1. Introduction
Over 70% of China’s permafrost is distributed in the Qinghai–Tibet Plateau, which is also the region with the highest altitude and the most extensive permafrost distribution in the middle and low latitudes of the world [
1,
2,
3]. Various large-scale engineering structures widely distributed in the plateau’s permafrost regions significantly affect heat transfer in the soil, ultimately changing the thermal balance and temperature field of the permafrost [
4,
5,
6]. This has led to noticeable permafrost degradation, presenting new problems and challenges to the construction and operation of large-scale linear engineering projects in this region [
7,
8,
9,
10]. In terms of practical engineering, the thermal conductivity of permafrost in negative temperature areas, which is close to the initial freezing temperature point (defined as the “near-phase-transition zone”), is the key physical parameter affecting the thermal stability of engineering structures. However, measuring the thermal conductivity of permafrost in the near-phase-transition zone requires temperature control and heat flow monitoring with an extremely high precision, which is limited by cost and complexity. To date, there are no well-established methods for testing the thermal conductivity of permafrost in the near-phase-transition zone, which is crucial for describing the distribution of the permafrost temperature field and ensuring project safety.
Recent investigations into soil thermal conductivity have led to significant progress in various aspects. Researchers have carried out a large number of studies on thermal conductivity test methods. For instance, Xu et al. [
11] utilized the calorimeter method, the heat flow meter method, and the probe method to measure the thermal conductivity of various frozen soils and established a series of widely used thermal parameter tables for frozen soil. Nusier et al. [
12] conducted laboratory testing using single- and dual-probe methods to determine the thermal conductivity of loam and found that the test results of the dual-probe method were more accurate than those of the single-probe method. Lu et al. [
13] and Zhang et al. [
14] also carried out a series of experimental studies on the thermal conductivity of aeolian sandy soil, clay, and silty clay using the transient hot wire method. With advancements in theoretical understanding and technological capabilities, numerous scholars have further improved and proposed new thermal conductivity testing techniques. For example, Alrtimi et al. [
15] incorporated a heat jacket into the frozen soil thermal conductivity testing apparatus used in the steady-state comparative method, significantly reducing the radial heat loss of the device and enhancing measurement accuracy. Kojima et al. [
16] designed an innovative dual-probe thermal pulse sensor for the measurement of the thermal properties of frozen soil.
To avoid costly and time-consuming experimental procedures and to facilitate engineering applications, researchers have conducted statistical analyses on a large amount of frozen soil thermal conductivity test data and proposed many data-driven empirical formulas [
17,
18]. Based on the results of testing 19 soil types, Kersten [
19] established initial empirical formulas for thermal conductivity using water content and dry density. Johansen [
20] proposed an interpolation calculation model of soil thermal conductivity using normalized coefficients and soil saturation. Bi et al. [
21] developed a general model for calculating the thermal conductivity of frozen soil which is based on the liquid water content, frost heave, porosity, and initial water content. Tian et al. [
22] established an empirical model of the relationship between thermal conductivity and porosity based on data from 28 partially frozen soils. Scholars have also made many improvements to classic thermal conductivity empirical models to broaden their applicability and increase their prediction accuracy. For example, He et al. [
23] established a new model similar to the Johansen model which simulates the relationship between various soil textures and their water contents and effective thermal conductivities. Zeng et al. [
24] compared and evaluated eight soil thermal conductivity models through actual measurements and proposed an improved thermal conductivity prediction model that considers particle size distribution parameters. Balland et al. [
25] expanded Johansen’s model and developed a thermal conductivity calculation model that is applicable to more soil conditions. Compared with traditional empirical fitting methods, machine learning methods can achieve more accurate estimations through training with a large amount of data, and they have also been widely used for the prediction of frozen soil thermal conductivity [
26,
27,
28]. For instance, Ren et al. [
29] analyzed the factors affecting the thermal conductivity of soil during the freeze–thaw process and established a prediction model for thermal conductivity using artificial neural network technology. Kardani et al. [
30] combined the firefly algorithm with the extreme learning machine method to establish a model for predicting the thermal conductivity of unsaturated soil.
