## 3 Case Studies
### Dams with a Low Groundwater Level
#### 3.1.3 Comparison between the Calculation Results  
In order to calculate the seepage through dams with a low groundwater level, a limited number of analytical methods were introduced by several scholars. The USBR code [31] recommends the following method to estimate seepage through the dam foundation:  
\[\mathrm{Q}_{t}=\overline{K_{x}}B\frac{TH_{1}}{L_{0}+\delta+0.43T}. \tag{36}\]  
[PERSON] and [PERSON] [32] approximated the profile of the concrete face as a triangle and derived a formula for calculating seepage through the dam body as follows:  
\[\mathrm{Q}_{c}=\frac{K_{c}L_{c}H_{1}^{2}}{6\delta sin\gamma}, \tag{37}\]  
where \(L_{c}\) is the crest length.  
[PERSON] et al. [44] proposed a simple method for dam body leakage calculations as follows:  
\[\mathrm{Q}_{c}=\sum_{1}^{n}\frac{K_{c}H_{n}^{2}}{2\delta sin\gamma}L_{n}. \tag{38}\]  
where \(L_{n}\) is the length of the panel, and \(H_{n}\) is the maximum head of the panel.  
Figure 6: Simplified calculation diagrams: (**A**) a typical cross section of the embankment dam; (**B**) front view of the simplified slope wall.  
[PERSON] et al. [39] also proposed a leakage calculation method for the dam body, which can be applied to different shapes of the slope wall. The formula is presented as:  
\[\mathrm{Q}_{c}=\int_{l_{lowest}}^{l_{H_{1}}}K_{c}\frac{H(l)}{\delta}WX(l)dl \tag{39}\]  
\[H(l)=\left\{\begin{array}{cc}H_{1}-z&z>H_{2}\\ H_{1}-H_{2}&z<H_{2}.\end{array}\right. \tag{40}\]  
To evaluate the application of the proposed method for engineering applications, the relative error (RE) and root mean squared error (RMSE) are introduced as:  
\[\mathrm{RE}=\frac{O_{i}-S_{i}}{S_{i}}\times 100\% \tag{41}\]  
\[\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(O_{i}-S_{i})^{2}}, \tag{42}\]  
where \(n\) is the number of data, and \(O_{i}\) and \(S_{i}\) are the analytical results and numerical results under different calculation conditions.  
Comparisons between these analytical results (using the above methods to calculate the seepage flux of the Fukang pumped storage power station) and the numerical results are shown in Table 2.  
It can be observed from Table 2 that the analytical results obtained by the proposed method are slightly larger than the numerical results, but their relative errors are less than 5%, which is acceptable. Small discrepancies may be caused by simplification of the slope wall and horizontal blanket and the assumptions of the vertical downward seepage direction in the proposed method. In contrast, the numerical method can simulate a more complicated and realistic seepage path.  
In addition, compared to the proposed method, the USBR method is relatively unreliable due to to its large RMSE and RE, as this method considers the blanket to be impermeable and the seepage direction of the dam foundation to be horizontal when the groundwater is low, which does not accord with reality. The results of the Deng Gang method provide as good an estimation as the proposed method. Nevertheless, there are few distinctions between these methods, since the shape of the slope wall is simplified in the proposed method. The results obtained using Casinader method differ significantly with the numerical results, as the Casinader method approximates the shape of the slope wall as a triangle, which means that the area of maximum water head only appears at the vertex of the triangle in the seepage calculations. The Chunjiang Fu method also suffers from instability, since this method regards the water head of each point on the slope wall as the maximum head. Thus, when there is a low water level, the seepage calculations are comparatively accurate. As the water level rises, the maximum water head and the errors also become larger.  
To conclude, the above analysis indicates that the proposed simplified calculation method is reasonably accurate and is reliable for engineering applications.