||One-dimensional consolidation in unsaturated soils under an external time-varying loading
||Department of Hydraulics & Ocean Engineering
Poroelasticity theory of consolidation
土壤壓密對於工程應用及災害防治常扮演關鍵的角色，尤其地層下陷的問題為台灣沿海及平原常見的地質災害。近年因乾旱所導致土壤壓密造成的災害更是需迫切解決的問題，因此應用數學解析解量化分析土壤壓密的結果，將能有效的提供工程師及決策者進行評估。本研究利用Lo et al. (2002)所發表的孔隙介質含兩項流體之動量方程式結合線性應力應變關係式(Lo et al., 2005)，並且依照前人研究土壤壓密的典型假設(Biot, 1941; Terzaghi, 1943)，將方程式的慣性項忽略所推導出的孔彈性壓密理論，並且根據Biot (1941)所提出的載重作用瞬間土體不排水的假設以及Lewallen and Wang(1998)應用載重效應(loading efficiency)所提出雙孔隙初始壓力而建立未飽和孔隙流體初始壓力，推導出土壤受到隨時間變化外部載重的孔隙流體壓力以及隨時間變化的總沉陷量。
Soil consolidation plays an important role in practical applications related to the field of engineering as well as in disaster prevention. In particular, subsidence is a common problem in coastal regions and in the alluvial fans in Taiwan. Drought-induced soil consolidation has also become a problem. As consolidation proceeds, the excess pore fluid pressure decreases, and the effective stress increases, reducing pore spaces and consolidating the soil grains. Time-varying loading, a common phenomenon in practical geotechnical engineering problems, is significant during the process of soil consolidation. Analytical solutions for quantifying soil consolidation can provide useful information to engineers and policymakers.
This study applies a set of coupled partial differential equations of momentum balance for two-phase fluid flows in a deformable porous medium developed by Lo et al. (2002) and linear stress-strain relations (Lo et al., 2005) to formulate a poroelasticity theory of consolidation. Closed-form analytical solutions describing the excess pore air and water pressure and the total settlement in response to an external loading under three types of boundary drainage conditions are formulated by employing the Laplace transform. To establish the initial conditions, Biot’s assumption (1941) that water is not allowed to escape when the loading is instantly applied on a porous medium is used. The effects of water content, soil texture, and the dimensionless frequency on one-dimensional consolidation in unsaturated soils are then determined.
The results show that the dissipation of excess pore water pressure is significantly sensitive to soil texture, which is almost completed in very short elapsed time in sand, followed by loamy sand, sandy loam, loam, sandy clay loam, silt loam, clay loam, sandy clay, silty clay loam, silt clay, and clay at water saturation equal to 0.9. This trend is consistent with the coefficient of consolidation for water. Irrespective of the elapsed time, the excess pore water pressure always dissipates faster under full-permeable boundaries than under a semi-permeable boundary. For a given soil texture, the dissipation rate of excess pore water pressure is higher in wetter soil. In the early stage of consolidation, the initial pore water pressure is affected by the loading efficiency for water, which is strongly controlled by the initial water saturation. For soil consolidation under time-varying loading, the amplitude of excess pore water pressure is not only controlled by the initial water saturation but also by the dimensionless frequency. The transient response decays immediately and produces an increase in the rate of change of water pressure immediately after the application of the time-varying loading. In reference to the time-dependent total settlement, it is found that it has a positive relationship with the inverse of the soil bulk modulus.
Table of Contents VI
List of Tables VIII
List of Figures IX
Chapter 1 Introduction 1
1.1 Motivation and objective 1
1.2 Literature review 2
1.3 Outline of dissertation 7
Chapter 2 Model equations 10
2.1 Momentum balance equations 10
2.2 Linear stress-strain relations 12
2.3 Generalized undrained modulus 14
2.4 The elasticity coefficients 17
Chapter 3 Theory of one-dimensional consolidation in unsaturated soils 19
3.1 Initial conditions 20
3.2 Analytical solutions with respect to full-permeable drainage condition 22
3.3 Analytical solutions with respect to semi-permeable drainage condition 28
3.4 Analytical solutions with respect to an opposite semi-permeable drainage condition 31
3.5 Reduction to one-dimensional consolidation in water-saturated soils 32
Chapter 4 Parameters for numerical simulation 39
4.1 Water retention curve 39
4.2 Relative permeability 40
4.3 Numerical simulation settings 40
Chapter 5 Computation results and discussion 45
5.1 Consolidation behaviors under constant loading 45
5.2 Consolidation behaviors under time-varying loading 50
Chapter 6 Conclusions and recommendations 126
2.Albers, B.,“Linear elastic wave propagation in unsaturated sands, silts, loams and clays,”Transp. Porous Media, Vol. 86, No. 2, pp. 567-587, 2011.
