Homogenization Analysis and Permeability of Porous Media

Abstract

The Navier-Stokes’ (NS) equations can be used to describe problems of fluid flow. Since these equations are scale-independent, flow in the microscale structure of a porous medium can also be described by a NS field. If the velocity on a solid surface is assumed to be null, the velocity field of a porous medium problem with a small pore size rapidly decreases (see Sect.5.3.2). We describe this flow field by omitting the convective term $\mathbf{v} \cdot \nabla \mathbf{v}$, which gives rise to the classical Stokes’ equation. We recall that Darcy’s theory is usually applied to describe seepage in a porous medium, where the scale of the solid skeleton does not enter the formulation as an explicit parameter. The scale effect of a solid phase is implicitly included in the permeability coefficient, which is specified through experiments. It should be noted that Kozeny-Carman’s formula (5.88) involves a parameter of the solid particle; however, it is not applicable to a geometrical structure at the local pore scale.