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Please use this identifier to cite or link to this item: http://hdl.handle.net/10373/163

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Title: The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives
Authors: Zhang, Xiaoxian
Lv, Mouchao
Crawford, John W.
Young, Iain M.
Affiliation: University of Abertay Dundee. Scottish Informatics, Mathematics, Biology and Statistics Centre
Keywords: Fractional advection-dispersion equation
Boundary conditions
Solute transport
Soil
Aquifer
Caputo derivatives
Numerical solution
Tracer experiment
Issue Date: May-2007
Publisher: Elsevier Science B.V., Amsterdam
Type: Journal Article
Refereed: peer-reviewed
Rights: Published version (c)Elsevier Science B.V., Amsterdam, available from 10.1016/j.advwatres.2006.11.002
Citation: Zhang, X., et al. 2007. The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives. Advances in Water Resources. 30(5): pp.1205-1217. [Online] Available from: DOI: 10.1016/j.advwatres.2006.11.002
Abstract: The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl− in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.
URI: http://hdl.handle.net/10373/163
ISSN: 0309-1708
Appears in Collections:SIMBIOS Collection

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