Influence of Convection Term on Temperature Field during Soil Freezing

Abstract

In recent years, the numerical model of frozen soil has mainly focused on the water-heat equation of unsaturated soil, which is of great significance to predicting engineering stability in frozen soil areas. In these numerical models, the change of freezing temperature is usually ignored, and 0 °C is often used as the freezing condition. In addition, most equations only consider the effect of latent heat released during water freezing on the frozen soil and ignore the effect of high-temperature water on the heat transfer of frozen soil when unfrozen water migrates to the frozen zone. Therefore, there will be deviations under long-time simulation. At the same time, due to excessive attention to the moisture field and neglect of the selection of the temperature field, there is no clear conclusion on when to choose the heat transfer equation with the convection term. The equation in this paper considered the change of freezing temperature under different initial saturation conditions and the convection effect during moisture migration. Through COMSOL 5.5 software simulation, they were combined with the experiment to verify. Three different hydrothermal equations were selected to analyze the effects of latent heat and the effects of convection on the temperature field under different initial saturation conditions. The results show that the convection term plays an essential role in the heat transfer equation for unsaturated soil with high initial saturation. Additionally, the frost heave occurs mainly above the ice front interface. This study provides a reference for when to choose the heat transfer equation with convective terms and can provide help for the construction and prediction of frozen soil in the future.

1. Introduction

The frozen soil area in China is vast, comprising 10% of the frozen soil resources in the world [1]. As a unique soil with rheological properties, frozen soil undergoes a freeze-thaw phase transition process caused by changes in external temperature, affecting the frozen soil’s mechanical structure [2]. In recent years, with the global temperature rising and showing a temporary irreversible trend [3], the occurrence of freezing and thawing in permafrost regions has become more intense. Furthermore, climate change has led to the gradual increase of thermokarst ponds [4], an increase in rainfall [5], and the intensified melting of glaciers and snow [6]. These natural environment changes increase the moisture content in the frozen soil, thus affecting more intense physical changes in the frozen soil [7]. Therefore, a more intense moisture migration effect will happen in the hydrothermal process with the ice-water phase transition of the practically frozen soil (unsaturated frozen soil). Therefore, the latent heat released by the soil during the freezing process and the heat transfer during the migration of high-temperature water to the freezing area will be amplified. The result is reflected in the freezing and thawing process of the soil, which will eventually evolve into engineering freeze-thaw hazards and ecological and environmental evolution problems. Therefore, the equation of moisture and heat of frozen soil is essential for predicting frozen soil. At present, there have been many studies on hydrothermal soil equations [8,9], but they have all focused on the field of moisture. Therefore, there needs to be a clear conclusion on whether the heat transfer equation should consider the latent heat term or whether to consider the convection term. Therefore, it is of great significance to pay attention to the study of the latent heat term and convection term in the heat transfer equation for the prediction of engineering stability, ecological environment evolution, and water conservation capacity evolution in frozen soil regions [10,11].

In the current numerical model, the frozen soil is generally considered saturated. In contrast, the soil in practical engineering is always unsaturated, and the moisture migration mechanism of the unfrozen moisture in the commonly frozen soil is considered to be similar to that in the unsaturated soil [9]. For example, Harlan [12] first proposed the hydrothermal equation in the freezing and thawing process, which describes the moisture migration mechanism and the latent heat effect of ice-water phase transition in the process without considering the discontinuous ice lens and upload pressure. Mao [13] was the first to apply the sensible heat capacity (solid phase increment method) to the temperature control equation to describe the latent heat effect caused by ice-water phase change. He used Darcy’s law to describe the water transfer mechanism to solve the frozen soil hydrothermal equation. However, most of these coupling theoretical mechanisms [14,15] only focus on the influence of the latent heat released during water freezing on the frozen soil temperature field and ignore the heat transfer between water and soil during the migration of unfrozen high-temperature water to the frozen area. Studies [16,17] have shown that the moisture migration process of saturated soil significantly impacts the temperature field results. Therefore, significant moisture migration must be accompanied by more vital latent heat and convection effects. Wei et al. [18] calculated the effect of convective heat on the freezing and thawing rate of saturated frozen soil under ideal conditions by COMSOL, and the results show that the effect of the convective term is significant. Hu et al. [19] calculated the temperature field calculation of saturated frozen soil by self-compiling with OpenFOAM software. By comparing the experimental results, it was found that the temperature control equation of frozen soil with convection term is more accurate than the experimental results. However, these studies have focused on saturated frozen soil and unsaturated frozen soil with low moisture content. There is a lack of discussion on unsaturated soil with high saturation. Therefore, there needs to be a more quantitative analysis of the influence of convection term on the temperature field during the freezing of unsaturated frozen soil with high temperature and moisture content.

