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Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?
Division of Hydrologic Sciences, Desert Research Institute, Las Vegas, NV 89119, USA
State Key Laboratory of Pollution Control and Resource Reuse, Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, China
Department of Hydrosciences, Nanjing University, Nanjing 210046, China
* Author to whom correspondence should be addressed.
Received: 30 August 2013; in revised form: 23 September 2013 / Accepted: 10 October 2013 / Published: 16 October 2013
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
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MDPI and ACS Style
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.
Zhang Y, Xu H, Lv X, Wu J. Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process? Entropy. 2013; 15(10):4376-4391.
Zhang, Yong; Xu, Hongxia; Lv, Xueyan; Wu, Jichun. 2013. "Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?" Entropy 15, no. 10: 4376-4391.