Multigrid Homogenization of Heterogeneous Porous Media

This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL); this report, however, reports on only two years’ research, since this project was terminated at the end of two years in response to the reduction in funding for the LDRD Program at LANL. The numerical simulation of flow through heterogeneous porous media has become a vital tool in forecasting reservoir performance, analyzing groundwater supply and predicting the subsurface flow of contaminants. Consequently, the computational el%ciency and accuracy of these simulations is paramount. However, the parameters of the underlying mathematical models (e.g., permeability, conductivity) typically exhibit severe variations over a range of significantly different length scales. Thus the numerical treatment of these problems relies on a homogenization or upscaling procedure to define an approximate coarse-scale problem that adequately captures the influence of ihe fine-scale structure, with a resultant compromise between the competing objectives of computational efficiency and num&ical accuracy. For h~mo~eni~ation in models of flow th~ough heterogeneous porous media, We have developed new, efficient, numerical, multilevel methods, that offer a significant improvement in the compromise between accuracy and efficiency. We recently combined this approach with the work of Dvor6k [6] to compute bounded estimates of the homogenized permeability for such flows and demonstrated the effectiveness of this new algorithm with numerical examples. Background and Research Objectives The flow of oil, water, chemicals, or heat in a porous underground formation can be modeled as a coupled system of nonlinear partial differential equations (PDEs) describing the flow, mass conservation and chemical or biological reactions of individual species. In many petroleum reservoir and aquifer computer models, a 2D fine mesh simulation has a resolution of 10-20 meters. This scale is much larger than the natural length scale of the permeability in the reservoir. In homogenizing or upscaling heterogeneous media, the macroscopic equations are averaged over the microscopic length scales and treated as locally homogeneous with variations on the much larger macroscopic length scale. *Principal Investigator, e-mail: [email protected]