It is generally believed that the unfrozen moisture content is related to temperature, and the relevant power function is obtained according to the experimental fitting. Many numerical calculations directly use this correlation when constructing the water-heat relationship, and the temperature is set to 0 °C while ignoring the effect of freezing temperature on the freezing process [20]. However, the actual freezing temperature varies with different soil conditions. Osterkamp et al. [21] found in their experiment that the freezing temperature of frozen soil with different initial moisture content will change, so they added the soil freezing temperature parameter to the hydrothermal correlation equation and improved it. Mckenzie et al. [22] believed that when the freezing temperature approached 0 °C, the unfrozen moisture content would have a singular value, so a continuous exponential correlation between the unfrozen moisture content and the initial moisture content and the freezing temperature was constructed. Zhang et al. [23] believed that the freezing temperature of the soil is related to the properties of the soil itself, particle size, the specific surface area of pores, initial liquid content, and liquid properties. Zhou et al. [24] believed that the freezing temperature of saturated and supersaturated frozen soil did not change much. In contrast, the freezing temperature of unsaturated frozen soil decreased with the moisture content. Therefore, it is not sufficient to treat the freezing temperature as a fixed value (0 °C) in the calculation of the freezing process of unsaturated permafrost. It can be seen that the soil freezing temperature is the key to solving the unfrozen moisture content, and it will also have a certain impact on the final temperature field.

In this paper, based on the seepage theory and heat transfer theory of unsaturated frozen soil, the influence of latent heat term and convection term on the temperature field is considered in the temperature control equation. The Richard equation [25] is used in the moisture control equation to describe the moisture migration in the unsaturated frozen soil process. Considering the change of freezing temperature under different initial saturation, combined with the “solid-liquid ratio” theory proposed by Bai [26], the relationship between water and heat is constructed. Through the PDE (different partial equation) mathematical module in the COMSOL Multiphysics software, the weak solution form of the established mathematical partial differential equation is solved. The experimental data and simulation results are used to verify the hydrothermal coupling equations set up in this paper. Then, the model of this paper is used to quantitatively analyze the temperature field changes during the soil freezing process under different initial saturation. The effects of latent heat terms and convection terms are discussed in the soil freezing process with different saturation. This paper can provide a particular reference for engineering design and numerical simulation calculation research of environmental evolution in permafrost regions.

The parameters used in this article are shown in

Table 1. Nomenclature.

