Thermal Effect in Nonlinear One-Dimensional Consolidation of Cold Region Soil

Abstract

The thermal effect can significantly influence the consolidation of the soil, especially in the cold region. Previous studies have established to research that the drops in the ambient temperature would slow down the consolidation process, resulting in the slow dissipation of excess pore water pressure. In addition, the previous studies neglect the final settlement because consolidation is also influenced by thermal effect. In this paper, a closed-form solution to the one-dimensional nonlinear consolidation of soil considering the thermal effect is proposed. In the mathematical framework, the influences of the thermal effect on the compression index, the permeability, and the elastic modulus of the soil are considered. The solution is fully verified by comparing it with the FDM solution neglecting the thermal effect and the classic Terzaghi’s solution. An analysis has been carried out to assess the influence of temperature, stress ratios, consolidation time, the ratio of compression index to permeability index, and the interface parameters on the consolidation process. Different from many previous studies overlooking the thermal effect on the modulus of the soil, a model has been developed which points out that the final settlement due to consolidation would vary significantly with the ambient temperature. Therefore, the thermal effect must be considered in the consolidation calculation of the freeze–thaw cycle soil in the cold region.

1. Introduction

The thermal effect has a significant influence on the consolidation of the soil [1,2,3]. Many studies reported that the increase in the temperature would accelerate the consolidation process. The thermal effect on the consolidation of soil in the cold region is often ignored [4,5]. The cold environment would not only slow down the consolidation of the cold region soil but also result in different consolidation-induced final settlements. Hence, it is essential to establish the fundamental consolidation theory to guide the consolidation analysis of the cold region soil [6,7,8].

Many publications can be found referring to the thermal effect on soil consolidation. For instance, Paaswell [9] found that the consolidation-induced settlement can vary differently with the temperature through experimental studies. Subsequently, Drnevich et al. [10] compared the temperature effect on pre-consolidation between the model test and field test. Booker and Savvidou [11] investigated the heat source based on Biot’s theories, which would cause the pore water pressure to dissipate. Later, an analytical solution was developed to solve the problem of a point heat source buried deep [12,13]. The scholars proposed the foundation treatment technology by thermal consolidation [14,15,16,17]. Meanwhile, an experimental test was carried out to justify the thermal effects on the mechanical [18]. Despite the influence of temperature on the soil constitutive model, it is necessary to investigate a half-space subjected to thermal loading [19]. Bai [20] introduced the thermos-osmosis effect and thermal filtration effect of saturated porous half-space under the variable thermal loading. Chen and Ledesma [21] used Laplace transform domain to solve the coupled T-H problem in the unsaturated clay barrier. Yu and Chen [22] investigated the importance of the coupled effect of thermally driven moisture transport that can alter the flow field in the low-permeability medium. Ai and Wang [23] applied the Laplace-Hankel transform to obtain an analytical layer element solution of the governing equations which considered axisymmetric thermal consolidation in multilayered porous thermoelastic media.

In addition to the thermal effect, the soil constitutive model and boundary conditions are also key factors influencing the consolidation process. Since Terzaghi [24] established the theory of one-dimensional linear consolidation based on Darcy’s law, scholars carried out a lot of research work on the load forms, the soil constitutive model, and drainage boundaries to satisfy various practical cases. Davis and Raymond [25] introduced the assumption that the soil permeability coefficient is constant to simplify the mathematical work. Based on the assumption, a closed-form solution was derived to solve the governing equation of the one-dimensional nonlinear consolidation which contains the logarithmic relationship between void ratio and effective stress. Chen et al. [26] further developed the layered nonlinear consolidation. Researchers proposed various solutions to incorporate the conditions e.g., layered soils under variable loading and ramp loading [27,28,29]. However, the consolidation coefficient did not always remain constant for most engineering scenes. Mesri and Rokhsar [30] developed nonlinear consolidation based on the assumption including the logarithmic relationship among void ratio, effective stress, and permeability coefficient. Subsequently, the scholars obtained different solutions through different methods such as the finite element method (FEM) and differential quadrature method (DQM) [31,32,33]. All these solutions are based on the Terzaghi drainage boundary [24]. It regarded the boundaries as fully permeable or impermeable. However, the real drainage surface lies between the permeable and the impermeable boundaries. To authentically model the drainage boundary conditions, Gay [34] developed the impeded drainage boundary. Schiffman and Stein [35] focused on the changes in soil permeability and compressibility with impeded drainage boundaries. Mesri [36] developed a closed-form solution to one-dimensional linear consolidation based on the impeded drainage boundary. It is complicated to solve the governing equation based on the nonlinear consolidation theory with impeded drainage boundary. Mei et al. [37,38] proposed the continuous drainage boundary that is convenient to obtain a closed-form solution. Zhou et al. [39] and Wang [40] introduced the continuous boundary into the unsaturated soil to obtain a semi-analytical solution. Huang et al. [41] further developed two-dimensional consolidation of unsaturated soil. Zong et al. [42] utilized the finite difference method to obtain the solution of one-dimensional nonlinear consolidation based on continuous drainage boundary.

