Discretizations of a Darcy's model for incompressible flows
\\ 
with pressure dependent porosity.

Professor Vivette Girault, 
Universit\'e Pierre et Marie Curie, Paris VI, Laboratoire
Jacques-Louis Lions


Joint work with A.Salgado and F.Murat.\\


We propose two finite-element schemes for discretizing  an incompressible steady Darcy's flow through a rigid porous medium in dimension two or three, when the frictional forces that the fluid encounters at the boundaries of the medium's pores vary exponentially with pressure, and inertial effects are ignored:
$$\alpha(p)\uu+\nabla p=\f, \quad \nabla\cdot\uu=0, $$
$$\alpha(\xi)=\alpha_{0}e^{\gamma\xi},$$
with positive constants $\alpha_{0}$ and $\gamma$. 
This is a simplified version of a model of enhanced oil recovery proposed by K.R.Rajagopal.  

When the drag coefficient $\alpha$ is truncated, the numerical analysis of a straightforward finite element scheme for approximating the solution can be done, either directly or by applying an implicit function theorem (a Brezzi-Rappaz-Raviart approach). But when $\alpha$ is not truncated, because of its exponential behavior, this analysis is still an open problem. We propose instead a splitting formulation, developped by F.Murat, that leads to two linear systems, and yields existence of a solution in particular situations when the pressure satisfies a Dirichlet boundary condition. The full analysis of the corresponding discrete scheme is still an open problem, but its numerical results  are very encouraging.