Implicit-Modal Discontinuous Galerkin Scheme for Two-Phase Flow with Discontinuous Capillary Pressure

Abstract

In this paper, we propose an implicit-modal discontinuous Galerkin scheme for solving an incompressible and immiscible oil-water flow through a heterogeneous reservoir with different rock types. The scheme uses the modal discontinuous Galerkin method for spatial discretization of both pressure and saturation equations and the backward Euler method for temporal discretization. In the modal discontinuous Galerkin method, we apply the well-localized bases that are constructed by both the nodal and modal basis functions. To find the approximate solution for the saturation equation, we propose the upwind technique. In the stability analysis of the scheme, we obtain the constant of the trace inequality for the approximate function and its first derivatives using the modal basis functions. In this way, we give suitable ranges for penalty terms. To guarantee the convergence of the approximate solution, we find suitable ranges of the time step. To demonstrate the applicability of the proposed scheme, we apply it to the dense nonaqueous phase liquids and the Versuchseinrichtung zur Grundwasser und Altlastensanierung infiltration problems. Numerical results are presented.