B. Depolarization factors

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:

$$N\_k = \frac{abc}{3} \int\_0^\infty \frac{d\sigma}{\left(\sigma + q\_k^2\right) \sqrt{\left(\sigma + a^2\right) \left(\sigma + b^2\right) \left(\sigma + c^2\right)}}\tag{162}$$

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

$$N\_{\mathbf{x}} + N\_{\mathbf{y}} + N\_{\mathbf{z}} = \mathbf{1}.\tag{163}$$

## C. Relations between two spherical coordinates systems

Consider the rectangular coordinate systems (x, y, z) and (x<sup>0</sup> , y<sup>0</sup> ,z0 ). We are looking for the relations between the spherical coordinates (r, θ, φ) and (r<sup>0</sup> , θ<sup>0</sup> , φ<sup>0</sup> ) associated with each of them. From

$$\mathbf{x}' = r' \sin \theta' \cos \phi' \qquad \mathbf{x} = r \sin \theta \cos \phi,\tag{164}$$

$$y' = r' \sin \theta' \sin \phi' \qquad y = r \sin \theta \sin \phi,\tag{165}$$

$$z' = r' \cos \theta' \qquad z = r \sin \theta,\tag{166}$$

one deduces

$$\begin{split} r'^2 &= \mathbf{x'}^2 + \mathbf{y'}^2 + \mathbf{z'}^2 \\ &= \frac{\mathbf{x}^2}{\kappa\_1} + \frac{\mathbf{y}^2}{\kappa\_2} + \frac{\mathbf{z}^2}{\kappa\_3} \\ &= r^2 \left( \frac{\sin^2 \theta \cos^2 \varphi}{\kappa\_1} + \frac{\sin^2 \theta \sin^2 \varphi}{\kappa\_2} + \frac{\cos^2 \theta}{\kappa\_3} \right) \end{split}$$

or

$$r' = r\Delta \quad \text{óu} \qquad \Delta = \sqrt{\frac{\sin^2 \theta \cos^2 \varphi}{\kappa\_1} + \frac{\sin^2 \theta \sin^2 \varphi}{\kappa\_2} + \frac{\cos^2 \theta}{\kappa\_3}}. \tag{167}$$

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

$$\begin{aligned} \chi' &= \frac{\chi}{\kappa\_1} \Rightarrow r' \sin \theta' \cos \varphi' &= \frac{r}{\kappa\_1} \sin \theta \cos \varphi'\\ \chi' &= \frac{\chi}{\kappa\_2} \Rightarrow r' \sin \theta' \sin \varphi' &= \frac{r}{\kappa\_2} \sin \theta \sin \varphi. \end{aligned}$$

By eliminating φ, one finds:

$$
\sin \theta' = \sin \theta \frac{\delta}{\Delta} \quad \text{avec} \quad \delta = \sqrt{\frac{\cos^2 \theta}{\kappa\_1} + \frac{\sin^2 \varphi}{\kappa\_2}}.\tag{168}
$$

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

$$\cos \theta' = \frac{1}{\Delta} \frac{\cos \theta}{\sqrt{\kappa\_3}}.\tag{169}$$

with

<sup>κ</sup><sup>11</sup> <sup>¼</sup> <sup>κ</sup><sup>1</sup> sin <sup>2</sup>

Tortuosity Perturbations Induced by Defects in Porous Media

<sup>κ</sup><sup>22</sup> <sup>¼</sup> <sup>κ</sup><sup>1</sup> cos <sup>2</sup>

Or, alternatively in the matrix form:

B ¼

0

B@

A ¼

93

0

B@

<sup>κ</sup><sup>12</sup> <sup>¼</sup> <sup>κ</sup><sup>1</sup> cos <sup>θ</sup> sin <sup>θ</sup> cos <sup>2</sup>

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

θ cos <sup>2</sup>

θ cos <sup>2</sup>

where I is the unit matrix 3 � 3 and A and B are given by:

sin <sup>2</sup>θ sin <sup>2</sup>φ cos θ sin θ sin <sup>2</sup>φ sin θ cos φ sin φ cos θ sin θ sin <sup>2</sup>φ cos <sup>2</sup>θ sin <sup>2</sup>φ cos θ cos φ sin φ sin θ cos φ sin φ cos θ cos φ sin φ cos <sup>2</sup>θ

> cos <sup>2</sup><sup>θ</sup> � cos <sup>θ</sup> sin <sup>θ</sup> <sup>0</sup> � cos <sup>θ</sup> sin <sup>θ</sup> sin <sup>2</sup><sup>θ</sup> <sup>0</sup> 0 00

<sup>κ</sup><sup>33</sup> <sup>¼</sup> <sup>κ</sup><sup>1</sup> sin <sup>2</sup>

<sup>φ</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> sin <sup>2</sup>

<sup>φ</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> cos <sup>θ</sup> sin <sup>θ</sup> sin <sup>2</sup>

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

θ sin <sup>2</sup>

θ sin <sup>2</sup>

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

<sup>φ</sup> <sup>þ</sup> <sup>κ</sup><sup>3</sup> cos <sup>2</sup>

<sup>φ</sup> <sup>þ</sup> <sup>κ</sup><sup>3</sup> sin <sup>2</sup>

κ<sup>21</sup> ¼ κ12, (179)

κ<sup>31</sup> ¼ κ13, (180)

κ<sup>32</sup> ¼ κ23: (181)

1

κ<sup>13</sup> ¼ ð Þ κ<sup>2</sup> � κ<sup>1</sup> sin θ cos φ sin φ, (175)

κ<sup>23</sup> ¼ ð Þ κ<sup>2</sup> � κ<sup>1</sup> cos θ cos φ sin φ, (177)

k ¼ κ1I þ ð Þ κ<sup>2</sup> � κ<sup>1</sup> A þ ð Þ κ<sup>3</sup> � κ<sup>1</sup> B, (182)

θ, (173)

θ, (176)

φ � κ<sup>3</sup> cos θ sin θ, (174)

φ, (178)

1

CA, (183)

CA: (184)

Similar relationships between angles φ and φ<sup>0</sup> are deduced from (164) and (165):

$$\sin \phi' = \frac{1}{\delta} \frac{\sin \phi}{\sqrt{\kappa\_2}},\tag{170}$$

$$\cos \phi' = \frac{1}{\delta} \frac{\cos \phi}{\sqrt{\kappa\_1}}.\tag{171}$$
