Three-dimensional consolidation theory of vertical drain based on continuous drainage boundary
Abstract
To remedy the limitation that the conventional drainage boundary only considers two extreme cases of pervious and impervious boundaries, the consolidation theory of vertical drain is derived by applying the continuous drainage boundary, and its validity is also proven. Based on the obtained solutions, the excess pore water pressure and the average degree of consolidation under the continuous drainage boundary condition are analyzed, and the effect of the drainage capacity of the top surface, the smear effect and the well resistance on consolidation are explored. Furthermore, the practicality of this theory is also validated by the comparison with experimental data. Results confirm that the complete and continuous process of the ground top surface can be changed from no drainage to a complete drainage by adjusting the value of the interface parameter b. Higher value of the interface parameter b means a stronger water permeability of the foundation, resulting in a faster dissipation of excess pore water pressure and a faster consolidation. Meanwhile, the vertical drainage of the vertical drain cannot be neglected in calculation even though vertical drains are based on a horizontal seepage. Moreover, the smear effect and the well resistance play an important role on consolidation.
Keyword : consolidation, vertical drain, continuous drainage boundary, interface parameter, smear effect, well resistance
How to Cite
Zhang, Y., Wu, W., Mei, G., & Duan, L. (2019). Three-dimensional consolidation theory of vertical drain based on continuous drainage boundary. Journal of Civil Engineering and Management, 25(2), 145-155. https://doi.org/10.3846/jcem.2019.8071
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Cai, Y. Q., Liang, X., Zheng, Z. F., & Pan, X. D. (2003). One-dimensional consolidation of viscoelastic soil layer with semi-permeable boundaries under cycle loadings. China Civil Engineering Journal, 36(8), 86-90. https://doi.org/10.15951/j.tmgcxb.2003.08.017
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Gibson, R. E. (1958). The progress of consolidation in a clay layer increasing in thickness with time. Geotechnique, 8(4), 171-183. https://doi.org/10.1680/geot.1958.8.4.171
Hansbo, S., Jamiolkowski M., & Kok, L. (1981). Consolidation by vertical drains. Geotechnique, 31(1), 45-66. https://doi.org/10.1680/geot.1981.31.1.45
Ho, L., Fatahi, B., & Khabbaz, H. (2015). A closed form analytical solution for two-dimensional plane strain consolidation of unsaturated soil stratum. International Journal for Numerical and Analytical Methods in Geomechanics, 39(15), 1665-1692. https://doi.org/10.1002/nag.2369
Ho, L., Fatahi, B., & Khabbaz, H. (2016). Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions. Computers and Geotechnics, 74, 102-121. https://doi.org/10.1016/j.compgeo.2015.12.019
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Indraratna, B., Rujikiatkamjorn, C., & Sathananthan, I. (2005). Radial consolidation of clay using compressibility indices and varying horizontal permeability. Canadian Geotechnical Journal, 42(5), 1330-1341. https://doi.org/10.1139/T05-052
Karim, M. R., & Oka, F. (2010). An automatic time increment selection scheme for simulation of elasto-viscoplastic consolidation of clayey soils. Geomechanics and Geoengineering, 5(3), 153-177. https://doi.org/10.1080/17486020903576226
Karlsson, M., Emdal, A., & Dijkstra, J. (2016). Consequences of sample disturbance when predicting long-term settlements in soft clay. Canadian Geotechnical Journal, 53(12), 1965-1977. https://doi.org/10.1139/cgj-2016-0129
Kim, R., Hong, S. J., Lee, M. J., & Lee, W. (2011). Time dependent well resistance factor of PVD. Marine Georesources and Geotechenology, 29(2), 131-144. https://doi.org/10.1080/1064119X.2010.525145
Li, C. X., Xu, C., & Xie, K. H. (2017). Nonlinear consolidation of clayed soil considering non-Darcy flow and stress history. Rock and Soil Mechanics, 38(1), 91-100. https://doi.org/10.16285/j.rsm.2017.01.012
Lu, M. M., Wang, S. Y., & Sloan, S. W. (2015). Nonlinear radial consolidation of vertical drains with coupled radial-vertical flow considering well resistance. Geotextiles and Geomembranes, 43(2), 182-189. https://doi.org/10.1016/j.geotexmem.2014.12.001
Mei, G. X., Xia, J., & Mei, L. (2011). Terzaghi’s one-dimensional consolidation equation and its solution based on asymmetric continuous drainage boundary. Chinese Journal of Geotechnical Engineering, 33(1), 28-31.