In summary, the above-mentioned studies fail to reveal the authentic one-dimensional nonlinear consolidation considering the negative thermal effect. Hence, it is necessary to establish a rigorous nonlinear mathematical framework to guide the consolidation analysis of the soil in the cold region.

2. Mathematical Model and Assumptions

2.1. Mathematical Model

The mathematical model established is depicted in Figure 1. Only the soil deformation and the water flow that occurred in the vertical direction are considered. To account for the thermal effect, the relationships between the compressibility and permeability of the soil and the ambient temperature are considered. The surcharge load is uniformly subjected at the ground surface.

2.2. General Assumptions

The main assumption adopted in the present study are listed as

(1)

The soil is homogeneous, isotropic and fully saturated.

(2)

The volumes of soil particles are incompressible.

(3)

The deformation of the soil caused by the consolidation is small.

(4)

The seepage flow of pore water inside the soil obeys Darcy’s law.

(5)

The drainage condition is modeled by the continuous drainage boundary.

3. Governing Equations and Solutions

3.1. Governing Equation

According to Quan et al. [43], the empirical formula for the compression index can be expressed as

$$C_{\mathrm{c}}=A+\chi \frac{T}{T_0}$$

(1)

where, $T$ is the temperature, $T_0=20 °\mathrm{C}$. According to the field tests by Eriksson et al. [44], $A$ and $\chi$ are constants, which are related to the tests.

According to Mesri and Rokhsar [30], the void ratio change due to the variation of effective stress is

$$e=e_0-C_{\mathrm{c}}\lg \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}$$

(2)

$$e=e_0+C_{\mathrm{k}}\lg \frac{k_{\mathrm{v}}}{k_{\mathrm{v}0}}$$

(3)

where, $e_0$ is initial void ratio. ${\sigma }^{\prime }$ and ${{\sigma }^{\prime }}_0$ are effective stress and initial effective stress, respectively. $C_{\mathrm{k}}$ represents the penetration index. $k_{\mathrm{v}}$ and $k_{\mathrm{v}0}$ represent the permeability coefficient and initial the permeability coefficient, respectively.

Substitution Equation (1) into Equation (2), yields

$$e=e_0-\left(A+\chi \frac{T}{T_0}\right)\lg \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}$$

(4)

According to Wang et al. [45], the relationship between the permeability coefficient and the dynamic viscosity coefficient of water is

$$k_{\mathrm{v}T}=k_{\mathrm{vR}}\frac{{\eta }_{\mathrm{R}}}{{\eta }_T}$$

(5)

where, $k_{\mathrm{vR}}$ is the permeability coefficient at temperature $\mathrm{R}=20 °\mathrm{C}$. ${\eta }_{\mathrm{R}}$ and ${\eta }_T$ are the dynamic viscosity coefficients of water at temperature $T$ and $\mathrm{R}$, respectively.

