**4.2 Capturing the time-effect in tunneling using a new numerical approach**

An axisymmetric parametric analysis was performed within FLAC software. The geometry of the model and the excavation sequence characteristics are shown in **Figure 22**. A circular tunnel of 6 m diameter and 400 m length was excavated in isotropic conditions. Full-face excavation was adopted. Two cases of a drill and blast (Case 1: D&B) and a TBM (Case 2: TBM) were assumed depending on the excavation step 1 m and 3 m, respectively. The rock mass was set to behave as an elasto-visco-elastic material using the CVISC model. The analysis aimed to examine the contribution of primary and secondary creep. In this regard, the Maxwell body's viscous dashpots within the CVISC model were deactivated and reactivated,

#### **Figure 22.**

*(Left) Schematic illustration of the excavation sequence used within the numerical axisymmetric analysis; case 1 refers to drill and blast method with 3 m excavation step per cycle; case 2 refers to TBM (Tunneling boring machine) method with 1 m excavation per cycle. (Right) Parameters used for CVISC model.*

accordingly. Besides, two different sets of visco-elastic (creep) parameters were used then for both cases shown in **Figure 22**. Furthermore, the excavation cycle duration was also simulated to consider both the time-dependent component and the tunnel advance representing the real conditions in a tunnel problem and varied from 2 to 8 hours. In addition, two supplementary analyses were performed: with the Kelvin-Voigt model and with the elastic model to validate the numerical models and compared with analytical solutions.

It should be stated that the visco-elastic parameters were chosen according to the analytical solution (Eq. (5)) of the Kelvin-Voigt model developed by [51].

$$\mu\_r = \frac{\sigma\_o r}{2G\_o} + \frac{\not{o}\_o r}{2G\_K} \left[ \mathbf{1} - \exp\left(-\frac{t}{T\_K}\right) \right] \tag{5}$$

(where: σ0 is the in-situ stress conditions, r is the tunnel radius, G0 the elastic shear modulus, GK is the Kelvin shear Modulus, ηK is Kelvin's viscosity and TK is known as retardation time and it is the ratio of Kelvin's viscosity over the Kelvin Shear Modulus and is indicator of when the model will convergence and reach a constant value.)

The selected retardation time (TK) varies one order of magnitude between the two sets as it controls the curvature of Kelvin's model behavior. The following **Figures 24** and **25** 'x' is the distance from the tunnel face, R is the tunnel radius, ur is the absolute radial tunnel wall displacement, uremax is the maximum elastic displacement and ur∞max is the maximum visco-elastic displacement of the Kelvin-Voigt model. Gray and black lines are the elastic and the zero-viscosity KV models respectively.

#### *4.2.1 Primary stage of creep, KELVIN-VOIGT (KV)*

The Kelvin-Voigt model was assumed to represent the primary stage of creep and was used to simulate an elasto-visco-elastic rock mass's mechanical behavior. The results for both cases are presented in **Figure 23**. They imply that increased cycle time or excavation delay exacerbates the rock mass's mechanical behavior; as in all models, an increase of the ultimate total displacement was observed. This increase depends on the visco-elastic parameters of the Kelvin-Voigt model. The increase of the retardation time will increase the time required by the model to reach a constant value and become time-independent.

The deviatoric stress was related to the displacement data normalized to the maximum displacement of the Kelvin-Voigt model (ur∞max). Time-dependent behavior starts for both cases when the deviatoric stress reaches a critical value (qcr) shown in **Figure 23b**. This critical value is attained after one excavation step at the point which the time-dependent LDPs deviate from the elastic LDP. In the drill and blast case, this is 3 m away from the tunnel, whereas for the TBM case it is 1 m.

#### *4.2.2 Secondary stage of creep, Burgers (B)*

The second stage of this analysis was to investigate the influence of both primary and secondary creep behavior stages using the Burgers model. The results presented in **Figure 24** show the maximum strains due to the secondary stage (Maxwell) are effectively infinite. This is also observed on **Figure 21**. In this part, it was noticed that the magnitude of the total displacements between the two cases varied significantly. The excavation method influences the accumulated displacements.

*Time-Dependent Behavior of Rock Materials DOI: http://dx.doi.org/10.5772/intechopen.96997*

#### **Figure 23.**

*(Left) Numerical results of LDPs, (right) closer representation of the data; relating the deviatoric stress (q) to the tunnel wall displacement normalized to the maximum displacement of the KELVIN-VOIGT model (ur*∞*max) for: (a) the drill and blast case (DB) and (b) the TBM case.*

In the drill and blast case, all two sets of parameters exhibited less displacement than the TBM case for the same duration of the excavation cycles. During a TBM tunnel excavation, the tunnel excavation requires more time than a drill and blast excavation for the same excavation cycle. For instance, a TBM that excavates 1 m every 6 hours, the elapsed time is three times longer than the drill and blast case of 3 m excavation per cycle. In the TBM case, the time for the excavation of the same length tunnel will result in an accumulation of displacement increase. However, this may not always represent real conditions as TBMs are commonly preferable since they tend to achieve better excavation rates; if proven affordable. Suppose the latter is the case, then a TBM excavation of a two-hour excavation cycle. In that case, it is shown that the surrounding rock mass represented by SET#1 exhibits less displacement than an eight-hour excavation cycle using drill and blast.

#### **Figure 24.**

*(Left) Numerical results of LDPs for: (a) the drill and blast (DB) and (b) the TBM case of the BURGERS (B) analysis (the hours on the legend denote hours per excavation cycle), (right) closer representation of the data.*

**Figure 25.**

*(a) Geometry and mesh conditions of the model used, rp denotes the radius of the plastic zone, and incremental reduction of (b) intact rock strength according to long-term strength and (c) Young's modulus.*