Parsa-Pajouh, A., Fatahi, B., & Khabbaz, H. (2016). Experimental and numerical investigations to evaluate two-simensional modeling of vertical drain–assisted preloading. International Journal of Geomechanics, 16(1), 1-14. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000507
Parsa-Pajouh, A., Fatahi, B., Vincent, P., & Khabbaz, H. (2014a). Trial embankment analysis to predict smear zone characteris- tics induced by prefabricated vertical strain installation. Geotechnical & Geological Engineering, 32(5), 1187-1201. https://doi.org/10.1007/s10706-014-9789-9
Parsa-Pajouh, A., Fatahi, B., Vincent, P., & Khabbaz, H. (2014b). Analyzing consolidation data to predict smear zone characteristics induced by vertical drain installation for soft soil improvement. Geomechanics and Engineering, 7(1), 105-131. https://doi.org/10.12989/gae.2014.7.1.105
Rujikiatkamjorn, C., & Indraratna, B. (2009). Design procedure for vertical drains considering a linear variation of lateral permeability within the smear zone. Canadian Geotechnical Journal, 46(3), 270-280. https://doi.org/10.1139/T08-124
Sales, M. M., Prezzi, M., Salgado, R., Choi, Y. S., & Lee, J. (2017). Load-settlement behaviour of model pile groups in sand under vertical load. Journal of Civil Engineering and Management, 23(8), 1148-1163. https://doi.org/10.3846/13923730.2017.1396559
Sun, J., Xie, X. Y., & Xie, K. H. (2007). Analytical theory for consolidation of double-layered composite round under impeded boundaries. Journal of Zhejiang University (Engineering Science), 41(9), 1467-1471.
Taslimian, R., Noorzad, A., & Javan, M. R. M. (2015). Numerical simulation of liquefaction in porous media using nonlinear fluid flow law. International Journal for Numerical and Analytical Methods in Geomechanics, 39(3), 229-250. https://doi.org/10.1002/nag.2297
Wang, L., Sun, D. A., Li, L. Z., Li, P. C., & Xu, Y. F. (2017). Semianalytical solutions to one-dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary. Computers and Geotechnics, 89, 71-80. https://doi.org/10.1016/j.compgeo.2017.04.005
Wang, Z. F., Shen, S. L., Ho, C. E., & Kim, Y. H. (2013). Investigation of field-installation effects of horizontal twin-jet grouting in Shanghai soft soil deposits. Canadian Geotechnical Journal, 50(3), 288-297. https://doi.org/10.1139/cgj-2012-0199
Xie, K. H., & Zeng, G. X. (1989). Consolidation theories for drain wells under equal strain condition. Journal of Geotechnical Engineering, 11(2), 3-17.
Xie, K. H., Wang, K., & Wang, Y. L. (2010). Analytical solution for one-dimensional consolidation of clayey soils with a threshold gradient. Computers and Geotechnics, 37(4), 487-493. https://doi.org/10.1016/j.compgeo.2010.02.001
Yoshikuni, H., & Nakanodo, H. (1974). Consolidation of soils by vertical drain wells with finite permeability. Soils and Foundations, 14(2), 35-46. https://doi.org/10.3208/sandf1972.14.2_35
Zhang, Y. G., Xie, K. H., & Wang, Z. (2006). Consolidation analysis of composite ground improved by granular columns considering variation of permeability coefficient of soil. In ASCE GeoShanghai International Conference Special Publication, Ground Modification and Seismic Mitigation (GSP 152) (pp. 135-142). https://doi.org/10.1061/40864(196)19
Zhou, Y., & Chai, J. C. (2017). Equivalent ‘smear’ effect due to non-uniform consolidation surrounding a PVD. Geotech nique, 67(5), 410-419. https://doi.org/10.1680/jgeot.16.P.087
Barron, R. A. (1948). Consolidation of fine-grained soils by drain wells. Transactions of American Society of Civil Engineers, 74(6), 718-742.
