Abstract
A robust and efficient simulation technique is developed based on the extension of the mimetic finite volume method to multiscale hierarchical hexahedral (corner-point) grids via use of the multiscale mixed finite element method. The implementation of the mimetic subgrid discretization method is compact and generic for a large class of grids, and thereby, suitable for discretizations of reservoir models with complex geologic architecture. Flow equations are solved on a coarse grid where basis functions with subgrid resolution accurately account 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 utilizes an adaptive local-global 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 to that 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 explicitly represent 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 adaptive local-global velocity basis functions are compared to 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.