An Efficient Implementation of the Discontinuous Galerkin Method for Multiphase Flows through Heterogeneous Porous Media

Abstract

Summary One of the main challenges in immiscible multiphase flows lies in getting an accurate representation of the strong coupling between the unavoidable heterogeneity of the porous medium and instabilities of immiscible multiphase flows appearing near the interface of the fluids. We propose an approach to improve the accuracy of the simulation of immiscible flows in heterogeneous porous media using a Discontinuous Galerkin (DG) method. The main objective of this work is to achieve both accuracy and computational efficiency by dynamically decomposing the domain and implementing different solution strategies in different flow regions. An important advantage of DG methods is the ability to approximate the solution by discontinuous polynomials of various degrees in various elements. Thanks to this feature, local flow details near the front may be taken into account by increasing the order of polynomial approximations in the elements of this flow region. To overcome the increased computational cost associated with high-order DG methods, a finite volume scheme is used far from the front. To this aim, we have also developed a front tracking method to model the position of the fluids interface. This method solves a simplified two-phase flow problem to identify the grid blocks in which the front is present. Knowing the position of the front using this fast computation, allows us to identify these different flow regions that are treated separately. Far from the front, the flow is mainly single-phase and the finite volume scheme proved to be satisfactory. In the vicinity of the front, high-order DG is used to capture the instabilities and complexities of the immiscible flow. In this work, the accuracy and computational efficiency of the results are presented in comparison to flow simulations where a high-order DG scheme is used over the whole domain.