## 3 Case Analysis and Model Verification
### Model Calculation  
Initially, the grout pressure in the circumferential diffusion process was calculated by the constructed model based on the parameter values in relevant literature and construction cases. Since the circumferential diffusion process is short in duration, it is suitable for calibrating the initial viscosity. If the calculated circumferential pressure values deviate significantly from the measured values, parameters are adjusted and recalculated until the calculated values match the measured values (with an error less than 5%).  
Once the calculated results of the circumferential pressure meet the accuracy requirements, the same parameters are applied to calculate the grout pressure of the radial and longitudinal diffusion processes in a specific time to calibrate the time-varying coefficient of viscosity. If the calculated values do not meet the requirements (with an error exceeding 10%), the grout pressure in the circumferential, radial, and longitudinal diffusion process would be recalculated until the results meet the accuracy requirements of the entire diffusion process. Then, both the model and the parameters are considered effective.  
The grout pressure during synchronous grouting in the Sophia railroad tunnel is calculated using the grout pressure model. The average grout flux can be obtained according to Equation (18):  
\[Q_{i}=\frac{m\pi\big{(}{R_{0}}^{2}-{R_{1}}^{2}\big{)}v_{d}}{2N}=\frac{1.9 \times 3.14\times 0.00072}{2\times 6}\left(4.885^{2}-4.725^{2}\right)=5.50 \times 10^{-4}\;\text{m}^{3}/\text{s}\]  
During the circumferential filling phase, the grout takes at least 100 s to completely fill the shield tail void. According to Equation (4), the thickness of the annular void along the longitudinal direction can be calculated as follows:  
\[\delta=v_{d}\cdot t_{y}=0.00072\times 100=0.072\;\text{m}\]  
When the shield tail void is completely filled, \(t_{z}\) is equal to \(t_{y}\) and the grouting rate can be obtained according to Equation (3):  
\[q=\frac{\pi\delta\big{(}{R_{0}}^{2}-{R_{1}}^{2}\big{)}}{t_{y}}=\frac{3.14 \times 0.072}{100}\left(4.885^{2}-4.725^{2}\right)=3.5\times 10^{-3}\;\text{m}^{ 3}/\text{s}\]  
According to the relationship between liquid permeability and porosity in the soil by [PERSON] et al. (2020), when the permeability coefficient \(K_{w}\) was \(5\times 10^{-4}\;\text{m}/\text{s}\), the porosity \(\varphi\) was \(0.3\), and the viscosity of water \(\mu_{w}\) at \(20\;^{\circ}\)C was \(1.01\times 10^{-3}\;\;\text{Pa}\cdot\text{s}\)[38].  
\begin{table}
\begin{tabular}{c c c} \hline
**Grouting Hole** & **Location (\({}^{\circ}\))** & **Injection Pressure (MPa)** \\ \hline
1 & 0 & 0.2 \\
2 & 55 & 0.23 \\
3 & 125 & 0.34 \\
4 & 180 & 0.37 \\
5 & 235 & 0.34 \\
6 & 305 & 0.23 \\ \hline \end{tabular}
\end{table}
Table 2: Sophia railroad tunnel grouting hole location and injection pressure.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(R_{1}\) (m) & \(R_{0}\) (m) & \(R\) (m) & \(b\) (m) & \(v_{d}\) (m/s) & \(\rho\) (kg/m\({}^{3}\)) & \(\tau_{0}\) (Pa) & \(m\) (\%) \\ \hline
4.725 & 4.885 & 4.805 & 0.16 & 0.00072 & 2190 & 100 & 190 \\ \hline \end{tabular}
\end{table}
Table 3: Sophia railroad tunnel information and grout parameters.  
The \(\beta_{r}\) can be obtained by substituting \(t_{1}\) (60 min) into Equation (29):  
\[\beta_{r}==\frac{0.907\big{(}e^{0.0107\times 60}-1\big{)}}{1.01\times 10^{-3} \times 0.0107\times 60}=1259.29\]  
According to Equation (26), the fitting coefficient \(c\) in the Sophia railroad tunnel is 14.99.  
\[c=\frac{30.4}{281.3^{0.622}\times 6.667^{0.533}\times 1259.29^{-0.534}}=14.99\]  
The grout pressure formed during the circumferential filling phase is the initial value of the longitudinal and radial pressure dissipation. Based on Equations (24), (25), (45), (46), (33), and (34), the grout pressure during synchronous grouting in the Sophia railroad tunnel can be calculated.