HP Primal Discontinuous Galerkin Finite Element Methods for Two-Phase Flow in Porous Media

The understanding and modeling of multiphase flow has been a challenging research problem for many years. Among the important applications of the two-phase flow problem are simulation of the oil recovery and environmental protection. The two-phase flow problem in porous media is mathematically modeled by a nonlinear system of coupled partial differential equations that express the conservation laws of mass and momentum. In general, these equations can only be solved by the use of numerical methods.The research in the thesis mainly focuses on the numerical simulation and analysis of different models of incompressible two-phase flow in porous media using primal Discontinuous Galerkin (DG) finite element methods.First, in our work we derive sharp computable lower bounds of the penalty parametersfor stable and convergent symmetric interior penalty Galerkin methods (SIPG) applied to the elliptic problem. In particular, we obtain the explicit dependence of the coercivity constants with respect to the polynomial degrees and the angles of the mesh elements. These bounds play an important role in the derivation of the stability bounds for the SIPG method applied to the the two-phase flow problem. Next, we consider three different implicit pressure-saturation formulations for two-phase flow. We study both h- and p-versions, i.e. convergence is obtained by either refining the mesh or by increasing the polynomial degree. We develop numerical analysis for one of the pressure-saturation formulations. Numerical tests which confirm our theoretical results are presented. Some validation of the proposed schemes, comparison between numerical solutions which are obtained by different schemes and numerical simulations of benchmark problems also given

A linear domain decomposition method for partially saturated flow in porous media

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.Comment: 34 pages, 13 figures, 7 table