Semi-analytical solutions to two-dimensional plane strain consolidation for unsaturated soil

Computers and Geotechnics 101 (2018) 100–113

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Research Paper

Semi-analytical solutions to two-dimensional plane strain consolidation for unsaturated soil

T

⁎

Lei Wanga,b, Yongfu Xua, , Xiaohe Xiaa, De'an Sunc a

Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China College of Urban Railway Transportation, Shanghai University of Engineering Science, Shanghai 201620, China c Department of Civil Engineering, Shanghai University, Shanghai 200444, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Semi-analytical solution Unsaturated soil Two-dimensional plane strain consolidation Homogeneous and mixed boundaries Finite sine transform Laplace transform

This paper presents semi-analytical solutions to two-dimensional plane strain consolidation of unsaturated soils under different initial and boundary conditions. By applying the finite sine and Laplace transforms, the partial differential equations are converted to the ordinary differential equations. The semi-analytical solutions are obtained by using the finite sine inversion transform. Crump’s method is adopted to perform the inverse Laplace transform to obtain analytical solutions in time domain. The present solutions are more general and can be degenerated into conventional solutions to one-dimensional consolidation of unsaturated soils. Several examples are provided to investigate two-dimensional plane strain consolidation behavior of unsaturated soils.

1. Introduction In general, consolidation for soil is the process of the dissipation of excess pore pressures and the corresponding reduction in volume under long term static loads [1,2]. Since the inception of classical soil mechanics, Terzaghi’s one-dimensional (1D) consolidation theory for saturated soils has formed an extremely useful conceptual framework in geotechnical engineering. As a case, vertical drains have been widely used to accelerate the consolidation process of fine-grained soils in preloading ground improvement projects. Barron [3] and Hansbo [4] have derived the consolidation solution for the sand drainage well, and then various analytical solutions were developed for consolidation of soft foundation by vertical drains [5–8]. However, sometimes the settlement cannot be estimated by existing consolidation theory very well in engineering practice. This is partly because the soil deposits above the phreatic line are unsaturated [2], and the solutions to the consolidation of unsaturated soils are more general. In addition, the consolidation behavior of unsaturated soils during vertical drain should be investigated, and it is essential to extent the concept of consolidation of unsaturated soils in two-dimensional (2D) problem [9]. As a common issue in geotechnical engineering, a considerable number of studies on the consolidation theory for unsaturated soil has been conducted, and a great progress has been achieved. Such as, Scott [10] estimated the consolidation of unsaturated soils with occluded air bubbles. Biot [11] proposed a general consolidation theory that is suitable for analyzing unsaturated soil with occluded air bubbles. Barden

⁎

[12] presented an analysis of the 1D consolidation of compacted unsaturated clay. By assuming that the air and water phases are continuous, Fredlund and Hasan [13] proposed a 1D consolidation theory, in which two partial differential equations (PDEs) were employed to describe the dissipation processes of excess pore-air and pore-water pressures in unsaturated soils. This theory is now widely accepted. Dakshanamurthy and Fredlund [14] then developed 2D plane strain governing equations of unsaturated soils based on the heat diffusion concept for the two-dimensional condition. Later, Dakshanamurthy et al. [15] extended the 1D consolidation theory of unsaturated soils proposed by Fredlund and Hasan [13] to analyze three-dimensional (3D) case. By assuming all the soil parameters remain constant during consolidation, Fredlund et al. [2] presented a simplified form of the 1D consolidation equations for unsaturated soils. Since the inceptions of 1D, 2D plane strain and axisymmetric consolidation theories of unsaturated soils, several analytical approaches have been frequently developed by Conte [16,17] Qin et al. [18,19], Zhou et al. [20], Shan et al. [21] and Ho et al. [22]. Conte [16] investigated coupled as well as uncoupled solutions about consolidation in unsaturated soil by using the Fourier transform, then Conte [17] extended this study and proposed a general formulation that can deal with coupled consolidation with plane strain as well as axial symmetry. Using the Laplace transformation and Cayley-Hamilton techniques, Qin et al. [18,19] presented analytical solutions to the 1D consolidation for a single-layer unsaturated soil subjected to instantaneous and exponentially time-dependent loadings, respectively. Zhou et al. [20]

Corresponding author. E-mail address: [email protected] (Y. Xu).

https://doi.org/10.1016/j.compgeo.2018.04.015 Received 9 November 2017; Received in revised form 16 March 2018; Accepted 16 April 2018 0266-352X/ © 2018 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 101 (2018) 100–113

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Nomenclature

m1s

aa

m2s

aw ba bw Ca Cw Cvax

Cvaz Cvwx

Cvwz g h k ax k az k wx k wz l M m1a

m2a

changing rate of initial excess pore-air pressure along depth changing rate of initial excess pore-water pressure along depth initial excess pore-air pressure at top surface initial excess pore-water pressure at top surface interactive constant with respect to air phase interactive constant with respect to water phase coefficient of volume change with respect to air phase in xdirection coefficient of volume change with respect to air phase in zdirection coefficient of volume change with respect to water phase in x-direction coefficient of volume change with respect to water phase in z-direction gravitational acceleration thickness of soil layer coefficient of air permeability in x-direction coefficient of air permeability in z-direction coefficient of water permeability in x-direction coefficient of water permeability in z-direction length of soil layer molecular mass of air coefficient of air volume change with respect to a change in σ −ua coefficient of air volume change with respect to a change in ua−u w

m1w m2w n n0 q0 R Sr0 s T t ua uatm ua0 uw u w0 w w∗ x z γw εv Θ

employed two alternative terms, ϕ1 and ϕ2, which consist of the excess pore-air and pore-water pressures, to convert the nonlinear inhomogeneous PDEs into traditional homogeneous PDEs, and then solutions for the single and double drainage conditions were obtained using the separation of variable method. Shan et al. [21] also deduced an exact solution to the 1D consolidation for unsaturated soils by adopting the separation of variable method. However, the final equations have been left undisclosed as the result of cumbersome derivation, and it is difficult to be followed by engineers. By adopting eigen-function expansion method and Laplace transform techniques, Ho et al. [22] discussed a simple yet precise analytical solution to the 1D consolidation of an unsaturated soil deposit under homogeneous boundary condition subjected to an instantaneous loading. These analytical solutions are for unsaturated soils with homogeneous boundary conditions that are permeable or impermeable to both air and water phases. In addition, Fredlund et al. [2] named a type of mixed boundary conditions, in which the boundary is permeable to air and impermeable to water at the top or bottom surface. For the mixed boundary conditions, Qin et al. [23] and Shan et al. [24] gave semi-analytical and analytical solutions by using their mathematical methods mentioned above, respectively. Furthermore, Zhou and Zhao [25] obtained a numerical solution to the 1D consolidation of unsaturated soils under various initial and boundary conditions, and complex time-dependent loadings by the differential quadrature method (DQM). However, these mentioned solutions are only used to predict the 1D consolidation problem. Afterwards, Ho and Fatahi [9,26] gave an analytical solution to the 2D plane strain consolidation of an unsaturated soil stratum under homogeneous boundary conditions subjected to instantaneous and time-dependent loadings using the same method as Ho et al. [22]. And as a further work, Ho and Fatahi [27,28] also gave an analytical solution to the axisymmetric consolidation of an unsaturated soil stratum under instantaneous and time-dependent loadings. On the other hand, the dissipation of excess pore pressures is

coefficient of volume change with respect to a change in σ −ua coefficient of volume change with respect to a change in ua−u w coefficient of water volume change with respect to a change in σ −ua coefficient of water volume change with respect to a change in ua−u w natural number initial porosity external loading universal gas constant initial degree of saturation complex number frequency parameter integral range of the finite sine and cosine transforms time excess pore-air pressure atmospheric pressure initial excess pore-air pressure excess pore-water pressure initial excess pore-water pressure settlement normalized settlement investigated distance investigated depth unit weight of water total volumetric strain absolute temperature.

