Characteristics and Prediction of the Thermal Diffusivity of Sandy Soil

Abstract

Revealing the variation law of thermal diffusivity of sandy soil can provide a theoretical basis for the engineering design and construction in cold and arid regions. Based on experimental data of sandy soil samples, the distribution characteristics and influence of dry density and moisture content on thermal diffusivity are analyzed in this work. Then, the prediction model based on the empirical fitting formula and RBF neural network method for thermal diffusivity of frozen and unfrozen sandy soil is established, and the prediction accuracy of different prediction methods is compared. The results show that (1) thermal diffusivity of sandy soil is positively correlated with the particle size. With the increase of sand size, thermal diffusivity of sandy soil increases significantly. (2) Partial correlation among natural moisture content, dry density, and thermal diffusivity varies with different frozen and unfrozen conditions. (3) For unfrozen sandy soil, the binary RBF neural network prediction model is obviously better than that of the binary empirical fitting formula model. (4) The ternary prediction model has significantly higher prediction accuracy than that of the binary prediction model for frozen sandy soil, and the ternary RBF neural network model has the best prediction effect among the four methods.

1. Introduction

Sandy soils are widely distributed in China, especially in the provinces of Inner Mongolia, Gansu, Xinjiang, Tibet, etc., which occupy more than 50% of China’s land surface [1]. Compared to other soil types, the thermal diffusivity of sandy soil is relative large, which turns into a faster rate of heat transfer and a larger temperature variation [1,2]. As one of the important thermophysical parameters of soil, thermal diffusivity has great significance in the application of frozen soil engineering, geological prospecting, land hydrothermal cycle, coal mining, and other engineering fields [3]. Thus, research on the variation feature and parameter influence mechanism of sandy soil thermal diffusivity is of great importance and helps to promote the stability and sustainability of engineering applications.

As a main thermodynamic parameter of soil, thermal diffusivity reflects the synergistic effect of thermal conductivity and specific heat capacity in the heat transfer process [4,5]. Thermal diffusivity describes the instantaneous process of heat transport under certain temperature boundary conditions [6], which can be obtained by theoretical calculation and an experimental testing method. The theoretical calculation is usually based on the assumption that the soil is a semi-unbounded medium with constant thermal diffusivity, and the calculation formula is obtained based on the one-dimensional heat conduction equation. At present, the calculation methods of soil thermal diffusivity mainly include the phase method, amplitude method, numerical method, heat conduction method, numerical method, harmonic method, arctangent method, and logarithmic method, among others [7,8,9]. Li et al. [8] calculated thermal diffusivity of Taklimakan desert soil at depths of 5 to 20 cm using four computation methods (harmonic method, phase method, amplitude method, and heat conduction convection method) based on the observation data from an atmospheric environment observatory. The results showed that the harmonic method had the highest calculation accuracy, and the amplitude method and phase method had large errors. Diniz et al. [10] adopted the amplitude method, arctangent method, and logarithmic method to estimate the thermal diffusivity of the soil, and concluded that the arctangent method is not suitable for estimating the thermal diffusivity of the soil layer with a depth between 5 and 50 cm. Gao et al. [6] calculated the soil thermal diffusivity based on the Laplace transform method and harmonic method, and the results showed that the soil thermal diffusivity obtained by the Laplace transform was slightly smaller than the value calculated by the harmonic method. In addition, the harmonic method was easier to implement than the Laplace transform. Danelichen et al. [11] compared and analyzed the difference in the estimation of soil thermal diffusivity by the amplitude method, logarithmic method, arctangent method, and phase method. The study showed that the logarithmic method was the best method, followed by the phase method. Oscar et al. [12] used the thermal wave amplitude method to measure the thermal change of soil temperature from a depth of 0.05 to 0.65 m, and estimated the thermal diffusivity to be between 2.26 × 10⁻⁷ and 8.71 × 10⁻⁷ m²·s⁻¹.

