Estimation of Interface Parameter for One-Dimensional Consolidation with Continuous Drainage Boundary Conditions
Publication: International Journal of Geomechanics
Volume 22, Issue 3
Abstract
In conventional consolidation theories, a boundary is simulated as either perfectly pervious or impervious, where the time-dependent drainage capacity at the boundary is ignored. Using the time-dependent impeded boundary condition, it is very difficult to derive solutions analytically. A continuous drainage boundary is proposed in this investigation to characterize the time-dependent drainage behavior at the boundary. The interface parameter is of physical significance, which depends on the properties of both the consolidating soil and the adjacent medium. In this study, two methods are developed to estimate the interface parameter. Back-analysis can be conducted to evaluate the interface parameter based on the variations of excess pore-water pressure from experimental or field measurements. Alternatively, an empirical approach is derived to correlate the interface parameter with the ratio of the coefficient of consolidation and the thickness ratio between adjacent media. The solution is further employed to analyze layered soils with a horizontal drain. It is found that both the plane of maximum excess pore-water pressure and the optimal position of horizontal drain move toward the boundary with a lower drainage capacity with time. A simplified design chart is finally presented to optimize the layout of the horizontal drain in layered clay–drain systems.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 41867034, 51878185, 52008124, 52078506, 52168046, and 52178321), the China Postdoctoral Science Foundation (Grant No. 2020M683210), the Natural Science Foundation of Guangxi Province (Grant Nos. 2016GXNSFGA380008 and 2019GXNSFBA185028), the Bagui Scholars Program (Grant No. 2016A31), the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Engineering Safety (Project No. 2019ZDK041), and the Project of Guangdong Key Laboratory of Oceanic Civil Engineering (Project No. LMCE202108).
Notation
The following symbols are used in this paper:
- a, k, r, and s
- fitting coefficients;
- b and c
- interface parameters;
- Cv
- coefficient of consolidation in the vertical direction;
- C
- error covariance matrix;
- ds
- thickness of sand cushion;
- Es
- elastic modulus of sand cushion;
- H
- thickness of soil;
- kc
- hydraulic conductivity of horizontal sand cushion in the vertical direction;
- ksh
- hydraulic conductivity of horizontal sand cushion in the horizontal direction;
- kcv
- hydraulic conductivity of soil in the vertical direction;
- L
- length of the horizontal sand blanket;
- p
- loading;
- t
- time;
- Tv
- time factor;
- u
- excess pore-water pressure;
- dimensionless excess pore-water pressure;
- dimensionless excess pore-water pressure below the horizontal drain;
- dimensionless excess pore-water pressure above the horizontal drain;
- U
- average degree of consolidation;
- W
- diagonal weighting matrix with entries that equal to the inverse of the data variance;
- z
- vertical coordinate;
- Z
- dimensionless vertical coordinate;
- α
- dimensionless interface parameter at the top boundary;
- β
- dimensionless interface parameter at the bottom boundary;
- γw
- unit weight of water;
- λ
- characteristic factor that governs the drainage behavior of sand;
- ξ
- layout position of the horizontal drain in the vertical direction;
- ζn
- nth eigenvalue;
- η
- thickness ratio between adjacent media; and
- κ
- ratio of the coefficient of consolidation between adjacent media.
References
Barden, L. 1965. “Consolidation of clay with non-linear viscosity.” Géotechnique 15 (4): 345–362. https://doi.org/10.1680/geot.1965.15.4.345.
Biot, M. A. 1941a. “Consolidation settlement under a rectangular load distribution.” J. Appl. Phys. 12 (5): 426–430. https://doi.org/10.1063/1.1712921.
Biot, M. A. 1941b. “General theory of three-dimensional consolidation.” J. Appl. Phys. 12 (2): 155–164. https://doi.org/10.1063/1.1712886.
Bo, M. W., A. Arulrajah, and H. Nikraz. 2007. “Preloading and prefabricated vertical drains design for foreshore land reclamation projects: A case study.” Proc. Inst. Civ. Eng. Ground Improv. 11 (2): 67–76. https://doi.org/10.1680/grim.2007.11.2.67.
