# 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  
The analysis focuses on a horizontal circular tunnel excavated in a porous saturated rock. The induced state of stress at the wall of the tunnel excavated along a principal direction (Figure 3) is calculated with the Kirsch solution when \(\sigma^{\prime}{}_{\theta}>\sigma^{\prime}{}_{r}\):  
\[\sigma^{\prime}{}_{\theta}=S-P_{o}-u\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot \sigma^{\prime}{}_{r}=P_{o}-u, \tag{5}\]  
where: \(\sigma^{\prime}{}_{\theta}\) is the effective tangential stress, \(\sigma^{\prime}{}_{r}\) is the effective radial stress, \(P_{o}\) is the support pressure and corresponds to \(\sigma_{r}=\sigma_{3}\), \(u\) is the pore water pressure and \(S\) is the induced state of stress (tangential stress) at the wall:  
\[S=\sigma_{v}[(1+K)+2(1-K)\cos 2\theta] \tag{6}\]  
where \(K<1\) defines the anisotropy of the far field stresses (Figure 3). In many porous rocks, the permeability is generally low, and immediately after the excavation (t = 0+), the condition near to the wall can result undrained. According to Skempton, in an isotropic linear elastic soil, in axial symmetry conditions, the excess pore pressure \(\Delta u\) is equal to the variation of the total mean stress \(p=1/3\) (\(\sigma_{1}+\sigma_{2}+\sigma_{3}\)):  
\[\Delta u=B\Delta p \tag{7}\]  
where \(B\) is the [PERSON]'s pore pressure parameter, which is a function of the bulk moduli of the solid skeleton, solid grains, and water. When the far field stresses are unequal, i.e., anisotropic far field stresses (\(K\
eq 1\)), the excavation induces volumetric strains [43; 44; 45], which are positive (compaction) and negative (dilatancy). Correspondingly, the undrained condition results in a change of pore pressure, which can be calculated with Equation (7). By using the complete Kirsch solution and imposing plane strain conditions, the mean stress variation \(\Delta p\) and the excess pore pressure \(\Delta u\) in a horizontal circular tunnel are:  
\[ \begin{array}{l}p_{final}=\frac{1}{3}(\sigma_{r}+\sigma_{\theta}+\sigma_{axis} )\\ p_{initial}=\frac{1}{3}(\sigma_{v}+2K\sigma_{v})\\ \Delta u=B\Delta p=\frac{2B(1+\
u_{u})}{3}(\sigma_{v}-K\sigma_{v})\frac{R^{2} }{r^{2}}\cos 2\theta\end{array}  \tag{8}\]  
where: \(\sigma_{axis}\) is the induced stress parallel to the hole axis (plane strain condition), and \(\
u_{u}\) is the undrained Poisson ratio. Equation (8) shows that the excess pore pressure is positive \(\Delta u>0\) in the direction of the minimum far field stress (where the tangential stress is \(\sigma_{max}\)) and is negative \(\Delta u<0\) in the direction of the maximum far field stress (where the tangential stress is \(\sigma_{theta}\)). The excess pore pressure at \(\theta=45^{\circ}\) is \(\Delta u=0\). The amount of excess pore pressure depends on the far field stress magnitude and anisotropy (\(K\)) and on the elastic parameters (\(B\) and \(\
u_{u}\)). In practical applications, drained  
Figure 3: Far field stresses (\(\sigma_{v},\sigma_{h}\)), radial distance (\(r\)), tunnel radius (\(R\)), azimuth (\(\theta\)), and support pressure (\(P_{o}\)) in a horizontal circular tunnel. and undrained conditions depend on the relative magnitude of permeability, stiffness, and rate of loading. The undrained condition in soils is considered isochoric, while in rocks a change in volume can occur. The undrained condition in rocks is defined as a \"no change of the pore fluid mass\" at the time t = 0+. In this general case, the excess pore pressure can be \(\Delta u\) = 0 (at any azimuth \(\vartheta\)) or \(\Delta u\
eq 0\) (with variation at different azimuth \(\vartheta\) in anisotropic far field state of stress). Practically, the pore pressure can increase, decrease, or remain constant immediately after the excavation, but drainage does not occur over the time scales of loading for a given time. This fact indicates that \(u\
eq 0\) at the excavation boundary in the short term.