2. Defect model

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

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

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 ρ<sup>f</sup> . 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 ρ<sup>f</sup> τ0, where τ<sup>0</sup> is

<sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>&</sup>lt;v<sup>2</sup> <sup>&</sup>gt;

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.

where < :> denotes averaging over the pore fluid volume Vf . Thereafter, in this

<sup>&</sup>lt;v><sup>2</sup> , (3)

different definitions of tortuosity, each with its own interpretation.

the static tortuosity for a constant flow (ω ¼ 0) defined by:

chapter, we adopt this definition of tortuosity.

greater tortuosity.

62

Figure 1.

Acoustics of Materials

Some tortuous paths through a porous medium.

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 Ω characterized by its shape and own parameters: intrinsic permeability kð Þ<sup>i</sup> and porosity ϕð Þ<sup>i</sup> . Intrinsic permeability is expressed in darcy: 1<sup>D</sup> <sup>¼</sup> <sup>0</sup>:<sup>97</sup> � <sup>10</sup>�12m2. The porous media we are interested in have permeabilities of the order of 10D. Moreover, it is supposed that the fluid saturating the inclusion Ω is the same (with viscosity coefficient η) as that flowing in the porous medium. Thereafter the mobility of the fluid defined by κ ¼ k=η is 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 ϕð Þ<sup>o</sup> and permeability kð Þ<sup>o</sup> . The saturating fluid is subject to action of a uniform pressure gradient ∇pð Þ <sup>0</sup> . 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 v and the pressure gradient ∇p are related by the Darcy's law:

$$\mathbf{v}^{(m)}(\mathbf{x}) = \frac{k^{(m)}(\mathbf{x})}{\eta} \nabla p^{(m)}(\mathbf{x}),\tag{4}$$

where m ¼ i if x∈ Ω and m ¼ o if x is in the host medium (x∉ Ω). These equations are subject to the following boundary conditions on ∂Ω:

• continuity of fluid flow

$$
\phi^{(o)}\mathbf{v}^{(o)}.\mathbf{n}^{(o)} = \phi^{(i)}\mathbf{v}^{(i)}.\mathbf{n}^{(i)},\tag{5}
$$

Figure 2. Oriented inclusion in a porous medium.

• continuity of normal stress component

$$
\pi^{(o)}\_{\vec{\imath}\vec{\jmath}} \mathfrak{n}^{(o)}\_{\vec{\jmath}} = \mathfrak{r}^{(i)}\_{\vec{\imath}\vec{\jmath}} \mathfrak{n}^{(i)}\_{\vec{\jmath}} \tag{6}
$$

<sup>v</sup>ð Þ <sup>m</sup> <sup>¼</sup> <sup>k</sup>ð Þ <sup>m</sup> η

Tortuosity Perturbations Induced by Defects in Porous Media

ð Þ<sup>i</sup> ð Þ¼ <sup>r</sup> <sup>¼</sup> <sup>a</sup> <sup>ϕ</sup>ð Þ<sup>i</sup> vr

In spherical coordinates, this equation is written as:

þ 1 sin θ

l

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

Að Þ <sup>m</sup> <sup>l</sup> r

Plð Þ <sup>x</sup> being the Legendre polynomial of degree <sup>l</sup>. The coefficients <sup>A</sup>ð Þ <sup>m</sup>

are related by the boundary conditions (8), those at r ¼ 0 and when r ! ∞. Inside the inclusion, the pressure must be finite at <sup>r</sup> <sup>¼</sup> 0. This condition leads to <sup>B</sup>ð Þ<sup>i</sup>

> l Að Þ<sup>i</sup> l r l

<sup>Δ</sup><sup>p</sup> <sup>¼</sup> <sup>∂</sup> ∂r r <sup>2</sup> ∂p ∂r 

not depend on φ. It follows that its solution is:

<sup>p</sup>ð Þ <sup>m</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup> <sup>∑</sup>

with the boundary conditions at r ¼ a:

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

ϕð Þ<sup>i</sup> vr

where

Eq. (4) becomes

for all l. So pð Þ<sup>i</sup> becomes:

Figure 3.

65

Spherical inclusion in a fluid flow.

∇pð Þ <sup>m</sup> where

τrr ¼ ϕ �p þ 2η

vr being the radial component of the velocity. For an incompressible fluid,

∂p <sup>∂</sup><sup>θ</sup> sin <sup>θ</sup>

The spherical symmetry of the problem (Figure 3) implies that the solution does

<sup>l</sup> <sup>þ</sup> <sup>B</sup>ð Þ <sup>m</sup> <sup>l</sup> r

ð Þ<sup>i</sup> ð Þ <sup>r</sup> <sup>¼</sup> <sup>a</sup> , <sup>τ</sup>ð Þ<sup>o</sup>

m ¼ i if r<a m ¼ o if r> a

rr <sup>r</sup>¼<sup>a</sup> <sup>¼</sup> <sup>τ</sup>ð Þ<sup>i</sup>

<sup>Δ</sup>pð Þ <sup>m</sup> <sup>¼</sup> <sup>0</sup> <sup>m</sup> <sup>¼</sup> i, o: (10)

þ

�ð Þ <sup>l</sup>þ<sup>1</sup> Plð Þ cos <sup>θ</sup> , (12)

1 sin <sup>2</sup>θ ∂2 p ∂2

Plð Þ cos θ : (13)

∂p ∂θ 

∂vr ∂r

rr

, (9)

<sup>r</sup>¼<sup>a</sup> <sup>∀</sup>θ, (8)