In recent years, researchers have also studied measurement methods and predictive models of the soil thermal conductivity in the near-phase-transition zone. For example, Zhao et al. [
31] combined the steady-state method with the transient method and proposed a method for measuring the thermal conductivity of frozen soil near 0 °C. After an analysis of various models for estimating characteristic soil freezing curves, Bi et al. [
32] proposed a new method for predicting the thermal conductivity of frozen soil based on the geometric mean model. This model can effectively predict the thermal conductivity of frozen soil and how it varies in the temperature range of −20 °C to 0 °C. He et al. [
33] used the steady-state method to measure the thermal conductivity of frozen and thawed lime soil and found that the phase change in the soil mainly occurs between −3 and −2 °C. Firat et al. [
34,
35] proposed an artificial neural network prediction model based on the internal and external factors of the soil, which can effectively predict the thermal conductivity of sandy soil at temperatures ranging from −7 °C to 4 °C. Although extensive research has been conducted on the thermal conductivity of various types of frozen soil [
36], studies on the thermal conductivity characteristics of frozen soil in the near-phase-transition zone are relatively scarce. Existing thermal conductivity data of frozen soil were mostly measured at low temperatures (e.g.,
T < −4 °C), and there is a lack of test data near the phase change point [
37,
38]. Furthermore, current models for predicting the thermal conductivity of frozen soil in the near-phase-transition zone have limitations in terms of accuracy and applicability [
39,
40]. To address this, in the present work, a method for testing the thermal conductivity of fine sandy soil in the near-phase-transition zone is proposed. This method involves measuring thermal conductivity using the transient plane heat source method and determining the volumetric specific heat capacity via weighing the unfrozen water content. The unfrozen water content of sand specimens in the near-phase-transition zone was determined using nuclear magnetic resonance (NMR) technology, and corresponding empirical fitting formulas were established. Based on the test results, the temperature variation characteristics and parameter influence laws of thermal conductivity in the near-phase-transition zone were investigated. Additionally, thermal conductivity prediction models based on multiple regression (MR) and a radial basis function neural network (RBFNN) were established.
4. Conclusions
In the present work, a method of determining the thermal conductivity of sandy soils in the near-phase-transition zone was proposed, and the temperature variation trends and parameter influence laws of thermal conductivity were investigated. Moreover, prediction models based on MR and RBFNN methods were established, and the prediction accuracies of the different models were compared. The conclusions are as follows:
- (1)
The average error between the proposed method and the steady-state heat flow method for soil thermal conductivity in the near-phase-transition zone is 7.25% and the maximum error is 10.1%, which verifies the reliability of the proposed test method.
- (2)
The unfrozen water content in fine sandy soil changes drastically in the near-phase-transition zone, and the amplitude of this change in the range of 0~−3 °C accounts for over 80% of the total change in the entire negative temperature range. The changes in unfrozen water content and thermal conductivity in fine sandy soil exhibit similar trends, and the near-phase-transition zone temperature interval can be divided into a drastic phase transition zone and a stable phase transition zone.
- (3)
Increases in the thermal conductivity of fine sandy soil mainly occur in the drastic phase transition zone; these increases account for about 60% of the total increase in thermal conductivity across the entire negative temperature region. With the increase in density and total water content, the rate of the increase in thermal conductivity in the drastic phase transition zone gradually decreases.
- (4)
The prediction models based on the MR and RBFNN methods both have a high accuracy, which highlights their engineering application value. The R2, MAE, and RSME of the RBFNN model in the drastic phase transition zone are 0.991, 0.011, and 0.021, respectively, and are better than those of the MR prediction model.
This study can help researchers to understand trends in the thermal conductivity of sandy soils with temperature in the near-phase-transition zone and also provides basic thermal data for the design of the future Qinghai–Tibet expressway. However, it should be noted that the MR fitting formula of thermal conductivity in the near-phase-transition zone is only applicable to fine sandy soil, and when the dry density of the sample is relatively small, there may be large errors in the predicted values in the drastic phase transition zone. In addition, although the RBFNN prediction model has a high accuracy in both the drastic and stable phase transition zones, it is naturally a data-driven method and its prediction accuracy depends on the amount of test data; thus, a certain amount of data are required for engineering designers to utilize it.