Parameter	Unit	Definition	Parameter	Unit	Definition
${\rho }_{s\left(\theta \right)}$	$(\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3})$	Comprehensive density of soil	$S$		Relative saturation of frozen soil
${\rho }_i$	$(\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3})$	Density of ice	${\theta }_r$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Residual moisture content
${\rho }_w$	$(\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3})$	Density of water	${\theta }_s$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Saturated moisture content
${\rho }_s$	$(\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3})$	Density of soil	$a$		Empirical constants related to soil properties
$c_{s\left(\theta \right)}$	$(\mathrm{k}\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{K}}^{-1})$	Comprehensive specific heat capacity of soil	$b$		Empirical constants related to soil properties
$c_i$	$(\mathrm{k}\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{K}}^{-1})$	Specific heat capacity of ice	$T_f$	$(\mathrm{K})$	Freezing temperature of soil
$c_w$	$(\mathrm{k}\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{K}}^{-1})$	Specific heat capacity of water	$T$	$(\mathrm{K})$	Instantaneous temperature of soil
$c_s$	$(\mathrm{k}\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{K}}^{-1})$	Specific heat capacity of soil	$t$	$(\mathrm{s})$	Time
${\lambda }_{}$	$(\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1})$	Comprehensive thermal conductivity of soil	$w_p$	%	Plastic limit moisture content
${\lambda }_i$	$(\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1})$	Thermal conductivity of ice	$w$	%	Total moisture content
${\lambda }_w$	$(\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1})$	Thermal conductivity of water	$B_i$		Solid-liquid ratio
${\lambda }_s$	$(\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1})$	Thermal conductivity of soil	S0		Initial saturation of soil
$L$	$\mathrm{k}\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Latent heat of ice-water phase change	${\theta }_0$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Initial volume moisture content
$\theta$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Volume content of moisture	A		Literature [27]
${\theta }_i$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Volume content of pore ice	B		Literature [27]
${\theta }_u$	$(\mathrm{c}{\mathrm{m}}^3\cdot \mathrm{c}{\mathrm{m}}^{-3})$	Volume content of unfrozen water	$k_{\left({\theta }_u\right)}$	$(\mathrm{m}\cdot {\mathrm{s}}^{-1})$	Permeability of unsaturated soil [28]
$D_{\left({\theta }_u\right)}$	$({\mathrm{m}}^2\cdot {\mathrm{s}}^{-1})$	Diffusivity of moisture in frozen soil	$c_{\left({\theta }_u\right)}$	$1\cdot {\mathrm{m}}^{-1}$	Specific moisture capacity
∇		Laplacian	$I$		Impedance factor [29]
$k_{g\left({\theta }_u\right)}$	$(\mathrm{m}\cdot {\mathrm{s}}^{-1})$	Permeability coefficient of unsaturated soil in the direction of gravity acceleration	$a_0$		Constitutive parameters of soil mass
$k_s$	$(\mathrm{m}\cdot {\mathrm{s}}^{-1})$	Permeability coefficient of saturated soil	$l$		Constitutive parameters of soil mass
$u_w$	$(\mathrm{m}\cdot {\mathrm{s}}^{-1})$	Migration velocity of moisture	$m$		Constitutive parameters of soil mass

.

2. Model Processing and Selection

This section mainly relates to the derivation of the equation. Because the relationship between unfrozen water and temperature is not well described, the solution of the formula has a singular value, such as Equation (10), so the “solid-liquid ratio” was introduced. At the same time, the solid-liquid ratio equation was improved, and the freezing temperature varying with the initial saturation was introduced. Finally, a weak solution form of a partial differential equation with only two unknowns, B_i and S, was derived.

The following assumptions are made for the water-heat coupling model of frozen soil:

(1)

Frozen soil is regarded as an isotropic continuum.

(2)

The influence of salinity in frozen soil is ignored.

(3)

During the freezing process, moisture migration only considers the part that migrates in liquid form.

(4)

Only the convection caused by the migration of liquid moisture is considered.

2.1. Temperature Field Equation

Based on the solid heat transfer model, considering the thermal parameters of the soil medium [30], the latent heat term of ice-water phase transition [31], and the convection term between the water and soil during the migration of high-temperature water to the frozen area [32]. Governing equation:

$${\rho }_s{c}_s\frac{\partial T}{\partial t}=\nabla [\lambda \cdot \nabla T]+L\cdot {\rho }_i\frac{\partial {\theta }_i}{\partial t}-{\rho }_w{c}_w\nabla \left(u_w\cdot T\right)$$

(1)

The constitutive equation of the temperature field is as follows:

$${\rho }_s{c}_s={\theta }_u\cdot {\rho }_w\cdot c_w+{\theta }_i\cdot {\rho }_i\cdot c_i+(1-{\theta }_s)\cdot {\rho }_s\cdot c_s$$

(2)

$$\lambda ={\theta }_u\cdot {\lambda }_{ w}+{\theta }_i\cdot {\lambda }_{ i}+(1-{\theta }_s)\cdot {\lambda }_{ s}$$

(3)

2.2. Temperature Field Equation

In the process of moisture migration, the Richard equation [25] suitable for moisture migration in unsaturated frozen soil was selected, and the hindering effect of ice crystals on moisture migration in the process of water-ice phase transition was considered [33]. As a result, the moisture control equation of the moisture migration process in the phase transition process is as follows:

$$\frac{\partial \theta }{\partial t}=\frac{\partial {\theta }_u}{\partial t}+\frac{{\rho }_i}{{\rho }_w}\cdot \frac{\partial {\theta }_i}{\partial t}=\nabla \left[D_{\left({\theta }_u\right)}\cdot \nabla {\theta }_u+k_{g\left({\theta }_u\right)}\right]$$

