Two-phase Flows Involving Discontinuities on the Capillary Pressure

Clément Cancès (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We consider a immiscible two-phase flow through an one-dimensional heterogeneous porous medium made of an apposition of several homogeneous porous media. This leads to a nonlinear degenerate parabolic problem, with monotonous transmission conditions between the different homogeneous subdomains. We give an implicit finite volume scheme for such a two-phase flow, and we prove the convergence of the inducted discrete solutions to a weak solution. Under assumption on the initial condition, i.e. if it yields bounded flux, and if the total flow-rate belongs to BV (0, T), then the discrete solution obtained via the scheme converges toward a solution with bounded flux. We prove a L 1-contraction principle for such a bounded flux solution. This uniqueness result is extended using a SOLA approach in the case where there are no particular regularity assumptions on the initial data, and we check that the discrete solution obtained via the implicit finite volume scheme converges to this SOLA.

Mots clés: flows in porous media; finite volume method; discontinuous capillarity pressures; nonlinear PDE of parabolic type

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Clément Cancès. Two-phase Flows Involving Discontinuities on the Capillary Pressure. FVCA5 - 5th International Symposium on Finite Volumes for Complex Applications , Jun 2008, Aussois, France. ⟨hal-01713566⟩