A Multiscale Adaptive Local-Global Method for Modeling Flow in Stratigraphically Complex Reservoirs

Abstract

Summary A robust and efficient simulation technique is developed on the basis of the extension of the mimetic finite-difference method (MFDM) to multiscale hierarchical-hexahedral (corner-point) grids by use of the multiscale mixed finite-element method (MsMFEM). The implementation of the mimetic subgrid-discretization method is compact and generic for a large class of grids and, thereby, is suitable for discretizations of reservoir models with complex geologic architecture. Flow equations are solved on a coarse grid where basis functions with subgrid resolution account accurately for subscale variations from an underlying fine-scale geomodel. The method relies on the construction of approximate velocity spaces that are adaptive to the local properties of the differential operator. A variant of the method for computing velocity basis functions is developed that uses an adaptive local-global (ALG) algorithm to compute multiscale velocity basis functions by capturing the principal characteristics of global flow. Both local and local-global methods generate subgrid-scale velocity fields that reproduce the impact of fine-scale stratigraphic architecture. By using multiscale basis functions to discretize the flow equations on a 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 accuracy and efficacy of the multiscale method is compared with those of fine-scale models and of coarse-scale models with no subgrid treatment for several two-phase-flow scenarios. Numerical experiments involving two-phase incompressible flow and transport phenomena are carried out on high-resolution corner-point grids that represent explicitly example stratigraphic architectures found in real-life shallow-marine and turbidite reservoirs. The multiscale method is several times faster than the direct solution of the fine-scale problem and yields more accurate solutions than coarse-scale modeling techniques that resort to explicit effective properties. The accuracy of the multiscale simulation method with ALG-velocity basis functions is compared with that of the local velocity basis functions. The multiscale simulation results are consistently more accurate when the local-global method is employed for computing the velocity basis functions.