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Dynamic response of a partially sealed tunnel in porous rock under inner water pressure Liu Gan-bin a,b,*, Xie Kang-he b, Liu Xiaohu a a b

College of Architectural, Civil Engineering and Environment, Ningbo University, Ningbo 315211, China Institution of Geotechnical Engineering, Zhejiang University, Hangzhou 310027, China

a r t i c l e

i n f o

Article history: Received 14 April 2008 Received in revised form 22 January 2010 Accepted 1 February 2010 Available online 26 February 2010 Keywords: Pressure tunnel Surrounding rock Lining Hydraulic–mechanical coupling

a b s t r a c t In this paper, the dynamic response of the surrounding rock and structure of a tunnel that is subjected to an inner water pressure is investigated by taking hydraulic–mechanical coupling into account. A dimensionless permeable parameter deﬁnes the ﬂow capacity of the lining, is introduced by considering the relative permeability of the lining of the tunnel and the surrounding rock. Further more, a dimensionless loading coefﬁcient depending on the porosity of the medium, is introduced to determine approximately the quantity of the inner water pressure supported by the solid and the pore water at the boundary of tunnel. Therefore, the coupling property of partial sealing, porosity of tunnel material and geometry is developed. The analytical solutions of stress, displacement and pore pressure are derived in the Laplace transform domain with and without considering the stiffness of lining. Numerical results in time domain are obtained by Durbin’s inverse Laplace transform and are used to analyze the inﬂuence of the loading coefﬁcient, permeable parameter, relative stiffness and thickness on stress, displacement and pore pressure in the rock mass. The available result without considering the coupling properties of partial sealing and porosity is only an extreme case of this paper. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction Many underground structures are built in saturated rock material. To study the deformation and stability of such structures, it is obviously necessary to take hydraulic–mechanical coupling into account. Biot’s poroelastic theory for both isotropic and anisotropic porous media deﬁned the fundamental for the model of such a coupling (Biot, 1941, 1955). More recent works have been performed to complete and generalize the initial works of Biot (Detournay and Cheng, 1993; Rice and Cleary, 1976), particularly for anisotropic porous media. A great number of experimental, analytical and numerical research works have been carried out to characterize and model the mechanical behavior of rock. A reformulation of anisotropic and poroelastic equation was presented by Thompson and Willis (Thompson and Willis, 1991) and, the relationships between the macroscopic poroelastic constants and the properties of porous media constituents have been established. To better elucidate the physical meaning of poroelastic constants, Cheng (1997) proposed a comprehensive methodology for the determination of anisotropic poroelastic constants from easily realizable laboratory tests. * Corresponding author. Address: College of Architectural, Civil Engineering and Environment, Ningbo University, Ningbo 315211, China. Tel./fax: +86 574 87600316. E-mail address: [email protected] (G.-b. Liu).

Detournay and Cheng (1988) were concerned with the analysis of various coupled poroelastic processes triggered by the drilling of a vertical borehole in a saturated formation subjected to a non-hydrostatic in situ stress. Schmitt et al. (1993) presented a time-dependent analytic solution for the pore pressure within a permeable and porous hollow cylinder, and was used to estimate the laboratory experiment results. Analytical and numerical solutions using anisotropic and poroelastic (or poroviscoelastic) theory were also proposed by Abousleiman et al. (1993, 1996) for the model of a generalized Mandel’s problem and of an inclined borehole one. All these models deal with the poroelastic and poroviscoelastic behaviour of porous materials with initial constant isotropy and anisotropy. Recently, Liu et al. (2005) analyzed the scattering of plane harmonic wave by a partially permeable cylindrical shell embedded in the poroelastic medium to model the effect of primary wave on the tunnel. Measurements of poroelastic constants and hydraulic ﬂow parameters of tight rock are important for modeling many geological processes. Bemer et al. (2004) studied the behavior of this clayey rock within the framework of Biot’s mechanics of ﬂuid saturated porous solids. Drained and undrained oedometric tests (i.e. uniaxial strain tests) were performed to determine the poroelastic parameters for different stress levels. Hart and Wang (2001) presented a method for determining three independent poroelastic constants: the drained bulk compressibility; the undrained bulk

