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.
Get full access to this article
View all available purchase options and get full access to this article.
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.
References
Aris, R. 1990. Vector, tensors and the basic equations of fluid mechanics. New York: Dover Publications.
Azevedo, M. M. 2016. “Performance of geotextiles with enhanced drainage.” Ph.D. thesis, University of Texas at Austin.
Bear, J. 1972. Dynamics of fluids in porous media. New York: Dover Publications.
Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert. 2000. “The fractional-order governing equation of Lévy Motion.” Water Resour. Res. 36 (6): 1413–1423. https://doi.org/10.1029/2000WR900032.
Bhattacharya, R. N., V. K. Gupta, and G. Sposito. 1976. “On the stochastic foundations of the theory of water flow through unsaturated soil.” Water Resour. Res. 12 (3): 503–512. https://doi.org/10.1029/WR012i003p00503.
Cavalcante, A. L. B., L. P. Borges, and J. G. Zornberg. 2019. “New 3D analytical solution for modeling transient unsaturated flow due to wetting and drying.” Int. J. Geomech. 19 (7): 04019077. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001461.
Cavalcante, A. L. B., and P. V. S. Mascarenhas. 2021. “Efficient approach in modeling the shear strength of unsaturated soil using soil water retention curve.” Acta Geotech. 16: 3177–3186. https://doi.org/10.1007/s11440-021-01144-6.
Cavalcante, A. L. B., and J. G. Zornberg. 2017a. “Efficient approach to solving transient unsaturated flow problems. I: Analytical solutions.” Int. J. Geomech. 17 (7): 04017013. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000875.
Cavalcante, A. L. B., and J. G. Zornberg. 2017b. “Efficient approach to solving transient unsaturated flow problems. II: Numerical solutions.” Int. J. Geomech. 17 (7): 04017014. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000876.
Cosenza, P., and D. Korošak. 2014. “Secondary consolidation of clay as an anomalous diffusion process.” Int. J. Numer. Anal. Methods Geomech. 38 (12): 1231–1246. https://doi.org/10.1002/nag.2256.
Costa, M. B. A., and A. L. B. Cavalcante. 2020. “Novel approach to determine soil–water retention surface.” Int. J. Geomech. 20 (6): 04020054. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001684.
Costa, M. B. A., and A. L. B. Cavalcante. 2021. “Bimodal soil–water retention curve and k-function model using linear superposition.” Int. J. Geomech. 21 (7): 04021116. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002083.
Cushman, J. H., and D. O’Malley. 2015. “Fickian dispersion is anomalous.” J. Hydrol. 531: 161–167. https://doi.org/10.1016/j.jhydrol.2015.06.036.
de Oliveira, E. C., and T. J. Machado. 2014. “A review of definitions for fractional derivatives and integral.” Math. Prob. Eng. 2014: 1–6. https://doi.org/10.1155/2014/238459.
Hall, C. 2007. “Anomalous diffusion in unsaturated flow: Fact or fiction?” Cem. Concr. Res. 37 (3): 378–385. https://doi.org/10.1016/j.cemconres.2006.10.004.
Ibe, O. C. 2013. Elements of random walk and diffusion processes. Hoboken, NJ: Wiley.
Kavvas, M. L., A. Ercan, and J. Polsinelli. 2017. “Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time.” Hydrol. Earth Syst. Sci. 21 (3): 1547–1557. https://doi.org/10.5194/hess-21-1547-2017.
Li, X. M., Q. Q. Zhang, R. F. Feng, J. G. Qian, and H. W. Wei. 2020. “Long-term deformation analysis for a vertical concentrated force acting in the interior of fractional derivative viscoelastic soils.” Int. J. Geomech. 20 (5): 04020040. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001649.
Li, X. M., Q. Q. Zhang, and S. W. Liu. 2021. “Semianalytical solution for long-term settlement of a single pile embedded in fractional derivative viscoelastic soils.” Int. J. Geomech. 21 (2): 04020246. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001906.
Mascarenhas, P. V. S., R. M. Moraes, and A. L. B. Cavalcante. 2019. “Using a shifted Grünwald–Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium.” Int. J. Numer. Anal. Methods Geomech. 43 (11): 1956–1977. https://doi.org/10.1002/nag.2936.
Meerschaert, M. M., and A. Sikorskii. 2012. Stochastic models for fractional calculus. Berlin: De Gruytier.