3.Ausilio, E., and Conte, E.,“Settlement rate of foundations on unsaturated soils,”Can. Geotech. J., Vol. 36, No. 5, pp. 940-946, 1999.
4.Barden, L.,“Consolidation of compacted and unsaturated clays,”Geotechnique, Vol. 15, No. 3, pp. 267-286, 1965.
5.Biot, M.A.,“General theory of three-dimensional consolidation,”J. Appl. Phys.,Vol 12, pp. 155-164, 1941.
6.Biot, M.A., and Willis, D.G.,“The elastic coefficients of the theory of consolidation,”J. Appl. Mech., Vol. 24, pp. 594-601, 1957.
7.Biot, M.A.,“Mechanics of deformation and acoustic propagation in porous media,” J. Appl. Phys., Vol. 33, No. 4, pp. 1482-1498, 1962.
8.Bishop, A.W.,“The effective stress principle,”Teknisk Ukeblad, Vol. 93, pp. 859-863, 1959.
9.Bishop, A.W., and Blight, G.E.,“Some aspects of effective stress in saturated and unsaturated soil,”Geotechnique, Vol. 13, No.3, pp. 177-197, 1963.
10.Brooks, R. H., and Corey, A. T., “Hydraulic properties of porous media,”Hydrol. Pap. 3, Civ. Eng. Dep., Colo. State Univ., Fort Collins, 1964.
11.Carcione, J.M., Helle, H.B., Santos, J.E., and Ravazzoli, C.L.,“A constitutive equation and generalized Gassmann modulus for multimineral porous media,”Geophysics, Vol. 70, No. 2, pp. N17-26, 2005.
12.Chen, Z.H.,“Consolidation theory of unsaturated soil based on the theory of mixture,” Appl. Math. Mech., Vol. 14, pp. 721-733, 1993
13.Conte, E.,“Consolidation analysis for unsaturated soils,”Can. Geotech. J. , Vol. 41, pp. 599-612, 2004.
14.Crone, T.J., and Wilcock, W.S.D.,“Modeling the effects of tidal loading on mid-ocean ridge hydrothermal systems,”Geohem. Geophys. Geosyst., Vol. 6, No. 7, pp. , 2005.
15.Fredlund, D.G., and Morgenstern, N.R.,“Constitutive relations for volume change in unsaturated soils,” Can. Geotech. J., Vol. 13, pp. 261-279, 1976.
16.Fredlund, D.G., and Hasan, J.U.,“One-dimensional consolidation theory: unsaturated soils,”Can. Geotech. J., Vol. 16, No. 3, pp. 521-531, 1979.
17.Fredlund, D.G., Rahardjo, H., Fredlund, M.D., Unsaturated Soil Mechanics in Engineering Practice. Wiley & Sons; New Jersey, 2012.
18.Gary, W. G., “General conservation equations for multi-phase system: 4. Constitutive theory including phase change,” Adv. Water Resour., Vol. 6, pp. 130-140, 1983.
19.Geertsma, J., “Problems of rock mechanics in petroleum production engineering,” Proc. First Int. Contr. ISRM, Lisbon, Vol. 1, pp. 585-594, 1966.
20.Gutierrez, M.S., and Lewis, R.W.,“Coupling of fluid flow and deformation in underground formations,”J. Eng. Mech., Vol. 128, No. 7, pp. 779–787, 2002.
21.Guenther, R.B., and Lee, J.W., Partial differential equations of mathematical physics and integral equations, Prentice Hall, Englewood Cliffs, NJ, 1988.
22.Jacob, C.E.,“On the flow of water in an elastic artesian aquifer,”Eos. Trans. AGU, Vol. 21, pp. 574-586, 1940.
23.Kameo, Y, Adachi, T,and Hojo, M, “Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading,” J. Mech. Phys. Solids, Vol. 56, pp. 1794-1805, 2008.
24.Lewallen, K.T., and Wang, H.F.,“Consolidation of a double-porosity medium,”Int. J. solids structures, Vol. 35, pp. 4845-4867, 1998.
25.Lo, W.C., Sposito G., and Majer, E.,“Immiscible two-phase fluid flows in deformable porous media, ” Adv. Water Resour., Vol. 25, No. 8-12, pp. 1105-1117, 2002.
26.Lo, W.C., Sposito, G., and Majer, E.,“Wave propagation through elastic porous media containing two immiscible fluids,”Water Resour. Res., Vol. 41, W02025, 2005.
