About the Relevance of Downscaling for Nonlinear Problems in Porous Media

Abstract:

In multiscale systems, scale separation is a major challenge in many industrial applications. The resulting complexity was, in the beginning, dealt with using phenomenological models and macroscopic approaches. Improvements in upscaling methods then allowed deriving macroscopic models from micro-scale transport models, greatly improving the understanding of experimentally observed phenomena. However, the investigation of many problems involving highly nonlinear phenomena (e.g. high-temperature heat transfer, chemistry, high-concentration mass transfer, etc.) remains out of the reach of current upscaling methods, even though the associated physics can be described with reasonable accuracy at the microscopic scale, mainly because the effects of nonlinearity can often not be fully passed from the microscopic scale to the macroscopic one without knowing the state of the medium at the microscopic scale.

From this observation comes the idea of using a multiscale approach to investigate problems requiring exchange of information between scales. While in upscaling, information goes from the micro-scale to the macro-scale, downscaling does the opposite and allows the reconstruction of information in a limited region of the micro-scale, based on macro-scale information. Used together, upscaling and downscaling allow the exchange of information between both scales. This multiscale approach facilitates the investigation of highly nonlinear problems or that of cases with evolving micro-geometry.

This presentation first aims at showing the relevance of a multiscale approach for transport in porous media and shows promising results yielded by the downscaling methodology for nonlinear heat transfer.