believed to be predominantly influenced by the lateral drainage during the 2D consolidation for unsaturated soils by vertical drains [9], and the dissipation of excess pore-air is easier than that of excess pore-water at the top or bottom boundary. That is to say, there is mixed boundary condition for the 2D consolidation for unsaturated soils, and the various initial and boundary conditions are more general and practical for consolidation problem of unsaturated soils by vertical drains. Therefore, as an attempt, this paper presents semi-analytical solutions to predict the dissipation of the excess pore-air and pore-water pressures, and settlement of unsaturated soil deposit using the 2D plane strain consolidation theory proposed by Dakshanamurthy and Fredlund [14] with the single, double and mixed drainage boundaries under different initial conditions. To obtain final solutions, the finite sine and Laplace transforms are used to convert the PDEs to the ordinary differential equations (ODEs), which are solved by applying the substitution method. It is found that the current solutions are more general and in a good agreement with the existing solutions in the literature, they also can be degenerated into that of 1D consolidation for unsaturated soils. Finally, examples are given to illustrate the 2D plane strain consolidation behavior of unsaturated soils. Changes are sufficiently demonstrated in the excess pore pressures and settlement under varying the parameter values of the ratios of air-water, horizontal-vertical permeability coefficients, depth and horizontal distance.

2. Mathematical model 2.1. Governing equations Based on the 2D plane strain consolidation equations for unsaturated soils proposed by Dakshanamurthy and Fredlund [14], Ho et al. [9] proposed a referential profile of an unsaturated soil stratum with a finite depth h (in z-direction) and length l (in x-direction) under vertical loading q0, as shown in Fig. 1. The soil system of interest 101

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L. Wang et al.

q0 x

Top boundary

z

Cw =

m2w /2m1w−1 m2w /2m1w

(3d)

Cvwx =

k wx γm m2w

(3e)

Unsaturated

h

soil stratum

Vertical

Cvwz =

Vertical drain

drain

k wz γm m2w m1a

(3f)

m2a

where and are the coefficients of air volume change in a soil element with respect to a change in net normal stress and suction; m1w and m2w are the coefficients of water volume in a soil element change with respect to a change in net normal stress and suction. k ax and k wx are coefficients of air and water permeability in x-direction (m/s), respectively; k az and k wz are coefficients of air and water permeability in z-direction (m/s), respectively. γw is the unit weight of water, that is γw = 9.8 kN/m3; g is the acceleration of gravity, that is g = 9.8 m/s2, and Sr0 and n 0 are the initial degree of saturation and initial porosity. M is the molecular mass of air, that is M = 0.029 kg/mol, ua0 is the initial excess pore-air pressure, uatm is the atmospheric pressure. R is the universal gas constant, R = 8.314 J/mol/K, and Θ is the absolute temperature, Θ = 293 K.

Bottom boundary l Fig. 1. A referential model of two-dimensional consolidation in unsaturated soils.

indicates that the dissipation of the excess pore-air and pore-water pressures can occur along simultaneously in the horizontal (x-direction) and vertical (z-direction) directions. The main assumptions for the 2D consolidation of unsaturated soils are made as follows: (1) The soil stratum is assumed to be homogeneous. (2) The flows of air and water are assumed to be continuous and independent. (3) Solid particle and water phase are incompressible; (4) Effects of environmental factors such as air diffusion and temperature change can be disregarded; (5) Deformations of a soil stratum happen along x-direction and z-direction; (6) The coefficients of permeability with respect to air and water phases and volume change for the soil remain constant throughout the consolidation process.

2.2. Initial condition The initial conditions are expressed by

ua (x ,z ,0) = ua0 (z ) = aa z + ba

(4a)

u w (x ,z ,0) = u w0 (z ) = a w z + bw

(4b)

where aa and aw are the changing rate of initial excess pore-air and pore-water pressures along the depth, ba and bw are the initial excess pore-air and pore-water pressures at the top surface. 2.3. Boundary condition

It is noted that the above assumptions are not completely accurate for all cases [14]. The coefficients of permeability with respect to air and water phases are functions of both water content and degree of saturation, and the moduli for the soil structure and water phases are non-linear. However, it may be acceptable to assume that these parameters are constant during the transient process for a small stress increment, and the corresponding assumption is made to obtain the solutions for 2D consolidation equations of unsaturated soils more easily. The governing equations for water and air phases after the application of the instantaneous loading q0 are written as follows [14]:

∂ua ∂u ∂ 2u a ∂ 2u a = −Ca w −Cvax −Cvaz ∂t ∂t ∂x 2 ∂z 2

The top and bottom boundaries are considered to be permeable or impermeable to air and water phases. The lateral drainage takes place through two side boundaries defined by vertical sand drains, so the lateral boundary is considered to be permeable to both air and water phases. The possible boundaries are as follows: Top boundary:

ua (x ,0,t ) = 0 or

∂ua (x ,0,t ) =0 ∂z

(5a)

u w (x ,0,t ) = 0 or

∂u w (x ,0,t ) =0 ∂z

(5b)

(1)

Bottom boundary:

∂u w ∂u ∂ 2u w ∂ 2u w = −Cw a −Cvwx −Cvwz ∂t ∂t ∂x 2 ∂z 2

(2)

where ua and u w are the excess pore-air and pore-water pressures; Ca and Cw are interactive constants with respect to the air and water phases, respectively; Cvax and Cvwx are the consolidation coefficients for air and water phases in x-direction; Cvaz and Cvwz are the consolidation coefficients for air and water phases in z-direction. The consolidation parameters are expressed as follows:

Ca =

Cvax =

Cvaz =

k ax RΘ gM [(ua0 + uatm )(2m1a−m2a)−n 0 (1−Sr 0 )]

u w (x ,h,t ) = 0 or

∂u w (x ,h,t ) =0 ∂z

(6b)

(7)

The method used in this paper can solve Eqs. (1) and (2) under any combination of above top, bottom and lateral drainage boundary conditions.

