## 2 Mathematical Model
### Circumferential Grout Pressure Distribution
When the total number of grouting holes is \(N\), \(Q_{\rm{i1}}\) can be calculated as follows:  
\[Q_{\rm{i1}}=\frac{m\pi\big{(}{R_{0}}^{2}-{R_{1}}^{2}\big{)}v_{d}}{2N} \tag{18}\]  
where \(m\) is the grouting volume ratio, which represents the ratio of the actual grouting volume to the theoretical grouting volume, usually ranging from 130% to 250% [36]. Equation (12) is substituted into Equation (17) as follows:  
\[A^{3}-\bigg{(}\frac{12Q_{ij}\mu R}{\delta b^{3}}+\frac{3\tau_{0}R}{b}\bigg{)} A^{2}+\frac{4\tau_{0}^{3}R^{3}}{b^{3}}=0 \tag{19}\]  
Equation (6) is substituted into Equation (19) to obtain the equation considering the temporal variation in grout viscosity:  
\[A^{3}-\bigg{(}\frac{12Q_{ij}\mu_{0}e^{\xi\big{[}\frac{\Delta b\beta}{4q}(R_{0 }^{2}-{R_{1}}^{2})\big{]}R}}{\delta b^{3}}+\frac{3\tau_{0}R}{b}\bigg{)}A^{2} +\frac{4\tau_{0}^{3}R^{3}}{b^{3}}=0 \tag{20}\]  
Defining \(-\rho gRcos\alpha-\frac{dP}{da}\) as \(A\), we can deduce as follows:  
\[\frac{dP}{da}=-A-\rho gRcos\alpha \tag{21}\]Therefore, when the grout flow direction is upward, the circumferential filling pressure of the grout can be obtained by integrating Equation (21), combining the boundary conditions (\(\alpha=\alpha_{i0}\) and \(P=P_{i}\)):  
\[P=P_{i}-A(\alpha-\alpha_{i0})+\rho gR(sina_{i0}-sina) \tag{22}\]  
\(\alpha\) in Equation (22) is replaced with \(\theta(\beta_{1}\leq\theta\leq\beta)\) to illustrate the grout pressure during the circumferential phase as shown in Figures 1b and 2a (\(\alpha=\frac{\pi}{2}-\theta,\alpha_{i0}=\frac{\pi}{2}-\beta\)). \[P=P_{i}-A(\beta-\theta)+\rho gR(\cos\beta-\cos\theta) \tag{23}\]  
where \(\theta\) is the angle between the grout element and the positive direction of the \(y\)-axis, \(\beta\) is the angle between the grouting hole \(i\) and the positive direction of the \(y\)-axis, \(\beta_{1}\) is the angle of upward grouting from the grouting hole \(i\), \(\beta_{1}=\frac{\pi}{2}-\alpha_{i2}\). Force analysis of the grout element during circumferential grout filling when the grout flow direction is downward is shown in Figure 4.  
When the grout flow direction is downward, the circumferential filling pressure of the grout is:  
\[P=P_{i}+A(\beta-\theta)+\rho gR(\cos\beta-\cos\theta) \tag{24}\]  
Therefore, the circumferential filling pressure distribution of the grout can be expressed as follows:  
\[Q_{ij}=\overline{v}S=\frac{\delta}{\mu}\bigg{(}-\frac{\tau_{0}b^{2}}{4}+\tau_ {0}{r_{p}}^{2}+\frac{Ab^{3}}{12R}-\frac{2{A{r_{p}}^{3}}}{3R}\bigg{)} \tag{17}\]0  
The circumferential filling pressure distribution of the grout in the case of six grouting holes can be calculated according to \[Q_{ij}=\overline{v}S=\frac{\delta}{\mu}\bigg{(}-\frac{\tau_{0}b^{2}}{4}+\tau_ {0}{r_{p}}^{2}+\frac{Ab^{3}}{12R}-\frac{2{A{r_{p}}^{3}}}{3R}\bigg{)} \tag{17}\]1 \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Grouting Hole Locations**} & **Grout Pressure** & **\# Range (Rad)** \\  & **(Pa)** & \\ \hline \(\beta=0\)\({}^{\circ}\) & \(P=P_{i}-A\theta+\rho gR(1-\cos\theta)\) & \(0\leq\theta\leq\frac{11\pi}{36}\) \\ \(\beta=55\)\({}^{\circ}\) & \(P=P_{i}-A\left(\frac{11\pi}{36}-\theta\right)+\rho gR\left(\cos\frac{11\pi}{36} -\cos\theta\right)\) & \(0\leq\theta\leq\frac{11\pi}{36}\) \\  & \(P=P_{i}+A\left(\frac{11\pi}{36}-\theta\right)+\rho gR\left(\cos\frac{14\pi}{36} -\cos\theta\right)\) & \(\frac{11\pi}{36}\leq\theta\leq\frac{25\pi}{36}\) \\ \(\beta=125\)\({}^{\circ}\) & \(P=P_{i}-A\left(\frac{25\pi}{36}-\theta\right)+\rho gR\left(\cos\frac{25\pi}{36} -\cos\theta\right)\) & \(\frac{11\pi}{36}\leq\theta\leq\frac{25\pi}{36}\) \\  & \(P=P_{i}+A\left(\frac{25\pi}{36}-\theta\right)+\rho gR\left(\cos\frac{25\pi}{36} -\cos\theta\right)\) & \(\frac{25\pi}{36}\leq\theta\leq\pi\) \\ \(\beta=180\)\({}^{\circ}\) & \(P=P_{i}-A(\pi-\theta)+\rho gR(-1-\cos\theta)\) & \(\frac{25\pi}{36}\leq\theta\leq\pi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Circumferential filling pressure distribution of the grout in the case of six grouting holes.