Using the Fractional Advection Dispersion Equation to Improve the Scale Effect in the Sampling Process of Column Tests with Lateritic Soils

Abstract

Breakthrough curves (BTCs) obtained from column tests in heterogeneous soils are not satisfactorily simulated with the advection-dispersion equation (ADE) for some heavy tailed cases. Furthermore, the dispersion coefficient calculated with the ADE for heavy tailed BTCs are scale dependent when simulating columns of soil larger than the original test depth. In this paper we compare the usage of a fractional ADE (FADE) and the classical ADE to fit column tests BTCs made with Brazilian lateritic soils, discussing both contaminant transport theories and underlying stochastic models. The FADE can more accurately simulate heavy tailed BTCs, and when applying the adjusted FADE parameters to longer depths of soil, the FADE also predicts a more realistic scenario of contaminant transport through heterogeneous soil. The addition of fractional calculus in the advection-dispersion equation proves to improve contaminant transport predictions based on column tests over the classical ADE, with the use of a constant fractional dispersion coefficient that is scale independent.

Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

“…Recently, the fractional solution was shown to be able to better simulate the behavior of laboratorial breakthrough curves from heterogeneous soils, resulting in a dispersion coefficient that is less susceptible to scale changes …”

“…Recently, the fractional solution 20 was shown to be able to better simulate the behavior of laboratorial breakthrough curves from heterogeneous soils, resulting in a dispersion coefficient that is less susceptible to scale changes. 17,18 Solutions to the fractional advection-dispersion are not easily obtainable or may have a complicated form. In order to avoid these obstacles, we employ another numerical method, which is validated mainly by studies on the field of fractional differential equations, the L2 method.…”

“…The field of contaminant transport has significantly advanced in the last two decades using fractional calculus, such as seen in Benson, Benson et al, and Schumer et al These studies provide the physical and mathematical formulation of the fractional advection‐dispersion equation to study contaminant transport in a porous medium as a superdispersion phenomenon. Notably, the fractional advection‐dispersion equation results in a dispersion coefficient that is invariant with respect to the scale of experimental observation . This feature makes the fractional approach especially useful to model flux in heterogeneous and lateritic soils…”

“…Notably, the fractional advection-dispersion equation results in a dispersion coefficient that is invariant with respect to the scale of experimental observation. 17 This feature makes the fractional approach especially useful to model flux in heterogeneous and lateritic soils. 18 The fractional solution to the advection-dispersion equation is obtained with the use of Fourier transforms in the Riemann-Liouville sense, which falls into an α-stable Levy distribution.…”