<sup>φ</sup> : (11)

<sup>l</sup> and <sup>B</sup>ð Þ <sup>m</sup> l

<sup>l</sup> ¼ 0

(7)

where nð Þ<sup>o</sup> and nð Þ<sup>i</sup> are 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 U<sup>∞</sup> under the action of the pressure gradient along the Ox axis. 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 ϕð Þ<sup>o</sup> and ϕð Þ<sup>i</sup> and permeability kð Þ<sup>o</sup> and kð Þ<sup>i</sup> which have constant values. A static incident pressure with a constant gradient along Ox axis 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 ¼ a, centered at the origin of axes with a constant permeability kð Þ<sup>o</sup> . Using the spherical coordinates (r, θ, φ), Eqs. (4)–(6) become

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

$$\mathbf{v}^{(m)} = \frac{k^{(m)}}{\eta} \nabla p^{(m)} \quad \text{where} \begin{cases} m = i & \text{if } r < a \\ m = o & \text{if } r > a \end{cases} \tag{7}$$

with the boundary conditions at r ¼ a:

$$\phi^{(i)}\boldsymbol{v}\_{r}^{(i)}(\boldsymbol{r}=\boldsymbol{a})=\phi^{(i)}\boldsymbol{v}\_{r}^{(i)}(\boldsymbol{r}=\boldsymbol{a}),\qquad\tau\_{rr}^{(o)}\big|\_{\boldsymbol{r}=\boldsymbol{a}}=\tau\_{rr}^{(i)}\big|\_{\boldsymbol{r}=\boldsymbol{a}}\quad\forall\theta,\tag{8}$$

where

• continuity of normal stress component

Oriented inclusion in a porous medium.

Figure 2.

Acoustics of Materials

when defects are present in the medium.

2.2 Isotropic homogeneous defect

2.2.1 Spherical defect

64

τ ð Þo ij <sup>n</sup>ð Þ<sup>o</sup> <sup>j</sup> ¼ τ ð Þi ij <sup>n</sup>ð Þ<sup>i</sup>

[7]. Figure 2 represents an oriented inclusion in a fluid in motion.

where nð Þ<sup>o</sup> and nð Þ<sup>i</sup> are 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

A porous medium of infinite extension is considered in which a viscous fluid flows at a constant uniform velocity U<sup>∞</sup> under the action of the pressure gradient along the Ox axis. We want to determine the local changes of the fluid velocity

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.

In this section, we assume that the background and the defect (embedded inclusion) are homogeneous each with its own parameters: porosity ϕð Þ<sup>o</sup> and ϕð Þ<sup>i</sup> and permeability kð Þ<sup>o</sup> and kð Þ<sup>i</sup> which have constant values. A static incident pressure with a constant gradient along Ox axis 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.

The simplest type of inclusion is the homogeneous spherical one, and we consider a porous sphere of radius r ¼ a, centered at the origin of axes with a constant permeability kð Þ<sup>o</sup> . Using the spherical coordinates (r, θ, φ), Eqs. (4)–(6) become

<sup>j</sup> (6)

$$
\pi\_{rr} = \phi \left( -p + 2\eta \frac{\partial v\_r}{\partial r} \right),
\tag{9}
$$

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

$$
\Delta p^{(m)} = 0 \quad m = i, o. \tag{10}
$$

In spherical coordinates, this equation is written as:

$$
\Delta p = \frac{\partial}{\partial r} \left( r^2 \frac{\partial p}{\partial r} \right) + \frac{1}{\sin \theta} \frac{\partial p}{\partial \theta} \left( \sin \theta \frac{\partial p}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2 p}{\partial^2 \rho}. \tag{11}
$$

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

$$\mathbf{p}^{(m)}(r,\theta) = \sum\_{l} \left( A\_{l}^{(m)} r^{l} + B\_{l}^{(m)} r^{-(l+1)} \right) P\_{l}(\cos \theta),\tag{12}$$

Plð Þ <sup>x</sup> being the Legendre polynomial of degree <sup>l</sup>. The coefficients <sup>A</sup>ð Þ <sup>m</sup> <sup>l</sup> and <sup>B</sup>ð Þ <sup>m</sup> l are related by the boundary conditions (8), those at r ¼ 0 and when r ! ∞. Inside the inclusion, the pressure must be finite at <sup>r</sup> <sup>¼</sup> 0. This condition leads to <sup>B</sup>ð Þ<sup>i</sup> <sup>l</sup> ¼ 0 for all l. So pð Þ<sup>i</sup> becomes:

Að Þ<sup>i</sup>

l

$$p^{(i)}(r, \theta) = \sum\_{l} A\_{l}^{(0)} r! P\_{l}(\cos \theta). \tag{13}$$

$$\begin{array}{ccc} & & & \textbf{y} & & \\ \hline \longrightarrow & & & & & \\ \hline \longrightarrow & & & & & \\ \hline \longrightarrow & & & & & \\ \hline \longrightarrow & & & & & \\ \hline \end{array} \qquad \begin{array}{ccc} & & & & \\ & & & & \\ & & & & \\ & & & & \\ & & & & \\ & & & & \\ \hline \end{array} \longrightarrow \begin{array}{ccc} & & & & \\ & & & & \\ & & & & \\ \hline \end{array} \longrightarrow x$$

Figure 3. Spherical inclusion in a fluid flow.

Outside, far from the inclusion, the pressure is:

$$p^{(o)} \sim -\frac{1}{\kappa^{(o)}} U\_{\infty} r cos \theta. \tag{14}$$

• 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

U∞κð Þ<sup>i</sup>

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

1

1

κð Þ<sup>i</sup> <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>

κð Þ<sup>i</sup> <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>

Psph ¼ αð Þ �U<sup>∞</sup> , (27)

! !

