# Theoretical Approach to Predicting the Diffusion Radius of Fracture Grouting in Soil-Rock Mixtures
## 4 Results and Discussion
### The Fracture Diffusion Formula of Bingham Fluid in the Soil-Rock Mixtures  
It is generally accepted that the slurry flow pattern transforms with the change in the water-cement ratio. When the water-cement ratio is relatively large, the viscosity is small, and the slurry fluidity is strong. At this time, the slurry is a Newtonian fluid. With the decrease in the water-cement ratio, the slurry viscosity increases and is gradually transformed into a Bingham fluid. When the water-cement ratio of the slurry continues to decrease, and the viscosity increases to a certain extent, the slurry transforms into a power-law fluid. That is, when the water-cement ratio of the slurry is less than 0.6, it belongs to a power-law fluid; when the water-cement ratio is greater than 1.0, the slurry is transformed into a Newtonian fluid, and the Bingham fluid is between them. The grouting diffusion models for each group of tests were classified, and the results are shown in Table 6.  
Groups #6 and #7 belong to the Newtonian body permeation grouting case; #3, #6, and #13 belong to the Newtonian body fracture grouting case; #4, #5, #8, and #9 belong to the Bingham fluid permeation grouting case; and #1, #2, #4, #5, #11, and #12 belong to the Bingham fluid fracture grouting case. Since groups #7, #8, and #9 are mainly permeation diffusion, the fracture diffusion depth is small; #3, #6, and #13 belong to the Newtonian body fracture grouting case, so they are not discussed. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\mu\) & \(\eta_{p}\) (MPa-s) & \(\beta\) & \(p_{t}\) (kPa) & \(E_{s}\) (MPa) & \(\varphi\) & \(\sigma_{c}\) (kPa) & \(\mathbf{M}\) & \(\mathbf{m}\) \\ \hline
0.35 & 118 & 116.8 & 2.492 & 8.3 & 13\({}^{\circ}\) & 107 & 1.58 & 2 \\ \hline \end{tabular}
\end{table} 
Table 5: Soil parameters of Group #10. Therefore, the theoretical diffusion radius of each experimental group can be deduced according to the parameter values and the Bingham fluid fracture grouting diffusion formula (Equation (14)). For those that cannot be solved directly, the squeeze theorem is used to approximate the unique real numerical solution continuously. The calculated value is compared with the measured value, and the comparison results are shown in Figure 18.  
As the test effects of groups #1, #2, #11, and #12 are dominated by fracture diffusion, permeation diffusion has less influence on their diffusion processes, so their permeation diffusion is not considered in the calculated values. The tests of groups #4 and #5 carry out both permeation diffusion and fracture diffusion processes, and both affect the diffusion effect of the slurry together, so both should be considered. Groups #1 and #2 were carried out in a single fracture grouting process, and the calculated results were very close to the measured values, with an error range of 3%. Although groups #11 and #12 were also carried out in a single fracture grouting process, the field test was easily disturbed by external conditions, so the error between the calculated results and the measured values was larger than that of groups #1 and #2 (the error value increased by approximately 2%), and the error range of groups #11 and #12 were within 5%. Compared with groups #1 and #2, the error values of the diffusion radius of groups #4 and #5 were also larger due to both permeation grouting and fracture grouting. The above results also show that when a single fracture grouting is carried out in a grouting test, the results calculated by the diffusion radius formula are more accurate, while when permeation grouting and fracture grouting are carried out simultaneously in soil-rock mixtures, the results will be more inaccurate. This is because when permeation grouting and fracture grouting are carried out simultaneously, the two processes share grouting pressure, grouting time, and grouting volume, and it is not easy to segment the two processes quantitatively.