## 4 Analytical Formula for Grouting Pressure
where \(\theta\) can be calculated from Equation (11):  
Figure 15:0  
Equation (14) can be written as  
Figure 15:1  
By adding Equations (16) and (12), the following can be obtained:  
\[ \begin{array}{l}p=\left(12\mu v\left(\frac{k\cdot\Sigma C/l}{e\cdot b}\right) ^{2}+3\tau_{0}\frac{k\cdot\Sigma C/l}{e\cdot b}\right)\cdot\frac{vt}{e}\\ +2\rho gR\cos(\frac{\pi}{2}-\frac{vt}{2 eR})\sin(\alpha-\frac{vt}{2 eR})\end{array}  \tag{17}\]  
Based on the proposed formula, the grouting pressure can be estimated using the given parameters. The estimated results are compared with the numerical results, as shown in Figure 17, where the solid line represents the numerical simulation results, and the dashed line represents the results estimated by the formula. At the initial stage of grouting, the pressure value obtained from the formula is slightly higher than the simulated value. When the grouting port is at 30\({}^{\circ}\), the dashed line is closer to the solid line. In contrast, when the grouting port is at 150\({}^{\circ}\), there is a significant difference between the dashed and solid lines. At \(t\) = 2 s, the formula-calculated value is only 40% of the simulated value. The analytical formula for grouting pressure developed in this study represents an extension of existing theoretical models and provides a preliminary analysis of pressure variations in pea gravel backfill layers. However, several limitations remain. First, in practical engineering applications, the grouting velocity is typically variable rather than constant, indicating that the formula should be modified to better reflect real-world conditions. Second, the formula does not account for the dynamic variation in grout viscosity over time or with flow conditions, which may compromise its accuracy during long-duration or large-volume grouting operations. Additionally, the current formula was derived under two-dimensional assumptions and neglects the axial diffusion of the grout; thus, its applicability and accuracy in three-dimensional contexts require further investigation and validation. Figure 17: Simulated grouting pressure versus values predicted by formula. To investigate the influence of porosity variation in the filling layer on grouting performance, two new pore structure models were constructed with porosities of 0.3 and 0.5, respectively. In both models, the grouting port was positioned at \(\alpha=150^{\circ}\), with the corresponding sums of grain perimeters being 65.168 m for the 0.3-porosity model and 49.201 m for the 0.5-porosity model. Figure 18 presents the time-dependent grouting pressure curves for the models with porosities of 0.3 and 0.5. The numerical simulation results indicate that the grouting pressure in both models exhibits an approximately linear increase over time. Specifically, the simulated pressure values for the 0.3-porosity model are consistently higher than the predicted values, with the predicted value at 2 s being approximately 65% of the simulated result. In contrast, the simulated values for the 0.5-porosity model show a higher degree of agreement with the theoretical predictions. Moreover, the pressure in the 0.3-porosity model increases at a faster rate than in the 0.5-porosity model, indicating that a lower porosity leads to greater resistance during grouting and thus results in higher pressure development. To analyze the influence of grout injection velocity on grouting performance, two additional models were constructed with injection velocities set to 0.1 m/s and 0.3 m/s, respectively. In both models, the grouting port was positioned at \(\alpha=150^{\circ}\). Figure 19 presents the time-dependent grouting pressure curves under the two injection velocity conditions. The numerical simulation results demonstrate that an increase in injection velocity leads to a significant rise in grouting pressure. Moreover, as the injection velocity increases, the discrepancy between the simulated and predicted pressure values becomes more pronounced.