There are many factors affecting the thermal diffusivity of soil, such as soil mineral composition, temperature, particle size, depth, dry density, and natural moisture content. Xiong et al. [2] used the thermal probe method to conduct an experimental study on the variation law of thermal conductivity of medium sand and coarse sand with different temperatures and volume moisture content. The results showed that the contribution of moisture content to thermal conductivity was greater than that of temperature, and the optimal volume moisture content was given to make the thermal conductivity of medium sand and coarse sand reach the ultimate point. Liu et al. [13] calculated the thermal diffusivity of soils with different textures through an unsteady one-dimensional heat conduction equation, and found that the soil heat transfer rate was positively correlated with soil texture and negatively correlated with soil entropy conditions. Zhou et al. [14] tested the changeable rule of thermal conductivity with different particle size and moisture content, and concluded that the particle size had little effect on thermal conductivity, and that thermal conductivity increased with the increase of moisture content. Diniz [10] et al. used soils at different depths to estimate the thermal diffusivity and found that the temperature gradually increased as the depth increased, and the maximum value appeared at 50 cm, but as the depth increased, it was found that the thermal change tended to decrease. Zhen et al. [15] used the transient plane heat source method to study the influence of dry density and moisture content on the thermal conductivity of sandy soil, and the results showed that the thermal diffusivity of remolded sandy soil was positively correlated with moisture content. Furthermore, the moisture content reached the extreme value at 15%, after which the thermal diffusivity tended to stabilize or slightly decrease. In addition, the moisture content had a greater influence on the thermal diffusivity than the dry density. Wang et al. [16] used measured soil temperature data to calculate the heterogeneous soil thermal conductivity, and the results showed that the soil thermal conductivity had a significant tendency to increase with the increase in depth. Zhou et al. [17] used the soil temperature data at 0.8 and 3.2 m measured by 39 observation stations on the Qinghai–Tibet Plateau to calculate the soil thermal diffusivity of each observation station using the heat conduction convection method, and gave the characteristics of soil thermal diffusivity changing in time and space by using the heat conduction and convection method. Song et al. [18] tested the thermal diffusivity of the main rock formations in Guizhou and found that the thermal diffusivity increases with increased mineral particle size, and the moisture content had a greater influence on the thermal diffusivity. Ma et al. [19], based on observation data of soil moisture content and soil thermal properties in the Loess Plateau, researched the thermal properties of the soil and its change characteristics and the influence of precipitation on the thermal properties of the soil in this area. The results showed that the soil thermal diffusivity did not increase linearly with the increase of depth, and the soil thermal diffusivity decreased with the increase of precipitation. Roxy et al. [20] studied the relationship between soil moisture and soil thermal diffusivity based on observational data, and the study showed that the soil thermal diffusivity first increased and then decreased with the increase of soil moisture. Tatiana et al. [21] used the unsteady state method to measure the thermal diffusivity of the undisturbed soil, and established a curve model of thermal diffusivity and moisture content.

Overall, it can be seen that the current research on soil thermal diffusivity mainly focuses on the comparison of test methods and the analysis of influencing factors, while the research on the prediction model of soil thermal diffusivity is relatively rare. Thus, based on the test results of 206 groups of frozen and unfrozen sandy soil thermal diffusivity, the prediction models of the empirical fitting formula method and the RBF neural network method are proposed in this paper. Additionally, the prediction effects of different models are compared and analyzed. The research results can help to provide data reference for the thermal design of engineering structures with sandy soil foundation.

2. Materials and Methods

2.1. Source of Test Specimens

The soil specimens were sourced from the Qinghai–Tibet engineering corridor (Xidatan to Tanggula mountain section) and obtained by the drilling sampling method. Total number of sandy soil test specimens was 206, which included 38 groups of silt soil specimens, 78 groups of medium-fine sand specimens, and 90 groups of gravel sand specimens. Maximum sampling depth of the borehole was 40 m, and each drilling hole was sampled at different depths. The detail sampling depth distribution is shown in Figure 1.

Table 1. Statistics of soil sample physical parameters.

Statistical Parameters	Maximum	Minimum	Median	Average Value
moisture content/%	83	0.6	17.4	19.7
dry density/g·cm−3	2.34	0.88	1.71	1.71

shows the statistics of the physical parameters of the soil samples. It can be seen that the moisture content of the frozen–unfrozen sandy soil samples was 0.6~83%, and the dry density was 0.88~2.34 g·cm⁻³. As shown in Figure 2 and Figure 3, the distribution of moisture content and dry density of sand, respectively, are given. The samples with moisture content of 5%, 15%, and 25% were in the majority, and then decrease successively. Samples with moisture content of more than 20% accounted for about 40% of the total number of samples. The soil samples tested and reported on in this paper had a wide range of moisture content, including dry soil, saturated soil, and supersaturated soil. Meanwhile, it is noted that the dry density of the sandy soil samples tested was relatively high, with an average value of 1.71 g·cm⁻³. The dry density of the samples was mainly concentrated in the 1.6~2.0 g·cm⁻³ range, which accounted for 77.67% of the total number of samples. Correspondingly, only 16.02% and 6.31% of the samples had dry densities less than 1.6 g·cm⁻³ and greater than 2.0 g·cm⁻³.

