## 3 Results and Discussion
### Effect of Pulsating Parameters on Penetration Distance  
There are mainly two pulsation parameters: one is the pulsating frequency, and the other is the pressure pulse amplitude. When comparing the performance of the two grouting methods, the average value of the sinusoidal pressure is taken as the pressure for steady pressure grouting, and all other parameters are kept consistent.  
Figure 5: Steam line of steady grouting.  
Figure 6: Steam line of pulsating grouting.  
Figure 7 illustrates the impact of pulsation frequency on the slurry diffusion for both grouting methods. The average pressure is 4 MPa, the porosity is 0.3, the average particle size is 1 mm, the viscosity of the slurry is 0.18 Pa-s, the pressure pulse amplitude is 0.5 MPa, and the back pressure is set to 2 MPa. The purple line represents the lateral penetration distance, and the orange line represents the vertical penetration distance. The solid lines represent the penetration distances for pulsating grouting at different frequencies, and the dashed line represents the penetration distance for steady pressure grouting. Since steady pressure grouting does not have a pulsating frequency, the dashed line is a straight line. It can be observed that the penetration distance of pulsating grouting is always greater than that of steady pressure grouting. And with the increase in frequency, the penetration distance of pulsating pressure grouting also increases; this phenomenon is similar to the results observed by researchers [5]. Both the lateral and vertical penetration distance follows an exponential distribution, and the expressions are Y = exp (5.52 + 0.138X \(-\) 0.016X2) and Y = exp (4.94 + 0.25X \(-\) 0.037X2), respectively. At a frequency of 3.5 Hz, the lateral and vertical diffusion distances increased by 70 mm and 60 mm compared to steady pressure grouting.  
Figure 8 shows the effect of pressure pulse amplitude on penetration distance. The average pressure is 4 MPa, the porosity is 0.3, the average particle size is 1 mm, the viscosity of the slurry is 0.18 Pa-s, the pulsating frequency is 2.5 Hz, and the back pressure is 2 MPa. It can be found that the penetration distance of pulsating grouting is still greater than that of steady pressure grouting. However, both the lateral distance and vertical distance increase initially and then decrease with the increase in pressure amplitude, with the turning point occurring at the pressure amplitude of 2 MPa. This is because the average pressure of the sinusoidal pressure is 4 MPa (which is consistent with steady pressure), and the back pressure at the outlet is 2 MPa. When the pressure amplitude exceeds 2 MPa, a part of the pressure pulsation is less than the back pressure. This means that the pressure inside the annulus will be lower than the pressure at the outlet, which hinders the upward movement of the slurry. The lateral penetration distance follows an exponential distribution, with Y = exp (5.6 + 0.23X \(-\) 0.057X2), while the vertical penetration follows Y = exp (5.1 + 0.29X \(-\) 0.007X2). When the pressure pulse amplitude is 2 MPa, the lateral and vertical diffusion distances increased by 72 mm and 50 mm, compared to steady pressure grouting.  
Figure 7: Effect of pulsating frequency on penetration distance.