CVD-MPFA Based Multiscale Formulation on Structured and Unstructured Grids

Subsurface reservoirs generally have complex geological and geometrical features, such as faults fractures, pinchouts, shales and layers defined on varying length scales. In addition the effect of heterogeneity leads to further multiscale features that cannot be modelled with desired precision on relatively coarse meshes. This has lead to development of multiscale methods over the last decade. This paper focuses on methods for fine scale modelling and presents development of multiscale methods in an unstructured grid framework with particular emphasis on the numerical flux approximation. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy’s law together with mass conservation. The schemes are control-volume distributed (CVD) with pressure and rock properties sharing the same location in a given control-volume and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale formulation applicable to both structured and unstructured grids in two-dimensions. Performance of the Darcy-flux approximations are compared in the multiscale modelling environment on a range of grid types resulting from both structured and unstructured grids. The methods are applied to domains with homogeneous and heterogeneous permeability fields involving a range of test cases. The effects of quadrature range of the schemes is tested. Boundary condition constraints and consequences of basis function formulation, together with implications of scheme and grid type are presented. The development of a CVD-MPFA based multiscale formulation leads to a novel approach for fine scale modelling. The results demonstrate the benefits of the new formulation.