*2.6. Numerical Modelling*

To verify the results of the empirical analysis, a 3D numerical analysis was developed. FLAC 3D commercial software was applied to obtain the deformations and the failure states considering the excavation of the network of tunnels and the support system [9,15,35]. Figure 3 shows the geometry of *2.6. Numerical Modelling* 

the models of the lower reservoir in the form of a network of tunnels that have been selected to conduct the numerical analysis. The size of the transversal tunnels model (Figure 3a) is 82.5 m long, 12.5 m wide and 69 m high, and the central tunnel–junction zone model (Figure 3b) is 174 m long, 12.5 m wide and 80 m high. It is assumed that there are roller boundaries at the bottom and along the sides and there is an unconstrained boundary at the top of model for application of uniform vertical stress, in order to simulate the primary stress field. The mesh was refined close to the contour of the excavations and gradually was coarser at positions outwards, for increasing the accuracy of the calculations. of the models of the lower reservoir in the form of a network of tunnels that have been selected to conduct the numerical analysis. The size of the transversal tunnels model (Figure 3a) is 82.5 m long, 12.5 m wide and 69 m high, and the central tunnel–junction zone model (Figure 3b) is 174 m long, 12.5 m wide and 80 m high. It is assumed that there are roller boundaries at the bottom and along the sides and there is an unconstrained boundary at the top of model for application of uniform vertical stress, in order to simulate the primary stress field. The mesh was refined close to the contour of the excavations and gradually was coarser at positions outwards, for increasing the accuracy of the calculations.

*Appl. Sci.* **2020**, *10*, x FOR PEER REVIEW 5 of 13

excavation of the network of tunnels and the support system [9,15,35]. Figure 3 shows the geometry

To verify the results of the empirical analysis, a 3D numerical analysis was developed. FLAC 3D

**Figure 3.** Geometry of the models of the network of tunnels: (**a**) transversal tunnels; (**b**) central tunneljunction zone. **Figure 3.** Geometry of the models of the network of tunnels: (**a**) transversal tunnels; (**b**) central tunnel-junction zone.

### *2.7. Simulation Procedure 2.7. Simulation Procedure*

For the models of the network of tunnels, the solution steps included: (1) establishment of the initial stress field by applying vertical and horizontal stresses and once equilibrium was reached deformations are reset, (2) excavation of the central and transversal tunnels, and (3) installation of the support system immediately after excavation. Failure states and total displacements caused by excavations and installation of the support system were analyzed. The rock mass is considered as homogenous and linearly elastic-perfectly plastic, according to the Mohr-Coulomb (M-C) field criterion. Plastic failure occurs when the shear stress on a certain plane reaches a limit called the shear yield stress. The M-C failure criterion is expressed in Equation (5). For the models of the network of tunnels, the solution steps included: (1) establishment of the initial stress field by applying vertical and horizontal stresses and once equilibrium was reached deformations are reset, (2) excavation of the central and transversal tunnels, and (3) installation of the support system immediately after excavation. Failure states and total displacements caused by excavations and installation of the support system were analyzed. The rock mass is considered as homogenous and linearly elastic-perfectly plastic, according to the Mohr-Coulomb (M-C) field criterion. Plastic failure occurs when the shear stress on a certain plane reaches a limit called the shear yield stress. The M-C failure criterion is expressed in Equation (5).

$$\frac{1}{2}(\sigma\_1 - \sigma\_3) = c \cos \phi - \frac{1}{2}(\sigma\_1 + \sigma\_3) \sin \phi \tag{5}$$

where *σ*1 and *σ*3 are the maximum and minimum principal stresses, respectively, in MPa, *c* is the cohesion, in MPa, and *ϕ* is the friction angle, in degrees. The initial primary stress field at the tunnels' depth of 450 m is 10.35 × 106 Pa in the vertical direction and 5.17 × 106 Pa in the horizontal direction, considering a density of 23 KN m−3. The in-situ horizontal stresses were obtained from the empirical equations from global data and a compilation of such data within the ACCB. where σ<sup>1</sup> and σ<sup>3</sup> are the maximum and minimum principal stresses, respectively, in MPa, *c* is the cohesion, in MPa, and φ is the friction angle, in degrees. The initial primary stress field at the tunnels' depth of 450 m is 10.35 <sup>×</sup> <sup>10</sup><sup>6</sup> Pa in the vertical direction and 5.17 <sup>×</sup> <sup>10</sup><sup>6</sup> Pa in the horizontal direction, considering a density of 23 KN m−<sup>3</sup> . The in-situ horizontal stresses were obtained from the empirical equations from global data and a compilation of such data within the ACCB.