27.Lo, W.C., Yeh, C.L., and Tsai, C.T.,“Effect of soil texture on the propagation and attenuation of acoustic wave at unsaturated conditions,”J. Hydrology, Vol. 338, pp. 273-284, 2007.
28.Lo, W.C., Yeh, C.L., and Jan, C.D.,“Effect of soil texture and excitation frequency on the propagation and attenuation of acoustic wave at saturated conditions,”J. Hydrology, Vol, 357, pp. 270-281, 2008.
29.Lo, W.C. , Sposito, G., and Huang, Y. H., “Modeling seismic stimulation: Enhanced non-aqueous fluid extraction from saturated porous media under pore-pressure pulsing at low frequencies”, J. Appl. Geophys, Vol. 78, pp. 77-84, 2012.
30.Lo, W.C., and Sposito, G.,“Acoustic waves in unsaturated soils,”Water Resour. Res., Vol. 49, No. 9, pp. 5674-5684, 2013.
31.Lo, W.C., Sposito, G., and Chu, H.,“Poroelastic theory of consolidation in unsaturated soils,”Vadose Zone J., Vol. 13, No. 5, 2014.
32.Mualem, Y.,“A new model for predicting the hydraulic conductivity of unsaturated porous media,”Water Resour. Res., Vol. 12, No. 3, pp. 513-522, 1976.
33.Pride, S.R., Gangi, A.F., and Morgan, F.D.,“Deriving the equations of motion for porous isotropic media,” J. Acoust. Soc. Am, Vol. 92, No. 6, pp. 3278-3290, 1992.
34.Rahal, M.A. and Vuez, A.R., “Analysis of settlement and pore pressure induced by cyclic loading of silo”, J. Geotech. Geoenviron. Eng., Vol. 124, No. 12, pp. 1208-1210, 1998.
35.Rawls, W.J., Ahuja, J.R., and Brakensiek, D.L.,“Estimating soil hydraulic properties from soils data.” Proceedings of Workshop on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, Riverside, CA, pp. 329-341, 1992.
36.Rice, J.R. and Cleary, M.P.“Some basic stress diffusion solutions for fluid-saturated porous media with compressible constituents.” Rev. Geophys. Space Phys., Vol. 14, pp. 227-241, 1976.
37.Shan, Z.D., Ling, D.S.,and Ding, H.J., “Exact solutions for one-dimensional consolidation of single-layer unsaturated soil”, Int J Numer Anal Methods Geomech, Vol. 36, pp. 708–722, 2012.
38.Terzaghi, K., Erdbaumechanik auf Bodenphysikalischer, Deutichke, Vienna, 1925.
39.Terzaghi, K., Theoretical soil mechanics, John Wiley, New York, 1943.
40.Tuncay, K., and Corapcioglu, M.Y.,“Consolidation of elastic porous media saturated by two immiscible fluids,” J. Eng. Mech., Vol. 122, No. 11, pp. 1077-1085, 1996.
41.Tuncay, K., and Corapcioglu, M.Y.,“Wave propagation in poroelastic media saturated by two fluids”, J. Appl. Mech., Vol. 64, No. 2, pp. 313-320, 1997.
42.Tuncay, K., kambham, K.K.R.,and Corapcioglu, M.Y., “Self-weight subsidence of saturated soft porous media”, J. Eng. Mech, Vol. 24, No. 6, pp. 630-638, 1998.
43.van der Kamp, G. and Gale, J.E.,“Theory of earth tide and barometric effects in porous media,”Water Resour. Res., Vol. 31, pp. 3103-3106, 1983.
44.van Genuchten, M. T.,“A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,”Soil Sci. Soc. Am. J., Vol. 44, No. 5, pp. 892-898, 1980.
45.Verruijt, A.,“Approximations of cyclic pore pressures caused by sea waves in a poro-elastic half-plane,”In: Pande, G. N., Zienkiewicz, O. S. (Eds.), Soil mechanics - Transient and cyclic loads, Wiley, New York, pp. 37–51, 1982.
46.Wang, H.F., Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press, Princeton, 2000.
47.Wang, K., and Davis, E.E.,“Theory for the propagation of tidally induced pore pressure variations in layered subseafloor formations,” J. Geo. Res., Vol. 101, No. B5, pp. 11483-11496, 1996.
48.Whitaker, S.,“Transport equations for multiphase system,”Chem. Eng. Sci., Vol.28, No. 1, pp. 139-147, 1973.
49.Wilson, N.E. and Elgohary, M.M., “Consolidation of soil under cyclic loading,” Can. Geotech. J., Vol. 11, pp. 420-423, 1974.