Cai, Y., H. Qiao, J. Wang, X. Geng, P. Wang, and Y. Cai. 2017. “Experimental tests on effect of deformed prefabricated vertical drains in dredged soil on consolidation via vacuum preloading.” Eng. Geol. 222: 10–19. https://doi.org/10.1016/j.enggeo.2017.03.020.
Cai, Y. Q., L. Xu, and S. M. Wu. 2004. “One-dimensional consolidation of layered soils with impeded boundaries under time-dependent loadings.” Appl. Math. Mech. 25 (8): 937–944. https://doi.org/10.1007/BF02438802.
Chai, J., S. Horpibulsuk, S. Shen, and J. P. Carter. 2014. “Consolidation analysis of clayey deposits under vacuum pressure with horizontal drains.” Geotext. Geomembr. 42 (5): 437–444. https://doi.org/10.1016/j.geotexmem.2014.07.001.
Chai, J., K. Matsunaga, A. Sakai, and S. Hayashi. 2009. “Comparison of vacuum consolidation with surcharge load induced consolidation of a two-layer system.” Géotechnique 59 (7): 637–641. https://doi.org/10.1680/geot.8.T.020.
Chai, J.-C., and N. Miura. 1999. “Investigation of factors affecting vertical drain behavior.” J. Geotech. Geoenviron. Eng. 125 (3): 216–226. https://doi.org/10.1061/(ASCE)1090-0241(1999)125:3(216).
Chan, A. H. C. 2003. “Determination of the coefficient of consolidation using a least squares method.” Géotechnique 53 (7): 673–678. https://doi.org/10.1680/geot.2003.53.7.673.
Chen, Z., P. Ni, Y. Chen, and G. Mei. 2020. “Plane-strain consolidation theory with distributed drainage boundary.” Acta Geotech. 15 (2): 489–508. https://doi.org/10.1007/s11440-018-0712-z.
Chu, J., and W. Guo. 2016. “Land reclamation using clay slurry or in deep water: Challenges and solutions.” Jpn. Geotech. Soc. Spec. Publ. 2 (51): 1790–1793. https://doi.org/10.3208/jgssp.TC217-02.
Chu, J., S. Yan, and B. Indraranata. 2008. “Vacuum preloading techniques—Recent developments and applications.” In GeoCongress 2008: Geosustainability and Geohazard Mitigation, Geotechnical Special Publication 178, edited by K. R. Reddy, M. V. Khire, and A. N. Alshawabkeh, 586–595. Reston, VA: ASCE.
Conte, E. 2004. “Consolidation analysis for unsaturated soils.” Can. Geotech. J. 41 (4): 599–612. https://doi.org/10.1139/t04-017.
Conte, E., and A. Troncone. 2006. “One-dimensional consolidation under general time-dependent loading.” Can. Geotech. J. 43 (11): 1107–1116. https://doi.org/10.1139/t06-064.
Davis, E. H., and G. P. Raymond. 1965. “A non-linear theory of consolidation.” Géotechnique 15 (2): 161–173. https://doi.org/10.1680/geot.1965.15.2.161.
Feng, J., P. Ni, and G. Mei. 2019. “One-dimensional self-weight consolidation with continuous drainage boundary conditions: Solution and application to clay-drain reclamation.” Int. J. Numer. Anal. Methods Geomech. 43 (8): 1634–1652. https://doi.org/10.1002/nag.2928.
Fredlund, D. G., and J. U. Hasan. 1979. “One-dimensional consolidation theory: Unsaturated soils.” Can. Geotech. J. 16 (3): 521–531. https://doi.org/10.1139/t79-058.
Fukuda, M., Y. Ito, S. Suwa, T. Nomura, and F. Hashizume. 2011. “Presentation of two layers horizontal drain model and design method for reduction of consolidation delay.” Geosynth. Eng. J. 26: 13–18. https://doi.org/10.5030/jcigsjournal.26.13.
Gibson, R., and G. C. Shefford. 1968. “The efficiency of horizontal drainage layers for accelerating consolidation of clay embankments.” Géotechnique 18 (3): 327–335. https://doi.org/10.1680/geot.1968.18.3.327.