2.2. Calculation Method of Thermal Diffusivity

Thermal diffusivity is defined as the ratio of thermal conductivity to volumetric specific heat, which is a comprehensive physical parameter reflecting the speed of temperature increase of an object. The calculation formula follows:

$$\alpha =\frac{\lambda }{C}$$

(1)

where α is the thermal diffusivity of the soil; λ is the thermal conductivity of the soil, W/(m·K); and C is the volumetric specific heat, kJ/(m³·K).

In this work, the thermal conductivity was measured by the transient plane source method, and the theoretical value of specific heat was calculated by the mass weighting method based on the test result of the sample’s dry density and natural moisture content [22]. Then, the thermal diffusivity of every soil specimen can be calculated by Equation (1).

The TPS1500s Thermal Conductivity Analyzer (Hot Disk Co., Ltd., Uppsala, Sweden) was utilized as the experimental apparatus for determining the soil sample thermal conductivity (as shown in Figure 4), which had the advantages of analysis speed, unaffected by contact thermal resistance, and low requirements for test samples. The basic technical parameters of the Hot Disk1500 follow: thermal conductivity range: 0.005~500 W/(m·K); temperature range: 10~1000 K; test accuracy: ±3%; probe size: 2.0~29.4 mm.

The natural density and moisture content of the soil samples were measured in situ and sealed for preservation. Then, the indoor remodeling test was carried out according to the geotechnical standard testing method, which mainly involved sample preparation, pre-treatment before the test, and the thermal property test (as shown in Figure 5). Samples in which the natural moisture content was below saturated moisture content were prepared by directly mixing water to make unfrozen soil samples. Saturated or supersaturated soil samples were made into frozen soil samples by adding smoothies to the unfrozen soil sample. Furthermore, in order to improve test accuracy, samples larger than 80 mm in diameter and 30 mm in height were fabricated using sampler and hydraulic press demudding, and circular Kapton probe with diameter of 29.4 mm was applied in the test. The Kapton probe was clamped in a flat sample room and fixed in a sample rack. The unfrozen sample was tested at room temperature, while the frozen soil sample was tested in a more complicated process, which required pre-freezing, leveling, and constant negative temperature experimental procedures.

The volume specific heat was calculated according to the formula given in the literature [22]. This study assumed that the soil specific heat was the mass-weighted average result of the multiphase components (soil skeleton, water, and ice), and the specific heat of frozen and unfrozen soil was significantly different due to the existence of solid ice. The specific heat calculation formulas follow:

$$C_u=\frac{C_{su}+W\cdot C_w}{1+W}{\rho }_u$$

(2)

$$C_f=\frac{C_{sf}+\left(W-W_u\right)\cdot C_i+W_u\cdot C_w}{1+W}{\rho }_f$$

(3)

In the formula, C_u and C_f are the volume specific heat of unfrozen soil and frozen soil, kJ/(m³·K); C_su, C_sf, C_w, and C_i are the mass specific heat of unfrozen soil skeleton, frozen soil skeleton, water, and ice, respectively, KJ/(kg·K); W and W_u are total moisture content and unfrozen moisture content, respectively; ρ_u and ρ_f are the natural density of unfrozen soil and frozen soil, kg/m³. The specific heat values of the soil skeleton, water, and ice used in the calculation are shown in

Table 2. Specific heat values of soil skeleton and water mass.

Category	Unfrozen Soil	Frozen Soil	Water	Ice
Specific heat/kg·(kg·K)−1	0.84	0.73	4.18	2.09

.

