## 1 Introduction  
Geological environment conditions are getting more complicated and changeable with the effect of rainfall, rock weathering and changes in in situ stress in engineering construction [1]. Engineering practice shows that grouting technology, as a method to improve the physical properties of rock and soil, is often used to deal with a complex geological environment [2]. Grout is injected into the cracks and pores in rocks and soil to plug water and reinforce the intensity of the injected body using grouting technology. Therefore, it has been widely used in several application domains such as construction engineering, mining engineering, water conservancy and hydropower engineering and civil engineering [3], for example. Several investigations have been conducted in penetration grouting mechanism and grouting experiments to study diffusion laws in porous media. For instance, [PERSON] [4] proposed an analytical expression for the relative penetration of unsaturated soil, which could accurately predict the influence of unsaturated characteristics on penetration. [PERSON] [5] derived a driving-water penetration grouting mechanism of Newtonian fluid in sand layers. The actual results showed that the mechanism developed by [PERSON] was closer to the engineering practice than the Maag formula. [PERSON] [6] proposed a penetration grouting mechanism of gravel soil reinforced by power-law fluid. This theoretical mechanism was validated by comparing results from theoretical analyses, indoor experiments and numerical simulations. [PERSON] [7] developed a spherical penetration and diffusion mechanism of Bingham fluid. He obtained grouting pressure that satisfied the design requirements of the diffusion radius of the Bingham grout and the variation laws of the diffusion radius with a formation depth. [PERSON] [8] analyzed the penetration process of Bingham grout in fully saturated soil. He then proposed the penetration theory of immiscible two-phase fluid for grout-groundwater in porous media. The theoretical results were in good agreement with the experimental results, which indicated that the theory could provide support for construction design. [PERSON] and [PERSON] [9; 10] studied the influence of the filtration effect on penetration and diffusion in grouting. They proposed a spatial-temporal variation relationship between the penetration coefficient and porosity, which deepened the understanding of the microscopic penetration grouting mechanism in porous media. [PERSON] [11; 12] obtained a propagation law of miscible grout in saturated deformed porous media, based on the spatial volume average statistical model of [PERSON]. From the perspective of conservation of quality, [PERSON] and [PERSON] [13] developed an iterative numerical calculation method considering the penetration effect and the corresponding grid model. Based on this model, the numerical solution of the sand column grouting model considering the percolation effect was obtained. [PERSON] [14] studied the spherical diffusion laws for cement and sodium silicate mixed grout while considering the time-dependency and space effect of viscosity. He also analyzed the influence of C-S viscosity parameters on the grout diffusion radius. The influence of viscosity parameters on grout diffusion distance was also discussed. [PERSON] [15; 16; 17] established the spherical permeation and diffusion model of Bingham and power-law fluid under pulsating pressure and developed the model using the Comsol Multiphysics numerical simulation software. The model and the practical results were compared and analyzed to verify its efficiency. [PERSON] [18] developed the penetration grouting diffusion model for C-S grout considering the spatiotemporal variation of viscosity, and the theory was verified through one-dimensional simulation experiments. [PERSON] et al. [19; 20] established a permeation grouting model considering the tortuous effect of the diffusion path of cement grout in porous media.