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United States Department of Agriculture

Agricultural Research Service

Title: Fractional Advective-Dispersive Equation as a Model of Solute Transport in Porous Media

item Pachepsky, Yakov
item San Jose Martinez, Fernando
item Rawls, Walter

Submitted to: Springer Verlag
Publication Type: Book / Chapter
Publication Acceptance Date: 8/18/2006
Publication Date: 7/20/2007
Citation: Pachepsky, Y.A., San Jose Martinez, F., Rawls, W.J. 2007. Fractional Advective-Dispersive Equation as a Model of Solute Transport in Porous Media. In: Sabatier, J., Agrawal, O., Machado, J., editors. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Dordrecht, The Netherlands:Springer Verlag. pp. 218-231.

Interpretive Summary: Understanding solute transport in soils is critical for managing the fate of agricultural chemicals in environment, crop productivity, and agricultural sustainability. Successful models of solute transport become an essential decision-support tool. Recent data re-analysis has shown that existing models cannot in some cases correctly describe spreading, or dispersion, of solutes in soils. We are developing and applying model of fractional solute dispersion in soil. This model assumes that the solute particles can stay trapped for a long time and also travel large distance with a relatively large speed. These assumptions are based on observations of structure of field soils in which large stagnant zones border with small pathways of preferential flow. We applied the fractional dispersion model theory leads to simulate solute transport in two field studies, and found that it simulated solute transport in field soil better than the classical convective-dispersive equation. The fractional dispersion model is a promising enhancement in the hydrologist’s toolbox.

Technical Abstract: Understanding and modeling transport of solutes in porous media is a critical issue in the environmental protection. The common model is the advective-dispersive equation (ADE) describing the superposition of the advective transport and the Brownian motion in water-filled pore space. Deviations from the advective-dispersive transport have been documented and attributed to the physical heterogeneity of natural porous media. It has been suggested that the solute transport can be modeled better assuming that the random movement of solute is the Lévy motion rather than the Brownian motion. The corresponding fractional advective-dispersive equation (FADE) was derived using fractional derivatives to describe the solute dispersion. We present and discuss an example of fitting the FADE numerical solutions to the data on chloride transport in columns of structured clay soil. The constant concentration boundary condition introduced a substantial mass balance error then the solute flux boundary condition was used. The FADE was a much better model compared to the ADE to simulate chloride transport in soil at low flow velocities.

Last Modified: 09/21/2017
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