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

fluid flow. In this case, the excitation field is:

leading to the following κeff expression:

leading, for spherical defects, to expression:

Sets of ellipsoidal inclusions: (a) aligned and (b) randomly oriented.

sions in different geometries (Figure 12):

where E ¼ �∇p and < � > is the averaging operation. The mean values of the filtration rate and the pressure gradient are given by:

$$<\mathbf{v}>\quad = f < \kappa^{(i)}\mathbf{E}^{(i)}> + (1 - f) < \kappa^{(o)}\mathbf{E}^{(o)}>,\tag{129}$$

$$\mathbf{k} < \mathbf{E} > \ \ = f < \mathbf{E}^{(i)} > + (\mathbf{1} - f) < \mathbf{E}^{(o)} > , \tag{130}$$

where <sup>f</sup> is the volumic fraction of the defect. Putting <sup>E</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>A</sup>Eð Þ<sup>o</sup> we show that:

$$\kappa\_{\sharp f'} = \frac{f\mathcal{A}\kappa^{(i)} + (\mathbf{1} - f)\kappa^{(o)}}{f\mathcal{A} + (\mathbf{1} - f)}. \tag{131}$$

For spherical defects, <sup>A</sup> <sup>¼</sup> <sup>3</sup>κð Þ<sup>o</sup> <sup>=</sup> <sup>2</sup>κð Þ<sup>o</sup> <sup>þ</sup> <sup>κ</sup>ð Þ<sup>i</sup> , Eq. (131) leads to the result:

$$\kappa\_{\sharp f} = \kappa^{(o)} \frac{f \frac{3\mathbf{x}^{(i)}}{2\mathbf{x}^{(o)} + \mathbf{x}^{(i)}} + (\mathbf{1} - f)}{f \frac{3\mathbf{x}^{(o)}}{2\mathbf{x}^{(o)} + \mathbf{x}^{(i)}} + (\mathbf{1} - f)}. \tag{132}$$

When <sup>f</sup> ! 0, then <sup>κ</sup>eff � <sup>κ</sup>ð Þ<sup>o</sup> , and when <sup>f</sup> ! 1, then <sup>κ</sup>eff � <sup>κ</sup>ð Þ<sup>i</sup> . Finally when f < <1, then

$$
\kappa\_{\sharp f} \sim \kappa^{(o)} + \mathfrak{F}\kappa^{(o)} \frac{\kappa^{(i)} - \kappa^{(o)}}{\kappa^{(i)} + 2\kappa^{(o)}}.\tag{133}
$$

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

$$ = \kappa\_{\mathfrak{gl}^\circ, \mathfrak{i}\mathfrak{j}} < E\_{\mathfrak{j}}>. \tag{134}$$

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

$$ \, = \kappa\_{\rm eff,j} < E\_j>, \quad j=x,y,z,\tag{135}$$

where κeff,j are the eigenvalues of κeff . Taking into account that Eð Þ<sup>i</sup> <sup>j</sup> <sup>¼</sup> <sup>A</sup>jEð Þ<sup>o</sup> j , Eq. (135) shows that:

$$\kappa\_{\sharp\overline{f}^{\circ},j} = \frac{f\mathcal{A}\_{\circ}\kappa^{(i)} + (1-f)\kappa^{(o)}}{f\mathcal{A}\_{\circ} + (1-f)},\tag{136}$$

with

$$\mathcal{A}\_{j} = \frac{\mathbf{3}\kappa^{(o)}}{2\kappa^{(o)} + \kappa\_{j}^{(i)}}, \quad j = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{137}$$

When the environment has several defects, the calculation of keff is more complicated because their mutual influence must be taken into account. The excitation pressure gradient E<sup>e</sup> defined from the filtration rate is introduced by the equation:

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

<v> ¼ κeff <E >, (128)

<sup>f</sup><sup>A</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> : (131)

<sup>2</sup>κð Þ<sup>o</sup> <sup>þ</sup>κð Þ<sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> : (132)

<sup>κ</sup>ð Þ<sup>i</sup> <sup>þ</sup> <sup>2</sup>κð Þ<sup>o</sup> : (133)

<sup>j</sup> <sup>¼</sup> <sup>A</sup>jEð Þ<sup>o</sup>

j ,

<vi > ¼ κeff,ij <Ej >: (134)

<sup>f</sup>A<sup>j</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> , (136)

, j ¼ 1; 2; 3: (137)

