Keywords

1 Introduction

Seasonal frost in temperature fluctuations can pose safety hazards to the stability of airport subgrades, making the composition, structure, and mechanical properties of soil in seasonal frost regions more complex than ordinary soils [1]. Cheng [2] analysed the embankment temperature field using both steady-state and transient thermal analysis methods, explaining the solution process of the embankment temperature field and proposing research ideas for frozen soil embankments. Feng [3] conducted numerical simulations of the highway embankment temperature field using Finite Element Method (FEM), and the results showed that the temperature change in the embankment lags behind the air temperature change. The aforementioned studies have proposed various methods for analysing the temperature field characteristics of frozen soil embankments, but research on modelling methods based on thermal characteristic parameters is relatively limited. Building upon existing theoretical research, this paper utilizes COMSOL finite element software to analyse the characteristic changes in the temperature field of a natural embankment.

2 Theoretical Analysis

2.1 Fundamentals of Temperature Field Description

Temperature Field.The temperature field is a collection of temperatures at various points in space that describe the thermal distribution of the studied object at a specific moment. It is typically represented as a function of both space and time, and its expression is as follows:

$$ T = f(x,y,z,t) $$
(1)

where T represents temperature in degrees Celsius (°C); x, y, z are spatial coordinates in a Cartesian coordinate system; t stands for time.

Temperature Gradient.For the studied object, within the same temperature field at a specific moment, temperatures are equal on the same isothermal surface. Heat transfer occurs between different isothermal surfaces, with the maximum heat transfer and temperature change occurring in the normal direction. The temperature change along the normal direction is commonly referred to as the temperature gradient (grad T), as shown in the following equation:

$$ gradT = \frac{\partial T}{{\partial n}}n $$
(2)

where ∂T/∂n represents the temperature directional derivative in the normal direction; n stands for the unit vector in the normal direction, with x, y, z components.

Fourier’s Law.

According to Fourier’s theorem, the heat flux vector and temperature gradient can be combined to establish the following equation:

$$ q_n = - \lambda \frac{\partial T}{{\partial n}} $$
(3)

where \(\lambda\) is the thermal conductivity coefficient (W·m − 1·K − 1).

2.2 Governing Temperature Field Equation

Based on Fourier’s Law, to calculate the heat flux vector, it is necessary to determine the temperature gradient, as illustrated in Fig. 1. In this study, by employing the first law of thermodynamics (the law of energy conservation and transformation), a connection between temperatures at various points within an object is established. The heat generated by frozen soil phase change is treated as a heat source, ultimately leading to the formulation of the soil thermal conduction equation.

Fig. 1.
figure 1

Thermal Conduction Within the Unit Cell

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Due to the fact that the temperature and moisture of the airport pavement subgrade change primarily in the vertical direction due to freeze-thaw cycles and settlement, it can be treated as a two-dimensional problem. Neglecting the influence of factors such as moisture evaporation, the non-steady-state heat conduction equation can be expressed using Eq. (4):

$$ \rho C(\theta )\frac{\partial T}{{\partial t}} = \lambda (\theta )\nabla^2 T + L \cdot \rho_I \frac{\partial \theta_I }{{\partial t}} $$
(4)

where C represents the specific heat capacity of the soil (J.kg−1.K−1); ρ represents the density of the soil (kg.m−3); \(\lambda\) represents the thermal conductivity coefficient (W.m−1.K−1); L represents the latent heat of water phase change, usually taken as 334.5 J/m3; \(\rho_I\) represents the density of ice (kg.m−3); \(\theta_I\) represents the volumetric content of ice (%).

Since the subgrade soil within the pavement is in a negative temperature state, the water inside the soil consists of both ice and pore water. Due to the different densities of unfrozen water and ice, the unfrozen water content is defined as follows:

$$ \theta_I = (\theta - \theta_u ) \cdot \frac{\rho_1 }{{\rho_w }} $$
(5)

where \(\theta\) represents the volumetric water content (%); \(\theta_u\) represents the volumetric unfrozen water content (%); \(\rho_w\) represents the density of water (kg m−3).

