# Prediction of Surface Deformation Induced by Ultra-Shallow-Buried Pilot Tunnel Construction
## 1 Introduction
### Calculation of Surface Deformation Considering Non-Uniform Convergence  
In reality, the cross-section with area \(\Omega\) does not collapse completely after excavation but shrinks to area \(\omega\). According to the superposition principle, the surface settlement value after adjusting the integration range is  
\[W(X)\ =\ \iint\limits_{\Omega-\omega}\frac{\tan\beta}{\eta}\exp\left[-\frac{ \pi\tan^{2}\beta}{\eta^{2}}(X-\xi)^{2}\right]\mathrm{d}\xi\mathrm{d}\eta \tag{9}\]  
The process of tunnel cross-section shrinkage from \(\Omega\) to \(\omega\) is the focus of current research on stochastic medium theory. Initially, [PERSON] [28] simplified the cross-section shrinkage process to uniform shrinkage when proposing stochastic medium theory, as shown in Figure 3a. After analyzing the tunnel deformation mechanism with measured data, [PERSON] [29] proposed that elliptical deformation is superimposed on uniform radial shrinkage, as shown in Figure 3b. [PERSON] [30] first suggested that shield tunnels undergo additional overall subsidence during shrinkage, as shown in Figure 3c. Figure 3: A schematic diagram of the shrinkage form of the tunnel section. (**a**) Uniform shrinkage; (**b**) elliptic deformation occurs on the basis of uniform radial shrinkage; (**c**) integral subsidence occurs during shrinkage. Figure 2: Schematic diagram of pilot tunnel excavation. The cross-sections of ultra-shallow-buried pilot tunnels in excavation sections are mostly horseshoe-shaped. Lateral ground settlement induced by construction is calculated based on the plane strain problem, and the solution is subsequently obtained by integrating stochastic medium theory. The method for determining the integration boundary follows the concept of shallow-buried tunnels [22], involving two deformations: The first deformation is the uniform shrinkage of the soil body. The arch part of the initial horseshoe section, which is originally circular, remains circular during shrinkage. The second deformation is the non-uniform deformation of the initial section. The arch part transforms from a circle to an ellipse, while the rectangular portion below remains a rectangle for both deformations. The shrinkage ratios of the short and long sides differ, with specific assumptions as follows:  
1. After excavating the pilot tunnel and until the initial support gains strength, the excavation face is in a critical state and the surrounding stratum stress redistributes. At this time, the excavation face shrinks uniformly, as shown in Figure 4a. 2. Before the strength of the second lining is formed, the initial support deforms under the action of surrounding soil-water pressure. The second deformation occurs after uniform shrinkage, exhibiting a deformation trend of lateral expansion and vertical shrinkage, as shown in Figure 4b. Both deformation parameters (\(\Delta\)R and \(\delta\)) incorporate the influence of construction grouting on lining deformation. 3. The sectional area remains constant before and after the second deformation. 4. The construction-induced vertical ground displacement results from the superposition of the above two deformation components, as shown in Figure 4c. The surface settlement due to the uniform shrinkage of the section is given by Equation (9):  
\[W_{1}(x)\ =\ \iint\limits_{\Omega-\omega}\frac{\tan\beta}{\eta}\exp\left[- \frac{\pi\tan^{2}\beta}{\eta^{2}}(x-\xi)^{2}\right]\mathrm{d}\xi\mathrm{d}\eta \tag{10}\]  
where \(\Omega\) is the integration region before shrinkage, \(\omega\) is the integration region after shrinkage, \(x\) is the horizontal distance from the axis of the pilot tunnel and \(\beta\) is the main influence angle of the surrounding rock above the pilot tunnel. Setting  
\[w(x,\xi,\eta)\ =\ \frac{\tan\beta}{\eta}\exp\left[-\frac{\pi\tan^{2}\beta}{ \eta^{2}}(x-\xi)^{2}\right] \tag{11}\]  
Figure 4: Shrinkage modes of horseshoe-shaped sections in ultra-shallow-buried subsurface excavation. (**a**) First-stage shrinkage mode; (**b**) second-stage shrinkage mode; (**c**) final-stage shrinkage mode.