(3a)

3. Derivation of semi-analytical solutions (3b)

3.1. General semi-analytical solutions

k a z RΘ gM [(ua0 + uatm )(2m1a−m2a)−n 0 (1−Sr 0 )]

(6a)

Lateral boundary:

n 0 (1 − Sr 0) (ua0 + uatm )

∂ua (x ,h,t ) =0 ∂z

ua (0,z ,t ) = u w (l,z ,t ) = 0

m2a 2m1a−m2a−

ua (x ,h,t ) = 0] or

In this section, the finite Fourier and Laplace transforms are applied to obtain the semi-analytical solutions for the problem described in

(3c) 102

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Section 2. Since the studied region is a 2D finite rectangle, and the lateral boundary is permeable to both air and water phases, the corresponding finite Fourier transform should take the form of the finite sine transform. Barry and Mercer [29] gave the elemental properties of the finite sine and cosine transforms. Here the definitions and properties of the finite sine transform are rewritten as follows:

Sxn {f (x )} =

2 f (x ) = T

∫0

T

f (x )sin(λn x ) dx = f1 (n)

General solutions of four-order ordinary differential equation (14) is

u w (n,z ,s ) = C1 e ξz + C2 e−ξz + D1 e ηz + D2 e−ηz−

f1 (n)sin(λn x )

(8c)

Sxn {f ′ (x )} = −λn f0 (n)

(8d)

where T is the integral range of the finite sine and cosine transforms, λn = nπ / T , n = 0, 1, 2, … In general, the Laplace transform is defined as

∫0

−(a2 +

η=

−(a2− a22−4a1 a3 )/2a1 ,

and C1, C2, D1 and D2 are arbitrary functions of n and s, which can be determined from the boundary conditions. Taking the first and second derivatives of Eq. (15) with respect to z, respectively, leads to

(8b)

Sxn {f ″ (x )} = −λn2 f1 (n) + λn f (0)−(−1)nλn f (T )

∼ L {f (t )} = f (s ) =

a22−4a1 a3 )/2a1 ,

ξ= (8a)

n=1

∞

f (t ) e−st dt

(9)

∂u w (n,z ,s ) a = C1 ξe ξz−C2 ξe−ξz + D1 ηe ηz−D2 ηe−ηz− 4 ∂z a3

(16)

∂2u w (n,z ,s ) = C1 ξ 2e ξz + C2 ξ 2e−ξz + D1 η2e ηz + D2 η2e−ηz ∂z 2

(17)

Combining Eqs. (15) and (17) with Eq. (12) gives

where s is a complex number frequency parameter. The transformed variables are defined as

ua (n,z ,s ) = C1 a6 e ξz + C2 a6 e−ξz + D1 a7 e ηz + D2 a7 e−ηz + a8 z + a9

ua (n,z ,s ) = LSxn {ua (x ,z ,t )},

where

u w (n,z ,s ) = LSxn {u w (x ,z ,t )},

a6 =

(Cvxw λn2−s )−ξ 2Cvzw , sCw

a7 =

(Cvxw λn2−s )−η2Cvzw , sCw

a8 =

(C w λ 2−s ) a4 ⎤ 1 ⎡ a w + Cw aa− vx n , ⎥ sCw ⎢ a3 ⎦ ⎣

a9 =

(C w λ 2−s ) a5 ⎤ 1 ⎡ bw + Cw ba− vx n . ⎥ sCw ⎢ a3 ⎦ ⎣

w (n,z ,s ) = LSxn {w (x ,z ,t )}. Applying the finite sine and Laplace transforms to Eqs. (1) and (2) leads to

sua (n,z ,s )−ua0 (z ) = −Ca [su w (n,z ,s )−u w0 (z )] + Cvax λn2 ua (n,z ,s ) −Cvaz

∂2ua (n,z ,s ) ∂z 2

∂2u w (n,z ,s ) ∂z 2

Taking the first derivative of Eq. (18) with respect to z leads to

(11)

∂ua (n,h,s ) = C1 ξa5 e ξh−C2 ξa5 e−ξh + D1 ηa6 e ηh−D2 ηa6 e−ηh + a8 ∂z

Eq. (11) can be rewritten as

Cvzw

∂2u w (n,z ,s ) ua (n,z ,s ) = − + sCw ∂z 2

Cvxw λn2−s sCw

u w (n,z ,s ) +

u w0 (z )

Cw ua0 (z )

+ sCw

Taking the second derivatives of Eq. (12) with respect to z leads to

Cvzw ∂ 4u w (n,z ,s ) C w λ 2−s ∂2u w (n,z ,s ) ∂2ua (n,z ,s ) = − + vx n ∂z 2 sCw ∂z 4 sCw ∂z 2

(13) 3.2. Solutions of boundary value problems

Combining Eqs. (12) and (13) with Eq. (10) gives

∂ 4u w (n,z ,s ) ∂2u w (n,z ,s ) + a2 + a3 u w (n,z ,s ) + a4 z + a5 = 0 4 ∂z ∂z 2

Since the boundary is symmetric during the consolidation of unsaturated soils, it may be meaningful to consider homogeneous or mixed top boundary and homogeneous bottom boundary. Therefore, the other possible boundary conditions are only following four cases.

(14)

where

a1 =

Cvza Cvzw , sCw

a2 = −

a3 =

a4 =

Case 1. The top boundary is permeable to air and water phases, the bottom boundary is impermeable to air and water phases, and the lateral boundary is permeable to air and water phases. Applying the finite sine and Laplace transforms to the boundary conditions for Case 1 leads to Top boundary:

a 2 Cvza (Cvxw λn2−s ) + Cvzw (Cvx λn −s ) , sCw

a 2 (Cvx λn −s )(Cvxw λn2−s ) −sCa, sCw a 2 (Cvx λn −s )(a w

sCw

+ Cw aa )

(19)

In conclusion, Eqs. (15) and (18) are general solutions of ua and uw in the domains of the finite sine and Laplace transforms, and Eqs. (16) and (19) are the first derivative of ua and uw in the domains of the finite sine and Laplace transforms, which will be used in obtaining solutions of the boundary value problems in next section.

(12)

a1

(18)

(10)

su w (n,z ,s )−u w0 (z ) = −Cw [sua (n,z ,s )−ua0 (z )] + Cvwx λn2 u w (n,z ,s ) −Cvwz

(15)

where

∞

∑

a4 z + a5 a3

ua (n,0,s ) = 0, u w (n,0,s ) = 0

(20a)

Bottom boundary:

+ aa + Ca a w,

∂ua (n,h,s ) ∂u (n,h,s ) = 0, w =0 ∂z ∂z

(C a λ 2−s )(bw + Cw ba) a5 = vx n + ba + Ca bw. sCw

Substituting Eq. (20) into Eqs. (15), (16), (18) and (19) gives 103

(20b)

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ua (n,z ,s ) = −

a6 (χ1 + χ2 ) a7 (χ3 + χ4 ) − + a8 z + a9 cosh[ξh] cosh[ηh]

(21a)

χ + χ4 a4 z + a5 χ1 + χ2 − 3 − cosh[ξh] cosh[ηh] a3

(21b)

u w (n,z ,s ) = −

λ1 =

a3 a8 + a4 a7 (a7 ξ cosh[ηh]cosh[ξz ]−a6 ηsinh[ηh]sinh[ξz ]), ξa3 (a6−a7)

a λ2 = ⎛a9 ηsinh[ηh] + 4 a7cosh[ηh]−λ2 a7 η⎞ cosh[ξ (h−z )], a3 ⎝ ⎠ ⎜

where

λ3 =

a a + a5 a7 χ1 = 3 9 cosh[ξ (h−z )], a3 (a6−a7)

⎟

a3 a8−a4 a6 (a6 ηcosh[ξh]cosh[ηh]−a7 ξ sinh[ξh]sinh[ηh]), ηa3 (a6−a7)

a λ 4 = ⎛a9 ξ sinh[ξh] + 4 a6cosh[ξh]−λ1 a6 ξ ⎞ cosh[η (h−z )] a3 ⎝ ⎠ ⎜

a a + a4 a 7 χ2 = 3 8 sinh[ξz ], ξa3 (a6−a7)