! !

A: (23)

A cosð Þθ : (24)

, (25)

: (26)

0 @

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>

> a3 r3

r3

0 @

<sup>P</sup>sph <sup>¼</sup> <sup>4</sup>πa<sup>3</sup>

<sup>σ</sup>sphð Þ¼ <sup>θ</sup> <sup>4</sup>πa<sup>3</sup>

Tortuosity Perturbations Induced by Defects in Porous Media

vð Þ<sup>o</sup>

v ð Þo

host medium. In our case, the dipole moment is

For a spherical inclusion of volume V, the susceptibility α is:

its stream lines.

Figure 4.

67

The corresponding density of induced "hydraulic" surface charges is

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

<sup>r</sup> ¼ �U<sup>∞</sup> cos θ 1 þ 2

<sup>θ</sup> <sup>¼</sup> <sup>U</sup><sup>∞</sup> sin <sup>θ</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup>

Figure 4 represents the seepage rate in the porous medium for κð Þ<sup>i</sup> > κð Þ<sup>o</sup> . Figure 4a shows the levels of the amplitude of the velocity, while Figure 4b shows

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

where α is a susceptibility which assesses the polarizability, i.e., the capacity of the porous inclusion to induce a dipole <sup>P</sup>sph under the action of excitation �U∞=κð Þ<sup>o</sup> .

Fluid flow through an inclusion in a porous medium: (a) velocity levels and (b) streamlines (κð Þ<sup>i</sup> =κð Þ<sup>o</sup> >1).

U∞κð Þ<sup>i</sup>

moment Psph of which is

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

It follows that the condition

$$p^{(o)}(r=a,\theta) = p^{(i)}(r=a,\theta) \quad \forall \theta,\tag{15}$$

which implies that only the term with l ¼ 1 remains in the sum. Thus, we get:

$$A\_1^{(o)} = -\frac{1}{\kappa^{(o)}} U\_\infty. \tag{16}$$

The expressions of the pressure are then:

$$p^{(i)} = A\_1^{(i)} r \cos \theta \quad \text{if} \ r < a,\tag{17}$$

$$p^{(o)} = -\frac{U\_{\infty}}{\kappa^{(o)}} r \cos \theta + B\_1^{(o)} r^{-2} \cos \theta \quad \text{if} \quad r > a,\tag{18}$$

Að Þ<sup>i</sup> <sup>1</sup> and <sup>B</sup>ð Þ<sup>o</sup> <sup>1</sup> been given by the conditions (8). Finally, these expressions are:

$$p^{(i)}(r,\theta) = -\frac{\phi^{(o)}}{\phi^{(i)}} \frac{U\_{\infty}}{\kappa^{(o)}} \left(\frac{\mathfrak{Z} + \mathfrak{1}\mathfrak{2}\frac{k^{(o)}}{a^2}}{\mathfrak{2} + \frac{k^{(i)}}{k^{(o)}} + \mathfrak{1}\mathfrak{2}\frac{k^{(i)}}{a^2}}\right) r\cos\theta, \quad r < a \tag{19}$$

$$p^{(o)}(r,\theta) = -\frac{U\_{\infty}}{\kappa^{(o)}}r\cos\theta + \frac{U\_{\infty}}{\kappa^{(o)}}\frac{a^3}{r^2}\left(\frac{\frac{k^{(i)}}{k^{(o)}} - \mathbf{1}}{2 + \frac{k^{(i)}}{k^{(o)}} + \mathbf{1}2\frac{k^{(i)}}{a^7}}\right)\cos\theta, \quad r > a \tag{20}$$

For defects with radius <sup>r</sup> � <sup>10</sup>�<sup>2</sup> , 10�<sup>3</sup> m, the quantities kð Þ<sup>i</sup> =a<sup>2</sup> and kð Þ<sup>o</sup> =a<sup>2</sup> are very small compared to unit (≃ 10�<sup>5</sup> , 10�6) 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

$$w^{(i)} = U\_{\infty} \frac{\mathfrak{Z}}{\mathbf{1} + \mathfrak{Z}^{\kappa^{(o)}}\_{\kappa^{(i)}}}.\tag{21}$$

If κð Þ<sup>i</sup> > κð Þ<sup>o</sup> , then vð Þ<sup>i</sup> > 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 κð Þ<sup>i</sup> <κð Þ<sup>o</sup> . When <sup>κ</sup>ð Þ<sup>o</sup> ! 0, i.e., for low permeability <sup>k</sup>ð Þ<sup>o</sup> , then <sup>v</sup>ð Þ<sup>i</sup> � <sup>3</sup>U∞. If <sup>κ</sup>ð Þ<sup>i</sup> ! 0, then <sup>v</sup>ð Þ<sup>i</sup> ! 0. Eq. (21) can also be written in the form:

$$\boldsymbol{w}^{(i)} = \boldsymbol{U}\_{\infty} \frac{\boldsymbol{\kappa}^{(i)}}{\kappa^{(o)} + \frac{1}{3} (\kappa^{(i)} - \kappa^{(o)})} \tag{22}$$

which will be generalized for the ellipsoidal inclusion.

