Adaptive higher order discontinuous Galerkin methods for porous-media multi-phase flow with strong heterogeneities

Abstract

In this thesis, we develop, analyze, and implement adaptive discontinuous Galerkin (DG) finite element solvers for the efficient simulation of porous-media flow problems. We consider 2d and 3d incompressible, immiscible, two-phase flow in a possibly strongly heterogeneous and anisotropic porous medium. Discontinuous capillarypressure functions and gravity effects are taken into account. The system is written in terms of a phase-pressure/phase-saturation formulation. First and second order Adams-Moulton time discretization methods are combined with various interior penalty DG discretizations in space, such as the symmetric interior penalty Galerkin (SIPG), the nonsymmetric interior penalty Galerkin (NIPG) and the incomplete interior penalty Galerkin (IIPG). These fully implicit space time discretizations lead to fully coupled nonlinear systems requiring to build a Jacobian matrix at each time step and in each iteration of a Newton-Raphson method. We provide a stability estimate of the saturation and the pressure with respect to initial and boundary data. We also derive a-priori error estimates with respect to the L2(H1) norm for the pressure and the L∞(L2)∩L2(H1) norm for the saturation. Moving on to adaptivity, we implement different strategies allowing for a simultaneous variation of the element sizes, the local polynomial degrees and the time step size. These approaches allow to increase the local polynomial degree when the solution is estimated to be smooth and refine locally the mesh otherwise. They also grant more flexibility with respect to the time step size without impeding the convergence of the method. The aforementioned adaptive algorithms are applied in series of homogeneous, heterogeneous and anisotropic test cases. To our knowledge, this is the first time the concept of local hp-adaptivity is incorporated in the study of 2d and 3d incompressible, immiscible, two-phase flow problems. Delving into the issue of efficient linear solvers for the fully-coupled fully-implicit formulations, we implement a constrained pressure residual (CPR) two-stage preconditioner that exploits the algebraic properties of the Jacobian matrices of the systems. Furthermore, we provide an open-source DG two-phase flow simulator, based on the software framework DUNE, accompanied by a set of programs including instructions on how to compile and run them.