## 5 Conclusions and Discussion  
This research elucidates the diffusion behavior of ultrafine cement grouting materials with varying water-to-cement ratios in the secondary lining cracks of tunnels. This study combines experimental testing with theoretical analysis, resulting in the following conclusions:  
1. As the water-cement ratio increases from 0.5 to 1.0, the initial viscosity of the ultrafine cement slurry decreases from 33.6 mPa-s to 1.80 mPa-s. By changing the water-cement  
Figure 7: Maximum diffusion radius in different water-cement ratio: (**a**) R–t variation relation; (**b**) R–\(\xi\), variation relation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Equation** & & & **R(t) = D\({}_{1}\)+A\({}_{1}\)e\({}^{-\frac{k}{\lambda}}\)**–A\({}_{2}\)e\({}^{-\frac{k}{\beta}}\)** & & & & **R(\(\xi\)) = D\({}_{2}\)+A\({}_{3}\)e\({}^{-\frac{k}{\lambda}}\)** \\ \hline
**Direction** & **D\({}_{1}\)** & **A\({}_{1}\)** & **\(\alpha\)** & **A\({}_{2}\)** & **\(\beta\)** & **Correlation Coefficient** & **D\({}_{2}\)** & **A\({}_{3}\)** & **\(\gamma\)** & **Correlation Coefficient** \\ \hline x+ & 18.39 & \(-\)13.75 & 19.62 & \(-\)4.14 & 1.15 & 1 & 18.64 & \(-\)30.28 & 0.25 & 0.98 \\ x\(-\) & \(-\)17.59 & 4.00 & 1.04 & 13.09 & 16.27 & 1 & \(-\)18.34 & 51.55 & 0.18 & 0.96 \\ y+ & 12.78 & \(-\)3.81 & 1.59 & \(-\)8.48 & 12.73 & 1 & 0.083 & 14.76 & 0.70 & 0.91 \\ y\(-\) & \(-\)30.12 & 24.98 & 27.31 & 5.03 & 2.23 & 1 & \(-\)32.82 & 80.38 & 0.27 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter Values of grouting diffusion equation in different water–cement ratio. ratio of the cement grouting material, the initial viscosity parameters of the slurry have changed. Through the grouting test, it can be found that the diffusion of ultrafine cement slurry in cracks is mainly divided into three diffusion forms: circular diffusion zone, excessive diffusion zone, and elliptical diffusion zone. It is a circular diffusion zone within 0 to 2.5 s; 2.5 s to 5 s is the excessive diffusion zone; 5 s to 25 s is the elliptic diffusion zone. At the initial stage of grouting at 0 to 2.5 s, the slurry is not affected by self-weight, and the diffusion form is circular. After that, with the increase in time under the action of self-weight, the asymmetric elliptic characteristics of the diffusion form become more and more obvious, and the larger the water-cement ratio, the more obvious the overall diffusion asymmetry. 2. The analysis of the maximum diffusion radius can be seen in the y + direction of the maximum diffusion radius, but with the increase in the water-cement ratio, there is a tendency to increase and then decrease, which is subject to viscous stress, slurry pressure, and the common effect of self-weight, and there is a maximum diffusion radius of the water-cement ratio of the critical value of the diffusion radius through the analysis of the experimental data to obtain the diffusion radius of the extreme value of the critical water-cement ratio for \(\xi\) = 0.8. 3. The value of the maximum radius of diffusion R\({}_{\max}\) increases with the water-cement ratio (W/C) (or is inversely proportional to the viscosity). As the water-cement ratio increases from 0.5 to 0.6, the maximum diffusion radius, R\({}_{\max}\), exhibits a significant increase. Within the water-cement ratio range of 0.6 to 1.0, the growth of R\({}_{\max}\) becomes relatively stable. However, within the range of 0.8 to 1.0, the diffusion radius in the y+ direction is notably influenced by viscosity and gravitational effects, leading to a decreasing trend in R\({}_{\max}\). 4. Through the collection and analysis of test data, the diffusion radius R and time t change function equation is obtained as R(t) =D\({}_{1}\) + A\({}_{1}\)e\({}^{-t/\alpha}\) - A\({}_{2}\)e\({}^{-t/\beta}\); the maximum diffusion radius value of R\({}_{\max}\) and the change in the water-cement ratio function equation is R(\(\xi\)) =D\({}_{2}\) + A\({}_{3}\)e\({}^{-\xi/\Lambda}\). The next step is to study the diffusion law of oblique crack grouting.