## 3 Modelling
### A Unified Thermal Conductivity and Water Retention Model
In unsaturated soils, it can be applied on the water and vapour phases coexisting in pore space, respectively, in terms of water content [44]:  
\[p_{w}=p_{0}\exp(\frac{\Delta u_{w}}{RT})=p_{0}\exp(\frac{\Delta u_{w}^{f}S_{w }+\Delta u_{w}^{0}}{RT})=p_{0}\exp(\frac{\Delta u_{w}^{0}}{RT})\exp(\alpha S _{w}) \tag{11a}\] \[p_{v}=p_{0}\exp(\frac{\Delta u_{v}}{RT})=p_{0}\exp(\frac{\Delta u_{v}^{f}(1-S_{w })+\Delta u_{w}^{0}}{RT})=p_{0}\exp(\frac{\Delta u_{w}^{0}}{RT})\exp(\beta(1- S_{w})) \tag{11b}\]  
where \(\Delta u_{w}\) and \(\Delta u_{v}\) are the average intrinsic molar potential change (the average LJ potential energy) of the water and vapour phases, respectively. \(\Delta u_{w}^{f}\) and \(\Delta u_{v}^{f}\) are the average LJ potential energy of the  
Figure 10: Water film adsorbed on a solid substrate. water and vapour phases at the two states when the soil is fully water saturated and when all the pore water has just been fully drained, respectively. They depend on the average pore size and the nature of the respective interfacial interaction of the pore water and vapour phases with the solid phase at the pore wall surface. \(\alpha=\frac{\Delta u_{w}^{\prime\prime}}{RT}\) and \(\beta=\frac{\Delta u_{w}^{\prime\prime}}{RT}\). \(\Delta u_{w}^{0}\) is the initial chemical potential change of the first bulk water starting to fill the initially empty pore space under the effect of capillary condensation (i.e., the chemical potential change of the initial water molecular film on the totally empty pore surface just at the start filling pores ([PERSON] 2010)). \(S_{w}\) is the degree of water saturation. Equation (11) defines the water and vapour states at the pore scale. Introducing in the volume average theorem to upscale the two state equations from pore scale to bulk material scale, the average pore pressure of the water and vapour phases in unsaturated soils can be described as:  
\[ \begin{array}{ll}\langle p_{w}\rangle&=\frac{1}{V_{pore}}\int_{0}^{V_{w}}p_{ w}dV_{w}\\ &=\frac{1}{V_{pore}}\int_{0}^{V_{pore}S_{w}}p_{0}\exp\Bigl{(}\frac{\Delta u_{w }^{0}}{RT}\Bigr{)}\exp(\alpha S_{w})d(V_{pore}S_{w})\\ &=\int_{0}^{S_{w}}p_{0}\exp\Bigl{(}\frac{\Delta u_{w}^{0}}{RT}\Bigr{)}\exp( \alpha S_{w})dS_{w}\\ &=\lambda\frac{1}{\alpha}[\exp(\alpha S_{w})-1]\end{array}  \tag{12a}\] \[ \begin{array}{ll}\langle p_{v}\rangle&=\frac{1}{V_{pore}}\int_{0}^{V_{w}}p_{ v}dV_{v}\\ &=\frac{1}{V_{pore}}\int_{0}^{V_{pore}(1-S_{w})}p_{0}\exp\Bigl{(}\frac{\Delta u _{w}^{0}}{RT}\Bigr{)}\exp(\beta(1-S_{w}))d(V_{pore}(1-S_{w}))\\ &=\int_{0}^{(1-S_{w})}p_{0}\exp\Bigl{(}\frac{\Delta u_{w}^{0}}{RT}\Bigr{)}\exp (\beta(1-S_{w}))d(1-S_{w})\\ &=\lambda\frac{1}{\beta}[\exp(\beta(1-S_{w}))-1]\end{array}  \tag{12b}\]  
where \(\langle p_{w}\rangle\) and \(\langle p_{v}\rangle\) are the volume average pore pressure of the water and vapour phases in unsaturated soils, respectively. \(\lambda=p_{0}\exp\Bigl{(}\frac{\Delta u_{w}^{0}}{RT}\Bigr{)}\) is a redefined constant. The Equation (12) can be used to describe the water retention characteristic of unsaturated soils, the constitutive relation between the matrix suction, or capillary pressure and the water content. In conventional, matric suction is taken as the pressure different between the pore air (vapour) and pore water phases coexisting in soils, i.e.,  
\[ \begin{array}{ll}\langle s\rangle&=-\langle p_{c}\rangle=-(\langle p_{w} \rangle-\langle p_{v}\rangle)\\ &=-\Bigl{(}\lambda\frac{1}{\alpha}[\exp(\alpha S_{w})-1]-\lambda\frac{1}{ \beta}[\exp(\beta(1-S_{w}))-1]\Bigr{)}\\ &=\frac{\lambda}{\alpha}-\frac{\lambda}{\beta}-\lambda\Bigl{(}\frac{1}{\alpha} \exp(\alpha S_{w})-\frac{1}{\beta}\exp(\beta(1-S_{w}))\Bigr{)}\end{array}  \tag{13}\]  
where \(\langle s\rangle\) is the matric suction, \(\langle p_{c}\rangle\) is called capillary pressure of bulk soils.