According to Guo et al. [46], the relationship between the dynamic viscosity coefficient of water and the temperature is linear, which can be given as

$$\frac{{\eta }_{\mathrm{R}}}{{\eta }_{\mathrm{T}}}=\frac{T+T_0}{2T_0}$$

(6)

Substituting Equation (6) into Equation (5), yields

$$k_{\mathrm{v}T}=k_{\mathrm{vR}}\frac{T+T_0}{2T_0}$$

(7)

According to Mesri and Rokhsar [30], one can obtain

$$k_{\mathrm{vR}}=k_{\mathrm{v}0}{\left(\frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}\right)}^{\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$$

(8)

Substituting Equation (8) into Equation (7), yields

$$k_{\mathrm{v}T}=\frac{T+T_0}{2T_0}{k}_{\mathrm{v}0}{\left(\frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}\right)}^{\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$$

(9)

The consolidation coefficient can be expressed as

$$C_{\mathrm{v}}=\frac{k_{\mathrm{v}T}}{m_{\mathrm{v}}{\gamma }_{\mathrm{w}}}$$

(10)

According to Darcy’s law, the seepage is

$$v=k_{\mathrm{v}}i=-\frac{k_{\mathrm{v}T}}{{\gamma }_{\mathrm{w}}}\frac{\partial u}{\partial z}$$

(11)

where, ${\gamma }_{\mathrm{w}}$ is the unit weight of water, and $u$ presents the excess pore water pressure. z is the variable of space in the vertical direction.

Substituting Equation (9) into Equation (11) yields

$$\frac{\partial v}{\partial z}\mathrm{d}z=-\frac{\partial }{\partial z}\left[\frac{T+T_0}{2T_0}{\left(\frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}\right)}^{\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}\frac{k_{\mathrm{v}0}}{{\gamma }_{\mathrm{w}}}\frac{\partial u}{\partial z}\right]\mathrm{d}z$$

(12)

According to assumption (1), $V_{\mathrm{v}}=V_{\mathrm{w}}$; thus

$$\frac{\partial V_{\mathrm{w}}}{\partial t}=-\frac{\partial }{\partial t}\left(\frac{e}{1+e_0}\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)$$

(13)

Based on the relationship between the strain and stress, it can be expressed as

$$v=k_{\mathrm{v}}i=-\frac{k_{\mathrm{v}T}}{{\gamma }_{\mathrm{w}}}\frac{\partial u}{\partial z}$$

(14)

Substituting Equation (14) into Equation (13), yields

$$\frac{\partial V_{\mathrm{w}}}{\partial t}=\frac{0.434C_{\mathrm{c}}}{\left(1+e_1\right){\sigma }^{\prime }}\frac{\partial {\sigma }^{\prime }}{\partial t}\mathrm{d}x\mathrm{d}y\mathrm{d}z$$

(15)

The reduction in pore volume in the soil unit is equal the amount of water flowing out the unit.

$$\frac{\partial V_{\mathrm{w}}}{\partial t}\mathrm{d}t=\frac{\partial v}{\partial z}\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t$$

(16)

Combining Equations (15) and (16), yields

$$\frac{\partial v}{\partial z}=\frac{0.434C_{\mathrm{c}}}{\left(1+e_1\right){\sigma }^{\prime }}\frac{\partial {\sigma }^{\prime }}{\partial t}$$

(17)

Substituting Equation (17) into Equation (12), one obtains

$$\frac{0.434C_{\mathrm{c}}}{\left(1+e_0\right){\sigma }^{\prime }}\frac{\partial {\sigma }^{\prime }}{\partial t}=-\frac{\partial }{\partial z}\left(\frac{T+T_0}{2T_0}{\left(\frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}\right)}^{\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}\frac{k_{\mathrm{v}0}}{{\gamma }_{\mathrm{w}}}\frac{\partial u}{\partial z}\right)$$

(18)

According to the principle of the effective stress, the thermal consolidation equation can be expressed as

$$\frac{\partial u}{\partial t}=\frac{\partial }{\partial z}\left(\frac{T+T_0}{2T_0}{c}_{\mathrm{v}0}{\left(\frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}\frac{\partial u}{\partial z}\right)$$

(19)

where, $c_{\mathrm{v}0}=\frac{k_{\mathrm{v}0}\left(1+e_0\right){{\sigma }^{\prime }}_0}{0.434C_{\mathrm{c}}{\gamma }_{\mathrm{w}}}$.

3.2. Boundary Conditions

(1)

The upper and bottom boundary conditions are modelled by the continuous drainage boundary:

$$z=0,u=q_0{\mathrm{e}}^{-\alpha \frac{c_{\mathrm{v}0}}{H}t}$$

(20)

$$z=H,u=q_0{\mathrm{e}}^{-\beta \frac{c_{\mathrm{v}0}}{H}t}$$

(21)

where, $q_0$ is the surcharge load. $\alpha$ and $\beta$ are the interface parameters, which are related to the permeability of the soil [37,38]. $H$ is the thickness of the soil.