λ5 = a6 ηcosh[ξh]sinh[ηh]−a7 ξ sinh[ξh]cosh[ηh]

a a −a a χ3 = 3 9 5 6 cosh[η (h−z )], a3 (a6−a7) χ4 =

Case 4. The top boundary is impermeable to air phase and permeable to water phase, the bottom boundary is impermeable to water and air phases, and the lateral boundary is permeable to water and air phases. Applying the finite sine and Laplace transforms to the boundary conditions for Case 4 leads to

a3 a8−a4 a6 sinh[ηz ]. ηa3 (a6−a7)

Case 2. The top, bottom and lateral boundaries are permeable to both air and water phases. Applying the finite sine and Laplace transforms to the boundary conditions for Case 2 leads to

Top boundary:

∂ua (n,0,s ) = 0, u w (n,0,s ) = 0 ∂z

Top boundary:

ua (n,0,s ) = 0, u w (n,0,s ) = 0

∂ua (n,h,s ) ∂u (n,h,s ) = 0, w =0 ∂z ∂z

(22b)

a7 (γ3 + γ4 ) a6 (γ1 + γ2) + + a8 z + a9 sinh[ξh] sinh[ηh]

u w (n,z ,s ) = −

γ + γ4 γ1 + γ2 a z + a5 + 3 − 4 sinh[ξh] sinh[ηh] a3

ua (n,z ,s ) =

a6 (−ψ1 + ψ2) + a7 (ψ3−ψ4 ) ψ5

(23a)

u w (n,z ,s ) =

−ψ1 + ψ2 + ψ3−ψ4 ψ5

(23b)

−

+ a8 z + a9

a4 z + a5 a3

(27a)

(27b)

where

where

ψ1 = λ1 [(a6 ξ cosh[ηh]cosh[ξz ]−a7 ηsinh[ηh]sinh[ξz ])],

a3 a9 + a5 a7 sinh[ξ (h−z )], γ2 a3 (a6−a7) a (a h + a9) + a7 (a4 h + a5) sinh[ξz ], = 3 8 a3 (a6−a7)

γ1 =

γ3 = −

(26b)

Substituting Eq. (26) into Eqs. (16), (18) and (19) gives

Substituting Eq. (22) into Eqs. (15) and (18) gives

ua (n,z ,s ) = −

(26a)

Bottom boundary:

(22a)

Bottom boundary:

ua (n,h,s ) = 0, u w (n,h,s ) = 0

⎟

a ψ2 = ⎡a8cosh[ηh]−a7 η ⎛λ2 + 5 sinh[ηh] ⎞ ⎤ cosh[ξ (h−z )], ⎢ a3 ⎝ ⎠⎥ ⎣ ⎦ ⎜

ψ3 = λ2 [−a6 ξ sinh[ξh]sinh[ηz ] + a7 ηcosh[ξh]cosh[ηz ]],

a3 a9−a5 a6 sinh[η (h−z )], a3 (a6−a7)

a ψ4 = ⎡a8cosh[ξh]−a6 ξ ⎛λ1 + 5 sinh[ξh] ⎞ ⎤ cosh[η (h−z )], ⎢ a3 ⎝ ⎠⎥ ⎣ ⎦ ⎜

a (a h + a9) + a6 (a4 h + a5) γ4 = 3 8 sinh[ηz ]. a3 (a6−a7)

Based on the 2D stress-state variable approach for unsaturated soils [14], the constitutive model for 2D plane strain deformation can be rewritten as

Top boundary:

∂ε v ∂u ∂u = (m2s−2m1s ) a −m2s w ∂t ∂t ∂t

where ε v is the volumetric strain, m1s = m1a + m1w , m2s = m2a + m2w . By applying the finite sine and Laplace transform to Eq. (28), we have:

(24b)

u u ε v (n,z ,s ) = (m2s−2m1sk ) ⎜⎛ua− a ⎟⎞−m2s ⎜⎛u w− w ⎞⎟ s ⎠ s ⎠ ⎝ ⎝

0

Substituting Eq. (24) into Eqs. (15), (16) and (19) gives

ua (n,z ,s ) =

a6 (λ1−λ2)−a7 (λ3−λ 4 ) + a8 z + a9 λ5

u w (n,z ,s ) =

λ1−λ2−λ3 + λ 4 a4 z + a5 − λ5 a3

(28)

(24a)

Bottom boundary:

∂ua (n,h,s ) ∂u (n,h,s ) = 0, w =0 ∂z ∂z

⎟

ψ5 = ξa6sinh[ξh]cosh[ηh]−ηa7cosh[ξh]sinh[ηh].

Case 3. The top boundary is permeable to air phase and impermeable to water phase, the bottom boundary is impermeable to both air and water phases, and the lateral boundary is permeable to both air and water phases. Applying the finite sine and Laplace transforms to the boundary conditions for Case 3 leads to

∂u w (n,0,s ) ua (n,0,s ) = 0, =0 ∂z

⎟

0

(29)

The settlement of unsaturated soil layer in the finite sine and Laplace transformed domains can be calculated by

(25a)

w (n,h,s ) = (25b)

∫0

h

ε v (n,z ,s ) dz

(30)

The solutions in the Laplace transformed domain are obtained by using the finite sine inversion transform to Eqs. (21), (23), (25), (27)

where 104

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intermediate variables become

and (30) as follows: ∞

∼ (x ,z ,s ) = 2 u a l

∑

ξ2 = −

(31)

n=1 ∞

∼ (x ,z ,s ) = 2 u w l

∼ (x ,h,s ) = 2 w l

ua (n,z ,t )sin(λn x )

∑

u w (n,z ,t )sin(λn x )

n=1

w (n,h,t )sin(λn x )

n=1

χ1 =

0 w (1−Ca Cw ) ⎡ 0 u w (s + η2Cvz ) ⎤ ua − , ⎥ ⎢ sCw Cw ⎦ ⎣

χ2 =

0 w (1−Ca Cw ) ⎡ 0 u w (s + ξ 2Cvz ) ⎤ ua − , ⎢ ⎥ sCw Cw ⎣ ⎦

χ3 =

(1−Ca Cw )(η2−ξ 2) Cvzw . (Cw )2

∞

2

⎧ ⎡⎢Cvza ⎛ 1w + ξ ⎞ ua0 + Ca uw0 ⎤⎥ cosh[ξ (h − z )] ⎫ ⎦ ⎪ ⎣ ⎝ Cvz s ⎠ ⎪ ⎪ ⎪ u0 cosh[ξh] 1 ∼ ua (z ,s ) = 2 2 a + a (η −ξ ) Cvz ⎨ ⎡C a ⎛ 1 + η2 ⎞ u 0 + C u 0 ⎤ cosh[η (h − z )] ⎬ s ⎪ ⎢⎣ vz ⎝ Cvzw s ⎠ a a w⎥⎦ ⎪ − ⎪ ⎪ cosh[ηh] ⎩ ⎭ ⎜

⎟

(34)

⎜

⎧ ⎪− ⎪ 1 ∼ (z ,s ) = u w (η2−ξ 2) Cvzw ⎨ ⎪ ⎪+ ⎩

⎟

(35)