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compressibility; Skempton’s B coefﬁcient, and two hydraulic ﬂow parameters: the hydraulic conductivity and the three-dimensional unconstrained speciﬁc storage, from a single test. The traditional dynamic and static analysis method of a tunnel subjected to inner water pressure is either to idealize the pressure as axisymmetric loading (Xu, 1982; Xie et al., 2004) without considering the porosity of solid, or to idealize the pressure as ﬂuid pressure (Xie et al., 2004; Senjuntichai and Rajapakse, 1993) when the porosity of solid is considered. In which there are only two extreme cases due to the assumption of porosity of the lining and the rock mass. In fact, a part of the inner water pressure is supported by the solid, the others are supported by the pore water at the boundary of tunnel, the proportion is then depended on the area coefﬁcient of pore water related to the porosity of tunnel material and rock. Therefore, the inner water pressure can’t be idealized simply as axisymmetric loading or ﬂuid pressure. A dimensionless loading coefﬁcient C, which can be used to denote the value of inner water pressure supported by the pore water and the rock material, respectively, was proposed by Liu et al. (2005), and an analysis was carried out. Since the proportionality constant j that depends on the porosity of the lining and the rock material was presented by Li (1999) to deﬁne the partial permeability property of tunnel and is introduced directly herein, then the coupling property of partial sealing and porosity of the tunnel material and geometry is developed, and the dynamic response of the lining and rock in a partially sealed pressure tunnel is studied in this article. The analytical solution of the interaction of lining and rock is derived in the Durbin (1974) Laplace transform, and by inverting the Laplace transform, numerical results in the time domain are obtained and are used to discuss the inﬂuence of the loading coefﬁcient C, the porous and geometry constants that indicate the property of the lining and rock on the stress, displacement and pore pressure. 2. Poroelastic model and general solution

p¼

2GBð1 þ mu Þ 2GB2 ð1 2mÞð1 þ mu Þ2 eþ n 3ð1 2mu Þ 9ðmu mÞð1 2mu Þ

3ðmu mÞ Bð1 2mÞð1 þ mu Þ

ð4Þ

Darcy’s law

ks

cw

p;i ¼ qi

ð5Þ

and the continuity equation for the ﬂuid phase

@n þ qi;i ¼ 0 @t

ð6Þ

where ks is the intrinsic permeability of rock; cw is the unit weight of the pore water; qi is the speciﬁc discharge. Let us consider now a pressure tunnel (shown in Fig. 1) embedded in an inﬁnite porous elastic rock with inner and outer radius r1 and r2, respectively. Prior to removal of the material, the porous mass is in situ stress state and that the deformation and redistribution of stress induced by the excavation tends to be stable is assumed before a time-dependent inner water pressure q(t) is applied on the surface of the lining. So, the effect of the in situ stress may not be taken into account. Since the lining and the rock are completely in contact, and the thickness of lining (h = r2 r1) is so small with respect to the radius of tunnel, there is no need to distinguish whether the loading is applied at r = r1 or r = r2. As a result, the inner water pressure q(t) can be thought of as acting on the interface of lining and rock. In addition, the tunnel is assumed to be long enough that it can be considered as a plane strain axially symmetric problem. To obtain the solutions of stress, displacement and pore pressure-induced by inner water pressure acting on the surface of the tunnel in an inﬁnite rock, the ﬁeld quantities have the form p = p(r, t), rij = rij(r, t), ur = ur(r, t). The diffusion equation can be derived by combining the governing Eqs. (2), (5), and (6):

cw

! @ 2 p 1 @p @e 1 @p ¼a þ þ @r 2 r @r @t M @t 2

ð7Þ

2

ð12mÞð1þmu Þ where M ¼ 2GB , is Biot’s modulus. 9ðmu mÞð12mu Þ With the help of Eqs. (1), (2), and (4), the relationship of volumetric strain and excess pore pressure can be obtained

@e a @p ¼ @r k þ 2G @r

ð8Þ

ð1Þ ð2Þ

h

where rij and p denote the increase in total stress components and pore pressure over the initial equilibrium, respectively. k, G are the Lame constants of rock material; dij is the Kronecker symbol; m and mu (the range of mu is m 0.5) are the drained and undrained Poisson’s ratios; B (ranges from 0 to 1) is the Skempton’s pore pressure coefﬁcient. The parameters B and mu can be used to account for the poroelastic coupling of deformation and ﬂow processes; a is the coefﬁcient of Biot effective stress, the realistic range of variation for a is 0–1, the expression of parameter a is

a¼

rij;j ¼ 0

ks

The poroelastic theory was ﬁrstly introduced by Biot (1941, 1955), and reformulated by Rice and Cleary (1976) in terms of easily identiﬁable quantities and material constants. Following the Biot’s original theory, the basic dynamic variable in the governing equations are the total stress rij (note that tension is here negative), the excess pore pressure p, the solid strain eij and the variation of the ﬂuid content per unit reference volume n with the corresponding conjugate kinematic quantities. The constitutive model can be written in terms of these quantities as (Rice and Cleary, 1976):

rij ¼ 2Geij þ kdij e adij p

Besides the constitutive Eq. (1), the governing equations for poroelasticity consist of the equilibrium equations:

