## 2 Mathematical Model
### Circumferential Grout Pressure Distribution
 \tag{7}\]  
where \(\rho\) is the grout density (kg/m\({}^{3}\)), \(g\) is the gravity acceleration (m/s\({}^{2}\)), \(P\) is the grout pressure (Pa), \(\tau\) is the shear stress (Pa), \(R\) is the tunnel radius at grout element (m), \(R=R_{1}+b/2\), \(b\) is the radial thickness of the shield tail void (m), \(\alpha\) is the grout element angle from the positive direction of the \(x\)-axis (rad). Since \(dr\) is much smaller than \(R\), \(R\pm dr/2\) is simplified as \(R\), and \(sin(d\alpha/2)\) is equal to 0, while \(cos(da/2)\) is equal to 1. Equation (7) is simplified as follows:  
\[\frac{d\tau}{dr}=-\frac{1}{R}\left(\rho gKcos\alpha+\frac{dP}{d\alpha}\right) \tag{8}\]  
Combining the boundary conditions (\(r=0\) and \(\tau=0\)), Equation (8) is integrated to:  
\[\tau=-\frac{r}{R}\left(\rho gR\cos\alpha+\frac{dP}{d\alpha}\right) \tag{9}\]  
Defining \(-\rho gKcos\alpha-\frac{dP}{da}\) as \(A\), then Equation (9) is expressed as:  
\[\tau=\frac{r}{R}A \tag{10}\]  
Equations (1) and (10) are combined as follows:  
\[\frac{dv}{dr}=\frac{1}{\mu}\left(\tau_{0}-\frac{rA}{R}\right) \tag{11}\]  
The \(\mu\) in Equation (11) is a simplified symbol of \(\mu_{(t_{2})}\), which is adopted in the following section. The velocity distribution of the Bingham fluid is shown in Figure 3.  
When the absolute value of the fluid height \(r\) is not greater than the flow core height \(r_{p}\) (\(|r|\leq r_{p}\)), the adjacent layers are relatively stable (\(dv/dr=0\)). The grout flow velocity is the maximum, and the grout shear stress is the minimum. When the fluid height \(r\) is between \(r_{p}\) and \(b/2\) (\(r_{p}<|r|\leq b/2\)), the grout flow velocity decreases from the maximum velocity value owing to the shear stress. According to Equation (10), when \(r=r_{p}\) (\(\tau=\tau_{0}\)), \(r_{p}\) can be expressed as follows:  
\[r_{p}=\frac{\tau_{0}R}{A} \tag{12}\]  
Figure 3: Flow velocity distribution of Bingham fluid. Integrating Equation (11) with the boundary conditions (\(r=b/2\) and \(v=0\)), the flow velocity distribution of the grout (\(r_{p}<|r|\leq b/2\)) can be obtained as follows:  
\[v=\frac{1}{\mu}\bigg{[}\tau_{0}\bigg{(}r-\frac{b}{2}\bigg{)}-\frac{A}{2R}\bigg{(} r^{2}-\frac{b^{2}}{4}\bigg{)}\bigg{]} \tag{13}\]  
When \(r=r_{p}\) (\(v=v_{p}\)), the flow velocity distribution of the grout (\(|r|\leq r_{p}\)) is expressed as follows:  
\[v_{p}=\frac{1}{\mu}\bigg{[}\tau_{0}\bigg{(}r_{p}-\frac{b}{2}\bigg{)}-\frac{A}{2 R}\bigg{(}{r_{p}}^{2}-\frac{b^{2}}{4}\bigg{)}\bigg{]} \tag{14}\]  
Therefore, the flow velocity distribution of the grout can be obtained by combining Equations (13) and (14):  
\[\left\{ \begin{array}{l}v=\frac{1}{\mu}\bigg{[}\tau_{0}\bigg{(}r-\frac{b}{2} \bigg{)}-\frac{A}{2R}\bigg{(}{r_{p}}^{2}-\frac{b^{2}}{4}\bigg{)}\bigg{]}\qquad \big{(}r_{p}<|r|\leq b/2\big{)}\\ v_{p}=\frac{1}{\mu}\bigg{[}\tau_{0}\bigg{(}r_{p}-\frac{b}{2}\bigg{)}-\frac{A}{2 R}\bigg{(}{r_{p}}^{2}-\frac{b^{2}}{4}\bigg{)}\bigg{]}\qquad\big{(}|r|\leq r_{p} \big{)}\end{array} \right. \tag{15}\]  
According to Equation (15), the average fluid velocity and flux of the grout can be obtained as follows:  
\[\overline{v}=\frac{1}{b}\int_{-\frac{b}{2}}^{\frac{b}{2}}vdr=\frac{2}{b\mu} \bigg{(}-\frac{\tau_{0}b^{2}}{8}+\frac{1}{2}\tau_{0}{r_{p}}^{2}+\frac{Ab^{3}} {24R}-\frac{{A{r_{p}}^{3}}}{3R}\bigg{)} \tag{16}\]  
\[Q_{ij}=\overline{v}S=\frac{\delta}{\mu}\bigg{(}-\frac{\tau_{0}b^{2}}{4}+\tau_ {0}{r_{p}}^{2}+\frac{Ab^{3}}{12R}-\frac{2{A{r_{p}}^{3}}}{3R}\bigg{)} \tag{17}\]  
where \(Q_{ij}\) is the average grout flux of the \(i\) grouting hole (m\({}^{3}\)/s), \(i\) denotes the number of the grouting hole, \(j=1,~{}2\) represent upward and downward, respectively, \(S\) is the sectional area of grouting (m\({}^{2}\)). It is assumed that the upward average grout flux is equal to the downward average grout flux of the \(i\) grouting hole.