∞

e αt j ⎡ 1 ∼ ∼ ⎛x ,z ,a + kπi ⎞ ⎞ cos kπt j w (x ,z ,a)− ∑ ⎧Re ⎛w ⎨ τ ⎢2 τ ⎠⎠ τ ⎝ ⎝ k=1 ⎩ ⎣ ⎜

⎟

∼ ⎛x ,z ,a + kπi ⎞ ⎞ sin kπt j ⎫ ⎤ −Im ⎛w ⎥ τ ⎠⎠ τ ⎬ ⎝ ⎝ ⎭⎦ ⎜

⎟

0 2 w ⎡C u 0 − uw (s + η Cvz ) ⎤ cosh[ξ (h − z )] ⎢ w a ⎥ s ⎣ ⎦ cosh[ξh]

⎫ ⎪ ⎪ u0 + w 0 w 2 s ⎡C u 0 − uw (s + ξ Cvz ) / s⎤ cosh[η (h − z )] ⎬ ⎢ w a ⎥ ⎪ s ⎣ ⎦ ⎪ cosh[ηh] ⎭

(36)

The computation program is developed, and the corresponding details of the Crump’s method can be found in Appendix A. 3.3. Verification It is noted that the governing Eqs. (1) and (2) can be degenerated to the 1D consolidation equations for unsaturated soils. For a 1D unsaturated soil stratum, the coefficients of permeability for water and air phases in the x-direction are equal to zero, and then the parameters in Eqs. (1) and (2) become Cvax = 0 and Cvwx = 0 . The corresponding parameters in Eqs. (14) and (18) become a1 = Cvza Cvzw / sCw , a3 = s (1−Ca Cw )/ Cw , a2 = (Cvza + Cvzw )/ Cw , a4 = u w0 (Ca Cw−1)/ Cw , w w 2 2 a5 = −(s + ξ Cvz )/ sCw , a6 = −(s + η Cvz )/ sCw and a7 = ua0 / s , and the

4. Example and discussion In the case study, the parameters are assumed as follows: l = 2 m,

1

1

(b) 0.8

0.6

0.6

0 101

uw/uw0

ua/ua0

(a) 0.8

0.2

Current solution Ho L.et al.[9] Degenerate solution Qin et al.[18]

102

103 104 Time (s)

0.4 0.2

105

(37b)

It is seen that Eqs. (37a) and (37b) are similar to the solutions to the 1D consolidation equations for unsaturated soils [18]. That is to say, the present semi-analytical solutions can be degenerated to the 1D condition of unsaturated soil. As shown in Fig. 2, the validation is conducted against the existing analytical solutions given by Ho et al. [9] and Qin et al. [18], respectively. In this comparison, the parameters ka/kw = 10 is adopted. It can be found the degenerate solutions from this paper show good agreement with the existing solution in literature [18], the calculating results of the present solution is consistent with that proposed by Ho et al. [9]. Therefore, it is concluded that the proposed solution is reliable and the present semi-analytical solutions are general solutions for an unsaturated soil stratum from the 1D to 2D plane strain consolidation problems.

⎟

0.4

(37a)

⎟

∼ ⎛x ,z ,a + kπi ⎞ ⎞ sin kπt j ⎫ ⎤ −Im ⎛u w ⎥ τ ⎠⎠ τ ⎬ ⎝ ⎝ ⎭⎦ w≈

⎟

⎜

e αt j ⎡ 1 ∼ ∼ ⎛x ,z ,a + kπi ⎞ ⎞ cos kπt j u w (x ,z ,a)− ∑ ⎧Re ⎛u w ⎨ τ ⎢2 τ ⎠⎠ τ ⎝ ⎝ k=1 ⎩ ⎣ ⎜

s,

At last, Eqs. (21a) and (21b) become

∞

u w (t j ) ≈

2Cvza Cvzw

s,

⎟

∼ ⎛x ,z ,a + kπi ⎞ ⎞ sin kπt j ⎫ ⎤ −Im ⎛u a ⎥ τ ⎠⎠ τ ⎬ ⎝ ⎝ ⎭⎦ ⎜

(Cvza + Cvzw )− (Cvza + Cvzw )2−Cvza Cvzw (1−Ca Cw )

(33)

e αt j ⎡ 1 ∼ ∼ ⎛x ,z ,a + kπi ⎞ ⎞ cos kπt j ua (x ,z ,a)− ∑ ⎧Re ⎛u a ⎢ ⎨ τ 2 τ ⎠⎠ τ ⎝ ⎝ k=1 ⎩ ⎣ ⎜

2Cvza Cvzw

η2 = −

∼ (x ,z ,s ) and w ∼ (x ,z ,s ) , u ∼ (x ,h,s ) In conclusion, the expressions of u w a were obtained for the 2D plane strain consolidation of unsaturated soils under four different drainage boundary conditions. By adopting the Crump’s method [30] to perform the Laplace inversion, analytical solutions of the excess pore-air and pore-water pressures and soil settlement are obtained in the time domain as follows: ua (t j ) ≈

(Cvza + Cvzw )2−Cvza Cvzw (1−Ca Cw )

(32)

∞

∑

Cvza + Cvzw +

106

0 103

Current solution Ho L.et al.[9] Degenerate solution Qin et al.[18]

104

105

106 Time (s)

107

Fig. 2. Variations in excess pore-air and pore-water pressures in 1D and 2D plane strain consolidation. 105

108

109

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m1s = − k w = 10−10 m/s, h = 5 m, n0 = 50%, Sr0 = 80%, 2.5 × 10−4 kPa−1, m1w = −0.5 × 10−4 kPa−1, m2s = −1.0 × 10−4 kPa−1, m2w = −2 × 10−4 kPa−1, ua0 = 20.0 kPa, u w0 = 40 kPa, uatm = 101.3 kPa, q0 = 100 kPa. The computational process of the initial excess pore-air and water pressures under the loading are provided in Appendix B. Based on the proposed solutions, homogeneous and mixed drainage boundaries and different initial conditions are simulated and discussed in next examples. Furthermore, the investigation is carried out to study the effects of the ratios of air to water permeability (ka/kw) and horizontal to vertical permeability (kx/kz) on changes in the excess pore-air and pore-water pressures and normalized settlement (w∗ = w / m1s q0 h ) under instantaneous loading. In addition, parametric studies are conducted to investigate the effects of depth and distance on the changes in the excess pore pressures. On the other hand, it is important to point out that when the effects of air to water permeability ratio (ka/kw), depth (z) and distance (x) are investigated on the 2D plane strain consolidation behavior under homogenous and mixed drainage boundary conditions, the initial excess pore pressures are uniform in the soil deposit (i.e. ua0 = 20.0 kPa and u w0 = 40 kPa), and the ratio of air to water permeability (ka/kw) ranges from 1 to 100, in which ka is a variable while kw keeps constant (i.e. kwx = kwz = 10−10 m/s). Similarly, the investigation is carried out to study the effects of the ratios of horizontal to vertical permeability (kx/kz), here it is assumed that kx/kz = kax/kaz = kwx/kwz, and the ratio kx/kz changes from 0.1 to 10, as kx changes while kz remains unchanged (i.e. kaz = kwz = 10−10 m/s). In investigating the 2D plane strain consolidation process under different initial conditions, it is assumed that the value of the air permeability is equal to that of water permeability (i.e. ka = kw = 10−10 m/s).

