Existence of solutions to various rock types odel model of two-phase flow in porous media

Abstract

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.

Keywords:

• heterogeneous porous media
• multiphase flow
• nonlinear diffusion convection
• mixed formulation

^*Laboratoire de Mathdmatiques Appliqudes, URA-CNRS 1204, Universitd de Pau et des Pays de l'Adour, Av. de l'université, 64000 Pau, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France

^*Laboratoire de Mathdmatiques Appliqudes, URA-CNRS 1204, Universitd de Pau et des Pays de l'Adour, Av. de l'université, 64000 Pau, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France

Notes

^*Laboratoire de Mathdmatiques Appliqudes, URA-CNRS 1204, Universitd de Pau et des Pays de l'Adour, Av. de l'université, 64000 Pau, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France

^**Equipe d'Analyse Numdrique, URA-CNRS 740, UFR Sciences, Université J. Monnet,23 Rue du Dr. Paul Michelon, 42023 Saint-Etienne, France