MsMFEM

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 subscale 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. The Multiscale Mixed Finite Element Methods (MsMFEM) has shown to be a particularly versatile multiscale method and is one of the key simulation technologies developed in the GeoScale projects.

Goal Fast, accurate, and robust pressure solvers for highly heterogeneous porous media to be used for direct simulation on high-resolution grid models with multimillion cells fast simulation of multiple (stochastic) realisations of reservoir heterogeneity Basic Idea Use a mixed finite-element method on a coarse scale with special basis that satisfy local flow problems and thereby account for subgrid variations Features Solution of flow on coarse scale gives conservative finte-scale velocities Allows for accurate capture of fine-scale saturation movement Robust and accurate calculation of coarse-grid fluxes when used as an upscaling methods Current inhouse solver(s): slightly compressible two-phase black-oil flow on Cartesian, tetrahedral and corner-point grids. Experimental versions for three-phase flow.  Advantages of MsMFEM Easy to build on-top-of existing pressure solvers High efficiency Scalable and easy to parallelise Flexible with respect to grids No need for dual grid cells Selected Project Results Multiscale methods - a robust and accurate alternative to upscaling: The SPE 10 benchmark Comparison with local-global upscaling and other multiscale methods Flexibility with respect to grids: Corner-point grids Nonuniform (Cartesian) grids Triangular grids

Coarse and fine grid, basis function, fine-scale velocities through subresolution in basis functions Coarse-grid cells = arbitrary connected collection of fine-grid cells. Here: uniform partition in index space of a corner-point grid model Corner-point grid of a Y-shaped synthetic reservoir

References:

1. J. E. Aarnes, S. Krogstad, and K.-A. Lie. Multiscale mixed/mimetic methods on corner-point grids. Computational Geosciences, Special issue on multiscale methods, 2008. DOI: 10.1007/s10596-007-9072-8
2. J. E. Aarnes, S. Krogstad, and K.-A. Lie. A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids, Multiscale Modelling and Simulation, Vol. 5, No. 2, pp. 337-363, 2006. DOI: 10.1137/050634566.