# Experimental and Numerical Analyses of Freezing Behavior of an Embankment in Cold Regions

Authors: Xiaoxiao Luo; Qinguo Ma; Fujun Niu; Wenji Su; He Hu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Frost heave is the prevailing damage to the embankment in cold regions. It is a challenge to ascertain frost damage behavior of the embankment due to the complication of freezing-thawing process involving water migration, heat convection process of water, ice-water phase transition, and frost heave. To investigate the freezing behavior of the embankment, a hydro-thermo-mechanical numerical model is deduced, and an embankment model test is carried out. Finally, the moisture, temperature, and deformation during the freezing-thawing process are analyzed. The results show that (1) there exist two warm frozen layers and a frozen layer at the bottom of the embankment at the time of the minimum air temperature and at the time of the maximum thaw depth, respectively. (2) Under the drive of temperature gradient, the water migrates and the redistributions occur. The soil in the freezing-thawing front is filled with unfrozen water and ice, and its water content is high, which directly lead to frost heave. (3) The horizontal deformation at the shoulder is larger than those in other zones, which easily leads to denudation damage. Meantime, the deformation difference between the shoulder and middle will lead to the longitudinal cracks and consequently embankment failures. The study will provide a theoretical basis and reference for the design, maintenance and research of embankment in cold regions.


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## Body

## 1. Introduction

Cold region areas account for about 50% of Earth’s land surface [1]. Many embankment projects have been constructed in cold regions, such as the Qinghai-Tibet Railway and Highway. For the regions where the temperature changes to zero °C, water in the soil voids turns to ice particles with the procedure known as freezing. In the case of above zero °C, it is known as thawing. Freezing and thawing action of soil is a dynamic process coupling heat, moisture, and deformation [2, 3]. Under the action of freezing and thawing, water migrates through the soil pores toward the freezing front, leading to the increase of water content in the freezing-thawing front. This freezing-thawing cycle could adversely impose a huge modification of soil characteristics on the cold place countries annually [4–7]. In such places, soils are subjected to freezing and frosting heave in the winters and thaw settlement and weakening in the springs (Figure 1). Thus, this kind of freezing and thawing damage causes engineering problems including cracking and deformation of embankment [8, 9]. Therefore, it is important to analyze the hydro-thermo-mechanical process of embankment and investigate the frost behavior of embankment in cold regions.Figure 1
Frost heave and thaw settlement of embankment in cold regions: (a) frost heave damage and (b) thaw settlement damage.
(a) (b)To reduce frost damage and extend the service life of the embankment in cold regions, some in situ temperature and deformation states of embankment were measured and analyzed [10–12]. At the same time, model test on the embankment was also conducted to investigate the damage of embankment during freezing-thawing process [13]. They were useful for analyzing the embankment deformation. However, most of them investigated the heat and deformation of embankment separately. The deformation of embankment in cold regions is related to heave of soil, decrease in ground temperature [14–16]. Some scholars established the hydro-thermo-mechanical model to depict the stability of the embankment in cold regions. Mao et al. [17] researched the mechanical behavior of the embankment taking no account of the temperature-related mechanical parameter and moisture migration. Tai et al. [18] established a hydro-thermo coupling differential equation using the frost heave ratio of the frozen soil, but the frost heave and thaw settlement mechanism were not illuminated. Zhang et al. [19, 20] estimated the deformation of warm frozen soil by layerwise summation calculation, but the changes of moisture and temperature were incompletely coupled. Li et al. [21, 22] and Zhang et al. [23] proposed hydro-thermo-mechanical coupled models by substituting the change of moisture into heat transfer equation, respectively. However, those models ignored the heat convection process of water on heat transfer in freezing soil.Therefore, the objective of this paper is to analyze the hydro-thermo-mechanical process and investigate the frost heave mechanism of embankment in cold regions. At first, a hydro-thermo-mechanical numerical model, including moisture migration, convection process of water, and phase transition based on the balance equations of energy, mass, and momentum, is developed. Then, a model test on embankment is done in three freezing-thawing cycles and the moisture, temperature, and deformation variations of the embankment during the freezing-thawing process are analyzed. At the same time, the mathematical model is assessed by comparing the simulated results with the measured data. Finally, the freezing behavior of the embankment is investigated. From this study, the freezing-thawing process of the embankment is clarified and the susceptible deformation zone is also indicated. This study can provide a theoretical basis and reference for the design and maintenance of the embankment in cold regions.


## 2. Mathematical Model and Equations

### 2.1. Moisture Migration Equation

The generalized moisture transport equation, including ice-water phase transition during freezing, can be expressed as [24, 25](1)∂θu∂t+ρiρw∂θi∂t=∇·Dθu∇θu+∂Kθu∂ywhere θu and θi are volumetric content of unfrozen water and volumetric ice content, respectively; ρw and ρi are water density and ice density, respectively; Dθu is moisture diffusivity coefficient; Kθu denotes hydraulic conductivity;t represents time. In the unfrozen soil, θu is water content θw, but in the frozen soil it is expressed as a temperature function [26, 27].(2)θu=a1TT0b1where a1 and b1 are experimental constants; T0 is the soil freezing temperature; T0=-0.25°C in this study.