consolidation. The plateau periods of the dissipation curves of excess pore water pressure in 2D consolidation are higher than that in 1D consolidation, that is because when the plateau period of the dissipation curves of excess pore-water pressure appearance, the dissipation of excess pore-air pressure accomplished, and the excess pore-air pressure dissipates by the vertical and lateral boundaries in 2D consolidation, while the dissipation of excess pore-air only through the vertical boundary in 1D consolidation. Similar to the results from Qin et al. [18,23], a bigger value of ka/kw accelerates the dissipation of the excess pore-air pressure at the later stage whilst the excess pore-water pressure at the intermediate stage, respectively, meanwhile it lengthens the plateau period of dissipation curve of the excess pore-water pressure. At last, the dissipation curves of the excess pore-water pressure at different values of ka/kw merge into one as the excess pore-air pressure dissipates completely. Compared with the dissipation curves of the excess pore-air and pore-water pressures, there are the same influential time areas at different values of ka/kw, that is because all differences in Fig. 3 are caused by the increase of ka. Fig. 4 depicts the dissipation processes of the excess pore-air and pore-water pressures at different depths under the single drainage boundary for 1D and 2D plane strain consolidations of unsaturated soils, the permeability ratio of ka/kw = 10 is adopted in this case. Compared with the results of 1D and 2D plane strain consolidations of unsaturated soils, it is found that the dissipation of the excess pore pressures has a similar process for the 1D and 2D plane strain consolidations, and they all dissipate more slowly as the point of interest is further away from the top surface, and are prone to dissipate completely at almost the same time at different depths. What makes it different is that when the depth is bigger than 1 m, the dissipation routes of pore pressures for the 2D plane strain consolidation are close to each other, while there is a clear distinction for those of the 1D consolidation. Meanwhile, there are no obvious plateau periods in the dissipation curves of the excess pore-water pressure for 2D plane strain consolidation. That is because the dissipations of pore pressures take place simultaneously in the horizontal and vertical directions. Since the 2D plane strain consolidation is affected by the horizontal and vertical permeability at the same time, the dissipation processes of the excess pore-air and pore-water pressures are significantly influenced by the value of kx/kz. Fig. 5 depicts the dissipation process of the excess pore-air and pore-water pressures at different values of kx/kz under the single drainage boundary. Obviously, a higher value of kx/kz will result in a faster dissipation rate of pore pressures. By comparing the results of Figs. 3 and 5, it can be found that when considering the values of ka/kw and kx/kz are both 10 (i.e., kax = kaz = 10−9 m/s, kwx = kwz = 10−10 m/s in Fig. 3, and kax = kwx = 10−9 m/s, kaz = kwz = 10−10 m/s in Fig. 5), the dissipation of the excess pore-air

4.1. Consolidation under homogenous drainage boundary The homogenous drainage boundary is all permeable or impermeable to air and water phases at the top or bottom surface. Based on the semi-analytical solutions for Case 1, the variations in the excess porewater and pore-air pressures and settlement with time at x = 1 m and z = 4 m are computed for the single drainage boundary. In order to compare with the 1D consolidation for unsaturated soils, the results from Qin et al. [18] by using the same parameters are also outlined for the single drainage boundary under instantaneous loading. Fig. 3 presents the variations in the excess pore-air and pore-water pressures at different values of ka/kw under the single drainage boundary for 1D and 2D plane strain consolidations of unsaturated soils. It is found that the excess pore pressures for the 2D plane strain consolidation dissipate more quickly than those for the 1D consolidation, and the dissipation time of the excess pore pressures for the 1D consolidation is almost 25 times of that for the 2D plane strain

1

1

(b)

0.8

0.8

0.6

0.6

uw/uw0

ua/ua0

(a)

0.4 0.2 0 100

ka/kw 1D 1 10 100

101

0.4

2D

0.2

102

103 104 Time (s)

105

106

107

0 102

ka/kw 1D 1 10 100

2D

103

104

105 106 Time (s)

107

108

109

Fig. 3. Variation in the excess pore pressures with time at different values of ka/kw under homogeneous drainage boundary for instantaneous loading: (a) excess poreair pressure, and (b) excess pore-water pressure. 106

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Fig. 4. Variation in the excess pore pressures with time at different depths under homogeneous drainage boundary for instantaneous loading: (a) excess pore-air pressure, and (b) excess pore-water pressure.

1

1

(b)

0.8

0.8

0.6

0.6 uw/uw0

ua/ua0

(a)

0.4 0.2 0 101

kx/kz 0.1 1 10

102

0.4 0.2

103

104 Time (s)

105

106

0 102

107

kx/kz 0.1 1 10

103

104

105 106 Time (s)

107

108

109

Fig. 5. Variation in the excess pore pressures with time at different values of kx/kz under homogeneous drainage boundary for instantaneous loading: (a) excess poreair pressure, and (b) excess pore-water pressure.

0

0

(a)

(b) 0.2

0.4

0.4

w

w*

0.2

0.6 0.8 1 100

0.6 ka/kw 1D 1 10 100

101

102

kx/kz

2D

0.8

103

104 105 Time (s)

106

107

108

109

1 101

0.1 1 10

102

103

104

105 106 Time (s)

107

108

109

Fig. 6. Variations in settlement with time under homogeneous drainage boundary at different values of (a) ka/kw with kx/kz = 1, and (b) kx/kz with ka/kw = 10.

pressure is almost identical, whereas the dissipation of the excess porewater pressure is markedly different. This can be explained that the lateral drainage provided by vertical drains play a major role in the dissipation processes of the excess pore-air and pore-water pressures, and the results at different values of ka/kw are changed by the increase of ka, while the outcomes at different values of kx/kz are demonstrated by the changing horizontal permeability (kx).

Fig. 6(a) illustrates the changes in normalized settlement at different ratios ka/kw under the single drainage boundary for the 1D and 2D plane strain consolidations of unsaturated soil. Similar to the results of the dissipation of the excess pore pressures in Fig. 3, the higher value of ka/kw leads to an increase in the normalized settlement rate at the beginning of the consolidation. When the excess pore-air pressure dissipates completely, the dissipation curves of the excess pore-water 107

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0

0 4

2.0*10 s

0.2

7

2.0*10 s

0.2

4

7

1.0*10 s

0.4

uw/uw0

ua/ua0

1.0*10 s

4

0.5*10 s

0.6 0.8

0.2*104s 0.1*104s

1 0

0.2

0.4

0.6

0.4 0.6

0.5*107s

0.8

0.2*107s

(a)

0.8

1

x/l

1

0.1*107s

0

0.2

0.4

0.6

0.8

(b)

1

x/l

Fig. 7. Excess pore pressure isochrones with distance (x) at depth of 4 m under the single drainage boundary: (a) excess pore-air pressure, and (b) excess pore-water pressure.

Fig. 8. Variation in the excess pore pressures with time at different values of ka/kw under the single and mixed drainage boundaries for instantaneous loading: (a) excess pore-air pressure, and (b) excess pore-water pressure.