(4)

Converting ice into an equivalent volume of moisture yields the following equation:

$$\theta ={\theta }_u+{\rho }_i/{\rho }_w\cdot {\theta }_i$$

(5)

Due to the surface activation energy of soil particles, there is always a portion of water that cannot be frozen. This portion is called residual moisture content. Then, the relative saturation is:

$$S=\frac{{\theta }_u-{\theta }_r}{{\theta }_s-{\theta }_r}$$

(6)

The flow rate under the unit gradient in the equation is expressed as follows [25]:

$$k_{\left({\theta }_u\right)}=k_s\cdot S^l{\left(1-{\left(1-S^{1/m}\right)}^m\right)}^2$$

(7)

The moisture content change caused by matrix potential change is as follows [25]:

$$c_{\left({\theta }_u\right)}=a_{0}m/\left(1-m\right)\cdot S^{1/m}{\left(1-S^{1/m}\right)}^m$$

(8)

The retarding effect of porous ice on the migration of unfrozen water is as follows [33]:

$$I={10}^{-10{\theta }_i}$$

(9)

2.3. Ice-Water Phase Transition Relationship

Since the above-mentioned hydrothermal equations have two equations, and we need to solve three unknowns, it is necessary to introduce one equation to establish an equation system of three equations and three unknowns so that the equation has a unique solution. Usually, a functional relationship between the three is established [34]:

$${\theta }_u=a{\left|T-T_f\right|}^{-b}$$

(10)

However, when the formula is used, the unfrozen moisture content will tend to be infinite when the soil temperature is very close to the freezing temperature. Therefore, the model in this paper adopts the formula and combines the concept of the “solid-liquid ratio” proposed by Bai [29] to establish a function between the water and temperature relationship, which avoids this problem.

$$B_i=\frac{{\theta }_i}{{\theta }_u}=\left\{\begin{array}{c}\frac{{\rho }_w}{{\rho }_i}\cdot {\frac{{\theta }_0\left|T-T_f\right|}{A}}^B-\frac{{\rho }_w}{{\rho }_i}(T<T_f)\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(T\ge T_f)\end{array}\right.$$

(11)

Generally, the freezing temperature of the soil is determined by the experimental model of Kozlowski [35]. In addition, the soil type used in this experiment applies to this model, and the plastic limit of soil in this paper is 26.4%:

$$T_f=-0.0729w_p{}^{2.462}{w}^{-2}$$

(12)

Because the freezing temperature is often set to 0 °C, this affects the calculation accuracy. Equation (12) was then added into Equation (11). At the same time, it can be determined from Equation (11) that:

$${\theta }_i=B_i(T)\cdot {\theta }_u$$

(13)

2.4. Processing and Selection of Hydrothermal Coupling Model

Adding Equation (13) into $\frac{\partial {\theta }_i}{\partial t}$ can obtain:

$$\frac{\partial {\theta }_i}{\partial t}=\frac{\partial [B_i(T)\cdot {\theta }_u]}{\partial t}=({\theta }_s-{\theta }_r)\cdot [\frac{\partial B_i(T)}{\partial t}\cdot S+B_i(T)\cdot \frac{\partial S}{\partial t}]$$

(14)

COMSOL software was used for numerical simulation. Due to the introduction of multiple equations, in order to simplify the formal expression, the equation was derived into a partial differential equation with only two unknowns, and the coefficient partial differential equation in the PDE module was used to solve it. Equation (14) was substituted into Equations (1) and (4), the form of coefficient partial differential equations in COMSOL was sorted, and finally the expression form of hydrothermal equations as shown in Equations (15) and (16) was obtained. The equations derived in this paper are highly nonlinear. The transient term is discretized by the implicit Euler backward difference method and solved by the nonlinear iterative modified damping Newton method.