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

Outside, far from the inclusion, the pressure is:

The expressions of the pressure are then:

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

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

<sup>p</sup>ð Þ<sup>o</sup> ¼ � <sup>U</sup><sup>∞</sup>

ϕð Þ<sup>i</sup>

It follows that the condition

Acoustics of Materials

Að Þ<sup>i</sup>

66

<sup>1</sup> and <sup>B</sup>ð Þ<sup>o</sup>

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

gradient) with value

For defects with radius <sup>r</sup> � <sup>10</sup>�<sup>2</sup>

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

very small compared to unit (≃ 10�<sup>5</sup>

<sup>p</sup>ð Þ<sup>o</sup> � � <sup>1</sup>

Að Þ<sup>o</sup> <sup>1</sup> ¼ � <sup>1</sup>

<sup>p</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>A</sup>ð Þ<sup>i</sup>

U<sup>∞</sup> κð Þ<sup>o</sup>

U<sup>∞</sup> κð Þ<sup>o</sup> a3 r2

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

<sup>v</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>U</sup><sup>∞</sup>

<sup>v</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>U</sup><sup>∞</sup>

which will be generalized for the ellipsoidal inclusion.

<sup>κ</sup>ð Þ<sup>o</sup> rcos<sup>θ</sup> <sup>þ</sup> <sup>B</sup>ð Þ<sup>o</sup>

0 @

which implies that only the term with l ¼ 1 remains in the sum. Thus, we get:

<sup>1</sup> r �2

<sup>1</sup> been given by the conditions (8). Finally, these expressions are:

<sup>3</sup> <sup>þ</sup> <sup>12</sup> <sup>k</sup>ð Þ<sup>o</sup> a2

> <sup>k</sup>ð Þ<sup>o</sup> <sup>þ</sup> <sup>12</sup> <sup>k</sup>ð Þ<sup>i</sup> a2

> > <sup>k</sup>ð Þ<sup>o</sup> <sup>þ</sup> <sup>12</sup> <sup>k</sup>ð Þ<sup>i</sup> a2

, 10�6) and can be neglected.

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

<sup>2</sup> <sup>þ</sup> <sup>k</sup>ð Þ<sup>i</sup>

• 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

If κð Þ<sup>i</sup> > κð Þ<sup>o</sup> , then vð Þ<sup>i</sup> > 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 κð Þ<sup>i</sup> <κð Þ<sup>o</sup> . When <sup>κ</sup>ð Þ<sup>o</sup> ! 0, i.e., for low permeability <sup>k</sup>ð Þ<sup>o</sup> , then <sup>v</sup>ð Þ<sup>i</sup> � <sup>3</sup>U∞. If <sup>κ</sup>ð Þ<sup>i</sup> ! 0, then <sup>v</sup>ð Þ<sup>i</sup> ! 0.

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> <sup>1</sup>

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

κð Þ<sup>i</sup>

1

1

, 10�<sup>3</sup> m, the quantities kð Þ<sup>i</sup> =a<sup>2</sup> and kð Þ<sup>o</sup> =a<sup>2</sup> are

<sup>2</sup> <sup>þ</sup> <sup>k</sup>ð Þ<sup>i</sup>

0 @

<sup>κ</sup>ð Þ<sup>o</sup> <sup>U</sup>∞rcosθ: (14)

<sup>κ</sup>ð Þ<sup>o</sup> <sup>U</sup>∞: (16)

cosθ if r>a, (18)

Ar cos θ, r<a (19)

A cos θ, r>a (20)

: (21)

<sup>3</sup> <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ (22)

<sup>1</sup> rcosθ if r<a, (17)

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼ <sup>r</sup> <sup>¼</sup> <sup>a</sup>; <sup>θ</sup> <sup>p</sup>ð Þ<sup>i</sup> ð Þ <sup>r</sup> <sup>¼</sup> <sup>a</sup>; <sup>θ</sup> <sup>∀</sup>θ, (15)

• 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 Psph of which is

$$\mathcal{P}\_{\text{sph}} = 4\pi a^3 U\_{\infty} \kappa^{(i)} \left( \frac{\frac{k^{(i)}}{k^{(o)}} - \mathbf{1}}{2 + \frac{k^{(i)}}{k^{(o)}}} \right). \tag{23}$$

The corresponding density of induced "hydraulic" surface charges is

$$\sigma\_{sph}(\theta) = 4\pi a^3 U\_\infty \kappa^{(i)} \begin{pmatrix} \frac{k^{(i)}}{k} - \mathbf{1} \\ \mathbf{2} + \frac{k^{(i)}}{k^{(o)}} \end{pmatrix} \cos\left(\theta\right). \tag{24}$$

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

$$v\_r^{(o)} = -U\_\infty \cos\theta \left(\mathbf{1} + 2\frac{a^3}{r^3} \left(\frac{\frac{\kappa^{(i)}}{\kappa^{(o)}} - \mathbf{1}}{2 + \frac{\kappa^{(i)}}{\kappa^{(o)}}}\right)\right),\tag{25}$$

$$\left(v\_{\theta}^{(o)}\right) = \; U\_{\infty} \sin \theta \left(\mathbf{1} - \frac{a^3}{r^3} \left(\frac{\frac{\kappa^{(i)}}{\kappa^{(o)}} - \mathbf{1}}{2 + \frac{\kappa^{(i)}}{\kappa^{(o)}}}\right)\right). \tag{26}$$

Figure 4 represents the seepage rate in the porous medium for κð Þ<sup>i</sup> > κð Þ<sup>o</sup> . 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

$$\mathcal{P}\_{sph} = a(-U\_{\infty}),\tag{27}$$

where α is a susceptibility which assesses the polarizability, i.e., the capacity of the porous inclusion to induce a dipole <sup>P</sup>sph under the action of excitation �U∞=κð Þ<sup>o</sup> . For a spherical inclusion of volume V, the susceptibility α is:

Figure 4. Fluid flow through an inclusion in a porous medium: (a) velocity levels and (b) streamlines (κð Þ<sup>i</sup> =κð Þ<sup>o</sup> >1).

$$a = \Im V \kappa^{(i)} \begin{pmatrix} \frac{k^{(i)}}{k^{(o)}} - \mathbf{1} \\ \mathbf{2} + \frac{k^{(i)}}{k^{(o)}} \end{pmatrix}. \tag{28}$$

a2 <sup>1</sup> � <sup>a</sup><sup>2</sup>

Tortuosity Perturbations Induced by Defects in Porous Media

pð Þ¼ r p0ð Þ¼ x Ex ¼ E

following expression of the scattered pressure:

equation, it must verify the differential equation:

d2 F <sup>d</sup>ξ<sup>2</sup> <sup>þ</sup>

Thus, the pressure outside the inclusion is

Figure 6.

69

Ellipsoidal inclusion in a fluid flow.

F1ð Þþ ξ F2ð Þξ , where F1ð Þ¼ x A is a constant and F2ð Þ x is

dF dξ d

F2ð Þ¼ ξ

<sup>p</sup>ð Þ¼ <sup>r</sup> <sup>p</sup>0ð Þ <sup>x</sup> <sup>A</sup> � <sup>B</sup>

ellipsoid are given in Appendix A.

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

<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>2</sup>

<sup>1</sup> � <sup>b</sup><sup>2</sup> <sup>2</sup> ¼ c 2 <sup>1</sup> � c 2

where ai, bi, and ci are their respective semiaxes. Some results related to the

Let pð Þr be the pressure with a constant gradient directed along the Ox axis applied to the porous medium (Figure 6). In absence of defect, its expression is:

where <sup>E</sup> ¼ �U∞=κð Þ<sup>o</sup> . In this relation, <sup>ξ</sup>, <sup>η</sup>, and <sup>ζ</sup> are the ellipsoidal coordinates given in Appendix A. Here, the field p0ð Þ x can 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

where Fð Þξ is a proportional coefficient. For pscð Þr to be a solution of the Laplace

where <sup>R</sup>ð Þ¼ <sup>σ</sup> <sup>a</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>σ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup> � � <sup>c</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>σ</sup> . <sup>F</sup>ð Þ<sup>ξ</sup> is then the sum of the two functions

2 ð<sup>∞</sup> ξ

dσ

dσ σ þ a<sup>2</sup> ð ÞRð Þ σ

ð<sup>∞</sup> ξ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að Þ <sup>2</sup> þ ξ að Þ <sup>2</sup> þ η að Þ <sup>2</sup> þ ζ <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 <sup>s</sup>

pscð Þ¼ r p0ð Þ x Fð Þξ : (33)

<sup>d</sup><sup>ξ</sup> ln <sup>R</sup>ð Þ<sup>ξ</sup> <sup>ξ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> � � � � � � <sup>¼</sup> <sup>0</sup> (34)

<sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð ÞRð Þ <sup>σ</sup> : (35)

� �: (36)

<sup>2</sup> (31)

(32)

Often, the quantity <sup>χ</sup> <sup>¼</sup> <sup>α</sup>=κð Þ<sup>i</sup> <sup>V</sup> which does not depend on the volume of the inclusion is more relevant. Its variations as function of the ratio <sup>κ</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> are shown in Figure 5. They range from �3/2 when <sup>κ</sup>ð Þ<sup>i</sup> << <sup>κ</sup>ð Þ<sup>o</sup> to 3 when <sup>κ</sup>ð Þ<sup>i</sup> >>κð Þ<sup>o</sup> . In the following, it is convenient to put the external pressure in the form:

$$p^{(o)}(r,\theta) = -\frac{U\_{\infty}}{\kappa^{(o)}}r\cos\theta + \frac{P\_{sph}}{4\pi}\frac{\cos\theta}{r^2} \tag{29}$$

## 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 c aligned with the axes of the Cartesian coordinates system and centered at the origin is described by the following equation:

$$\frac{x^2}{u+a^2} + \frac{y^2}{u+b^2} + \frac{z^2}{u+c^2} = \mathbf{1},\tag{30}$$

where x, y, and z are the position coordinates of any point on the surface of the ellipsoid. Eq. (30) has three roots ξ, η, and ζ which define the surfaces coordinates: surfaces with constant ξ are ellipsoids, while surfaces with constant η or ζ are hyperboloids. Surfaces of confocal ellipsoids are described by adjusting the scalar u. So, two ellipsoids defined by (30) with u ¼ u<sup>1</sup> and u ¼ u<sup>2</sup> are called confocal if their semi axes obey to the conditions

Figure 5. Hydraulic polarizability <sup>χ</sup> <sup>¼</sup> <sup>α</sup>=κð Þ<sup>i</sup> <sup>V</sup> of a spherical inclusion vs log (k).

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

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

the following, it is convenient to put the external pressure in the form:

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

x2 <sup>u</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup>

Hydraulic polarizability <sup>χ</sup> <sup>¼</sup> <sup>α</sup>=κð Þ<sup>i</sup> <sup>V</sup> of a spherical inclusion vs log (k).

2.2.2 Ellipsoidal defect

Acoustics of Materials

described by the following equation:

semi axes obey to the conditions

Figure 5.

