Application of Fractional Differential Equations in Analysis of Seepage Line in Coarse Porous Media

Document Type : Research Paper

Authors

1 Department of Irrigation and Reclamation Engineering, Faculty of Agricultural Engineering and Technology, University College of Agriculture and Natural Resources, University of Tehran, P. O. Box 4111, Karaj, 31587-77871, Iran.

2 Associate Professor, Department of Irrigation and Reclamation Engineering, University College of Agriculture and Natural Resources, University of Tehran, P. O. Box 4111, Karaj, 31587-77871, Iran

3 Associate Professor, Soil Science Department, College of Agriculture, Yasouj University, P.O.BOX: 353, Yasouj, 75918-74831, Iran.

Abstract

In this study, the fractional-order differential equations in range of (0,1) were used to model the water surface profile under Darcy's law condition in porous medium for a fully developed turbulent flow. The developed equation is solved analytically. The laboratory model used in this study consists of a coarse-grained porous medium with 6.4 m length, 0.8 m width and 1 m height, including rounded corner materials, which are tested for different flow rates and three longitudinal slopes of 0, 4, 20.3%. Then, parameters of model and porous media were calibrated based on laboratory data. In order to evaluate the proposed analytical solution, the obtained results from fractional-order differential model were compared with the laboratory data. The results showed a satisfactory agreement with experimental data of water surface profile (seepage line) in all three slopes. The maximum error of the proposed model is 3.5% compared to the experimental data. It can be concluded that the proposed method can provide better description of water surface profile analysis under non-Darcy flow conditions as compared to Darcy model in porous media.

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References

Bari, R., and Hansen, D. (2002). Application of gradually-varied flow algorithms to simulate buried streams. Journal of Hydraulic Research, 40(6), 673-683.
Bazargan, J., and Shoaei, S. M. (2006). Application of gradually varied flow algorithms to simulate buried streams.advection‐dispersion equation. Water resources research, 36(6), 1403-1412.
Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000). Application of a fractional advection dispersion equation. Water resources research, 36(6), 1403-1412.
Cooke, R. A., Badiger, S., and Garcı́a, A. M. (2001). Drainage equations for random and irregular tile drainage systems. Agricultural Water Management, 48(3), 207-224.
Ding, Z., Xiao, A., and Li, M. (2010). Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients. Journal of Computational and Applied Mathematics, 233(8), 1905-1914.
Harr, M. E. (1963). Groundwater and seepage. Soil Science, 95(4), 289.
Huang, Q., Huang, G., and Zhan, H. (2008). A finite element solution for the fractional advection–dispersion equation. Advances in Water Resources, 31(12), 1578-1589.
Kavvas, M. L., and Ercan, A. (2014). Fractional governing equations of diffusion wave and kinematic wave open-channel flow in fractional time-space. I. Development of the equations. Journal of Hydrologic Engineering, 20(9), 04014096.
Martinez, F. S. J., Pachepsky, Y. A., and Rawls, W. J. (2010). Modelling solute transport in soil columns using advective–dispersive equations with fractional spatial derivatives. Advances in engineering software, 41(1), 4-8.
Moutsopoulos, K. N. (2009). Exact and approximate analytical solutions for unsteady fully developed turbulent flow in porous media and fractures for time dependent boundary conditions. Journal of Hydrology, 369(1-2), 78-89.
Oldham, K., and Spanier, J. (1974). The fractional calculus theory and applications of differentiation and integration to arbitrary order (Vol. 111). Elsevier.
Parkin, A. K. (1963). Rockfill Dams with Inbuilt Spillways: Hydraulic Characteristics. Water Research Foundation of Australia.
Pavlovsky, N. N. (1956). Collected works, Izd. AN SSSR Moscow–Leningrad, USSR.
Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). Elsevier.
Reddi, L. N. (2003). Seepage in soils: principles and applications. John Wiley & Sons.
Sarkhosh, P., Samani, J. M. V., and Mazaheri, M. (2017). A one-dimensional flood routing model for rockfill dams considering exit height. Proceedings of the Institution of Civil Engineers-Water Management 171(1), 42-51.
Schumer, R., Benson, D. A., Meerschaert, M. M., and Wheatcraft, S. W. (2001). Eulerian derivation of the fractional advection–dispersion equation. Journal of contaminant hydrology, 48(1-2), 69-88.
Sedghi-Asl, M., and Ansari, I. (2016). Adoption of Extended Dupuit–Forchheimer Assumptions to Non-Darcy Flow Problems. Transport in Porous Media, 113(3), 457-469.
Sedghi-Asl, M., Farhoudi, J., Rahimi, H., and Hartmann, S. (2014b). An analytical solution for 1-D non-Darcy flow through slanting coarse deposits. Transport in porous media, 104(3), 565-579.
Sedghi-Asl, M., Rahimi, H., Farhoudi, J., Hoorfar, A., and Hartmann, S. (2014a). One-dimensional fully developed turbulent flow through coarse porous medium. Journal of Hydrologic Engineering, 19(7), 1491-1496.
Stephenson, D. J. (1979). Rockfill in hydraulic engineering (Vol. 27). Elsevier.
Wheatcraft, S. W., and Meerschaert, M. M. (2008). Fractional conservation of mass. Advances in Water Resources, 31(10), 1377-1381.
Wilkins, J. K. (1955). Flow of water through rock fill and its application to the design of dams. New Zealand Engineering, 10(11), 382.
Zhou, H. W., and Yang, S. (2018). Fractional derivative approach to non-Darcian flow in porous media. Journal of hydrology, 566, 910-918.
Iranian Journal of Soil and Water Research
Volume 51, Issue 3
June 2020
Pages 563-574

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