3. Analysis of the Distribution Characteristics and Influencing Factors of the Thermal Diffusivity of Frozen and Unfrozen Soil

3.1. Distribution Characteristics of Thermal Diffusivity of Frozen and Unfrozen Soil

The probability distribution of thermal diffusivity of frozen and unfrozen soil with different particle sizes is given in Figure 6 and Figure 7. It can be seen that the distribution characteristics of frozen and unfrozen soils are similar. The main distribution interval of thermal diffusivity is distributed in the order of silt, medium-fine sand, and gravel sand, while the corresponding main distribution interval values increase sequentially. In order to facilitate comparison and analysis, the average thermal diffusivity coefficients of the frozen and unfrozen soils of various sandy soils are calculated. The average thermal diffusivity coefficients of silty sand, medium-fine sand, and gravel sand unfrozen soil were 0.615 × 10⁻⁶, 0.706 × 10⁻⁶, and 0.746 × 10⁻⁶ m²/s, respectively. The average values of the thermal diffusivity of frozen soil were 1.117 × 10⁻⁶, 1.220 × 10⁻⁶, and 1.222 × 10⁻⁶ m²/s, respectively. It can be shown that the thermal diffusivity of sandy soil is positively correlated with the particle size. With the increase of sand size, thermal diffusivity of sandy soil increases significantly.

In addition, statistics were analyzed for the main distribution intervals of the thermal diffusivity of the frozen–unfrozen sandy soil. To eliminate sampling random errors and calculation errors, the cumulative distribution probability 20~80% interval statistics indicate that the thermal diffusivity of unfrozen soil is 0.577 × 10⁻⁶~0.826 × 10⁻⁶ m²/s, and the thermal diffusivity of frozen soil is 0.999 × 10⁻⁶~1.402 × 10⁻⁶ m²/s. Figure 8 and Figure 9, respectively, show the distribution relationship between thermal diffusivity and dry density and the moisture content of frozen and unfrozen soil samples. It can be seen that the corresponding distribution characteristic of thermal diffusivity under different dry density and moisture content is generally discrete and random in a natural state. There is no obvious linear relationship between thermal diffusivity and dry density or moisture content. As mentioned above, this is because the thermal diffusivity of frozen soil is determined by the characteristics of many factors (e.g., mineral skeleton composition, particle size, geothermal, depth, dry density, and moisture content).

3.2. Analysis of Influencing Factors of Thermal Diffusivity of Frozen and Unfrozen Soil

The partial correlation among natural moisture content, dry density, and thermal diffusivity of frozen–unfrozen sandy soil is analyzed statistically, and the results are shown in

Table 3. Partial correlation analysis results of thermal diffusivity and influencing factors.

Variable	Related Parameters	Moisture Content	Dry Density	Thermal Diffusivity of Unfrozen Soil	Thermal Diffusivity of Frozen Soil
Moisture content	correlation	1.000	−0.565	−0.132	−0.058
	significant	-	0	0.139	0.418
Dry density	correlation	−0.565	1.000	0.224	−0.012
	significant	0	-	0.011	0.864
Thermal diffusivity of unfrozen soil	correlation	−0.132	0.224	1.000	0.879
	significant	0.139	0.011	-	0

. It can be seen from the table that the thermal diffusivity of frozen soil is negatively correlated with natural moisture content and dry density, and the thermal diffusivity of frozen soil is also negatively correlated with natural moisture content but positively correlated with dry density. At the same time, it can also be found that a strong negative correlation exists between the dry density and natural moisture content, and a significant positive relation exists between the thermal diffusivity of frozen and unfrozen sandy soil.

Figure 10 exhibits the distribution relationship for the thermal diffusive coefficient of frozen and unfrozen soil. It can be seen that the thermal diffusive coefficient of frozen soil has a wider distribution range, concentrating in the range 0.8 × 10⁻⁶~1.6 × 10⁻⁶ m²·s⁻¹, while the thermal diffusive coefficient of unfrozen soil is concentrated in the range of 0.6 × 10⁻⁶~0.9 × 10⁻⁶ m²·s⁻¹. Additionally, the thermal diffusivity of frozen soil has a significant linear relationship with the thermal diffusivity of unfrozen soil.