Gray, H. 1945. “Simultaneous consolidation of contiguous layers of unlike compressive soils.” Am. Soc. Civ. Eng. 110 (1): 1327–1356.
Hansbo, S. 1997. “Aspects of vertical drain design: Darcian or non-Darcian flow.” Géotechnique 47 (5): 983–992. https://doi.org/10.1680/geot.1997.47.5.983.
Hu, Y.-Y., W.-H. Zhou, and Y.-Q. Cai. 2014. “Large-strain elastic viscoplastic consolidation analysis of very soft clay layers with vertical drains under preloading.” Can. Geotech. J. 51 (2): 144–157. https://doi.org/10.1139/cgj-2013-0200.
Jang, W. Y., and M. Mimura. 2005. “Effect of permeability and compressibility of sandwiched gravelly sand layers on subsequent settlement of Pleistocene deposits.” Soils Found. 45 (6): 111–119. https://doi.org/10.3208/sandf.45.111.
Li, C. X., X. U. Chao, and K. H. Xie. 2017. “Nonlinear consolidation of clayed soil considering non-Darcy flow and stress history.” Rock Soil Mech. 38 (1): 91–100.
Li, L. H., Q. Wang, N. X. Wang, and J. P. Wang. 2009. “Vacuum dewatering and horizontal drainage blankets: A method for layered soil reclamation.” Bull. Eng. Geol. Environ. 68 (2): 277–285. https://doi.org/10.1007/s10064-009-0200-7.
Liu, J. C., and Q. Ma. 2013. “One-dimensional consolidation of soft ground with impeded boundaries under depth-dependent ramp load.” In Int. Symp. on Pavement and Geotechnical Engineering for Transportation Infrastructure, Geotechnical Special Publication 8, edited by B. Huang, B. F. Bowers, G.-X. Mei, S.-H. Luo, and Z. Zhang, 127–134. Reston, VA: ASCE.
Mei, G., J. Xia, and L. Mei. 2011. “Terzaghi’s one-dimensional consolidation equation and its solution based on asymmetric continuous drainage boundary.” Chin. J. Geotech. Eng. 33 (1): 28–31.
Mei, G.-X., T. M. H. Lok, J. Xia, and S. S. Wu. 2014. “One-dimensional consolidation with asymmetrical exponential drainage boundary.” Geomech. Eng. 6 (1): 47–63. https://doi.org/10.12989/gae.2014.6.1.047.
Mesri, G. 1973. “One-dimensional consolidation of a clay layer with impeded drainage boundaries.” Water Resour. Res. 9 (4): 1090–1093. https://doi.org/10.1029/WR009i004p01090.
Mikasa, M., N. Takada, A. Oshima, and M. Kiyama. 1998. “Nonlinear consolidation theory for nonhomogeneous clay layers and its application.” Soils Found. 38 (4): 205–212. https://doi.org/10.3208/sandf.38.4_205.
Mohamedelhassan, E., and J. Q. Shang. 2002. “Vacuum and surcharge combined one-dimensional consolidation of clay soils.” Can. Geotech. J. 39 (5): 1126–1138. https://doi.org/10.1139/t02-052.
Ni, P., S. Mangalathu, G. Mei, and Y. Zhao. 2017. “Permeable piles: An alternative to improve the performance of driven piles.” Comput. Geotech. 84: 78–87. https://doi.org/10.1016/j.compgeo.2016.11.021.
Ni, P., S. Mangalathu, G. Mei, and Y. Zhao. 2018. “Laboratory investigation of pore pressure dissipation in clay around permeable piles.” Can. Geotech. J. 55 (9): 1257–1267. https://doi.org/10.1139/cgj-2017-0180.
Ni, P., G. Mei, and Y. Zhao. 2019a. “Surcharge preloading consolidation of reclaimed land with distributed sand caps.” Mar. Georesour. Geotechnol. 37 (6): 671–682. https://doi.org/10.1080/1064119X.2018.1482389.
Ni, P., K. Xu, G. Mei, and Y. Zhao. 2019b. “Effect of vacuum removal on consolidation settlement under a combined vacuum and surcharge preloading.” Geotext. Geomembr. 47 (1): 12–22. https://doi.org/10.1016/j.geotexmem.2018.09.004.