ð3Þ

To obtain some limiting cases, the micromechanical parameters should be dealt with, i.e. the upper bounds for B, mu and a are simultaneously reached for cases in which both the ﬂuid and the solid constituents are incompressible.

r

θ

q (t )

r1 r2 Fig. 1. Geometry of a pressure tunnel.

G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

409

Substituting Eq. (8) into the diffusion Eq. (7) gives a homogeneous uncoupled diffusion equation

3. Solution without considering the stiffness of lining

@ 2 p 1 @p 1 @p þ ¼ @r 2 r @r c @t

The average permeability coefﬁcient of concrete lining is about 1 108 m/s, and has great effect on the amount of ground water inﬁltration of tunnel (Fernández, 1994; Fernández et al., 1994; Fitzpatrik et al., 1981). Due to the relative permeability of lining and rock, the tunnel is partially sealed and the ﬂow boundary condition is partial permeable. The partially permeable property of the tunnel can be denoted approximately with a dimensionless permeability parameter j, i.e. j = kl/ks/(ln r2 ln r1), in which kl is the intrinsic permeability of lining (Li, 1999). Theoretically, the constant j ranges between zero and inﬁnite. As j approaches to zero, an impermeable lining is recovered and as j approaches to a very large value, a permeable lining is obtained. As the material of lining and rock is assumed to be porous elastic medium, the solid and the ﬂuid constituents share the applied loading at boundary of the tunnel. The general cases in which the pressure was idealized as axisymmetric loading (Xu, 1982; Xie et al., 2004) and as ﬂuid pressure (Xie et al., 2004; Senjuntichai and Rajapakse, 1993) without considering the porosity of rock, are only two extreme cases and are not in accordance with practical engineering. As a result, the dimensionless loading coefﬁcient C (C ranges from 0 to 1) that depends on the porosity n of rock was proposed by Liu et al. (2005) to estimate the water pressure supported by the solid and the pore water, respectively. The coefﬁcient C is deﬁned as 1 gc, and gc, calculated approximately as gc = n2/3, is the area coefﬁcient of pore water on the surface of the tunnel due to the poroelastic property of lining and rock. For the concrete material, gc = 2/3 1; and the fracture rock gc 1 (Ling and Cai, 2002). Neglecting the inﬂuence of thickness of a thin lining, the value of pressure supported by the solid at the boundary of tunnel can be ascertained approximately with the help of loading coefﬁcient C, i.e. Cq(t), and that supported by the pore water is (1 C)q(t). To analyze the inﬂuence of loading coefﬁcient, we consider ﬁrstly, the case of completely ﬂexible lining, that is, the stiffness El of lining as it approaches zero. Coupling the property of partial sealing and porosity of tunnel material and geometry, the stress and pore pressure boundary condition of pressure in the tunnel can thus be deﬁned as:

ð9Þ

where c is deﬁned as the generalized diffusion coefﬁcient, 1 a2 þ 1 . ¼ ckws kþ2G c M It is shown in Eq. (9) that the excess pore pressure can be solved independently of the other quantities. After obtaining the solution of p, the displacement can be given by solving Eq. (8) (Detournay and Cheng, 1988), i.e.

ur ¼

/ðtÞ a 1 þ r k þ 2G r

Z

r

rp dr

ð10Þ

r2

where /(t) is the function of time which can be ascertain with the help of the boundary condition. Deﬁning the function of Laplace transform, f(r, t) as

f ðr; sÞ ¼

Z

1

0

@ n f ðr; tÞ st e dt @t n

ð11aÞ

and the inversion by

f ðr; tÞ ¼

1 2pi

Z

cþi1

ci1

@ n f ðr; sÞ st e ds @tn

ð11bÞ

pﬃﬃﬃﬃﬃﬃﬃ in which s is the Laplace transform variable; i ¼ 1; the parameter c is so selected that the line Re(s) = c is to the right of all singularities of f ðr; sÞ. Applying Laplace transform to Eq. (9) yields

