Consolidation Analysis for Soft Clay Reinforced by a Granular Column with Non-Darcian Flow
Publication: International Journal of Geomechanics
Volume 22, Issue 10
Abstract
The relationship between flow speed and pore pressure gradient does not follow the classical Darcy’s law for seepage within high-permeability and low-permeability media. Thus, considering the non-Darcian flow described by exponential relation in soft clay and Forchheimer equation in granular column, the governing equation for the axisymmetric model of composite ground is established. A finite-difference scheme with central and phantom nodes is used to obtain the numerical solution. Then, the effects of two kinds of non-Darcian flow and geometrical parameters on the consolidation behaviors are drawn from the parametric study. The consolidation rate of surrounding soil by considering the exponential flow is less than that of Darcian flow in the same case. The consolidation rate of the granular column calculated by the Forchheimer equation is less than that of Darcian flow. Moreover, increasing the area replacement ratio is recommended to reduce these effects of non-Darcian flow.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data and models that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This research was supported by the National Nature Science Foundation of China (No. 51978612, 51778572). These supports are gratefully acknowledged.
References
Ahmed, N., and D. K. Sunada. 1969. “Nonlinear flow in porous media.” Journal of the Hydraulics Division 95 (6): 1847–1858. https://doi.org/10.1061/JYCEAJ.0002193.
Balaam, N. P., and J. R. Booker. 1981. “Analysis of rigid rafts supported by granular piles.” Int. J. Numer. Anal. Methods Geomech. 5 (4): 379–403. https://doi.org/10.1002/nag.1610050405.
Barksdale, R. D., R. C. Bachus, and R. D. Barksdale. 1983. Design and construction of stone columns. Washington, DC: US Dept. of Transportation, Federal Highway Administration.
Barron, R. A. 1948. “Consolidation of fine-grained soils by drain wells by drain wells.” Transactions of the American Society of Civil Engineers 113 (1): 718–742. https://doi.org/10.1061/TACEAT.0006098.
Castro, J. 2016. “An analytical solution for the settlement of stone columns beneath rigid footings.” Acta Geotech. 11 (2): 309–324. https://doi.org/10.1007/s11440-014-0358-4.
Castro, J., and C. Sagaseta. 2009. “Consolidation around stone columns. Influence of column deformation.” Int. J. Numer. Anal. Methods Geomech. 33 (7): 851–877. https://doi.org/10.1002/nag.745.
Chaudhary, K., M. B. Cardenas, W. Deng, and P. C. Bennett. 2011. “The role of eddies inside pores in the transition from Darcy to Forchheimer flows.” Geophys. Res. Lett. 38: 24. https://doi.org/10.1029/2011GL050214.
Chen, Y., A.-F. Hu, S.-L. Xie, Y.-Y. Chen, and Z.-Q. Gong. 2021a. “Consolidation theory for the stone column reinforced ground with time-dependent drainage boundary considering the foundation stiffness.” Comput. Geotech. 136: 104218. https://doi.org/10.1016/j.compgeo.2021.104218.
Chen, D., P. Ni, X. Zhang, Z. Chen, G. Mei, and J. Feng. 2021b. “Consolidation theory of unsaturated soils with vertical drains considering well resistance and smear effect under time-dependent loading.” J. Eng. Mech. 147 (9): 04021055. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001966.
Chen, Y.-F., J.-Q. Zhou, S.-H. Hu, R. Hu, and C.-B. Zhou. 2015. “Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures.” J. Hydrol. 529: 993–1006. https://doi.org/10.1016/j.jhydrol.2015.09.021.
Crank, J., and P. Nicolson. 1947. “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type.” Math. Proc. Cambridge Philos. Soc. 43 (1): 50–67. https://doi.org/10.1017/S0305004100023197.