4. Prediction of Thermal Diffusivity of Frozen and Unfrozen Soil

4.1. Prediction Models of Thermal Diffusivity of Frozen and Unfrozen Soil

4.1.1. Binary Empirical Formula Fitting

By the curve estimation of the fitting relationship between the dry density, natural moisture content, and thermal diffusivity of frozen and unfrozen sandy soil, it is found that the fitting curve of thermal diffusivity is roughly in the form of a functional polynomial. The general fitting formula of thermal diffusivity of frozen and unfrozen sandy soil is obtained as follows:

$$\alpha =a+b\omega +c{\omega }^2+d{\omega }^3+e{\rho }_d+f{\rho }_d^2$$

(4)

where α is the thermal diffusivity of the soil, 10⁻⁶ m²/s; ω is the natural moisture content, %; ${\rho }_d$ is the dry density, g/cm³; a, b, c, d, e, and f are all undetermined fitting coefficients, and the fitting values of each soil type are shown in

Table 4. Related parameters of thermal diffusivity binary fitting.

Parameter	a	b	c	d	e	f
Unfrozen soil	0.572	0.046	−0.002	3.176 × 10−5	−0.354	0.175
Frozen soil	2.149	0.045	−0.001	9.072 × 10−6	−1.688	0.529

.

4.1.2. Ternary Fitting of Frozen Soil Based on Thermal Diffusivity of Unfrozen Soil

Compared with unfrozen soil, the sampling and testing procedures of frozen soil samples are more complicated. For example, thermal diffusivity of unfrozen soil may reflect the information of particle size and soil composition to some extent, and the partial correlation analysis results reported above also show that the thermal diffusivity of frozen and unfrozen soil has a strong positive linear correlation. Therefore, the thermal diffusivity of frozen soil was fitted by parameters of natural moisture content, dry density, and thermal diffusivity of unfrozen soil. The ternary fitting formula is given as follows:

$${\alpha }_f=-0.75+1.757{\alpha }_u+0.271\omega +0.037{\rho }_d$$

(5)

where ${\alpha }_f$ and ${\alpha }_u$ are the thermal diffusivity of frozen and unfrozen soil, 10⁻⁶ m²/s.

4.1.3. Radial Basis Function (RBF) Neural Network

RBF neural network is a three-layer pre-feedback neural network, which has the characteristics of strong local approximation ability, fast learning convergence speed, small calculation amount, and strong generalization ability. The strong nonlinear approximation ability of RBF neural network has a wide range of applications in many fields [23,24]. JMP V13.2.0 Pro software was used to take moisture content, dry density, and unfrozen sample thermal diffusivity (ternary model) as the input layer of the RBF neural network, and the pre-feedback model was established. The model uses a Gaussian function as the hidden layer activation function and the hidden layer is set to 2 layers with 12 nodes in each layer (as shown in Figure 11).

Because of the small number of test samples, the prediction model may have errors caused by underfitting. Meanwhile, to minimize the error, the optimal data processing strategy is to increase the number of training sets. Therefore, the test results of thermal diffusivity of unfrozen soil are randomly divided into blocks according to 9:1, in which 90% of the sample data are used as the training samples of the neural network model, and the remaining 10% are used to verify the prediction ability of the neural network. Furthermore, samples with large deviation in prediction results are removed in order to effectively improve prediction accuracy. Figure 12 illustrates the prediction results of the RBF neural network model for thermal diffusivity of unfrozen soil. It can be seen that the predicted value closely matches the calculated value (R² = 0.79).

4.2. Comparison of Prediction Models for Thermal Diffusivity of Frozen and Unfrozen Soil

The prediction results of the binary fitting model and RBF neural network prediction model of unfrozen soil were calculated, as shown in Figure 13. It is noted that most of the predictive points are distributed within ±10% relative error range. The proportion of predictive points for the binary fitting model within the ±10% error range is 65.69%, while the proportion of predictive points for the RBF neural network model within the ±10% error range is 75.41%. The predicted values obtained by the two prediction models are in general agreement with the calculated values, which proves the effectiveness and engineering application value of the two prediction models. In addition, by comparing the two prediction methods, it clearly shows that the prediction effect of the RBF neural network model is better, which has about 10% more predictive points within the ±10% error range and has more predictive points at both ends of the thermal diffusivity distribution range.

