## 2 Tunnel-Soil-Pipeline Interaction
### Free-Field Movements Induced by Ground Volume Loss  
[PERSON] [24] proposed the virtual image method to predict the displacements and stress fields due to a point sink in the elastic half-space. Based on the cavity expansion/contraction theory and superposition method, the closed-form solutions of the displacement and stress related to the ground volume loss are given by [PERSON], et al. [34]:  
\[u_{x}=\frac{Vx}{4\pi}[\frac{1}{R_{1}^{3}}+\frac{3-4\mu}{R_{2}^{3}}-\frac{6z(z +H)}{R_{2}^{5}}] \tag{1}\]  
Figure 2: Interaction between shield tunnel, soil, and existing pipeline.  
\[u_{y}=\frac{Vy}{4\pi}[\frac{1}{R_{1}^{3}}+\frac{3-4\mu}{R_{2}^{3}}-\frac{6z(z+H)}{R_ {2}^{5}}] \tag{2}\]  
\[u_{z}=\frac{V}{4\pi}[\frac{z-H}{R_{1}^{3}}+\frac{2z}{R_{2}^{3}}-\frac{(3-4\mu)(z +H)}{R_{2}^{3}}-\frac{6z(z+H)^{2}}{R_{2}^{5}}] \tag{3}\]  
where \(u_{x}\), \(u_{y}\), and \(u_{z}\) are the three-dimensional displacement components; \(\mu\) is the Poisson's ratio; \(H\) is the distance between the ground surface and point sink; \(V\) is the ground volume loss; the parameters \(R_{1}\) and \(R_{2}\) are expressed as:  
\[R_{1}=\sqrt{x^{2}+y^{2}+(z-H)^{2}} \tag{4}\]  
\[R_{2}=\sqrt{x^{2}+y^{2}+(z+H)^{2}} \tag{5}\]  
The ground volume loss should be integrated along the direction of tunnel construction shown in Figure 2 to analyse the spatiotemporal characteristic of ground movements. According to Equations (1)-(3), the three-dimensional ground movements can be given by integrating the Green functions along the direction of tunnel construction from \(L_{1}\) to \(L_{2}\):  
\[u_{x}^{V}=\int_{L_{1}}^{L_{2}}u_{g}(x,z)f_{x}(x,y-\eta,z)d\eta \tag{6}\]  
\[u_{y}^{V}=\int_{L_{1}}^{L_{2}}u_{g}(x,z)f_{y}(x,y-\eta,z)d\eta \tag{7}\]  
\[u_{z}^{V}=\int_{L_{1}}^{L_{2}}u_{g}(x,z)f_{z}(x,y-\eta,z)d\eta \tag{8}\]  
where \(L_{1}\) and \(L_{2}\) are the distance from the intersection point O (Figure 2) to the launching shaft of the new tunnel and the shield tunnel face, respectively; \(u(x,z)\) is the ground loss distribution of the inner tunnel boundary; \(f_{x}\), \(f_{y}\), and \(f_{z}\) are the Green functions determined by assuming \(V=1\) in Equations (1)-(3). It means the ground movement induced by unit volume loss; \(d\eta\) is the unit length of tunnel excavation.  
If the ground loss distribution is uniform radial, \(u_{g}(x,z)\) will be a constant \(u_{0}\). However, [PERSON] and [PERSON] (2016) indicated that the ground loss distribution is a non-uniform pattern instead of a uniform one and proposed the formula:  
\[u_{g}(x,z)=\frac{4 gR+g^{2}}{4R^{2}}\exp(-\frac{1.38x^{2}}{\left(H+R\right)^{2 }}-\frac{0.69z^{2}}{H^{2}}) \tag{9}\]  
\[V=\frac{4 gR+g^{2}}{4R^{2}} \tag{10}\]  
where \(g\) is the gap parameter; \(R\) is the radius of the shield tunnel; \(V\) is the equivalent ground loss parameter.  
The gap parameter \(g\) is related to the tunnelling machine and lining geometry, the 3D elastoplastic ground movement at the tunnel face, and the over-excavation of soil around the tunnel shield periphery. Some scholars ([PERSON] _et al._, 2016; [PERSON] _et al._, 2017; [PERSON] _et al._, 2018) have investigated how to determine the parameter \(g\) reasonably.