## 3 Improved Resistance Coefficient Method and Solution of Water Level Outside the Curtain
### Solution of Water Level Outside the Curtain  
Based on the improved resistance coefficient method, this study divides the groundwater seepage path, as shown in Figure 3, into three sections: I, II, and III:  
Section I is the seepage of groundwater from the water level outside the curtain to the bottom of the waterproof curtain, which is the same as the form of the inlet and outlet seepage section summarized by [PERSON] [31]. Therefore, the resistance coefficient of Section I is calculated directly using the calculation formula of the resistance coefficient of the inlet and outlet section:  
\[\zeta_{1}=1.5(\frac{S_{1}}{T_{1}})^{1.5}+0.441, \tag{15}\]  
Figure 3: Schematic diagram of foundation pit seepage section division. \(S_{1}\)—vertical depth of penetration outside the curtain; \(S_{2}\)—vertical depth of penetration inside the curtain; \(T_{1}\)—calculated depth of the foundation; \(d\)—width of the waterproof curtain; I, II, and III represent three seepage sections, respectively. Section II is the seepage of groundwater from the outside of the foundation pit to the inside of the foundation pit at the bottom of the waterproof curtain. The calculation formula of the resistance coefficient of the horizontal seepage section is as follows:  
\[\zeta_{2}=\frac{d-0.7(S_{1}+S_{2})}{T_{1}}, \tag{16}\]  
where \(d\) is the width of the waterproof curtain; \(S_{1}\) is the vertical depth of penetration outside the curtain, which is equal to the difference between the water level outside the curtain and the thickness of the flow cross-section: \(S_{1}\)\(=h_{w}-M\); \(S_{2}\) is the vertical depth of penetration inside the curtain, which is equal to the difference between the water level of the pit and the thickness of the flow cross-section: \(S_{2}\)\(=h_{n}-M\); and \(T_{1}\) is the calculated depth of the foundation, which is equal to the water level outside the curtain \(h_{w}\). The units used for the above parameter are m.  
In the suspended waterproof curtain foundation pit, the length of the horizontal seepage section is relatively short. Obviously, \(d<0.7\) (\(S_{1}+S_{2}\)). According to the calculation formula of the horizontal seepage section resistance coefficient, the calculation result is negative; thus, take \(\zeta_{2}=0\). That is, the hydraulic head loss of this seepage section can be regarded as zero, and the hydraulic head of the starting and ending points is equal. In the subsequent calculation, this seepage section can be ignored, and the hydraulic head of the starting and ending points of the horizontal seepage section is \(h_{0}\). Section III is the seepage of groundwater from the bottom of the waterproof curtain to the bottom of the foundation pit. This section is located in the seepage field of the foundation pit. According to Equation (11), the loss relating to the hydraulic head of this section can be obtained. According to the relationship between the resistance coefficient and the water inflow of the foundation pit, the value of the resistance coefficient can be directly obtained. The hydraulic head loss of Section I and Section III is \(\Delta h_{1}\)\(=h_{w}-h_{0}\), \(\Delta h_{3}\)\(=h_{0}-h_{n}\) respectively, and the total hydraulic head loss is \(\Delta H=h_{w}-h_{n}\). In conjunction with Equation (14), the calculation formula of the Section III resistance coefficient can be obtained as follows:  
\[\zeta_{3}=\frac{\Delta h_{3}K}{Q}=\frac{4(h_{0}-h_{n})\sum\limits_{p=1}^{\infty }\frac{1-\sin A_{p}L}{A_{p}^{3}(1-e^{2A_{p}b})}[1+e^{2A_{p}b}]}{(h_{n}-H_{0}) h_{n}M^{2}}, \tag{17}\]  
According to Equation (14), there are the following relations:  
\[\frac{\Delta h_{1}}{\Delta h_{3}}=\frac{h_{w}-h_{0}}{h_{0}-h_{n}}=\frac{\zeta _{1}}{\zeta_{3}}, \tag{18}\]  
After \(\zeta_{3}\) is obtained using Formula (17), it is substituted into Equation (18), and in conjunction with Formula (15), the following can be obtained:  
\[\frac{h_{w}-h_{0}}{h_{0}-h_{n}}=\frac{1.5\Big{(}\frac{h_{w}-M}{h_{w}}\Big{)}^{ 1.5}+0.441}{\zeta_{3}}, \tag{19}\]  
Equation (19) is an implicit equation about \(h_{w}\).