# Failure in the Tension Zone around a Circular Tunnel Excavated in Saturated Porous Rock
## 3 Identification of the Limit Pore Pressure around a Circular Tunnel in a Saturated Brittle Rock
In the elastoplastic case, the magnitude of the excess pore pressure also depends on the strength parameters of the rock and on the stress redistribution related to the spread of plasticity. In moderately weak brittle rocks, the excess pore pressure can be \(\Delta u\) = 0 or \(\Delta u\
eq 0\), but the total pore pressure is always positive and not negligible, at least at some azimuths. In the following analysis, the total pore pressure, immediately after the excavation, at the boundary of the tunnel is assumed \(u>0\) (with \(\Delta u\) = 0 or \(\Delta u\
eq 0\)). By coupling the Kirsch solution with the H&B criterion (Equation (1)), the limit support pressure \(P_{o}=\sigma_{r}\), when \(\sigma_{\vartheta}>\sigma_{axis}>\sigma_{r}\) at the wall of the tunnel is:  
\[P_{o}=\frac{4S+m\sigma_{c}-\left[\sigma_{c}^{2}\left(m^{2}+16\right)+8m\sigma _{c}S-16m\sigma_{c}u\right]^{0.5}}{8}, \tag{9}\]  
The pore pressure \(u\) in Equation (9) is the total pore pressure, which can also take into account the excess pore pressure (\(\Delta u\
eq 0\)). The parameter \(m\) can be in the tension zone or in the compression zone in dependence of the type of stress condition of the analysis. Equation (9) indicates whether the rock is self-supporting (\(P_{o}\leq 0\)) or needs a support pressure (\(P_{o}>0\)). The purpose of this study is the identification of a limit pore pressure \(u_{lim}\) that induces failure in the tension zone in a given real case, characterized by rock strength properties and far field stresses. Consequently, the support pressure must be set \(P_{o}=0\). The comparison of this limit pore pressure \(u_{lim}\) with the in-site pore pressure indicates if failure is expected to occur in a given real case. If the in-site pore pressure is higher than the limit pore pressure, countermeasures are necessary, such as drainage or pre-reinforcements. By imposing \(P_{o}\) = 0 in Equation (9) and introducing \(X\) = \(S/\sigma_{c}\), the limit condition of the pore pressure \(u_{lim}\) is:  
\[u_{\rm lim}=\frac{1}{m\sigma_{c}}\left(\sigma_{c}^{2}-S^{2}\right)=\frac{ \sigma_{c}}{m}\left(1-X^{2}\right), \tag{10}\]  
The sign of the term in the right side of Equation (10) depends on the parameter \(X\), which expresses the relative magnitude of \(S\) and \(\sigma_{c}\). The equation indicates that \(u_{lim}\) increases with increasing \(\sigma_{c}\). An overestimate of \(u_{lim}\) occurs if \(m\) is fitted in the compression zone at high confinement (\(m=m_{c}\)) because the limit pore pressure decreases with increasing \(m\). Equation (10) indicates that the limit pore pressure is \(u_{lim}\) = 0 when \(X\) = \(S/\sigma_{c}=1\). In this case, failure can already occur in dry conditions, because \(P_{o}\) = 0. Consequently, a support pressure \(P_{o}\) > 0 is always required when \(X\geq 1\). If Equation (4) holds (\(m_{t}\) fitted in the tension zone), the expression of \(u_{lim}\) becomes:  
\[u_{\rm lim}=\frac{\sigma_{c}}{m_{t}}\left(1-X^{2}\right)=\sigma_{t}\left(1-X^ {2}\right), \tag{11}\]  
when \(S\) is very small compared to \(\sigma_{c}\) (tunnel at shallow depth/strong rock), the term \(X\) becomes negligible, and the limit pore pressure in Equation (11) can be approximated by:  
\[u_{\rm lim}\approx\sigma_{t}, \tag{12}\]  
Equation (12) represents the occurrence of \"pure\" tensile failure with \(\sigma^{\prime}{}_{1}\approx 0\). Equations (10)-(12) show the importance of the choice of design input parameters for the analysis of tunnel stability: far field stresses and strength parameters of the rock. On the other hand, the strength and deformability parameters of the rock also affect the generation of undrained conditions and excess pore pressure \(\Delta u\). Furthermore, a reliable estimation of the in-site pore pressure is necessary, for the calculation of \(\Delta u\) and the subsequent comparison of the in-site total pore pressure \(u\) with \(u_{lim}\). This study aims at investigating the role of each parameter of Equation (10) in the variation of the limit pore pressure.