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Time-Dependent Behavior of the Tunnels in Squeezing Ground: An Experimental Study

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The presence of squeezing ground conditions often poses significant challenges in predicting tunnel response over time and to the design of an adequate support system to stabilize the tunnel. Many analytical, empirical, observational, and numerical models have been proposed over the years for the design of tunnels in squeezing ground conditions, but they all have some limitations. The study presented in this paper focused on improving the understanding of tunnel squeezing via a unique physical model test that simulated tunnel boring machine (TBM) excavation in squeezing clay-rich rocks. The physical model included a large true-triaxial cell, a miniature TBM, a laboratory-prepared synthetic test specimen having properties similar to natural mudstone, and instrumentations to monitor deformations around the tunnel boundary during and after the excavation. Experiments were conducted at realistic in-situ stress levels to study time-dependent tunnel convergence in three-dimensions. A tunnel was excavated using the miniature TBM in a cubical rock specimen loaded in the true-triaxial cell; then the confining stress was increased in stages to values above the rock unconfined compressive strength. Strain gauges embedded in the rock specimen and a digital borehole caliper monitored tunnel wall deformations with time. The degree of tunnel squeezing was characterized using a classification system based on tunnel radial strain. A model for time-dependent tunnel longitudinal displacement profile (LDP) was proposed using measurements of the tunnel convergence at different times and different stress levels.

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The authors gratefully acknowledge the financial support of the University Transportation Center for Underground Transportation Infrastructure (UTC-UTI) at the Colorado School of Mines under Grant No. 69A3551747118 from the U.S. Department of Transportation (DOT). The opinions expressed in this paper are those of the authors and not of the DOT.

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Appendix A: Validation of the Methodology to Embed Strain Gauge

The methodology is discussed in detail by Arora et al. (2020). The strain gauge, glued on a 1 mm thick 10 × 10 × mm2 Teflon sheet, was embedded longitudinally in the cylindrical synthetic mudstone specimen, as shown in Fig. 

Fig. 14
figure 14

Validation of the embedded strain gauge technique by embedding strain gauge in cylindrical synthetic specimen and comparing the measurements with LVDT in uniaxial compression test

14. The specimen was loaded uniaxially and the strain measurement from the embedded strain gauges were compared with the average strain obtained from the linear variable differential transformer (LVDT) monitoring the axial deformation of the specimen. It can be seen from Fig. 14 that the strains from the embedded strain gauges were very close to the strains recorded by LVDT with the difference limited 15%. Hence, this technique was shown to be entirely instrumental in determining the strain inside the synthetic mudstone specimen.

Appendix B: Creep Tests and Qualitative Comparison with Model Tests

Arora (2020a) developed a synthetic mudstone for laboratory scale physical model and determined the physical, mechanical and creep properties by carrying out uniaxial compression test (UCT), triaxial tests and creep tests. The elastic properties and the Mohr–Coulomb parameters are already discussed in Table 2. Arora (2020a) also carried out unconfined compression creep tests on the cylindrical specimen of the synthetic mudstone. The tests were conducted at stress levels of 0.40σc, 0.50σc and 0.75σc as shown in Fig. 

Fig. 15
figure 15

a Creep test setup with a loaded cylindrical mudstone specimen and Sketch of the Burger’s model considering viscoelastic behavior, and Creep strain with time along with best linear fit for steady-state creep (b) 0.40σc, (c) 0.50σc, and (c) 0.75σc

15. The creep test setup used by Arora et al. (2020a) cannot be used to apply stress higher than 0.75σc.

The creep behavior of synthetic mudstone specimen is described using the Burger’s viscoelastic model as given in Fig. 15a and defined by the following equation:

$${\varepsilon }_{1}=\frac{{\sigma }_{1}}{{E}^{M}}+\frac{{\sigma }_{1}}{{\eta }^{M}}t+\frac{{\sigma }_{1}}{{E}^{K}}\left[1-exp\left(-\frac{{E}^{K}}{{\eta }^{K}}t\right)\right]$$

where EM, EK, ηM, and ηK are the Burger’s model parameter, and σ1 and ε1 are the stress and strain in the axial direction of the cylindrical specimen in the uniaxial compression creep test. Figure 15b through C shows the increase in ε1 with time for the three creep tests at stress levels of 0.40σc, 0.50σc, and 0.75σc, respectively. Burger’s model parameters obtained from the three tests and average value are provided in Table 

Table 5 Burger’s model parameters for the creep behavior of synthetic mudstone


The differences and similarities between the physical model test observations and the creep test results can be summarized as follows:

  1. 1.

    In the case of creep tests, the maximum stress was 0.75σc, and hence the strains developed predominantly due to viscous and elastic behavior of the material. On the other hand, in the physical model test, the maximum stress level was 2.0σc and the strains developed due to viscous, elastic, and plastic behavior of the material.

  2. 2.

    In the physical model test, considering all lading stages and tunnel cross-sections, the tunnel strain u/R at the end of the loading stage was 1.5–2.4 times the u/R value at the beginning of the loading stage. Similarly, for the three creep tests, ε1 at the end of the test was 2.0–2.9 times that of the beginning of the test.

  3. 3.

    The trend of increase in the strain with time, for both the physical model tests and the creep tests, can be defined by the exponential functions (Eqs. (1) and (8)).

  4. 4.

    In both physical model test and creep tests, the strain value reached the steady state after a certain time (see Figs. 9 through 12 and 15).

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Arora, K., Gutierrez, M., Hedayat, A. et al. Time-Dependent Behavior of the Tunnels in Squeezing Ground: An Experimental Study. Rock Mech Rock Eng 54, 1755–1777 (2021).

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