# Estimating the Thermal Conductivity of Unsaturated Sand
## 1 Introduction  
In recent decades, with the development of energy engineering, soil science, and agricimatology, the study of the thermal conductivity of soil (\(\lambda\), Wm\({}^{-1}\)c\({}^{-1}\)) has always occupied a crucial place in soil thermal property research [1; 2; 3]. As the direct impact factor in initial temperature changes of soil, the soil mechanical properties and soil ecosystem are affected by the temperature gradient distribution, which is associated with the thermal conductivity of soil [4]. A number of studies have been conducted on the experimental measurement of the thermal conductivity of soil. [PERSON] et al. [5] measured the variation of four sand thermal conductivities \(\lambda\) with differences in porosity and found that \(\lambda\) was a function of moisture content under both transient drainage or drying and wetting conditions. [PERSON] [6] published experimental results for the thermal conductivity \(\lambda\) of four kinds of quartz sands with different particle gradations by the transient thermal probe method. The thermal conductivity \(\lambda\) of sands tended to diminish with increasing porosity; in contrast, it increased with increasing moisture content. A series of thermal conductivity experiments was performed by [PERSON] et al. [7], who tested six soils and three rocks from the region around Melbourne, Australia, and measured the impacts of moisture content, dry density, mineralogical composition, and particle size on the sample thermal conductivities. [PERSON] et al. [8] used the heat-pulse method to measure the thermal conductivity \(\lambda\) of six soils under wide ranges of moisture contents and bulk densities and proposed a new model that could accurately describe the trend of thermal conductivity \(\lambda\). Therefore, the majority of the experimental research investigated the effects of physical characteristics, including water content, bulk and particle densities, compositional factors, and gradation, on thermal conductivity [9; 10; 11; 12; 13; 14]. There are a variety of models that can predict the soil thermal conductivity \(\lambda\). Three types of models can be distinguished in the present literature: theoretical, empirical, and mixture models [15]. Theoretical models were frequently derived from simplistic mathematical models and analyses based on heat transfer mechanisms [16], soil grain geometries [17; 18], and other properties, e.g., dielectric permittivity. However, the theoretical models generally applied homogenization assumptions to consider the effective thermal property of a given component, and none of the theoretical models considered the microstructure of the soil [19]. The empirical soil thermal conductivity models were developed using datasets of experimental measurements of soil. [PERSON] [20] proposed a kr-Sr relationship between normalized thermal conductivity and soil physical properties, such as soil type, porosity, degree of saturation, and mineral component. [PERSON] [21] proposed an empirical model based on a series of laboratory measurements of 19 types of soils, and the empirical model described the relationships between thermal conductivity, moisture content, and dry density. However, empirical models are only persuasive for specific soils and lack a clear physical foundation because of differences between the physical properties of each natural soil [19]. The most common mixture models are based on classical mixing laws of the series thermal conductivity (STC) model and parallel thermal conductivity (PTC) model [19]. Compared to theoretical and empirical models, the advantages of mixing models are more obvious since they are based on the physical and heat transport properties of porous media [22]. Two basic mixture models, STC and PTC models, determine the volume percentage and thermal conductivity of each phase in media [14; 23; 24; 25; 26]. [PERSON] et al. [27] developed a numerical model that combines a heat transport model and a heat exchanger model to predict the heat exchange rates for a ground source heat pump system.