## 4 Analytical Formula for Grouting Pressure  
[PERSON] [26] established a grouting pressure model for porous media under the assumption of linear grout flow paths. In this study, the model is extended to the circumferential direction to represent grout infiltration within the annular pea gravel backfilling layer. To verify the validity of the numerical model, theoretical analysis was conducted, and an analytical solution for the grouting pressure was proposed. The gravel layer is regarded as a bundle of capillaries. The equivalent width \(b\) of the porous medium can be estimated, and the calculation formula is as follows:  
\[b=\frac{e\cdot b_{0}}{k\cdot\Sigma C/l} \tag{4}\]  
where \(e\) is the porosity of the porous medium, \(b_{0}\) is the width of the pea gravel layer, \(\Sigma C\) is the specific surface area per unit area, defined by the perimeter of all particles, \(l\) is the length of the fluid domain, and \(k\) is an empirical parameter affected by the particle distribution, which is set to 2.7 in this calculation. The calculation results of the equivalent width \(b\) of the numerical model in this study are shown in Table 2.  
Taking a fluid element within the capillary tube for force analysis, as shown in Figure 15, the force balance equation is  
\[2\tau\mathrm{d}l+2h\mathrm{d}p=0 \tag{5}\]  
Therefore, the distribution of the shear stress on the fluid element is given by  
\[\tau=-h\frac{\mathrm{d}p}{\mathrm{d}l} \tag{6}\]  
The velocity distribution of the grout is given by [27]  
\[v=\left\{ \begin{array}{ll}\frac{1}{\mu}\left[-\frac{b^{2}-4h^{2}}{16}\frac{ dp}{dl}-\tau_{0}(b/2-h)\right]&(h_{0}\leq h\leq b)\\ \frac{1}{\mu}\left[-\frac{b^{2}-4h^{2}_{0}}{16}\frac{dp}{dl}-\tau_{0}(b/2-h_{0 })\right]&(h\leq h_{0})\end{array} \right. \tag{7}\]  
\[h_{0}=-\tau_{0}\left(\frac{dp}{dl}\right)^{-1} \tag{8}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sum of Grain** & **Length of Fluid** & **Width of Fluid** & **Equivalent** \\
**Perimeter** & **Domain** & **Domain** & **Width** \\ \hline
55.414 m & 3.456 m & 0.2 m & 0.00185 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation results of the equivalent width. Figure 15: Force analysis of fluid microelement. \(\tau_{0}\) is the yield shear stress, which is set to 7 Pa. The average velocity is expressed as  
\[\overline{v}=\frac{1}{b}\int_{-b/2}^{b/2}v\mathrm{d}h \tag{9}\]  
By substituting into Equation (7), we can obtain  
\[\overline{v}=\frac{b^{2}}{12\mu}\left[-\frac{dp}{dl}-3\lambda-4\frac{\lambda^{ 3}}{\left(dp/dl\right)^{2}}\right],\lambda=\frac{\tau_{0}}{b} \tag{10}\]  
Since \(dp/dl\) is much larger than \(\lambda\), the last term in the parentheses can be omitted, and the equation can be directly integrated. When the grouting velocity is constant, the relationship between \(v\) and \(l\) can be expressed as Equation (11) based on the principle of mass conservation:  
\[l=\frac{vt}{e} \tag{11}\]  
Through integrating Equation (10) and substituting Equations (4) and (11), the relationship between grouting pressure and grouting time can be obtained:  
\[p=\left(12\mu v\left(\frac{k\cdot\Sigma C/l}{e\cdot b}\right)^{2}+3\tau_{0} \frac{k\cdot\Sigma C/l}{e\cdot b}\right)\cdot\frac{vt}{e} \tag{12}\]  
Assuming that the grout diffuses at the same velocity in both directions, the velocity \(v\) in Equation (12) should be taken as half of the injection speed. The grout column formed after a certain period is shown in Figure 16.  
Given that \(\beta=\alpha-\theta/2\), the result can be derived using trigonometric functions:  
\[h_{g}=2R\cos(\frac{\pi}{2}-\frac{\theta}{2})\sin(\alpha-\frac{\theta}{2}) \tag{13}\]  
When considering the self-weight of the grout, if it is assumed that the gravitational pressure at point M is equal to that at the injection port, then the pressure at the injection port caused by the upward-diffusing grout column can be expressed as  
\[p_{g}=2\rho gR\cos(\frac{\pi}{2}-\frac{\theta}{2})\sin(\alpha-\frac{\theta}{2}) \tag{14}\]  
Figure 16: Analysis of grout diffusion.