A Closed-Form Equation for Effective Stress of Unsaturated Saline Clay Considering Capillary and Osmosis Effects
Publication: International Journal of Geomechanics
Volume 24, Issue 6
Abstract
The interaction mechanism between matric suction and osmotic suction was clarified through the analysis of certain physicochemical effects on unsaturated saline clay. The Young–Laplace equation for unsaturated saline clay was then derived by considering the contribution of the chemical potential to the Gibbs free energy at the air–solution interface. An effective stress equation for unsaturated saline clay considering the capillary and osmotic effects was obtained by utilizing a soil skeleton stress equilibrium analysis method. The definition and calculation formula for the osmotic efficiency parameter was determined by analyzing the chemical osmotic effect. Finally, the volume deformation and shear strength of unsaturated saline silty clay were tested under different salt content and matric suction conditions, and the validity of the effective stress equation was verified through tests on expansive clay saturated with salt solution. The results indicated that the effective stress–based generalized effective stress equation for unsaturated saline clay, taking into account the effect of the salt solution on the pore liquid pressure, usw, was related to osmotic suction and the osmotic efficiency parameter. The osmotic efficiency parameter is the ratio of change in the surface tension at the air–solution interface to the change in the corresponding osmotic suction, which reflects the strength of the chemomechanical coupling in the micropore space and its smooth transition to the macroscopic scale. In addition, the validation demonstrated that the proposed effective stress equation was able to uniformly describe the mechanical phenomena of unsaturated saline clay. Our findings provide a convenient way to simulate the mechanical behavior of unsaturated clay in saline environments.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author upon request.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 12362032, 11962016, and 51978320), the Foundation for Innovation Groups of Basic Research in Gansu Province (No. 20JR5RA478), and the Natural Science Foundation of the Gansu Province of China (No. 20JR10RA471). The authors are grateful to the reviewers for their insightful and constructive comments.
Notation
The following symbols are used in this paper:
- A
- area of air–solution interface;
- effective electrostatic attraction between clay particles;
- aw
- activity of salt solution;
- D
- fractal dimension;
- e
- void ratio;
- eC
- void ratio of sample containing sodium chloride solution;
- eS
- void ratio of sample containing sodium sulfate solution;
- eW
- void ratio of sample containing distilled water;
- shape factor;
- pore scale within aggregate;
- pore scale within clay platelet;
- material constant;
- m
- molality;
- N
- number of moles in air–solution interface system;
- NAir
- number of moles of pore air;
- NLiquid
- number of moles of pore solution;
- number of moles of the interfacial, or excess number of moles of the air–solution, system;
- nl
- volume fraction of salt solution in pore;
- p
- mean principal stress;
- pf
- peak mean principal stress;
- p′
- Terzaghi’s effective stress;
- qf
- peak shear stress;
- R
- gas constant;
- effective electrostatic repulsion between clay particles;
- r
- mean radius of curvature;
- S
- entropy of air–solution interface system;
- effective saturation;
- Sr
- saturation;
- residual saturation;
- entropy of air–solution interface;
- s
- matric suction;
- entropy per unit area of air–solution interface;
- T
- absolute temperature;
- U
- internal energy of air–solution interface system;
- internal energy of gas–solution interface;
- u
- pressure;
- ua
- pore air pressure;
- usw
- pore liquid pressure caused by salt solution;
- uw
- pore water pressure;
- real pore water pressure of unsaturated saline clay;
- V
- volume of air–solution interface system;
- volume of aqueous solution in the macroscopic pore space;
- Vg
- volume of air;
- volume of salt solution absorbed in the micropore space;
- volume of salt solution absorbed by each unit on surface of clay platelet;
- volume of aqueous solution in the microporous space;
- Vr
- volume of representative element volume;
- Vs
- volume of clay particles;
- molar volume of water;
- volume of air–solution interface;
- α
- VG model parameters associated with air entry value;
- parameter related to contribution of osmotic suction;
- β
- VG model parameter related to curve shape;
- γ
- surface tension at the air–water interface;
- surface tension at the air–solution interface;
- unit tensor;
- κb(r)
- mean curvature function;
- μ
- chemical potential of interface system;
- μa
- chemical potential of air–water interface system;
- μb
- chemical potential of air–solution interface system;
- ν
- total number of ions after solute ionization;
- Π
- generalized osmotic pressure;
- π
- osmotic suction;
- πf
- final osmotic suction;
- πr
- reference osmotic suction;
- π0
- initial osmotic suction;
- total stress tensor;
- mean soil skeleton stress tensor;
- σ1
- vertical stress;
- vertical effective stress;
- effective stress tensor;
- normal stress;
- τf
- peak shear strength;
- τr
- residual shear strength;
- ϕ
- osmotic coefficient of solute;
- φs
- capillary pressure;
- χo
- osmotic efficiency parameter;
- ψ
- Gibbs free energy of air–solution interface system;
- Gibbs free energy of air–solution interface; and
- ω
- hydrodynamic membrane efficiency.
