## 3 Numerical Analysis  
This study aims to reveal the spatiotemporal variation in the porosity of porous media from a microscopic perspective. Therefore, the model derived in this paper was numerically analyzed. The influence of grouting parameters (Table 1), such as the water-cement ratio and the fractal dimension of particle mass on the temporal and spatial variation mechanism of grout diffusion in porous media, was studied under the constant pressure condition. In order to verify the applicability of the theoretical model, the design of test parameters and results was based on the study by [PERSON] [45]. Therefore, the numerical analysis results were compared with the experimental results in the reference. Figure 1 depicts the relationship between porous media porosity and cement particle diffusion distance at each test point of the model and experimental results [45] under different fractal dimensions of the pore. The results show that: the porosity of the media in the test section is close to that at the injection port in the sand with a fractal dimension (\(D_{\text{m}}\)) of particle mass of 2.1, and the porosity is low and generally unchanged. When it diffuses to 1.05 m, the porosity slowly increases. When it reaches 1.35 m, the porosity rapidly increases. At a \(D_{\text{m}}\) of 1.5, the porosity increases slowly with the increase in the distance from the injection port. Comparing the predicted and measured relationship between the porosity and the diffusion distance of the porous medium shows that the experimental results are consistent with the model prediction. However, there is a slight difference in quantity. This is because during grouting diffusion, massive cement slurry particles are gathered on the unit section close to the injection port, resulting in a high retention rate of cement particles at the injection port. Thus, it is easy to cause a rapid decline in porosity and even block pores. When the porosity of the injected medium approach to the injection port is reduced to the threshold value or the pore channel is completely blocked, the retention rate of cement particles increases gradually near the injection port. The medium will be blocked. Then, close to the injection source, the retention rate of cement particles rapidly falls. Additionally, the porosity of the porous medium approach to the injection port is large. Thus, it is difficult for cement particles to stay in the porous medium, and the blockage will not occur immediately. As a result, the diffusion distance of cement slurry is relatively large for a porous medium with a low \(D_{\text{m}}\). In a porous medium, cement particles are larger and more evenly dispersed. The overall trend of the model and the experiment is the same. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Water–Cement Ratio of** & **Fractal Dimensions of** & **Other Parameters** \\  & **Cement Slurry (w:c)** & **the Pore/\(D_{\text{m}}\)** & **Other Parameters** \\ \hline \multirow{2}{*}{Value} & \(1.0\) & \(1.5\) & \multirow{2}{*}{T = 20 \({}^{\circ}\)C} \\  & \(1.5\) & \(2.1\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selection of model parameters. Due to the accuracy of the experiment operation, the difference lies within the allowable error range. Figure 2 shows the variation law of porous medium porosity with distance at each test point and model at different water-cement ratios of the cement slurry. Figure 2 demonstrates that the porosity of each measurement point increases with the increase in slurry diffusion distances. The variation trend of the porosity in a low-porosity medium is relatively slow. This is because the cement concentration is large at a low water-cement ratio of the slurry. The proportion of cement particles per unit volume is relatively large. The cement particles stay in a certain section of the porous medium instantaneously so that the slurry diffusion stops, and the porosity of each test point increases rapidly with distance.