Models of two phase flows in porous media, used in petroleum
engineering, lead to a system of two coupled equations with elliptic
and parabolic degenerate terms, and two unknowns,
the saturation and the pressure.
For the purpose of their approximation, a coupled scheme, consisting in
a finite volume method together with
a phase-by-phase upstream weighting scheme, is used in the industrial setting.
This paper presents a mathematical analysis of this coupled scheme, first showing
that it satisfies some a priori estimates:
the saturation is shown to remain in a fixed interval, and
a discrete L2(0,T;H1(Ω)) estimate is proved for both the pressure
and a function of the saturation. Thanks to these properties,
a subsequence of the sequence of approximate solutions is shown to
converge to a weak solution
of the continuous equations
as the size of the discretization tends to zero.