Doan, S., and B. Fatahi. 2020. “Analytical solution for free strain consolidation of stone column-reinforced soft ground considering spatial variation of total stress and drain resistance.” Comput. Geotech. 118: 103291. https://doi.org/10.1016/j.compgeo.2019.103291.
Dubin, B., and G. Moulin. 1986. “Influence of a critical gradient on the consolidation of clays.” In Consolidation of soils: Testing and evaluation, edited by R. N. Yong, and F. C. Townsend, 354–377. West Conshohocken, PA: ASTM.
Forchheimer, P. 1901. “Wasserbewegung durch boden.” Z. Ver. Dtsch. Ing. 45: 1782–1788.
Han, J., and S.-L. Ye. 2001. “Simplified method for consolidation rate of stone column reinforced foundations.” J. Geotech. Geoenviron. Eng. 127 (7): 597–603. https://doi.org/10.1061/(ASCE)1090-0241(2001)127:7(597).
Han, J., and S. L. Ye. 2002. “A theoretical solution for consolidation rates of stone column-reinforced foundations accounting for smear and well resistance effects.” Int. J. Geomech. 2 (2): 135–151. https://doi.org/10.1061/(ASCE)1532-3641(2002)2:2(135).
Hansbo, S. 1960. “Consolidation of clay with special reference to influence of vertical sand drains.” Swedish Geotechnical Institute Proceeding 18: 45–50.
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.
Hansbo, S. 2001. “Consolidation equation valid for both Darcian and non-Darcian flow.” Géotechnique 51 (1): 51–54. https://doi.org/10.1680/geot.2001.51.1.51.
Hansbo, S., M. Jamiolkowski, and L. Kok. 1981. “Consolidation by vertical drains.” Géotechnique 31 (1): 45–66. https://doi.org/10.1680/geot.1981.31.1.45.
Leo, C. J. 2004. “Equal strain consolidation by vertical drains.” J. Geotech. Geoenviron. Eng. 130 (3): 316–327. https://doi.org/10.1061/(ASCE)1090-0241(2004)130:3(316).
Li, C.-x., and K.-h. Xie. 2013. “One-dimensional nonlinear consolidation of soft clay with the non-Darcian flow.” Journal of Zhejiang University Science A 14 (6): 435–446. https://doi.org/10.1631/jzus.A1200343.
Liu, W., X. Fei, and J. Fang. 2012. “Rules for confidence intervals of permeability coefficients for water flow in over-broken rock mass.” International Journal of Mining Science and Technology 22 (1): 29–33. https://doi.org/10.1016/j.ijmst.2011.06.003.
Lu, M.-M., K.-H. Xie, and B. Guo. 2010. “Consolidation theory for a composite foundation considering radial and vertical flows within the column and the variation of soil permeability within the disturbed soil zone.” Canadian Geotechnical Journal 47 (2): 207–217. https://doi.org/10.1139/T09-086.
Ma, Z. G. 2003. Studies on characteristics of water seepage in crushed rock mass of gob. [In Chinese.] Xuzhou, China: China Univ. of Mining and Technology.
Miller, R. J., and P. F. Low. 1963. “Threshold gradient for water flow in clay systems.” Soil Sci. Soc. Am. J. 27 (6): 605–609. https://doi.org/10.2136/sssaj1963.03615995002700060013x.
Mishra, A., and N. R. Patra. 2019. “Long-term response of consolidating soft clays around a pile considering non-Darcian flow.” Int. J. Geomech. 19 (6): 04019040. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001392.
Morton, K. W., and D. F. Mayers. 2005. Numerical solution of partial differential equations: An introduction. Cambridge, UK: Cambridge University Press.
Moutsopoulos, K. N., I. N. E. Papaspyros, and V. A. Tsihrintzis. 2009. “Experimental investigation of inertial flow processes in porous media.” J. Hydrol. 374 (3-4): 242–254. https://doi.org/10.1016/j.jhydrol.2009.06.015.
