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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Ilya Mishev, Exxon Mobil Corporation
Mixed Multiscale Finite Volume Methods for Reservoir Simulation


  • Ilya Mishev
  • Lijian Jiang


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 (MMsFV) method 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 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 MMsFV method is the explicit approximation of the velocity, with a new multiscale basis being constructed to approximate the pressure with piecewise constants. The velocity basis functions can be calculated with either local information (local MMsFV) or global information (global MMsFV). We demonstrate the improved accuracy of the global MMsFV compared to the local version on several problems including the SPE 10 comparative solution problem. The error of the global MMsFv is usually several times smaller than that of the local method.

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