One-Phase Newtonian Flow

Abstract

This section is devoted to the derivation of Darcy’s law for an incompressible viscous fluid flowing in a porous medium. Starting from the steady Stokes equations in a periodic porous medium, with a no-slip (Dirichlet) boundary condition on the solid pores, Darcy’s law is rigorously obtained by periodic homogenization using the two-scale convergence method. The assumption of the periodicity of the porous medium is by no means realistic, but it allows casting this problem in a very simple framework and proving theorems without too much effort. We denote by e the ratio of the period to the overall size of the porous medium. It is the small parameter of our asymptotic analysis because the pore size is usually much smaller than the characteristic length of the reservoir. The porous medium is contained in a domain Q, and its fluid part is denoted by TE. From a mathematical point of view, 513E is a periodically perforated domain, i.e., it has many small holes of size e which represent solid obstacles that the fluid cannot penetrate.