Nash, D. F. T., and S. J. Ryde. 2001. “Modelling consolidation accelerated by vertical drains in soils subject to creep.” Géotechnique 51 (3): 257–273. https://doi.org/10.1680/geot.2001.51.3.257.
Priebe, H. J. 1995. “The design of vibro replacement.” Ground Engineering 28 (10): 31.
Ranganadha, R. P., and C. Suresh. 1970. “Discussion of “nonlinear flow in porous media”.” Journal of the Hydraulics Division 96 (8): 1732–1734. https://doi.org/10.1061/JYCEAJ.0002649.
Rujikiatkamjorn, C., and B. Indraratna. 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.
Ryde, S. J. 1997. “The performance and backanalysis of embankments on soft estuarine clay.” Ph.D. thesis, Dept. of Civil Engineering, Univ. of Bristol.
Sidiropoulou, M. G., K. N. Moutsopoulos, and V. A. Tsihrintzis. 2007. “Determination of Forchheimer equation coefficients a and b.” Hydrol. Processes 21 (4): 534–554. https://doi.org/10.1002/hyp.6264.
Swartzendruber, D. 1962. “Modification of Darcy’s law for the flow of water in soils.” Soil Sci. 93 (1): 22–29. https://doi.org/10.1097/00010694-196201000-00005.
Vincent, M. C., C. M. Pearson, and J. Kullman. 1999. Non-Darcy and multiphase flow in propped fractures: Case studies illustrate the dramatic effect on well productivity. Richardson, TX: OnePetro.
Wang, L., A. Zhou, Y. Xu, and X. Xia. 2020. “Consolidation of unsaturated composite ground reinforced by permeable columns.” Comput. Geotech. 125: 103706. https://doi.org/10.1016/j.compgeo.2020.103706.
Xie, K.-H., M.-M. Lu, and G.-B. Liu. 2009. “Equal strain consolidation for stone columns reinforced foundation.” Int. J. Numer. Anal. Methods Geomech. 33 (15): 1721–1735. https://doi.org/10.1002/nag.790.
Xie, K. H., and G. X. Zeng. 1989. “Consolidation theories for drain wells under equal strain condition.” Chinese J. Geotech. Eng. 11 (2): 3–17.
Xu, C., Z. Liu, J. Zhang, and J. Huang. 2021. “Analysis of large-strain elastic viscoplastic consolidation for soft clay with vertical drains considering non-Darcian flow.” Appl. Math. Modell. 92: 770–784. https://doi.org/10.1016/j.apm.2020.11.038.
Yang, T., J. Z. Yang, and J. Ni. 2014. “Analytical solution for the consolidation of a composite ground reinforced by partially penetrated impervious columns.” Comput. Geotech. 57: 30–36. https://doi.org/10.1016/j.compgeo.2014.01.001.
Yu, C., A. Zhang, Y. Wang, and W. Ren. 2020. “Analytical solution for consolidation of combined composite foundation reinforced with penetrated impermeable columns and partially penetrated permeable stone columns.” Comput. Geotech. 124: 103606. https://doi.org/10.1016/j.compgeo.2020.103606.
Zeng, Z., and R. Grigg. 2006. “A criterion for non-Darcy flow in porous media.” Transp. Porous Media 63 (1): 57–69. https://doi.org/10.1007/s11242-005-2720-3.
Zhang, Y. G., K. H. Xie, and Z. Wang. 2006. “Consolidation analysis of composite ground improved by granular columns considering variation of permeability coefficient of soil.” In Ground Modification and Seismic Mitigation, Geotechnical Special Publication 152, edited by A. Porbaha, S.-L. Shen, J. Wartman, and J.-C. Chai, 135–142. Reston, VA: ASCE.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 22 • Issue 10 • October 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Dec 17, 2021
Accepted: Apr 10, 2022
Published online: Jul 21, 2022
Published in print: Oct 1, 2022
Discussion open until: Dec 21, 2022
ASCE Technical Topics:
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.