## 3 Modelling
### A Unified Thermal Conductivity and Water Retention Model  
In order to investigate the intrinsic relationship between thermal conductivity and water retention characteristic--it is noted that both are linked to the pore water distribution and the pore size distribution in soils--this paper tries to use an analytical water retention model proposed by [42; 43; 44; 45] to describe the thermal conductivity of soils at different water contents. The analytical water retention model takes account of the multiphase interaction in pores, and describes the respective behaviour of the coexisting water phase and vapor phase in a pore system in terms of the fundamental interfacial physics and the classical capillary theory. Figure 8: The Fitting Result of the Parametric Functions, (**a**) Equation (4b); (**b**) Equation (4c); (**c**) Equation (4d). Figure 9: The Fitting Result of the Parametric Functions, (**a**) Equation (7b); (**b**) Equation (7c); (**c**) Equation (7d); (**d**) Equation (7e). In terms of surface phenomena, fluids retained in porous media could be regarded as being adsorbed on pore wall surfaces because chemical and physical interactions exit at their interfacial region due to the inter atomic and molecular attraction forces, known as the Van de Waals forces [42; 44; 45]. These forces will modify the chemical potential of the adsorbates (the fluids) at the interfacial region referring to their free bulk phase value. The chemical potential change is called Lennard-Jones (LJ) potential energy [46], which can be approximately defined as a function of the distance from adsorbed fluid molecule to the substrate surface (pore wall surfaces) and takes a pair-wise summation of all interaction forces between the fluid molecule and the solid atoms. Figure 10 illustrates that pore water resides on pore surface. The average LJ potential energy of the water film on the substrate surface can be expressed as a function of the fluid thickness above the substrate surface as [47]:  
\[\Delta u_{w}=\frac{A^{m}}{h^{3}}+\frac{A^{e}}{h^{2}}+\frac{A^{s}}{h} \tag{9}\]  
where \(\Delta u_{w}\) is the LJ potential energy of the water film on substrate surface, and \(A^{m}\), \(A^{e}\), and \(A^{s}\) are three constants related to the nature of the interfacial interaction. On the other hand, the chemical potential change (the LJ potential energy) of the adsorbate fluids on a substrate surface can also be evaluated using the Kelvin equation. Finally, the intrinsic pressure of the adsorbed fluids on substrates can be expressed in terms of the following equation [43]:  
\[p_{w}=p_{\Delta u_{w}}^{d}+p_{v}^{g}=\frac{\Delta u_{w}}{V_{wm}}+p_{0}\exp( \frac{\Delta u_{w}}{RT}) \tag{10}\]  
where \(p_{w}\) is the average intrinsic pressure of the water on the surface, \(\Delta u_{w}\) is the average intrinsic molar potential change (the LJ potential energy) of the fluid of a thickness \(h\), which is defined as \(\Delta u_{w}=\frac{1}{h}\int_{0}^{h}\Delta u_{w}^{z}(z)dz\) (or estimated by Equation (9)), where \(\Delta u_{w}^{z}(z)\) is the water molar molecule potential change at position \(z\) above the substrate surface, and \(V_{wm}\) is the molar volume of the water phase. \(R\) is the gas constant; \(T\) is the temperature, \(p_{0}\) is a normal pressure. The \(p_{w}\) consists of two components, \(p_{\Delta u_{w}}^{d}=\frac{\Delta u_{w}}{V_{wm}}\) is called the disjoining pressure [48], and \(p_{v}^{g}=p_{0}\exp(\frac{\Delta u_{w}}{RT})\) is called the gauge pressure which is equal to the pressure of the surrounding equilibrium vapour. [PERSON] et al. [45] suggested the use of the gauge pressure to describe the state of the fluids residing in porous media at the pore scale.