## 3 Modelling
### Comparison of Two Major Types of Empirical Model  
[PERSON] et al. [40] used the following Equation (2) to model the thermal conductivity of bentonite,  
\[K=K_{dry}\{1+[(9.75n-0.706)S_{w}]^{0.285n+0.731}\} \tag{2}\]  
where \(K_{dry}\) is the dry soil thermal conductivity, and \(n\) and \(S_{w}\) are the porosity and the degree of water saturation, respectively. Later, [PERSON] [18] proposed a similar model in the form of Equation (3) for sandy soils.  
\[K=K_{w}^{n}K_{s}^{1-n}[(1-b)S_{w}+b]^{cn} \tag{3}\]where \(K_{s}\) is the solid thermal conductivity, and \(K_{w}\) is the water thermal conductivity. Mathematically, these two forms may be characterised into a general expression as Equation (4).  
\[K =K_{0}\Big{[}1+(aS_{w})^{b}\Big{]} \tag{4a}\] \[K_{0} =k_{0}k_{1}^{n}\] (4b) \[a =a_{1}n+a_{0}\] (4c) \[b =b_{1}n+b_{0} \tag{4d}\]  
where \(K_{0}\) and \(a\) and \(b\) are three parameters related to porosity, \(n\), and \(k_{0}\), \(k_{1}\), \(a_{0}\), \(a_{1}\), \(b_{0}\), and \(b_{1}\) are six constants, which can be regarded to depend all together upon the soil nature, elementary material properties, pore structure, and pore water configuration.  
[PERSON] et al. [41] proposed the following exponential formula for the thermal conductivity of different soils:  
\[K=K_{w}^{n}K_{s}^{1-n}\exp\Big{[}-3.08n(1-S_{w})^{2}\Big{]} \tag{5}\]  
Later, [PERSON] et al. [39] proposed a similar model, given by Equation (6) below, and tested it on a wide range of soils,  
\[K=\Big{[}K_{w}^{n}K_{s}^{1-n}-(b-an)\Big{]}\exp\Big{[}c(1-S_{w}^{c-1.33}) \Big{]}+(b-an) \tag{6}\]  
These two models may also be characterised into a general mathematical formula as:  
\[K=K_{0}\exp\Big{[}a(1-S_{w})^{b}\Big{]}+c \tag{7a}\] \[K_{0}=k^{n}+c\] (7b) \[a =a_{1}n+a_{0}\] (7c) \[b =b_{1}n+b_{0}\] (7d) \[c =c_{1}n+c_{0} \tag{7e}\]  
where \(K_{0}\) and \(a\), \(b\), and \(c\) are four parameters related to porosity, \(n\), and \(k\), \(a_{1}\), \(a_{0}\), \(b_{1}\), \(b_{0}\), \(c_{1}\), and \(c_{0}\) are seven constants, which depend together on the soil nature, elementary material properties, pore structure, and pore water configuration.  
Figure 7 compares the modelling results using the two characterized general forms to fit the measured thermal conductivity of the sands in this work at varied degrees of pore water saturation and porosities. The degrees of water saturation are calculated in terms of the results in Figure 6 and the intrinsic relationship:  
\[S_{w}=\frac{w\rho_{dry}}{n\rho_{w}} \tag{8}\]  
where \(w\) is the water content, \(\rho_{dry}\) is the dry density of samples, and \(\rho_{w}\) is the water density. Figures 8 and 9 show the fitting results using the parameter functions, Equation (4b-d) and Equation (7b-e), to describe the variation of parameters, \(K_{0}\), \(a\), \(b\), and \(c\), against the porosity.  
Figure 7 shows that both Equations (4) and (7) have produced comparable results of the modelling thermal conductivity against water saturation, for all the four porosity magnitudes, in terms of the agreement with the experimental measurements. Figure 8 shows that the predicted variation of the introduced parameters in Equation (4) against the porosity has presented good accuracy. However, for Equation (7), Figure 9 shows that the variation of the introduced parameters against the porosity does not display an accountable correlation, where the parameter formulas are unable to fit all of the four parameters at different porosity magnitudes. The comparison result suggests that the form of Equation (4) is more simple and flexible than Equation (7), and may work for a wider range of soils.  
Figure 8: \(Cont\).  
Figure 7: The Modelling Results using Equations (4a) and (7a), (**a**) porosity \(n=0.507\); (**b**) porosity \(n=0.479\); (**c**) porosity \(n=0.471\); (**d**) porosity \(n=0.466\).