(2)

The initial condition can be express as

$$t=0,u=q_0$$

(22)

3.3. Approximate Solutions for the Governing Equations

In order to facilitate the mathematical derivation, the following assumption is introduced

$$\omega ={\left(N_{\mathrm{q}}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-{\left(\frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$$

(23)

Substituting Equation (22) into Equation (19), yields

$$\frac{\partial \omega }{\partial t}=\frac{T+T_0}{2T_0}{c}_{\mathrm{v}0}\omega \frac{{\partial }^2\omega }{\partial z^2}$$

(24)

The dimensionless parameters are introduced herein as:

$$T_{\mathrm{v}}=\frac{c_{\mathrm{v}0}}{H^2}t=\frac{B}{\left(A+\chi T/T_0\right)H^2}t,Z=\frac{z}{H},\overline{u}=\frac{u}{q_0}$$

(25)

where, $B=\frac{k_{\mathrm{v}0}\left(1+e_0\right){{\sigma }^{\prime }}_0}{0.434{\gamma }_{\mathrm{w}}}$.

Substituting Equation (25) into Equation (24), $\omega$ is replaced by the mean ${\omega }_0$.

$$\frac{\partial \omega }{\partial T_{\mathrm{v}}}=\frac{T+T_0}{2T_0}{\omega }_0\frac{{\partial }^2\omega }{\partial Z^2}$$

(26)

where, ${\omega }_0=\frac{1}{2}\left[1+N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}\right]$ [47].

The boundary condition can satisfy

$$T_{\mathrm{v}}=0,\omega ={\left(N_{\mathrm{q}}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-1$$

(27)

$$Z=0,\omega ={\omega }_{\alpha }\left(T_{\mathrm{v}}\right)$$

(28)

$$Z=1,\omega ={\omega }_{\beta }\left(T_{\mathrm{v}}\right)$$

(29)

where ${\omega }_{\alpha }\left(t\right)={\left(N_{\mathrm{q}}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-{\left[N_{\mathrm{q}}-\left(N_{\mathrm{q}}-1\right){\mathrm{e}}^{-\alpha T_{\mathrm{v}}}\right]}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$, ${\omega }_{\beta }\left(t\right)={\left(N_{\mathrm{q}}\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-{\left[N_{\mathrm{q}}-\left(N_{\mathrm{q}}-1\right){\mathrm{e}}^{-\beta T_{\mathrm{v}}}\right]}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$.

After homogenizing the boundary conditions, one obtains

$$\omega =v+Z\left[{\omega }_{\beta }\left(T_{\mathrm{v}}\right)-{\omega }_{\alpha }\left(T_{\mathrm{v}}\right)\right]+{\omega }_{\alpha }\left(T_{\mathrm{v}}\right)$$

(30)

Substituting Equation (30) into Equation (26), yields

$$\frac{\partial v}{\partial T_{\mathrm{v}}}=\frac{T+T_0}{2T_0}{\omega }_0\frac{{\partial }^{2}v}{\partial Z^2}+f\left(Z,T_{\mathrm{v}}\right)$$

(31)

where,

$$f\left(Z,T_{\mathrm{v}}\right)=Z\left(\frac{\mathrm{d}{\omega }_{\beta }\left(T_{\mathrm{v}}\right)}{\mathrm{d}{T}_{\mathrm{v}}}-\frac{\mathrm{d}{\omega }_{\alpha }\left(T_{\mathrm{v}}\right)}{\mathrm{d}{T}_{\mathrm{v}}}\right)+\frac{\mathrm{d}{\omega }_{\alpha }\left(T_{\mathrm{v}}\right)}{\mathrm{d}{T}_{\mathrm{v}}}$$

(32)

$$\frac{\mathrm{d}{\omega }_{\beta }\left(T_{\mathrm{v}}\right)}{\mathrm{d}{T}_{\mathrm{v}}}=-\beta D{\mathrm{e}}^{-\beta T_{\mathrm{v}}}\left[1+{\displaystyle \sum _{m=1}^{\infty }{C}_m}{\mathrm{e}}^{-\beta m{T}_{\mathrm{v}}}\right]$$

