Study of full implicit petroleum engineering finite volume scheme for   compressible two phase flow in porous media
  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. 
Bilal Saad and Mazen Saad
math.NA