Asymptotic Determination of the Liquefaction Depth for Short and Long Water Waves
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 147, Issue 3
Abstract
In this work, an analytical solution that models the dynamic response of a poroelastic soil, induced by the action of water waves in a stratified fluid, is obtained. We propose a reduction of the generalized governing equations for the limit when the horizontal displacements of the porous soil are very small compared with the vertical displacements. The results obtained are similar to those calculated with the governing equations in their complete form. We consider that the instantaneous liquefaction depth is unknown; therefore, the boundary value problem is solved as an eigenvalue problem. The instantaneous liquefaction depth is calculated for a wide range of wave frequencies, from long waves to short waves. It is found that liquefaction depth increases with increasing wavelength. For large values of soil shear modulus, the liquefaction depth increases because the pore pressure magnitude is larger than for the case of soils with low shear modulus. This work can be used as a benchmark to identify the effects that the physical parameters of a poroelastic soil have on the liquefaction depth.
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Acknowledgments
This research was supported by project 20195909 from the Instituto Politécnico Nacional.
Notation
The following symbols are used in this paper:
- G
- soil shear modulus, N/m2;
- H
- total water depth, m;
- h1, h2
- water depths, m;
- k
- wavenumber, m−1;
- ks
- Darcy’s permeability, m/s;
- Lz
- liquefaction depth, m;
- n
- porosity;
- p
- pore pressure, N/m2;
- dimensionless pore pressure;
- Sρ1/ρ2
- ratio of fluid densities;
- and
- dimensionless displacements;
- u
- horizontal soil displacement, m;
- ratio between characteristic horizontal and vertical soil displacements;
- w
- vertical soil displacement, m;
- αγwβnω/ks k2
- dimensionless parameter;
- β0(kLz)2
- eigenvalue;
- Γγwω/G ks k2
- dimensionless parameter;
- γs
- specific weight of soil, N/m3;
- γw
- specific weight of water, N/m3;
- λ
- wavelength, m;
- ν
- Poisson’s ratio of soil; and
- ω
- angular wave frequency.
References
Arcos, E., E. Bautista, and F. Méndez. 2017. “Dynamic response of a poro-elastic soil to the action of long water waves: Determination of the maximum liquefaction depth as an eigenvalue problem.” Appl. Ocean Res. 67: 213–224. https://doi.org/10.1016/j.apor.2017.07.010.
Biot, M. A. 1941. “General theory of three-dimensional consolidation.” J. Appl. Phys. 12 (2): 155–164. https://doi.org/10.1063/1.1712886.
Biot, M. A. 1956. “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range.” J. Acoust. Soc. Am. 28 (2): 179–191. https://doi.org/10.1121/1.1908241.
Chen, C., and J. Hsu. 2005. “Interaction between internal waves and a permeable seabed.” Ocean Eng. 32 (5–6): 587–621. https://doi.org/10.1016/j.oceaneng.2004.08.010.
Chowdhury, B., G. R. Dasari, and T. Nogami. 2006. “Laboratory study of liquefaction due to wave-seabed interaction.” J. Geotech. Geoenviron. 132 (7): 842–851. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:7(842).
De Groot, M., M. Bolton, P. Foray, P. Meijers, A. Palmer, R. Sandven, A. Sawicki, T. Teh. 2006. “Physics of liquefaction phenomena around marine structures.” J. Waterway, Port, Coastal, Ocean Eng. 132 (4): 227–243. https://doi.org/10.1061/(ASCE)0733-950X(2006)132:4(227).
Guo, Z., D. Jeng, and W. Guo. 2014. “Simplified approximation of wave-induced liquefaction in a shallow porous seabed.” Int. J. Geomech. 14 (4): 1–5. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000366.
Hsu, C.-J., Y.-Y. Chen, and C.-C. Tsai. 2019. “Wave-induced seabed response in shallow water.” Appl. Ocean Res. 89: 211–223. https://doi.org/10.1016/j.apor.2019.05.016.
Hsu, J., and D. Jeng. 1994. “Wave-induced soil response in an unsaturated anisotropic seabed of finite thickness.” Int. J. Numer. Anal. Methods Geomech. 18 (11): 785–807. https://doi.org/10.1002/(ISSN)1096-9853.
Jeng, D.-S. 2012. Porous models for wave-seabed interactions. Berlin: Springer.
