## 2 Problem Description  
According to the causes of volume loss in slurry pressure balance shield tunnelling [10; 24; 30], volume loss can be categorized into three components, as shown in Figure 1:  
\(V_{lf}\): Volume loss at the tunnel face. The stability of the working face is achieved by the support pressure provided by a pressurized mixture of bentonite or clay and water (i.e., slurry). However, the support pressure is different from the lateral earth pressure. In general, the soil tends to move towards the excavation chamber and the ground movement will cause the subsidence of the ground surface before the face.  
\(V_{ls}\): Volume loss along the shield. To facilitate the advancement of the shield machine, the radius at the head of the shield is bigger than that at the tail. This will cause the overcutting of the soil and the subsidence of the ground surface above the shield.  
Figure 1: Sketch of proposed problem.  
\(V_{h}\): Volume loss at the tail. The voids between the excavation surface and precast segment cause volume loss at the tail. Volume loss at the tail can be compensated by the appropriate grout at the tail.  
In this paper, we focus on investigating the volume loss at the tunnel face (\(V_{Lf}\)) under different support pressure supplied by the slurry. The other components of volume loss are assumed to be zero, and the displacement of the tunnel opening is fixed. The same assumption has been adopted by previous researchers [31; 32; 33]. \(V_{L}\) refers to the volume loss at the ground surface which is defined as the ratio of the integral of the vertical displacement over the three-dimensional settlement trough to the cross-sectional area of the tunnel per unit length. \(V_{lf}\) is defined as the ratio of the integral of the horizontal displacement over the tunnel face to the cross-sectional area of the tunnel per unit length. Due to difficulties in considering the volume changes of soil, most researchers assume that the moved soil has a constant volume and \(V_{L}\) is equal to \(V_{f}\). In our method, we simply consider the volume change.  
As shown in Figure 1, we assumed a circular tunnel of diameter \(D\) at finite depth \(H\). The depth of the cover soil is denoted by \(C\). The stress of the soil increases with the depth, and the Buoyant unit weight of the soil is \(\rho^{\prime}\). The lateral pressure coefficient of the soil is \(k\). The initial horizontal earth pressure at the crown of the tunnel is noted by \(\sigma_{u}\), as shown in Figure 1.  
When tunnelling, the soil ahead of the face tends to move into the excavation chamber. The support pressure supplied by the slurry is important in controlling the ground movement. As shown in Figure 1, the support pressure at the tunnel crown is denoted by \(P_{u}\), and it can be given by Equation (1):  
\[P_{u}=\alpha\cdot\sigma_{w}=\alpha\cdot\left(k\sigma_{u}^{\prime}+\sigma_{w}\right) \tag{1}\]  
where \(\alpha\) is the support pressure ratio; \(\sigma_{u}\) is the sum of the lateral earth pressure (\(\sigma_{u}^{\prime}\)) and the static pore pressure (\(\sigma_{w}\)) at the tunnel crown.  
Due to the weight of the slurry, the support pressure (\(P\)) linearly increases along \(h\) and increases to \(P_{b}\) at the tunnel invert (Equation (2)). The support pressure can be given as follows:  
\[P=P_{u}+\gamma h \tag{2}\]  
where \(\gamma\) is the unit weight of the slurry, and \(h\) is the distance from the tunnel crown.