References
Barbour, S. L., and D. G. Fredlund. 1989. “Mechanisms of osmotic flow and volume change in clay soils.” Can. Geotech. J. 26 (4): 551–562. https://doi.org/10.1139/t89-068.
Bartoli, F., R. Philippy, M. Doirisse, S. Niquet, and M. Dubuit. 1991. “Structure and self-similarity in silty and sandy soils: The fractal approach.” J. Soil Sci. 42 (2): 167–185. https://doi.org/10.1111/j.1365-2389.1991.tb00399.x.
Bishop, A. W. 1959. “The principle of effective stress.” Teknisk Ukeblad. 39: 859–863.
Burton, G. J., J. A. Pineda, D. Sheng, and D. Airey. 2015. “Microstructural changes of an undisturbed, reconstituted and compacted high plasticity clay subjected to wetting and drying.” Eng. Geol. 193: 363–373. https://doi.org/10.1016/j.enggeo.2015.05.010.
Callahan, K. M., N. N. Casillas-Ituarte, M. Xu, M. Roeselová, H. C. Allen, and D. J. Tobias. 2010. “Effect of magnesium cation on the interfacial properties of aqueous salt solutions.” J. Phys. Chem. A 114 (32): 8359–8368. https://doi.org/10.1021/jp103485t.
Cheng, G., and M. J. Hendry. 2014. “Chemico-osmosis in geologic membranes: Role of membrane potential gradient.” Geochim. Cosmochim. Acta 141: 270–280. https://doi.org/10.1016/j.gca.2014.06.017.
Diamond, S. 1970. “Pore size distributions in clays.” Clays Clay Miner. 18 (1): 7–23. https://doi.org/10.1346/CCMN.1970.0180103.
Duan, X., L. Zeng, and X. Sun. 2019. “Generalized stress framework for unsaturated soil: Demonstration and discussion.” Acta Geotech. 14 (5): 1459–1481. https://doi.org/10.1007/s11440-018-0739-1.
Fredlund, D. G., and N. R. Morgenstern. 1978. “Stress state variables for unsaturated soils.” J. Geotech. Geoenviron. 103 (5): 447–466.
Gavish, N., and K. Promislow. 2016. “Dependence of the dielectric constant of electrolyte solutions on ionic concentration: A microfield approach.” Phys. Rev. E 94 (1): 012611. https://doi.org/10.1103/PhysRevE.94.012611.
Gens, A. 2010. “Soil–environment interactions in geotechnical engineering.” Géotechnique 60 (1): 3–74. https://doi.org/10.1680/geot.9.P.109.
Graham, J., J. M. Oswell, and M. N. Gray. 1992. “The effective stress concept in saturated sand–clay buffer.” Can. Geotech. J. 29 (6): 1033–1043. https://doi.org/10.1139/t92-121.
Guendouzi, M. E., A. Mounir, and A. Dinane. 2003. “Water activity, osmotic and activity coefficients of salt solutions of Li2SO4, Na2SO4, K2SO4, (NH4)2SO4, MgSO4, MnSO4, NiSO4, CuSO4, and ZnSO4 at T = 298.15 K.” J. Chem. Thermodyn. 35 (2): 209–220. https://doi.org/10.1016/S0021-9614(02)00315-4.
Houlsby, G. T. 1979. “The work input to a granular material.” Géotechnique 29 (3): 354–358. https://doi.org/10.1680/geot.1979.29.3.354.
Houlsby, G. T. 1997. “The work input to an unsaturated granular material.” Géotechnique 47 (1): 193–196. https://doi.org/10.1680/geot.1997.47.1.193.
Jungwirth, P., and D. J. Tobias. 2006. “Specific ion effects at the air/water interface.” Chem. Rev. 106: 1259–1281. https://doi.org/10.1021/cr0403741.
Kedem, O., and A. Katchalsky. 1963. “Permeability of composite membranes. Part 1.—Electric current, volume flow and flow of solute through membranes.” Trans. Faraday Soc. 59: 1918–1930. https://doi.org/10.1039/TF9635901918.