(33)

$$\frac{\mathrm{d}{\omega }_{\alpha }\left(T_{\mathrm{v}}\right)}{\mathrm{d}{T}_{\mathrm{v}}}=-\alpha D{\mathrm{e}}^{-\alpha T_{\mathrm{v}}}\left[1+{\displaystyle \sum _{m=1}^{\infty }{C}_m}{\mathrm{e}}^{-\alpha m{T}_{\mathrm{v}}}\right]$$

(34)

$$D=\left(1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}\right)\left(N_{\mathrm{q}}-1\right){\left(N_{\mathrm{q}}\right)}^{-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}$$

(35)

$$C_m=\frac{{(-1)}^m}{m!}{\left(1-\frac{1}{N_{\mathrm{q}}}\right)}^m\left(-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}\right)\left(-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}-1\right)\cdot \cdot \cdot \left(-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}-m+1\right)$$

(36)

The initial and boundary conditions are

$$v\left(0,Z\right)=0$$

(37)

$$v\left(0,T_{\mathrm{v}}\right)=0$$

(38)

$$v\left(1,T_{\mathrm{v}}\right)=0$$

(39)

According to the intrinsic function system, the solution of the Equation (31) can be expressed as

$$v\left(Z,T_{\mathrm{v}}\right)={\displaystyle \sum _{n=1}^{\infty }{v}_n}\left(T_{\mathrm{v}}\right)\sin \left(n\pi Z\right)$$

(40)

Substituting Equation (39) into Equation (30), yields

$${\displaystyle \sum _{n=1}^{\infty }{v^{\prime }}_n}\left(T_{\mathrm{v}}\right)\sin \left(n\pi Z\right)-{\displaystyle \sum _{n=1}^{\infty }{f}_n}\left(T_{\mathrm{v}}\right)\sin \left(n\pi Z\right)=-{\left(n\pi \right)}^2\frac{\mathrm{T}+{\mathrm{T}}_0}{2{\mathrm{T}}_0}{\omega }_0{\displaystyle \sum _{n=1}^{\infty }{v}_n}\left(T_{\mathrm{v}}\right)\sin \left(n\pi Z\right)$$

(41)

where, $f_n\left(T_{\mathrm{v}}\right)=2{\displaystyle {\int }_0^{1}f\left(Z,T_{\mathrm{v}}\right)}\sin \left(n\pi Z\right)\mathrm{d}Z$.

In other words:

$${v^{\prime }}_n\left(T_{\mathrm{v}}\right)+{f^{\prime }}_n\left(T_{\mathrm{v}}\right)+{\left(n\pi \right)}^2\frac{T+T_0}{2T_0}{\omega }_0{v}_n\left(T_{\mathrm{v}}\right)=0$$

(42)

Subsequently, applying Laplace transform to Equation (42), one obtains

$$s{v}_n\left(s\right)+{\left(n\pi \right)}^{2}M{v}_n\left(s\right)=F_n\left(s\right)$$

(43)

where, $M=\frac{T+T_0}{2T_0}{\omega }_0$, $L\left[{v^{\prime }}_n\left(T_{\mathrm{v}}\right)\right]=s{v}_n\left(s\right)-v_n\left(0\right)=s{v}_n\left(s\right)$, $L\left[v_n\left(T_{\mathrm{v}}\right)\right]=v_n\left(s\right)$, $L\left[f_n\left(T_{\mathrm{v}}\right)\right]=F_n\left(s\right)$, $L\left[ \right]$ represents the Laplace transform.

Obviously, the Equation (43) can be expressed as

$$v_n\left(s\right)=\frac{F_n\left(s\right)}{s+{\left(n\pi \right)}^{2}M}$$

(44)

