**4.3 Predicting the long-term behavior of rock masses in tunneling**

Although there are time-dependent models available to predict rock materials' rheological behavior potential, it is commonly observed that in two-dimension (2D) modeling, time-dependent behavior is not directly simulated using selected 2D coded software with noted limitations [1]. It is, therefore, necessary a method to be developed and 'pseudo' simulate this type of behavior. For instance, the plastic zone can be used as an indicator of the overall time-dependent displacements and calibrated to in-situ measurements or laboratory testing [1].

[25] proposed a new methodology for predicting rock masses' long-term behavior using the information derived when testing rock materials under constant loading, which results in strength degradation by 'pseudo-simulating' numerically this behavior. [25] examined two main sets of numerical models in plane strain conditions in-plane RS2 (Rocscience). The models' main difference was that the material included in the plastic zone changed parameters with time-steps in the one set of models. The first aimed at pseudo-simulating time-dependent behavior by using the Long-Term Strength (LST) according to strength-degradation Eq. (6) of the limestone bases on the laboratory data previously shown in **Figure 18**.

$$\text{(\(\sigma/UCS) = - 0.022\ln(t) + 0.95)}\tag{6}$$

In addition, 19 stages were simulated, as shown in **Figure 25a**. Where in each stage, a new σci (strength of intact rock) was assigned only to the material of the plastic zone, according to Eq. (6). Each strength reduction represented a specific time from 1 second to 1000 years. The second set of analyses were based on Young's modulus (Ei) reduction from the initial 40 GPa to 12 MPa of 10% in every modeling stage (**Figure 25b**). It should be highlighted that the decrease in both strength and Young's Modulus reduction was applied to the plastic zone, assuming that the rest model behaves as an elastic material.

Every increment on the strength-degradation models was related to a time according to the lab results and time to failure graph shown in **Figure 18**, such as the YMR models can be associated with a specific time. For instance, a reduction of the intact strength of 21% (0.79 σci) can simulate the deformation acquired in 1 day and reflects the deformation of the 30% reduction of Young's Modulus (**Figure 26**). Moreover, to simulate the rock mass's deformation around the tunnel after a 2-year period, one can either reduce the intact strength to 0.71 σci or reduce the Young's Modulus to 50% (**Figure 26**).

**Figure 26.**

*Numerical resluts of total displacements of LST models (left colummn) and YMR (right colummn).*

Relating the strength-degradation (or the LTS) with the YMR to time for specific lithologies can produce a database that one can use to capture the time effect on the rock mass behavior, as shown in **Figure 27a**. The yellow triangles reflect the YMR with the time, whereas the green circles the limestone's overall behavior based on the laboratory data (blue and light blue diamonds and squares). When the YMR method is used, the reduction factor can be estimated using Eq. (7), where t is time, and E/Ei is Young's Modulus-ratio.

$$\text{YMS of limestone} : \text{(E / Ei)} = -0.028 \ln(\text{t}) + 1.014 \tag{7}$$

Using this YMR approach proposed by [25] for the granite (**Figure 27b**) the estimated reduction of the Young's Modulus is given in Eq. (8).

$$\text{YMS of graphite}: \text{(E/}E\text{)} = -0.03\ln\left(\text{t}\right) + \text{1.018} \tag{8}$$

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

#### **Figure 27.**

*Driving stress-ratio and Young's modulus-ratio in relation to time to failure from static load test data performed at room temperature in dry conditions, (where the driving stress-ratio is the stress level at failure to unconfined compressive strength of the material), for a. limestone and b. granite.*

It should be stated that the analyses presented herein can be used for values of at least 0.5σci and higher as the below this threshold (CI), no failure is anticipated, below this threshold, the observed behavior is considered to be linear elastic.

### **5. Conclusions**

This research work and the resultant publications presented in this Chapter have contributed to a better understanding of the "Time-dependent behaviour of rock materials". This section summarizes the key findings of this study.

