Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media

TL;DR

This paper introduces a distributed order Hausdorff diffusion model to effectively describe complex non-Fickian diffusion behaviors in heterogeneous porous media, capturing both accelerating and decelerating anomalous diffusion phenomena.

Contribution

The work develops a novel distributed order Hausdorff diffusion model that is mathematically equivalent to a nonlinear diffusion equation and can describe various non-Fickian diffusion types.

Findings

Model captures both accelerating and decelerating anomalous diffusion.

Successfully describes tracer transport in pore spaces and prefractal media.

Fits experimental data of sub-diffusion in porous structures.

Abstract

Many theoretical and experimental results show that solute transport in heterogeneous porous media exhibits multi-scaling behaviors. To describe such non-Fickian diffusions, this work provides a distributed order Hausdorff diffusion model to describe the tracer transport in porous media. This model is proved to be equivalent with the diffusion equation model with a nonlinear time dependent diffusion coefficient. In conjunction with the structural derivative, its mean squared displacement (MSD) of the tracer particles is explicitly derived as a dilogarithm function when the weight function of the order distribution is a linear function of the time derivative order. This model can capture both accelerating and decelerating anomalous and ultraslow diffusions by varying the weight parameter c. In this study, the tracer transport in water-filled pore spaces of two-dimensional Euclidean is demonstrated as a decelerating sub-diffusion, and can well be described by the distributed order Hausdorff diffusion model with c = 1.73. While the Hausdorff diffusion model can accurately fit the sub-diffusion experimental data of the tracer transport in the pore-solid prefractal porous media.