Combined Effect of Non-Darcian Flow and Semipermeable Drainage Boundaries on One-Dimensional Consolidation of Unsaturated Soil
Publication: International Journal of Geomechanics
Volume 24, Issue 8
Abstract
The present article aims to understand the consolidation of unsaturated soil enclosed by semipermeable drainage boundaries and subjected to nonlinear water flow. The transient process of fluid flow is governed by the classical diffusion equations of air and water phases. The impeded drainage boundaries are modeled as a combination of the field variables and their gradient. The nonlinear water flow is dictated by an exponent and the threshold gradient-based non-Darcian model. The fluid phases are assumed to be continuous, inviscid, and flowing in the vertical direction. The deformation modulus of the air and water phases with respect to the net normal stress and matric suction are duly incorporated into the formulations. The numerical simulations are carried out by employing the Crank–Nicolson implicit scheme of the finite-difference technique. The nonlinear differential equations are linearized by evaluating the space gradient term within the exponent in the previous (known) time step. The results are presented and analyzed in the form of normalized isochrones, consolidation curves at specific spatial points, and in the global (average) scale. The flow rule impacts the dissipation process at the later stage of consolidation.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions. The function files for incorporating the non-Darcian flow law and the variable parameters are confidential.
Acknowledgments
The corresponding author acknowledges the support of “Science and Engineering Research Board (SERB), Government of India” under Grant Number SRG/2019/000149.
References
Barden, L. 1965. “Consolidation of compacted and unsaturated clays.” Geotechnique 15 (3): 267–286. https://doi.org/10.1680/geot.1965.15.3.267.
Barden, L. 1974. “Consolidation of clays compacted dry and wet of optimum water content.” Geotechnique 24 (4): 605–625. https://doi.org/10.1680/geot.1974.24.4.605.
Biot, M. A. 1941. “General theory of three-dimensional consolidation.” J. Appl. Phys. 12 (2): 155–164. https://doi.org/10.1063/1.1712886.
Blight, G. E. 1971. “Flow of air through soils.” J. Soil Mech. Found. Div. 97 (4): 607–624. https://doi.org/10.1061/JSFEAQ.0001577.
Dakshanamurthy, V., and D. G. Fredlund. 1980. “Moisture and air flow in an unsaturated soil.” In Expansive soils, 514–532. Reston, VA: ASCE.
Darkshanamurthy, V., D. G. Fredlund, and H. Rahardjo. 1984. “Coupled three-dimensional consolidation theory of unsaturated porous media.” In Proc., 5th Int. Conf. on Expansive Soils Preprints of Papers. Sydney, Australia: Institution of Engineers.
Dubin, B., and G. Moulin. 1986. “Influence of a critical gradient on the consolidation of clays.” In Consolidation of soils testing and evaluation. West Conshohocken, PA: ASTM International.
Duncan, J. M. 1993. “Limitations of conventional analysis of consolidation settlement.” J. Geotech. Eng. 119 (9): 1333–1359. https://doi.org/10.1061/(ASCE)0733-9410(1995)121:6(513)
Elnaggar, H. A., R. J. Krizek, and G. M. Karadi. 1973. “Effect of non-Darcian flow on time rate of consolidation.” J. Franklin Inst. 296 (5): 323–337. https://doi.org/10.1016/0016-0032(73)90212-3.
Fick, A., 1855. “Über diffusion.” Poggendorff's Ann. Phys, 94 (1855): 59–86.
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.
Fredlund, D. G., and N. R. Morgenstern. 1976. “Constitutive relations for volume change in unsaturated soils.” Can. Geotech. J. 13: 261–276. https://doi.org/10.1139/t76-029.
Fredlund, D. G., and H. Rahardjo. 1993. Soil mechanics for unsaturated soils. Hoboken, NJ: John Wiley & Sons.
Gray, H. 1945. “Simultaneous consolidation of contiguous layers of unlike compressible soils.” Trans. ASCE 110: 1327–1356.
