A Numerical IMPES Discontinuous Galerkin method for Immiscible Groundwater Contaminations Flow Using Lax-Wendroff scheme

Abstract The numerical modeling of the immiscible flows in the porous media is one of the issues which have always been considered by researchers due to their application in the monitoring of the groundwater pollutions, water and oil behavior in the petroleum reservoirs and hydrology sciences. In this study, we present a two-dimensional discontinuous Galerkin numerical model of immiscible flows in a porous media using the high order implicit pressure-explicit saturation (IMPES) strategy for governing equations. Here, the primary unknowns are wetting phase-pressure and saturation. In this hybrid numerical scheme, for the first time we developed the second-order Lax-Wendroff method to solve the water saturation equation which is considered as the main novelty of this paper. For the numerical modeling of immiscible groundwater pollutions, it has been utilized the local conservative discontinuous Galerkin scheme as the spatial discretization. The backward Euler and second-order Lax-Wendroff scheme are applied as temporal discretization for pressure and saturation equations respectively. Also, we stabilized the exchanging numerical flux and used projection of the velocity field in the H (div) vectorial interpolation space for improvement of results at the heterogeneities.at the end of each time step, non-physical oscillations omitted using modified Chaven-Jaffre slope limiter and the results are stabilized. The second-order Lax-Wendroff scheme based on the Taylor expansion and the high order time derivatives is comparable with conventional IMPES strategy schemes such as multi stage Runge-kutta Method (RKDG) while has less computation cost than multi stage schemes. However, the time step size and the Courant number have some restrictions with respect to the explicit solving of the saturation equation. In order to validation of the model, the Buckley-Leverett benchmark problem is considered. The results of the developed model are compared with of other authors and a good agreement is observed between them. Also, model efficiency and ability have been evaluated with two test cases for high heterogeneous aquifers. Also employing various techniques improved the discontinuities resolution in highly heterogeneous media. Numerical models showed good non-oscillatory resolution of saturation around the less permeable subdomains and frontal interface between the wetting and nonwetting phases. In this study, the penalty parameter varies between 50 and 100. In SWIP version of DG method, the penalty parameter should be chosen greater than 50 while in OBB-DG method zero values could be allocated. The sensitivity analysis of the model has been considered for various effective parameters in modeling.