A Multilevel Multiscale Finite-Volume Method

Abstract

The Multiscale Finite-Volume (MsFV) method has been developed over the last decade to efficiently solve large reservoir models. The method projects the original pressure problem onto a second coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine-scale grid. One characteristic of the MsFV algorithm is to allow the reconstruction of an approximate but fully conservative velocity field from the prolongated pressure. This makes the method particularly attractive for applications involving the solution of transport problems. Here, we present an extension of the MsFV method (MMsFV) that can employ multiple levels of coarsening instead of the single coarse level used in the standard algorithm. Whereas the coarse problem and the prolongation operators can be easily obtained by recursive application of the MsFV method, formulating an efficient reconstruction of the conservative velocity is not trivial. We devise a nested reconstruction procedure that is novel and has computational cost comparable with the MsFV reconstruction. By analyzing the computational complexity of the algorithm we show that the MMsFV method allows obtaining a conservative approximation of the fine-scale velocity more efficiently than the MsFV method. However, the accuracy of the solution deteriorates and MMsFV errors are larger than MsFV errors. By means of numerical test cases we demonstrate that, when the MMsFV operator is used as preconditioner in GMRES, the number of iterations necessary to achieve the same accuracy is larger than with the MsFV operator. For highly-heterogeneous permeability fields more than several hundreds of iterations might be required. Such a large number of iterations might be practically intractable. Therefore, we propose two ideas that can be used to control the number of iterations. The first is to combine the MMsFV coarse operator with an appropriate smoother in a two-step preconditioner; the second is to modify the localization assumptions used to solve edge problems. Our numerical tests show that both strategies lead to a significant reduction of the iterations and suggest that efficient MMsFV methods can be obtained by identifying optimal smoothers or devising better localizations for the edge problems.