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Entropy 2013, 15(10), 4376-4391; doi:10.3390/e15104376
Article

Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?

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Received: 30 August 2013 / Revised: 23 September 2013 / Accepted: 10 October 2013 / Published: 16 October 2013
(This article belongs to the Special Issue Dynamical Systems)
Download PDF [428 KB, 24 February 2015; original version 24 February 2015]

Abstract

Solute transport through homogeneous media has long been assumed to be scale-independent and can be quantified by the second-order advection-dispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silica-sand columns exhibits sub-diffusion growing with the spatial scale. Only at a very small travel distance (approximately 10 cm) and a relatively short temporal scale can the transport be approximated by normal diffusion. This is also the only spatiotemporal scale where the fundamental concept of the “representative element volume” (which defines the scale of homogeneous cells used by the ADE-based hydrologic models) is valid. The failure of the standard ADE motivated us to apply a tempered-stable, fractional advection-dispersion equation (TS-FADE) to capture the transient anomalous dispersion with exponentially truncated power-law late-time tails in CuSO4 breakthrough curves. Results show that the tempering parameter in the TS-FADE model generally decreases with an increase of the column length (probably due to the higher probability of long retention processes), while the time index (which is a non-local parameter) remains stable for the uniformly packed columns. Transport in sand columns filled with relatively homogeneous silica sand, therefore, is scale-dependent, and the resultant transient sub-diffusion can be quantified by the TS-FADE model.
Keywords: fractional dynamics; fractional-derivative models; scale; homogeneous fractional dynamics; fractional-derivative models; scale; homogeneous
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhang, Y.; Xu, H.; Lv, X.; Wu, J. Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process? Entropy 2013, 15, 4376-4391.

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