## 3 Case Analysis and Model Verification
### Model Verification  
As shown in Figure 6, the grout pressure calculated using the grout pressure model at the end of the circumferential filling phase is compared with the field measurements. The calculation results of the grout pressure model considering the temporal variation in grout viscosity are consistent with the field measurements. The grout pressure during the circumferential filling phase of the Sophia railroad tunnel shows an irregular annular distribution. Compared with the grout pressure at the bottom part of the tunnel, the grout pressure at the top is lower owing to the gravity and initial injection pressure. The gradient of grout pressure variation is relatively small within the range of the top (0\({}^{\circ}\)-30\({}^{\circ}\)) and bottom (150\({}^{\circ}\)-180\({}^{\circ}\)) of the tunnel. The grout pressure changes from 200.0 kPa and 360.2 kPa to 203.9 kPa and 370 kPa, respectively. The circumferential gradient of grout pressure is 1.3 kPa/m and 3.9 kPa/m, respectively. However, in the range of the tunnel shoulder to waist (\(\theta=60^{\circ}\)-90\({}^{\circ}\)), the variation of the grout pressure is obvious. The grout pressure changes from 230.0 kPa to 281.3 kPa, and the circumferential gradient of grout pressure is approximately 20.7 kPa/m. Similarly, [PERSON] et al. (2021) used a semi-elliptical surface grout diffusion model to calculate the circumferential variation gradient of the grout pressure [32]. The circumferential variation gradient of grout pressure near the top and bottom areas of the tunnel was 5-10 kPa/m. In comparison, the circumferential variation gradient of grout pressure from the tunnel shoulder to the waist area was approximately 20 kPa/m. When the grout flows downward from the grouting hole, gravity has a positive effect on the grout flow. However, when the grout flows upward, gravity has a negative effect. Regardless of the grout flow direction, the grout viscous resistance has a negative effect on the grout flow. In the areas near the top (\(\theta=0^{\circ}\)) and bottom (\(\theta=180^{\circ}\)) of the tunnel, the gravity effect on the grout pressure is not obvious, and the grout pressure changes slowly. In contrast, from the tunnel shoulder to the waist area, the gravity effect plays an important role in the  
Figure 6: Theoretical values and field measurements of the grout pressure during circumferential filling phase. grout pressure, and the grout pressure changes faster. However, the calculation results of the grout pressure model without considering the temporal variation in grout viscosity are also close to the field measurements. Therefore, during the circumferential filling phase, the temporal variation in grout viscosity has little effect on the grout pressure owing to the short duration. The grout pressure calculated using the model and the field measurements at 1 h during the circumferential filling process is shown in Figure 7.  
The theoretical values of the grout pressure fit well with the field-measured values, which can reflect the variation of the grout pressure. The maximum relative error (Er\({}_{\text{max}}\)) and root mean square error (RMSE) are used as comprehensive indicators to evaluate the deviation between the results of the grout pressure model and field-measured values. The Er\({}_{\text{max}}\) and RMSE of the calculation results without considering the temporal variation in grout viscosity are 16.13% and 22.11 kPa, respectively. In contrast, the Er\({}_{\text{max}}\) and RMSE of the grout pressure model considering the temporal variation in grout viscosity are 5.29% and 9.83 kPa, respectively. [PERSON] et al. (2021) investigated the distribution of the grout pressure in a quasi-rectangular tunnel over approximately 1 h [32]. The established model did not consider the temporal variation in grout viscosity. The calculated results showed an error of 11.35% compared to the measured values. The root mean square error was 14.24 kPa.