Incorporating inverse Laplace transform into Equation (44), one can obtain

$$\begin{array}{l}{v}_n\left(T_{\mathrm{v}}\right)=-2\frac{{\left(-1\right)}^{n-1}}{n\pi }\beta D\frac{{\mathrm{e}}^{-\beta T_{\mathrm{v}}}-{\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}}{{\left(n\pi \right)}^{2}M-\beta }-2\frac{{\left(-1\right)}^{n-1}}{n\pi }\beta D{\displaystyle \sum _{m=1}^{\infty }{C}_m}\frac{{\mathrm{e}}^{-\beta \left(m+1\right)T_{\mathrm{v}}}-{\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}}{{\left(n\pi \right)}^{2}M-\left(m+1\right)\beta }\\ -\frac{2}{n\pi }\alpha D\frac{{\mathrm{e}}^{-\alpha T_{\mathrm{v}}}-{\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}}{{\left(n\pi \right)}^{2}M-\alpha }-\frac{2}{n\pi }\alpha D{\displaystyle \sum _{m=1}^{\infty }{C}_m}\frac{{\mathrm{e}}^{-\alpha \left(m+1\right)T_{\mathrm{v}}}-{\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}}{{\left(n\pi \right)}^{2}M-\left(m+1\right)\alpha }\end{array}$$

(45)

Substituting Equations (45) and (30) into Equation (23), the dimensionless excess pore water pressure can be expressed as

$$u={{\sigma }^{\prime }}_{\mathrm{f}}-{{\sigma }^{\prime }}_0{\left[{\left(N_q\right)}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-v\right]}^{\frac{C_{\mathrm{k}}}{C_{\mathrm{k}}-C_{\mathrm{c}}}}$$

(46)

According to the empirical formula by Li et al. [48], the relationship between the elastic modulus, the temperature, and the strain rate can be expressed by the following equation.

$$E=\left(153.59\left|T\right|+766.53\right)·{\left(\epsilon /{\epsilon }_0\right)}^m$$

(47)

where, $\epsilon$ and ${\epsilon }_0=1s^{-1}$ represent the strain rate and the dimensionless reference strain rate, respectively. $m$ is the empirical parameter. When $-15 °\mathrm{C}\le T\le -2 °\mathrm{C}$, $m=0.178$. Therefore, the soil consolidation settlement can be expressed as:

$$s=\frac{{\displaystyle {\int }_0^H{u}_0-u_t\mathrm{d}z}}{\left(153.59\left|T\right|+766.53\right)\cdot {\left(\epsilon /{\epsilon }_0\right)}^{0.178}}$$

(48)

The average degree of consolidation defined by excess pore water pressure can be established as:

$$U_{\mathrm{p}}=1-{\displaystyle {\int }_0^1\frac{u}{q_0}\mathrm{d}Z}$$

(49)

The average degree of consolidation defined by settlement can be expressed as:

$$U_{\mathrm{s}}=\frac{{\displaystyle {\int }_0^H\epsilon \mathrm{d}z}}{{\displaystyle {\int }_0^H{\epsilon }_{\mathrm{f}}\mathrm{d}z}}$$

(50)

where, $\epsilon =\frac{C_{\mathrm{c}T}}{1+e_0}\lg \left(\frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}\right)$ and ${\epsilon }_{\mathrm{f}}=\frac{C_{\mathrm{c}T}}{1+e_0}\lg \left(\frac{{{\sigma }^{\prime }}_{\mathrm{f}}}{{{\sigma }^{\prime }}_0}\right)$ are the soil strain and the maximum soil strain in vertical direction.

4. Model Verifications

4.1. Comparisons with the Solution by Quan et al.

In this section, the present solution is compared with the solution based on the Terzaghi boundary. As the interface parameters approach infinity ($\alpha =\beta =\mathrm{10,000}$), the continuous drainage boundary can be degenerated into Terzaghi drainage boundary.

$$Z=0,\omega ={\omega }_{\alpha }\left(T_{\mathrm{v}}\right)=0$$

(51)

$$Z=1,\omega ={\omega }_{\beta }\left(T_{\mathrm{v}}\right)=0$$

(52)

$$v_n\left(T_{\mathrm{v}}\right)=\left[\frac{2}{n\pi }-2\frac{{\left(-1\right)}^{n-1}}{n\pi }\right]\left(N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-1\right){\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}$$

(53)

Substituting Equation (53) into Equation (30), based on the Terzaghi double-sided drainage boundary, the $\omega$ can be obtained as

$$\omega =N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-\left(N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-1\right){\displaystyle \sum _{n=1,3,5\cdot \cdot \cdot }^{\infty }\frac{4}{n\pi }}\sin \left(n\pi Z\right){\mathrm{e}}^{-{\left(n\pi \right)}^{2}M{T}_{\mathrm{v}}}$$

