Experimental and numerical investigation of time-dependent consolidation coefficient using spectral-finite difference method with Chebyshev cardinal functions

Havvaei, Arash; Ejlali, Reza Goli; Lakestani, Mehrdad; Behrooz Sarand, Fariba

doi:10.22034/cmde.2026.70597.3517

Experimental and numerical investigation of time-dependent consolidation coefficient using spectral-finite difference method with Chebyshev cardinal functions

Document Type : Research Paper

Authors

¹ Department of Civil Engineering, Ta. C., Islamic Azad University, Tabriz, Iran.

² 1. Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran. 2. Software Engineering Department, Faculty of Engineering and Natural Sciences, Istinye University, 34396 Sariyer, Istanbul, Turkey.

10.22034/cmde.2026.70597.3517

Abstract

In this study, modifications were made to the conventional oedometer apparatus by installing a pore water pressure sensor at the base of the soil specimen cell. This setup enabled the direct measurement of pore water pressure during consolidation at different times, providing valuable insight into the internal behavior of soil under loading. Based on the recorded data, a time-dependent consolidation coefficient was defined as a function of pore water pressure variation. To accurately represent the pore pressure dissipation trend, a double-exponential function was employed, ensuring high precision in curve fitting. Subsequently, the one-dimensional Terzaghi consolidation equation was solved in a nonlinear framework using the spectral-finite difference method in combination with Chebyshev cardinal functions. The obtained results demonstrated excellent accuracy, surpassing the conventional finite difference method and providing predictions closer to realistic soil behavior. This approach highlights the advantages of integrating advanced mathematical techniques with experimental modifications to enhance the understanding of soil consolidation processes.

Keywords

Main Subjects

Spectral methods

References

[1] H. Aboshi and H. Yoshikuni, The Consolidation of Soft Clay Improved by Sand Drains, in: Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering, 1 (1972), 21–27.
[2] A. Afarideh, F. D. Saei, M. Lakestani, and B. N. Saray, Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions, Phys. Scr., 96(12) (2021), 125267.
[3] A. W. Bishop and D. J. Henkel, The Measurement of Soil Properties in the Triaxial Test, Edward Arnold, London, 1962.
[4] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, 2000.
[5] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, 1988.
[6] R. P. Chen, W. Zhou, H. H. Z. Wang, and Y. M. Chen, One-dimensional nonlinear consolidation of multi-layered soil by differential quadrature method, Comput. Geotech., 32(5) (2005), 358–369.
[7] J. C. Chai, N. Miura, M. Asaoka, and T. S. R. Lo, Consolidation analysis considering vertical drain well resistance and smear effect, Int. J. Numer. Anal. Methods Geomech., 28(4) (2004), 365–390.
[8] X. Gong, J. Yin, and Y. Gao, Investigation of the mechanical behavior of unsaturated soils under cyclic loading, Acta Geotech., 11(6) (2016), 1353–1365.
[9] I. Hashim, O. Abdulaziz, and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14(3) (2009), 674–684.
[10] A. Havvaei, R. G. Ejlali, M. Lakestani, and F. B. Sarand, Solving the exact nonlinear consolidation equation using Chebyshev cardinal functions, Appl. Comput. Math., 24(3) (2025), 469–485.
[11] B. Indraratna and I. Redana, Dynamic response of railway ballast under cyclic loading, J. Geotech. Geoenviron. Eng., 123(12) (1997), 1107–1117.
[12] M. Khodaparast, A. Taheri, and R. Khodaparast, Application of spectral methods to nonlinear Terzaghi consoli-dation problems in soil mechanics, Int. J. Geomech., 17(4) (2017), 04016109.
[13] M. Lakestani and M. Dehghan, Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions, Appl. Math. Model., 36(2) (2012), 605–617.
[14] M. Lakestani and J. Manafian, Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations, Optical and Quantum Electronics, 50(396), (2018).
[15] K. K. Lee, K. H. Xie, and Y. K. Cheung, Consolidation of layered soil systems, Geotechnique, 42(3) (1992), 363–372.
[16] S. Leroueil and D. Marques, Slope instability in sensitive clays, Can. Geotech. J., 33(2) (1996), 241–256.
[17] S. Liu and G. Lei, Consolidation analysis of multilayered soil with partially drained and impermeable boundaries, Comput. Geotech., 43 (2012), 22–30.
[18] J. Manafian and M. Lakestani, Interaction among a lump, periodic waves, and kink solutions to the fractional generalized CBS–BK equation, Mathematical Methods in the Applied Sciences, 44(1), (2021), 1052–1070.
[19] S. Momani and K. Al-Khaled, A numerical solution for systems of fractional differential equations by the decom-position method, Appl. Math. Comput., 162(3) (2005), 1351–1365.
[20] S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decompo-sition method, Appl. Math. Comput., 177(2) (2006), 488–494.
[21] F. Nourian, M. Lakestani, S. Sabermahani, and Y. Ordokhani, Touchard wavelet technique for solving time-fractional Black-Scholes model, Comput. Appl. Math., 41(4) (2022), 150.
[22] R. L. Schiffman and S. Stein, One-dimensional consolidation of layered soil systems, J. Soil Mech. Found. Div., 96(SM5) (1970), 1551–1568.
[23] Z. Shafinejhad and M. Zarebnia, Numerical solution of fractional Volterra integro-differential equations using flatlet oblique multiwavelets, Comput. Methods Differ. Equ., 12(2) (2024), 374–391.
[24] A. Singh, S. Kumar, and H. Ramos, An efficient computational method based on exponential B-splines for a class of fractional sub-diffusion equations, Comput. Methods Differ. Equ., 12(4) (2024), 719–740.
[25] C. Tang and K. Onitsuka, Undrained behavior of overconsolidated clays under triaxial loading, Soils Found., 37(3) (1997), 1–12.
[26] K. Terzaghi, Theoretical Soil Mechanics, John Wiley & Sons, New York, 1943.
[27] M. Valizadeh, Y. Mahmoudi, and F. Dastmalchi Saei, On fractional linear multi-step methods for fractional order multi-delay nonlinear pantograph equation, Comput. Methods Differ. Equ., 12(3) (2024), 511–522.
[28] X. Wang, J. Li, Y. Zhang, and H. Chen, Mechanical behavior of geosynthetic-reinforced soil under cyclic loading, J. Geotech. Geoenviron. Eng., 146(11) (2020), 04020128.
[29] K. H. Xie and R. L. Schiffman, One-dimensional consolidation of layered systems, Geotechnique, 37(4) (1987), 391–400.
[30] H. Y. Xu, A. Aljohani, Y. H. Xu, H. Li, and J. A. Shali, Results on solutions for several systems of the first order nonlinear PDEs and PDDEs in C2, TWMS J. Pure Appl. Math., 15(2) (2024), 228–245.
[31] J. H. Yin and J. Graham, Elastic visco-plastic modeling of one-dimensional time-dependent behavior of clays, Geotechnique, 49(5) (1999), 631–642.
[32] O. C. Zienkiewicz and T. Shiomi, Dynamic analysis of saturated porous media: The generalized Biot formulation and its numerical solution, Int. J. Numer. Anal. Methods Geomech., 8(1) (1984), 71–96.