Colloquium

The development of numerical algorithms for simulations of flow processes in large-scale highly heterogeneous porous formations is challenging because properties of natural geologic porous formations (e.g., permeability) display high variability and complex spatial correlation structures which can span a hierarchy of length scales. It is usually necessary to resolve a wide range of length and time scales, which can be prohibitively expensive, in order to obtain accurate predictions of the flow, mechanical deformation, and transport processes under investigation. In practice, some types of coarsening (or upscaling) of the detailed model are usually performed before the model can be used to simulate complex processes. Many approaches have been developed and applied successfully when a scale separation adequately describes the spatial variability of the subsurface properties (e.g., permeability) that have bounded variations. The quality of these approaches deteriorates for complex heterogeneities without scale separation and high contrast. In this talk, I will describe multiscale model reduction techniques that can be used to systematically reduce the degrees of freedoms of fine-scale simulations and discuss applications to preconditioners and coupling to global model reduction tools. Numerical results will be presented that show that one can improve the accuracy of multiscale methods by systematically adding new coarse basis functions, obtain contrast-independent preconditioners for complex heterogeneities, and get reduced order models at low cost.