The Muskat Problem

Abstract

In this chapter a free boundary problem describing a joint filtration of two immiscible incompressible liquids is derived from the homogenization theory. We start with a mathematical model at the microscopic level, which consist of the stationary Stokes system for an incompressible inhomogeneous viscous liquid, occupying a pore space, the stationary Lamé’s equations for an incompressible elastic solid skeleton, coupled with corresponding boundary conditions on the common boundary “solid skeleton—pore space”, and transport equation for the unknown liquid density. Next we prove the solvability of this model and rigorously fulfil the homogenization procedure as the dimensionless size of pores tends to zero, while the porous body is geometrically periodic. As a result, we prove the solvability of the Muskat problem for a viscoelastic filtration.