We present a new, high-order, control-volume-finite-element (CVFE) method with discontinuous Nth-order representation for pressure and (N+1)th-order for velocity. The method conserves mass and ensures that the extended Darcy equations for multi-phase flow are exactly enforced, but does not require the use of control volumes (CVs) that span domain boundaries. We demonstrate that the approach, amongst other features, accurately preserves sharp saturation changes associated with high aspect ratio geologic features such as fractures and mudstones, allowing efficient simulation of flow in highly heterogeneous models. Moreover, in conjunction with dynamic mesh optimization, in which the mesh adapts in space and time to key solution fields such as pressure, velocity or saturation whilst honoring a surface-based representation of the underlying geologic heterogeneity, accurate solutions are obtained at significantly lower computational cost than an equivalent fine, fixed mesh and conventional CVFE methods. The work presented is significant for two reasons. First, it resolves a long- standing problem associated with the use of classical CVFE methods to model flow in highly heterogeneous porous media; second, it reduces computational cost/increases solution accuracy through the use of dynamic mesh optimization without compromising parallelization.


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