An experimental study on scale dependency of fractional dispersion coefficient


Six series of solute transport experiments were conducted with an objective to compare scale decency of fractional dispersion coefficient (Df) in space fractional advection–dispersion equation (s-FADE) with that of dispersion coefficient (D) in advection–dispersion equation (ADE) at three flow rates. The experiments were performed in a 2.5 m × 0.1 m × 0.6 m (length × width × height) sandbox filled with a homogeneous or heterogeneous soil. The parameters of the s-FADE and ADE were estimated using an inverse method. The results indicated that the scale dependency of Df was significantly less than that of D in the heterogeneous soil. The maximum variations of Df and D for the heterogeneous soil were observed at the flow rate of 200 cm3 min–1, in which the ratios of the maximum Df and D to the minimum ones were 6.009 and 39.039, respectively. However, in the homogeneous soil, the scale dependency of Df was relatively similar to that of D. In this soil, the ratios of the maximum Df and D to the minimum ones were 3.109 and 5.041, respectively, which were observed at the flow rate of 548 cm3 min–1. Contrary to Df and D, the v values of the s-FADE and ADE for both soils varied in relatively similar ranges at the flow rates studied. Also, the fractional differentiation order (α) of the s-FADE described properly the heterogeneity degrees of the soils used. In a nutshell, the s-FADE is an efficient solute transport model to explain the solute transport process in natural porous media with different heterogeneity degrees.

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Advection–dispersion equation


Space fractional advection-dispersion equation

C :


\( \overline{C} \) :

Average of measured concentrations

v :

Average pore-water velocity

D :

Dispersion coefficient

x :

Spatial coordinate (distance)

t :

Temporal coordinate (time)

L :

Size of spatial domain

OF :

Objective function

n :

Positive integer number

N :

Total number of measurement points


Root mean square error

r 2 :

Determination coefficient

meas :


calc :


f :



Order of fractional differentiation (heterogeneity degree)

ξ :

Auxiliary variable

τ :

Integration variable

Γ(⋅) :

Gamma function


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The authors would like to express thanks for financial support from University of Kurdistan (UOK). The authors also greatly appreciate anonymous reviewers for their valuable comments and suggestions to improve the manuscript.

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Correspondence to Behrouz Mehdinejadiani.

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Responsible Editor: Broder J. Merkel

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Moradi, G., Mehdinejadiani, B. An experimental study on scale dependency of fractional dispersion coefficient. Arab J Geosci 13, 409 (2020).

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  • Fractional dispersion coefficient
  • Fractional order
  • Non-Fickian transport
  • Non-locality
  • Scale dependency