Keywords: fractional derivatives, fractional advective-dispersive equation, solute transport, water quality, porous media, miscible displacement experiments in soils.

In the vadose zone, the unsaturated part of soil between the soil surface and the aquifers, solute transport is interpreted as the result of the joint action of advection and dispersion. The parabolic advective-dispersive equation (ADE) has been the theoretical framework to model the fate and transport of chemicals stemming from agricultural practices or waste disposal operations to address critical environmental issues in the last decades. The underlying assumption of the ADE is that solute particles undergo Brownian motions. This model fails to capture some important features of solute transport. On the one hand, the dispersion coefficient tends to increase with the distance of solute concentration observations (scale effects on the dispersion process). On the other hand, breakthrough curves which depict solute concentrations at the outlet of the soil columns as functions of time, display concentrations approaching the asymptotic values more slowly than predicted by the ADE (i.e. BTC have heavy tails). This behaviour is sometime referred to as the anomalous or the non-Fickian dispersion.

The complexity of pore spaces in natural porous media makes the hypothesis of Brownian motion too restrictive in some cases. It has been suggested that, in the soil matrix, high velocity regions tend to be spatially continuous at all scales, and then a solute particle travelling faster than the mean is much more likely to do so over large distances. Similarly, the slower particles experience long waiting times before they have a chance to move in the general direction of the flow. The Lévy motion is a broader framework that allows for persistence in movements of solute particles, and includes the Brownian motion as a specific case. This makes Lévy motion an attractive generalization of Brownian motion when describing solute transport in porous media.

Similarly to the ADE that can be derived assuming Brownian motion, an equation using fractional derivatives can be derived for Lévy motions. The one-dimensional version of the fractional advective-dispersive equation (FADE) with symmetric dispersion for conservative tracers and steady water flow is

Here, D_f is the fractional dispersion coefficient, the superscript is the order of fractional differentiation, , c is the solute concentration, v is the flow velocity, x is the distance and t is the time. Our objective was to consider the FADE as a unified framework and to compare the performance of both the classical model (ADE) and the fractional one (FADE).

The shifted Grünwald approximations

and

of the left and right-sided fractional derivatives of the FADE were used to develop an explicit finite difference scheme for FADE

Here, g_k stands for the Grünwald weights. Boundary conditions have been chosen to ensure mass conservation.

We have assembled a database on published solute transport experiments in soil and evaluated the FADE as the transport model in comparison with classical advective-dispersive equation or ADE. For selected values of (from 1 to 2 with the step of 0.05), D_f and v were estimated using a version of the Marquardt-Levenberg algorithm to fit simulated breakthrough curves to measured ones

Parameter varied with solute transport experimental conditions, i.e. type of soil, type of tracer, flow velocity and saturation degree. Trends of the increase in values of with the increase in saturation and in flow velocity were been observed for this particular dataset. The differences in values of parameter presumably reflected different degrees of complexity of the movement of solute particles in soil which might be caused by the differences in the hierarchical structure of soil pore space for each particular case.

The FADE, as a general model that includes ADE, accurately simulates experimental breakthrough curves. From the 53 experimental BTC's considered, 28 have been better fitted with smaller than 2.00, i.e. with the fractional model, and 25 have been best fitted with , i.e. the classical ADE model. This suggests that FADE rather than ADE should be used as a general framework to study solute transport in soil. The fractional advective-dispersive equation as a generalization of the classical advective-dispersive equation is a promising enhancement in the hydrologist toolbox.

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