Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media

Abstract

In this paper, we are concerned with the convergence analysis of a positive control volume finite element scheme (CVFE) for a degenerate compressible two-phase flow model in anisotropic porous media. For this, we consider the global pressure saturation formulation. We next use an implicit Euler scheme in time and a CVFE discretization in space. This approach rests on a particular choice of the mean value of the gas density on the interfaces, a centered scheme of the total mobility, and the upwind approximation of fractional fluxes according to the total velocity. Thus, the maximum principle is fulfilled without any constraint on the stiffness coefficients. Moreover, uniform estimates on the discrete gradient of the global pressure and the dissipative term are derived. As the mesh size is sent to zero, we establish that the sequence of approximate solutions converges to a weak solution of the continuous problem. Numerical tests are presented in two dimensions to exhibit the behavior of the gas pressure and the water saturation through the medium.