## 4 Comparison between the Theoretical and Experimental Results  
In order to analyze the applicability and accuracy of the mechanism, the relevant penetration grouting experimental results obtained by [PERSON] [30] are used. The experimental device of [PERSON] is shown in Figure 3. Experimental results and theoretical calculation values which were obtained respectively from Equations (18) and (19) are compared. Two kinds of cement grout with water-cement ratios of 0.90, 1.00 and 1.25 [30; 31] are considered to perform the penetration grouting experiments. It can be seen from the literature of [PERSON] that they are typical Bingham fluids. The experimental scheme is shown in Table 1. The rheological equations of two cement grouts are shown in Table 2. The water viscosity is \(\mu_{w}\) = 0.001 Pa\(\cdot\)s, and the stratum parameters are shown in Table 1[27]. The theoretical calculation values of the diffusion radius can be obtained by the mechanism (cf. Equation (18)) and column penetration grouting formula (cf. Equation (19)) of Bingham fluid without considering the displacement effect. The results are compared with the experimental values, as shown in Table 3.  
It can be deduced from Table 3 that the theoretical calculation values of diffusion radius obtained by the mechanism are smaller than those obtained by the column penetration and diffusion formula of Bingham fluid without considering the displacement effect. The former is closer to the experimental values than the latter. Therefore, it can be seen that the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & **Water-Cement** & **Grouting** & **Grouting Time/s** & **Penetration** & \multirow{2}{*}{**Porosity**} \\  & **Ratio (w/c)** & **Pressure/MPa** & **Gouting Time/s** & **Coefficient** & **k/cm\(\cdot\)s\({}^{-1}\)** & \\ \hline G1 & 0.90 & 0.12 & 26.50 & 3.45 & 0.4519 \\ G2 & 1.00 & 0.10 & 24.90 & 3.09 & 0.4391 \\ G3 & 1.25 & 0.08 & 23.50 & 2.89 & 0.4414 \\ G4 & 1.00 & 0.05 & 22.50 & 3.90 & 0.4524 \\ G5 & 1.00 & 0.06 & 21.70 & 2.11 & 0.4505 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values in the penetration grouting experiments [30]. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Water-Cement Ratio**} & **Rheological Equation** \\ \hline
1.25 & \(\tau=0.1136+0.0159\)\(\gamma\) \\
1.00 & \(\tau=0.8593+0.0169\)\(\gamma\) \\
0.90 & \(\tau=1.7876+0.0194\)\(\gamma\) \\
0.75 & \(\tau=3.2130+0.0203\)\(\gamma\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rheological equations of two kinds of cement grout [30]. Figure 3: Grouting experimental device [30] (1—pressure supply equipment, 2—connecting pipe, 3—grout storage box, 4—electronic scale, 5—capillary channel, 6—experimental box, 7—gravel soil). mechanism is better than the column penetration and diffusion formula of Bingham fluid without considering the displacement effect, in order to reflect the diffusion morphology and laws of Bingham fluid in porous media. It can also be seen from Table 3 that there is still a certain deviation between the theoretical calculated values of the diffusion radius obtained by the mechanism and the experimental values. The main reasons for this deviation are summarized as follows: (1) the grouting theory deduced in this study only considers the influence of the displacement effect on the penetration diffusion morphology. Several studies show that different factors affect the grouting diffusion effect. For instance, [PERSON], [PERSON], [PERSON] and [PERSON] et al. [14; 18; 25; 32] deduced that the rheological parameters of cement grout have an important influence on the diffusion radius. [PERSON] [24] believes that the self-weight of grout will affect its diffusion radius in different penetration directions. In addition, [PERSON]'s study [32] showed that the cement grout in the process of grouting will continue to water, which results in a change in viscosity and the increase in the difficulty of grouting.