Pre-asymptotic Transport Upscaling in Inertial and Unsteady Flows Through Porous Media

In most classical formulations of flow and transport through porous media Reynolds numbers are assumed to be small (Re<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Re}<1$$\end{document}), meaning that the role of inertia is considered negligible. However, many examples of practical relevance exist where this is not the case and inertial effects can be important leading to changes in flow structure and even giving rise to unsteady and turbulent flows as Reynolds numbers become larger. This change in flow structure can have a profound impact on how solutes are transported through the porous medium, influencing how effective large-scale transport should be modeled. Here we simulate, using high-resolution numerical models, flow and transport through an idealized porous medium for flow conditions over a range of Reynolds numbers, including steady and unsteady flows. For all these conditions we propose and test three upscaled models for transport—an advection dispersion equation, an uncorrelated spatial model (USM) and a spatial Markov model (SMM). The USM and SMM fall into the wider and more general family of continuous time random walk models. We test these models by their ability to reproduce pre-asymptotic and asymptotic plume second centered moments and breakthrough curves. We demonstrate that for steady flows where inertial effects are strong, the spatial Markov model outperforms the other two, faithfully capturing many of the non-Fickian features of transport, while for unsteady flows the uncorrelated spatial model performs best, due to the fact that unsteadiness in the flow field dampens the role of correlation on large scale transport. We conclude that correlation must be accounted for to properly upscale transport in steady flows, while it can be neglected in unsteady flows.