A fully coupled hydraulic-mechanical solution of a circular tunnel in strain-softening rock masses

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Abstract

This paper presents a fully coupled hydraulic-mechanical strain-softening model considering Biot’s effective stress. Both the hydraulic and mechanical parameters are considered as functions of the confining pressure and plastic shear strain. Using the proposed fully coupled hydraulic-mechanical model, the displacement, stress and pore-water pressure around a circular tunnel in both Mohr-Coulomb and Hoek-Brown strain-softening rock masses are derived using a non-associated flow rule. The proposed solutions are validated by the conventional hydraulic-mechanical elasto-brittle-plastic and strain softening rock mass using analytical and numerical methods, and the examples of the fully coupled hydraulic-mechanical strain-softening rock mass are investigated. The results show that the seepage force enlarges the displacement, residual and plastic radii. With the increase in the initial pore-water pressure, the displacement, water inflow, residual and plastic radii increase in an approximately linear manner. The increasing bulk modulus of the solid constituent increases the above four variables, which reach the approximate limit values, corresponding to the Darcy’s effective stress solutions, when Ks = 10 K0. The increasing permeability of the plastic rock mass reduces the increasing rate of the pore-water pressure but increases the water inflow. For the proposed fully coupled examples, both the hydraulic parameters of permeability and Biot’s coefficient show a non-linear decrease with the increasing radius.

Introduction

Underground engineering is commonly conducted below the groundwater table in a pervious rock mass. In such a saturated stratum, the excavation not only induces a mechanical response but also water seepage in the pores of the surrounding rock. During the deformation and seepage process, the hydraulic behavior affects the mechanical response and vice versa. This is the so-called fully coupled hydraulic-mechanical coupling effect. For hydraulic-mechanical coupling engineering, the failure and deformation of the surrounding rock are induced by the variations in the effective stress, which depends on the changes in the total stress and pore pressure. Therefore, the predictions on the water inflow, displacement and plastic region are very important to the stability evaluation, design of the support and drainage systems.

In the past decades, many achievements have been made regarding the change in effective stress induced by seepage around a cavity. Early analytical solutions for estimating the effective stress around a circular tunnel in a pressurized conduit were proposed by Bouvard and Pinto (1969) and by Schleiss (1986). Considering the influence of the seepage force, Fernandez and Alvarez (1994) presented an approach that takes into account the seepage force effect by treating the fractured rock mass as a continuous porous elastic medium by using the image well method proposed by Harr (1962). The distribution of the effective stresses along a radius intersecting the springline of the tunnel was estimated from the closed-form solution derived for the simplified condition that neglects the circumferential seepage force. On the basis of conformal mapping, Kolymbas and Wagner (2007) proposed an analytical expression for the estimation of the steady-state groundwater ingress into a deep and/or shallow drained tunnel with a circular cross-section. Park et al., 2008a, Park et al., 2008b derived a simple closed-form analytical solution of the seepage force along the circular tunnel circumference for both zero water pressure and constant total head boundary conditions using the conformal mapping technique. Ming et al. (2010) derived the analytical solutions of pore water pressure for a steady groundwater flow into a horizontal tunnel in a fully saturated isotropic semi-infinite aquifer. Considering the Biot’s effective stress and the pore-water pressure distribution proposed by Kolymbas and Wagner, 2007, Fahimifar and Zareifard, 2013 derived a closed-form solution of an unlined pressure tunnel. Among these solutions, the surrounding rock was considered to be an isotropic and homogeneous elastic rock medium, and the permeability was assumed to be constant by neglecting its changes due to fracture. In this regard, the previous elastic seepage induced stress and deformation solutions were only suitable for shallow cavities, in which the stress was small and the surrounding rock mainly works at the elastic state.

For the deep underground engineering, the initial ground stress is very high, and it usually induces rock mass fracture, which thereafter affects the seepage of groundwater. So far, many achievements have been made for the cavities in the drained stratum using the elasto-brittle (perfectly)-plastic model (Sharan, 2003, Park and Kim, 2006, Zhang et al., 2012a, Zhang et al., 2012b), strain-softening model (Brown et al., 1983, Alejano et al., 2010, Lee and Pietruszczak, 2008, Park et al., 2008a, Park et al., 2008b, Wang et al., 2010, Zhang et al., 2012b, Zhang et al., 2019, Cui et al., 2015) and elasto-plastic coupling model (Zhang et al., 2018). For the cavities in the saturated stratum, the seepage force acted as a body force on the surrounding rock and caused significant difficulties in the analytical derivations of displacement and stress. To overcome this issue, two methods, i.e., total stress and the superposition concept, were commonly used. Brown and Bray (1982) considered the effects of the hydraulic-mechanical coupling on the fracture zone analysis. However, the seepage effect on the elastic region was negligible. Assuming that the failure was induced by Terzaghi’s effective stress, Zareifard and Fahimifar, 2014a, Zareifard and Fahimifar, 2014b proposed the elastic-brittle-plastic solution of a circular tunnel by adopting Biot’s effective stress and constant hydraulic parameters. However, both the failure and deformation of the surrounding rock were induced by the change of the effective stress. Moreover, the pore pressure distributions of steady state seepage were obtained with the assumption of constant permeability of the surrounding rock, and this was inconsistent with the experimental results, where the permeability depends on both the confining pressure and fracture distribution (Wang et al., 2003, Chen et al., 2007, Chen et al., 2014, Yang et al., 2015, Xu and Yang, 2016).

