Abstract
Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid, and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features like faults, fractures, facies, or other geobodies; or a particular computational challenge like propagating displacement and chemical fronts, wells being turned on or off, etc. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a non-overlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse grid block. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators – computed on different coarse grids and possibly also by different basis-function formulations – to be combined into one iterative procedure.
Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators having different spatial resolution and that each individual operator can be designed to target, among others, challenging grids including faults, pinchouts and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.
Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators having different spatial resolution and that each individual operator can be designed to target, among others, challenging grids including faults, pinchouts and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.