Laloui, L., and M. Nuth. 2009. “On the use of the generalised effective stress in the constitutive modelling of unsaturated soils.” Comput. Geotech. 36 (1–2): 20–23. https://doi.org/10.1016/j.compgeo.2008.03.002.
Liu, S. H., and D. A. Sun. 2002. “Simulating the collapse of unsaturated soil by DEM.” Int. J. Numer. Anal. Methods Geomech. 26 (6): 633–646. https://doi.org/10.1002/nag.215.
Lu, N. 2016. “Generalized soil water retention equation for adsorption and capillarity.” J. Geotech. Geoenviron. Eng. 142 (10): 04016051. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001524.
Lu, N., J. W. Godt, and D. T. Wu. 2010. “A closed-form equation for effective stress in unsaturated soil.” Water Resour. Res. 46 (5): 567–573.
Lu, N., and W. J. Likos. 2006. “Suction stress characteristic curve for unsaturated soil.” J. Geotech. Geoenviron. Eng. 132 (2): 131–142. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:2(131).
Ma, T., C. Wei, X. Xia, and P. Chen. 2016. “Constitutive model of unsaturated soils considering the effect of intergranular physicochemical forces.” J. Eng. Mech. 142 (11): 04016088. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001146.
Ma, T., C. Yao, Y. Dong, P. Yi, and C. Wei. 2019. “Physicochemical approach to evaluating the swelling pressure of expansive soils.” Appl. Clay Sci. 172: 85–95. https://doi.org/10.1016/j.clay.2019.02.011.
Marine, I. W., and S. J. Fritz. 1981. “Osmotic model to explain anomalous hydraulic heads.” Water Resour. Res. 17 (1): 73–82. https://doi.org/10.1029/WR017i001p00073.
Musso, G., E. Romero, and G. D. Vecchia. 2013. “Double-structure effects on the chemo-hydro-mechanical behaviour of a compacted active clay.” Géotechnique 63 (3): 206–220. https://doi.org/10.1680/geot.SIP13.P.011.
Osipov, V. I. 2014. “The 2012 Hans Cloos lecture: Physicochemical theory of effective stress in soils.” Bull. Eng. Geol. Environ. 73 (4): 903–915. https://doi.org/10.1007/s10064-014-0605-9.
Rao, S., G. B. Deepak, P. Raghuveer Rao, and P. Anbazhagan. 2017. “Influence of physico-chemical components on the consolidation behavior of soft kaolinites.” Acta Geotech. 12: 441–451. https://doi.org/10.1007/s11440-016-0478-0.
Rao, S. M., and T. Thyagaraj. 2007. “Swell–compression behaviour of compacted clays under chemical gradients.” Can. Geotech. J. 44 (5): 520–532. https://doi.org/10.1139/t07-002.
Song, M.-M., L.-L. Zeng, and Z.-S. Hong. 2017. “Pore fluid salinity effects on physicochemical-compressive behaviour of reconstituted marine clays.” Appl. Clay Sci. 146 (09): 270–277. https://doi.org/10.1016/j.clay.2017.06.015.
Sridharan, A., and G. V. Rao. 1973. “Mechanisms controlling volume change of saturated clays and the role of the effective stress concept.” Géotechnique 23 (3): 359–382. https://doi.org/10.1680/geot.1973.23.3.359.
Sridharan, A., and G. Venkatappa Rao. 1979. “Shear strength behaviour of saturated clays and the role of the effective stress concept.” Géotechnique 29 (2): 177–193. https://doi.org/10.1680/geot.1979.29.2.177.
Sun, W., C. Liu, D. Yang, and D. Sun. 2022. “Evaluation of hydro-mechano-chemical behaviour of bentonite-sand mixtures.” J. Rock Mech. Geotech. Eng. 14 (2): 637–652. https://doi.org/10.1016/j.jrmge.2021.10.008.
Tao, G., Y. Chen, H. Xiao, Q. Chen, and J. Wan. 2019. “Determining soil-water characteristic curves from mercury intrusion porosimeter test data using fractal theory.” Energies 12 (4): 752. https://doi.org/10.3390/en12040752.
Terzaghi, K. 1936. “The shearing resistance of saturated soils and the angle between planes of shear.” In Proc., 1st Int. Conf. on Soil Mechanics and Foundation Engineering, 54–56. Cambridge, MA: Harvard University Press.
van Genuchten, M. T. 1980. “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils.” Soil Sci. Soc. Am. J. 44 (5): 892–898. https://doi.org/10.2136/sssaj1980.03615995004400050002x.