### 2.2. Heat Transfer Equation

The evaporation process of water is very little and can be ignored. Taking heat convection process of water and ice-water phase transition into account, the governing equation of heat transfer is as follows [24]:(3)cρ∂T∂t+vw·∇ρwcwT=Lρi∂θi∂t+∇·λ∇Twhere c, T, ρ, and λ are the specific heat capacity, temperature, density, and thermal conductivity of soil, respectively; vw is the seepage velocity of water; cw is the specific heat capacity of water;Lis the latent heat of ice-water phase transition.The seepage velocity of water varies with the volumetric content of unfrozen water [27]. Thus,(4)vw=-Dθu∇θu-KθuThe relationship between volumetric content of unfrozen water and temperature can be expressed as(5)∂θu∂t=∂θu∂T∂T∂tSubstituting (1), (4), and (5) into (3), it is obtained as(6)C-∂T∂t+-Dθu∇θu-Kθu·∇ρwcwT=∇λ-∇T+Lρw∂Kθu∂ywhere C- is equivalent heat capacity, and C-=cρ+Lρw∂θu/∂T; λ- is equivalent thermal conductivity, and λ-=λ+LρwDθu∂θu/∂T.


### 2.3. Stress-Strain Equation

#### 2.3.1. Equilibrium Equation

The volumetric deformation is assumed to be positive under compression condition based on the general assumption in geotechnical engineering. The local equilibrium equation can be written as(7)∂Tσ-F=0where ∂ is the differential operator matrix of strain, and ∂=∂/∂x0∂/∂y0∂/∂y∂/∂xT; σ is the total stress vector, and σ=σxσyτxyT; F is the body force vector, and F=FxFyT.


#### 2.3.2. Strain Displacement Equation

Based on linear small-deformation hypothesis of classical mechanics, the strain-displacement relation of soil can be written as(8)ε=-∂uwhere ε is the strain vector, and ε=εxεyτxyT; u is the displacement vector, and u=uxuyT.


#### 2.3.3. Constitutive Equation

An instantaneous deformation and a time-dependent deformation occur when the frozen soil is under external load. The stress-strain relationship can be given in an incremental function as follows [28, 29]:(9)Δσ=DTΔε-Δεv-Δεvpwhere Δσ is the stress increment vector; DT is the elastic matrix related to temperature, andDT=ET1-νT/1-νT1-2νT1νT/1-νT0νT/1-νT10001-2νT/21-νT; ET and νT are elastic modulus and Poisson’s ratio, respectively, which are both related to temperature [22]; Δε, Δεv, and Δεvp represent the stress increment vector, the frost heave strain increment vector, and the viscoplastic strain increment vector, respectively.Frost heave of soil is caused by the ice-water phase transition of in situ water and migrating water during the process of freezing. The increase in soil volume can be written as [30, 31](10)εv=1+βθ0+Δθ-θu+θu-nwhere β is the in situ frost heave rate; θ0 is the initial water content; Δθ is the volumetric content of migrating water; n represents the porosity of soil. Hence, frost heave strain increment under plane strain condition can be written as(11)Δεv=εv31+νT1+νT0TIn complicated stress state, the viscoplastic strain rate is given as [29](12)ε˙vp=γTϕF∂Q∂σwith(13)ϕF=0F≤0ϕF=ϕFF>0where γT is the viscosity parameter; Q is the plastic potential function; F is the yield stress. The Drucker-Prager yield rule is applied in this study, and(14)∂Q∂σ=∂F∂σ=∂F∂σx∂F∂σy∂F∂τxyTEquations (1)-(14) make up the hydro-thermo-mechanical coupled numerical model for freezing-thawing soil. The model can meet the requirement to describe the deformation behavior of the embankment in cold regions [23, 32].


## 2.1. Moisture Migration Equation

The generalized moisture transport equation, including ice-water phase transition during freezing, can be expressed as [24, 25](1)∂θu∂t+ρiρw∂θi∂t=∇·Dθu∇θu+∂Kθu∂ywhere θu and θi are volumetric content of unfrozen water and volumetric ice content, respectively; ρw and ρi are water density and ice density, respectively; Dθu is moisture diffusivity coefficient; Kθu denotes hydraulic conductivity;t represents time. In the unfrozen soil, θu is water content θw, but in the frozen soil it is expressed as a temperature function [26, 27].(2)θu=a1TT0b1where a1 and b1 are experimental constants; T0 is the soil freezing temperature; T0=-0.25°C in this study.


## 2.2. Heat Transfer Equation

The evaporation process of water is very little and can be ignored. Taking heat convection process of water and ice-water phase transition into account, the governing equation of heat transfer is as follows [24]:(3)cρ∂T∂t+vw·∇ρwcwT=Lρi∂θi∂t+∇·λ∇Twhere c, T, ρ, and λ are the specific heat capacity, temperature, density, and thermal conductivity of soil, respectively; vw is the seepage velocity of water; cw is the specific heat capacity of water;Lis the latent heat of ice-water phase transition.The seepage velocity of water varies with the volumetric content of unfrozen water [27]. Thus,(4)vw=-Dθu∇θu-KθuThe relationship between volumetric content of unfrozen water and temperature can be expressed as(5)∂θu∂t=∂θu∂T∂T∂tSubstituting (1), (4), and (5) into (3), it is obtained as(6)C-∂T∂t+-Dθu∇θu-Kθu·∇ρwcwT=∇λ-∇T+Lρw∂Kθu∂ywhere C- is equivalent heat capacity, and C-=cρ+Lρw∂θu/∂T; λ- is equivalent thermal conductivity, and λ-=λ+LρwDθu∂θu/∂T.


