Application of Differential Transform Method for Solving Free-Surface Seepage Problem of One-Dimensional Porous Media

Document Type : Original Article

Authors

Mohammad Saleh Baradaran ¹

Ahmad Aftabi Sani ²

Saeed abrishami ²

¹ Department of Civil Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

² Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

10.22067/jfcei.2024.82170.1226

Abstract

Several numerical methods, such as finite difference, finite volume, finite element, and so on, have been utilized to solve the problem of the free surface seepage in one-dimensional and two-dimensional domains. The governing equation of the seepage problem in porous media is the well-known Laplace equation. The finite difference and finite element methods are based on meshing the desired surface and obtaining the head values at this level; and repeating this process until convergence is reached. In this paper, a new method is proposed to solve the one-dimensional free-surface seepage problem in one-dimensional water column based on the Taylor-series expansion, entitled Differential Transform Method (DTM). It should be mentioned, up to now, DTM is employed to solve the various form of differential equations, but it has not been used in seepage problems. For some verification purposes, the obtained results are compared with the well-known finite difference method, and the good agreement is observed. Moreover, the analytical solution of the problem is attained by the use of mathematical techniques and it is reported in the paper as an appendix.

Keywords

Differential transform method

porous media

Laplace equation

free surface seepage

extended pressure method

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References

 [1] USACE. Seepage analysis and control for dams. (1993)

 [2] M. Baudin, C. Berthon, F. Coquel, M. Roland, “A relaxation method for two-phase flow models with hydrodynamic closure law,” Numerische Mathematik, vol. 99, no. 3, pp. 411-440, (2005).

[3] S. S. Motsa, P. G. Dlamini and M. Khumalo, “Spectral relaxation method and spectral quasilinearization method for solving unsteady boundary layer flow problems.” Advances in Mathematical Physics, (2014).

[4] M. T. Ayvaz, M. Tuncan, H. Karahan, A. Tuncan, “An extended pressure application for transient seepage problems with a free surface,” Journal of Porous Media, vol. 8, no. 6, pp. 613-625 (2005).

[5] Southwell, R. V. Relaxation methods. (1946).

 [6] F. Coquel, E. Goquel. B. Perthame. A. In, P. Rascle, “Some new Godunov and relaxation methods for two-phase flow problems,” Godunov Methods, Springer, vol. 1, pp. 179-188, (2001).

[7] Harr, M. E. Groundwater and seepage. Courier Corporation. (2012).

 [8] K. J. Bathe, M. R. Khoshgoftaar, M. R, “Finite element free surface seepage analysis without mesh iteration,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 3, no, 1, pp. 13-22, (1979).

[9] M. H. Bazyar, A. Graili, “A practical and efficient numerical scheme for the analysis of steady state unconfined seepage flows,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 36, no. 16, pp. 1793-1812, (2012).

[10] K. Chaiyo, P. Rattanadecho, S. Chantasiriwan, “The method of fundamental solutions for solving free boundary saturated seepage problem,” International Communications in Heat and Mass Transfer, vol. 38, no. 2, pp. 249-254, (2011).

[11] J. T. Chen, C. C. Hsiao, Y. P.Chiu, Y. T. Lee, “Study of free‐surface seepage problems using hypersingular equations,” Communications in numerical methods in engineering, vol. 23, no. 8, pp. 755-769, (2007).

[12] A. Cividini and G. Giancarlo, “Finite-element approach to the erosion and transport of fine particles in granular soils,” International Journal of Geomechanics. vol. 4, no. 3, pp. 191-198, (2004).

[13] Fengzhi, L. Scaled Boundary Finite-element Method for Seepage Free Surfaces Analysis [J]. Chinese Journal of Computational Physics, 5 (2009).

 [14] M. I. Herreros, M, Mabssout, M. Pastor, “Application of level-set approach to moving interfaces and free surface problems in flow through porous media,” Computer methods in applied mechanics and engineering, vol. 195, no, 1-3, pp. 1-25, (2006).

[15] M. J. Kazemzadeh-Parsi, F. Daneshmand, “Three dimensional smoothed fixed grid finite element method for the solution of unconfined seepage problems,” Finite Elements in Analysis and Design, vol. 64, pp. 24-35, (2013).

[16] M. J. Kazemzadeh‐Parsi, F. Daneshmand, “Unconfined seepage analysis in earth dams using smoothed fixed grid finite element method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 36, no. 6, pp. 780-797, (2012).

[17] S. S. Athani, Shivamanth, C. H. Solanki, G. R. Dodagoudar, “Seepage and stability analyses of earth dam using finite element method,” Aquatic Procedia, vol. 4, pp. 876-883, (2015).

[18] G. Li, J. Ge, J, Y. Jie, “Free surface seepage analysis based on the element-free method,”. Journal of Tsinghua University (Science and Technology) ((Mechanics research communications)), p. 11, (2002).

[19] M. Bazyar, A. Talebi, “Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite‐element method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 39, no. 1, pp. 1-22, (2015).

[20] Y. Jie, G. Jie, Z. Mao, G. Li, “Seepage analysis based on boundary-fitted coordinate transformation method,” Computers and Geotechnics, vol. 31, no. 4, pp. 279-283, (2004).

[21] J. Zhang, Q. Xu, Z. Chen, “Seepage analysis based on the unified unsaturated soil theory,” Mechanics Research Communications, vol. 28, no. 1, pp. 107-112, (2001).

[22] H. Zheng, H. C. Dai, D. F. Liu, “A variational inequality formulation for unconfined seepage problems in porous media,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 437-450, (2009).

[23] K. Chaiyo, P. Rattanadecho, S. Chantasiriwan, “The method of fundamental solutions for solving free boundary saturated seepage problem,” International Communications in Heat and Mass Transfer, vol. 38, no. 2, pp. 249-254, (2011).

[24] M, Li., Diao, N-r. Diao, Z-H. Fang, Analysis of seepage flow in a confined aquifer with a standing column well, Journal of Hydrodynamics, Ser. B, vol. 19, no. 1, pp. 84-91, (2007).

[25] A. Leontiev, W. Huacasi, “Mathematical programming approach for unconfined seepage flow problem,” Engineering analysis with boundary elements, vol. 25, no. 1, pp. 49-56, (2001).

[26] A. Naji, D. Ouazar, D, A. H. D. Cheng, “Locating the saltwater-freshwater interface using nonlinear programming and h-adaptive BEM,” Engineering analysis with boundary elements, vol. 21, no. 3, pp. 253-259, (1998).

[27] J. K. Zhou, “Differential transformation and its applications for electrical circuits,” In: China, Wuhan, Huazhong University Press, (1986).

 [28] S. H. Ho, C. K. Chen, “Analysis of general elastically end restrained non-uniform beams using differential transform,” Applied Mathematical Modelling, vol. 22, no. 4-), pp. 219-234, (1998).

[29] C. K. Chen, S. H. Ho, “Solving partial differential equations by two-dimensional differential transform method,” Applied Mathematics and computation, vol. 106, no. 2-3, pp. 171-179, (1999).

[30] F. Ayaz, “On the two-dimensional differential transform method,” Applied Mathematics and computation, vol. 143, no. 2-3, pp. 361-374, (2003).

[31] F, Ayaz., “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and computation, vol. 147, no. 2, pp. 547-567, (2004).

[32] J.-P. Bardet, T. Tobita, “A practical method for solving free-surface seepage problems,” Computers and Geotechnics, vol. 29, no. 6, pp. 451-475, (2002).