# Discrete Element Analysis of Grouting Reinforcement and Slurry Diffusion in Overburden Strata
## 4 Discrete Element Grouting Diffusion Results
### Influence of Formation Confining Pressure on Slurry Diffusion Radius  
Investigating the influence of grouting depth on diffusion distance provides a theoretical basis for optimizing grouting parameter design. To accurately define diffusion distance under different grouting pressures, the following concepts are introduced: (1) Minimum diffusion radius (\(R_{1}\)): defined as the radius of a dynamically expanding concentric detection zone centered on the grouting hole, at the point when it is first completely filled with cement particles. (2) Maximum diffusion radius (\(R_{2}\)): the radius of the detection zone when it fully encompasses the outermost extent of all migrated cement particles. In the grouting simulations, grouting pressures are set to 500 kPa, 750 kPa, 1 MPa, 1.25 MPa, 1.5 MPa, 2 MPa, 2.5 MPa, and 3 MPa, respectively. (1) Law of slurry diffusion distance  
Based on the discrete element simulation results, the extent of slurry diffusion is outlined in Figure 10a, and the corresponding diffusion distances are plotted in Figure 10b. The analysis shows that with the increase of grouting pressure, the maximum and minimum diffusion distances show a trend of increasing rapidly and then increasing slowly, and the increasing trend of the diffusion radius of the whole slurry is nonlinear. For example, the maximum diffusion distances corresponding to grouting pressures of 500 kPa, 750 kPa, 1 MPa, 1.25 MPa, 1.5 MPa, 2 MPa, 2.5 MPa, and 3 MPa are 6.41, 6.55, 6.92, 7.28, 7.44, 7.74, 8.48, and 8.67 cm, respectively. (2) Formula of slurry diffusion distance  
Based on the test data presented in Figure 10, the influence of grouting pressure on the slurry diffusion radius is analyzed using a single-factor regression method. In conjunction with the mathematical model established in previous studies [24], the following assumptions are proposed: it is assumed that the dependent variable is the diffusion distance \(R\), and the independent variable is the grouting pressure \(p\), and these variables are assumed to follow a power function relationship, as expressed in Equation (2). \[R=a\cdot p^{b}m^{c}K^{d}t^{e} \tag{2}\]  
In the formula, \(a\), \(b\), \(c\), \(d\), and \(e\) represent undetermined regression coefficients. [PERSON] et al. [24] calibrated the formula using experimental data from grouting tests conducted in a sand-gravel layer, and the resulting expression is presented as Equation (3):  
\[R=19.953\cdot p^{0.412}m^{0.121}100K^{0.429}60t^{0.437} \tag{3}\]  
In this study, the water-cement ratio m (W/C = 0.7), permeability coefficient \(K\) (1 \(\times\) 10\({}^{-4}\) cm/s), and grouting time t (2.4 min, i.e., 144 s) are set as fixed parameters. By substituting these parameters into Equation (2) and performing regression analysis using the model test data from Figure 10, Equation (4) is obtained, with a coefficient of determination \(R^{2}\) of 0.9495. \[R_{\text{min}}=2024.96\cdot p^{0.1865}0.7^{0.0379}0.0001^{0.6636}2.3^{0.5611} \tag{4}\]  
To verify and compare the applicability of the proposed formula, several commonly used slurry diffusion models are summarized, including Equations (5)-(7). The Maag theoretical model is regarded as one of the earliest and most representative studies in grouting theory in porous media. This model is based on the flow behavior of Newtonian fluids in porous media. It assumes that when slurry is injected into a uniform and isotropic formation as a point source, it diffuses spherically within the medium, as described by Equation (5). Building on the Maag model, some researchers further derived a cylindrical diffusion model for slurry, as shown in Equation (6). [PERSON] et al.