Research Paper
A fractional viscoplastic model to predict the time-dependent displacement of deeply buried tunnels in swelling rock

https://doi.org/10.1016/j.compgeo.2020.103901Get rights and content

Abstract

This study focused on tunneling under challenging conditions, particularly with regard to the stress distribution and deformation in the humidity stress field. The swelling phenomenon during tunneling was treated as a quasi–humidity–mechanics coupled process, where the humidity diffusion and stress dilatancy were considered together to obtain the stress and deformation fields for tunnels crossing formations with high swelling potential. A solution to the nonstationary process of humidity transfer was derived according to Fick’s second law. First, the humidity distribution was obtained from the surrounding rock mass to determine the stress field. Then, a modified fractional viscoplastic model was developed to describe the whole creep process of soft rock swelling, as well as the damage and swelling deformation due to water immersion. Next, closed-form analytical solutions were derived for the displacement in the viscoplastic region, with more focus on the tunnel wall, based on a non-associated flow rule. The sensitivity of time-independent parameters was also analyzed. The solution was found to reasonably represent the tunnel convergence versus time when compared with the numerical simulation and monitoring data from the Nabetachiyama Tunnel.

Introduction

Swelling rocks exhibit volumetric expansion and changes in mechanical properties when exposed to water (Alonso et al., 2013, Bilir et al, 2013, Wang et al., 2017). This often causes serious damage to tunnels in regions with high swelling potentials, such as the Bologna–Florence railway (Boldini et al., 2004), Caneva–Stevenà Quarry (Barla, 2008), southern Ontario (Hawlader et al., 2005), and Xiaotun Coal Mine (Zhang et al., 2017). Rocks swell when clay minerals, anhydrite, or pyrite/marcasite are present (Anagnostou, 1993, Butscher et al., 2011, Papagiannakis et al., 2014). Weak rocks rich in anhydrite or clay minerals are characterized by considerable variations in volume and time-dependent deformations with changing water content. This swelling phenomenon originates from physicochemical reactions involving water and stress relief (ISRM Characterization of swelling rock, 1983). Water is a major contributor, but the rapid release of stored energy caused by tunnel excavation develops a path and expands the space for the swelling strain because of the emergence of an unloading surface.

Research on the swelling of weak rock exposed to water has been ongoing for many years. Existing rock swelling theories are based on specific swelling test models, among which the one-dimensional (Gysel, 1987, Gysel, 2002) and three-dimensional swelling theories (Wittke-Gattermann and Wittke, 2004) are the most representative. Recent studies (Berdugo de Moya et al., 2009, Parsapour and Fahimifar, 2016) have shown that considerable pressure develops when the swelling strain is prevented. However, research on swelling strain has mostly relied on numerical simulations (Anagnostou, 1993, Tang and Tang, 2012, Butscher et al., 2017) and laboratory or in situ tests (Bilir et al, 2013, Pimentel, 2013, Zhang et al., 2017) rather than analytical solutions. Hawlader (Hawlader et al., 2005) introduced swelling components to a constitutive model and considered the three-dimensional stress effect and long-term swelling with anisotropic swelling potential to derive an accurate closed-form solution for the swelling deformation of the Darlington cold-water intake tunnel. Graziani (Graziani and Boldini, 2012) and Boldini (Boldini and Graziani, 2012) studied the stress and displacement distribution caused by the excavation of deeply buried argillaceous tunnels by considering the hydraulic coupling effect and decomposed the mechanical response of the surrounding rock into the instantaneous stress and long-term stability for the analysis of the seepage state. Such previous studies provided a basic understanding of the deformation mechanism of tunnels excavation in swelling rock. However, the models above cannot consider water diffusion and seepage in the swelling rock.

The main reason for swelling deformation is attributable to suction changes in clay minerals in rocks, based on which, Anagnostou (Anagnostou, 1993) proposed the anhydrite theory and an improved computational model that treats the swelling of the rock surrounding a tunnel as a hydraulic–mechanical coupled process to simulate the observed floor heaves more realistically. Similarly, Tang and Tang (Tang and Tang, 2012) used the humidity stress field theory to model the observed floor heaves and inward movements realistically, without needing to consider the complex chemical processes induced by water–rock interaction in light of the damage and fracture theories and investigated the mechanism and deformation response. These papers provide a verification that water swelling (Anagnostou, 1993) and creep (Tang and Tang, 2012) play a significant role in tunnel deformation in swelling rock from a numerical point of view. Masoudian and Hashemi (Masoudian and Hashemi, 2016) considered the swelling/shrinkage effect and provided a novel analytical solution for a circular opening in the elastic–brittle–plastic rock by employing the non-associated flow rule, although they did not consider the creep of swelling rock. But the analytical studies within the realm of merely rheological theory with consideration of the effect of water have much difficulty in fully explaining the phenomena.

