Abstract

Flow and transport in fractured porous media are, very often, processes not dominated by diffusion.  This makes the mathematical problem almost hyperbolic, which naturally develops sharp features in  the solution. Classical numerical methods produce a solution that either lacks stability, resulting in nonphysical oscillations, or accuracy, by showing excessive numerical diffusion. Development of novel numerical methods for the complete equations of multiphase compositional flow in multidimensions must necessarily start from simplified models in one space dimension. These reduced model problems should display, however, the key features which pose difficulties in obtaining satisfactory numerical solutions such as, for instance, wild nonlinearity, shocks or near-shocks, boundary layers and degenerate diffusion. The key point of the proposed formulation is a multiscale decomposition of the variable of interest into resolved (or grid) scales and unresolved (or subgrid) scales, which acknowledges the fact that the fine-scale structure of the solution cannot be captured by any mesh.  However, the influence of the subgrid scales on the resolvable scales is not negligible. A novel idea of subgrid stabilization by means of the concave hull of the flux function is introduced. By accounting for the subgrid scales, the oscillatory behavior of classical Galerkin is drastically reduced and confined to a small neighborhood containing the sharp features, while the solution is high-order accurate where the solution is smooth. This ensures that the numerical solution is not globally deteriorated. The  method does not emanate from a monotonicity argument and, therefore, it does not rule out small overshoots and undershoots near the sharp layers. To prevent this situation, a subscale-driven shock-capturing mechanism is presented. The generality of the proposed formulation makes it amenable to further extensions.