Soil slope diseases in seasonally frozen regions are mostly related to water migration and frost heave deformation of the soil. Based on the partial differential equation defined using the COMSOL Multiphysics software, a thermohydromechanical coupling model considering water migration, icewater phase change, ice impedance, and frost heave is constructed, and the variations in the temperature field, migration of liquid water, accumulation of solid ice, and deformation of frost heave in frozen soil slopes are analysed. The results show that the ambient temperature has a significant effect on the temperature and moisture field of the slope in the shallow area. In addition, the degree of influence gradually weakens from the outside to the inside of the slope, and the number of freezethaw cycles in deep soil is less than that in shallow soil. During the freezing period, water in the unfrozen area rapidly migrates to the frozen area, and the total moisture content abruptly changes at the vicinity of the freezing front. The maximum frozen depth is the largest at the slope top and the smallest at the slope foot. During the melting period, water is enriched at the melting front with the frozen layer melting; the slope is prone to shallow instability at this stage. The melting of the frozen layer is bidirectional, so the duration of slope melting is shorter than that of the freezing process. The slope displacement is closely related to the change in temperature—a relation that is in agreement with the phenomenon of thermal expansion and contraction in unfrozen areas and reflects the phenomenon of frost heave and thaw settlement in frozen areas.
Seasonally frozen soil is a special soilwater system wherein ice and water coexist. The seasonally frozen soil region in China accounts for 53.5% of the total area of China [
The coupling of water, heat, and force is a key problem in the study of frozen soil in seasonally frozen regions and is also at the frontier of international research in this field. Throughout the years, many studies have proposed various frozen soil models. Harlan [
In this study, a THM coupling model is constructed considering water migration, icewater phase change, ice impedance, and frost heave. The coupling simulation is realized using the finiteelement analysis software COMSOL Multiphysics, and the variations in the slope temperature field, migration of liquid water, accumulation of solid ice, and deformation of frost heave are analysed under freezing and thawing environments.
Based on the main physical processes in seasonally frozen soil, several hypotheses have been proposed: the freezethaw soil medium of the slope is incompressible, homogeneous, and isotropic; there is only movement of liquid water in frozen soil, while the ice remains stationary; the effect of water vapour migration on unfrozen water and heat flow migration is ignored; the effect of water migration caused by temperature gradients and convection heat transfer is ignored; the unfrozen water content in frozen soil is in dynamic equilibrium with the negative temperature of the soil. Based on these assumptions, the THMcoupled model in seasonally unsaturated frozen soils is established.
For plane problems, the law of water migration in unsaturated frozen soils can be expressed by Richards’ equation with a phase transition [
The heat conduction equation for the latent heat of the phase transition as the internal heat source is expressed as follows [
The equilibrium equation of isotropic linear thermoelastic materials can be expressed as follows:
The Cauchy equation is expressed as
The stress continuity equation can be expressed as follows [
By using the above static equilibrium equation and Cauchy equation, the THMcoupled stresscontrol equation can be expressed as
The content of unfrozen water in the pores of the soil can be expressed as
The relation between the content of unfrozen water and the negative temperature is always in a dynamic equilibrium [
The water in soil is composed of pore ice and pore water, the volume content of unfrozen water
For frozen zones, because of the ice in the soil slope, the impedance coefficient
The impedance coefficient is mainly determined by experience, Taylor and Luthin [
Using this treatment method, the value of the impedance coefficient is very arbitrary. Chen et al. [
The saturated unfrozen water content in the frozen soil is
Equation (
The twodimensional Heaviside step function is introduced to characterise the icewater transition process during freezing. The expression of the Heaviside step function is as follows:
The Heaviside step function is shown in Figure
Heaviside step function.
Taking a typical soil slope of the Monghua Railway in a seasonally frozen region as an example, the height of the slope is 15 m and the angle of the slope is 30°; the slope lithology is mainly silty clay. The geometry of the slope is shown in Figure
Slope geometric dimensions diagram.
The diffusivity and hydraulic conductivity of unsaturated soil in the frozen area are divided by the impedance coefficient
Main computational parameters.

