The latest mathematical and empirical models to calculate the thermal conductivity of the soils

The paper presents the latest models for calculating the thermal conductivity of soil. Precise determination of this parameter is necessary for the correct and safe location of geoengineering objects, underground infrastructure such as cables or ground heat exchangers. A universal model that is easy to apply and gives the most accurate results has not yet been developed. New models are constantly being developed. The aim of this work is to present the latest models for calculating thermal conductivity, so that knowing the properties of the soil, it is possible to select an appropriate model to calculate its conductivity.


Introduction
Correct determination of the thermal parameters of the ground is of great importance for the safe and ergonomic location of the geoconstruction, underground infrastructure such as cables, pipelines, ground heat exchangers. The basic thermal properties of soil are thermal conductivity, thermal diffusivity and heat capacity [1]. The main mechanism by which heat flows through the ground is conduction. Convection plays a significant role only in very well-drained soils, such as gravel [2]. Radiation is relevant only for surface soils. The basic parameter describing the heat transfer in the soil is the thermal conductivity of the soil. It is a measure of a soil's ability to transfer heat due to a temperature difference and depends on many factors such as humidity, dry density, mineralogy, particle size distribution, time and temperature [1]. Despite the fact that there are many methods of measuring the thermal conductivity of the soil (e.g. the heat plate method, the thermal pulse method, the linear source method), the direct measurement of the thermal conductivity of the soil may turn out to be impractical in large-scale studies due to the time-consuming, labor-intensive and relatively high costs. During the last few decades, many models for calculating the thermal conductivity of the soil have been developed [3]. These models are empirical or theoretical/mathematical, with most theoretical models having been empirically modified, resulting in semi-empirical models [4]. Empirical models (e.g., de Vries (1963); Gemant (1952); Johansen (1975); Kersten (1949); Kunii and Smith (1960); McGaw (1969); Mickley (1951); Smith (1942); Van Rooyen and Winterkorn (1959); Woodside and Messmer (1961)) are based on the analysis of the measured thermal conductivity of the soil as a function of easily measurable soil parameters such as soil water * Corresponding author: agnieszka.drzyzga@doktorant.pk.edu.pl saturation and density [3]. Theoretical/ mathematical models treat the soil as a three-phase system consisting of air, water and a mineral part [5]. These models are based on other physical models designed to describe other soil properties and have a complex computational process [1]. The aim of this article is to review the latest mathematical and empirical models for calculating the thermal conductivity of the soil.

Models of thermal conductivity of soil
There are many models available in the literature for calculating the thermal conductivity of the soil. Comparisons of the available methods of determining the thermal conductivity of soils were undertaken, among others, by Farouki [6], He et al. [7], Rerak [8], Zhang [9], Różański [10]. Farouki [6] compiled and compared 11 methods for determining the thermal conductivity of soil, incl. Kersten [11], de Vries and Afgan [12,13] and Johansen [14]. He took into account soils of various grain size (fine-grained and coarsegrained) and water content in the soil. According to Farouki [6], the Johansen method [14] works best for fine-grained, unfrozen soils, while with a saturation below 20%, the original Johansen method lowers the thermal conductivity by about 5-15% [15]. In such a case, better results are obtained by applying the modified Johansen method described by Peters-Lidard et al. [16]. Zhang [9] analyzed 13 predictive models and assessed their performance for sands under varying humidity conditions. The best results were obtained when the Zhang et al. [17] model were used, Chen [18], and Haigh [19]. Różański [10] compiled 15 models, as well as different approaches used for the Kersten number and the conductivity of dry and saturated soil presented by Cote and Konrad [20,21], Lu et al. [22], He  [23]. Despite the fact that there are many models for calculating the thermal conductivity of the soil, more are still being developed in order to create a model that is universal, easy to apply and gives the most accurate forecasts.
3 Review of the latest models of soil thermal conductivity The latest calculation models for the thermal conductivity of soils are presented below. This compilation includes mathematical and empirical models. Numerical models are also included.

Empirical models
Empirical models are created based on the numerical or analytical analysis of experimental data between the effective thermal conductivity and easily measurable soil properties such as moisture, bulk density or degree of saturation. Their credibility largely depends on the accuracy of the experimental data. However, the application of empirical models should be narrowed down to the soil types or geomorphological regions on which they were developed, as they are difficult to apply to soils outside the studied range of saturation levels, bulk density or textures [23].

