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In this article, a numerical solution of incompressible two-phase flow in isothermal condition, based on wetting pressure-wetting saturation formulation (Pw,Sw) using high order primal discontinuous Galerkin (DG) methods is considered which can capture the shock fronts of two-phase flow in heterogeneous porous media. In this presented model, the velocity field is reconstructed by a H(div) post-process in lowest order of Raviart-Thomas space (RT0). Also in this study, the scaled penalty and weighted average (harmonic average) formulation significantly improve the especial discretization formulation of governing equations which cause to reduce the instabilities in heterogamous media. The modified MLP slope limiter is used to remove the non-physical saturation values at end of each time step. In this study, the slope limiter should be considered as one of the main novelties due to the impressive effects in results stabilization. The proposed model is verified by pseudo 1D Buckley-Leverett and Mcwhorter problems. Two test cases, a problem for modeling the secondary recovery of petroleum reservoirs and other one a problem for detecting immiscible contamination are used to show the abilities of shock capturing two phases interface in porous media.

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The present paper aims to evaluate a class of discontinuous Galerkin methods for modeling of coupled flow and mass transport equations in porous medium. Various combinations of primal discontinuous Galerkin methods were used for discretization of the coupled nonlinear system of flow and mass transport equations in a saturated porous medium and a fully implicit backward Euler scheme was applied for temporal discretization. The primal DGs have been developed successfully for density-dependent flows by applying both Cauchy and Dirichlet boundary conditions to the mass transport equation. To avoid the errors arising from non-compatible selection of DG methods for flow and mass transport equations, only compatible combinations were applied. To linearize the resulting nonlinear systems, Picard iterative technique was applied and a slope limiter was used to eliminate the nonphysical oscillations appeared in solution. For the purpose of consistent velocity approximation, Frolkovic-Knabner method was used. Three benchmark problems were simulated for validation and verification of the numerical code, which the results from the simulations show a good accuracy and low numerical dispersion for the model. Finally, to highlight the significance of consistent velocity approximation, a hydrostatic test problem was prepared.