Hansbo, S. 2003. “Deviation from Darcy's law observed in one–dimensional consolidation.” Géotechnique 53 (6): 601–605. https://doi.org/10.1680/geot.2003.53.6.601.
Ho, L., and B. Fatahi. 2016. “One–dimensional consolidation analysis of unsaturated soils subjected to time-dependent loading.” Int. J. Geomech. 16 (2): 04015052. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000504.
Ho, L., B. Fatahi, and H. Khabbaz. 2014. “Analytical solution for one-dimensional consolidation of unsaturated soils using eigenfunction expansion method.” Int. J. Numer. Anal. Methods Geomech. 38 (10): 1058–1077. https://doi.org/10.1002/nag.2248.
Huang, M. H., and M. H. Zhao. 2020. “A general analytical solution for one-dimensional consolidation of unsaturated soil incorporating impeded drainage boundaries.” Comp. Geotech. 128: 103801. https://doi.org/10.1016/j.compgeo.2020.103801
Ing, T. C., and N. Xiaoyan. 2002. “Coupled consolidation theory with non-Darcian flow.” Comput. Geotech. 29 (3): 169–209. https://doi.org/10.1016/S0266-352X(01)00022-2.
Kutilek, M. 1967. “Temperature and non-Darcian flow of water.” In Vol. 1 of Proc., Int. Soil Water Symp., 35–49. Vienna, Austria: International Atomic Energy Agency (IAEA).
Lei, G. H., C. W. Fu, and C. W. W. Ng. 2016. “Vertical-drain consolidation using stone columns: An analytical solution with an impeded drainage boundary under multi-ramp loading.” Geotext. Geomembr. 44 (1): 122–131. https://doi.org/10.1016/j.geotexmem.2015.07.003.
Li, C.-X., and K.-H. Xie. 2013. “One–dimensional nonlinear consolidation of soft clay with the non-Darcian flow.” J. Zhejiang Univ. Sci. A 14 (6): 435–446. https://doi.org/10.1631/jzus.A1200343.
Li, C.-X., K.-H. Xie, A.-F. Hu, and B.-X. Hu. 2012. “One–dimensional consolidation of double-layered soil with non-Darcian flow described by exponent and threshold gradient.” J. Cent. South Univ. 19 (2): 562–571. https://doi.org/10.1007/s11771-012-1040-3.
Li, C.-X., K.-H. Xie, and K. Wang. 2010. “Analysis of 1D consolidation with non-Darcian flow described by exponent and threshold gradient.” J. Zhejiang Univ.-Sci. A 11 (9): 656–667. https://doi.org/10.1631/jzus.A0900787.
Liakopoulos, A. C. 1965. “Theoretical solution of the unsteady unsaturated flow problems in soils.” Int. Assoc. Sci. Hydrol. Bull. 10 (1): 5–39. https://doi.org/10.1080/02626666509493368.
Liu, J.-C., and G. H. Lei. 2013. “One–dimensional consolidation of layered soils with exponentially time-growing drainage boundaries.” Comput. Geotech. 54: 202–209. https://doi.org/10.1016/j.compgeo.2013.07.009.
Liu, Z. Y., L. Y. Sun, J. C. Yue, and C. W. Ma. 2009. “One–dimensional consolidation theory of saturated clay based on non-Darcy flow.” Chin. J. Rock Mech. Eng. 28 (5): 973–979.
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.
Moradi, M., A. Keshavarz, and A. Fazeli. 2019. “One dimensional consolidation of multi-layered unsaturated soil under partially permeable boundary conditions and time-dependent loading.” Comput. Geotech. 107: 45–54. https://doi.org/10.1016/j.compgeo.2018.11.020.
Olsen, H. W. 1965. “Deviations from Darcy's law in saturated clays.” Soil Sci. Soc. Am. J. 29 (2): 135–140. https://doi.org/10.2136/sssaj1965.03615995002900020009x.
