## 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.

##### Department

##### Description

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