In view of the fully coupled effect between the hydraulic and mechanical parameters, i.e., pore pressure influences the effective stress, and mechanical failure enlarges the permeability of the surrounding rock, many efforts have considered strain and/or stress-dependent permeability (Brown and Bray, 1982, Min et al., 2004). Fazio and Ribacchi, 1984, Carosso and Giani, 1989 considered the effects of the hydraulic-mechanical coupling effect by assuming that the fraction of the permeability ratio for the plastic and elastic zones varies from zero to one. However, the method for defining the proposed fraction was not given. Considering the prior given analytical distribution of the pore-water pressure and strain-dependent permeability proposed by Brown and Bray, 1982, Fahimifar and Zareifard, 2009, Zareifard and Fahimifar, 2014b, Zou et al., 2016 derived the theoretical hydraulic-mechanical solutions of circular tunnels with the assumption of superposition in displacements induced by hydraulic and mechanical behavior for strain-softening and elasto-brittle-plastic rock masses, respectively. Unfortunately, a single variable of the absolute volume strain implies an increasing permeability for both the contraction and dilation of the rock mass. This is inconsistent with the decreasing permeability with the increasing confining pressure (Wang et al., 2003). The permeability of fractures is much larger than that of a porous rock matrix (Liu et al., 2016), and the fracture degree can be represented by the plastic strain according to the elasto-plastic theory. Similar to the evolutions of mechanical parameters for strain-softening rock mass, the hydraulic parameters, i.e., Biot’s coefficient and permeability, vary with the confining pressure and plastic strain during the formation and through the cracking process (Hu et al., 2010).

In view of the limitations of the previous hydraulic-mechanical solutions of circular tunnels, this paper first analyzed the variations in permeability and Biot’s coefficient during the progressive failure process according to the experimental tests and proposed the fully coupled hydraulic-mechanical strain-softening model (HMSS). Then, the analytical solutions of stress and displacement for the surrounding rock in both the Mohr-Coulomb (MC) and Hoek-Brown (HB) HMSS rock masses were derived. Finally, the proposed solutions were validated using analytical and numerical methods for drained and saturated tunnels, and the influences of pore-water pressure, Biot’s coefficient and permeability of fully coupled HMSS rock mass on the plastic radius, displacement and water inflow were further studied.

Section snippets

Proposal of a fully coupled HMSS model

Many triaxial compression tests demonstrate the strain-softening behavior in the post-failure region for various rock masses (Martin and Chandler, 1994, Alejano and Alonso, 2005, Zhao and Cai, 2010, Zhang et al., 2013). With the increase in deviatoric stress, first the initial cracks are compacted, then new micro-cracks occur and gradually extend, leading to a decrease first and then an increase in the apparent porosity (Chen et al., 2014, Wang et al., 2003). Fig. 1(a) shows the typical

Hydraulic solutions

The general solution of p(r)w can be obtained by the integrals of Eq. (4). Using the second pore-water pressure boundary condition in Eq. (14), the pore-water pressure of each annulus can by uniformly formulated as follows:p(i)w=pi-1w+γwQi2πkilnrRi-1where γw is the unit weight of water, and Qi is the water inflow through the periphery of the ith annulus.

During the seepage process, the water flow obeys the continuity equation, which requires a constant water inflow for each annulus, i.e., Q1

Solution for the elastic rock mass

For the seepage-unaltered elastic region, the pore-water pressure remains unchanged, leading to a zero seepage force. Using the second and the fourth boundary conditions in Eq. (14), the stress and displacement can be obtained according to Lame’s solution as follows:σr,(n)=-p0-σr,nRn2/r2+p0σθ,(n)=p0-σr,nRn2/r2+p0u(n)=Rn2p0-σr,n/2Gr

By combining the radial and tangential kinematic equations of Eq. (8), the deformation compatibility equation can be obtained as follows:dεθ,(i)dr=εr,(i)-εθ,

Calculation procedure

Although the stress and displacement expressions have been derived, the radial stress and displacement of the MC rock mass and the radius of the HB rock mass cannot be obtained immediately due to the variations of the hydraulic and mechanical parameters with the confining pressure and plastic shear strain. The solutions can be theoretically obtained by solving the 2n-order equations; however, the equations are complex and the exact results are difficult to obtain if the number n is very large.

Verifications for the elasto-brittle (perfectly)-plastic rock mass

To verify the validity of the proposed solutions, two groups of typical input data for both the MC and HB rock masses that were used by many researchers (Park et al., 2008a, Park et al., 2008b, Wang et al., 2010, Zhang et al., 2018) are considered. The input data are listed in Table 1, and the confining pressure-dependent effect is not considered. To guarantee the calculation precision, the concerned region of [R1, 10R1] is discretized into n1 = 1000 annuli and that of [10R1, Ru] is n2 = 5000

Conclusions

Considering the evolutions of permeability and Biot’s coefficient during the progressive failure process, this paper proposed a fully coupled HMSS model of rock mass, in which both the hydraulic and mechanical property parameters were taken into account as variables of plastic shear strain and confining pressure. Assuming that the material parameters are constants in a very small region, a recursive method was employed to derive the hydraulic-mechanical solutions according to the ideal

CRediT authorship contribution statement

Qiang Zhang: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Data curation, Writing - review & editing. Cong Shao: Validation, Supervision, Writing - review & editing, Visualization. Hong-Ying Wang: Validation, Formal analysis, Visualization. Bin-Song Jiang: Resources, Writing - review & editing, Supervision. Yu-Jing Jiang: Resources, Writing - review & editing, Supervision. Ri-Cheng Liu: Resources, Supervision, Data curation, Writing - review & editing.

Declaration of Competing Interest

The authors declared that they have no conflicts of interest to this work.

Acknowledgements

This study has been partially supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2017XKQY048).

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