A hybridizable discontinuous Galerkin method for two‐phase flow in heterogeneous porous media

Abstract

We present a new method for simulating incompressible immiscible two-phase flow in porous media. The semi-implicit method decouples the wetting phase pressure and saturation equations. The equations are discretized using a hybridizable discontinuous Galerkin method. The proposed method is of high order, conserves global/local mass balance, and the number of globally coupled degrees of freedom is significantly reduced compared to standard interior penalty discontinuous Galerkin methods. Several numerical examples illustrate the accuracy and robustness of the method. These examples include verification of convergence rates by manufactured solutions, common one-dimensional benchmarks, and realistic discontinuous permeability fields. KEYWORDSdiscontinuous Galerkin, incompressible flow, transport INTRODUCTIONMultiphase flows in porous media are fundamental processes in geophysics. For instance, they characterize enhanced oil recovery, 1 hydrogeology, 2,3 and CO2 sequestration in geological formations. 4 The equations that govern two-phase flow form a coupled system of nonlinear partial differential equations. Numerous techniques have been proposed to resolve the nonlinearity inherent in the system, for instance, implicit-explicit (IMPES), semi-sequential, semi-implicit, and fully implicit methods. 5 The selection of spatial discretization is also a critical decision in the solution process. Accuracy, local conservation, mass balance, and efficiency of implementation are all important features. With respect to the wetting phase pressure equation, incorrect approximations to the phase velocity can cause oscillations and instability when used in the convection-dominated transport equation satisfied by the wetting phase saturation. Compatible discretizations (as defined in the work of Dawson et al 6 ) for flow and transport maintain local and global mass conservation, which provides stability and accuracy in the numerical methods.In this work, we discretize the pressure and saturation equations by the hybridizable discontinuous Galerkin (HDG) method. The HDG method has several interesting properties. In particular, discrete analogs of global conservation for flow and local conservation for transport are satisfied. This postprocessing is available since the normal component of the numerical flux for the HDG method is single valued. 7 Classical discontinuous Galerkin (DG) methods construct the velocity from the pressure, which means that the velocity (and subsequent its H(div) postprocessing) converges sub-optimally. 8 The DG method has received a lot of attention since the 1990s. These DG methods have a number of attractive features, eg, local mass conservation, hp-adaptation, their ability to handle nonconforming meshes, and the fact that they are well Int J Numer Methods Eng. 2018;116:161-177.wileyonlinelibrary.com/journal/nme

A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

“…To the best of our knowledge, there are very few papers on HDG for complex porous media flows. Recently, we applied the HDG method to two-phase flows in [20] Examples of standard DG methods for miscible displacement can be found in [21,6]. Classical primal non-compatible DG methods that are used for the Darcy system require special attention.…”

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

“…11 Recently, the HDG method has been applied in two-phase flow through porous media. [18][19][20] It has all the advantages of the discontinuous Galerkin formulations. This method introduces the trace of the scalar variable as a new unknown.…”

“…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. 11,15,[18][19][20] If the analytical solution is smooth enough, then the numerical solution obtained with a method of order k converges to the analytical one as h k e in L 2 -norm, being h e the element size of the mesh. [21][22][23] Hence, it has been shown that high-order spatial discretization methods can be more accurate than low-order ones for the same mesh resolution.…”

“…For these reasons, high-order HDG formulation has been recently applied in porous media flow problems. [17][18][19][20] To exploit the advantages of the high-order HDG formulation, we propose to perform a high-order fully implicit RK method to control the temporal error. These RK schemes are unconditionally stable for any combination of element size, polynomial degree and time-step.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…HDG simulations of immiscible incompressible two-phase flows in heterogeneous porous media were first proposed in [130] and coupled with high-order diagonally implicit Runge-Kutta (DIRK) time integrators in [107]. Moreover, in [168] a linear degenerate elliptic problem modelling two-phase mixture is approximated using a hybridised DG approach.…”