## 2.3. Stress-Strain Equation

### 2.3.1. Equilibrium Equation

The volumetric deformation is assumed to be positive under compression condition based on the general assumption in geotechnical engineering. The local equilibrium equation can be written as(7)∂Tσ-F=0where ∂ is the differential operator matrix of strain, and ∂=∂/∂x0∂/∂y0∂/∂y∂/∂xT; σ is the total stress vector, and σ=σxσyτxyT; F is the body force vector, and F=FxFyT.


### 2.3.2. Strain Displacement Equation

Based on linear small-deformation hypothesis of classical mechanics, the strain-displacement relation of soil can be written as(8)ε=-∂uwhere ε is the strain vector, and ε=εxεyτxyT; u is the displacement vector, and u=uxuyT.


### 2.3.3. Constitutive Equation

An instantaneous deformation and a time-dependent deformation occur when the frozen soil is under external load. The stress-strain relationship can be given in an incremental function as follows [28, 29]:(9)Δσ=DTΔε-Δεv-Δεvpwhere Δσ is the stress increment vector; DT is the elastic matrix related to temperature, andDT=ET1-νT/1-νT1-2νT1νT/1-νT0νT/1-νT10001-2νT/21-νT; ET and νT are elastic modulus and Poisson’s ratio, respectively, which are both related to temperature [22]; Δε, Δεv, and Δεvp represent the stress increment vector, the frost heave strain increment vector, and the viscoplastic strain increment vector, respectively.Frost heave of soil is caused by the ice-water phase transition of in situ water and migrating water during the process of freezing. The increase in soil volume can be written as [30, 31](10)εv=1+βθ0+Δθ-θu+θu-nwhere β is the in situ frost heave rate; θ0 is the initial water content; Δθ is the volumetric content of migrating water; n represents the porosity of soil. Hence, frost heave strain increment under plane strain condition can be written as(11)Δεv=εv31+νT1+νT0TIn complicated stress state, the viscoplastic strain rate is given as [29](12)ε˙vp=γTϕF∂Q∂σwith(13)ϕF=0F≤0ϕF=ϕFF>0where γT is the viscosity parameter; Q is the plastic potential function; F is the yield stress. The Drucker-Prager yield rule is applied in this study, and(14)∂Q∂σ=∂F∂σ=∂F∂σx∂F∂σy∂F∂τxyTEquations (1)-(14) make up the hydro-thermo-mechanical coupled numerical model for freezing-thawing soil. The model can meet the requirement to describe the deformation behavior of the embankment in cold regions [23, 32].


## 2.3.1. Equilibrium Equation

The volumetric deformation is assumed to be positive under compression condition based on the general assumption in geotechnical engineering. The local equilibrium equation can be written as(7)∂Tσ-F=0where ∂ is the differential operator matrix of strain, and ∂=∂/∂x0∂/∂y0∂/∂y∂/∂xT; σ is the total stress vector, and σ=σxσyτxyT; F is the body force vector, and F=FxFyT.


## 2.3.2. Strain Displacement Equation

Based on linear small-deformation hypothesis of classical mechanics, the strain-displacement relation of soil can be written as(8)ε=-∂uwhere ε is the strain vector, and ε=εxεyτxyT; u is the displacement vector, and u=uxuyT.


## 2.3.3. Constitutive Equation

An instantaneous deformation and a time-dependent deformation occur when the frozen soil is under external load. The stress-strain relationship can be given in an incremental function as follows [28, 29]:(9)Δσ=DTΔε-Δεv-Δεvpwhere Δσ is the stress increment vector; DT is the elastic matrix related to temperature, andDT=ET1-νT/1-νT1-2νT1νT/1-νT0νT/1-νT10001-2νT/21-νT; ET and νT are elastic modulus and Poisson’s ratio, respectively, which are both related to temperature [22]; Δε, Δεv, and Δεvp represent the stress increment vector, the frost heave strain increment vector, and the viscoplastic strain increment vector, respectively.Frost heave of soil is caused by the ice-water phase transition of in situ water and migrating water during the process of freezing. The increase in soil volume can be written as [30, 31](10)εv=1+βθ0+Δθ-θu+θu-nwhere β is the in situ frost heave rate; θ0 is the initial water content; Δθ is the volumetric content of migrating water; n represents the porosity of soil. Hence, frost heave strain increment under plane strain condition can be written as(11)Δεv=εv31+νT1+νT0TIn complicated stress state, the viscoplastic strain rate is given as [29](12)ε˙vp=γTϕF∂Q∂σwith(13)ϕF=0F≤0ϕF=ϕFF>0where γT is the viscosity parameter; Q is the plastic potential function; F is the yield stress. The Drucker-Prager yield rule is applied in this study, and(14)∂Q∂σ=∂F∂σ=∂F∂σx∂F∂σy∂F∂τxyTEquations (1)-(14) make up the hydro-thermo-mechanical coupled numerical model for freezing-thawing soil. The model can meet the requirement to describe the deformation behavior of the embankment in cold regions [23, 32].


