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Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary

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Abstract

The semi-analytical solutions to Fredlund and Hasan’s one-dimensional (1D) consolidation for unsaturated soils with a semi-permeable drainage boundary are presented. Two variables are introduced to transform the two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations (PDFs), which are easily solved by the Laplace transform method. Then, the pore-water pressure, pore-air pressure, and soil settlement are obtained in the Laplace domain. The Crump method is adopted to perform the inverse Laplace transform in order to obtain the semi-analytical solutions in the time domain. It is shown that the proposed solutions are more applicable to various types of boundary conditions and agree well with the existing solutions from the literature. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with single, double, mixed, and semi-permeable drainage boundaries. The changes in the pore-air and pore-water pressures and the soil settlement with the time factor at different values of the semi-permeable drainage boundary parameters are illustrated. In addition, parametric studies are con- ducted on the pore-air and pore-water pressures at different ratios (the air permeability coefficient to the water permeability coefficient) and depths.

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Abbreviations

C a :

interactive constant with respect to the air phase

C w :

interactive constant with respect to the water phase

C av :

coefficient of volume change with respect to the air phase

C wv :

coefficient of the volume change with respect to the water phase

C a σ :

consolidation coefficient for the air phase

C w σ :

consolidation coefficient for the water phase

g :

gravitational acceleration

h :

soil layer thickness

h 0 :

top boundary thickness

k a :

coefficient of the air permeability

k w :

coefficient of the water permeability

k a0 :

coefficient of the air permeability at the top boundary

k w0 :

coefficient of the water permeability at the top boundary

M :

molecular mass of air

m a1k :

coefficient of the air volume change with respect to a change in (σ - ua)

m a2 :

coefficient of the air volume change with respect to a change in (u a − u w)

m w1k :

coefficient of the water volume change with respect to a change in (σ − u a)

m w2 :

coefficient of the water volume change with respect to a change in (u a − u w)

n 0 :

initial porosity

q 0 :

initial surcharge

R :

universal gas constant

Q(s):

result of the Laplace transform of \(\frac{{{\partial _q}\left( t \right)}}{{\partial t}}\)

R a :

parameter of the semi-permeable drainage for the air at the top bound- ary

R w :

parameter of the semi-permeable drainage for the water at the top boundary

S r0 :

initial degree of the saturation

T :

absolute temperature

t :

time

u a :

pore-air pressure

u atm :

atmospheric pressure

\(\overline u _a^0\) :

absolute pore-air pressure

u 0a :

initial excess pore-air pressure

u w :

pore-water pressure

u 0w :

initial excess pore-water pressure

w :

settlement

w*:

normalized settlement

z :

depth

γ w :

unit weight of water

ε v :

volumetric strain

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Author information

Authors and Affiliations

  1. School of Urban Railway Transportation, Shanghai University of Engineering Science, Shanghai, 201620, China

    Lei Wang

  2. Department of Civil Engineering, Shanghai University, Shanghai, 200444, China

    Lei Wang & De’an Sun

  3. Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yongfu Xu

Authors
  1. Lei Wang
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  2. De’an Sun
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  3. Yongfu Xu
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Corresponding author

Correspondence to De’an Sun.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 41630633 and 11672172)

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Wang, L., Sun, D. & Xu, Y. Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary. Appl. Math. Mech.-Engl. Ed. 38, 1439–1458 (2017). https://doi.org/10.1007/s10483-017-2243-6

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2010 Mathematics Subject Classification

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