<vj > ¼ κeff,j <Ej >, j ¼ x, y, z, (135)

<sup>&</sup>lt; <sup>v</sup><sup>&</sup>gt; <sup>¼</sup> <sup>f</sup> <sup>&</sup>lt;κð Þ<sup>i</sup> <sup>E</sup>ð Þ<sup>i</sup> <sup>&</sup>gt; <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> <sup>&</sup>lt; <sup>κ</sup>ð Þ<sup>o</sup> <sup>E</sup>ð Þ<sup>o</sup> <sup>&</sup>gt; , (129)

<sup>&</sup>lt; <sup>E</sup><sup>&</sup>gt; <sup>¼</sup> <sup>f</sup> <sup>&</sup>lt;Eð Þ<sup>i</sup> <sup>&</sup>gt; <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> <sup>&</sup>lt; <sup>E</sup>ð Þ<sup>o</sup> <sup>&</sup>gt;, (130)

where E ¼ �∇p and < � > is the averaging operation. The mean values of the

where <sup>f</sup> is the volumic fraction of the defect. Putting <sup>E</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>A</sup>Eð Þ<sup>o</sup> we show that:

<sup>2</sup>κð Þ<sup>o</sup> <sup>þ</sup>κð Þ<sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup>

<sup>κ</sup>eff <sup>¼</sup> <sup>f</sup>Aκð Þ<sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> <sup>κ</sup>ð Þ<sup>o</sup>

For spherical defects, <sup>A</sup> <sup>¼</sup> <sup>3</sup>κð Þ<sup>o</sup> <sup>=</sup> <sup>2</sup>κð Þ<sup>o</sup> <sup>þ</sup> <sup>κ</sup>ð Þ<sup>i</sup> , Eq. (131) leads to the result:

f <sup>3</sup>κð Þ<sup>o</sup>

When <sup>f</sup> ! 0, then <sup>κ</sup>eff � <sup>κ</sup>ð Þ<sup>o</sup> , and when <sup>f</sup> ! 1, then <sup>κ</sup>eff � <sup>κ</sup>ð Þ<sup>i</sup> . Finally when

<sup>κ</sup>eff � <sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> <sup>3</sup><sup>f</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup>

where κeff,j are the eigenvalues of κeff . Taking into account that Eð Þ<sup>i</sup>

<sup>A</sup><sup>j</sup> <sup>¼</sup> <sup>3</sup>κð Þ<sup>o</sup> 2κð Þ<sup>o</sup> þ κ

<sup>κ</sup>eff,j <sup>¼</sup> <sup>f</sup>Ajκð Þ<sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>f</sup> <sup>κ</sup>ð Þ<sup>o</sup>

ð Þi j

When the environment has several defects, the calculation of keff is more complicated because their mutual influence must be taken into account. The excitation pressure gradient E<sup>e</sup> defined from the filtration rate is introduced by

For anisotropic inclusion, mobility is a second rank tensor defined by the rela-

<sup>κ</sup>eff <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>f</sup> <sup>3</sup>κð Þ<sup>i</sup>

filtration rate and the pressure gradient are given by:

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

f < <1, then

Acoustics of Materials

tionship:

Eq. (135) shows that:

with

the equation:

86

$$
\kappa^{(o)}\mathbf{E}^{\epsilon} = \kappa^{(o)}\mathbf{E} + L \cdot P. \tag{138}
$$

In this relationship, L is an operator that takes into account the shape of the defect and its orientation with respect to the fluid flow, and P is due to the induced polarization in the inclusion. P defined 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 n identical defects per unit volume, P ¼ np, p being the dipole moment of each defect, and since p is proportional to the applied field (<sup>p</sup> <sup>¼</sup> <sup>α</sup>E<sup>e</sup> ), we have <sup>P</sup> <sup>¼</sup> <sup>n</sup>αE<sup>e</sup> . For an ellipsoidal defect, L is reduced to depolarization factors, i.e., L ¼ Nk, k ¼ x, y, z, which takes into account the direction of fluid flow. In this case, the excitation field is:

$$E' = \frac{E}{1 - naN\_k/\kappa^{(o)}},\tag{139}$$

leading to the following κeff expression:

$$\kappa\_{\mathcal{G}\overline{f}} = \kappa^{(o)} + \frac{na}{1 - naN\_k/\kappa^{(o)}},\tag{140}$$

leading, for spherical defects, to expression:

$$
\kappa\_{\mathcal{G}\overline{f}} = \kappa^{(o)} + \frac{na}{1 - na/3\kappa^{(o)}}.\tag{141}
$$

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 x of the fluid flow:

$$\kappa\_{\sharp f} = \kappa^{(o)} + \frac{na\_{\mathfrak{x}}}{1 - na\_{\mathfrak{x}}N\_{\mathfrak{x}}/\kappa^{(o)}}.\tag{142}$$

Figure 12. Sets of ellipsoidal inclusions: (a) aligned and (b) randomly oriented.

