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We propose and analyze a combined finite volume--nonconforming finite elementscheme on general meshes to simulate the two compressible phase flow in porousmedia. The diffusion term, which can be anisotropic and heterogeneous, isdiscretized by piecewise linear nonconforming triangular finite elements. Theother terms are discretized by means of a cell-centered finite volume scheme ona dual mesh, where the dual volumes are constructed around the sides of theoriginal mesh. The relative permeability of each phase is decentred accordingthe sign of the velocity at the dual interface. This technique also ensures thevalidity of the discrete maximum principle for the saturation under a nonrestrictive shape regularity of the space mesh and the positiveness of alltransmissibilities. Next, a priori estimates on the pressures and a function ofthe saturation that denote capillary terms are established. These stabilitiesresults lead to some compactness arguments based on the use of the Kolmogorovcompactness theorem, and allow us to derive the convergence of a subsequence ofthe sequence of approximate solutions to a weak solution of the continuousequations, provided the mesh size tends to zero. The proof is given for thecomplete system when the density of the each phase depends on the own pressure.

A combined finite volume--nonconforming finite element scheme for compressible two phase flow in porous media