Statistics of random walks in geologically relevant conductance fields

Area of Science:

• Physics
• Geophysics
• Applied Mathematics

Background:

• Random walks model diffusion in various media, with established power laws for distinct sites visited and return probabilities.
• Previous studies assumed uniform bond conductances, limiting applicability to heterogeneous systems like porous media.

Purpose of the Study:

• To investigate how broadly distributed hydraulic conductances affect random walk statistics in porous media.
• To determine if universal power laws hold for diffusion in heterogeneous geomedia.

Main Methods:

• Extensive Monte Carlo simulations of random walks on a 2D lattice.
• Consideration of five geologically relevant conductance distributions: normal, log-normal, fractional Brownian motion (FBM), log-FBM, and stable distributions.
• Analysis of key diffusion statistics: mean number of distinct sites visited S(t), mean probability of return to origin P₀(t), and mean-squared displacements .

Main Results:

• Random walk statistics, including exponents p and ζ for S(t) and P₀(t), depend on the conductance distribution.
• Diffusion is anomalous for at least three distributions, with deviating from linear time dependence.
• Exponents differ from those found in homogeneous lattices and fractal structures.

Conclusions:

• Diffusion in highly heterogeneous porous media is anomalous.
• The statistics of random walks are sensitive to the underlying conductance distribution.
• Anomalous diffusion in geomedia can be described by fractional partial differential equations whose orders depend on conductance distribution details.