It is widely accepted that significant research contributes to studying the timedependent behavior of geo-materials and their effects in geoengineering applications by developing models. However, these models mainly focus on simulating the visco-elastic creep behavior and are developed based on the back-analysis of existing datasets. In many cases, these models can replicate sufficiently creep behavior during the primary and secondary stage when appropriate parameters are derived and used. Furthermore, the applicability of such models is commonly broader when dealing with weak rock masses. As a result, there is a knowledge gap when dealing with time-dependent behavior in brittle rock materials. Its effect is considered limited and usually neglected and covered by other progressive damage mechanisms. Nevertheless, this study has shown the importance of taking into consideration brittle time-dependent behavior, and it is recommended that engineers, scientists and practitioners utilize the existing models to simulate time-dependent behavior with appropriate parameters as with a few modifications, these models can capture the behavioral trend as long as the appropriate parameters are utilized.

In this study relaxation, the decrease of applied load at constant deformation was investigated and re-defined. It was shown that relaxation could be considered as an inversion of a creep behavior. It was concluded herein that axial straincontrolled tests are less sensitive to testing challenges during a relaxation test than a creep test. Single-step and multi-step tests have been performed in this study. It was shown that single-step is easier to perform, but there is sample consuming in order to obtain a complete dataset to cover the total stress spectrum. From the results, it was also shown that relaxation takes places when cracks initiate and propagate during the sample. It attains a constant value (asymptote) when axial crack stabilization is reached. The most important outcome of this work was identifying the existence of three distinct stages that occur during time-dependent stress relaxation. These three stages (RI, RII, and RIII) were introduced and clearly defined. The first two stages are similar to the first two stages in creep behavior. In contrast, the third stage differs as the sample reaches a stable condition compared to tertiary creep where it reaches failure.

Another set of tests, static load, both single-step and multi-step, are presented here. This time the axial load (stress) was kept constant, focusing on timedependent behavior over time. Once again, it is shown that multi-step tests might be advantageous in terms of deriving visco-elastic parameters in different target stress levels using only one specimen; however, when considering time-to-failure single-step tests are preferred. In this section, two types of limestone (Jurassic and Cobourg) were investigated, and the time-to-failure behavior was compared to other rock types from data published in the literature. It was concluded that an overall trend does exist. This general trend was scrutinized at a second stage based on rock types providing specific trends for sedimentary, metamorphic, igneous, which can be used to predict time-to-failure for laboratory samples:


where σ/UCS is the driving stress-ration and t refers to time.

The limestone dataset was compared to the widely used dataset of Lac du Bonnet granite, showing that the limestone's long-term strength is higher than the granite. The latter means that the limestone can withstand longer time-depended behavior than the granite. The latter can be explained by the fact that these two rock types

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

differ in their mineralogical structures. The granite's increased heretegoeneity contributes to different creep rates of the various grains (quartz, feldspars, mica). The latter generates incompatible strains over time, causing micro-cracking. Similar processes do occur within the limestone, but due to its homogeneity, creep is constrained in calcite (monomineralic) and is associated with less damage increasing the time up to failure. It was also observed that both limestones that failed at a stress threshold above 0.8 UCS failed within the first 60 minutes. On the contrary, below CI threshold, no failure was observed, and between 0.5 to 0.8 UCS, failure will take place at some point between the first hours to months, depending on the rock type. Another outcome of this work was the identification of Maxwell's viscosity threshold as an indicator of failure. This observation can explain why some specimens fail and some others did not (yet).

Time-dependent behavior during tunneling can play an important role in the project success in the design and, most notably, in the construction process. This fourth dimension (time-effect) in tunneling was investigated numerically by performing an axisymmetric parametric analysis. From the research was concluded that current conventional methods adopted to predict the Longitudinal Displacement Profile of tunnel displacements have limited applications and fail to capture the overall displacement over time. It was also presented that both the excavation methods and excavation rate (tunnel advancement rate) can affect (deteriorate) the mechanical behavior of the surrounding rock mass. In this work, only creep behavior was considered a contributor to time-dependent deformation and was simulated with the modified purely visco-elastic CVISC model assuming the rock mass as a visco-elastic medium. It was further concluded that the retardation time (in the Kelvin-Voigt model) does control the timing at which the maximum tunnel displacement is reached during the primary stage of creep.

Finally, a new but yet simple tool that can be used to predict the long-term behavior of brittle materials as limestone using either the Long-Term Strength (LTS) approach (strength-degradation) and the Young's Modulus-Ratio (YMR) was presented. It should be stated that the proposed methodology should be used as a first estimate to relate the strength-deterioration of the rock material over time. Furthermore, input parameters can be derived in the plastic zone around an underground opening using this approach that can then be used in numerical analyses similar to the one presented herein.
