Diffusion and Convection in Porous Media

Abstract

In this chapter the correct mathematical models of a diffusion-convection in porous media is derived from the homogenization theory. The mathematical model at the microscopic level consist of the Stokes system for an incompressible viscous liquid occupying a pore space and the Lame’s system for an incompressible solid skeleton, coupled with a diffusion-convection equation for the admixture. The problem is ended by the boundary conditions on the common boundary “pore space—solid skeleton”, boundary conditions on the outer boundary, and initial conditions. We prove the global in time existence of a weak solution for the corresponding to microscopic model initial boundary-value problem and rigorously fulfil the homogenization procedure as the dimensionless size of pores tends to zero, while the porous body is geometrically periodic. As the basic models at the microscopic level we consider the mathematical model ${\mathbb {M}}_{27}$ of diffusion-convection in an absolutely rigid skeleton, and the mathematical model ${\mathbb {M}}_{25}$ of diffusion-convection in poroelastic media.