(54)

Replacing the $H$ in the double-sided drainage with $2H$, the consolidation corresponding to the Terzaghi drainage condition can be obtained as

$$\omega =N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-\left(N_{\mathrm{q}}{}^{1-\frac{C_{\mathrm{c}}}{C_{\mathrm{k}}}}-1\right){\displaystyle \sum _{n=1}^{\infty }\frac{4}{\left(2n-1\right)\pi }}\sin \left(\frac{\left(2n-1\right)\pi }{2}Z\right){\mathrm{e}}^{-{\left(\frac{\left(2n-1\right)\pi }{2}\right)}^{2}M{T}_{\mathrm{v}}}$$

(55)

During the comparison with the solution by Quan et al. [43], $C_{\mathrm{c}}/C_{\mathrm{k}}=1$. In addition, the effect of temperature on the soil compressive modulus is not considered. According to the test by Eriksson et al. [44], the soil parameters are provided in

Table 1. Soil profiles given by Eriksson [44].

$\mathit{A}$	$\mathit{\chi }$	$\mathit{H}$ (m)	${\mathbf{k}}_{\mathbf{v}0} (\mathbf{m}/\mathbf{s})$	${{\mathit{\sigma }}^{\prime }}_{\mathbf{0}} \left(\mathbf{kPa}\right)$	${{\mathit{\sigma }}^{\prime }}_{\mathit{f}} \left(\mathbf{kPa}\right)$	${\mathit{e}}_0$
0.275	0.005	1	10−8	1	101	1.2

. Unless otherwise stated, the soil parameters remain the same in the following paragraphs.

A comparison of consolidation degree defined by settlement between the degenerated solution and the solution by Quan et al. [43] is shown in Figure 2. Generally, the present degradation is in good fitness with the solution by Quan et al. [43]. But the continuous drainage boundary can reflect the change of excess pore water pressure on the ground. With the increase of the temperature, the consolidation rate and the dissipation of the excess pore water pressure would increase and be accelerated.

4.2. Comparisons with the Finite Difference Solution Neglecting the Thermal Effect

To verify the rationality of the present solution, the present solution is compared with the finite difference method by Zong et al. [42], in which the thermal effect is neglected. When $T$ is equal to $T_0=20 °\mathrm{C}$, the present solution is degenerated, which means that the compressibility coefficient and permeability coefficient of the soil decrease with the decrease of the void ratio. As shown in Figure 3, the degree of consolidation defined by excess pore water pressure in present solution is in good agreement with the finite difference solution, which justifies the reliability of the present solution. It is also observed that the dissipation of the excess pore water pressure would slow down with the increase of $C_{\mathrm{c}}/C_{\mathrm{k}}$.

5. Parametric Study

To further analyze the parameters related to the model proposed by this study, a parametric study is conducted. The soil adopted herein is listed in Tabs. 1 given by Eriksson [44].

5.1. Influence of Temperature

The ambient temperature around the soil can be different at different cold regions. Therefore, it is necessary to analyze the effect on the consolidation rate under different temperature conditions. As shown in Figure 4, it presents the influence of the temperature on the excess pore water pressure and the degree of the consolidation defined by the settlement and excess pore water pressure. The growth of the temperature would increase the dissipation rate of excess pore water pressure. As the temperature increases, the compressibility index of the soil would increase, because of which the permeability of the soil would also increase. The temperature has a greater effect on the dissipation of excess pore water pressure as shown in Figure 4a. Although the time is the same, the excess pore water pressure of the drainage interface is different. This is because the temperature has an influence on the initial consolidation coefficient, resulting in an incompletely aligned drainage boundary. It is also interesting to find that the effect of temperature on the dissipation of pore pressure is especially pronounced at low temperature ($T<T_0=20 °\mathrm{C}$).