The migration and redistribution of moisture in the soil, as well as the distribution of heat and temperature, are interrelated. Considering the water-heat coupling issue during soil freeze-thaw processes [4], the latent heat of phase change is treated as an internal heat source in the established heat conduction equation:

$$ \rho C_{vs} \frac{\partial T}{{\partial t}} = \lambda \nabla T^2 + L\rho_I \frac{\partial \omega_I }{{\partial t}} $$
(6)

\(\nabla\) represents the spatial differential operator; \(\rho\) is the density of soil in kg m−3; \(\rho_I\) is the density of ice in kg m−3; \(C_{vs}\) is the specific heat capacity of soil in J·(kg ℃)−1; \(\lambda\) is the thermal conductivity of soil in W·(m ℃)−1; \(\omega_I\) is the ice content in soil, expressed as a percentage; L denotes the latent heat of phase change, taken as 335,000 J kg−1; t stands for time in seconds; and T represents the temperature of the soil in degrees Celsius.

3 Finite Element Model Establishment

3.1 Geometric Parameters

The numerical simulation analysis is performed using a full cross-sectional two-dimensional model as shown in Fig. 2. The airport runway has a width of 45 m, with 15 m-wide shoulders on each side. Therefore, the model length is set to 75 am, and the depth is taken as 30 m below the natural ground surface.

Fig. 2.
figure 2

Numerical Model Geometry Dimensions.

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3.2 Basic Parameters

This study adopts soil thermal characteristic parameters as the representation parameters for the temperature field model. The soil thermal characteristic parameters include the specific heat capacity and thermal conductivity of the soil. From a perspective of material composition, Xu [5] proposed a method for determining thermal characteristic parameters with mass-weighted average properties. Applying the step function to the additive model, we obtain the following expression:

$$ \left\{ \begin{gathered} C = \rho_d C_s + \rho_w C_w \theta_u + \rho_i C_i \theta_i \hfill \\ C_s = C_{sf} + (C_{su} - C_{sf} )H(T) \hfill \\ \end{gathered} \right. $$
(7)
$$ \left\{ \begin{gathered} \lambda = \rho_d \lambda_s + \rho_w \lambda_w \theta_u + \rho_i \lambda_i \theta_i \hfill \\ \lambda_s = \lambda_{sf} + (\lambda_{su} - \lambda_{sf} )H(T) \hfill \\ \end{gathered} \right. $$
(8)

where Cu and Cf represent the specific heat capacity of unfrozen and frozen soil, respectively, measured in J kg−1 ·K−1; Csu and Csf denote the specific heat capacity of the unfrozen and frozen soil skeleton, respectively, measured in J kg−1 K−1; \(\lambda_u\) and \(\lambda_f\) stand for the thermal conductivity of unfrozen and frozen soil, respectively, measured in W m−1 K−1; \(\lambda_{su}\) and \(\lambda_{sf}\) represent the thermal conductivity of the unfrozen and frozen soil skeleton, respectively, measured in W m−1 K−1; \(\theta\), \(\theta_u\), and \(\theta_i\) refer to the total water content, unfrozen water content, and ice content, respectively, expressed as a percentage; Cw and Ci indicate the specific heat capacity of water and ice, respectively, measured in J kg−1 K−1; ρd, ρw and ρi represent the density of soil, water, and ice, respectively, measured in kg m−3.

In the additive model, the relevant parameters of water and ice vary at a relatively small rate with changes in the external environment and can be treated as constants. The values are given in the Table 1 below:

Table 1. Values of Parameters in the Additive Model.
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The initial moisture content is 18.7%. However, the specific heat capacity and thermal conductivity of the soil skeleton primarily depend on the mineral composition and organic matter content, and are temperature-dependent [6]. The measured values of similar soils may also vary.

