An elliptic-hyperbolic system, which models termodynamical evolution of the hydrocarbon components during the pressure drop due to the extraction of oil and the mass transfers in the oil reservoir, is investigated in this paper. The article is focused on the case oil and water are assumed to be incompressible immiscible fluid with a common pressure and the reservoir is supposed to be horizontal homogeneous isotropic domain. From mathematical point of view we obtain the elliptic-hyperbolic system of two unknown functions: common pressure \(u\) and saturation \(s\) of the water phase, with nonlinear functions \(\gamma(s),\) “fractional flow” and \(\lambda(s)\) “total mobility” in divergence term. The problem is investigated from numerical point of view, where the convergence of the two coupled numerical solutions for non-constant term \( \lambda(s)\) is presented for elliptic problem. For discretization of this system couple finite volume scheme is used, more precise, convergence of approximate solution for pressure in \(L^2(\Omega)\) to the solution of elliptic problem is proved, whereas the approximate saturation converges only in a weak sense in \(L^\infty(\Omega)\) for weak* topology. An extension of the H-convergence background to a discrete settings has been performed. This H-limit of a sequence of finite volume approximation for the general meshes is done for two unknowns which are computed in the same grid. In general case the H-limit of the discrete diffusion is not the diffusion of elliptic problem.

Reviewer: Angela Handlovičová (Bratislava)