The thesis focuses on developing numerical schemes for two-phase flow in rigid and deformable porous media problems. We present a stable and efficient sequential Discontinuous Galerkin (DG) method for solving the linear poroelasticity equations, which characterize two-phase flow within a deformable porous media. More precisely, we approximate the pressure of the wetting phase, the pressure of the non-wetting phase, and the displacement of the solid skeleton in three dimensions by a high-order interior penalty discontinuous Galerkin (IPDG) spatial discretization combined with a backward Euler discretization in time. The proposed work is based on previous developments in single fluid flow in deformable porous media. The numerical scheme solves the coupled equations sequentially while keeping each equation implicitly with respect to its unknown. The equations are fully decoupled in this sequential approach, which significantly reduces the computational cost compared to the implicit and iterative approaches. Numerical experiments show the convergence of the scheme is optimal. Finally, we apply the sequential DG scheme to a variety of physical problems with realistic data including common benchmarks, heterogeneous porous media with discontinuous permeability, porosity and capillary pressure, and porous media subjected to load.

The second part of this thesis proposes an adaptive hybrid numerical scheme for solving two-phase flow in rigid porous media problems. The spatial discretization for transport phenomena problem in heterogeneous porous media requires locally mass conservative methods, such as finite volume methods and discontinuous Galerkin methods. The numerical scheme uses discontinuous Galerkin methods in regions of interest where high accuracy is needed and uses finite volume methods in the rest of the domain. The proposed schemes take advantage of the high accuracy of the discontinuous Galerkin method due to its local mesh adaptivity and local choice of polynomial degree. Finite volume methods are only first-order accurate but computationally efficient and robust for general geometries with structured mesh. We develop adaptive indicators to dynamically identify the regions of each method. By using such an adaptive indicator we are able to find the optimal balance between accuracy and computational cost.