Advective–Dispersive Equation with Spatial Fractional Derivatives Evaluated with Tracer Transport Data

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The classical model used to describe solute transport in soil is based on the advective–dispersive equation (ADE) in which an analog of Fick's law is used to model dispersion. The fractional ADE (FADE) has been proposed to address discrepancies between experimental solute concentrations and those predicted with the ADE. The order of the fractional derivative or Lévy exponent, α, characterizes the deviation of the FADE solutions from the ADE, which is a specific case of the FADE for α = 2. The objective of this work was to test the hypothesis that using the FADE with values of α other than 2 can provide more accurate simulations of solute transport in soils. We fitted the FADE to 47 published data sets on tracer breakthrough in disturbed and undisturbed soil columns. The FADE was solved numerically with a mass-conserving boundary condition. While 19 breakthrough curves were best fitted with the ADE, the rest were fitted better using the FADE with α < 2. The RMSEs of the FADE and ADE were close when the FADE α was >1.5. In contrast, the FADE RMSEs were, on average, 1.5 times smaller than the ADE RMSE when the FADE was fitted with α < 1.5. The value of α apparently reflected the structure of the void space available for flow and transport. Considered as a generalization of the classical ADE, the FADE can be a useful model if tails of the breakthrough curve are of special interest.