$$\left[{\rho }_{s\left(\theta \right)}-L\cdot {\rho }_i\cdot \left[\left({\theta }_s-{\theta }_r\right)\cdot S+{\theta }_r\right]\cdot \frac{\partial B_i}{\partial T}\right]\cdot \frac{\partial T}{\partial t}+\nabla \cdot \left(-\lambda \cdot \nabla T+{\rho }_w{c}_w\cdot \left(u_w\cdot T\right)\right)=L\cdot {\rho }_i\cdot \left({\theta }_s-{\theta }_r\right)\cdot B_i\cdot \frac{\partial S}{\partial t}$$

(15)

$$\left({\theta }_s-{\theta }_r\right)\cdot \left[1+\frac{{\rho }_i}{{\rho }_w}\cdot B_i\right]\cdot \frac{\partial S}{\partial t}+\left({\theta }_s-{\theta }_r\right)\cdot \frac{{\rho }_i}{{\rho }_w}\cdot \frac{\partial B_i}{\partial t}\cdot S=\nabla \cdot \left[D_{\left(S\right)}\cdot \nabla S+k_{g\left(S\right)}\right]$$

(16)

3. Experimental and Model Validation

This section mainly introduces the experiment. The purpose of the experiment was to verify the model of this paper. At the same time, the setting of boundary conditions and numerical parameters in the simulation software was given. Finally, the reliability of the equation was verified by comparing the experimental data with the simulation results.

3.1. Experimental Principle

As shown in Figure 1, the unidirectional freezing experiment was conducted on the prepared soil. A thermistor and data-logging system measured the soil temperature at different depths during the experiment. At the end of the experiment, the loop-cutting method was used to cut and weigh the soil to obtain the moisture state. Furthermore, the data obtained were used to verify the above equation.

Since the data collected by the data acquisition instrument is resistant, it needs to be converted into temperature according to the following formula:

$$T=\alpha \cdot {\left[(\frac{\gamma }{\ln (R-R_o)}-\beta )-273.15\right]}^2+\xi \cdot \left[(\frac{\gamma }{\ln (R-R_o)}-\beta )-273.15\right]+Z$$

(17)

where $\alpha$, $\gamma$, $\beta$, $\xi$ and $Z$ are the parameters of each probe, $R$ is the measured resistance, and $R_o$ is the line resistance.

3.2. Experimental Results

As shown in Figure 2, three sets of repeated one-way freezing experiments were conducted. The data trends of the three groups of experiments at ordinary moments are almost the same, indicating that the freezing process is consistent. The error of the temperature field may be caused by the soil’s inhomogeneity and the calculation error when the resistance is converted into a temperature. The error of the moisture field may be caused by human error during the weighing operation. The maximum error of the three data sets was less than 3%, so the experiment can be proven to be effective. The experimental mean value is taken as the comparison condition of simulation calculation. The experiment lasted for 2830 min.

The thermal physical parameters of the material are shown in

Table 2. Physical parameters.

Parametric	Value	Parametric	Value
${\rho }_i$	918	$c_s$	0.89
${\rho }_w$	1000	${\lambda }_i$	2.31
${\rho }_s$	1500	${\lambda }_w$	0.63
$c_i$	2.1	${\lambda }_s$	1.38
$c_w$	4.2	$L$	334.56

, which will be used as a material parameter in the simulation.

3.3. Definite Solution Conditions

In order to prove the reliability of the model used in this paper, the vertical soil column freezing experiment was used to compare and verify the model in this paper. As shown in Figure 3, the model’s geometric, initial, and boundary conditions are given. ${\gamma }_d$ is the dry bulk density of the soil, ${\theta }_0$ is the initial uniform moisture content of the soil, $G_0$ is the initial saturation of the soil, and $T_0$ is the initial temperature distribution of the soil at the beginning of the experiment. Among them, the initial temperature and boundary temperature are the average values of the three groups of experiments. Since the soil material is assumed to be uniform, the symmetry axis calculation can save resources. The temperature boundary of the soil column is the Dirichlet boundary at the top and bottom, the side is the adiabatic boundary, and the soil column’s moisture boundary is impervious at the top and bottom.

As shown in Figure 4, values were assigned to parameters and variables instead of material selection.

3.4. Grid Independence

As shown in Figure 5, in order to verify the grid independence, four types of grids, namely extremely fine, extra fine, finer and normal, were selected for analysis.

