## 3 Results and Discussion
### Effect of Working Conditions on Penetration Distance  
In this section, the effect of various working conditions (porosity, average particle size, viscosity, grouting pipe length, and back pressure) on the penetration distance is discussed. Figures 9 and 10 show the effect of porosity on penetration distance. The percentage increase in distance is defined as  
\[\frac{\text{distance of pulsating grouting - distance of steady grouting}}{\text{distance of steady grouting}} \tag{17}\]  
As shown in Figure 9, it can be observed that with the increase in porosity, the lateral penetration distances of both pulsating grouting and steady grouting increase linearly. The expressions are Y = 1484X \(-\) 123 and Y = 1450X \(-\) 164, respectively. However, as the porosity increases, the percentage increase in distance gradually decreases from 29.7-19.7%, following the expression Y = exp (2.52 + 8.98X \(-\) 23.35X\({}^{2}\)). This indicates that pulsating grouting performs better under high resistance. As shown in Figure 10, the vertical penetration distance of pulsating grouting and steady grouting also increases linearly, with the expressions Y = 996X \(-\) 106.6 and Y = 932X \(-\) 115.6, respectively. However, the percentage increase in distance exhibits a trend of initially increasing and then decreasing. This is because when the porosity is low, the slurry tends to flow towards the annular area with lower flow resistance rather than into the soil below the grouting pipe. As a result, when the porosity decreases from 0.25 to 0.2, due to the faster diff  
Figure 8: Effect of pressure pulse amplitude on penetration distance. Figure 9: Effect of porosity on lateral penetration distance. the influence of pulsation, more slurry flows into the annular space, which reduces the percentage increase in distance. With the increase in porosity, the flow resistance decreases, and the patterns for lateral and vertical penetration behavior become similar. Figures 11 and 12 display the influence of the particle size of soil on penetration behavior. The penetration patterns of the lateral and vertical directions are similar to the influence of porosity. The expressions for lateral penetration distance and particle size for pulsating grouting and steady pressure grouting are as follows: Y = 306.3X + 11.6 and Y = 295.8X - 32.8, respectively. The expressions for vertical penetration distance and particle size of the two grouting methods are as follows: Y = 216.28X - 34.5 and Y = 190.45X - 38.7, respectively. As the particle size increases, the percentage increase in lateral distance gradually decreases. And that of vertical distance also initially rises and then decreases. This is also due to the smaller particle size, which results in more fluid flowing into the annular region during the pulsating grouting process. Figure 11: Effect of particle size on lateral penetration distance. Figure 10: Effect of porosity on vertical penetration distance. Figures 13 and 14 present the effect of viscosity on penetration behavior. As viscosity increases, the lateral penetration distance of pulsating grouting decreases from 549 mm to 294 mm, with the expression Y = exp (6.78 \(-\) 9.09X + 18.696X2). Similarly, the penetration distance for steady pressure grouting decreases from 430 mm to 242 mm, with the expression Y = exp (6.52 \(-\) 8.92X + 19.18X2). With the increase in viscosity, the percentage increase in distance gradually increases, which also demonstrates that pulsating grouting can achieve better performance in high-resistance conditions. As shown in Figure 14, the vertical penetration distance of pulsating grouting and steady grouting exhibits an exponential decrease. However, the percentage increase in distance follows a pattern of initially increasing and then decreasing. This is also due to the low resistance in the annular space. When the slurry viscosity is high, more slurry flows into the annular space under the influence of pulsating pressure. Figure 12: Effect of particle size on vertical penetration distance.