The model of Lu and Dong [24]
Lu and Dong [24] proposed an innovative closed equation for thermal conductivity as a function of water content. They investigated the influence of all types of soil and humidity on thermal conductivity when the ambient temperature is within 20-25oC. In the proposed equation, Lu and Dong [24] took into account soil-water retention regimes. The equation was verified on the basis of 27 different types of soil presented from literature and research. It has the form: where k is the thermal conductivity of the soil [W/mK], kdry, ksat are the dry and saturated thermal conductivity, respectively [W/mK], Sr is the degree of saturation, Sf is the degree of saturation at which the cableway regime begins (mainly water desaturation occurs ) and m determines the connectivity of the fluid network and represents the rate of change of thermal conductivity with a water content. Sf and m are fit parameters that can be related to the behavior of water in the soil.

The model of Zhang et al. [25]
Based on the concept of normalized Ke thermal conductivity presented by Johansen [14], as well as the Ke-Sr relation taking into account the influence of the soil type proposed by Cote and Konrad [20,21], Zhang et al. [25] proposed a model of thermal conductivity for quartz sands: where ks is the thermal conductivity of the solid particles [W/mK], n is the porosity, к is the parameter depending on the soil texture, η and χ are the shape dependent coefficients.

The model of Zhang et al. [26]
Zhang et al. [26] proposed a generalized model of soil thermal conductivity based on laboratory tests. They considered pure sands with a quartz content of 99%, kaolin clays and mixtures of sand and kaolin clay in various proportions, with different porosity and degrees of saturation. Zhang et al. [26] presented a new relationship between kr -Sr as follows: where, q is the quartz (sand) content [%]. For kdry and ksat the following equations were proposed: where kq is the thermal conductivity of quartz and kkaolin is the thermal conductivity This model takes into account the influence of factors such as porosity and degree of saturation, as well as quartz content and soil type, on the thermal conductivity of soils.

The model of Tarnawski and Leong [27]
Tarnawski and Leong [27] developed an advanced geometric mean model (A-GMM) to calculate the effective thermal conductivity keff of unsaturated soils based on the thermal resistance coefficient of intermolecular contact, the degree of saturation of a small pore space and the thermal conductivity of the soil skeleton: where α is the coefficient of intermolecular contact resistance, ks is the thermal conductivity of the soil skeleton [W/mK], ka is the thermal conductivity of the air [W/mK].

The Model of Tong et al. [28]
Tong et al. [28] proposed a simple model for estimating the thermal conductivity of the soil, which requires only two input parameters such as porosity and water volume content: where θ is the volumetric humidity, a and b are empirical parameters, and the parameter c is 3.90. The application of the model is limited only to soils with porosity in the range of 0.40 to 0.55.

The model of He et al. [23]
He et al. [23] based on the models of van Genuchten (van Genuchten, 1980) and Lu et al. [22,24] proposed a new standardized model for estimating the thermal conductivity of soil. They proposed a linear function to calculate the thermal conductivity of dry soil on the basis of porosity. The Kersten number Ke is expressed as: Ke=0, θ=0 where A and B are the fit parameters. The new model is as follows: The thermal conductivity of saturated and dry soils is calculated from the equations: where a and b are empirical parameters.