Basack, S., & Nimbalkar, S. (2017). Free strain analysis of the performance of vertical drains for soft soil improvement. Geomechanics and Engineering, 13(6), 963-975.
Cai, Y. Q., Liang, X., Zheng, Z. F., & Pan, X. D. (2003). One-dimensional consolidation of viscoelastic soil layer with semi-permeable boundaries under cycle loadings. China Civil Engineering Journal, 36(8), 86-90. https://doi.org/10.15951/j.tmgcxb.2003.08.017
Carrillo, N. (1942). Simple two and three dimensional case in the theory of consolidation of soils. Studies in Applied Mathematics, 21(1-4), 1-5. https://doi.org/10.1002/sapm19422111
Deng, Y. B., Xie, K. H., Lu, M. M., Tao, H. B., & Liu, G. B. (2013). Consolidation by prefabricated vertical drains considering the time dependent well resistance. Geotextiles and Geomembranes, 36, 20-26. https://doi.org/10.1016/j.geotexmem.2012.10.003
Geng, X., & Yu, H. S. (2017). A large-strain radial consolidation theory for soft clays improved by vertical drains. Geotechnique, 67(11), 1020-1028. https://doi.org/10.1680/jgeot.15.T.013
Gibson, R. E. (1958). The progress of consolidation in a clay layer increasing in thickness with time. Geotechnique, 8(4), 171-183. https://doi.org/10.1680/geot.1958.8.4.171
Hansbo, S., Jamiolkowski M., & Kok, L. (1981). Consolidation by vertical drains. Geotechnique, 31(1), 45-66. https://doi.org/10.1680/geot.1981.31.1.45
Ho, L., Fatahi, B., & Khabbaz, H. (2015). A closed form analytical solution for two-dimensional plane strain consolidation of unsaturated soil stratum. International Journal for Numerical and Analytical Methods in Geomechanics, 39(15), 1665-1692. https://doi.org/10.1002/nag.2369
Ho, L., Fatahi, B., & Khabbaz, H. (2016). Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions. Computers and Geotechnics, 74, 102-121. https://doi.org/10.1016/j.compgeo.2015.12.019
Hu, A. F., Xia, C. Q., Cui, J., Li, C. X., & Xie, K. H. (2018). Nonlinear consolidation analysis of natural structured clays under time-dependent loading. International Journal of Geomechanics, 18(2), 1-16. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001059
Indraratna, B., Rujikiatkamjorn, C., & Sathananthan, I. (2005). Radial consolidation of clay using compressibility indices and varying horizontal permeability. Canadian Geotechnical Journal, 42(5), 1330-1341. https://doi.org/10.1139/T05-052
Karim, M. R., & Oka, F. (2010). An automatic time increment selection scheme for simulation of elasto-viscoplastic consolidation of clayey soils. Geomechanics and Geoengineering, 5(3), 153-177. https://doi.org/10.1080/17486020903576226
Karlsson, M., Emdal, A., & Dijkstra, J. (2016). Consequences of sample disturbance when predicting long-term settlements in soft clay. Canadian Geotechnical Journal, 53(12), 1965-1977. https://doi.org/10.1139/cgj-2016-0129
Kim, R., Hong, S. J., Lee, M. J., & Lee, W. (2011). Time dependent well resistance factor of PVD. Marine Georesources and Geotechenology, 29(2), 131-144. https://doi.org/10.1080/1064119X.2010.525145
Li, C. X., Xu, C., & Xie, K. H. (2017). Nonlinear consolidation of clayed soil considering non-Darcy flow and stress history. Rock and Soil Mechanics, 38(1), 91-100. https://doi.org/10.16285/j.rsm.2017.01.012
Lu, M. M., Wang, S. Y., & Sloan, S. W. (2015). Nonlinear radial consolidation of vertical drains with coupled radial-vertical flow considering well resistance. Geotextiles and Geomembranes, 43(2), 182-189. https://doi.org/10.1016/j.geotexmem.2014.12.001
Mei, G. X., Xia, J., & Mei, L. (2011). Terzaghi’s one-dimensional consolidation equation and its solution based on asymmetric continuous drainage boundary. Chinese Journal of Geotechnical Engineering, 33(1), 28-31.