## 3. Experimental Design

### 3.1. Experimental Equipment

The main parts of the experimental equipment are a heat-insulation box, a ventilation system, a temperature-controlling system, and a data-acquisition system, as shown in Figure2. The dimension of the box was 3.5 m × 2.0 m × 2.2 m. The modeling box was insulated from the outdoor ambient with 10 cm insulation layer. The ventilation system consisted of cooling fan, speeding fan, wind velocity controlling instrument, and a passage for wind circulation. The temperature-controlling system consisted of a double head SANYO compressor (7.5 kW), an automatic temperature controller (precision: ± 0.3°C) with a temperature sensor (precision: ± 0.1°C), Freon circulation pipes, and an evaporator. The inner ambient temperature was set near the design temperature through the auto temperature controller. The data-acquisition system was composed of temperature sensors (thermistor, precision: ± 0.05°C, operating temperature: -30-30°C), displacement sensors (manufactured by Shanghai TM Automation Instruments Co., Ltd. precision: ± 0.001 mm, measurement range: 0.001-50 mm, operating temperature: -20-80°C), three data loggers (DT500, Data taker Inc., Australia), and a desktop computer. Data were automatically collected by the data loggers at an interval of 20 min.Figure 2
Schematic of the experimental equipment.


### 3.2. Experimental Model

Figure3 is a photo of the test embankment. The embankment was designed according to related specification and an approximate similarity method [33]. The geometric size and time were 1/10 and 1/21.9 of the practical ones, respectively. The embankment structure size is illustrated in Figure 4. In order to investigate the variation characteristics of temperature and deformation, 27 temperature sensors and 1 displacement sensor were placed in the mid-cross sections of the test embankment, as shown in Figure 4. The embankment model is filled with silty clay with a dry density of 1.72 g/cm3 and volumetric water content of 28.0%. The liquid limit, plastic limit, and plasticity index of the silty clay are 33.1%, 19.2%, and 13.9, respectively. The particle size distribution was presented in Figure 5.Figure 3
A photo of the test embankment.Figure 4
Physical embankment model test and distribution of sensors (unit: m). (Note: circles indicate temperature measuring points and are named asTx, wherexdenotes the number of the temperature sensor. Inverted triangles indicate deformation measuring point and are named as Dz, where z denotes the number of the displacement sensor.)Figure 5
Particle size distribution of the silty clay.According to the long-term observed results of the in situ temperature conditions [34, 35], the ambient temperature in the modeling box was designed as (15)Tamb=-4.0+15sin⁡2πt-260400where t is the time (h) and Tamb is the ambient temperature (°C).In order to obtain uniform temperature distribution, the embankment was kept in the box for 72 h, and then the test was started and performed for three cycles. The ambient temperature in the modeling box was controlled by (15). The measured ambient temperature is shown in Figure 6.Figure 6
Ambient temperature variation in the modeling box.


## 3.1. Experimental Equipment

The main parts of the experimental equipment are a heat-insulation box, a ventilation system, a temperature-controlling system, and a data-acquisition system, as shown in Figure2. The dimension of the box was 3.5 m × 2.0 m × 2.2 m. The modeling box was insulated from the outdoor ambient with 10 cm insulation layer. The ventilation system consisted of cooling fan, speeding fan, wind velocity controlling instrument, and a passage for wind circulation. The temperature-controlling system consisted of a double head SANYO compressor (7.5 kW), an automatic temperature controller (precision: ± 0.3°C) with a temperature sensor (precision: ± 0.1°C), Freon circulation pipes, and an evaporator. The inner ambient temperature was set near the design temperature through the auto temperature controller. The data-acquisition system was composed of temperature sensors (thermistor, precision: ± 0.05°C, operating temperature: -30-30°C), displacement sensors (manufactured by Shanghai TM Automation Instruments Co., Ltd. precision: ± 0.001 mm, measurement range: 0.001-50 mm, operating temperature: -20-80°C), three data loggers (DT500, Data taker Inc., Australia), and a desktop computer. Data were automatically collected by the data loggers at an interval of 20 min.Figure 2
Schematic of the experimental equipment.


## 3.2. Experimental Model

Figure3 is a photo of the test embankment. The embankment was designed according to related specification and an approximate similarity method [33]. The geometric size and time were 1/10 and 1/21.9 of the practical ones, respectively. The embankment structure size is illustrated in Figure 4. In order to investigate the variation characteristics of temperature and deformation, 27 temperature sensors and 1 displacement sensor were placed in the mid-cross sections of the test embankment, as shown in Figure 4. The embankment model is filled with silty clay with a dry density of 1.72 g/cm3 and volumetric water content of 28.0%. The liquid limit, plastic limit, and plasticity index of the silty clay are 33.1%, 19.2%, and 13.9, respectively. The particle size distribution was presented in Figure 5.Figure 3
A photo of the test embankment.Figure 4
Physical embankment model test and distribution of sensors (unit: m). (Note: circles indicate temperature measuring points and are named asTx, wherexdenotes the number of the temperature sensor. Inverted triangles indicate deformation measuring point and are named as Dz, where z denotes the number of the displacement sensor.)Figure 5
Particle size distribution of the silty clay.According to the long-term observed results of the in situ temperature conditions [34, 35], the ambient temperature in the modeling box was designed as (15)Tamb=-4.0+15sin⁡2πt-260400where t is the time (h) and Tamb is the ambient temperature (°C).In order to obtain uniform temperature distribution, the embankment was kept in the box for 72 h, and then the test was started and performed for three cycles. The ambient temperature in the modeling box was controlled by (15). The measured ambient temperature is shown in Figure 6.Figure 6
Ambient temperature variation in the modeling box.