Figure 13. Domains used in numerical simulations for the evaluation of <vð Þ<sup>o</sup> <sup>2</sup> >.

• the ellipsoids are randomly oriented:

$$\kappa\_{\mathfrak{eff}} = \kappa^{(o)} + \frac{\mathbf{1}/\mathfrak{Z} \sum\_{i=\mathbf{x}, \mathbf{y}, \mathbf{z}} na\_i}{\mathbf{1} - \sum\_{i=\mathbf{x}, \mathbf{y}, \mathbf{z}} na\_i N\_i/\kappa^{(o)}}.\tag{143}$$

For ellipsoidal inclusion, the external pressure is:

Tortuosity Perturbations Induced by Defects in Porous Media

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

<sup>v</sup>ð Þ<sup>o</sup> <sup>2</sup> <sup>≈</sup> <sup>U</sup><sup>∞</sup> κð Þ<sup>o</sup> 2

<sup>&</sup>lt;vð Þ<sup>o</sup> <sup>2</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>U</sup><sup>2</sup>

Figure 14.

89

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼� <sup>r</sup>; <sup>θ</sup> <sup>U</sup><sup>∞</sup>

¼ � <sup>U</sup><sup>∞</sup>

inclusion. By keeping only the terms greater than or equal to r�2, one obtains:

<sup>1</sup> � <sup>2</sup><sup>α</sup>

that the dipolar effects are negligible. For a spherical inclusion, it results:

<sup>∞</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup>

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

< vð Þ<sup>o</sup> ><sup>2</sup> is calculated from the definition of effective mobility:

<sup>κ</sup>eff <sup>≈</sup>κð Þ<sup>o</sup> <sup>þ</sup> <sup>f</sup><sup>A</sup> <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> � � and <sup>&</sup>lt; <sup>E</sup><sup>&</sup>gt; <sup>≈</sup> <sup>U</sup><sup>∞</sup>

Evaluation of the tortuosity induced as a function of the distance to the defect.

<sup>κ</sup>ð Þ<sup>o</sup> <sup>r</sup> cos <sup>θ</sup> <sup>þ</sup>

<sup>κ</sup>ð Þ<sup>o</sup> <sup>r</sup> � <sup>α</sup>

where P<sup>d</sup> and α are, respectively, the dipol moment and the polarisability of the

<sup>r</sup><sup>3</sup> 2 cos <sup>2</sup>

The average value <vð Þ<sup>o</sup> <sup>2</sup> > is calculated by integration on the volume between two spheres of radius a (characteristic size of the defect) and R sufficiently large so

> <sup>∞</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup> R3 � � <sup>κ</sup>ð Þ<sup>i</sup>

r2

P<sup>d</sup> 4π

<sup>θ</sup> � sin <sup>2</sup>

<sup>θ</sup> � � � � (149)

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

!<sup>2</sup>

< v> ¼ κeff <E > (151)

cos θ

� � cos <sup>θ</sup> (148)

<sup>r</sup><sup>2</sup> (147)

: (150)

<sup>κ</sup>ð Þ<sup>o</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>f</sup><sup>A</sup> : (152)

#### 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 τ<sup>d</sup> is defined by:

$$\pi\_d = \frac{}{^2} \tag{144}$$

where vð Þ<sup>o</sup> is the perturbation of filtration rate due to the defects. When the ratio k=a<sup>2</sup> where a is 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 <v<sup>2</sup> > , 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 <vð Þ<sup>o</sup> <sup>2</sup> > is then:

$$ =  + \tag{145}$$

where

$$v\_r^{(o)} = -\kappa^{(o)} \frac{\partial p^{(o)}}{\partial r}, \text{ and } v\_\theta^{(o)} = -\kappa^{(o)} \frac{\partial p^{(o)}}{r \partial \theta}. \tag{146}$$

