## 1 Introduction  
Under the action of long-term load and environment, loosening is more likely to occur at the semi-rigid base of the pavement, leading to insufficient bearing capacity and decreased service life [1, 2]. Conventional restoration methods require excavation and re-placement of highways, which causes resource waste and impacts the environment. Although the micro-surface and thin cladding can extend pavement service life [3], it is difficult to cure deep semi-rigid base defects. Grouting is a convenient road maintenance method [4, 5]. In the process of underground excavation, grouting can also reinforce the soil to prevent collapse. Cement and chemical slurry are two kinds of grouting materials that are widely used [6, 7, 8]. Cement-based grouting materials have the characteristics of low cost and less impact on the environment, thus facilitating more extensive use [9, 10, 11]. The hydration and microcosmic properties of cement-based materials have been extensively studied [12, 13, 14]. However, the percolation effect on cement in the grouting process has not been fully considered, which will lead cement particles to be blocked in the pores of grouted media during diffusion. Then, the cement particles will gather near the grouting mouth. Consequently, the porosity of the injected medium near the grouting mouth is reduced, blocking the pore channel of the injected medium. Thus, it is difficult for cement particles to continue to diffuse, affecting the repair effect of grouting [15, 16]. More importantly, the performance of the consolidated body after grouting is closely related to the concentration of injected cement and the diffusion characteristics of cement particles in the medium. The strength of the consolidated body increases with the cement particles, indicating that the filtration effect will affect the repair effect of grouting [17; 18]. The influence of the filtration effect on the diffusion distance and grouting pressure of different types of cement grout and porous media has been studied. [PERSON] [15] established a one-dimensional slurry percolation diffusion numerical model by including the percolation effect and revealed the influence mechanism of the percolation effect on the grouting effect. Their findings provided a theoretical basis for improving the grouting method. [PERSON] [16] found that the uneven precipitation of suspended particles in water-saturated porous media was affected by the size and distribution of cement particles. [PERSON] [19] developed a grouting diffusion model considering the temporal variation in viscosity and the percolation effect and used the difference method to solve the theoretical model. [PERSON] [20] established a bentonite-slurry diffusion distance calculation model considering the slurry percolation effect. [PERSON] [21] conducted experiments to determine the mechanism responsible for the termination of penetration and diffusion of cement grout. [PERSON] [22] established diffusion models of single-hole grouting and multi-hole grouting considering the percolation effect. It was found that the percolation effect can cause the cement particles to gather at shorter distances. However, there are almost no cement particles at the edge of the diffusion distance. [PERSON] [23] established a grout diffusion model for grouting behind shield tunnel walls considering the percolation effect, and the results showed that the percolation effect significantly affects the grouting pressure. Then, the cylindrical and spherical infiltration diffusion model of power-law fluid grouts was developed by [PERSON] [24]. [PERSON] [25] verified that the percolation effect would reduce the diffusion distance of slurry through experiments. Meanwhile, the percolation effect can also restrict the diffusion of cement, affecting the distribution of cement particles in the pores of the medium. [PERSON] [26] proposed a numerical model for grouting penetration in porous media considering the filtration effect in order to study the grouting motion law.