Stochastic Foundation to Solving Transient Unsaturated Flow Problems Using a Fractional Dispersion Term
Publication: International Journal of Geomechanics
Volume 22, Issue 1
Abstract
Mathematical modeling of unsaturated water seepage through soils uses the core concepts of continuum mechanics. More precisely, it combines the continuum mass balance equation and Darcy–Buckingham Law into the Richards equation for water flow in an unsaturated porous medium. This paper proposes the possibility of an interpretation following another perspective: using the concepts of statistical mechanics and the movement of quasi-molecules of water in soils. It resumes an underexplored approach using the Langevin equation to describe the movement of the quasi-molecules. By replacing the term that accounts for the Gaussian white noise for a term that follows a stable distribution, the mathematical manipulations render a fractional partial differential equation that describes the water flow into unsaturated soils. A constructed analytical solution is proposed for the new equation. A set of experimental data for an unsaturated flow on a soil column is fitted using the new model. Its results are compared to a fit using the integer-order linearized Richards equation solution.
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Acknowledgments
This study was financed in part by the Coordination for the Improvement of Higher Education Personnel—Brasil (CAPES)—Finance Code 001. The authors also acknowledge the support of the National Council for Scientific and Technological Development (CNPq Grants 304721/2017-4, 435962/2018-3, and 305484/2020-6), the Foundation for Research Support of the Federal District (FAPDF) (Project Nos. 0193.002014/2017-68 and 0193.001563/2017), the CEB Geração S.A. (Project No. PD-05160-1904/2019), and the University of Brasília.
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Received: May 27, 2021
Accepted: Sep 21, 2021
Published online: Nov 10, 2021
Published in print: Jan 1, 2022
Discussion open until: Apr 10, 2022
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