## 4. Numerical Model

According to the ambient temperature and the adherent layer theory [36], the temperature at the top surface and the two lateral sides varies as follows:(16)T=-2.2+15sin⁡2πt-260400The vertical displacement at the bottom side is restrained. The top surface and two lateral sides are free boundaries without restraint.The thermal parameters and physical parameters are shown in Table1. In frozen soil, ice lenses grow within soil pores and thereby impede water migration paths, and the hydraulic conductivity and diffusivity are much smaller compared with those in unfrozen state. This phenomenon is often reflected by an impedance factor,I, and the diffusivity and hydraulic coefficients can be rewritten as follows [25, 37]:(17)D=a2θub2I=a2θub21010θi(18)K=a3θub3I=a3θub31010θiWhere a2, b2, a3, and b3 are experimental coefficients and given in Table 1.Table 1
Thermal parameters and physical parameters of the silty clay.
Physical Variable λ f λ u c f c u ρ θ 0 W/(m∙°C) W/(m∙°C) J/(kg∙°C) J/(kg∙°C) kg/m3 Silty clay 1.61 1.42 1150 1510 2000 0.28  Physical Variable a 1 b 1 a 2 b 2 a 3 b 3 m2/s m/s  Silty clay 0.2 -0.26 1.78×10−6 5.14 5.61×10−5 11.27The mechanical properties of soil are dependent on the unfrozen water, ice, and temperature changes. So, the mechanical parameters in (7)-(14) can be expressed as follows [21, 22, 31, 32]:(19)ET=a4+b4Tc4T≤T0d4θu2+e4θu+f4T>T0(20)νT=a5+b5Tc5T≤T0a5T>T0(21)cT=a6+b6Tc6T≤T0d6θu2+e6θu+f6T>T0(22)φT=a7+b7Tc7T≤T0d7θu2+e7θu+f7T>T0whereai (i=4~7),bi (i=4~7),ci (i=4~7),di (i=4, 6, 7),ei (i=4, 6, 7), andfi (i=4, 6, 7) are experimental coefficients and their values are listed in Table 2.Table 2
Mechanical parameters of the silty clay.
Physical Variable a 4 b 4 c 4 d 4 e 4 f 4 a 5 b 5 c 5 a 6 b 6 MPa MPa MPa MPa kPa kPa Silty clay 24.7 12.3 0.6 -53.1 34.7 17.4 0.28 -0.007 1.0 27.8 7.0  Physical Variable c 6 d 6 e 6 f 6 a 7 b 7 c 7 d 7 e 7 f 7 kPa kPa kPa ° ° ° °  Silty clay 1.24 -101 66.5 13.8 20.0 3.4 1.1 -64.5 42.2 11.1


## 5. Results and Analyses

### 5.1. Temperature Analysis

In order to evaluate the theoretical model, the ground temperatures of the test embankment were monitored during the freezing-thawing process and the comparison between the measured and simulated temperatures is given in Figure7. Obviously, the simulated temperatures are in good agreement with the measured temperatures at the positions of T1-4, T2-5 and T3-6. The simulated and measured temperatures at the positions of T1-4, T2-5, and T3-6 show the similar variation tendency during the freezing-thawing process. The maximum differences are 2.37°C and 3.61°C at the positions of T2-5 and T3-6, respectively. However, the maximum difference at the position of T1-4 is only 1.74°C. So, the numerical model can well simulate the freezing-thawing process of the embankment model.Figure 7
Comparisons between the simulated and measured temperatures at the different positions: (a) T1-4, (b) T2-5, and (c) T3-6.
(a) (b) (c)The temperature and deformation are the main factors resulting in the stability of the embankment. So, temperature distributions, water content distributions, and deformation states at the time of the minimum air temperature and at the time of the maximum thaw depth in the last cycle are analyzed, respectively.The temperature distributions at the time of the minimum air temperature and the maximum thaw depth are given in Figures8 and 9, respectively. Compared with perfect boundary conditions of numerical model, the boundary conditions of the model test on embankment are complex and limited. Therefore, there exist some differences between the simulated temperature distribution and the measured one.Figure 8
Temperature distributions of the embankment at the time of the minimum air temperature (unit: °C): (a) simulated temperature data and (b) measured data.
(a) (b)Figure 9
Temperature distributions of the embankment at the time of the maximum thaw depth (unit: °C): (a) simulated data and (b) measured data.
(a) (b)From Figure8, it can be found that the embankment is in a frozen state. The temperature increases with depth deepening in the embankment model, and the minimum and maximum temperature exist at the surface and the bottom of the embankment model, respectively. It is noteworthy that the temperatures of two frozen layers at the bottom of the embankment model are very high and close to 0°C, which implies that there are two warm frozen layers and the warm frozen layers possibly trend to thaw over time.Meanwhile, from Figure9, it can be found that the simulated and measured maximum thaw depths are, respectively, about 0.462 m and 0.487 m, and their difference is only 0.025 m. The simulated and measured areas of frozen core are 0.099 m2 and 0.073m2, and their difference is only 0.026 m2. So, the simulated temperature distribution of the embankment is approximately similar to the measured one. The numerical model can well simulate the freezing and thawing state of the test embankment to some extent. From Figure 9(b), there are a frozen core with a minimum temperature of -0.743°C and a freezing-thawing front in the embankment model. All of the ice lenses above the freezing-thawing front in the previous frozen soil disappear completely, and large thaw settlement will be possibly induced.


