# Numerical Investigation of Slurry Fracturing during Shield Tunneling under a Reservoir
## 3 Verification of Numerical Solution
### Analysis of Calculation Results and Comparative Validation
The fracturing pressure values of numerical calculation results and test results are provided in Table 2. It can be observed that the error of the fracture initiation pressure obtained by simulation under different confining pressures is basically within 5%, compared with the test results, which verifies the accuracy of the numerical simulation results. The simulation results of the failure shape of the specimens under different confining pressures are provided in Figure 10. As can be observed from the figure, the fractures of the specimen tend to be vertical when the confining pressure is less than the axial pressure; the fractures of the specimen tend to be horizontal when the confining pressure is greater than the axial pressure. The slurry fractures are deflected in the direction of the maximum principal stress. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Condition** & **Axial Pressure** & **Confining Pressure** & **Pressure** & **Pressure Test** & **Error** \\
**Number** & **(MPa)** & **(MPa)** & **Value (MPa)** & **Value (MPa)** & **(MPa)** & \\ \hline
1 & 0.35 & 0.2 & 0.36 & 0.382 & \(-\)5.76\% \\
2 & 0.35 & 0.3 & 0.45 & 0.457 & \(-\)1.53\% \\
3 & 0.35 & 0.4 & 0.58 & 0.557 & 4.13\% \\
4 & 0.35 & 0.5 & 0.63 & 0.652 & \(-\)3.37\% \\
5 & 0.35 & 0.6 & 0.75 & 0.735 & 2.04\% \\
6 & 0.35 & 0.7 & 0.83 & 0.841 & \(-\)1.31\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The setting of working conditions and fracture initiation pressure simulation results. Figure 10: Fracturing extension path simulation results of the samples under different confining pressures. (**a**) 0.2 MPa; (**b**) 0.3 MPa; (**c**) 0.4 MPa; (**d**) 0.5 MPa; (**e**) 0.6 MPa; (**f**) 0.7 MPa. The research conclusions of [PERSON] et al. suggest that there is a linear relationship between the unconfined compressive strength of cohesive soil and the slurry fracturing pressure [13; 14]. Based on laboratory tests of cylindrical samples with through boreholes, they proposed an empirical formula for estimating the fracturing pressure:  
\[P_{f}=\sigma_{3}+\alpha_{v}q_{u} \tag{12}\]  
where \(\sigma_{3}\) is the confining stress, \(\alpha_{v}\) is the coefficient related to the viscosity of the pressurized liquid, and \(q_{u}\) is the unconfined compressive strength. The influence of shield slurry viscosity on fracturing is more complicated, so \(\alpha_{v}\) needs to be valued according to experience, which limits the applicability of the formula. [PERSON] et al. tested the data of different unconfined compressive strengths and slurry fracturing pressures through the slurry fracturing test of blind-hole samples [17]. The same parameters were used to model and analyze the blind-hole samples with different unconfined compressive strengths, and the slurry fracturing pressure was obtained, which was compared with the test data, as illustrated in Figure 11. The calculation results show that the simulated value of slurry fracturing pressure is slightly larger than the test value. This is due to slight errors between the measured and true values of the clay material parameters. Usually, the determination of clay parameters in the laboratory is accurate. The preparation of blind-pored clay samples is relatively difficult, and it is easy to cause tiny defects and affect the strength of the material. Therefore, when the simulation calculation is accurate, the simulated value is often larger than the experimental value. The fitted relationship between unconfined compressive strength and slurry fracturing pressure demonstrates a linear positive correlation trend, illustrating that the XFEM can obtain consistent laws with the sample tests; the results are consistent with the point of view put forward by [PERSON]. The positive correlation between unconfined compressive strength and splitting pressure also reflects the coexistence of tensile and shear failure. The properties of soil have a great influence on the slurry fracturing pressure. Slurry viscosity is also a key factor affecting the slurry fracturing pressure of cohesive soils.