Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Abstract

Abstract The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two‐phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non‐degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.