Nogami, T., and M. Li. 2002. “Consolidation of system of clay and thin sand layers.” Soils Found. 42 (4): 1–11. https://doi.org/10.3208/sandf.42.4_1.
Nogami, T., and M. Li. 2003. “Consolidation of clay with a system of vertical and horizontal drains.” J. Geotech. Geoenviron. Eng. 129 (9): 838–848. https://doi.org/10.1061/(ASCE)1090-0241(2003)129:9(838).
Olson, R. E. 1977. “Consolidation under time-dependent loading.” J. Geotech. Geoenviron. Eng. 103 (1): 55–60.
Paul, M., and R. Sahu. 2013. “One dimensional consolidation under cyclic loading.” Int. J. Geotech. Eng. 6 (3): 395–401. https://doi.org/10.3328/IJGE.2012.06.03.395-402.
Rendulic, L. 1936. “Porenziffer and porenwasserdruck in tonen.” Der Bauingenieur 7 (1): 559–564.
Rujikiatkamjorn, C., and B. Indraratna. 2007. “Analytical solutions and design curves for vacuum-assisted consolidation with both vertical and horizontal drainage.” Can. Geotech. J. 44 (2): 188–200. https://doi.org/10.1139/t06-111.
Schiffman, R. L., and J. R. Stein. 1970. “One-dimensional consolidation of layered systems.” Soils Found. 96 (4): 1499–1504.
Schmidt, J. D., and R. A. Westmann. 1973. “Consolidation of porous media with non-Darcy flow.” J. Eng. Mech. Div. 99 (6): 1201–1216. https://doi.org/10.1061/JMCEA3.0001833.
Suleiman, M. T., L. Ni, and A. Raich. 2014. “Development of pervious concrete pile ground-improvement alternative and behavior under vertical loading.” J. Geotech. Geoenviron. Eng. 140 (7): 04014035. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001135.
Suzuki, K., and K. Yasuhara. 2007. “Increase in undrained shear strength of clay with respect to rate of consolidation.” Soils Found. 47 (2): 303–318. https://doi.org/10.3208/sandf.47.303.
Tan, S.-A., K.-M. Liang, K.-Y. Yong, and S.-L. Lee. 1992. “Drainage efficiency of sand layer in layered clay–sand reclamation.” J. Geotech. Eng. 118 (2): 209–228. https://doi.org/10.1061/(ASCE)0733-9410(1992)118:2(209).
Terzaghi, K. 1925. Erdbaumechanik and bodenphysikalischer grundlage. Wien: Fanz deuticke.
Tsai, T.-L. 2015. “A coupled one-dimensional viscoelastic-plastic model for aquitard consolidation caused by hydraulic head variations in aquifers.” Hydrol. Processes 29 (22): 4779–4793. https://doi.org/10.1002/hyp.10524.
Wang, J., Y. Cai, J. Ma, J. Chu, H. Fu, P. Wang, and Y. Jin. 2016. “Improved vacuum preloading method for consolidation of dredged clay-slurry fill.” J. Geotech. Geoenviron. Eng. 142 (11): 06016012. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001516.
Weber, T., M. Plötze, J. Laue, G. Peschke, and S. Springman. 2010. “Smear zone identification and soil properties around stone columns constructed in-flight in centrifuge model tests.” Géotechnique 60 (3): 197–206. https://doi.org/10.1680/geot.8.P.098.
Wilson, N. E., and M. M. Elgohary. 1974. “Consolidation of soils under cyclic loading.” Can. Geotech. J. 11 (3): 420–423. https://doi.org/10.1139/t74-042.
Xie, K.-H., D.-Z. Huang, Y.-L. Wang, and Y.-B. Deng. 2014. “Analytical theory for one-dimensional consolidation of soil induced by time-dependent pumping and loading.” Mar. Georesour. Geotechnol. 32 (4): 328–350. https://doi.org/10.1080/1064119X.2013.764555.
Xie, K.-H., X.-Y. Xie, and X. Gao. 1999. “Theory of one dimensional consolidation of two-layered soil with partially drained boundaries.” Comput. Geotech. 24 (4): 265–278. https://doi.org/10.1016/S0266-352X(99)00012-9.