1000 

2.32 

918 

1.2 
L (kJ/kg)  334.56 

1.3 

3.05 

25 

0.005  G (Pa)  2.22 

3 

0.35 
COMSOL Multiphysics is a finiteelement numerical simulation software that can couple multiphysical fields and solve nonlinear differential equations. Herein, on the basis of the module of the coefficient partial differential equation in the COMSOL software, the THM control equation of frozen soil is transformed into a unified general differential equation group provided by COMSOL. Then, the temperature, humidity, and displacement fields are obtained by applying certain boundary and initial conditions.
The surface layer of a slope is mainly affected by the external ambient temperature; a periodic change in the ambient temperature causes a change in the slope surface temperature, which can be estimated using a sinusoidal curve [
For example, the variation in the average annual air temperature in Harbin, China, from 1971 to 2000, can be expressed as follows:
The sinusoidal temperature function is applied to the upper surface of the slope, i.e., the temperature load is applied simultaneously on the three sides of AB, BC, and CD in Figure
The calculated values are compared with the experimental data to verify the calculation model. The test data are obtained from a temperaturemonitoring sensor placed in the subgrade slope of the the Monghua Railway test section. The temperature and humidity probe adopts the method of excavation and layered embedding, as shown in Figure
Temperature and humidity sensor embedded in the test section.
Comparison of calculated values with test values. (a) Temperature distribution with depth; (b) water content distribution with depth.
Based on the boundary conditions, the annual minimum temperature of the external environment is −18.3°C. We analysed the maximum frozen depth of the slope at three key positions: top, waist, and foot. Figure
Maximum frozen depth at different locations: (a) slope top; (b) slope waist; (c) slope foot.
The isoline of the slope temperature field is shown in Figure
Distribution of slope temperature isoline at different times. (a)
Temperature distribution of the slope waist along vertical distance from slope surface at different times.
The temperature distribution of the slope waist along vertical distance from slope surface is shown in Figure
The temperature of point J on the slope surface varies periodically and is completely consistent with the ambient temperature (as shown in Figure
Variation in the slope temperature at different times.
With an increase in the depth of the slope, the influence of the ambient temperature on the slope is gradually weakened. The peak temperature range of points K and L changes from 23.0°C to −18.3°C of ambient temperature from 19.1°C to −11.4°C and 16.1°C to −6.3°C. Point M has a peak temperature of 10.9°C – 1.9°C, which is already in the nonfreezethaw area and is close to the initial temperature of the slope for most of the time.
The isoline of the slope moisture field is similar to that of the temperature field, as shown in Figure
Distribution of the slope moisture isoline at different times. (a)
Moisture distribution of the slope waist along vertical distance from slope surface at different times.
However, with the melting of the surface layer of the slope, water that accumulated in the surface layer during the freezing period is enriched at the melting front. Thus, there is a peak value of the water content of the slope at the melting front. Because the frozen layer still exists in the lower part of the thawing layer for a relatively long time, it is very likely that the slope will lose its stability at the freezethaw interface during this stage.
Figure
Distribution of the slope surface displacement. (a) Vertical displacement; (b) horizontal displacement.
The slope temperature fluctuates periodically with the change in seasonal ambient temperature, and the influence of ambient temperature weakens gradually from the outside to the inside of the slope. During the freezing period, the slope gradually freezes from outside to inside. The freezing front moves inward continuously, and the freezing occurs only in the shallow range of the slope soil. The maximum frozen depth of the slope is the largest at the slope top and the smallest at the slope foot. During the melting period, the melting of the frozen layer is bidirectional. Compared with the freezing process of the slope, the melting process is shorter and faster. The number of freezethaw cycles in deep soil is less than that in shallow soil.
In the shallow area of the slope, the humidity is considerably affected by the ambient temperature. During the period of freezing, the water content of the unfrozen zone moves rapidly toward the frozen zone and the total moisture content abruptly changes at the vicinity of the freezing front. In the spring thaw period, with the shallow layer melting, the water is enriched at the melting front and the slope is prone to shallow instability at the freezethaw interface.
The variation in slope displacement is closely related to the change in the ambient temperature. The displacement corresponds to the phenomena of expansion with heat and contraction with cold in the unfrozen area and presents the phenomena of frost heaving and thaw settlement in the frozen area. The effect of frost heaving and thaw settlement is obviously greater than that of thermal expansion and contraction.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This research is supported by the Youth Innovation Promotion Association CAS (2015270), Outstanding Youth Fund of Hubei Province (2017CFA056), Jilin Transportation Science and Technology Project (201618), Natural Science Foundation of China (Nos. 41472286, 41472290, and 41672312), and Science and Technology Service Network Initiative (KFJSTSZDTP037). These financial supports are gratefully acknowledged.