# A Simple Model for Estimating the Hydraulic Conductivity of Unsaturated Soil
## 2 Methods
### Parameter Calibration
Furthermore, the slope of the tangent line through a point of SWCC (in a semi-logarithmic coordinate) can be derived as follows:  
\[\frac{dS_{r}}{d\left(\log\psi\right)}=-\psi\ln(10)\left\{\frac{\left\{\ln \left[e+\left(\psi/\alpha\right)^{n}\right]\right\}^{-m}}{C_{r}\left(1+\psi/ C_{r}\right)\ln\left(1+10^{6}/C_{r}\right)}+\frac{mn\left(\psi/\alpha\right)^{n-1} \left[1-\frac{\ln\left(1+\psi/C_{r}\right)}{\ln\left(1+10^{6}/C_{r}\right)} \right]}{\alpha\left[e+\left(\psi/\alpha\right)^{n}\right]\left\{\ln\left[e+ \left(\psi/\alpha\right)^{n}\right]\right\}^{m+1}}\right\} \tag{25}\]  
By substituting the value of \(\Psi_{f}\) in Equations (1), (2) and (25), the degree of saturation and the slope of the tangent line at the inflection point can be obtained by Equations (26) and (27):  
\[S_{rf}=\frac{1-\frac{\ln\left(1+\psi_{f}/C_{r}\right)}{\ln\left(1+10^{6}/C_{ r}\right)}}{\left[\ln\left(e+1\right)\right]^{m}} \tag{26}\]  
\[\frac{dS_{rf}}{d\left(\log\psi_{f}\right)}=-\psi_{f}\ln(10)\left\{\frac{\left[ \ln\left(e+\left(\psi_{f}/\alpha\right)^{n}\right)\right]^{-m}}{C_{r}\left(1+ \psi_{f}/C_{r}\right)\ln\left(1+10^{6}/C_{r}\right)}+\frac{mn\left[1-\frac{\ln \left(1+\psi_{f}/C_{r}\right)}{\ln\left(1+10^{6}/C_{r}\right)}\right]}{\alpha \left[e+\left(\psi_{f}/\alpha\right)^{n}\right]\left[\ln\left(e+\left(\psi_{ f}/\alpha\right)^{n}\right)\right]^{m+1}}\right\} \tag{27}\]where \(S_{rf}\) is the degree of saturation at the inflection suction \(\Psi_{f}\), and \(dS_{rf}/d(\log\Psi_{f})\) is the slope of the tangent line at the inflection point \(\Psi_{f}\). Moreover, the tangent line through the inflection point can be expressed by Equation (28):  
\[S_{r}=S_{rf}+\frac{dS_{rf}}{d\big{(}\log\psi_{f}\big{)}}\Big{(}\log\psi-\log \psi_{f}\Big{)} \tag{28}\]  
As suggested by [PERSON] and [PERSON] [41; 42], the straight line of the SWCC at the high suction is determined by the suction of 3000 kPa. By substituting \(\Psi\) = 3000 kPa in Equations (1), (2) and (25), the expression of this straight line can then be obtained as follows:  
\[S_{r}=S_{r3000}+\frac{dS_{r3000}}{d(\log\psi_{3000})}(\log\psi-\log 3000) \tag{29}\]  
\[S_{r3000}=\frac{1-\frac{\ln(1+3000/C_{r})}{\ln(1+10^{6}/C_{r})}}{\left\{\ln \big{[}e+(3000/\alpha)^{n}\big{]}\right\}^{m}} \tag{30}\]  
\[\frac{d^{2}S_{r}}{d\left(\log\psi\right)^{2}}=\ln(10)\psi\left[\frac{d^{2}S_{ r}}{d\psi^{2}}\ln(10)\psi+\frac{dS_{r}}{d\psi}\ln(10)\right]=0 \tag{22}\]0  
where \(S_{r3000}\) is the degree of saturation at the suction of 3000 kPa and \(dS_{r3000}/d(\log\Psi_{3000})\) is the slope of the tangent line at the suction of 3000 kPa. Furthermore, by combining Equations (20), (27) and (28), expressions for \(y_{a}\) and \(y_{r}\) are formed:  
\[\frac{d^{2}S_{r}}{d\left(\log\psi\right)^{2}}=\ln(10)\psi\left[\frac{d^{2}S_{ r}}{d\psi^{2}}\ln(10)\psi+\frac{dS_{r}}{d\psi}\ln(10)\right]=0 \tag{22}\]1  
\[\frac{d^{2}S_{r}}{d\left(\log\psi\right)^{2}}=\ln(10)\psi\left[\frac{d^{2}S_{ r}}{d\psi^{2}}\ln(10)\psi+\frac{dS_{r}}{d\psi}\ln(10)\right]=0 \tag{22}\]2  
where \(\Psi_{a}\), \(\Psi_{r}\) and \(\Psi_{f}\) are the air-entry value, residual suction and inflection point, respectively; \(S_{rf}\) and \(S_{r3000}\) are the degree of saturations at the inflection suction and the suction of 3000 kPa; \(dS_{rf}/d(\log\Psi_{f})\) and \(dS_{r3000}/d(\log\Psi_{3000})\) are the slopes of the tangent lines at the inflection point and the suction of 3000 kPa.