Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media
Provides theoretical guarantees for an industrial finite volume scheme used in petroleum engineering, addressing a known gap in convergence proofs for compressible two-phase flow.
The paper proves stability (maximum principle, energy estimate) and convergence of an implicit finite volume scheme for compressible two-phase flow in porous media, establishing that a subsequence converges to a weak solution as discretization size tends to zero.
An industrial scheme, to simulate the two compressible phase flow in porous media, consists in a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness. We show that the proposed scheme satisfy the maximum principle for the saturation, a discrete energy estimate on the pressures and a function of the saturation that denote capillary terms. These stabilities 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. The proof is given for the complete system when the density of the each phase depends on the own pressure.