The model of Zhao et al. [3]
Zhao et al. [3] proposed a new, generalized model for studying the thermal conductivity of soil, which takes into account the influence of porosity, degree of saturation, organic matter content and soil texture on its thermal conductivity. The model is applicable to soils of various textures and all degrees of water saturation and takes the form of a logarithmic equation: where A1 and B1 are parameters depending on the composition, structure and texture of the soil, ln is the natural logarithm function and Sr is the degree of saturation which takes into account the effect of the soil bulk density. The A1 parameter depends on the soil porosity and expresses the thermal conductivity of dry soil, while the B1 parameter depends on the texture and structure of the soil, as well as the difference between the dry and saturated thermal conductivity. The parameters A1 and B1 are derived directly from the lower and upper boundary conditions as follows: lower boundary condition: Sr = 0 => k =kdry i k = A1 => A1 = kdry upper boundary condition: Sr = 1 => k =ksat i k = kdry + B1 × ln(2) => B1 = (ksat -kdry)/ln(2) thus the thermal conductivity equation is written as follows: The two-phase model of the geometric mean by Cote and Konrad was used to estimate the kdry and the ksat [20,21].
3.1.8 The model of Wang et al. [29] With an increase in the amount of precipitated CaCO3, the dry density of the soil increases, as well as the thermal conductivity of the soil. In order to study the thermal conductivity of sands, Wang et al. [29] subjected them to a microbial calcite recovery (MICP) treatment. The thermal conductivity of both untreated and treated sands at various degrees of saturation in the drying process was measured. Since the k of the MICPtreated sand is higher than that of the crude sand, the improvement in thermal conductivity of sand I was calculated from the equation: where T0 is the surface temperature of the ground, Ta is the air temperature, w is soil moisture and WVP is the pressure of the water vapor surface. The second equation was found to be more convenient to use because the WVP measurement was more readily available.

The model of Tian et al. [31]
Tian et al. [31] presented the relationship between the thermal conductivity of partially frozen soils and the pores filled with air na. The proposed exponential relationship between k-na has the following form: The model is easy to use, it ignores the influence of temperature on the thermal conductivity of frozen soil and works best for soils with a temperature not greater than -4 o C.

The model of Tian et al. [32]
Tian et al. [32] presented the correlation between thermal conductivity and geophysical parameters such as wave velocity and wave density. Using exponential fit, they determined the relationship between thermal conductivity and wave velocity: where k is the thermal conductivity [W/m o C] and Vs represents the wave velocity [m/s]. They also determined the relationship between thermal conductivity and density: where ρ is the density at the appropriate depth. These correlations make it possible to determine the thermal conductivity quickly, economically and non-invasively of any zone in which no tests have been performed.

The model of Xiao et al. [33]
Xiao et al. [33] proposed an empirical equation taking into account the general relationship between the porosity coefficient and the relative density by taking into account the influence of the shape of the particles: where ID is the relative density, ke0, ��� � , ��� � , ��� � , ��� � are the fit parameters and OR is the particle shape parameter.

The model of Song et al. [34]
The model for determining the thermal conductivity of unsaturated clays presented by Song et al. [34] takes into account the shares of individual minerals in the soil as well as the content of water and air in the soil pores. The assumption of the model is an even distribution of solid particles water and air in the soil and the basic unit of soil is represented by spheres each of which consists of a solid, liquid and gas phase. The equation for calculating the thermal conductivity of unsaturated clay is as follows: where ks is the thermal conductivity of the solid phase and is � = ∏ � �

The model of Sun et al. [35]
Sun et al. [35] developed a model of thermal conductivity and electrical resistance for silty clays in the temperature range -20 o C to 10 o C. For frozen soil, the equation of thermal conductivity is as follows: where z is the water content at a given freezing degree, θ is the water content by volume and θ=nSr, A=ks(1-n), B= ki, C=kw/ki, where ki is the thermal conductivity of the ice. Sun et al. [35] showed that in the temperature range 0 o C to 10 o C, thermal conductivity decreases with electrical resistance and can be expressed by the equation: where rs, ra, ri are the electrical resistance of the solid part, air and ice respectively [Ωm]. In the temperature range -10 o C to -20 o C, the equation for thermal conductivity has the form: The thermal conductivity varies linearly with the electrical resistance here.

The model of He et al. [36]
He et al. [36] proposed a new model to predict the effective thermal conductivity keff of a soil from the matrix potential ψ (ie at high water content). Using the analogy between keff(ψ) and water retention in soil proposed a function depending on the air entry point ψb: where A2 and A3 are the fitting parameters, pF is the full matrix potential range. This model can be used across the entire range of the matrix potential.
3.1. 16 The model of He et al. [37] He et al. [37] proposed a new method to estimate the thermal conductivity of soil solids ks with different textures and water content. Knowledge of the mineralogical composition of the soil is not required. This method can be used for fine-grained soil and land for which there is no measured ksat. It then requires at least three measurements between ks and ksat: where the coefficients A and B and the thermal conductivity of the constant part ks are the fitting parameters.

