A Variational Multiscale Method for the Simulation of Porous Media Flow in Highly Heterogeneous Formations

Multiscale phenomena are ubiquitous to flow and transport in porous media. They manifest themselves through at least the following three facets: (1) effective parameters in the governing equations are scale dependent; (2) some features of the flow (especially sharp fronts and boundary layers) cannot be resolved on practical computational grids; and (3) dominant physical processes may be different at different scales. Numerical methods should therefore reflect the multiscale character of the solution. In this paper, we concentrate on the development of simulation techniques that account for the extreme heterogeneity present in realistic reservoirs, and have the ability to perform accurate predictions on coarse grids. Traditionally, this problem has been tackled by upscaling the fine-scale properties to coarse-scale effective coefficients. Recently, two related but distinct methods have been proposed: the multiscale finite element method, and the variational multiscale method. The organizing center of the variational multiscale (VMS) paradigm is that the original problem is split (rigorously) into a coarse-scale problem and a subgrid-scale problem. The framework is very flexible with respect to how each of these problems is approximated. In this paper, we develop a VMS method for the simulation of flow in highly heterogeneous reservoirs. The proposed VMS method employs a low-order mixed finite element method at the coarse scale, and a finite volume method at the subgrid scale. The method is therefore locally conservative at both the coarse and fine scales. We pay special attention to the definition of the local boundary conditions for the subgrid problems. In particular, we develop a well model, which accounts for subgrid heterogeneity and radial flow regime in a consistent fashion, without compromising the local mass conservation property. We present the application of the method to two-dimensional, highly heterogeneous problems. These results illustrate the applicability and enormous potential of the method. Finally, we discuss the extension of the method to nonlinear problems, three-dimensional systems, and complex (unstructured) grids.