Fig. 9. Variation in the excess pore pressures with time different depths under the mixed drainage boundary for instantaneous loading: (a) excess pore-air pressure, and (b) excess pore-water pressure.

pressures with different ka/kw are the same, and the normalized settlements also occur along the same route. In addition, it takes shorter time to reach the final settlement for the 2D plane strain consolidation than that for the 1D consolidation. Fig. 6(b) shows the variation in normalized settlement at different values of kx/kz under the single drainage boundary. Since the vertical drains have a significant influence on the 2D consolidation process, as

the value of the ratio kx/kz is bigger, the normalized settlement increases more quickly. It also can be observed that the settlement curves grow continuously without obvious plateau period, that means as long as the excess pore-air pressure dissipates completely, the excess porewater pressure immediately begins to dissipate. Investigating the horizontal distribution (in direction x) of the excess pore pressures for the 2D plane strain consolidation in unsaturated

108

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1

1

(b)

0.8

0.8

0.6

0.6 uw/uw0

ua/ua0

(a)

0.4 kx/kz 0.1 1 10

0.2 0 101

102

0.4 kx/kz 0.1 1 10

0.2

103

104 Time (s)

105

106

0 101

107

102

103

104

105 106 Time (s)

107

108

109

Fig. 10. Variation in the excess pore pressures with time at different values of kx/kz under the mixed drainage boundary conditions for instantaneous loading: (a) excess pore-air pressure, and (b) excess pore-water pressure.

0

0

(b) (a)

0.2

0.2 0.4 w*

w

*

0.4

0.6

0.6 ka/kw 1D 1 10 100

0.8 1 100

101

102

kx/kz 0.1 1 10

2D

0.8

103

104 105 Time (s)

106

107

108

1 101

109

102

103

104

105 106 Time (s)

107

108

109

Fig. 11. Variations in settlement with time under the mixed drainage boundary conditions at different ratios: (a) ka/kw, and (b) kx/kz.

0

0 4

2.0*10 s

0.2

7

2.0*10 s

0.2

4

7

1.0*10 s

0.4

uw/uw0

ua/ua0

1.0*10 s

4

0.5*10 s

0.6 0.8

0.2*104s 0.1*104s

1 0

0.2

0.4

0.6

(a)

0.8

1

0.4 0.6

0.5*107s

0.8

0.2*107s

1

0.1*107s

0

0.2

x/l

0.4

0.6

0.8

(b) 1

x/l

Fig. 12. Excess pore pressure isochrones with distance (x) at depth of 4 m under the mixed drainage boundary: (a) excess pore-air pressure, and (b) excess pore-water pressure.

lateral drainage path.

soils, Fig. 7 illustrates the isochrones of the excess pore-air and porewater pressures varying with distance (x) under the single drainage boundary. It can be found that the isochrones of pore pressures are all like parabolas along the x-direction, and finally turn into a straight line, as the excess pore pressures dissipate completely. The reason is that the excess pore pressures are zero at the left (x = 0) and right (x = l) boundaries, where the vertical sand drains are installed to provide the

4.2. Consolidation under the mixed drainage boundary The mixed drainage boundary is permeable to one of air and water phases and impermeable to another one at the top or bottom surface. Eqs (25) and (27) estimate the dissipation processes of the excess pore 109

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0

0

(b)

(a) 1

2

2

z(m)

z(m)

1

3

3

4

4

Air phase Water phase

Air phase Water phase

5 0

10

20 0 u (kPa)

30

5 0

40

10

20 u0(kPa)

30

40

Fig. 13. Initial distribution of excess pore pressures: (a) rectangular, and (b) triangular.

drainage boundaries (abbreviated as sin. and mix., respectively) for the 1D and 2D plane strain consolidations of unsaturated soils. Fig. 9 present the variations in the excess pore-air and pore-water pressures at different depths under the mixed drainage boundary for the 1D and 2D plane strain consolidations of unsaturated soils. Fig. 10 depicts the dissipation processes of the excess pore-air and pore-water pressures at different ratios kx/kz under the mixed drainage boundary for the 2D plane strain consolidation of unsaturated soils. Fig. 11(a) illustrates the change in normalized settlement at different values of ka/kw under the

pressures adopting two kinds of mixed boundaries, respectively. In this section, the variations in the excess pore-water and pore-air pressures and soil settlement at x = 1 m and z = 4 m are calculated for the mixed drainage boundaries considering the semi-analytical solutions of Case 3. In addition, comparing with the results of the 1D consolidation for unsaturated soils, the results from Qin et al. [23] at z = 4 m are also presented for the mixed drainage boundary under instantaneous loading. Fig. 8 presents the variations in the excess pore-air and porewater pressures at different values of ka/kw under the single and mixed

0

0.2

0.2

0.4

0.4

z/h

z/h

0

Time (104s) 0.5 1.0 2.0 5.0 10.0

0.6 0.8 1

0

0.2

0.4

0.6

6

Time (10 s) 0.1 1.0 5.0 10.0 20.0

0.6 0.8

0.8

1

1

0

0.2

0.4

ua/ua0

0.6

0.8

1

0.8

1

uw/uw0

(a) Single drainage 0

0

0.2

0.2 6

4

z/h

z/h 0.6

0.6 0.8

0.8 1

Time (10 s) 0.1 1.0 5.0 10.0 20.0

0.4

Time (10 s) 0.5 1.0 2.0 5.0 10.0

0.4

0

0.2

0.4

ua/ua0

0.6

0.8

1

1

0

0.2

0.4

0.6

uw/uw0

(b) Double drainage Fig. 14. Isochrones of excess pore pressures with depth (z) for the rectangular distribution of initial condition under the single (a) and double (b) drainage boundaries. 110

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0

0

6

4

Time (10 s) 0.5 1.0 2.0 5.0 10.0

z/h

0.4

0.4

0.6

0.6

0.8

0.8

1

0

0.2

0.4

0.6

0.8

Time (10 s) 0.1 1.0 5.0 10.0 20.0

0.2

z/h

0.2

1

1

0

0.2

0.4

ua/ua0

0.6 uw/uw0

0.8

1

(a) Single drainage 0

0 4

Time (10 s) 0.5 1.0 2.0 5.0 10.0

z/h

0.4

0.4

0.6

0.6

0.8

0.8

1

0

0.2

0.4

0.6

0.8

Time (10 6s) 0.1 1.0 5.0 10.0 20.0

0.2

z/h

0.2

1

1

0

ua/ua0

0.2

0.4

0.6 uw/uw0

0.8

1

(b) Double drainage Fig. 15. Isochrones of excess pore pressures with depth (z) for the triangular distribution of initial condition under the single (a) and double (b) drainage boundaries.

boundary, while the excess pore-water pressure no longer dissipates under the mixed boundary for 1D consolidation, as shown in Figs. 8(b) and 9(b). In turn, after the excess pore-air pressure dissipates completely, the settlements at different ka/kw grow gradually until the final values for the 2D plane strain consolidation, while they no longer increase for 1D consolidation, as shown in Fig. 11(a). In conclusion, there are almost identical consolidation behavior for 2D plane strain unsaturated soil deposit under the single and mixed drainage boundary conditions.

