## 3 Modelling  
### Previous Research  
Mathematical characterisation and modelling of soil thermal conductivity under different structural and physical conditions have been a consistent interest in research. [PERSON] et al. [26] reviewed all the major soil thermal conductivity models and compared their performance. These models were classified into three groups: mixing model, empirical model, and mathematical model. Mixing models take soils as a mixture of three elementary phase components, solid particles, water, and air. The effective bulk soil thermal conductivity depends on the individual thermal properties of these elementary components and their arrangement in soils, which takes two typical forms, in series and in parallel. The empirical models have little physical foundation, while the mathematical models compare the thermal conductivity to other physical properties, such as electrical conductivity [27],  
Figure 6: The thermal Conductivity vs. water content (**a**) and water saturation (**b**) at different porosities. In terms of the tests on varied soils, including sands, silts, and clay, it has been found that both the mixing and mathematical models demonstrated an underperformance in general, because it is difficult to adequately represent all the material characteristics related to the thermal conductivity of soils, such as the particle geometry, pore size distribution, and the configuration of the water bridges between particles or at the pore necks/throats. Recently, other new methods considering the involved underlying physics have also been investigated. [PERSON] and [PERSON] [28] put forward an improvement for the geometrical mean model proposed by [PERSON] and [PERSON] [19] by introducing an inter-particle thermal contact resistance factor to reflect the interaction between the involved three phases (i.e., soil solids, water, and air). [PERSON] and [PERSON] [29] presented a percolation-based effective-medium approximation (P-EMA) model, in terms of the properties of the elementary phases, a scaling exponent, and a percolation threshold. It has been concluded that the thermal conductivity under fully and partially saturated conditions conforms to non-universal behaviour. [PERSON] [15] predicted and compared the thermal conductivity and water retention curve of unsaturated soils based on the water configuration retained in an idealized unit representative pore formed in a simple cubic pack of same sized spheres. Taking soils as a fractal structure to estimate the thermal conductivity has also been an effective approach [30]. [PERSON] et al. [31] made new progress in this direction. So far, there has not been a comprehensive model able to effectively describe and generally define all the major underlying mechanisms involved in soil thermal conductivity. In practice, fitting empirical models to experimental data is still the most effective and popularly way employed for real world applications [32; 33]. [PERSON] [34] has given a comprehensive review on soil thermal conductivity empirical models. Later, [PERSON] et al. [35] and [PERSON] et al. [14] updated the list covering other new developments. All of these models, however, were reported and tested on specific soils. Each of them had a limit of application. So far there has not been an accepted model generally applicable on a wide range of soil types, porosities, and water contents. The recent review by [PERSON] et al. [26] has compared the empirical models proposed by [PERSON] [36], [PERSON] [37], [PERSON] and [PERSON] [38], [PERSON] et al. [39], and [PERSON] [18]. It showed that although these models had demonstrated better performance than the mixing and mathematical models, they generally largely underestimated the thermal conductivity of sands. In the next section of this paper, two of the principal types of empirical models for sands are compared against the experimental data of this work. One type has a power function form and the other type an exponential function form.