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Modeling flow with fractures can be challenging due to complex fracture geometries, strong variations in length and time scales as well as the need to combine multiple flow models. In this work, we propose a technique for overcoming these challenges based on the mimetic finite difference method (MFD) and the related multipoint flux mixed finite element method (MFMFE). These discretizations can be employed for solving problems over a set of general polyhedral meshes that can capture nonplanar fracture geometry. Our approach defines a physically and mathematically consistent technique for including internal boundary conditions. In our discretizations, flow in the reservoir and flow in the fracture are coupled using different physical models and numerical schemes. The respective flow models are locally conservative and are based on an implicit pressure/explicit saturation formulation and its iterative form.

We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L 2 -inner product of vectors in two H(Ω; div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω; div)-conforming Petrov-Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.Mathematics Subject Classification. 65M60, 65N08, 76S05.

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