Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media

Abstract

In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixedhybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena.Keywords. Brezzi-Douglas-Marini, Raviart-Thomas, mixed-hybrid, continuity of wetting phase pressure, penalty discontinuous Galerkin, discontinuous nonlinear interface condition, centroidal Voronoi Delaunay triangulation § 1 IntroductionIn the article we focus on the numerical methods for the model of fractional flow formulation of a two-phase flow problem. The fractional flow formulation has two equations, the pressure equation and the saturation equation which are coupled through the total velocity and the saturation affected coefficients. Depending on different combination of the basic equations of the two-phase flow

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

“…It can be observed that the capillary pressure functions are essentially in agreement with the experimental data. The capillary pressure curves clearly demonstrate that the capillary pressure functions are non-convex functions with respect to the saturation S w , and thus the conventional convex functions [2,[4][5][6][7]16] for the capillary pressure would not agree with the experimental data.…”

“…The classical two-phase flow models [2,[4][5][6][7] have been developed based on the single-phase flow model through introducing several new quantities, such as saturation, relative permeability, and capillary pressure. Relative permeability considerably effects on oil and water two-phase flows in porous media.…”

High‐order hybridizable discontinuous Galerkin formulation with fully implicit temporal schemes for the simulation of two‐phase flow through porous media

“…Several methods have been used to simulate two-phase flow, such as the finite element method, 2 the finite volume method, [2][3][4][5] mixed finite element methods, 2,6,7 and discontinuous Galerkin methods. 2,[8][9][10][11][12][13][14][15][16][17] Recently, many efforts have been focused on applying high-order methods to these kind of problems due to their advantages.…”

Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

“…In [29,30,44], decoupled solution using a mixed finite element method for the pressure equation and an explicit or semi-implicit DG method for the saturation equation were studied. In [36], decoupled DG methods with interior penalties and upwinding schemes were applied to the original formulation while the pressure equation is obtained by summing the discretized conservation equations of two phases.…”