Non-local formulation for multiscale flow in porous media

Abstract

Summary The multiscale nature of geological formations is reflected in the flow and transport behaviors of the pore fluids. For example, multiple pathways between different locations in the porous medium are usually present. The topology, length, and strength of these flow paths can vary significantly, and the total flow at a given location can be the result of contributions from a wide range of pathways between the points of interest. We use a high-resolution pore network of a natural porous formation as an example of the multiscale connectivity of the pore space. A single continuum model can capture the contributions from all the flow paths properly only if the control volume (computational cell) is much larger than the longest pathway. However, depending on the densities and lengths of these long pathways, choosing the appropriate size of the control volume that allows for a single continuum description of the properties, such as conductivity and transmissibility, may conflict with the desire to resolve the flow field properly. To capture the effects of the multiscale pathways on the flow, a non-local continuum model is described. The model can represent non-local effects, for which Darcy’s law is not valid. In the limit where the longest connections are much smaller than the size of the control volume, the model is consistent with Darcy’s law. The non-local model is used to describe the flow in complex pore networks. The pressure distributions obtained from the non-local model are compared with pore-network flow simulations, and the results are in excellent agreement. Importantly, such multiscale flow behaviors cannot be represented using the local Darcy law.