# Optimisation of Synchronous Grouting Mix Ratio for Shield Tunnels
## 3 Results and Discussion
### Uniform Test  
2.1 Effects of Water-Cement, Bentonite-Water, Cement-Sand, Water Reducer-Cement, and Late Powder-Cement Ratios on Fluidity, Segregation Rate, and Bleeding Rate  
The effects of the water-cement, bentonite-water, cement-sand, water reducer-cement, and latex powder-cement ratios on the fluidity, segregation rate, and bleeding rate of the slurry are shown in Tables 5-7, respectively. Because the rheological curve trends of each test group were roughly the same and the slurry in group 16 had better anti-segregation and anti-bleeding properties, the rheological curve of the group 16 slurry was selected for analysis. The shear stress \(\tau\), shear rate \(\gamma\), and apparent viscosity \(\eta\)-shear rate \(\gamma\) characteristic curves of the rheological test of the group 16 samples are shown in Figure 3a,b. The rheology of synchronous grouting slurry can reflect its flow and deformation patterns under different shear rates and express the corresponding relationship between the shear stress and rate [52]. Therefore, the rheological  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No. Test**} & \multicolumn{2}{c}{**Dry Weight of**} & \multicolumn{2}{c}{**Dry Weight of**} & \multicolumn{2}{c}{**Layered Segregation**} \\  & \multicolumn{2}{c}{**Upper Aggregate**} & \multicolumn{2}{c}{**Middle Aggregate**} & \multicolumn{2}{c}{**Lower Aggregate**} & \multicolumn{2}{c}{**Rate**} \\ \cline{2-5}  & \multicolumn{2}{c}{**[g]**} & \multicolumn{2}{c}{**[g]**} & \multicolumn{2}{c}{**[\%]**} \\ \hline
1 & 840.6 & 850.4 & 886.3 & 1.1 \\
2 & 848.7 & 912.6 & 920.7 & 7.1 \\
3 & 834.0 & 945  
parameters--yield shear stress \(\tau_{0}\) and viscosity coefficient \(K\)--can quantitatively describe the flow state of the synchronous grouting slurry. When the shear rate was lower than 10 s\({}^{-1}\), the shear stress decreased, and the apparent viscosity decreased rapidly as the shear rate increased. When the shear rate reached 10 s\({}^{-1}\), the shear stress of the grouting slurry increased linearly with increasing shear rate, and the viscosity was stable. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Initial Pulp** & **3 h Bleeding** & **3 h Expansion** & **24 h Bleeding** & **24 h Expansion** & **3 h Bleeding** & **24 h Bleeding** \\
**No. Test** & **Surface** & **Surface** & **Surface** & **Surface** & **Surface** & **Rate** & **Rate** \\ \cline{2-8}  & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[\%]** & **[\%]** \\ \hline
1 & 284 & 286 & 282 & 286 & 276 & 280 & 278  
To quantify the rheological parameters of a fresh slurry, a suitable mathematical model must be selected to characterise its rheological properties. Research has shown that the Newtonian fluid, according to [PERSON] [53], improved Bingham (M-B) [54] and H-B models [55] can be used for rheological performance analyses, as shown in Equations (1)-(4), respectively. However, the synchronous grouting slurry in shield tunnels possesses a yield stress, which makes the Newtonian fluid model unsuitable for describing its rheological properties. Therefore, the three other models are used in the article to mathematically fit the test points of 16 experimental groups, and the results show that the R\({}^{2}\) of the H-B model is better than the other two models in 11 groups. In addition, it can be seen from Figure 3a,b that the changing trends of the shear stress and viscosity of the slurry are consistent with the H-B law. For materials such as fresh concrete, mortar, and cement paste, the relationship between the shear rate and viscosity coefficient changes nonlinearly, which is consistent with the H-B model [17]. Therefore, this study used the H-B model to fit and analyse the relationship curve between the shear stress and rate. The \(\tau\)-\(\gamma\) curve fitting results are shown in Figure 3c, where the rheological index is 1.73.