68

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

<sup>κ</sup>ð Þ<sup>o</sup> rcos<sup>θ</sup> <sup>þ</sup>

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 c aligned with the axes of the Cartesian coordinates system and centered at the origin is

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

where x, y, and z are the position coordinates of any point on the surface of the ellipsoid. Eq. (30) has three roots ξ, η, and ζ which define the surfaces coordinates: surfaces with constant ξ are ellipsoids, while surfaces with constant η or ζ are hyperboloids. Surfaces of confocal ellipsoids are described by adjusting the scalar u. So, two ellipsoids defined by (30) with u ¼ u<sup>1</sup> and u ¼ u<sup>2</sup> are called confocal if their

z2

1

Psph 4π

cosθ

A: (28)

<sup>r</sup><sup>2</sup> (29)

<sup>u</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>, (30)

0 @

Often, the quantity <sup>χ</sup> <sup>¼</sup> <sup>α</sup>=κð Þ<sup>i</sup> <sup>V</sup> which does not depend on the volume of the inclusion is more relevant. Its variations as function of the ratio <sup>κ</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> are shown in Figure 5. They range from �3/2 when <sup>κ</sup>ð Þ<sup>i</sup> << <sup>κ</sup>ð Þ<sup>o</sup> to 3 when <sup>κ</sup>ð Þ<sup>i</sup> >>κð Þ<sup>o</sup> . In

$$a\_1^2 - a\_2^2 = b\_1^2 - b\_2^2 = c\_1^2 - c\_2^2\tag{31}$$

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

Let pð Þr be the pressure with a constant gradient directed along the Ox axis applied to the porous medium (Figure 6). In absence of defect, its expression is:

$$p(\mathbf{r}) = p\_0(\mathbf{x}) = E\mathbf{x} = E\sqrt{\frac{(a^2 + \xi)(a^2 + \eta)(a^2 + \zeta)}{(b^2 - a^2)(c^2 - a^2)}}\tag{32}$$

where <sup>E</sup> ¼ �U∞=κð Þ<sup>o</sup> . In this relation, <sup>ξ</sup>, <sup>η</sup>, and <sup>ζ</sup> are the ellipsoidal coordinates given in Appendix A. Here, the field p0ð Þ x can 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:

$$p\_{sc}(\mathbf{r}) = p\_0(\mathbf{x}) F(\xi). \tag{33}$$

where Fð Þξ is a proportional coefficient. For pscð Þr to be a solution of the Laplace equation, it must verify the differential equation:

$$\frac{d^2F}{d\xi^2} + \frac{dF}{d\xi}\frac{d}{d\xi}\left(\ln\left[R(\xi)\left(\xi + a^2\right)\right]\right) = 0\tag{34}$$

where <sup>R</sup>ð Þ¼ <sup>σ</sup> <sup>a</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>σ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup> � � <sup>c</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>σ</sup> . <sup>F</sup>ð Þ<sup>ξ</sup> is then the sum of the two functions F1ð Þþ ξ F2ð Þξ , where F1ð Þ¼ x A is a constant and F2ð Þ x is

$$F\_2(\xi) = \int\_{\xi}^{\infty} \frac{d\sigma}{(\sigma + a^2)R(\sigma)}.\tag{35}$$

Thus, the pressure outside the inclusion is

$$p(\mathbf{r}) = p\_0(\mathbf{x}) \left[ A - \frac{B}{2} \int\_{\xi}^{\infty} \frac{d\sigma}{(\sigma + a^2)R(\sigma)} \right]. \tag{36}$$

Figure 6. Ellipsoidal inclusion in a fluid flow.

The two constants A and B are determined by the boundary conditions at the surface ξ ¼ 0 of the inclusion:

1. continuity of fluid flow at ξ ¼ 0:

$$
\phi^{(e)}\kappa^{(e)}\frac{\mathbf{1}}{h\_{\xi}}\frac{\partial p^{(e)}}{\partial \xi} = \phi^{(i)}\kappa^{(i)}\frac{\mathbf{1}}{h\_{\xi}}\frac{\partial p^{(i)}}{\partial \xi},\tag{37}
$$

Here, V is the volume of the ellipsoidal inclusion. The factor Nx,

<sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi

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 Nz appear 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

From Eq. (163), it is possible to find the values of the depolarization factors of

Figure 7 is a plot of depolarization factors of a ellipsoidal inclusion having two

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 ¼ αaU∞, where α<sup>a</sup> is the eigenvalue of the tensor polarizability along its principal direction Ox which defines the polarizability along this axis. In Figure 8, we depict the variations of the susceptibility χ as

The expression of pscð Þ x given by (42) is not exact since it is a result of the approximation <sup>ξ</sup> � <sup>r</sup>2, i.e., far from the inclusion, where only the dipolar effects are

dσ

<sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>p</sup> <sup>Þ</sup> <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, (45)

Nx <sup>¼</sup> abc 3

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

Lifchitz [11]).

relevant.

Figure 7.

71

some particular inclusions such as:

• spherical inclusion Nx ¼ Ny ¼ Nz ¼ 1=3,

• oblong inclusion Nx ¼ Ny ¼ 1=2, Nz ¼ 0.

equal semiaxes according to their ratio.

• inclusion in disc form (axis Oz) Nx ¼ Ny ¼ 0, Nz ¼ 1, and

Depolarization factors for a ellipsoidal inclusion with semiaxes (a, a, b) as functions of a/b.