Figure 6 shows the result variation curve of this model from normal to extremely fine mesh. It can be seen that with the increase in the number of grids, the curve basically remained the same, and the parameter only tended to be more and more fixed. Grid sensitivity was not particularly strong. Because the geometric model in this paper is relatively simple and the number of grids is small, we selected the ultra-fine grid for calculation.

3.5. Comparison and Analysis

As shown in Figure 7, combined with the experiment and its calculation results, three typical moments were selected for temperature field comparison: at the beginning of the experiment (34 min), during the freezing process (528 min), and at the end of the experiment (2380 min). The moisture field at the end of the experiment was compared. As shown in Figure 7a, the distribution of simulated temperature field values and experimental values at three typical moments was basically consistent. At the end of the simulation, the height of the ice front interface (where the most ice surface is) was about 8 cm, which is basically consistent with the experimental value. As seen in Figure 7b, during the freezing process, the soil gradually freezes from top to bottom, and the surface of the freezing front keeps moving down. Due to the action of matrix potential, the water moves upward to the ice front interface. As the upper layer of ice gradually forms and the water continues to move upward, the obstructing effect of the ice becomes greater and greater. Therefore, the total water should gradually increase from top to bottom. At the end of the experiment, at the ice front interface, the obstruction caused by the formation of ice crystals was the largest here. The water generated by upward migration under the interface surface is blocked below the frozen surface, so the moisture content is the largest here, and the water in the lower layer is sharply reduced. There is still some ice formation from the ice front interface to the bottom of the frozen zone, but the moisture content is small and has little effect on frost heave. In the unfrozen zone, the moisture content is lower than the original due to the constant upward moisture migration. The minimum humidity measured by the experiment was 0.186 $\mathrm{c}{\mathrm{m}}^3/\mathrm{c}{\mathrm{m}}^3$, and the calculated value in this paper was 0.181 $\mathrm{c}{\mathrm{m}}^3/\mathrm{c}{\mathrm{m}}^3$. The maximum humidity measured in the experiment was 0.353 $\mathrm{c}{\mathrm{m}}^3/\mathrm{c}{\mathrm{m}}^3$, and the calculated value in this paper was 0.335 $\mathrm{c}{\mathrm{m}}^3/\mathrm{c}{\mathrm{m}}^3$. The maximum error between the model and the measured value was about 0.018 $\mathrm{c}{\mathrm{m}}^3/\mathrm{c}{\mathrm{m}}^3$, so the calculated value of the proposed model is basically consistent with the measured value, and the comparison with the calculated results of the other two models verifies the reliability of the proposed model. On the contrary, the area with lower moisture content is more likely to cause structural instability.

4. Influence of Moisture Migration and Accumulation on Temperature Field

The influence of moisture migration on the temperature field is mainly reflected in three aspects: first, the comprehensive thermal conductivity of soil changes due to moisture migration and accumulation and freezing, so the thermal conductivity of soil at different locations changes during freezing; second, the accumulation of moisture migration changes the moisture content at different locations, and then the phase change latent heat during freezing disturbs the temperature field; third, in the process of moisture migration, unfrozen moisture migrates to the frozen area. At this time, unfrozen water belongs to the high-temperature water body, and convection heat transfer will occur between the surrounding soil and unfrozen water. In order to discuss the influence of these three conditions on the temperature field in the freezing process, three different forms of hydrothermal equations are selected to compare the same experiment, as shown in

Table 3. Two contrasting models.

Comparison model I (No convection term in heat transfer equation)

$\left[{\rho }_s-L\cdot {\rho }_i\cdot \left[\left({\theta }_s-{\theta }_r\right)\cdot S+{\theta }_r\right]\cdot \frac{\partial B_i}{\partial T}\right]\cdot \frac{\partial T}{\partial t}+\nabla \cdot \left(-\lambda \cdot \nabla T\right)=L\cdot {\rho }_i\cdot \left({\theta }_s-{\theta }_r\right)\cdot B_i\cdot \frac{\partial S}{\partial t}$

$\left({\theta }_s-{\theta }_r\right)\cdot \left[1+\frac{{\rho }_i}{{\rho }_w}\cdot B_i\right]\cdot \frac{\partial S}{\partial t}+\left({\theta }_s-{\theta }_r\right)\cdot \frac{{\rho }_i}{{\rho }_w}\cdot \frac{\partial B_i}{\partial t}\cdot S=\nabla \cdot \left[D_{\left(S\right)}\cdot \nabla S+k_{g\left(S\right)}\right]$

Comparison model II (solid heat transfer)

${\rho }_s{c}_s\frac{\partial T}{\partial t}=\nabla [{\lambda }_{\left(\theta \right)}\cdot \nabla T]$

No moisture migration

.