Yao, R., P. Ni, G. Mei, and Y. Zhao. 2019. “Numerical analysis of surcharge preloading consolidation of layered soils via distributed sand blankets.” Mar. Georesour. Geotechnol. 37 (8): 902–914. https://doi.org/10.1080/1064119X.2018.1506529.
Yao, Y. P., L. Niu, and W. J. Cui. 2014. “Unified hardening (UH) model for overconsolidated unsaturated soils.” Can. Geotech. J. 51 (7): 810–821. https://doi.org/10.1139/cgj-2013-0183.
Yin, J.-H., and J. Graham. 1996. “Elastic visco-plastic modelling of one-dimensional consolidation.” Géotechnique 46 (3): 515–527. https://doi.org/10.1680/geot.1996.46.3.515.
Zhao, X. H., H. Sun, and K. W. Lo. 2002. “An elastoplastic damage model of soil.” Géotechnique 52 (7): 533–536. https://doi.org/10.1680/geot.2002.52.7.533.
Zhou, W.-H., and L.-S. Zhao. 2014. “One-dimensional consolidation of unsaturated soil subjected to time-dependent loading with various initial and boundary conditions.” Int. J. Geomech. 14 (2): 291–301. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000314.
Zhou, W.-H., L.-S. Zhao, A. Garg, and K.-V. Yuen. 2017. “Generalized analytical solution for the consolidation of unsaturated soil under partially permeable boundary conditions.” Int. J. Geomech. 17 (9): 04017048. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000942.
Zhu, G. F., and J. H. Yin. 1999. “Consolidation of double soil layers under depth-dependent ramp load.” Géotechnique 49 (3): 415–421.
Zhu, G. F., and J.-H. Yin. 2005. “Solution charts for the consolidation of double soil layers.” Can. Geotech. J. 42 (3): 949–956. https://doi.org/10.1139/t05-001.
Zong, M., W. Wu, M. H. El Naggar, G. Mei, P. Ni, and M. Xu. 2020. “Analytical solution for one-dimensional nonlinear consolidation of double-layered soil with improved continuous drainage boundary.” Eur. J. Environ. Civ. Eng. 1–22. https://doi.org/10.1080/19648189.2020.1813207.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 22 • Issue 3 • March 2022
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Dec 28, 2020
Accepted: Nov 2, 2021
Published online: Dec 20, 2021
Published in print: Mar 1, 2022
Discussion open until: May 20, 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
- Lin-Shuang Zhao, Wan-Huan Zhou, Shui-Long Shen, Semianalytical Solution for Dissipation Process of Partially Saturated Soils Considering Nonsmooth Boundary and Stress Level, Journal of Engineering Mechanics, 10.1061/JENMDT.EMENG-7048, 149, 9, (2023).
- Xiao-Qian Zhang, Ming-Guang Li, Jin-Jian Chen, One-Dimensional Reverse Consolidation Model for Basal Soil from Deep Excavation Based on the Continuous Drainage Boundary, International Journal of Geomechanics, 10.1061/IJGNAI.GMENG-8611, 23, 7, (2023).
- Mengfan Zong, Wenbing Wu, M. Hesham El Naggar, Guoxiong Mei, Yi Zhang, Analysis of One-Dimensional Consolidation for Double-Layered Soil with Non-Darcian Flow Based on Continuous Drainage Boundary, International Journal of Geomechanics, 10.1061/IJGNAI.GMENG-7853, 23, 3, (2023).
- Penglin Li, Jian-Hua Yin, Zhen-Yu Yin, Zejian Chen, One–dimensional nonlinear finite strain analysis of self–weight consolidation of soft clay considering creep, Computers and Geotechnics, 10.1016/j.compgeo.2022.105081, 153, (105081), (2023).
- Zheng Chen, Pengpeng Ni, Xun Zhu, Guoxiong Mei, Using Boundary Transform Method to Solve Geotechnical Problems with Mixed-Type Boundary Conditions, Journal of Geotechnical and Geoenvironmental Engineering, 10.1061/(ASCE)GT.1943-5606.0002921, 148, 12, (2022).