Mehdinejadiani, B., H. Jafari, and D. Baleanu. 2013. “Derivation of a fractional Boussinesq equation for modelling unconfined groundwater.” Eur. Phys. J. Spec. Topics 222 (8): 1805–1812. https://doi.org/10.1140/epjst/e2013-01965-1.
Moraes, R. M., and A. L. B. Cavalcante. 2015. “Using the fractional advection dispersion equation to improve the scale effect in the sampling process of column tests with lateritic soils.” Defect Diffus. Forum 365: 188–193. https://doi.org/10.4028/www.scientific.net/DDF.365.188.
Nelson, E. 2001. Dynamical theory of Brownian motion. Princeton, NJ: Princeton University Press.
Ochoa-Tapia, J. A., F. J. Valdes-Parada, and J. Alvarez-Ramirez. 2007. “A fractional-order Darcy’s law.” Physica A 374 (1): 1–14. https://doi.org/10.1016/j.physa.2006.07.033.
Ogata, A., and R. B. Banks. 1961. A solution of the differential equation of longitudinal dispersion in porous media: Fluid movement in earth materials. Fluid Movement in Earth Materials. Geological Survey Professional Paper 411-A, 1–7. Washington, DC: United States Government Printing Office.
Olsen, J. S., J. Mortensen, and A. S. Telyeakovskiy. 2016. “A two-sided fractional conservation of mass equation.” Adv. Water Resour. 91: 117–121. https://doi.org/10.1016/j.advwatres.2016.03.007.
Ozelim, L. C., and A. L. B. Cavalcante. 2018. “Representative elementary volume determination for permeability and porosity using numerical three-dimensional experiments in microtomography data.” Int. J. Geomech. 18 (2): 04017154. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001060.
Pachepsky, Y., D. Timlin, and W. Rawls. 2003a. “Generalized Richards’ equation to simulate water transport in unsaturated soils.” J. Hydrol. 272 (1–4): 3–13. https://doi.org/10.1016/S0022-1694(02)00251-2.
Pachepsky, Y., D. Timlin, and W. Rawls. 2003b. “Errata to “generalized Richards’ equation to simulate water transport in unsaturated soils” [J. Hydrol, 272:3–13].” J. Hydrol. 279 (1–4): 290. https://doi.org/10.1016/S0022-1694(03)00195-1.
Schumer, R., M. M. Meerschaert, and B. Baeumer. 2009. “Fractional advection–dispersion equations for modeling transport at the earth surface.” J. Geophys. Res. 114 (F4): F00A07. https://doi.org/10.1029/2008JF001246.
Su, N. 2017. “The fractional Boussinesq equation of groundwater flow and its applications.” J. Hydrol. 547: 403–412. https://doi.org/10.1016/j.jhydrol.2017.01.015.
Sun, Y., and Y. Shen. 2017. “Constitutive model of granular soils using fractional-order plastic-flow rule.” Int. J. Geomech. 17 (8): 04017025. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000904.
Sun, Y., Y. Gao, and C. Chen. 2019. “Critical-state fractional model and its numerical scheme for isotropic granular soil considering state dependence.” Int. J. Geomech. 19 (3): 04018202. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001353.
Szyemkiewicz, A. 2013. Modelling water flow in unsaturated porous medium. Berlin: Springer.
Wheatcraft, S. W., and M. M. Meerschaert. 2008. “Fractional conservation of mass.” Adv. Water Resour. 31 (10): 1377–1381. https://doi.org/10.1016/j.advwatres.2008.07.004.
Yanovsky, V. V., A. V. Chechkin, D. Schertzer, and A. V. Tur. 2000. “Lévy anomalous diffusion and fractional Fokker–Planck equation.” Physica A 282 (1–2): 13–34. https://doi.org/10.1016/S0378-4371(99)00565-8.
Zhuang, P., F. Liu, I. Turner, and Y. T. Gu. 2014. “Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation.” Appl. Math. Modell. 38 (15–16): 3860–3870. https://doi.org/10.1016/j.apm.2013.10.008.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 22 • Issue 1 • January 2022
Copyright
© 2021 American Society of Civil Engineers.
History
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
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Silvio Crestana, Paulo E. Cruvinel, Developing Spectroscopic and Imaging Techniques for Advanced Studies in Soil Physics Based on Results Obtained at Embrapa Instrumentation, Brazilian Journal of Physics, 10.1007/s13538-022-01202-8, 52, 6, (2022).