Transport Phenomena in Porous Media

Porous materials can be both natural and artificial. Suchmaterials are encountered in a wide range of problems dealing with engineering, agriculture and science: Natural materials include soil and rock, and artificial materials can include membrane materials and catalysts used in chemical engineering, etc.

The mechanics of a deformable body treated here is based on Newton’s laws of motion and the laws of thermodynamics.

We first review the principles of Classical Thermodynamics (see also Appendix D), and proceed to give an alternative formulation of Thermodynamics in the context of a true dynamical process.

In this Chapter we consider the virtual work equation of a static problem and its relationship to the variational method and energy principle.

In this chapter we develop the coupled diffusion and seepage problem using the theory of mixtures. It is clearly understood that the diffusion problem is strongly linked to the seepage problem through the mass conservation law. Adsorption on the solid surface is treated using the concept of an ‘adsorption isotherm’.

Soil is a mixture of a solid phase, a liquid phase (i.e., water) and a gaseous phase. Saturated soil is a two-phase material which consists of a solid phase and a liquid phase. The formulation of a theory for a saturated soil commenced with the concept of the effective stress of Terzaghi at the beginning of the twentieth century, and Terzaghi’s concept was extended by Biot to a three dimensional consolidation theory.

We outline the essential features of a multiscale homogenization analysis. A problem of a one-dimensional elastic bar is given as an example.

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.

We examine the problem of diffusion in a porous medium using a homogenization analysis (HA). Diffusion problems have important applications in environmental geosciences. We clarify the mechanism of diffusion, convective transport and adsorption in porous media at both the microscale and macroscale levels. Attention is particularly focused on diffusion processes in bentonite, which is an engineered geological barrier to be used to buffer the transport of radionuclides from deep geologic repositories.

Proposals for the geological disposal of heat emitting high-level radioactive wastes (HLW) have been put forward by many countries including Japan, Canada, Sweden, Switzerland, USA, Spain and others. The disposal concepts invariably involve underground multi-barrier schemes where bentonite clay is chosen for a number of desirable attributes including its swelling potential and ability to trap the majority of released radionuclides (JNC 1999; Chapman and McCombie 2003).