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

For ellipsoidal inclusion, the external pressure is:

$$p^{(o)}(r,\theta) = -\frac{U\_{\infty}}{\kappa^{(o)}}r\cos\theta + \frac{\mathcal{P}\_d}{4\pi}\frac{\cos\theta}{r^2} \tag{147}$$

$$=-\frac{U\_{\infty}}{\kappa^{(o)}} \left(r - \frac{a}{r^2}\right) \cos\theta \tag{148}$$

where P<sup>d</sup> and α are, respectively, the dipol moment and the polarisability of the inclusion. By keeping only the terms greater than or equal to r�2, one obtains:

$$v^{(o)2} \approx \frac{U\_{\infty}}{\kappa^{(o)}} \left[ 1 - \frac{2a}{r^3} \left( 2\cos^2\theta - \sin^2\theta \right) \right] \tag{149}$$

The average value <vð Þ<sup>o</sup> <sup>2</sup> > is calculated by integration on the volume between two spheres of radius a (characteristic size of the defect) and R sufficiently large so that the dipolar effects are negligible. For a spherical inclusion, it results:

$$ = U^2\_{\infty} + U^2\_{\infty} \left( 1 - \frac{a^3}{R^3} \right) \left( \frac{\frac{\kappa^{(i)}}{\kappa^{(o)}} - 1}{2 + \frac{\kappa^{(i)}}{\kappa^{(o)}}} \right)^2. \tag{150}$$

< vð Þ<sup>o</sup> ><sup>2</sup> is calculated from the definition of effective mobility:

$$<\upsilon> = \kappa\_{\tilde{\mathcal{U}}} < E>\tag{151}$$

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

$$
\kappa\_{\sharp \overline{\mathcal{U}}} \approx \kappa^{(o)} + f\mathcal{A}\left(\kappa^{(i)} - \kappa^{(o)}\right) \quad \text{and} \quad < E> \approx \frac{U\_{\infty}}{\kappa^{(o)}} (1 + f\mathcal{A}).\tag{152}
$$

Figure 14. Evaluation of the tortuosity induced as a function of the distance to the defect.

• the ellipsoids are randomly oriented:

Domains used in numerical simulations for the evaluation of <vð Þ<sup>o</sup> <sup>2</sup> >.

it is taken into account for that of < v> .

terms, the expression of <vð Þ<sup>o</sup> <sup>2</sup> > is then:

vð Þ<sup>o</sup>

<sup>r</sup> ¼ �κð Þ<sup>o</sup> <sup>∂</sup>pð Þ<sup>o</sup>

3.3 Induced tortuosity

Figure 13.

Acoustics of Materials

where

88

<sup>κ</sup>eff <sup>¼</sup> <sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup>

<sup>1</sup>=<sup>3</sup> <sup>∑</sup><sup>i</sup>¼x, y, <sup>z</sup> <sup>n</sup>α<sup>i</sup>

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 τ<sup>d</sup> is defined by:

<sup>τ</sup><sup>d</sup> <sup>¼</sup> <sup>&</sup>lt;vð Þ<sup>o</sup> <sup>2</sup> <sup>&</sup>gt;

where vð Þ<sup>o</sup> is the perturbation of filtration rate due to the defects. When the ratio k=a<sup>2</sup> where a is 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 <v<sup>2</sup> > , whereas

With the pressure scattered field by the inclusions being limited to the dipolar

<sup>&</sup>lt; <sup>v</sup>ð Þ<sup>o</sup> <sup>2</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>&</sup>lt;vð Þ<sup>o</sup> <sup>2</sup> <sup>r</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>&</sup>lt; <sup>v</sup>

<sup>∂</sup><sup>r</sup> , and <sup>v</sup>

ð Þo

<sup>1</sup> � <sup>∑</sup><sup>i</sup>¼x, y, <sup>z</sup> <sup>n</sup>αiNi=κð Þ<sup>o</sup> : (143)

<sup>&</sup>lt;vð Þ<sup>o</sup> <sup>&</sup>gt;<sup>2</sup> (144)

<sup>θ</sup> > (145)

<sup>r</sup>∂<sup>θ</sup> : (146)

