An HDG formulation for incompressible and immiscible two-phase porous media flow problems

Abstract

We develop a high-order hybridizable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridization procedure, provides built-in stabilization for arbitrary polynomial degrees and, if the error of the temporal discretization is low enough, the pressure, the saturation and their fluxes converge with order P + 1 in L 2 -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P + 2 in L 2 -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery. KEYWORDSTwo-phase flow; immiscible; incompressible; hybridizable discontinuous Galerkin method; diagonally implicit Runge-Kutta method; differential algebraic equations.

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.…”

“…Let l be the l-th iteration of the fix-point iterative method. Thus, we first solve Equation (20) for the oil saturation unknowns (S…”

“…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.…”