## 2 Experimental Studies
### Three-Dimensional Spatial Diffusion Morphology Model  
The above analysis shows that the diffusion of cement grout in gravel soil is approximately ellipsoidal, and the diffusion dimensions in three-dimensional space are not consistent. The diffusion dimensions in the vertical direction differ greatly from those in the horizontal transverse direction. Therefore, in order to theoretically reveal the three-dimensional spatial diffusion morphology of cement grout in gravel soil and provide some theoretical support for practical engineering, based on the experimental results, the mathematical ellipsoidal geometric trajectory is used this paper to reveal the three-dimensional spatial diffusion morphology of cement grout in gravel soil by penetration grouting, which can be seen in Figure 4.  
Where \(a\), \(b\), and \(c\), are respectively the half-line of \(x\)-axis, \(y\)-axis, and \(z\)-axis, of the approximated ellipsoidal morphology by the penetration diffusion of cement grout in the gravel soil. The O point at the center of its geometric trajectory is the location of the grout \begin{table}
\begin{tabular}{c c c} \hline
**Theoretical Values of** & **Equivalent Penetration** & \\
**Penetration Diffusion** & **Diffusion Sphere Radius** & \\
**Radius Obtained from the** & **Calculated from** & **Difference between the Two** \\
**Penetration Grouting** & **Experimental** & **Values/\%** \\
**Diffusion Mechanism/cm** & **Measurements/cm** & \\ \hline
6.9 & 10.4 & 33.65 \\
8.5 & 6.4 & 24.71 \\
37.9 & 11.7 & 32.48 \\
18.0 & 7.9 & 56.11 \\
13.7 & 12.3 & 10.22 \\
3.6 & 10.7 & 66.36 \\
27.2 & 12.7 & 53.31 \\
7.0 & 11.7 & 40.17 \\
7.7 & 6.6 & 14.29 \\ \hline \end{tabular}
\end{table} 
Table 2: Theoretical values of the penetration diffusion radius, the equivalent penetration diffusion sphere radius, and the difference. Figure 4: The geometric trajectory of the ellipsoid of the three-dimensional spatial diffusion dimension of cement grout into the gravel soil by penetration grouting. Combining the rectangular equation of the ellipsoid with the experimental results in Table 1, the following equation can be used to describe the three-dimensional spatial diffusion morphology of cement grout by penetration grouting in gravel soil,  
\[\frac{x^{2}}{\left(m_{1}R_{T}\right)^{2}}+\frac{y^{2}}{\left(m_{2}R_{T}\right)^ {2}}+\frac{z^{2}}{\left(m_{3}R_{T}\right)^{2}}=1 \tag{2}\]  
where \(m_{1}\) is the adjustment coefficient of half-line of \(x\)-axis, varying from 0.70 to 1.70, generally taken as 1.20; \(m_{2}\) is the adjustment coefficient of half-line of \(y\)-axis, varying from 0.80 to 1.80, generally taken as 1.30; \(m_{3}\) is the adjustment coefficient of half-line of \(z\)-axis, varying from 0.70 to 1.70, generally taken as 1.20; \(R_{T}\) is the theoretical value of the penetration diffusion radius calculated by the penetration grouting diffusion mechanism. Equation (2) is the three-dimensional spatial diffusion trajectory equation of cement grout in gravel soil by penetration grouting, which can be used as the calculation model of three-dimensional spatial diffusion dimension parameters. Therefore, this model can be used to predict the three-dimensional spatial diffusion dimension of cement grout in gravel soil by penetration grouting. Effect of Coupling Effect of Porosity and Cement Grout Weight on the Diffusion Morphology of Penetration Grouting in Gravel Soil  
As mentioned in the former research of the author [22], the most significant factor affecting the size of cement grout in gravel soil by penetration diffusion is the water-cement ratio. The second factor is gravel soil's porosity, while grouting pressure's effect on it is small. The grouting pressure provides the same force from the grouting hole to the cement grout in the surrounding gravel soil by penetration diffusion [1; 2]. Therefore, the coupling effect of porosity and self-weight of cement grout has a very important effect on the diffusion morphology of gravel soil infiltration grouting.