Multiscale Mixed Methods on Corner-Point grids: Mimetic versus Mixed Subgrid Solvers
Jørg Espen Aarnes, Stein Krogstad, Knut‐Andreas Lie · BIBSYS Brage (BIBSYS (Norway)) · 2006
Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid-models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating sub-scale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately-sized) coarse grid, one can retain the efficiency of an upscaling method, while at the same time produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has shown to be a particularly versatile approach. In this paper we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. To this end, we propose two different subsolvers that both are designed to give accurate and robust discretisations with minimal grid-orientation effects on grids with strong anisotropies and irregular geometries. Our first subsolver subdivides the hexahedral cornerpoint cells into a set of tetrahedra, on which a standard mixed finite-element method is used. Our second subsolver is based on mimetic discretisation principles and works directly on the hexahedral cells. A mimetic discretisation is a generalisation of mixed finite elements that gives a discrete inner product and allows for curved grid faces and polyhedral elements. The versatility and accuracy of the multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. In particular, we demonstrate how one can avoid the usual difficulties of resampling, when moving from a fine to a coarse grid, and vice versa, since the multiscale mixed formulation allows the cells in the coarse grid to be chosen as an (almost) arbitrary connected collection of cells in the underlying fine grid.