mixed drainage boundary for the 1D and 2D plane strain consolidations of unsaturated soils. Fig. 11(b) shows the variation in normalized settlement at different kx/kz under the mixed drainage boundary. Finally, Fig. 12 illustrates the isochrones of the excess pore-air and pore-water pressures varying with distance (x) under the mixed drainage boundary. It can be observed that the dissipation process of the excess pore-air pressure under the mixed drainage boundary, as shown in Figs. 8(a)–12(a), are identical to those under the single drainage boundary for the 2D plane strain consolidation. The reason is that there is the same drainage boundary for air phase. Therefore, there are the same computing results of the excess pore-air pressure under the single and mixed drainage boundaries. On the other hand, the results of the excess pore-water pressure, for instance as shown in Figs. 3(b) and 8(b), are close to each other under the single and mixed drainage boundaries for the 2D plane strain consolidation, where the water phase can flow along the horizontal and vertical directions under the single drainage boundary, but the excess pore-water pressure only dissipates along the horizontal direction under the mixed drainage boundary. In addition, as the dissipation paths of excess pore pressures in the lateral direction are shorter than that in the vertical direction, the lateral drainage plays a dominant role in the 2D plane strain consolidation. Therefore, even if there is no vertical drainage path for water phase, like the mixed drainage boundary of Case 3, the dissipation process of the excess porewater pressure is analogous to that under the single drainage boundary. In addition, when the dissipation of the excess pore-air pressure is completed, the excess pore-water pressure continues to dissipate under the mixed drainage boundary, like that under the single drainage

4.3. Consolidation under different initial conditions In order to examine the effect caused by the different initial conditions, different initial conditions are outlined by changing the values of parameters aa, aw, ba and bw in Eq. (4). In this section, the rectangular and triangular distributions of the initial excess pore pressures, as shown in Fig. 13, are applied to the solutions under the single and double drainage boundaries (the solutions of Case 1 and Case 2). Figs. 14 and 15 present the isochrones of the excess pore pressures against time t with the rectangular and triangular distributions of the initial excess pore pressures, respectively. It is observed that the initial condition has a noticeable effect on the distribution of the excess pore pressures along depth. At the early stage of consolidation, the distributions of the excess pore pressures along depth are similar with the distribution shapes of the initial condition. As the excess pore pressures dissipate, the isochrones of the excess pore pressures all tend to be consistent. On the other hand, it is obvious that the maximum values of the excess pore pressures along depth are different in the process of 111

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excess pore pressure, and the changing kx/kz will affect the whole consolidation process of the excess pore pressures and normalized settlement. (3) As the pore pressures dissipates simultaneously along the top and lateral surfaces under single drainage boundary, the significantly different dissipation routes of the excess pore pressures only occur at the interest point close to the top surface. The isochrones of the excess pore pressures are all like parabolas along the x-direction, and finally turn into a straight line as the excess pore pressures dissipate completely. (4) Compared with the results of excess pore pressures for 1D and 2D plane strain consolidation under different drainage boundary, the lateral drainage boundary play a dominant role in the consolidation process for the 2D plane strain unsaturated soil deposit, and the vertical drains accelerate obviously the consolidation speed of unsaturated soils. (5) The distributions of initial condition have significant influence on the isochrones of the excess pore pressures along depth, and as the excess pore pressures dissipate, the isochrones of the excess pore pressures changes in different distribution shapes.

consolidation under the single and double drainage boundaries, and the values under the single drainage boundary should be bigger than that under the double drainage boundary. But they are close to each other in this example, which means that the different vertical drainage boundary does not have a significant effect on the dissipation of the excess pore pressures. 5. Concluding remarks In this paper, a set of semi-analytical solutions to two-dimensional plane strain consolidation equations for unsaturated soils with different initial and boundary conditions under instantaneous loading were obtained by using the finite sine and Laplace transforms. The main conclusions are summarized as follows: (1) The present semi-analytical solutions are more general compared with the previous available solutions in the literatures and can be degenerated into the solutions to the one-dimensional consolidation for unsaturated soils. In addition, the semi-analytical solutions have been obtained under the single, double and mixed drainage boundaries. (2) A bigger ratio ka/kw or kx/kz all accelerates the dissipation rate of the excess pore pressures. The changes in the excess pore pressures and normalized settlement at different values of ka/kw only occur in the same influential part of consolidation process. For the anisotropic permeability, there are obvious different dissipation routes of

Acknowledgements This research was partially supported by the National Natural Science Foundation of China (Grant Nos. 41472250, 41630633 and 11672172).

Appendix A Given a function F(s) defined for complex values of s, the routine estimates values of its inverse Laplace transform by Crump’s method [30]. Crump’s method applies the epsilon algorithm to the summation in Durbin’s Fourier series approximation [31] ∞

f (t j ) ≈

kπt j e αt j ⎡ 1 kπi ⎞ ⎞ kπi ⎞ ⎞ kπt j ⎫ ⎤ −Im ⎛F ⎛a + F (a)− ∑ ⎧Re ⎛F ⎛a + cos sin ⎥ for j = 1,2,…,n. ⎨ τ ⎢2 τ τ τ ⎠⎠ τ ⎬ ⎠⎠ ⎝ ⎝ ⎝ ⎝ ⎭⎦ k=1 ⎩ ⎣ ⎜

⎟

⎜

⎟

(A.1)

In this study, the method for taking values of parameters is explained as follows: (i) τ = t fac × max(0.01, t j ) , where t fac = 0.8; (ii) α = αb−ln(0.1 × Er )/2τ , where αb should be specified equal to, or slightly larger than the value of α , and α has two alternative interpretations: ① α is the smallest number such that |f (t )| ⩽ m × exp(αt ) for large t; ② α is the real part of the singularity of F(s) with largest real part. Er is the required relative error in the values of the inverse Laplace transform, and thus Er must be in the range 0 ≤ Er < 1.0 ; (iii) The values of tj, for j = 1, 2, …, n, must be supplied in monotonically increasing order. The routine calculates the values of the inverse function f(tj) in decreasing order of j. Appendix B Considering isotropic loading conditions, the changes in the excess pore-air and pore-water pressures in response to the applied total loads can be expressed in terms of the change in the total minor principal stress:

Δua = Ba Δσ3

(B.1)

Δu w = B w Δσ3

(B.2)

where Δua and Δu w are changes in the excess pore-air and pore-water pressures due to applied loads, respectively; Δσ3 is the change in the minor principal stresses; and Ba and B w are tangent pore pressure parameters with respect to the air and water phases under an undrained compression, respectively. The parameters of interest, Ba and B w , can be further developed as follows:

Ba =

R2s R1a−R2a 1−R1s R1a

(B.3)

Bw =

R2s−R1s R2a 1−R1s R1a

(B.4)

where

R1s =

(m2s−m1s )−

(1 − Sr + hSr ) n ua + uatm

m2s + nSr βw

(B.5a)

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R2s =

m1s m2s + nSr βw

(B.5b)

R1a =

m2a (1 − S + hS ) n a a (m2 −m1 )− u +r u r a atm

(B.5c)

R2a =

m1a (1 − S + hS ) n a a (m2 −m1 )− u +r u r a atm

(B.5d)

βw = −

1 dVw Vw du w

(B.5e)

The term h in Eq. (B.5) is the volumetric coefficient of solubility. On the other hand, the solutions for Ba and B w require an iterative procedure as they contain the absolute pore-air pressure. Note that the increase in the total stress applied on a soil may result in the reduction of matric suction under the undrained loading condition. When the total stress becomes appreciably large, all the air bubbles dissolve in water and the pore-air and porewater pressures approach a single value, indicating the matric suction equals zero. In this situation, the tangent parameters Ba and B w will approach 1 as the change in the total stress is almost similar to the change in the pore-water pressure.

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