### 5.2. Moisture Analysis

Under the drive of temperature gradient, water in the embankment migrates and redistributes during the freezing-thawing process. Water distributions at the time of the minimum air temperature and the maximum thaw depth are shown in Figures10 and 11, respectively. Compared with the initial volumetric water content of 28% in the embankment model, the water content is changed a lot at these two moments.Figure 10
Water content distributions of the embankment at the time of the minimum air temperature: (a) unfrozen water content, (b) ice content, and (c) total water content.
(a) (b) (c)Figure 11
Water content distributions of the embankment at the time of the maximum thaw depth: (a) unfrozen water content, (b) ice content, and (c) total water content.
(a) (b) (c)At the time of the minimum air temperature, there is lots of ice-water mixture in the embankment model. The unfrozen water, determined by negative temperature, increase with depth deepening in the embankment model. The maximum unfrozen water exists in warm frozen layers at the bottom of the embankment and the volumetric content of unfrozen water is 27% (Figure10(a)). The maximum and minimum volumetric content of ice exist at the surface of the embankment and in warm frozen layers, and their contents are 22.6% and 0.04%, respectively (Figure 10(b)). The surface of the frozen core is filled with unfrozen water and ice, and the total water content is high due to water migration. The volumetric content of total water is over 32.0% and larger than the soil porosity (Figure 10(c)), which leads to the frost heave occurring in the embankment.At the time of the maximum thaw depth, all of the ice lenses above the freezing-thawing front in the previous frozen soil disappear completely, which leads to a dramatic increase in water content at the freezing-thawing front with a maximum volumetric unfrozen water of 28.3% (Figure11(a)), which is larger than the initial water content and reduce soil shear strength. The minimum volumetric content of unfrozen water in the frozen layer decreases to 15.2% (Figure 11(a)). The layer of ice lens is forming and growing in the embankment, and the thickness of thawed soil at the mid-cross section is about 0.467 m, which is in good agreement with the frozen depths in Figure 9. The maximum volumetric content of ice exists at bottom of frozen core with the value of 13.7% (Figure 11(b)). The total water content at the freezing-thawing front is the maximum, and the volumetric content is 31.8%.


### 5.3. Deformation Analysis

The mechanical response of soil and frost heave force depends on freezing, thawing, and the redistribution of water in the embankment. To evaluate the correctness of simulated displacement, the comparison between the measured and simulated deformation at the position of D1 (see Figure4) is given in Figure 12. Obviously, the simulated deformation and the measured data have similar variation tendency during the freezing-thawing process. Although there are some deviations of the simulated deformation from the measured deformation during the freezing and thawing action, the maximum difference is only 0.54 mm. So, the simulated deformation can simulate the actual deformation state in the embankment model.Figure 12
Comparison between the simulated and measured deformation at the position of D1.The deformation states of the embankment model at the time of the minimum air temperature and the maximum thaw depth are presented in Figures13 and 14, respectively. As much water redistributes in frozen layer, the frost heave deformation is considerable. Meanwhile, the free surfaces along the top and slopes of the embankment amplify the frost heave deformation. From Figure 13, it can be found that the horizontal deformation at the slope surface is larger than those in other zones and the maximum value is 2.38 mm at the shoulder of the embankment (Figure 13(a)). The vertical deformation shows layered distribution and its maximum value is 3.65 mm at the top surface of the embankment (Figure 13(b)). It is believed that denudation damage easily occurs at the shoulder, and the embankment suffers frost damage. At the time of the maximum thaw depth, most of the ice lenses above the freezing-thawing front in the previous frozen soil transformed into liquid water completely, and the water content at the freezing-thawing front is greater than the initial water content. The strength and stability of the soil decrease and the settlement deformation of the embankment is induced compared with that at the time of the minimum air temperature. The horizontal deformation at shoulder is 0.90 mm and is much larger than those at other positions. The vertical deformation at the top surface of the embankment is 0.70 mm, and the maximum vertical deformation is only 1.44 mm at the freezing-thawing front. It means that frost heave at the time of the minimum air temperature decreases with the thawing of the frozen layer, and thaw settlement will occur.Figure 13
Deformation states at the time of the minimum air temperature (unit: mm): (a) horizontal displacement and (b) vertical displacement.
(a) (b)Figure 14
Deformation states at the time of the maximum thaw depth (unit: mm): (a) horizontal displacement and (b) vertical displacement.
(a) (b)In order to fully reveal the deformation process of the embankment, the deformations at the shoulder and middle of the embankment during the freezing-thawing process are shown in Figure15. From Figure 15, it can be found that, during the whole experiment, the deformations at these two positions varied periodically, showing a significant hysteresis. The maximum frost heave and the maximum settlement deformation appear after the time of the minimum air temperature and the time of the maximum thaw depth, respectively. The cumulative frost heave increases with time. Based on the related deformation theories [27], segregation frost heave is the main factor resulting in the heave deformation of the embankment. Meanwhile, because of the frozen core in the embankment, the deformation at the shoulder is markedly different from that at the middle of the embankment, and the maximum difference at the two positions is 1.43 mm during the freezing-thawing process. It is the difference between the deformation at the shoulder and at the middle that leads to the longitudinal crack and consequently embankment failure [38].Figure 15
The deformations at the shoulder and middle of the embankment.


