Flow in Heterogeneous Porous Media: Fractures and Uncertainty Quantification

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This work addresses two problems commonly encountered when approximating flow in heterogeneous porous media. The first part presents several numerical methods for a discrete-fracture model dealing with single-phase Darcy flow based on Lagrange multiplier unknowns which couple the conductive fractures of codimension one with the surrounding medium (bulk). This allows for finite element discretizations where the mesh of the bulk domain does not need to be aligned with the fractures. The theoretical analysis on existence, uniqueness and convergence is sustained by various numerical experiments. The second part applies the hybrid stochastic Galerkin method to quantify uncertainties of heterogeneous two-phase flow scenarios. The intrusive method extends the concept of the polynomial chaos expansion for a multi-element decomposition of the stochastic space resulting in a partly decoupled, deterministic system. This hyperbolic-elliptic system is discretized with a central-upwind finite volume scheme for the hyperbolic part along with a mixed finite element method for the elliptic one. Accuracy and convergence is assessed numerically in comparison with non-intrusive methods.