Mathematical Modeling and Analysis Multigrid Homogenization of Heterogenous Porous Media

Introduction: The earth is perhaps the most obvious example of a porous medium. Flows in porous media are part of our everyday lives, from water in the garden, to oil in underground reservoirs. The mathematical modeling of flow in porous media, particularly with the ever increasing power of computers, is playing a fundamental and increasingly important role in the forecasting of petroleum reservoir performance, groundwater supply, and subsurface contaminant flow. A critical underlying problem in the numerical treatment of these models is the need to resolve the multiscale structure of heterogeneous geological formations. Unfortunately, the length scales observed in sedimentary laminae range from the millimeter scale upward, while the simulation domain may be on the order several kilometers. As a result, fully resolved simulations are computationally intractable, and yet the fine-scale variations of the model’s parameters (e.g., structure and orientation of laminae) significantly affect the coarse-scale properties of the solution (e.g., average flow rates). This complex interaction of significantly different length scales is not unique to flows in porous media, but arises in many other disciplines, and is currently studied by T-7 in several other important contexts; including composite materials, global atmospheric and ocean circulation models, and in solid-solid phase transitions.