## 5.1. Temperature Analysis

In order to evaluate the theoretical model, the ground temperatures of the test embankment were monitored during the freezing-thawing process and the comparison between the measured and simulated temperatures is given in Figure7. Obviously, the simulated temperatures are in good agreement with the measured temperatures at the positions of T1-4, T2-5 and T3-6. The simulated and measured temperatures at the positions of T1-4, T2-5, and T3-6 show the similar variation tendency during the freezing-thawing process. The maximum differences are 2.37°C and 3.61°C at the positions of T2-5 and T3-6, respectively. However, the maximum difference at the position of T1-4 is only 1.74°C. So, the numerical model can well simulate the freezing-thawing process of the embankment model.Figure 7
Comparisons between the simulated and measured temperatures at the different positions: (a) T1-4, (b) T2-5, and (c) T3-6.
(a) (b) (c)The temperature and deformation are the main factors resulting in the stability of the embankment. So, temperature distributions, water content distributions, and deformation states at the time of the minimum air temperature and at the time of the maximum thaw depth in the last cycle are analyzed, respectively.The temperature distributions at the time of the minimum air temperature and the maximum thaw depth are given in Figures8 and 9, respectively. Compared with perfect boundary conditions of numerical model, the boundary conditions of the model test on embankment are complex and limited. Therefore, there exist some differences between the simulated temperature distribution and the measured one.Figure 8
Temperature distributions of the embankment at the time of the minimum air temperature (unit: °C): (a) simulated temperature data and (b) measured data.
(a) (b)Figure 9
Temperature distributions of the embankment at the time of the maximum thaw depth (unit: °C): (a) simulated data and (b) measured data.
(a) (b)From Figure8, it can be found that the embankment is in a frozen state. The temperature increases with depth deepening in the embankment model, and the minimum and maximum temperature exist at the surface and the bottom of the embankment model, respectively. It is noteworthy that the temperatures of two frozen layers at the bottom of the embankment model are very high and close to 0°C, which implies that there are two warm frozen layers and the warm frozen layers possibly trend to thaw over time.Meanwhile, from Figure9, it can be found that the simulated and measured maximum thaw depths are, respectively, about 0.462 m and 0.487 m, and their difference is only 0.025 m. The simulated and measured areas of frozen core are 0.099 m2 and 0.073m2, and their difference is only 0.026 m2. So, the simulated temperature distribution of the embankment is approximately similar to the measured one. The numerical model can well simulate the freezing and thawing state of the test embankment to some extent. From Figure 9(b), there are a frozen core with a minimum temperature of -0.743°C and a freezing-thawing front in the embankment model. All of the ice lenses above the freezing-thawing front in the previous frozen soil disappear completely, and large thaw settlement will be possibly induced.


## 5.2. Moisture Analysis

Under the drive of temperature gradient, water in the embankment migrates and redistributes during the freezing-thawing process. Water distributions at the time of the minimum air temperature and the maximum thaw depth are shown in Figures10 and 11, respectively. Compared with the initial volumetric water content of 28% in the embankment model, the water content is changed a lot at these two moments.Figure 10
Water content distributions of the embankment at the time of the minimum air temperature: (a) unfrozen water content, (b) ice content, and (c) total water content.
(a) (b) (c)Figure 11
Water content distributions of the embankment at the time of the maximum thaw depth: (a) unfrozen water content, (b) ice content, and (c) total water content.
(a) (b) (c)At the time of the minimum air temperature, there is lots of ice-water mixture in the embankment model. The unfrozen water, determined by negative temperature, increase with depth deepening in the embankment model. The maximum unfrozen water exists in warm frozen layers at the bottom of the embankment and the volumetric content of unfrozen water is 27% (Figure10(a)). The maximum and minimum volumetric content of ice exist at the surface of the embankment and in warm frozen layers, and their contents are 22.6% and 0.04%, respectively (Figure 10(b)). The surface of the frozen core is filled with unfrozen water and ice, and the total water content is high due to water migration. The volumetric content of total water is over 32.0% and larger than the soil porosity (Figure 10(c)), which leads to the frost heave occurring in the embankment.At the time of the maximum thaw depth, all of the ice lenses above the freezing-thawing front in the previous frozen soil disappear completely, which leads to a dramatic increase in water content at the freezing-thawing front with a maximum volumetric unfrozen water of 28.3% (Figure11(a)), which is larger than the initial water content and reduce soil shear strength. The minimum volumetric content of unfrozen water in the frozen layer decreases to 15.2% (Figure 11(a)). The layer of ice lens is forming and growing in the embankment, and the thickness of thawed soil at the mid-cross section is about 0.467 m, which is in good agreement with the frozen depths in Figure 9. The maximum volumetric content of ice exists at bottom of frozen core with the value of 13.7% (Figure 11(b)). The total water content at the freezing-thawing front is the maximum, and the volumetric content is 31.8%.