The comparison model II is a simple solid heat transfer model. This model does not consider the moisture migration process nor the thermal influence caused by the ice-water phase transition. It simply considers the change of physical parameters after the ice-water phase transition. Compared with the comparative model II, the comparative model I considers the moisture migration of the convection effect and the latent heat of phase transition after the accumulation of moisture migration. Therefore, the comparative model I can better reflect the influence of the latent heat of phase transition caused by the accumulation of moisture migration on the temperature field. The model in this paper also considers the effect of heat transfer between water and soil on the temperature field during the migration of unfrozen water to the frozen area based on the comparison model I.

4.1. Influence of Different Hydrothermal Equations on Temperature Field

As shown in Figure 8 and Figure 9, three different models were calculated at four initial saturation. It can be seen that when the initial saturation is slight, the result changes little. However, different models give different results for the simulation of high initial saturation. Therefore, two experiments were added to verify the reliability of the calculated results.

When the initial saturation was 0.2, the calculation results of the three models on the temperature field tended to be consistent. It can be seen from the observation of the moisture field that the moisture content was small, and the moisture had almost no migration effect. When the initial saturation was 0.35, the disturbance generated by the moisture field can be seen. However, the amount of moisture migration was not large, and the blocking effect of water into ice was not significant. The moisture migration follows the action of matrix potential, and the blocking effect of ice can be ignored. Therefore, the final calculated temperature field results also tended to be consistent. Compared with the experimental group with initial saturation of 0.4968, it can be seen that when the initial saturation was slight, the temperature and moisture content were almost unchanged. Therefore, we could consider that if the initial saturation is slight, the effects of the latent heat term and convection term on the temperature field can be ignored, and the reverse influence of the temperature field on the moisture field can also be ignored. Finally, the influence on the stress field can also be ignored.

The calculation results of the two groups of high initial saturation had little error compared to the experiment. When the initial saturation was 0.65, the latent heat and convective heat transfer effects become apparent. Compared with Model II, the temperature difference between Model I and Model II at the ice front is about 0.43 °C, indicating that the latent heat of phase transition significantly impacts the temperature field. Similarly, the temperature difference between the proposed Model and Model I at the ice front is about 0.45 °C. This shows that the influence of convective heat transfer caused by moisture migration on the temperature field is also significant at this saturation. When the initial saturation is 0.8, the effect of latent heat of phase transition is 0.68 °C, and that of convective heat transfer is about 0.61 °C. These two thermal actions make the difference between the results calculated by the three models more obvious.

The influence of latent heat of phase transition is related to the frozen water volume, the influence of convective heat transfer is related to the migration rate of moisture, and the amount of moisture migration also affects the frozen water volume. However, too much water can make freezing ice more difficult. Therefore, the influence of the latent heat term and convection term on the calculation results should be addressed in this process.

In conclusion, with the increase of initial saturation, the thermal effects of latent heat and convective heat transfer on the freezing process cannot be ignored. This is consistent with Zhang’s [36] experimental conclusion in saturated soil. In addition, no matter which model was used, it can be seen that the upper part of the frozen soil begins to freeze at 528 min, and the freezing rate of the soil with higher initial saturation is lower than that of the frozen soil with lower initial saturation. It is because the comprehensive heat transfer coefficient of the soil with higher initial saturation is lower than that of the soil with lower initial saturation under the action of moisture migration. At 2830 min, the freezing rate of the soil with high initial saturation was higher than that of the frozen soil with low initial saturation, which was caused by the increase of the comprehensive thermal conductivity of the soil due to a large amount of moisture migration and freezing. By comparing the numerical results of the solid heat transfer model, it is also apparent that the freezing rate will increase after moisture migration occurs in the freezing process. This indicates that the freezing rate of the soil is low at the beginning of the freezing process and then gradually increases, and this phenomenon is more evident in unsaturated soil with high initial saturation. The upward moisture migration reduces the heat transfer coefficient in the unfrozen region. Hence, the temperature gradient from the ice front interface to the bottom is more significant than that in the frozen region. It can be seen that the freezing process of soil under different saturation can cause a significant temperature difference at the ice front interface. Therefore, in the simulation calculation, the selection of the heat transfer equation becomes particularly important.

