HP primal discontinuous Galerkin finite element methods for two-phase flow in porous media.

摘要

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 parameters for 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 are also given.