## 3 Case Studies
### Dams with High Groundwater Level
#### 3.2.3 Comparison between the Calculation Results  
In order to calculate the seepage through dams with a high groundwater level, several analytical methods were introduced by several scholars, as shown below.  
[PERSON] and [PERSON]. [47] proposed a method to estimate the seepage per width through a dam as follows:  
\[\mathrm{q}_{c}=K_{c}\frac{\left(1+cot^{2}\gamma\right)\left(H_{1}^{2}-H_{2}^{2 }\right)}{2\delta} \tag{43}\]  
\[\mathrm{q}_{t}=\overline{K_{x}}\frac{\alpha(H_{1}-H_{2})(T-t)}{th(\alpha L_{ 0})},\alpha=\sqrt{\frac{K_{t}}{\overline{K_{x}}t(T-t)}}. \tag{44}\]  
[PERSON] [33] computed dam leakage based on the subsection method:  
\[\mathrm{q}_{c}=K_{c}\frac{\left(H_{1}^{2}-h_{1}^{2}\right)}{2\delta sin\gamma} \tag{45}\]  
\[\mathrm{q}_{t}=\sqrt{\frac{\overline{K_{x}}K_{t}T}{t}}cth(AL_{0})(H_{1}-h_{1} ),A=\sqrt{\frac{K_{t}}{\overline{K_{x}}Tt}} \tag{46}\]  
\[\mathrm{q}_{2}=\frac{K_{b}\left(h_{1}^{2}-H_{2}^{2}\right)}{2L_{1}}+\frac{ \overline{K_{x}}(h_{1}-H_{2})T}{L+0.44T}. \tag{47}\]  
Table 4 shows the comparison results obtained from our proposed method, [PERSON] and [PERSON]'s method, [PERSON]'s method, [PERSON]'s method, and the numerical method. Table 4 shows that the analytical results obtained from the proposed method are slightly different from those of the numerical results, but their relative errors are less than 5%, which is acceptable.  
The small differences between these models are due to the following reasons: (1) the dam foundation and the shapes of the slope wall and blanket are simplified in the proposed method, while those simulated by the numerical method are more precise and closer to reality; (2) the water flow  
Figure 10: Simplified calculation diagrams: (**A**) a typical cross section of the Shangmo dam; (**B**) a front view of the simplified slope wall.  
direction in the proposed method is also simplified compared to the numerical simulation; (3) the pore water pressure might become negative in the area of the slope wall that is higher than the reservoir water level, and its distribution will be more complex as well, which is difficult to achieve using an analytical method.  
In addition, compared to the proposed method, [PERSON] and [PERSON]'s method is a conservative seepage calculation approach, whose results are clearly different from those of the numerical results because [PERSON] and [PERSON]'s method does not consider the head loss of the dam body and a relatively large hydraulic gradient may be concentrated in the slope wall, which may lead to large errors and uneconomical designs. [PERSON]'s method performs well for seepage calculations of the dam body but poorly in seepage calculations of the dam foundation, as it ignores seepage through areas higher than the water level behind the slope wall. [PERSON]'s method provide a good estimation of dam body seepage that can suit any slope wall shape, but the formula does not account for head loss of the dam body.  
The results indicate that the proposed method provides a good estimation compared to the numerical method and some other analytical methods introduced by scholars. The proposed method offers high precision under high groundwater level conditions and can be applied to engineering analyses.