Wang, L. Y., F. X. Zhou, and H. Qin. 2020. “Fractional creep model and experimental study of saturated saline soil.” Rock Soil Mech. 41 (2): 543–551.
Wang, L. Y., F. X. Zhou, L. Z. Zhou, and Y. W. Liang. 2024. “Stress-dependent properties of physicochemical interaction of unsaturated saline clay.” Chin. J. Geotech. Eng. 45 (1): 1–10.
Wang, Y., A. Zhang, W. Ren, and L. Niu. 2019. “Study on the soil water characteristic curve and its fitting model of Ili loess with high level of soluble salts.” J. Hydrol. 578: 124067. https://doi.org/10.1016/j.jhydrol.2019.124067.
Wei, C. 2014. “A theoretical framework for modeling the chemomechanical behavior of unsaturated soils.” Vadose Zone J. 13 (9): 1–21. https://doi.org/10.2136/vzj2013.07.0132.
Witteveen, P., A. Ferrari, and L. Laloui. 2013. “An experimental and constitutive investigation on the chemomechanical behaviour of a clay.” Géotechnique 63 (3): 244–255. https://doi.org/10.1680/geot.SIP13.P.027.
Xu, Y. 2019. “Peak shear strength of compacted GMZ bentonites in saline solution.” Eng. Geol. 251: 93–99. https://doi.org/10.1016/j.enggeo.2019.02.009.
Xu, Y., G. Xiang, H. Jiang, T. Chen, and F. Chu. 2014. “Role of osmotic suction in volume change of clays in salt solution.” Appl. Clay Sci. 101: 354–361. https://doi.org/10.1016/j.clay.2014.09.006.
Xu, Y. F., H. Matsuoka, and D. A. Sun. 2003. “Swelling characteristics of fractal-textured bentonite and its mixtures.” Appl. Clay Sci. 22 (4): 197–209. https://doi.org/10.1016/S0169-1317(02)00159-X.
Yao, C., P. Chen, T. Ma, X. Xia, and C. Wei. 2020. “Physicochemical effect on shear strength characteristics of clayey soils based on ring-shear experiment.” Can. Geotech. J. 57 (12): 1820–1831. https://doi.org/10.1139/cgj-2019-0513.
Yao, C., C. Wei, T. Ma, P. Chen, and H. Tian. 2021. “Experimental investigation on the influence of thermochemical effect on the pore–water status in expansive soil.” Int. J. Geomech. 21 (6): 04021080. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002020.
Yaroshchuk, A. E. 1995. “Osmosis and reverse osmosis in fine-porous charged diaphragms and membranes.” Adv. Colloid Interface Sci. 60 (1/2): 1–93. https://doi.org/10.1016/0001-8686(95)00246-M.
Yong, R. N. 1999. “Soil suction and soil-water potentials in swelling clays in engineered clay barriers.” Eng. Geol. 54 (1–2): 3–13. https://doi.org/10.1016/S0013-7952(99)00056-3.
Yong, R. N., and B. P. Warkentin. 1975. Soil properties and behavior. New York: Elsevier Scientific Publishing.
Zhang, L., D. Sun, and D. Jia. 2016. “Shear strength of GMZ07 bentonite and its mixture with sand saturated with saline solution.” Appl. Clay Sci. 132–133: 24–32. https://doi.org/10.1016/j.clay.2016.08.004.
Zhang, Y., W. Ye, Y. Chen, and B. Chen. 2017. “Impact of NaCl on drying shrinkage behavior of low-plasticity soil in earthen heritages.” Can. Geotech. J. 54 (12): 1762–1774. https://doi.org/10.1139/cgj-2016-0403.
Zhao, C., Z. Liu, P. Shi, J. Li, G. Cai, and C. Wei. 2016. “Average soil skeleton stress for unsaturated soils and discussion on effective stress.” Int. J. Geomech. 16 (6): D4015006. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000610.
Zhou, F. X., L. Y. Wang, and Y. M. Lai. 2020. “Review and research on osmotic suction of saturated saline soils.” Chin. J. Geotech. Eng. 42 (7): 1199–1210.
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International Journal of Geomechanics
Volume 24 • Issue 6 • June 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Jun 26, 2023
Accepted: Nov 28, 2023
Published online: Mar 21, 2024
Published in print: Jun 1, 2024
Discussion open until: Aug 21, 2024
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