Acknowledgements

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

## A. Ellipsoidal coordinates

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

$$\frac{x^2}{a^2+u} + \frac{y^2}{b^2+u} + \frac{z^2}{c^2+u} = 1.\tag{154}$$

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

$$\infty^2 = \frac{(a^2 + \xi)(a^2 + \eta)(a^2 + \zeta)}{(b^2 - a^2)(c^2 - a^2)},\tag{155}$$

$$y^2 = \frac{\left(b^2 + \xi\right)\left(b^2 + \eta\right)\left(b^2 + \zeta\right)}{\left(a^2 - b^2\right)\left(c^2 - b^2\right)},\tag{156}$$

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

$$x^2 = \frac{(c^2 + \xi)(c^2 + \eta)(c^2 + \zeta)}{(a^2 - c^2)(b^2 - c^2)},\tag{157}$$

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. The scalar factors are the vector norms:

$$h\_{q\_i} = \|\frac{\partial \mathbf{r}}{\partial q\_i}\| \quad \text{ðu} \quad q\_i = \xi, \eta, \zeta. \tag{158}$$

Their values are:

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

<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>

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

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

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

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

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

<sup>x</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>ξ</sup> <sup>a</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>η</sup> <sup>a</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>ζ</sup>

<sup>y</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ξ</sup> � � <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>η</sup> � � <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ζ</sup> � �

þ

z2

<sup>c</sup><sup>2</sup> <sup>þ</sup> <sup>u</sup> <sup>¼</sup> <sup>1</sup>: (154)

<sup>b</sup><sup>2</sup> � <sup>a</sup><sup>2</sup> � � <sup>c</sup><sup>2</sup> � <sup>a</sup><sup>2</sup> ð Þ , (155)

<sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � <sup>c</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � , (156)

<sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>u</sup>

x2 a<sup>2</sup> þ u 3

<sup>5</sup> <sup>1</sup> � <sup>2</sup><sup>f</sup> <sup>3</sup> <sup>κ</sup>ð Þ<sup>i</sup>

" #

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

: (153)

<sup>τ</sup><sup>d</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup>

4

field lines of Figure 4b.

Acoustics of Materials

expression of tortuosity

Acknowledgements

A. Ellipsoidal coordinates

00079.

90

4. Conclusion

R3 � � <sup>κ</sup>ð Þ<sup>i</sup>

!<sup>2</sup> 2

$$h\_{\xi} = \frac{1}{2} \sqrt{\frac{(\eta - \xi)(\xi - \xi)}{(a^2 - \xi)\left(b^2 - \xi\right)(c^2 - \xi)}},\tag{159}$$

$$h\_{\eta} = \frac{1}{2} \sqrt{\frac{(\xi - \eta)(\zeta - \eta)}{(a^2 - \eta)\left(b^2 - \eta\right)(c^2 - \eta)}},\tag{160}$$

$$h\_{\zeta} = \frac{1}{2} \sqrt{\frac{(\eta - \zeta)(\xi - \zeta)}{(a^2 - \zeta)(b^2 - \zeta)(c^2 - \zeta)}}. \tag{161}$$
