## 2 Methods
### A Simple Model for Estimating the Hydraulic Conductivity of Unsaturated Soil
(2018), the capillary component and adsorptive component in Figure 1b intersect roughly at the residual suction, which indicates that the hydraulic conductivity at the residual suction is approximately double the size of the adsorptive component. Additionally, Figure 1b shows that the hydraulic conductivity at very high suction depends almost entirely on the adsorptive water flow. Therefore, the hydraulic conductivities \(k_{uv}\) and \(k_{vcm}\) can be determined by the adsorptive water flow as follows:  
\[k_{uv}=2k_{ar} \tag{6}\]  
\[k_{vcm}=k_{am} \tag{7}\]  
where \(k_{uv}\) and \(k_{vcm}\) are hydraulic conductivities at the residual and maximum suction; \(k_{ar}\) is the adsorptive component of the hydraulic conductivity at the residual suction; and \(k_{am}\) is the adsorptive component of the hydraulic conductivity corresponding to the maximum suction. A classic hydraulic conductivity model of the adsorptive water film flow proposed by [PERSON] (2018, 2019) is adopted to calculate the adsorptive component of the hydraulic conductivity, which is presented as follows:  
\[k_{a}=\frac{4\rho_{w}g}{\pi\mu_{w}}\frac{1-n^{\prime}}{d_{e}}f^{3} \tag{8}\]  
where \(k_{a}\) is the adsorptive component of the permeability; \(\rho_{w}\) is the water density, \(\rho_{w}=1000\) kg/m\({}^{3}\); \(\mu_{w}\) denotes the water viscosity, \(\mu_{w}=1.005\)\(\times\)\(10^{-3}\) Pa\(\cdot\)s; \(g\) denotes the gravity acceleration, \(g=9.8\) m/s\({}^{2}\); \(n^{\prime}\) is the porosity; \(d_{e}\) is the equivalent diameter for soil particles; and \(f\) denotes the thickness of the adsorptive water film. Equation (8) shows that the key to estimating the adsorptive component of the hydraulic conductivity is to determine the thickness of the adsorptive water film \(f\) and the equivalent diameter for soil particles \(d_{e}\). The thickness of the adsorptive water film is mainly dependent on the potential of the adsorptive water film. For a planar thin water film, its potential at the high suction is approximated by the disjointing pressure, which is considered to be identical to the suction ([PERSON] et al., 2018). Although the potential of the adsorptive water film includes four components, only the ionic-electrostatic (i.e., electric double layer) and molecular (i.e., van der Waals)potentials are considered in the studies of the thin water film flow [25; 38], which can be expressed by the following equations [39; 40]:  
\[\Pi_{m}=-\frac{A_{svl}}{6\pi}\frac{1}{f^{3}} \tag{9}\]  
\[\Pi_{e}=\frac{\varepsilon\epsilon_{0}}{2}\bigg{(}\frac{\pi k_{B}T}{ze}\bigg{)}^ {2}\frac{1}{f^{2}} \tag{10}\]  
where \(\Pi_{m}\) denotes the molecular potential; \(\Pi_{e}\) denotes the ionic-electrostatic potential; \(f\) denotes the thickness of the adsorptive water film; \(\varepsilon\) denotes the static relative permittivity of water, \(\varepsilon\) = 78.54; \(\varepsilon_{0}\) denotes the permittivity of free space, \(\varepsilon_{0}\) = 8.85 \(\times\) 10\({}^{-12}\) C\({}^{2}\).J\({}^{-1}\).m\({}^{-1}\); \(T\) denotes the Kelvin temperature, \(T\) = 293.15 K; \(k_{B}\) denotes the Boltzmann constant, \(k_{B}\) = 1.381 \(\times\) 10\({}^{-23}\) J/K; \(z\) denotes the ion charge, \(z\) = 1; \(A_{svl}\) denotes the Hamaker constant, \(A_{svl}\) = \(-6\)\(\times\) 10\({}^{-20}\) J in this study; and \(e\) denotes the electron charge, \(e\) = 1.602 \(\times\) 10\({}^{-19}\) C.  
Furthermore, the ratio of the molecular potential to the ionic-electrostatic potential for various adsorptive water film thicknesses has been calculated and is presented in Figure 3. The molecular potential, \(\Pi_{m}\), is found to be much higher than the ionic-electrostatic potential, \(\Pi_{e}\), for a very thin water film. However, the value of \(\Pi_{m}/\Pi_{e}\) decreased rapidly for an increase in water film thickness, and the ionic-electrostatic potential becomes much larger than the molecular potential for a relative thick water film.