## 2 Methods
### A Simple Model for Estimating the Hydraulic Conductivity of Unsaturated Soil
This variation in \(\mathrm{P_{m}}/\mathrm{P_{e}}\) with adsorptive water film thickness is consistent with the results reported by [PERSON] and [PERSON] [37]. Since the adsorptive water film thickness increases with a decreasing suction, for simplicity, it is suggested that the potentials of the adsorptive water film at the residual suction and the highest suction are approximated by the ionic-electrostatic potential and the molecular potential, respectively. Therefore, according to Equations (9) and (10), the adsorptive water film thicknesses at the residual suction and at the highest suction can be expressed as follows:  
\[f_{r}=\sqrt{\frac{\varepsilon\epsilon_{0}}{2}}\bigg{(}\frac{\pi k_{B}T}{ze} \bigg{)}\frac{1}{\sqrt{\Psi_{r}}} \tag{11}\]  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]0  
where \(f_{r}\) is the thickness of adsorptive water film at the residual suction \(\Psi_{r}\); and \(f_{m}\) is the thickness of adsorptive water film at the maximum suction \(\Psi_{m}\). \[k_{wa}=S_{ra}k_{s} \tag{4}\]1 The variation in \(\Pi_{m}/\Pi_{e}\) with the adsorptive water film thickness. The parameters \(\Pi_{m}\) and \(\Pi_{e}\) are the molecular and ionic-electrostatic potentials, respectively. In addition, [PERSON] and [PERSON] [25] derived the expression of the equivalent diameter of the soil particles. The specific surface area of the polydisperse sample can be estimated in terms of the soil-water characteristic data at high suction as follows [21]:  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]2  
where \(A_{s}\) is the specific surface area; \(n^{\prime}\) is the porosity; \(A_{ssl}\) is the Hamaker constant; \(\Psi_{m,m}\) is the suction value, which corresponds to the situation where the capillary condensation caused by surface roughness is negligible, which is approximately taken as \(10^{4}\) kPa [21]; and \(S_{rm,m,m}\) is the degree of saturation at \(\Psi_{m,m}\), which can be calculated in the following equation:  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]3  
Furthermore, the specific surface area of the monodisperse sample composed of the identical spherical particles is approximated by \(6/d_{e}\)[25]. Assuming the polydisperse sample has the same specific surface area as the monodisperse sample, then the expression of the equivalent diameter of the polydisperse sample can be derived as follows [25]:  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]4  
As a consequence, combining Equations (6)-(8) and (11)-(15), equations for the hydraulic conductivities at the residual suction and at the highest suction are obtained as follows:  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]5  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]6  
where \(k_{wr}\) and \(k_{wm}\) are hydraulic conductivities at the residual and maximum suction. Notably, the parameters in the brackets \([\ldots]\) of Equations (16) and (17) are generally considered to be constants. Therefore, these equations can be furthermore simplified to the following formulas:  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]7  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]8  
As a consequence, by substituting Equations (4), (18) and (19) into Equation (3), a simple hydraulic conductivity model is achieved. \[k_{wa}=S_{ra}k_{s} \tag{4}\]9  
\[S_{ra}=\frac{1-\frac{\ln(1+\Phi_{a}/C_{r})}{\ln(1+10^{6}/C_{r})}}{\left\{\ln \left[e+\left(\psi_{a}/\alpha\right)^{n}\right]\right\}^{m}} \tag{5}\]0  
\[S_{ra}=\frac{1-\frac{\ln(1+\Phi_{a}/C_{r})}{\ln(1+10^{6}/C_{r})}}{\left\{\ln \left[e+\left(\psi_{a}/\alpha\right)^{n}\right]\right\}^{m}} \tag{5}\]1  
where the porosity \(n^{\prime}\) is a fundamental physical parameter; \(S_{ra}\) and \(S_{rm,m}\) can be calculated by Equations (5) and (14); \(\Psi_{m,m}\) is commonly set to be \(10^{4}\) kPa; \(\Psi_{s}\) is commonly set to \(0.01\) kPa, \(0.1\) kPa or \(1\) kPa representing the minimum suction of the measured SWCC. The air-entry value \(\Psi_{a}\) and the residual suction \(y_{r}\) can be conventionally calibrated by the measured SWCC. This proposed model can, thus, predict the hydraulic conductivity of unsaturated soil over a wide suction range.