Parsa-Pajouh, A., Fatahi, B., & Khabbaz, H. (2016). Experimental and numerical investigations to evaluate two-simensional modeling of vertical drain–assisted preloading. International Journal of Geomechanics, 16(1), 1-14. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000507
Parsa-Pajouh, A., Fatahi, B., Vincent, P., & Khabbaz, H. (2014a). Trial embankment analysis to predict smear zone characteris- tics induced by prefabricated vertical strain installation. Geotechnical & Geological Engineering, 32(5), 1187-1201. https://doi.org/10.1007/s10706-014-9789-9
Parsa-Pajouh, A., Fatahi, B., Vincent, P., & Khabbaz, H. (2014b). Analyzing consolidation data to predict smear zone characteristics induced by vertical drain installation for soft soil improvement. Geomechanics and Engineering, 7(1), 105-131. https://doi.org/10.12989/gae.2014.7.1.105
Rujikiatkamjorn, C., & Indraratna, B. (2009). Design procedure for vertical drains considering a linear variation of lateral permeability within the smear zone. Canadian Geotechnical Journal, 46(3), 270-280. https://doi.org/10.1139/T08-124
Sales, M. M., Prezzi, M., Salgado, R., Choi, Y. S., & Lee, J. (2017). Load-settlement behaviour of model pile groups in sand under vertical load. Journal of Civil Engineering and Management, 23(8), 1148-1163. https://doi.org/10.3846/13923730.2017.1396559
Sun, J., Xie, X. Y., & Xie, K. H. (2007). Analytical theory for consolidation of double-layered composite round under impeded boundaries. Journal of Zhejiang University (Engineering Science), 41(9), 1467-1471.
Taslimian, R., Noorzad, A., & Javan, M. R. M. (2015). Numerical simulation of liquefaction in porous media using nonlinear fluid flow law. International Journal for Numerical and Analytical Methods in Geomechanics, 39(3), 229-250. https://doi.org/10.1002/nag.2297
Wang, L., Sun, D. A., Li, L. Z., Li, P. C., & Xu, Y. F. (2017). Semianalytical solutions to one-dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary. Computers and Geotechnics, 89, 71-80. https://doi.org/10.1016/j.compgeo.2017.04.005
Wang, Z. F., Shen, S. L., Ho, C. E., & Kim, Y. H. (2013). Investigation of field-installation effects of horizontal twin-jet grouting in Shanghai soft soil deposits. Canadian Geotechnical Journal, 50(3), 288-297. https://doi.org/10.1139/cgj-2012-0199
Xie, K. H., & Zeng, G. X. (1989). Consolidation theories for drain wells under equal strain condition. Journal of Geotechnical Engineering, 11(2), 3-17.
Xie, K. H., Wang, K., & Wang, Y. L. (2010). Analytical solution for one-dimensional consolidation of clayey soils with a threshold gradient. Computers and Geotechnics, 37(4), 487-493. https://doi.org/10.1016/j.compgeo.2010.02.001
Yoshikuni, H., & Nakanodo, H. (1974). Consolidation of soils by vertical drain wells with finite permeability. Soils and Foundations, 14(2), 35-46. https://doi.org/10.3208/sandf1972.14.2_35
Zhang, Y. G., Xie, K. H., & Wang, Z. (2006). Consolidation analysis of composite ground improved by granular columns considering variation of permeability coefficient of soil. In ASCE GeoShanghai International Conference Special Publication, Ground Modification and Seismic Mitigation (GSP 152) (pp. 135-142). https://doi.org/10.1061/40864(196)19
Zhou, Y., & Chai, J. C. (2017). Equivalent ‘smear’ effect due to non-uniform consolidation surrounding a PVD. Geotech nique, 67(5), 410-419. https://doi.org/10.1680/jgeot.16.P.087