Nonlocal Formulation for Multiscale Flow in Porous Media

Abstract:

In porous media multiple pathways may exist between different locations. The length and strength of these pathways vary significantly, and the total flow at a given location is composed of contributions from both very short and long paths. If a pore network representation of such a medium is considered, there exist pores which get bypassed by long tubes. A local single continuum model can only capture the contributions from all paths properly, if the computational cells are larger than the longest connections. However, depending on the density and the lengths of these bypassing connections, choosing appropriately coarse grid blocks might be in conflict with the desired resolution.

In order to capture these non-local effects present due to long bypassing connections, a non-local continuum model has been proposed. Here, it is explained how the model can be derived from the fine-scale description of a porous medium, and it is shown that in the limit where the longest connections are much smaller than the size of the computational cells, the model is consistent with Darcy's law. The non-local model was applied to pore-networks generated from a Berea sand stone sample, and the results were compared with corresponding pore-network simulations. It is shown that the resulting pressure solutions are in very good agreement, yet differ significantly from the Darcy solution. At the field-scale, this method is relevant in connection with fracture networks.