@ 2 p 1 @p s p¼0 þ @r 2 r @r c

ð12Þ

In the Laplace transform domain, the Eq. (12) is a modiﬁed Bessel equation, the general solution can be written as:

p ¼ AK 0

rﬃﬃﬃ rﬃﬃﬃ s s r þ BI0 r c c

ð13Þ

where I0 (x) and K0 (x) are the modiﬁed Bessel functions of the ﬁrst and second kind of zero order, respectively. A, B are the variables that can be ascertained by the boundary conditions. In terms of the property of the Bessel function, I0 (x) approaches to an inﬁnite value when x ? 1. Therefore, the variable B is equal to zero for the limitation of pore pressure when r ? 1. So, the general solution of the pore pressure can be written as

rﬃﬃﬃ s p ¼ AK 0 r c

ð14Þ

From Eqs. (10) and (14), the polar coordinate stress component and displacement in the Laplace transform can be derived as following:

rr ¼ rh ¼

ur ¼

rﬃﬃﬃ Z r 2G Aa s r dr /ðsÞ þ rK 0 r2 c k þ 2G r2

rﬃﬃﬃ Z r 2G Aa s r dr /ðsÞ þ rK 0 r2 c k þ 2G r2 r ﬃﬃ ﬃ 2Ga s r AK 0 c k þ 2G /ðsÞ Aa 1 þ r k þ 2G r

Z

r

r2

rK 0

rﬃﬃﬃ s r dr c

ð15aÞ

rr ¼ CqðtÞ at r ¼ r2

ð16aÞ

@p j ¼ ½p ð1 CÞqðtÞ at r ¼ r 2 @r r 2

ð16bÞ

From Eq. (16), two extreme boundaries can be obtained, i.e. the inner water pressure is completely supported by the pore water when C ? 0 and is completely supported by the solid of rock when C ? 1. There exist also two extreme cases for the parameter j, i.e. when j ? 0, an impermeable lining is recovered and when j ? 1, a permeable case is obtained. The inner water pressure applied on the surface of lining tunnel, considered herein is assumed to be an axially symmetric radial traction with the step style. In the Laplace transform domain, the pressure can be expressed as:

ð1 es Þq0 s2

ð15bÞ

qðsÞ ¼

ð15cÞ

Substituting Eqs. (14) and (15a) into the boundary condition (16a) and (16b) yields:

For boundary value problem characterized by the application of constant boundary conditions, the solution can be solved by the Laplace transform and, the solution in the time domain can be obtained by using the numerically inverse Laplace transform technique.

ð17Þ

/ðsÞ ¼

r 22 CqðsÞ 2G

ð18aÞ

AðsÞ ¼

jð1 CÞqðsÞ fK 1 ðfÞ þ jK 0 ðfÞ

ð18bÞ

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G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

pﬃs

pﬃs

where x ¼ cr; f ¼ cr2 ; K0(x) and K1(x) are, respectively, the modiﬁed Bessel functions of the second kind of zero and ﬁrst order. The analytical solutions of stress, displacement and pore pressure of the pressure tunnel can be obtained by substituting Eqs. (18a) and (18b) into the general solution described in Section 2. This gives

p ¼ AðsÞK 0 ðxÞ

rr ¼ rh ¼

2G 2Ga 1 r 2 r 22 /ðsÞ þ ðxÞ K ðfÞ AðsÞ K 1 1 r2 k þ 2G f r r2

2G 2Ga 1 r 2 r22 /ðsÞ ðxÞ K ðfÞ AðsÞ K 1 1 r2 k þ 2G f r r2 2Ga K 0 ðxÞAðsÞ k þ 2G

ur /ðsÞ a 1 r2 ¼ K 1 ðxÞ K 1 ðfÞ AðsÞ rr2 r2 k þ 2G f r

ulr ¼ ur

ur jr¼r2 ¼

M ¼ 0;

ulh ¼ 0 ulr ,

ð19bÞ

rrs ¼

ð19cÞ

Then, one can obtain the stress rlr and, further more, the axisforce and deformation of lining from Eq. (20). In addition, the stress, displacement and pore pressure in the rock mass can be derived from Eq. (19) by replacing CqðsÞ with rsr , to give

