Reservoir Flow Simulation by Adaptive ADER Schemes

Abstract

In this chapter, an extension of ADER schemes is presented in order to solve both linear and nonlinear scalar conservation laws on unstructured triangulations. The proposed scheme is conservative and belongs to the class of finite volume schemes. It combines high order reconstruction techniques with a high order flux evaluation method to update cell average values through fluxes across cell interfaces. The ADER approach results in an explicit, one-step scheme based on the solution of generalized Riemann problems across cell interfaces. Moreover, the triangulation is adaptively modified during the simulation to effectively combine high order accuracy with locally refined meshes and therefore reduce the computational costs. The required adaption rules for the refinement and coarsening of the triangular mesh rely on a customized error indicator. Numerical experiments confirm the expected orders of accuracy and show the good performance of the proposed scheme for linear and nonlinear problems. Finally, the adaptive ADER schemes are applied to a test case from oil industry, which plays an important role in the modelling of fluid flow in petroleum reservoirs.