4.2. Effect of Different Initial Saturation on Temperature Field

As shown in Figure 10, the temperature field of the soil freezing process was compared when the initial saturation was 0.2, 0.35, 0.4968, 0.65, and 0.8. It can be seen that with the increase of initial saturation, the change in freezing rate is also apparent, which is ultimately reflected in the temperature difference at the ice front. This is because, with the increase of initial saturation, the moisture content in the soil is more extensive, and the moisture migration is more intense than in the soil with low saturation. A large amount of moisture migration is blocked by the ice front interface, converging here, which significantly influences the comprehensive thermophysical parameters of this place, resulting in a significant difference in the final temperature.

The choice of freezing temperature also apparently influences the effect of frozen soil. When the freezing temperature is set to 0 °C, the actual freezing height is less than the theoretical height. Meanwhile, the soil’s freezing height will also increase with the initial saturation increase. However, the freezing front interface and freezing height are different. The freezing front interface increases significantly when the initial saturation is low, and its position remains almost constant as the saturation increases gradually. As the initial saturation increases, the choice of freezing temperature becomes essential. The calculation affects the freezing rate and ice-forming process and plays a vital role in the final freezing result.

Ultimately, this was reflected in the project and plays a decisive role in predicting the stability of the project.

5. Conclusions

The simulation and prediction of frozen soil play an essential role in the stability of engineering and the early warning of geological disasters. Currently, China’s central performance is the protection of road and railway disasters in frozen areas and warning of geological landslide disasters; for example, the Qinghai-Tibet Highway. Due to the elevatory moisture content in China’s frozen soil in recent years, many thermal melt lakes have been formed. The overflow of thermal melt lakes has formed many natural thermal melt rivers, which pass through or from the vicinity of the original buildings. Furthermore, the high moisture content in the soil makes the foundation of the original buildings extremely unstable. This paper aims to analyze the soil’s moisture content’s influence on the temperature equation to obtain more accurate results in the last simulation prediction and ensure the stability of permafrost engineering.

This paper compared three different hydrothermal equations. By changing different initial soil saturation conditions, the effects of different water-heat coupling equations and initial soil saturation on the soil freezing process were simulated and analyzed, and the following conclusions were obtained:

(1)

In the process of soil freezing, the freezing rate above the freezing front is lower than below due to the small amount of moisture transfer at the beginning. Then, with the increase of moisture transfer and the accumulation of frozen water, the comprehensive thermal conductivity of the upper frozen area changes significantly, and the freezing rate above the freezing front starts to be greater than below the freezing front. Especially with the increase of initial saturation, this phenomenon is more evident in unsaturated soil freezing. The difference in freezing rate and the influence of periodic boundary will make the final result deviate.

(2)

During the freezing process of soils with different initial saturation, the heat transfer equation’s latent heat term and convection term have different disturbances to the temperature field. Therefore, the latent heat of phase transition and convective heat transfer caused by moisture migration can be ignored in the simulation prediction of unsaturated permafrost with slight initial saturation, which has almost no effect on the results. In this way, some computing resources can be saved in the complex multi-field simulation prediction of large time scales in the future. However, when the initial saturation is high, we must pay attention to the influence of these two terms on the entire freeze-thaw results.

(3)

The choice of freezing temperature also has some influence on the result. It is mainly represented by freezing rate, ice formation resistance, and freezing height. As a result, the comprehensive heat transfer coefficient will change. The more accurate the freezing temperature, the more reliable the simulation results. This paper only analyzes the effect of initial saturation on freezing temperature. In fact, many factors also affect freezing temperature, which will be analyzed in future work. In the simulation prediction, the change in freezing temperature can be considered, which can better predict the changing trend of the project.