Multiscale finite volume (MsFV) methods have been successfully applied to solving reservoir simulation problems with localized high heterogeneity (i.e., separable scales), but the accuracy decreases when this is not possible (non-separable scales and long-range features). We develop a mixed multiscale finite volume method (MMsFV) on a uniform mesh that can use global information in order to improve the accuracy and the robustness of the multiscale simulations of fluid flows in porous media with non-local features.

Our development starts with the observation that multi-point flux approximation (MPFA) methods implicitly approximate the velocity and therefore any multiscale generalization also has to do the same. MsFV uses multiscale approximation of the pressure and piecewise constant approximation of the velocity. The novelty of the mixed MsFV method is the explicit approximation of the velocity. We construct a new multiscale basis for the velocity and approximate the pressure with piecewise constants. The velocity basis functions can be calculated with either local information (local mixed MsFV) or global information (global mixed MsFV). We demonstrate the improved accuracy of the global mixed MsFV compared to the local version on several problems including the SPE 10 comparative solution problem. The error of the global mixed MsFV is usually much smaller than the error of the local method when porous media exhibit non-local features.

Using the same framework and the extra flexibility of the two approximation spaces other mixed multiscale finite volume methods can be derived including extensions to unstructured meshes.

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