## 3 Cylinder Diffusion Theory of Infiltration Grouting
### Calculation of Diffusion Radius  
Infiltration grouting is a complex mechanical behavior. The size of the diffusion radius is affected by many factors. The decisive factors are as follows. The first is the internal structure characteristics of the injected medium, including mutual cementation between the rock-soil particles or blocks, crack or hole porosity, and permeability coefficient. The second is the properties of the grouting materials, including viscosity and gel time. The third is the constructing conditions, including grouting pressure and grouting rate. In addition, it is also affected by ground stress conditions and groundwater occurrence. In engineering practice, the diffusion radius of infiltration grouting is generally estimated by the following empirical formulae [23]:  
\[r_{1}=\sqrt{\frac{2 Kh_{1}t}{n\beta\ln(r_{1}/r_{0})}} \tag{3}\]  
\[t=\frac{n\beta r_{1}^{2}\ln(r_{1}/r_{0})}{2 Kh_{1}} \tag{4}\]  
where \(K\) is the infiltration coefficient of the injected formation (cm/s); \(\beta\) is the viscosity ratio of slurry to water, \(\beta=\mu_{g}/\mu_{w}\); \(r_{0}\) is the radius of the grouting pipe (cm); \(r_{1}\) is the diffusion radius of the slurry (cm); \(h_{1}\) is the grouting pressure head (cm); \(t\) is the grouting time (s); \(n\) is the crack or hole clearance rate of the injected medium; \(\mu_{w}\) is the viscosity coefficient of water (s); and \(\mu_{g}\) is the viscosity coefficient of the slurry (s).  
The principle of infiltration grouting is not to damage the stratum structure, and its grouting pressure is difficult to control. In order to obtain the maximum allowable grouting pressure of infiltration grouting, the relationship between the grouting pressure and grouting volume is analyzed through construction test holes before formal grouting. When the grouting pressure rises to a certain value \(p_{t}\), the amount of grout absorption increases suddenly. It can be considered that the stratum structure has been damaged and the grout diffusion mode has changed from infiltration to splitting. Therefore, this inflection point value is defined as the extreme value of infiltration grouting (Figure 2).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multirow{2}{*}{**Lithology**} & \multirow{2}{*}{**Density**} & \multirow{2}{*}{**Moisture**} & \multirow{2}{*}{**Porosity**} & \multirow{2}{*}{**Infiltration**} & \multicolumn{3}{c}{**Accumulative Screen Residue/(\%)**} \\  & & & & & & & **Coefficient** & **10** & **50** & **90** & **95** \\ \hline \multirow{2}{*}{1} & Crushed stone & 1.82 & 3.24 & 0.466 & 0.962 & \multirow{2}{*}{\begin{tabular}{c} \textless{}9.5 \\ mm \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \textless{}37.5 \\ mm \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} \textless{}75 mm \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} \textless{}90 mm \\ \end{tabular} } \\  & Completely weathered & & & & & & & \\  & porphyritic granite & & & & & & & \\  & Strongly weathered & & & & & & & \\  & porphyritic granite & 1.76 & 5.26 & 0.439 & \(5.8\times 10^{-2}\) & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic parameters of grouting stratum rock and soil mass.  
Figure 2: Relationship between grouting pressure and amount.