Numerical Approximation of Filtration Processes through Porous Media

Abstract

In this thesis, we studied numerical methods for the coupling of free fluid flow with porous medium flow. The free fluid flow is modelled by the Stokes equations while the flow in the porous medium is modelled by Darcy’s law. Appropriate conditions are imposed at the interface between the two regions. The weak formulation of the problem is based on mixed-formulation for Stokes and on a primal-mixed formulation for Darcy equation, incorporating in a natural way the interface conditions. The finite element discretization of the problem leads to large, sparse and ill-conditioned algebraic system to be solved for velocities in both domains, Stokes pressure and piezometric head in porous domain. The system is reduced to interface systems for the normal velocity and piezometric head by a Schur complement approach. We present numerical results for several solution methods based on different preconditioning techniques for the solution of the interface systems. We study the effectiveness of the preconditioners with respect to mesh refinement and physical parameters. An application to cross-flow membranes has been considered. Finally, we also assess the numerical accuracy of an uncoupled algorithm for transient problem, which uses different time steps in the Stokes and in the Darcy domains.