ð19dÞ

rr ¼

ð20aÞ ð20bÞ ð20cÞ

ulh

2G ur r2

ð21Þ

As the stiffness of thin lining is considered, a part of the inner water pressure is dissipated in the lining (denoted as rlr ). So, two normal radial stresses can be computed along the rock-lining interface, one from the solid of surrounding rock rsr and the other from the lining rlr (neglecting the effect of pore water in lining), and there exists the expression rlr þ rsr ¼ CqðtÞ. In the Laplace transform domain, the following stress equilibrium and displacement continuity conditions at the interface (r = r2) of the lining and rock are found

rlr þ rsr ¼ CqðsÞ at r ¼ r2

2G CqðsÞ El h=r 2 þ 2G

p ¼ AðsÞK 0 ðxÞ

where are the radial and circumferential displacement; N is the axis-force, (note that tension is here taken positive); M is the bending moment; El ¼ El =ð1 l2l Þ, El and ll are the elastic modulus and Poisson’s ratio of lining, respectively.In the Laplace transform domain, the relationship of stress and deformation at r = r2 can be obtained with the help of Eqs. (19b) and (19d), gives

rsr ¼

ð23Þ

Substituting Eqs. (23) into (21), the stress supported by the surrounding rock rsr can be obtained

In Section 3, the lining has been considered as inﬁnitely ﬂexible. However, the lining is not only porous but also with some stiffness. Although this simpliﬁcation made the analysis easy, it is well known that the lining stiffness affects the distribution of stresses, strains and pore pressure in the surrounding rock and, at the same time the structural design of the lining requires to take into account its ﬁnite ﬂexibility. In this section, we will consider the stiffness of the lining. The interaction effect has been incorporated assuming no shear stress and smooth rock-lining interface for any inner water pressure and considering only equilibrium for uniform radial pressure. The stress and displacement of lining can be calculated in terms of the elasticity theory of thin elastic shell (Yang, 1981). Under the axisymmetric distribution loading rlr , in the Laplace transform domain, the deformation and the internal force of lining can be expressed as

r22 l r El h r

r2 CqðsÞ El h=r 2 þ 2G

ð19aÞ

rh ¼

ulr ¼

ð22bÞ

The displacement at the interface of the lining and rock can be obtained by substituting Eqs. (20) and (21) into Eq. (22), yields

4. Solution corresponding to stiff lining

N ¼ r 2 rlr

at r ¼ r 2

ð22aÞ

ð24Þ

ð25Þ

2G 2Ga 1 r2 r22 u ðsÞ þ ðxÞ K ðfÞ AðsÞ K 1 1 r2 k þ 2G f r r2

2G 2Ga 1 r2 r 22 u ðsÞ ðxÞ K ðfÞ AðsÞ K 1 1 r2 k þ 2G f r r2 2Ga K 0 ðxÞAðsÞ k þ 2G

ur uðsÞ a 1 r2 ¼ K 1 ðxÞ K 1 ðfÞ AðsÞ rr 2 r2 k þ 2G f r

ð26Þ

ð27Þ

ð28Þ

r 22 C

where uðsÞ ¼ E h=r2 þ2G qðsÞ. l So far, the expressions of stress, displacement and pore pressure of rock mass in a pressure tunnel are obtained in the domain of Laplace transform. Due to the mathematical complexities of these expressions, it is difﬁcult to invert them analytically. One alternative is to use a numerical inversion technique. Among the various available methods, the one developed by Durbin (1974) is used. This method is shown to be accurate and easy to implement. 5. Numerical results and analysis In the preceding model, some assumptions have been made to take into account the interaction between the lining and the surrounding rock, i.e. the tunnel thickness is not large, the tunnel is partially permeable and deformable and, the ﬂow and deformation are coupled following Biot’s theory, the loading coefﬁcient C is introduced to approximately estimate the loading supported by the solid and the pore water at the boundary of the tunnel, respectively. The solutions corresponding to ﬂexible and stiff lining have been obtained in Laplace transform. The numerical results in the time domain can be obtained by applying an approximate Laplace inversion to the analytical expressions derived in the preceding section. There exist a great variety of numerical Laplace inversion methods. The results in time domain are obtained by using Durbin’s method (Durbin, 1974) for its accuracy, efﬁciency and stability. The formula of numerical inversion Laplace transform suggested by Durbin is given by:

"

N X 2ect 1 2p 2p cos nt RefFðcÞg þ Re F c þ in 2 T0 T0 T0 n¼0

in2p 2p Im F c þ sin nt ð29Þ T0 T0

f ðtÞ ¼

where cT0 = 5–10 gives good results for N ranging from 50 to 5000 (Durbin, 1974). The computation values of all parameters in Eq. (29) are taken T0 = 20, c = 0.25 and N = 1000 in this paper. We now selectively examine the effect of some material and geometry parame-

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G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

ters of ﬂuid, rock and lining, i.e. parameter C, permeability j, stiffness El/G and thickness h/r2. The material and geometry parameters used in calculation have the following values: m ¼ 0:2; mu ¼ 0:4; B ¼ 0:8; G ¼ 6GPa; k ¼ 4GPa, ks = 107 m/s.