ð<sup>∞</sup> 0

Tortuosity Perturbations Induced by Defects in Porous Media

2. continuity of the stress component τξξ in fluid at ξ ¼ 0:

$$\phi^{(\epsilon)}\left(-p^{(\epsilon)} + 2\eta \frac{\mathbf{1}}{h\_{\xi}} \frac{\partial v\_{\xi}^{(\epsilon)}}{\partial \xi}\right) = \phi^{(i)}\left(-p^{(i)} + 2\eta \frac{\mathbf{1}}{h\_{\xi}} \frac{\partial v\_{\xi}^{(i)}}{\partial \xi}\right) \tag{38}$$

where v<sup>ξ</sup> is the normal component of the seepage flow to the surface ξ ¼ 0 given by the Darcy's law:

$$
v\_{\dot{\lambda}\_i} = -\kappa \frac{1}{h\_{\dot{\lambda}\_i}} \partial\_{\dot{\lambda}\_i} p, \quad \dot{\lambda}\_i = \xi, \eta, \zeta. \tag{39}
$$

In these relations, the coefficients hλ<sup>i</sup> are the scale factors of ellipsoidal coordinates given in Appendix A. We obtain the values of the pressure such that:

1. within the inclusion:

$$p^{(i)}(\mathbf{x}) = \frac{\kappa^{(o)}}{\kappa^{(o)} + N\_{\mathbf{x}}(\kappa^{(i)} - \kappa^{(o)})} p\_{\mathbf{0}}(\mathbf{x}),\tag{40}$$

2. outside the inclusion:

$$p^{(o)}(\mathbf{x}) = p\_0(\mathbf{x}) + p\_\varkappa(\mathbf{x}) = p\_0(\mathbf{x}) - \frac{abc\left(\kappa^{(i)} - \kappa^{(o)}\right)}{\kappa^{(o)} + N\_\varkappa(\kappa^{(i)} - \kappa^{(o)})} p\_0(\mathbf{x}) F\_2(\xi). \tag{41}$$

Far from the center of the inclusion, ξ≈r2, the scattered pressure can be approximate by:

$$\begin{split} p\_{\kappa}(\mathbf{x}) &\approx \frac{abc\left(\kappa^{(i)} - \kappa^{(o)}\right)}{\kappa^{(o)} + N\_{\text{x}}(\kappa^{(i)} - \kappa^{(o)})} p\_{0}(\mathbf{x}) \Big|\_{r^{2}} \frac{dr}{r^{5/2}} \\ &\approx \frac{abc\left(\kappa^{(i)} - \kappa^{(o)}\right)}{\kappa^{(o)} + N\_{\text{x}}(\kappa^{(i)} - \kappa^{(o)})} \frac{\mathcal{X}}{3r^{3}} E. \end{split} \tag{42}$$

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:

$$v^{(i)} = U\_{\infty} \frac{\kappa^{(o)}}{\kappa^{(o)} + N\_{\text{x}}(\kappa^{(i)} - \kappa^{(o)})},\tag{43}$$

the dipole moment is then

$$\mathcal{P}\_{\text{ell}i,a} = -\text{V}U\_{\infty}\kappa^{(i)}\frac{\left(\kappa^{(i)} - \kappa^{(o)}\right)}{\kappa^{(o)} + N\_{\text{x}}(\kappa^{(i)} - \kappa^{(o)})}.\tag{44}$$

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

The two constants A and B are determined by the boundary conditions at the

<sup>∂</sup><sup>ξ</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup> <sup>κ</sup>ð Þ<sup>i</sup> <sup>1</sup>

where v<sup>ξ</sup> is the normal component of the seepage flow to the surface ξ ¼ 0 given

κð Þ<sup>o</sup>

Far from the center of the inclusion, ξ≈r2, the scattered pressure can be

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Nx <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ <sup>p</sup>0ð Þ <sup>x</sup>

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

<sup>P</sup>elli,a ¼ �VU∞κð Þ<sup>i</sup> <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> � �

κð Þ<sup>o</sup>

pscð Þ <sup>x</sup> <sup>≈</sup> abc <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> � �

<sup>v</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>U</sup><sup>∞</sup>

<sup>≈</sup> abc <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> � � κð Þ<sup>o</sup> þ Nx κð Þ<sup>i</sup> � κð Þ<sup>o</sup> ð Þ

hξ

<sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup> �pð Þ<sup>i</sup> <sup>þ</sup> <sup>2</sup><sup>η</sup>

∂pð Þ<sup>i</sup>

1 hξ ∂v ð Þi ξ ∂ξ

p, λ<sup>i</sup> ¼ ξ, η, ζ: (39)

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Nx <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ <sup>p</sup>0ð Þ <sup>x</sup> , (40)

ð r2

x <sup>3</sup>r<sup>3</sup> <sup>E</sup>:

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Nx <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ <sup>p</sup>0ð Þ <sup>x</sup> <sup>F</sup>2ð Þ<sup>ξ</sup> : (41)

dr r<sup>5</sup>=<sup>2</sup>

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Nx <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ, (43)

<sup>κ</sup>ð Þ<sup>o</sup> <sup>þ</sup> Nx <sup>κ</sup>ð Þ<sup>i</sup> � <sup>κ</sup>ð Þ<sup>o</sup> ð Þ: (44)

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

!

