# Discontinuous Galerkin finite element methods applied to two-phase, air-water flow problems

## Abstract

A set of discontinuous Galerkin (DG) finite element methods are proposed to solve the air-water, two-phase equations arising in shallow subsurface flow problems. The different time-splitting approaches detailed incorporate primal formulations, such as Oden-Baumann-Babuska DG (OBB-DG), Symmetric Interior Penalty Galerkin (SIPG), Non-Symmetric Interior Penalty Galerkin (NIPG), and Incomplete Interior Penalty Galerkin (IIPG); as well as a local discontinuous Galerkin (LDG) method applied to the saturation equation. The two-phase flow equations presented are split into sequential and implicit pressure/explicit saturation (IMPES) formulations. The IMPES formulation introduced in this work uses one of the primal DG formulations to solve the pressure equation implicitly at every time step, and then uses an explicit LDG scheme for saturation equation. This LDG scheme advances in time via explicit Runge-Kutta time stepping, while employing a Kirchoff transformation for the local solution of the degenerate diffusion term. As fluid saturations may be discontinuous at the interface between two material types, DG methods are a natural fit for this problem. An algorithm is introduced to efficiently solve the system of equations arising from the primal DG discretization of the model Poisson’s Equation on conforming grids. The eigenstructure of the resulting stiffness matrix is examined and the reliance of this system on the penalty parameter is detailed. This analysis leads to an algorithm that is computationally optimal and guaranteed to converge for the order of approximation p = 1. The algorithm converges independently of h and of the penalty parameter σ. Computational experiments show that this algorithm also provides an excellent preconditioning step for higher orders of approximation and extensions are given to 2D and 3D problems. Computational results are also shown for a more general second order elliptic equation, for example, cases with heterogeneous and non-isotropic K. The numerical schemes presented are verified on a collection of standard benchmark problems and the two-phase flow formulations are validated using empirical results from the groundwater literature. These results include bounded column infiltration problems in which the soil air becomes compressed and entrapped, as well as other shallow subsurface infiltration problems. It is shown that the IMPES approach introduced holds promise for the future, especially for problems with very small, or even zero, capillary pressure. Such problems are commonly found in the SPE literature. Finally, initial computational results are shown which relate to a simplified model of the CO2 sequestration problem.