Finite volume approximation of degenerate two‐phase flow model with unlimited air mobility

Abstract

Models of two-phase flows in porous media, used in petroleum engineering, lead to a coupled system of two equations, one elliptic, the other degenerate parabolic, with two unknowns: the saturation and the pressure. In view of applications in hydrogeology, we are interested at the singular limit of this model, as the ratio µ of air/liquid mobility goes to infinity, and in a comparison with the one-phase Richards model. We construct a robust finite volume scheme that can apply for large values of the parameter µ. This scheme is shown to satisfy a priori estimates (the saturation is shown to remain in a fixed interval, and a discrete L 2 (0, T ; H 1 (Ω)) estimate is proved for both the pressure and a function of the saturation) which are sufficient to derive the convergence of a subsequence to a weak solution of the continuous equations, as the size of the discretization tends to zero. At the limit as the mobility of the air phase tends to infinity, we obtain the two-phase flow model introduced in the work Henry, Hilhorst and Eymard [14] (see also [13]) which we call the quasi-Richards equation.

Positivity-preserving finite volume scheme for compressible two-phase flows in anisotropic porous media: The densities are depending on the physical pressures

“…These latter results, especially the first order accuracy, are somehow satisfactory and turn out the scheme in question a robust compromise for the discretization of challenging complex problems. This motivates the application of our idea to the diphasic compressible model since the mobilities of both phases are often approximated by first order upwind schemes as investigated in many works [4,6,7,29,32] and the references are therein.…”

Upwind discontinuous Galerkin methods with mass conservation of both phases for incompressible two‐phase flow in porous media

“…A number of methods simulating multiphase flow problems have been proposed and analyzed in theory, for example, the Refs. [25][26][27][28][29][30]. The difficulty of theoretical analysis for our numerical schemes is how to treat nonlinear intrinsic relationship between the pressures and the saturations.…”