Generalized Skewed Model for Spatial-Fractional Advective–Dispersive Phenomena

Abstract

The conventional mathematical model expressed by the advection–dispersion equation has been widely used to describe contaminant transport in porous media. However, studies have shown that it fails to simulate early arrival of contaminant, long tailing breakthrough curves and presents a physical scale-dependency of the dispersion coefficient. Recently, advances in fractional calculus allowed the introduction of fractional order derivatives to model several engineering and physical phenomena, including the anomalous dispersion of solute particles. This approach gives birth to the fractional advection–dispersion equation. This work presents new solutions to the fractional transport equation that satisfies the initial condition of constant solute injection in a semi-infinite medium. The new solution is derived based on a similarity approach. Moreover, laboratory column tests were performed in a Brazilian lateritic soil to validate the new solution with experimental data and compare its accuracy with the conventional model and other fractional solutions. The new solution outperforms the existing ones and reveals an interesting fractal-like scaling rule for the diffusivity coefficients.

Fractional advection diffusion asymmetry equation, derivation, solution and application

“…Contamination spreading, as one of the most crucial problems ranging from environment to agriculture, has attracted a lot of attention [68,82,83,85,86]. In real systems, how to quantity the contaminant transport is quite a challenge for researchers since the processes are so complex.…”