## 5.3. Deformation Analysis

The mechanical response of soil and frost heave force depends on freezing, thawing, and the redistribution of water in the embankment. To evaluate the correctness of simulated displacement, the comparison between the measured and simulated deformation at the position of D1 (see Figure4) is given in Figure 12. Obviously, the simulated deformation and the measured data have similar variation tendency during the freezing-thawing process. Although there are some deviations of the simulated deformation from the measured deformation during the freezing and thawing action, the maximum difference is only 0.54 mm. So, the simulated deformation can simulate the actual deformation state in the embankment model.Figure 12
Comparison between the simulated and measured deformation at the position of D1.The deformation states of the embankment model at the time of the minimum air temperature and the maximum thaw depth are presented in Figures13 and 14, respectively. As much water redistributes in frozen layer, the frost heave deformation is considerable. Meanwhile, the free surfaces along the top and slopes of the embankment amplify the frost heave deformation. From Figure 13, it can be found that the horizontal deformation at the slope surface is larger than those in other zones and the maximum value is 2.38 mm at the shoulder of the embankment (Figure 13(a)). The vertical deformation shows layered distribution and its maximum value is 3.65 mm at the top surface of the embankment (Figure 13(b)). It is believed that denudation damage easily occurs at the shoulder, and the embankment suffers frost damage. At the time of the maximum thaw depth, most of the ice lenses above the freezing-thawing front in the previous frozen soil transformed into liquid water completely, and the water content at the freezing-thawing front is greater than the initial water content. The strength and stability of the soil decrease and the settlement deformation of the embankment is induced compared with that at the time of the minimum air temperature. The horizontal deformation at shoulder is 0.90 mm and is much larger than those at other positions. The vertical deformation at the top surface of the embankment is 0.70 mm, and the maximum vertical deformation is only 1.44 mm at the freezing-thawing front. It means that frost heave at the time of the minimum air temperature decreases with the thawing of the frozen layer, and thaw settlement will occur.Figure 13
Deformation states at the time of the minimum air temperature (unit: mm): (a) horizontal displacement and (b) vertical displacement.
(a) (b)Figure 14
Deformation states at the time of the maximum thaw depth (unit: mm): (a) horizontal displacement and (b) vertical displacement.
(a) (b)In order to fully reveal the deformation process of the embankment, the deformations at the shoulder and middle of the embankment during the freezing-thawing process are shown in Figure15. From Figure 15, it can be found that, during the whole experiment, the deformations at these two positions varied periodically, showing a significant hysteresis. The maximum frost heave and the maximum settlement deformation appear after the time of the minimum air temperature and the time of the maximum thaw depth, respectively. The cumulative frost heave increases with time. Based on the related deformation theories [27], segregation frost heave is the main factor resulting in the heave deformation of the embankment. Meanwhile, because of the frozen core in the embankment, the deformation at the shoulder is markedly different from that at the middle of the embankment, and the maximum difference at the two positions is 1.43 mm during the freezing-thawing process. It is the difference between the deformation at the shoulder and at the middle that leads to the longitudinal crack and consequently embankment failure [38].Figure 15
The deformations at the shoulder and middle of the embankment.


## 6. Conclusions

In order to explore the complex freezing-thawing process of the embankment in cold regions, which involves heat conduction, water migration, heat convection process of water, ice-water phase transition, and frost heave, a hydro-thermo-mechanical numerical model is deduced and an embankment model test is carried out. The temperature, moisture, and deformation of the embankment during the freezing-thawing process are analyzed. Based on the study, the following conclusions can be obtained:(1) Both the simulated temperature distributions and deformation variations of the embankment model are approximately similar to the measured ones, which mean that the hydro-thermo-mechanical numerical model can simulate the actual thermal-mechanical state of the embankment model.(2) At the time of minimum air temperature, the whole embankment is in a frozen state, and the frost heave occurs due to the in situ frost heave and segregation frost heave. At the time of the maximum thaw depth, the soil above the freezing-thawing front is in thawed state, and the settlement appears. Meanwhile, at the bottom of the embankment, there exist two warm frozen layers at the time of minimum air temperature and a frozen core at the time of the maximum thaw depth, respectively.(3) Under the drive of temperature gradient, water in the embankment migrates and redistributes. At the time of minimum air temperature, the whole embankment is in a frozen state and the volumetric content of total water in some zones is larger than the soil porosity, which results in the frost heave.(4) Due to water redistribution and high ice content in frozen layer, the frost heave deformation is considerable. At the time of the minimum air temperature, the horizontal deformation at the shoulder and the vertical deformation at the top surface of the embankment are both larger than those in other zones. At the time of maximum thaw frost, the settlement deformation occurs. The horizontal deformation at the shoulder and the vertical deformation at the freezing-thawing front are both larger than those in other zones. So, it is believed that denudation damage easily occurs at the shoulder. Meanwhile, the deformation at the shoulder is markedly different from that at the middle of the embankment, and it is the difference that leads to the longitudinal crack and consequently embankment failure.



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*Source: 1437904-2019-03-10.xml*