Jeng, D. S., and B. R. Seymour. 1997. “Response in seabed of finite depth with variable permeability.” J. Geotech. Geoenviron. 123 (10): 902–911. https://doi.org/10.1061/(ASCE)1090-0241(1997)123:10(902).
Liu, Z., D. Jeng, A. Chan, and M. Luan. 2009. “Wave-induced progressive liquefaction in a poro-elastoplastic seabed: A two-layered model.” Int. J. Numer. Anal. Meth. Geomech. 33: 591–610. https://doi.org/10.1002/nag.v33:5.
Madsen, O. S. 1978. “Wave-induced pore pressures and effective stresses in a porous bed.” Geotechnique 28 (4): 377–393. https://doi.org/10.1680/geot.1978.28.4.377.
Mei, C. C., and M. A. Foda. 1981. “Wave-induced responses in a fluid-filled poro-elastic solid with a free surface-a boundary layer theory.” Geophys. J. R. Astron. Soc. 66 (3): 597–631. https://doi.org/10.1111/gji.1981.66.issue-3.
Sakai, T., K. Hatanaka, and H. Mase. 1992. “Wave-induced effective stress in seabed and its momentary liquefaction.” J. Waterway, Port, Coastal, Ocean Eng. 118 (2): 202–206. https://doi.org/10.1061/(ASCE)0733-950X(1992)118:2(202).
Sassa, H., H. Sekiguchi, and J. Miyamoto. 2001. “Analysis of progressive liquefaction as a moving-boundary problem.” Géotechnique 51 (10): 847–857. https://doi.org/10.1680/geot.2001.51.10.847.
Sassa, S., and H. Sekiguchi. 1999. “Wave-induced liquefaction of beds of sand in a centrifuge.” Géotechnique 49 (5): 621–638. https://doi.org/10.1680/geot.1999.49.5.621.
Sumer, B. 2014. Liquefaction around marine structures. Advances Series on Ocean Engineering. Singapore: World Scientific.
Tong, L., J. Zhang, K. Sun, Y. Guo, and J. Zheng. 2018. “Experimental study on soil response and wave attenuation in a silt bed.” Ocean Eng. 160: 105–118. https://doi.org/10.1016/j.oceaneng.2018.04.027.
Ulker, M., M. Rahman, and D.-S. Jeng. 2009. “Wave induced response of seabed: Various formulations and their applicability.” Appl. Ocean Res. 31: 12–24. https://doi.org/10.1016/j.apor.2009.03.003.
Wang, J., M. Karim, and P. Lin. 2007. “Analysis of seabed instability using element free Galerkin method.” Ocean Eng. 34 (2): 247–260. https://doi.org/10.1016/j.oceaneng.2006.01.004.
Yamamoto, T., H. L. Koning, H. Sellmeijer, and E. P. Van Hijum. 1978. “On the response of a poro-elastic bed to water waves.” J. Fluid Mech. 87 (1): 193–206. https://doi.org/10.1017/S0022112078003006.
Ye, G.-L., J. Leng, and D.-S. Jeng. 2018. “Numerical testing on wave-induced seabed liquefaction with a poroelastoplastic model.” Soil Dyn. Earthquake Eng. 105: 150–159. https://doi.org/10.1016/j.soildyn.2017.11.026.
Zen, K., and H. Yamasaki. 1990. “Mechanism of wave-induced liquefaction and densification in seabed.” Soils Found. 30 (4): 90–104. https://doi.org/10.3208/sandf1972.30.4_90.
Zhang, Y., D.-S. Jeng, F. Gao, and J.-S. Zhang. 2013. “An analytical solution for response of a porous seabed to combined wave and current loading.” Ocean Eng. 57: 240–247. https://doi.org/10.1016/j.oceaneng.2012.09.001.
Zhang, Y., and J. Li. 2010. “Analytical solution for wave-induced response of isotropic poro-elastic seabed.” Sci. China Technol. Sci. 53 (10): 2619–2629. https://doi.org/10.1007/s11431-010-4077-2.
Zienkiewicz, O. C., C. T. Chang, and P. Bettess. 1980. “Drained, undrained, consolidating and dynamic behaviour assumptions in soils.” Geotechnique 30 (4): 385–395. https://doi.org/10.1680/geot.1980.30.4.385.
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Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 147 • Issue 3 • May 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: May 13, 2020
Accepted: Nov 12, 2020
Published online: Feb 5, 2021
Published in print: May 1, 2021
Discussion open until: Jul 5, 2021
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