<sup>∂</sup><sup>ξ</sup> , (37)

(38)

(42)

∂pð Þ<sup>e</sup>

surface ξ ¼ 0 of the inclusion:

Acoustics of Materials

by the Darcy's law:

1. within the inclusion:

2. outside the inclusion:

the dipole moment is then

approximate by:

inclusion:

70

1. continuity of fluid flow at ξ ¼ 0:

<sup>ϕ</sup>ð Þ<sup>e</sup> �pð Þ<sup>e</sup> <sup>þ</sup> <sup>2</sup><sup>η</sup>

<sup>ϕ</sup>ð Þ<sup>e</sup> <sup>κ</sup>ð Þ<sup>e</sup> <sup>1</sup> hξ

2. continuity of the stress component τξξ in fluid at ξ ¼ 0:

!

vλ<sup>i</sup> ¼ �κ

<sup>p</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>x</sup>

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼ <sup>x</sup> <sup>p</sup>0ð Þþ <sup>x</sup> pscð Þ¼ <sup>x</sup> <sup>p</sup>0ð Þ� <sup>x</sup>

1 hλi ∂λi

In these relations, the coefficients hλ<sup>i</sup> are the scale factors of ellipsoidal coordinates given in Appendix A. We obtain the values of the pressure such that:

1 hξ ∂v ð Þe ξ ∂ξ

Here, V is the volume of the ellipsoidal inclusion. The factor Nx,

$$N\_{\mathbf{x}} = \frac{ab\boldsymbol{\sigma}}{\mathfrak{Z}} \int\_{0}^{\infty} \frac{d\boldsymbol{\sigma}}{(\sigma + a^2)\sqrt{\sigma + a^2} \left(\sigma + b^2\right) (\sigma + c^2)},\tag{45}$$

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 Nz appear 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:


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

The expression of pscð Þ x given by (42) is not exact since it is a result of the approximation <sup>ξ</sup> � <sup>r</sup>2, 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 ¼ αaU∞, where α<sup>a</sup> is the eigenvalue of the tensor polarizability along its principal direction Ox which defines the polarizability along this axis. In Figure 8, we depict the variations of the susceptibility χ as

Figure 7. Depolarization factors for a ellipsoidal inclusion with semiaxes (a, a, b) as functions of a/b.

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

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 kð Þ<sup>o</sup> , that of the outermost layer is kð Þ<sup>1</sup> , and so on to the central sphere whose

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 n is related to those in the layer number n þ 1 and n � 1 by the boundary conditions. It is assumed that the defect is a set of N concentric spherical layers in which the value of permeability is constant kð Þ <sup>n</sup> . Let κð Þ <sup>n</sup> denotes the ratio kð Þ <sup>n</sup> =η in the layer number n delimited by the spheres of radii rn and rnþ<sup>1</sup> such that rn >rnþ1. The core of the sphere has index i ¼ N (Figure 9). The determination of pressure and velocity in this type of inclusion consists in solving the Laplace equation <sup>Δ</sup>pð Þ <sup>n</sup> <sup>¼</sup> 0 in each layer and in connecting the solutions using the boundary conditions: continuity of fluid flow and that of the

<sup>p</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>A</sup>ð Þ <sup>n</sup> <sup>r</sup> cos <sup>θ</sup> <sup>þ</sup> <sup>B</sup>ð Þ <sup>n</sup> <sup>r</sup>

The coefficients A and B of two consecutive layers are connected by the follow-

<sup>r</sup> ð Þ¼ <sup>r</sup> <sup>¼</sup> rnþ<sup>1</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> <sup>v</sup>ð Þ <sup>n</sup>þ<sup>1</sup> <sup>r</sup> ð Þ <sup>r</sup> <sup>¼</sup> rnþ<sup>1</sup> , (48)

rr ð Þ¼ <sup>r</sup> <sup>¼</sup> rnþ<sup>1</sup> <sup>τ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> rr ð Þ <sup>r</sup> <sup>¼</sup> rnþ<sup>1</sup> : (49)

�<sup>2</sup> cos θ: (47)

varying function.

permeability is kð Þ<sup>i</sup> .

2.3.1 Layered spherical defect

radial component of the stress.

ing conditions in r ¼ rnþ1:

Figure 9. Layered sphere.

73

In the layer number n, the pressure is noted:

Tortuosity Perturbations Induced by Defects in Porous Media

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

ϕð Þ <sup>n</sup> vð Þ <sup>n</sup>

τð Þ <sup>n</sup>

#### Figure 8.

Hydraulic polarizability <sup>χ</sup> <sup>¼</sup> <sup>α</sup>x=<sup>V</sup> of an ellipsoidal inclusion vs. log <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> for different values of the depolarization factors (Nx, Ny, Nz) (�•�: Nx ¼ 0:1, Ny ¼ 0:1, Nz ¼ 0:8); (�⋄�: Nx ¼ 0:2, Ny ¼ 0:2, Nz <sup>¼</sup> <sup>0</sup>:6); (�◀�: Nx <sup>¼</sup> <sup>0</sup>:4, Ny <sup>¼</sup> <sup>0</sup>:4, Nz <sup>¼</sup> <sup>0</sup>:2); (�■�: Nx <sup>¼</sup> <sup>0</sup>:45, Ny <sup>¼</sup> <sup>0</sup>:45, Nz <sup>¼</sup> <sup>0</sup>:1); (–––: Nx ¼ 1=3, Ny ¼ 1=3, Nz ¼ 1=3).

function of κð Þ<sup>i</sup> =κð Þ<sup>o</sup> for different sets of the depolarization factors when incident pressure is along Ox.

The pressure outside the inclusion is then:

$$p^{(o)}(r,\theta) = -\frac{U\_{\infty}}{\kappa^{(o)}}r\cos\theta + \frac{\mathcal{P}\_{ellli,a}}{4\pi}\frac{\cos\theta}{r^2}.\tag{46}$$

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 F<sup>2</sup> is an approximate computation when ξ=r ≫ 1. 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.
