## 5 Fitting Analysis of Transverse Ground Settlement Data with Different Tunnel Spaces  
[PERSON] proposed the following Peck formula by analyzing a large amount of tunnel construction settlement data:  
\[S(x)=S_{\text{max}}\ \exp\biggl{(}\frac{-x^{2}}{2i^{2}}\biggr{)} \tag{1}\]  
\[S_{\text{max}}=\frac{V_{\text{loss}}}{\sqrt{2\pi i}}=\frac{\pi R^{2}\eta}{\sqrt {2\pi i}} \tag{2}\]  
where \(S_{\text{max}}\) is the maximum ground settlement value, \(V_{\text{loss}}\) is the soil loss per unit length, \(\eta\) is the soil loss rate, \(i\) is the width of the ground settlement trough caused by tunnel excavation, and \(S(x)\) is the ground settlement value at \(x\) in the abscissa. [PERSON] [35] measured and analyzed a large number of double-line tunnels, using the superposition principle, and obtained the calculation formula of ground transverse settlement deformation of double-line tunnel excavation:  
\[S(x)=S_{\text{max}}\exp\biggl{(}-\frac{(x-0.5l)^{2}}{2i_{\text{f}}^{2}} \biggr{)}+S_{\text{max}}\exp\biggl{(}-\frac{(x+0.5l)^{2}}{2i_{\text{s}}^{2}} \biggr{)} \tag{3}\]  
where \(S_{\text{max}}\) and \(S_{\text{max}}\) are the maximum ground settlement values caused by the excavation of the first tunnel and the later tunnel, respectively; \(i_{\text{f}}\) and \(i_{\text{s}}\) are the widths of ground settlement troughs caused by the excavation of the first tunnel and the later tunnel, respectively; \(l\) is the distance between the two tunnels. Since the two tunnels are close and the geological conditions are the same, it is assumed that the width \(i\) of the settlement troughs is the same.  
Since the ground transverse settlement curves under different tunnel longitudinal spaces are relatively close, only the ground settlement of double-line tunnels under different tunnel transverse spaces is analyzed. Origin software and Equation (4) are used to nonlinearly fit the lateral ground settlement of the model, and the ground settlement  
Figure 9: The comparison between the simulation results and the field settlement value under the condition of continuous excavation of the 1.0D tunnel: (**a**) fitting of the ground settlement due to left-line tunnel construction; (**b**) fitting of the ground settlement due to double-line tunnel construction.  
curves of double-line tunnels under different tunnel spaces (1.0D, 1.5D, 2.0D) are obtained as follows:  
\[S(x)=\left\{\begin{array}{l}4.058\exp(-\frac{(x+3.2)^{2}}{2\times 8.677^{2}})+1.008\exp(-\frac{(x-3.2)^{2}}{2\times 8.677^{2}})\\ 3.860\exp(-\frac{(x+4.8)^{2}}{2\times 8.805})+1.121\exp(-\frac{(x-4.8)^{2}}{2 \times 8.805})\\ 3.445\exp(-\frac{(x+6.4)^{2}}{2\times 9.324^{2}})+1.103\exp(-\frac{(x-6.4)^{2}}{2 \times 9.324^{2}})\end{array}\right\} \tag{4}\]  
It can be seen from the fitting function Equation (4) that the maximum ground settlement of the first tunnel gradually decreases with the increase in the transverse space of the tunnel. The superposition settlement of the later tunnel has little change, and the width of the ground settlement trough gradually increases. Setting \(S^{\prime}=S_{\rm{mass}}/S_{\rm{max}}\) as the settlement ratio, the settlement ratio \(S^{\prime}\) is found to be linearly related to the tunnel space D, as shown in Figure 10.  
The linear fitting of the settlement ratio \(S^{\prime}\) and tunnel space D is carried out, and the fitting function is shown in Formula (5). Hence, Formulas (4) and (5) can be combined to predict the ground settlement of the double-line tunnel with different horizontal spaces in the region.  
\[S^{\prime}=0.178+0.072\rm{D} \tag{5}\]