Mathematical models
Mathematical models are created from prediction models of other physical properties of soils, such as dielectric permittivity, magnetic permeability, electrical conductivity or hydraulic conductivity. They are calculated using specific mathematical algorithms. They take into account the thermal conductivity of individual soil components [24]. Mathematical models assume a simplified geometry of soil grains in order to obtain a mathematical relationship [9].

The model of Ofrikhter et al. [5]
Developed by Ofrikhter et al. [5] the cube model is used to estimate the thermal conductivity of the soil based on known soil parameters, such as soil porosity and the degree of water saturation. The soil is treated as a threephase medium consisting of air, water and a mineral part (Fig. 1) and there is no heat exchange due to heat and mass transfer through the groundwater. The model assumes that the heat flow is limited to a single direction parallel to the Z axis (Fig.2). The total heat flux Qtot passing through the cube is divided into individual fluxes Qi passing through a different combination of soil components. The model assumes 5 paths of heat flow (Fig.2). The formula for the effective thermal conductivity of the soil is as follows: where l is the length of the unit cube, ∆T is the temperature gradient. Miter cut [5] After substituting, the formula for effective thermal conductivity is: where ks is the effective conductivity of the mineral part, kw is the effective conductivity of the water, ka is the effective conductivity of the air. Thermal conductivity of water and air depends on temperature and is determined from the formulas: The thermal conductivity of the mineral part of the soil sample is determined from the measured thermal conductivity, temperature, porosity, and water saturation coefficient of the sample by any available means. The radius of the sphere reflecting the content of the mineral part and the water is determined from the formulas: � = 120,7 � − 207,9 � + 144,6 � − 52,55 � + 10,97 � − 1,73 + 0,755, This model is applicable to soils with porosity and water saturation within the specified limits: If the water saturation coefficient exceeds the value indicated in the roughness it is assumed that the soil is saturated, and the air contained in the soil has practically no effect on its thermal conductivity.

The model of Zhu [38]
Zhu [38] developed a new cell model to determine the effective thermal conductivity of an unsaturated porous medium. It treats the soil as a two-phase mixture consisting of a liquid part (which is a mixture of liquid and gas) and a solid part (Fig.3). The thermal conductivity of mixed liquid phases is expressed as the volume-weighted average thermal conductivity of a gas and a liquid: where kf is the thermal conductivity of the mixed liquid phase (water and air) [W/mK], ks is the thermal conductivity of the solid phase [W/mK], Rs is the radius of the solid particle and represents the average radius of the solid particles in unsaturated porous materials, Rf is the radius mixed fluid outer shell and is related to the porosity of unsaturated porous materials. The equation of effective thermal conductivity expressed as the ratio of keff to porosity has the form: It can also be expressed as the ratio of keff to the thermal conductivity of the liquid phase kl: This model takes into account the effect of saturation with the water phase on the effective thermal conductivity.

Numerical models
Wen et al. [39] developed a model based on multiple linear regression (MLR) and artificial neural networks (ANN) for soils in the full range of saturation. The input parameters in the input layer are clay content, silt content, sand content, thermal conductivity of solids, thermal conductivity of dried soils, thermal conductivity of saturated soils, quartz content and porosity. Rizvi et al. [40] developed an artificial neural network (ANN) model based on a fast learning (DL) algorithm. This model is applicable to the calculation of the effective thermal conductivity of unsaturated sandy soils. It has one input layer (which receives three parameters: porosity, degree of saturation and quartz content), three hidden and one output. Zhang et al. [41] developed the ANN model. They developed separate predictive models for clay, silt, silty sand, fine sand and coarse sand, as well as a generalized model covering five different soil types. The input parameters for individual models were humidity and dry density. For the generalized model, the input parameters were moisture, dry density, clay content, and quartz content (clay and quartz represent changes in gradation and mineralogy). Shrestha and Wuttke [42] used an artificial neural network approach to predict the thermal conductivity of geomaterials taking into account their gradation, porosity and mineralogy. The input parameters for the input layer are porosity, quartz content, curvature factor and uniformity factor.

Conclusions
There are many models of the thermal conductivity of the soil in the literature. However, none of them is universal enough to be used in any case. Therefore, there is a need to constantly develop new models. The latest models for calculating the thermal conductivity of soils are presented above, but they are also not universal models with a wide range of applications. However, this compilation will help you find an equation for