Abstract

In this chapter, we describe the effects of defects in a homogeneous saturated porous medium. Defects are modelized by inclusions which disturb the motion of the viscous fluid flowing in the pore space of the medium. The seepage rate of the fluid in the host medium and in the inclusion is given by the Darcy's law. Disturbances thus produced modify the shape of the stream lines from which we establish the tortuosity induced by the defects and its implications on the acoustic waves propagation in saturated porous media.

Keywords: tortuosity, defects, porous media, refractive index

## 1. Introduction

Among the essential physical parameters to describe the microstructure of porous media, tortuosity is one of the most important parameters. For a review, we can refer to the paper of Ghanbarian et al. [1].

Tortuosity was introduced as a correction to the permeability of Kozeny's model [2] of porous media defined by the Darcy's law relating the fluidic characteristics and pore space of the medium [3]:

$$\mathbf{v} = -\frac{k}{\eta} \nabla p,\tag{1}$$

where v is the seepage rate of the fluid, η the viscosity coefficient of the fluid, ∇p is the pressure gradient applied to the medium, and k is its permeability. The Kozeny's model was developed in the framework of straight and parallel streamlines in porous media. Carman has generalized it to neither straight nor parallel streamlines by introducing the hydraulic tortuosity τ defined by:

$$
\pi = \frac{<\lambda>}{L}. \tag{2}
$$

When a fluid flows through a porous medium from point A to point B distant from L (Euclidean distance) (Figure 1), it follows different paths whose mean length is < λ>, where λ is the length of the different paths connecting these two points. In isotropic media, the tortuosity is a scalar number greater than unit (<λ> ≥L), whereas for the low porous media, its values may be greater than 2; they range from 1 to 2 for high porosity media such as fibrous materials and some plastic foams.

Taking into account the presence of defects that change the permeability of

the porous medium leads to the notion of effective permeability (keff ). In general, the keff value is not unique but depends on the chosen model for the homogenization of the porous medium. The homogenization process only makes sense for lower scales than the spatial variations of incident excitation, which therefore justifies that mobility is calculated for a low-frequency filtration rate (quasi-static regime). These considerations lead us to be interested only in the instantaneous individual response of defects to external solicitations. Since in our case only media with low levels of defect are considered, it is legitimate to ignore

Tortuosity Perturbations Induced by Defects in Porous Media

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

The present chapter is organized as follows. Section 2 describes the mathematical model of the defects and gives the solution of the fluid flow in the presence of homogeneous and layered spherical and ellipsoidal defects. Then, the results are generalized to anisotropic defects. Finally, the hydraulic polarizability is introduced. Section 3 is relative to tortuosity. The expression of effective mobility is given for some particular defects. The induced tortuosity is deduced from the previous results and its effects on the wave propagation

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,

When the fluid flows through the porous medium, its motion is perturbed by the

<sup>k</sup>ð Þ <sup>m</sup> ð Þ <sup>x</sup> η

where m ¼ i if x∈ Ω and m ¼ o if x is in the host medium (x∉ Ω). These

<sup>∇</sup>pð Þ <sup>m</sup> ð Þ <sup>x</sup> , (4)

<sup>ϕ</sup>ð Þ<sup>o</sup> <sup>v</sup>ð Þ<sup>o</sup> :nð Þ<sup>o</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup> <sup>v</sup>ð Þ<sup>i</sup> :nð Þ<sup>i</sup> , (5)

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:

<sup>v</sup>ð Þ <sup>m</sup> ð Þ¼ <sup>x</sup>

equations are subject to the following boundary conditions on ∂Ω:

their mutual interactions.

are given.

2. Defect model

we use indifferently the words defect or inclusion.

2.1 Mathematical formulation

• continuity of fluid flow

63

Figure 1. Some tortuous paths through a porous medium.

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 greater tortuosity.

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 different definitions of tortuosity, each with its own interpretation.

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

$$
\pi\_0 = \frac{<\mathbf{v}^2>}{<\mathbf{v}>^2},
\tag{3}
$$

where < :> denotes averaging over the pore fluid volume Vf . Thereafter, in this chapter, we adopt this definition of tortuosity.

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.

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

Taking into account the presence of defects that change the permeability of the porous medium leads to the notion of effective permeability (keff ). In general, the keff value is not unique but depends on the chosen model for the homogenization of the porous medium. The homogenization process only makes sense for lower scales than the spatial variations of incident excitation, which therefore justifies that mobility is calculated for a low-frequency filtration rate (quasi-static regime). These considerations lead us to be interested only in the instantaneous individual response of defects to external solicitations. Since in our case only media with low levels of defect are considered, it is legitimate to ignore their mutual interactions.

The present chapter is organized as follows. Section 2 describes the mathematical model of the defects and gives the solution of the fluid flow in the presence of homogeneous and layered spherical and ellipsoidal defects. Then, the results are generalized to anisotropic defects. Finally, the hydraulic polarizability is introduced. Section 3 is relative to tortuosity. The expression of effective mobility is given for some particular defects. The induced tortuosity is deduced from the previous results and its effects on the wave propagation are given.
