  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.