The swelling of the rock mass due to water absorption is just one part of the puzzle: the other part is dilatancy (Masoudian and Hashemi, 2016). It should be noted that the rock dilatancy is not caused by a physical or chemical reaction, but a mechanical process. The swelling rock is originally impermeable, but cracks are produced, and therefore porosity increases as a result of stress dilatancy, which makes it easy for water content to penetrate the rock. Thus, the rock particles contact with water, causing the rock to swell. The more evident the dilatancy is, the easier it is to absorb water. Moreover, the more water is absorbed, the larger the volume of rock is, and the easier the rock is to be destroyed. In short, the swelling due to water absorption and the dilatancy under the action of deviator stress promote each other. Nonlinear mechanical dilation concomitantly occurs with stress redistribution in the surrounding rock. The rock surrounding deeply buried tunnels inevitably dilates owing to the unloading of the large in situ stress (Wu et al., 2018). Detournay (Detournay, 1986) deduced an analytical solution for tunnel excavation that considers the dilatancy of the surrounding rock and showed that it significantly impacts the deformation of the tunnel wall. Nilsen (Nilsen, 2011) stated that a high clay content may be favorable in many cases, even if smectite is present because this will reduce the permeability and completely seal the joints. That is, dilatancy is an important factor to be considered in the analysis of the deformation mechanism of swelling rock excavation. More than this, the microcracks caused by dilatancy accelerate the water entering the surrounding rock and exacerbate the physicochemical reactions of the swelling soft rock, which rapidly attenuates the rock strength. Therefore, the volume dilatancy of the surrounding rock needs to be considered when calculating the mechanics of deeply buried tunnels.

In the case of tunnels surrounded by swelling rock, the time-dependent deformation should be considered, where the stress–dilation effect is combined with the quasi–humidity–mechanics coupled process. Classical models for describing viscoelastic behavior from a macroscopic perspective combine spring elements and Newton dashpots in series or in parallel to form different structures and establish corresponding dynamic equations (Bonini et al., 2009, Sharifzadeh et al., 2013, Yang et al., 2017). Bonini et al. (Bonini et al., 2009) used two constitutive models to simulate the mechanical and time-dependent behavior of clay shales (CS): a viscoelastic-plastic model (CVISC) and an elastic-viscoplastic model (VIPLA). The Nishihara model and generalized Nishihara model are widely used to describe the whole creep curve of soft rock (Barla et al., 2011, Yang et al., 2014). CVISC/ VIPLA can describe accelerated creep but use a constant acceleration for the creep rate, which is inconsistent with the real creep law of deep soft rock in coal mines and tunnels. The deep rock contains individual structural planes, and the stress, confining pressure, and humidity vary with the depth; all of these factors affect deep soft rock (Tang and Tang, 2012, Liu et al., 2018). The Nishihara model and its modified version include too many elements (six or more), which causes difficulty with identification and a lack of clarity on the physical meaning. To simplify the calculation parameters and reflect the unsteady creep rate, this paper introduces the fractional-order element. Actually, the introduction of fractional calculus into the constitutive equation has led to breakthroughs in viscoelastic theory. Classical creep models such as the Maxwell model and the Kelvin–Voigt model were improved by the use of fractional derivatives (Zhou et al., 2011, Wu et al., 2015, Xu and Jiang, 2017, Wang et al., 2019, Zhang et al., 2019). Zhou et al. (Zhou et al., 2011) proposed a new constitutive equation for the creep of rock salt based on Riemann–Liouville fractional calculus and derived an analytical solution by replacing the Newtonian dashpot in the classical Nishihara model with the Abel dashpot. Zhang et al. (Zhang et al., 2019) substituted the fractional Poynting–Thomson (FPT) model (Xu and Jiang, 2017) into the improved Maxwell model and deduced a three-dimensional creep equation for the fractional viscoplastic (FVP) model by using the von Mises equivalent plastic stress–strain relationship (Lin and Ito, 1966). However, the model cannot adapt to the swelling deformation characteristics and, therefore, it must be improved for application to swelling rock.

In this study, the swelling phenomenon during tunneling was treated as a quasi–humidity–mechanics coupled process to consider the stress redistribution, humidity diffusion, and volume dilatancy simultaneously when calculating the mechanical behavior of deeply buried tunnels in a zone with swelling potential. A novel FVP model was developed that introduces a linear swelling factor and unsteady viscoplastic element to consider the additional swelling strain and damage to the rock mass caused by water immersion and diffusion. Closed-form solutions were derived for the plastic stress field and viscoplastic deformation, and a parametric analysis was performed.