3.3 Boundary Conditions

Zhu [7] proposed the “attachment layer principle,” which replaces the complex subgrade surface with a stable “attachment layer” as the upper temperature boundary condition. The expression is shown as follows.

$$ T(t) = 9.22 + 16.37\sin (\frac{2\pi t}{{365}} + \frac{\pi }{2}) + \frac{0.032t}{{365}} $$
(9)

4 Analysis of the Natural Subgrade Temperature Field

In this section, a natural subgrade with a depth of 30 m and a width of 75 m is taken as an example. The temperature boundary conditions, soil material parameters, and initial temperature conditions determined in the previous section are applied to establish the temperature field model of the natural subgrade. Figures 3 and 4 show the temperature field contour maps of the subgrade.

Fig. 3.
figure 3

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Fig. 4.
figure 4

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It can be observed that the significant variation of the natural subgrade temperature field throughout the entire freeze-thaw cycle mainly occurs within the depth range of −5 m. Beyond the −5 m depth range, the subgrade soil temperature remains around 10–12 ℃, and the 0 ℃ isotherm extends to approximately −2 m depth below the ground. Next, we focus on studying the temperature field variation within the depth range of −5 m of the natural subgrade, as shown in Table 2.

Table 2. Pattern of Temperature Variation.
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The variation pattern of the 0 ℃ isotherm with respect to time is obtained, as shown in Fig. 5.

Fig. 5.
figure 5

The distribution of the 0 ℃ isotherm in the natural subgrade.

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As depth increases, there is a lag phenomenon in temperature fluctuations. The temperature variation pattern of the shallow layer of the natural pavement subgrade is similar to that of the surface temperature. With increasing depth, the influence of atmospheric temperature gradually diminishes, and the geothermal curve becomes smoother. The maximum surface temperature is 26.46 ℃, and the maximum temperature at a depth of −5 m is 14.10 ℃, decreasing by 12.36 ℃. The minimum surface temperature is −6.26 ℃, and the minimum temperature at a depth of −5 m is 5.96 ℃, increasing by 12.22 ℃. The amplitude of surface temperature and the temperature at a position 5 m below the surface over an annual cycle are approximately 16.36 ℃ and 4.07 ℃, respectively, with a decrease of 12.29 ℃ in temperature amplitude.

By the end of November 2021, the surface temperature started to become negative. Around December 15th, the 0 ℃ isotherm line was at a depth of about −0.4 m underground. As the external temperature continued to drop, by January 15th, 2022, the 0 ℃ isotherm line had reached a depth of about −1.2 m in the natural pavement subgrade. With the ongoing migration of the freezing front into the subgrade, the 0 ℃ isotherm line continued to descend. By February 15th, 2022, the 0 ℃ isotherm line had descended to a depth of about −1.46 m, and the frozen part within the subgrade continued to increase. By March 10th, 2022, the external temperature had risen to positive values. A second 0 ℃ isotherm line appeared within the subgrade, with the first 0 ℃ isotherm line at a depth of about −1.6 m underground. At this point, the frozen soil was sandwiched between two warm soil bodies, resulting in bidirectional melting with a relatively rapid rate. By April 1st, 2022, this was the moment of maximum freezing depth for the year, with the freezing depth being approximately −1.81 m. By April 10th, 2022, as the temperature increased, the internal soil continued to warm. With the external environmental temperature continuously rising, the two 0 ℃ isotherm lines coincided, and the temperature inside the frozen soil was projected to transition entirely to a positive temperature state, resulting in the disappearance of the 0 ℃ isotherm line within the soil.

5 Conclusion

With increasing depth, there is a lag phenomenon in temperature fluctuations. The temperature variation pattern in the shallow layer of the natural subgrade is similar to that at the surface, but as the depth increases, the temperature variation curve tends to be smoother. During the cold season, when the atmospheric temperature is lower than the temperature of the natural subgrade, the surface temperature of the subgrade starts to decrease. When it reaches the freezing point, freezing begins at the surface and gradually extends to the interior of the subgrade, causing the 0 ℃ isotherm to gradually descend with freezing time. During the warming period, the atmospheric environment returns to positive temperatures, leading to a gradual recovery of the surface temperature of the subgrade. The thawing of the surface frozen soil begins, and due to the fact that the lower part of the frozen soil remains at positive temperatures, the thawed soil during the warming period will be sandwiched between two positive-temperature soil masses, leading to bidirectional thawing.