Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

Abstract

We describe the most important developments in the application of three theoretical tools to modeling of the morphology of porous media and flow and transport processes in them. One tool is percolation theory. Although it was over 40 years ago that the possibility of using percolation theory to describe flow and transport processes in porous media was first raised, new models and concepts, as well as new variants of the original percolation model are still being developed for various applications to flow phenomena in porous media. The other two approaches, closely related to percolation theory, are the critical‐path analysis, which is applicable when porous media are highly heterogeneous, and the effective medium approximation—poor man's percolation—that provide a simple and, under certain conditions, quantitatively correct description of transport in porous media in which percolation‐type disorder is relevant. Applications to topics in geosciences include predictions of the hydraulic conductivity and air permeability, solute and gas diffusion that are particularly important in ecohydrological applications and land‐surface interactions, and multiphase flow in porous media, as well as non‐Gaussian solute transport, and flow morphologies associated with imbibition into unsaturated fractures. We describe new applications of percolation theory of solute transport to chemical weathering and soil formation, geomorphology, and elemental cycling through the terrestrial Earth surface. Wherever quantitatively accurate predictions of such quantities are relevant, so are the techniques presented here. Whenever possible, the theoretical predictions are compared with the relevant experimental data. In practically all the cases, the agreement between the theoretical predictions and the data is excellent. Also discussed are possible future directions in the application of such concepts to many other phenomena in geosciences.

Prediction of Soil Formation as a Function of Age Using the Percolation Theory Approach

“…This theoretical description has been thoroughly reviewed in some recent books (Hunt et al, 2013;Hunt and Manzoni, 2016), as well as in Reviews of Geophysics (Hunt and Sahimi, 2017). This theoretical treatment so far does not incorporate directly changes in the medium associated with changing density, changing particle size (network scale), or changing depth due either to deposition or erosion, but such changes can, in principle, be introduced into the equation describing changes in soil depth as calculated from the erosion and soil production rates, a topic deferred to later.…”

“…Percolation theory generates a suite of properties relevant to flow, diffusion, and dispersion, as well as to structure, although it is sometimes necessary to choose which percolation results are appropriate, or whether an alternate formulation, such as the effective medium approximation (EMA), may be more suited to generating an accurate prediction in any specific case (Hunt and Sahimi, 2017). Nevertheless, the general theory of percolation is best discussed in terms of its topological implications first.…”

An Entrogram‐Based Approach to Describe Spatial Heterogeneity With Applications to Solute Transport in Porous Media

“…In the recent work of Rizzo and de Barros (), for instance, the connectivity of the K field is quantified with a static metric based on the identification of paths of least resistance (Tyukhova & Willmann, ), which is subsequently shown to be correlated to the early arrival times of a solute plume. Connectivity is closely related to the concepts of percolation and critical path analysis (e.g., Harter, ; Hunt & Sahimi, ; Stauffer & Aharony, ). For instance, by invoking the percolation threshold, Knudby and Carrera () associated the transmissivity along the paths of least resistance (critical path transmissivity) to the effective transmissivity of heterogeneous fields.…”

Comparison and Contrast in Soil Depth Evolution for Steady State and Stochastic Erosion Processes: Possible Implications for Landslide Prediction

“…The square root dependence of soil depth on duration of unsaturated conditions recalls Philip (1957Philip ( , 1967 infiltration theory for the infiltration depth, preserving even the dependence on the square root of K h (Hunt & Sahimi, 2017). The inference of the importance of unsaturated conditions, however, precludes using the Braun et al (2016) model for weathering experiments in (1D) columns under saturated conditions (Salehikhoo et al, 2013); White & Brantley, 2003) but produces the same proportionality to flow rate and the same time dependence of weathering rates as do field experiments.…”

“…Our model for soil formation derives from solute transport theory in porous media (Hunt & Skinner, 2008) developed from percolation theory (PT). Hunt and Sahimi (2017) have recently reviewed the application of percolation theory (PT) to solute transport, chemical weathering and soil formation. Within the framework of PT, solute velocity diminishes with increasing solute transport distance in porous media, and the scaling of travel time (t) on travel distance (x) is determined by the fractional dimensionality of the percolation backbone (D b ), t ∝x Db .…”