Homogenization Analysis and Permeability of Porous Media

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 v Vv, 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. 

If a homogenization analysis (HA) is applied to porous media flow, which is described by the Stokes equation, we can immediately obtain Darcy s formula and the seepage equation in a macroscale field while in the microscale field the distributions of velocity and pressure are specified (Sanchez-Palencia 1980). We can also apply HA for a problem with a locally varying viscosity. 

In this Section we first show that a local variation of viscosity in the pore water of a saturated smectitic clay such as montmorillonite or beidellite, which is a platelet crystal of about one nanometer (=10 m) thickness, can be calculated by a molecular dynamic (MD) simulation. Then, by applying the HA with the locally distributed viscosity, we can calculate the seepage field of the smectitic clay, which consists of stacks of clay minerals. Consequently, we apply a three-scale analysis of homogenization for a bentonite clay with quartz grains of about 10 [xm (1 fxm = 10 m). 

Ichikawa and A.P.S. Selvadurai, Transport Phenomena in Porous Media, 

1 Micro-inhomogeneous Porous Media and Stokes Equation 

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Homogenization Analysis and Permeability of Porous Media Montmorillonlte stack... 

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