Study of Full Implicit Petroleum Engineering Finite-Volume Scheme for Compressible Two-Phase Flow in Porous Media

An industrial scheme, to simulate the compressible two-phase flow in porous media, consists of a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness since the proposed scheme discretizes the equations with gravity and capillary terms. We show that the proposed scheme satisfies the maximum principle for the saturation, a discrete-energy estimate on the pressures, and a function of the saturation that denotes capillary terms. These stability results allow us to derive the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. To our knowledge, this is the first convergence result of a finite volume scheme in the case of two-phase compressible flow in several space dimensions. The proof is given for the complete system when the density of each phase depends on its own pressure.