Abstract
A large number of multiscale methods have been developed based on the same basic concept: Solve localized flow problems to estimate the local effects of fine-scale petrophysical properties. Use the resulting multiscale basis functions to pose a global flow problem a coarser grid. Reconstruct conservative fine-scale approximations from the coarse-scale solution. By extending the basic concept with iteration cycles and additional local stages, one can systematically drive the fine-scale residual towards machine precision. Posed algebraically, this can be seen as a set of restriction operators for computing a reduced global problem and a set of prolongation operators for constructing conservative fine-scale approximations. Such multiscale finite-volume methods have been extensively developed for Cartesian grids in the literature. The industry, however, uses very complex with unstructured connections and degenerate cell geometries to represent realistic structural frameworks and stratigraphic architectures. A successful multiscale method should therefore be able to handle unstructured polyhedral grids, both on the fine and coarse scale, and be as flexible as possible to enable automatic coarse partitionings that adapt to wells and geological features in a way that ensures optimal accuracy for a chosen level of coarsening. Herein, we will discuss a compare a set of prolongation operators that can be combined with finite-element or finite-volume restriction operators to form different multiscale finite-volume methods. We consider the MsFV prolongation operator (developed on a dual coarse grid with unitary at coarse block vertices), the more recent MsTPFA operator (developed on primal grid with unitary flux across coarse block faces), as well as a simplified constant prolongation operator. The methods will be compared on a variety of test cases ranging from simple synthetic grids to highly complex, real-world, field models. Our discussion will focus on flexibility wrt (coarse) grids and tendency of creating oscillatory approximations. In addition, we will look at various methods for improving the methods’ convergence properties when used as preconditioners, as well as for generating novel prolongation operators. This is relevant for oil recovery because: - Multiscale methods may provide a way to significantly speed up reservoir simulation and make previously intractable problems possible to solve. - The extension of such methods to industry standard grids used for reservoir modelling enables the evaluation of the methods on real world models - The construction of basis functions for multiscale methods may have direct connections to the process of upscaling rock derived properties such as transmissibility