5.2. Influence of the Ratio of Finial Effective Stress to Initial Effective $N_{\mathrm{q}}$

Figure 5 depicts the distribution of the excess pore water pressure along the depth. The permeability coefficient of the soil $k_{\mathrm{v}}$ and the compress coefficient of the soil $m_{\mathrm{v}}$ would decrease with the increase of $N_{\mathrm{q}}$. When $C_{\mathrm{c}}/C_{\mathrm{k}}<1$, $k_{\mathrm{v}}$ decreases slower than $m_{\mathrm{v}}$. Therefore, the main factor influencing the soil consolidation rate is the compressive modulus. The smaller the $m_{\mathrm{v}}$, the harder the soil is to compress, that makes the rate of the soil settling slow. When the ratio of the compress index to permeability index $C_{\mathrm{c}}/C_{\mathrm{k}}$ is less than 1, the pore pressure dissipation rate increases as $N_{\mathrm{q}}$ increases. However, the opposite occurs when $C_{\mathrm{c}}/C_{\mathrm{k}}>1$. With the increase of $N_{\mathrm{q}}$, the excess pore water pressure would increase under the same boundary conditions. That is because $k_{\mathrm{v}}$ is the main factor to change the dissipation of excess pore water pressure during consolidation when $C_{\mathrm{c}}/C_{\mathrm{k}}>1$. The increase in permeability would accelerate the dissipation of excess pore water pressure.

5.3. Influence of Time

Figure 6 presents the relationship between the distribution of excess pore water pressure and the consolidation time. The excess pore water pressure would decrease dramatically with the increase in time at the drainage boundary, that is one of the biggest differences from the Terzaghi drainage boundary. As is shown in Figure 6a, when $\alpha =\beta$, under which circumstance the ground surface and the bottom surface of the soil have the same permeability, the distribution of excess pore water pressure is symmetrical in the depth of the soil layer. The increase in time would only result in the decrease of excess pore water pressure amplitudes. The average excess pore water pressure in the upper side of the soil is significantly lower than that in the lower when $\alpha >\beta$, which is suggested by Figure 6b. It is evident that the permeability of the surfaces would significantly influence the excess pore water pressure distribution along the depth. Once the ground surface has stronger permeability than the bottom surface, the excess pore water pressure near the ground would dissipate faster than that in other places. In addition, the excess pore water on the drainage boundary decreases with time, which is an important difference between continuous drainage boundary and Terzaghi boundary.

5.4. Influence of $C_{\mathrm{c}}/C_{\mathrm{k}}$

The influence of $C_{\mathrm{c}}/C_{\mathrm{k}}$ on the excess pore water and degree of the consolidation is illustrated in Figure 7. Both excess pore water pressure and settlement of the soil would decrease with the increase of $C_{\mathrm{c}}/C_{\mathrm{k}}$. According to Equation (9), the increase of permeability coefficient $k_{\mathrm{v}}$ originates from the increase of $C_{\mathrm{c}}/C_{\mathrm{k}}$. However, the compress coefficient of the soil $m_{\mathrm{v}}$ would not change with the increase of $C_{\mathrm{c}}/C_{\mathrm{k}}$. Therefore, the increase of the $C_{\mathrm{c}}/C_{\mathrm{k}}$ can only result in the increase of the permeability coefficient, that accelerated soil consolidation.

5.5. Influence of Interface Parameter $\alpha$ and $\beta$

As is shown in Figure 8, the excess pore water pressure would dissipate dramatically over the time. The rate of the soil consolidation settlement increases with the increase of the interface parameter. When $\alpha =\beta$, the maximum of the excess pore water is in the middle of the soil depth. If the interface parameters $\alpha$ and $\beta$ are different, the maximum of excess pore water pressure would be found close to the drainage boundary with the larger interface parameters.

6. Conclusions

In this paper, a one-dimensional nonlinear consolidation solution considering the thermal effect is derived. The main findings can be concluded as follows:

(1)

The temperature has a greater effect on the dissipation of excess pore water pressure. As the temperature increases, the excess pore water pressure would dissipate faster. The soil settlement would decrease with the decrease in temperature in cold regions. That is because the decrease in temperature would make the compressibility index and the permeability of the soil decrease.

(2)

When $C_{\mathrm{c}}/C_{\mathrm{k}}>1$ the consolidation rate would increase with the increase in the ratio of final effective stress to the initial effective stress. It is interesting to find that when $C_{\mathrm{c}}/C_{\mathrm{k}}<1$, the change of the dissipation of excess pore water pressure is the opposite to that when $C_{\mathrm{c}}/C_{\mathrm{k}}>1$. The larger $N_{\mathrm{q}}$ is, the faster the excess pore water would be dissipated.