Traveling time and traveling length in critical percolation clusters

Abstract

We study traveling time and traveling length for tracer dispersion in two-dimensional bond percolation, modeling flow by tracer particles driven by a pressure difference between two points separated by Euclidean distance r. We find that the minimal traveling time t(min) scales as t(min) approximately r(1.33), which is different from the scaling of the most probable traveling time, t* approximately r(1.64). We also calculate the length of the path corresponding to the minimal traveling time and find l(min) approximately r(1.13) and that the most probable traveling length scales as l* approximately r(1.21). We present the relevant distribution functions and scaling relations.

Saturation dependence of dispersion in porous media

“…In the current adaptation to generate the saturation dependence of the dispersion, we had to make the calculations suitable for structural percolation controls, i.e., direct topological constraints. The theoretical development is strongly influenced by work of the Eugene Stanley group [59], which shows that characteristic system crossing times scale with system length to the fractal dimensionality of the backbone, D b . We did not use the scaling function for arrival times used in [59] and related publications, however.…”

“…The theoretical development is strongly influenced by work of the Eugene Stanley group [59], which shows that characteristic system crossing times scale with system length to the fractal dimensionality of the backbone, D b . We did not use the scaling function for arrival times used in [59] and related publications, however. That distribution was proposed for use in a binary medium, in which the constituents are either highly, or weakly, permeable.…”

“…Our existing development [12,[38][39][40][41] starts with a combination of critical path analysis and the cluster statistics of percolation theory to predict the probability density function, W p (g|x) that a given system of length x can be spanned by an interconnected cluster of conductances with minimum local conductance value, g, and cluster length equal to the system length x (ie, the clusters span the system). Then we use the theory for the topology of such clusters to predict the dependence of the cluster crossing time, t(g,x) on cluster length, x, as well as minimal conductance value, g. In this theory the dominant parameter is the fractal dimensionality of the percolation backbone [59]. From fundamental probability theory we can then predict the distribution of solute arrival times in terms of W p (g|x) and t (g,x).…”

“…For the former we referred to Ref. [59] that related the typical arrival time to the fractal dimensionality of the backbone.…”

Relevance of percolation theory to power‐law behavior of dynamic processes including transport in disordered media

“…Since this article was written we have learned of a paper by Lee et al [122] in which it is shown that the scaling of time with Euclidean length is governed by the mass fractal dimensionality of the backbone cluster, and this should be used to calculate an effective tortuosity. Numerical results show agreement with relevant simulations for two-dimensional diffusionless dispersion by Liu et al [123].…”