# Estimating the Thermal Conductivity of Unsaturated Sand
## 3 Model Development
### Equivalent sand Particle Thermal Conductivity \(\lambda_{\text{es}}\)  
When the sand is in a completely dry condition, conductive heat transfer is mainly controlled by contact between particles, which has been discussed previously. However, in the PTC model, the solid particle thermal conductivity \(\lambda_{s}\) is adopted to characterize the property of conductive heat transfer of the solid phase, which does not consider contacts between particles. Therefore, the influence of the contacts between particles could not be sufficiently reflected by the solid particle thermal conductivity \(\lambda_{s}\). Then, the formation of water bridges among particles remarkably enlarges the transfer paths of heat conduction in sand so that the thermal conductivity of sand increases sharply when the sand is under low degrees of saturation. Similarly, the effect of water bridges cannot be adequately represented in \(\lambda_{s}\). Thus, in this study, we proposed using the equivalent thermal conductivity of solid particles \(\lambda_{\text{es}}\) instead of solid particle thermal conductivity \(\lambda_{s}\) in the PTC model; this changed the PTC model to a nonlinear model, which was sufficient to present the effects of the contact features of particles and water bridges near the particle contacts. Detailed derivations of the equivalent thermal conductivity of solid particle \(\lambda_{\text{es}}\) are included below. First, the measured value of the thermal conductivity of sand was determined by testing in the dry state, which contained the effect of contacts between particles. Then, by substituting the measured value in the left side of Equation (2), we obtained the formula by a simple transformation of variables:  
\[\lambda_{\text{es}}=\lambda_{\text{es}}^{dry}=(\lambda_{\text{measured}}-n \lambda_{a})/(1-n)\ S_{r}=0 \tag{3}\]  
where \(\lambda_{\text{es}}^{dry}\) is the equivalent thermal conductivity of solid particles in the completely dry state, which represents the thermal conductivity of the solid phase of sand affected by contacts between particles and is typically less than \(\lambda_{s}\). Finally, the measured value of \(S_{r}\
eq 0\) was substituted in the left side of Equation (2), which was determined by testing, and the formula was obtained as:  
\[\lambda_{\text{es}}=[\lambda_{\text{measured}}-nS_{r}\lambda_{w}-n(1-S_{r}) \ \lambda_{a}]/(1-n)\ \ \ S_{r}\
eq 0 \tag{4}\]  
For example, with Sample A, Figure 3 indicates that the equivalent thermal conductivity of solid particle \(\lambda_{\text{es}}\) increased with \(S_{r}\) and was generally larger than the measured value of the sand thermal conductivity at the same \(S_{r}\). The change in the trend of \(\lambda_{\text{es}}\) with \(S_{r}\) was consistent with the trend of the measured value. The \(\lambda_{\text{es}}\) at the low range of \(S_{r}\) increases rapidly on top of \(\lambda_{\text{es}}^{dry}\) and then goes into a stable region as \(S_{r}\) approaches saturation. When \(S_{r}\
eq 0\), liquid in the pores exerts an effect on heat transfer, including increased contact area between particles, by the presence of a transitory water bridge at low \(S_{r}\) and heat conduction in liquid at higher \(S_{r}\). As a result, the effects of liquid were not captured in \(\lambda_{\text{es}}^{dry}\). Figure 3 shows that \(\lambda_{\text{es}}\) was a function of \(S_{r}\) when \(S_{r}\
eq 0\), and the relationship between \(\lambda_{\text{es}}\) and \(S_{r}\) was described by a hyperbolic expression. Thus, \(\lambda_{\text{es}}\) was expressed by the following equations:  
\[\lambda_{\text{es}} =\lambda_{\text{es}}^{dry}+f(S_{r}) \tag{5}\] \[f(S_{r}) =S_{r}/(A\cdot S_{r}+B) \tag{6}\]  
where \(A\) and \(B\) are sand-dependent parameters. The lower and upper limit conditions of Equation (5) are as follows:  
Lower limit condition: \(S_{r}=0\rightarrow\lambda_{es}=\lambda_{es}^{dry}\);  
Upper limit condition: \(S_{r}=1\rightarrow\lambda_{es}=\lambda_{es}^{dry}+1/(A+B)\).