## 4 Parameter Sensitivity Analysis
### Radial and Longitudinal Grout Diffusion Phase
#### 4.2.2 Time-Varying Coefficients of the Grout Viscosity \(\xi\)  
The characteristics of the grout with different flow patterns are different. The predominant grout flow forms are mostly power-law and Bingham fluids. The viscosity of Bingham grout is primarily influenced by three parameters. The initial viscosity represents the viscosity or resistance characteristics of the grout at the beginning of the grout flow, which is a physical quantity to measure the difficulty of the initial grout flow. The time-varying coefficient of viscosity is a parameter that measures the change rate of the grout viscosity changing with time. The third parameter is the grouting time.  
Figure 13: The effect of porosity \(\varphi\) and permeability coefficient \(K_{w}\) on grout pressure \(P\) ((**a**) tunnel section, (**b**) rectangular coordinate system).  
The pressure model is constructed to simulate the distribution of grout pressure with Bingham fluid. Therefore, the time-varying coefficients of the grout viscosity (0.0236 min\({}^{-1}\), 0.0270 min\({}^{-1}\), and 0.0304 min\({}^{-1}\)) are adopted to conduct parameter sensitivity analysis [52]. The variation of the grout pressure \(P\) with the time-varying coefficient of the grout viscosity \(\xi\) is shown in Figure 14.  
The increase in the time-varying coefficient of the grout viscosity leads to an increase in grout viscosity. The increase in the grout viscosity causes the reduction of the diffusion distance, which reduces the pressure loss during the grout flow process. The positive correlation between grout pressure dissipation value and diffusion distance has been proved in tunnel engineering [33]. As \(\xi\) increases from 0.0107 min\({}^{-1}\) to 0.0304 min\({}^{-1}\), the grout pressure at the top and bottom of the tunnel increases from 167.9 kPa and 320.8 kPa to 191.1 kPa and 352.9 kPa, respectively. The grout pressure increases by 13.8% and 10.0%. The grout diffusion range can be expanded by reducing the grout viscosity [35].  
When the viscosity of the grout is low, the grout can flow into the soil quickly. The retention rate in the stratum is low, which results in the loss of a large amount of grout. In addition, the uneven distribution of the grout has influence on the grouting reinforcement effect. The stratum stability cannot be maintained with a low grout viscosity. On the other hand, the retention rate of the grout in the soil can be improved with high-viscosity grout, but the diffusion distance and grouting reinforcement range are limited. Therefore, the grout with higher viscosity should be adopted in a larger-porosity stratum to improve the retention rate and control the diffusion range. Conducting shield tunnel engineering with improper grout viscosity may influence the grouting effect and increase the cost [53].  
In fact, during the grouting process, the grout adheres to the surface of the soil skeleton or fills the pores between soil particles by permeation diffusion. Therefore, the design of viscosity should take the soil properties into account. The grout diffusion was difficult with high-viscosity grout in the stratum with poor permeability. In the high-permeability stratum, the retention rate of the grout with low viscosity is low, which makes it difficult to maintain the stability of the stratum.  
In the process of grout design, parameters such as the density, viscosity, and yield shear stress of the grout can be obtained through laboratory tests. The distribution of grout pressure and the range of reinforcement can be simulated by the model combined with the geological parameters. The calculation results are compared with the expected reinforcement range and stability requirements. Then, the grout mix ratios are repeatedly adjusted according to the results, and the relevant parameters are retested and recalculated until the results meet the engineering requirements.  
Figure 14: The effect of the time-varying coefficients of the grout viscosity \(\xi\) on grout pressure \(P\) ((**a**) tunnel section, (**b**) rectangular coordinate system).