Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
A fundamental challenge in modeling flow through porous media is the need to resolve the multiscale structure of geological formations, which contain length scales from millimeters to kilometers. Consequently, fully resolved flow simulations are likely to remain intractable for the foreseeable future; however, current upscaling techniques, needed to generate computationally tractable coarse-scale models, are either extremely costly or fail to capture the true influence of fine-scale heterogeneous structure on the flow. To address this critical weakness, we explore a new multilevel upscaling (MLUPS) methodology that accurately and efficiently treats the multiscale properties of the underlying porous medium and flow model. This approach leverages key components of robust variational multigrid solvers to generate a complete hierarchy of coarse-scale models, which naturally defines multiscale basis functions that not only capture the influence of fine-scale structure on the coarse scale, but may also be used to recover additional fine-scale flow information as well. Thus, our approach is naturally related to the Multiscale Finite Element Method (MSFEM) and, indeed, attains similar accuracy in the metrics considered here, while being an order of magnitude faster.