The prediction effect of the four prediction models for frozen soil was calculated and compared; the results are shown in Figure 14. Generally, the four models all have good prediction performance. The determination coefficient (R²) of the binary fitting, binary RBF neural network, ternary fitting, and ternary RBF neural network prediction methods is 0.26, 0.69, 0.86, and 0.97, respectively. Additionally, the proportion of predictive points within the error range of ±10% are 52.50%, 57.75%, 74.14%, and 95%, respectively, indicating that the prediction accuracy of the four prediction models improved successively. However, by comparing the number of predictive points within ±10% error range and the differences between the calculated values and the predictive values, it can be found that, by using the unfrozen soil thermal diffusivity as a regression parameter, the ternary fitting method has higher accuracy than the binary prediction method (only using ω and ${\rho }_d$). Moreover, the prediction effect of the RBF neural network method is obviously better than that of the polynomial fitting method, especially for both ends of the interval.

4.3. Application Comparison of Thermal Diffusivity Prediction Model

Table 5. Comparison of prediction results of different models.

Evaluation Parameters	Binary Fitting		Binary RBF Neural Network		Ternary Fitting		Ternary RBF Neural Network
	Unfrozen Soil	Frozen Soil	Unfrozen Soil	Frozen Soil	Unfrozen Soil	Frozen Soil	Unfrozen Soil	Frozen Soil
R2	0.53	0.26	0.79	0.69	-	0.86	-	0.97
$\theta$	8.42	10.90	7.08	9.35	-	7.08	-	1.76
$P_{\pm 10\%}$	65.69	52.50	75.41	57.75	-	74.14	-	95.00
Fitting parameters	${\rho }_d$, ω		${\rho }_d$, ω		${\rho }_d$$, \omega , {\alpha }_u$		${\rho }_d$$, \omega , {\alpha }_u$

compares the results of R², mean relative error ($\theta$), and proportion of error within ±10% ($P_{\pm 10\%}$) for the different thermal diffusivity prediction models. It can be seen from the table that among the two prediction methods of thermal diffusibility of unfrozen soil, the prediction results of the RBF neural network model are better than that of the empirical formula fitting model. The R² and proportion of predictive points within ±10% error of the binary RBF neural network model for unfrozen soil are 0.79 and 75.41%, respectively, while the binary fitting model are only 0.53 and 65.69%.

In addition, by comparing the four models of thermal diffusibility of frozen soil, it can be seen that the proportion of the ternary model predictive points within ±10% error lines is about 20% higher than that of the binary model. Moreover, the R² is also significantly higher than that of the binary model, which further indicates that the thermal diffusivity of unfrozen soil contains extremely important soil composition information, which can effectively compensate for the defects of its incomplete parameter regression model. Meanwhile, by comparing the ternary RBF neural network method with the ternary empirical formula fitting method, it can be found that the accuracy of the ternary RBF neural network method is higher than that of the ternary empirical formula fitting method. The R² and $P_{\pm 10\%}$ of the ternary RBF neural network model for frozen soil are as high as 0.97 and 95%, indicating that the RBF neural network method can more effectively capture the characteristics of the relationship between the thermal diffusivity and the influencing factors, thus improving the prediction accuracy.

5. Conclusions

Revealing the variation law of thermal diffusivity of sandy soil can provide a theoretical basis for engineering design and construction in cold and arid regions, which is sensitive to the temperature variation. In present work, the distribution characteristics and influence law of various parameters of thermal diffusivity were analyzed based on the experimental data of drilled sandy soil samples. The prediction model based on the empirical fitting formula and RBF neural network method for the thermal diffusivity of frozen and unfrozen sandy soil was established, and the prediction accuracy of different prediction methods compared. The main conclusions follow:

• Dry density and natural moisture content have significant correlations with thermal diffusivity of sandy soil, but the correlations have differences in the frozen and unfrozen states. Additionally, the thermal diffusivity of the frozen and unfrozen soil has a significant positive linear relationship.
• The binary fitting method only needs dry density and natural moisture content as fitting parameters, which has the advantage of easy access to data and can meet the basic requirements of engineering estimation. However, the binary fitting method has a narrow application range and relatively large error on both sides of the thermal diffusivity distribution interval.
• For unfrozen sandy soil, the binary RBF neural network prediction model is obviously better than that of the binary empirical fitting formula model. The ternary prediction model has significantly higher prediction accuracy than that of the binary prediction model, and the ternary RBF neural network model has the best prediction effect among the four methods, despite that it is only limited for frozen soil.