# A Simple Model for Estimating the Hydraulic Conductivity of Unsaturated Soil
## 3 Results  
In this section, the hydraulic conductivities of different soils have been estimated by using the proposed hydraulic conductivity model. The experimental data of different soils were collected from previous studies. The porosities, saturated hydraulic conductivities and references of these soils are presented in Table 1. Also, the validation of the proposed model has been evaluated by using the coefficient of determination (\(R^{2}\)) for the estimated values versus experimental data. \(R^{2}\) has been widely used to evaluate the quality of fitting of models in practice, and its expression is presented as follows [43]:  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{N}\left[\log k_{wi}-\log\hat{k}_{wi}\right] ^{2}}{\sum\limits_{i=1}^{N}\left[\log k_{wi}-\log\overline{k}_{wi}\right]^{2}} \tag{34}\]  
where \(N\) is the number of experimental data points; \(\hat{k}_{wi}\) and \(k_{wi}\) are the estimated and measured hydraulic conductivities corresponding to the experimental data point \(i\); and \(\overline{k}_{wi}\) is the average of the measured hydraulic conductivities. The closer the value of \(R^{2}\) is to 1, the better the predictive effect of the model. In the present study, the detailed calculation procedure is as follows: (i) the measured SWCC is fitted by Equations (1) and (2), and the parameters of the SWCC model are obtained; (ii) the values of \(\Psi_{a}\) and \(\Psi_{r}\) are calculated by Equations (21)-(33); (iii) by substituting the auctions of \(\Psi_{a}\) and \(\Psi_{m,m}\) in Equations (5) and (14), their corresponding degree of saturation (\(S_{na}\) and \(S_{m,m}\)) can be calculated; (iv) the estimated hydraulic conductivity is acquired by substituting the parameters \(n^{\prime}\), \(S_{m}\), \(S_{m,m}\), \(\Psi_{m,m}\), \(\Psi_{s}\), \(\Psi_{a}\), \(\Psi_{r}\) and \(k_{s}\) in Equation (20). The fitted parameters of the SWCC for the different soils are presented in Table 2.  
As shown in Figure 4, the \(R^{2}\) values are larger than 0.98 and the fitted SWCCs are highly consistent with the experimental data, which verifies the good applicability of [PERSON] and [PERSON]'s equation to different soils. Figure 5 shows the estimated hydraulic conductivities together with the corresponding experimental data. It can be seen that the measured hydraulic conductivity curves of different soils are approximately in the form of three segments, which are consistent with the proposed hydraulic conductivity model. Note that the inflection points of estimated hydraulic conductivities are almost exactly coincident with the experimental results, which indicates that the proposed parameters calibration method can accurately determine the air-entry value and residual suction and proves again that it is reasonable to divide the hydraulic conductivity curve by air-entry value and residual suction. Furthermore, it can be seen that the estimated hydraulic conductivities agree quite well with the experimental data, with an \(R^{2}\) value larger than 0.83. This good agreement exists throughout the whole suction range. This result demonstrates a good performance of the proposed model in the estimation of the hydraulic conductivity of sand, silt and clay over a wide suction range. Moreover, it can be seen that even though the suction range of the measured SWCCs is relatively small, the estimated hydraulic conductivity curve also shows good agreement with the experimental data over the wide suction range. For example, Figure 5g shows a well-estimated hydraulic conductivity curve in the whole suction range, while the suction of measured SWCC in Figure 4g is less than 300 kPa.