# Research on Control Technology of Large-Section Water-Bearing Broken Surrounding Rock Roadway
## 2 Engineering Summary and Analysis of Surrounding Rock Failure Characteristics
### Theoretical Model Establishment and Analysis of Roadway
The rock mass obeys the Mohr-Coulomb criterion, and the two-phase medium satisfies the Mohr-Coulomb equation:  
\[\sigma_{\theta}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{r}+\frac{2\sin \varphi}{1-\sin\varphi}(C\cot\varphi-\eta p_{w}) \tag{7}\]  
The simultaneous Equations (1), (6), and (7) are formulated as follows:  
\[\frac{rd\sigma_{r}}{dr}=\frac{2\sigma_{r}\sin\varphi+2C\cdot\cos\varphi}{1- \sin\varphi}-\frac{\eta p_{w}}{\ln(R_{s}/R_{0})}+\frac{2\eta p_{s}\cdot\sin \varphi}{(1-\sin\varphi)\ln(R_{s}/R_{0})}\ln(\frac{R_{0}}{R_{s}}) \tag{8}\]After the roadway is supported, the boundary of the roadway is subjected to uniform pressure of the rock mass. At this time, \(r=R_{0}\) and \(\sigma_{r}=p_{i}\) are taken as the definite solutions to solve the differential Equation (8). \[\sigma_{r}=C\cdot\cot\varphi+\frac{1-\sin\varphi}{2\sin\varphi}\cdot\alpha\left[ \left(\frac{r}{R_{0}}\right)^{\beta}-1\right]+\left[p_{i}+(1-\eta)p_{w}\right] \left(\frac{r}{R_{0}}\right)^{\beta} \tag{9}\]  
In this equation, \(\alpha=\frac{\eta p_{w}}{\ln(R_{\
u}/R_{0})}\) and \(\beta=\frac{2\sin\varphi}{1-\sin\varphi}\). Substituting Equation (9) into Equation (7), we can obtain the following:  
\[\sigma_{\theta}=\left(C\cdot\cot\varphi+\frac{1-\sin\varphi}{2\sin\varphi} \cdot\alpha\right)\left[\frac{1+\sin\varphi}{1-\sin\varphi}\cdot\left(\frac{r }{R_{0}}\right)^{\beta}-1\right]+\left[p_{i}+(1-\eta)p_{w}\right]\left(\frac{ r}{R_{0}}\right)^{\beta}-\beta \tag{10}\]  
The radial stress at \(r=R_{p}\) is \(\sigma_{p}\). According to [PERSON]'s theorem and Equation (1), the stress expression in the elastic zone can be obtained as follows:  
\[ \begin{split}\sigma_{r}&=p_{0}\left(1-\frac{R_{p} ^{2}}{r^{2}}\right)+\sigma_{p}\frac{R_{p}^{2}}{r^{2}}+\eta p_{w}\\ \sigma_{\theta}&=p_{0}\left(1+\frac{R_{p}^{2}}{r^{2} }\right)-\sigma_{p}\frac{R_{p}^{2}}{r^{2}}+\eta p_{w}\end{split}  \tag{11}\]  
After the excavation of the roadway, the surrounding rock of the roadway produces a broken zone and an elastic zone, as shown in Figure 3. With the gradual increase in r, the radial stress also increases. According to the strength characteristics of rock mass under three-dimensional stress, the strength of the rock mass will increase with the increase in confining pressure such that the stress in the rock mass gradually transitions to an elastic stress state. Therefore, the stress at a certain point in the surrounding rock will be the junction point of the stress in the broken zone and the elastic zone; that is, the stress of the surrounding rock at \(r=R_{p}\) satisfies the stress equation of the broken zone and the stress equation of the elastic zone. From the simultaneous Equations (9)-(11), the radius of the broken zone \(R_{p}\) can be obtained. \[R_{p}=R_{0}\left[\frac{(1-\sin\varphi)(p_{0}+C\cdot\cot\varphi-\alpha/\sin \varphi)}{p_{i}+C\cdot\cot\varphi-2\alpha/\beta}\right]^{\frac{1}{p}} \tag{12}\]  
According to the Equation (12), the radius of the fracture zone is not only related to the strength and properties of the rock mass but is also affected by the initial ground stress \(p_{0}\), the radius of the roadway \(R_{0}\), the support force \(p_{i}\), and the pore water pressure \(p_{w}\). By substituting the Equation (12) into the Equations (9) and (10), respectively, the surrounding rock stress at \(r=R_{p}\) is obtained as follows:  
\[ \begin{split}\sigma_{r}&=\left[p_{0}+C\cdot\cot \varphi+(1-2\eta)p_{w}\right](1-\sin\varphi)-C\cdot\cos\varphi\\ \sigma_{\theta}&=\left[p_{0}+C\cdot\cot\varphi+(1-2 \eta)p_{w}\right](1+\sin\varphi)+C\cdot\cos\varphi\end{split}  \tag{13}\]  
The displacement \(u_{p}\) of the fracture zone is as follows:  
\[u_{p}=\frac{(1+\mu)R_{p}^{2}}{E\cdot r}\left\{\left[p_{0}+(1-2\eta)p_{w} \right]\sin\varphi+2C\cdot\cos\varphi\right\} \tag{14}\]  
In the equation,  
C--Cohesion;  
E--Elastic modulus;  
\(\varphi\)--Angle of internal friction;\(\mu\)--Poisson's ratio.