ð Þo 2

<sup>θ</sup> ¼ �κð Þ<sup>o</sup> <sup>∂</sup>pð Þ<sup>o</sup>

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

$$\tau\_d = \left[ \mathbf{1} + \left( \mathbf{1} - \frac{a^3}{R^3} \right) \left( \frac{\frac{\kappa^{(i)}}{\kappa^{(o)}} - \mathbf{1}}{2 + \frac{\kappa^{(i)}}{\kappa^{(o)}}} \right)^2 \right] \left[ \mathbf{1} - \mathbf{2} f \frac{\mathbf{3} \frac{\kappa^{(i)}}{\kappa^{(o)}}}{2 + \frac{\kappa^{(i)}}{\kappa^{(o)}}} \right]. \tag{153}$$

<sup>z</sup><sup>2</sup> <sup>¼</sup> <sup>c</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>ξ</sup> <sup>c</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>η</sup> <sup>c</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>ζ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ η � ξ ð Þ ζ � ξ <sup>a</sup>ð Þ <sup>2</sup> � <sup>ξ</sup> <sup>b</sup><sup>2</sup> � <sup>ξ</sup> � � <sup>c</sup>ð Þ <sup>2</sup> � <sup>ξ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ ξ � η ð Þ ζ � η <sup>a</sup>ð Þ <sup>2</sup> � <sup>η</sup> <sup>b</sup><sup>2</sup> � <sup>η</sup> � � <sup>c</sup>ð Þ <sup>2</sup> � <sup>η</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ η � ζ ð Þ ξ � ζ <sup>a</sup>ð Þ <sup>2</sup> � <sup>ζ</sup> <sup>b</sup><sup>2</sup> � <sup>ζ</sup> � � <sup>c</sup>ð Þ <sup>2</sup> � <sup>ζ</sup>

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.

dσ

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð Þ <sup>σ</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � <sup>σ</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> ð Þ <sup>q</sup> (162)

, y<sup>0</sup> ,z0 ). We are

, θ<sup>0</sup> , φ<sup>0</sup> )

Nx þ Ny þ Nz ¼ 1: (163)

cos φ<sup>0</sup> x ¼ rsin θ cos φ, (164)

sin φ<sup>0</sup> y ¼ rsin θ sin φ, (165)

cos θ<sup>0</sup> z ¼ rsin θ, (166)

subject to the conditions �ξ<c<sup>2</sup> <sup>&</sup>lt; � <sup>η</sup><b<sup>2</sup> <sup>&</sup>lt; � <sup>ζ</sup> <sup>&</sup>lt;a2.

hqi <sup>¼</sup> <sup>∥</sup> <sup>∂</sup><sup>r</sup> ∂qi

s

s

s

<sup>h</sup><sup>ξ</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>h</sup><sup>η</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>h</sup><sup>ζ</sup> <sup>¼</sup> <sup>1</sup> 2

The scalar factors are the vector norms:

Tortuosity Perturbations Induced by Defects in Porous Media

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

Their values are:

B. Depolarization factors

Nk <sup>¼</sup> abc 3

associated with each of them. From

91

x<sup>0</sup> ¼ r 0 sin θ<sup>0</sup>

y<sup>0</sup> ¼ r 0 sin θ<sup>0</sup>

> z<sup>0</sup> ¼ r 0

ð<sup>∞</sup> 0

σ þ q<sup>2</sup> k

C. Relations between two spherical coordinates systems

Consider the rectangular coordinate systems (x, y, z) and (x<sup>0</sup>

looking for the relations between the spherical coordinates (r, θ, φ) and (r<sup>0</sup>

where k ¼ x (resp. y, z) et qk ¼ a, (resp. b, c) and satisfy the relation:

Their expression is:

<sup>a</sup><sup>2</sup> � <sup>c</sup><sup>2</sup> ð Þ <sup>b</sup><sup>2</sup> � <sup>c</sup><sup>2</sup> � � , (157)

∥ òu qi ¼ ξ, η, ζ: (158)

, (159)

, (160)

: (161)

Results of numerical simulations: The results of a numerical simulation for <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> <sup>¼</sup> 10 and <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> <sup>¼</sup> 100 are shown in Figure 14. The tortuosity value is calculated on square domains around the inclusion (Figure 13). Inside the inclusion, τ<sup>b</sup> ¼ 1. As x increases, the tortuosity increases to reach its maximum value at x ¼ 1:7 for <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> <sup>¼</sup> 10 and <sup>x</sup> <sup>¼</sup> <sup>1</sup>:6 when <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> <sup>¼</sup> 10. For larger values of <sup>x</sup>, 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.