Olsen, H. W. 1985. “Osmosis: A cause of apparent deviations from Darcy's law.” Can. Geotech. J. 22 (2): 238–241. https://doi.org/10.1139/t85-032.
Pascal, F., H. Pascal, and D. W. Murray. 1981. “Consolidation with threshold gradients.” Int. J. Numer. Anal. Methods Geomech. 5 (3): 247–261. https://doi.org/10.1002/nag.1610050303.
Qin, A.-F., G.-J. Chen, Y.-W. Tan, and D.-A. Sun. 2008. “Analytical solution to one–dimensional consolidation in unsaturated soils.” Appl. Math. Mech. 29: 1329–1340. https://doi.org/10.1007/s10483-008-1008-x.
Qin, A.-F., D.-A. Sun, and Y.-W. Tan. 2010. “Semi-analytical solution to one–dimensional consolidation in unsaturated soils.” Appl. Math. Mech. 31: 215–226. https://doi.org/10.1007/s10483-010-0209-9.
Schiffman, R. L., and J. R. Stein. 1970. “One–dimensional consolidation of layered systems.” J. Soil Mech. Found. Div. 96 (4): 1499–1504. https://doi.org/10.1061/JSFEAQ.0001453.
Scott, R. F. 1963. Principle of soil mechanics. San Francisco, CA: Addison-Wiesly Pub. Company.
Shan, Z., D. Ling, and H. Ding. 2012. “Exact solutions for one-dimensional consolidation of single-layer unsaturated soil.” Int. J. Numer. Anal. Methods Geomech. 36 (6): 708–722. https://doi.org/10.1002/nag.1026.
Shan, Z.-D., D.-S. Ling, and H.-J. Ding. 2013. “Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition.” J. Zhejiang Univ. Sci. A 14: 61–70. https://doi.org/10.1631/jzus.A1200167.
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.
Terzaghi, K. 1925. Erdbaumechanik auf bodenphysikalischer Grundlage. Deuticke, Wien.
Wang, L., D. Sun, L. Li, P. Li, and Y. Xu. 2017a. “Semi-analytical solutions to one–dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary.” Comput. Geotech. 89: 71–80. https://doi.org/10.1016/j.compgeo.2017.04.005.
Wang, L., D. Sun, and A. Qin. 2018a. “Semi-analytical solution to one–dimensional consolidation for unsaturated soils with exponentially time-growing drainage boundary conditions.” Int. J. Geomech. 18 (2): 04017144. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001056.
Wang, L., D. A. Sun, A. F. Qin, and Y. F. Xu. 2017b. “Semi-analytical solution to one–dimensional consolidation for unsaturated soils with semi-permeable drainage boundary under time-dependent loading.” Int. J. Numer. Anal. Methods Geomech. 41: 1636–1655. https://doi.org/10.1002/nag.2694.
Wang, S., W. Zhu, K. Fei, C. Xu, and N. Zhang. 2018b. “Study on non-darcian flow sand-clay mixtures.” Appl. Clay Sci. 151: 102–108. https://doi.org/10.1016/j.clay.2017.10.028.
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.
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.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 24 • Issue 8 • August 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Sep 10, 2023
Accepted: Jan 31, 2024
Published online: May 30, 2024
Published in print: Aug 1, 2024
Discussion open until: Oct 30, 2024
ASCE Technical Topics:
- Consolidated soils
- Domain boundary
- Drainage
- Engineering fundamentals
- Flow (fluid dynamics)
- Fluid dynamics
- Fluid flow
- Fluid mechanics
- Geomechanics
- Geotechnical engineering
- Hydrologic engineering
- Irrigation engineering
- Mathematics
- Soil mechanics
- Soil properties
- Soil water
- Soils (by type)
- Unsaturated soils
- Water and water resources
- Water flow
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.