Section snippets

Humidity stress field theory

The complex engineering properties of swelling rock can induce various severe and widely distributed geological disasters, which makes it one of the most significant concerns of tunnel engineering. Swelling behavior is a humidity–mechanics coupled process that involves water and stress relief (ISRM Characterization of swelling rock, 1983). The stress relief during tunnel excavation typically causes a damaged zone in the unsealed area comprising discontinuities along sedimentation planes and

Solution for the stress field

The closed-form solutions presented herein refer to a plane-strain circular opening with a radius R0 in an infinite elastoplastic medium with the uniform and isotropic in situ total stress p0 (Fig. 1). The opening surface is subjected to an internal pressure pi. The linear Mohr-Coulomb yield criterion was assumed to govern the plastic flow. A plastic annulus forms around the tunnel wall after plastic yielding. Compressive stress is denoted by a positive value, while tensile stress is negative.

A

Abel dashpot

The classical rheological constitutive model combines springs and Newton dashpots in series or parallel to form different structures and establishes the corresponding dynamic equations in the form of integer differentials or integrals (Bonini et al., 2009, Parsapour and Fahimifar, 2016). In contrast, fractional-order calculus has a better global correlation and can better reflect the historical dependence of the developing system to overcome the disagreement between the classical model and

Viscoelastic plastic deformation solution

The stress fields given in Section 3 can be used to solve deformation fields. As shown in Fig. 1, the surrounding rock is divided into the elastic zone and viscoplastic zone, where the viscoplastic behavior of the surrounding rock complies with the 3D FVP model. In addition, the non-associated flow rule (Zhang et al., 2019, Zhao et al., 2019) is used to describe the dilation phenomenon of a rock mass:du̇rvpdr+λu̇rvpr=ε̇r(t)+λε̇θ(t)where the overdot denotes the time derivative; uvp r is the

Stress field analysis

To investigate the effect of the humidity field on the stress field, the results considering the swelling stress and without it were compared. In addition, the sensitivity of the maximum humidity change wmax was analyzed. The swelling stress coefficient κ was set to κ = Eα/(1 2v) to determine whether or not the swelling pressure/strain was considered. Eq. (15c) degenerated into the modified Fenner formula when κ = 0 (i.e., the swelling pressure/strain is not considered). This assumes that the

Comparison with the numerical simulation

Tang and Tang (Tang and Tang, 2012) proposed a numerical model to simulate the floor heave processes of a swelling rock tunnel exposed to high humidity and considered the time-dependent deformation and failure processes. Section 7.1 compares their numerical model and the model in this study for different water contents. According to the equal-area method, a circular cross-section with a radius of 13.75 m is equivalent to an inverse U-shaped cross-section with a radius of 16 m and a height of

Conclusion

This study aimed to predict the time-dependent displacement of the tunnel in the swelling surrounding rock following an excavation. For this, an elastic–plastic solution of the stress field at the circular opening was attained based on the humidity stress field theory, and a modified four-element fractional viscoplastic model was established to describe the creep behavior. Combined with the non-associated flow rule, the viscoplastic deformation of the swelling rock tunnels was captured by the

CRediT authorship contribution statement

Geng-Yun Liu: Conceptualization, Methodology, Data curation, Writing - original draft. You-Liang Chen: Conceptualization, Methodology. Xi Du: Validation, Writing - review & editing. Rafig Azzam: Validation.

Declaration of Competing Interest

The authors declared that there is no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 51608323) and Key Program in Soft Science Research in Shanghai (Grant No. 18692106100). The authors would like to thank Dr. Rafig Azzam for helpful discussions and the research team under Dr. Chen Youliang for financing this study.

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      Therefore, it is of great significance for the development of underground space engineering to systematically study the mechanism of the deformation and failure of soft rock roadways induced by humidity diffusion. Since the 1970 s, theoretical models were widely used to study the characteristics of rock swelling deformation (Bian et al., 2018; Bizjak and Petkovšek, 2004; Ji and Zhang, 2020; Miao et al., 1993; Sun, 2017) and the mechanical behavior of soft rock roadway (Deng et al., 2021; Grob, 1975; Gysel, 1987; Liu et al., 2021; Yu et al., 2020; Zhao et al., 2019). For example, Grob (1975) first proposed the approximate calculation method to predict the vertical swelling deformation of the roadway floor in 1972.

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