## 1 Introduction
However, during grouting, the porosity of the injected medium changes dynamically. If the porosity is regarded as constant, the error of diffusion distance and diffusion effect will be affected. Nevertheless, the above research does not fully consider real-time porosity changes. Since [PERSON] published his famous paper on fractal geometry1967, the fractal theory has been applied in various fields [27]. For example, it can be applied to the vulnerability assessment of water network disasters, damage diagnosis for offshore wind turbine foundations [28; 29], prediction of the permeability and strength of multi-scale pores and fractures in different media [30; 31], evaluation of the landform, surface roughness, and pore and fractal structure characteristics of porous media [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. Therefore, fractal theory can be used to characterize the distribution law of pore channel diameter and a zigzag degree in porous media [43; 44]. However, considering the influence of the percolation effect, there are few studies on applying fractal theory to characterize the dynamic variation in porosity when grout diffuses in the medium [45]. Moreover, the percolation effect will cause the slurry to spread to a certain area of the porous medium. However, due to the low concentration of the slurry, the strength of the area cannot reach the design strength. This is because the diffusion range of cement particles will affect the porosity of the medium after injection, thus changing the medium strength. [PERSON] [46] studied the percolation stability and strength of cement slurries at various temperatures. [PERSON] [25] studied the changing law of the compressive strength of grout at different diffusion distances and found that the strength of the consolidated body decreased with the increase in grouting distance. [PERSON] [47] analyzed the mechanical properties of the consolidated body at different distances from the grouting pipe through model tests and found that the mechanical properties of the samples obtained at different locations of the consolidated body were different. Nevertheless, the influence of the percolation effect on the strength of the medium after grouting has not been quantitatively studied. During grouting construction, the diffusion effect of grout in the medium and the distribution characteristics of particles are the key factors that affect the effect of grouting reinforcement. Consequently, understanding the spatial distribution of cement slurry injected into porous media is beneficial to reveal the repair and reinforcement effect of cement grouting. Therefore, considering the influence of the percolation effect, the fractal theory is introduced into the diffusion model of cement grouting. This paper focuses on the porosity variation in the sand medium during grouting and develops a porosity variation model for the injected porous medium. Experimental results in the literature were selected to verify the influence of injection parameters such as cement content, pore characteristics and spatial distribution characteristics of cement strength on the grouting effect in the theoretical model. The findings of this study can provide a guide for the design of injection parameters of cement grouting and help to overcome the shortcomings of empirical grouting parameter selection.