1 =0 =0.4 =0.7 =1.0

0.8

5.1. Effect of the loading coefﬁcient C

0.6 r / r2 = 1

0.4

pressure and stress are examined ﬁrstly and four values of C, i.e. C ¼ 0; 0:4; 0:7 and 1 are discussed. The results of two extreme

κ = 100

0.2

0 0

2

4

6

8

10

t* Fig. 3. Histories of normalized pore pressure for different values of coefﬁcient C.

0

-0.2

-0.4

El → 0

r /q0

cases that the inner water pressure is supported completely by the pore water in the poroelastic medium when C is equal to zero and by the solid of the poroelastic medium when C = 1 is also plotted. Assuming the lining to be of ﬂexible material, the effect of lining and pore water in the lining is neglected. Eq. (19) is used for the calculation when a certain permeability parameter j is equal to 100 (an approximately permeable boundary). The numerical results are shown in Figs. 2–7. The pore pressure isochrones and histories are plotted in Figs. 2 and 3 for four values of the coefﬁcient C ¼ 0; 0:4; 0:7 and 1. It is noted that the effect of C is remarkable. The pore pressure decreases with the increasing of C and dissipates rapidly along the radial direction of the tunnel, for the permeability of rock is deﬁned as ks = 107 m/s. As a result, the pore pressure approaches zero when r=r 2 is equal to 2. The pore pressure at the boundary of the tunnel increases with the increasing of t ðt ¼ ct=r22 Þ, and tends to an asymptotic value when t P 1. Assuming tension is here negative, the stress induced by the inner water pressure in the rock is negative and the isochrones and histories of stress via the loading coefﬁcient C are plotted in Figs. 4 and 5. The stress increases with the increasing of C. At the boundary of the tunnel, i.e. r/r2 = 1, the pressure is completely supported by the solid of rock when C is equal to 1 and there is no pressure in the solid of rock when C ? 0. Although there is no stress on the surface of the tunnel when C ? 0, there exists tension stress in the rock and the distribution of stress is identical with the result of Ling and Cai (2002) for the area coefﬁcient of pore water gc = 100. At the boundary of tunnel, the stress increases with the increasing of t* and tends to an asymptotic value when t P 1 (Fig. 5). Figs. 6 and 7 show the isochrones and histories of the radial displacement for various C. As shown in both ﬁgures, the stress boundary condition symbolized by C has great inﬂuence on the

El → 0

p/q0

The effect of the loading coefﬁcient C on displacement, pore

t* = 1 κ = 100

-0.6

=0 =0.4

-0.8

=0.7 =1.0

-1 1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 4. Isochrones of normalized stress variation with radius for different values of coefﬁcient C.

0 =0 =0.4

1

-0.2

=0.7

=0

=1.0

=0.4

0.8

-0.4

=1.0

0.6

El → 0

r /q0

=0.7

El → 0

r / r2 = 1

κ = 100

-0.6

p/q0

t* = 1 κ = 100

0.4

-0.8

0.2

-1 0

2

4

6

8

10

t* 0 1

2

3

4

5

6

7

8

9

10

Fig. 5. Histories of normalized stress for different values of coefﬁcient C.

r / r2 Fig. 2. Isochrones of normalized pore pressure variation with radius for different values of coefﬁcient C.

normalized radial displacement at the vicinity of the tunnel. For certain time instant t* = 1 (Fig. 6), it can be seen that the radial var-

412

G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

1.4

( +2G)ur/r2q0

=0

1.2

=0.4

1

=0.7 =1.0 El → 0 t* = 1 κ = 100

0.8 0.6 0.4 0.2 0 1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 6. Isochrones of normalized radial displacement variation with radius for different values of coefﬁcient C.

the response of displacements, pore pressure and stress have strong relation to the parameter C. To analyze the effect of permeability independently, the parameter C = 0.4 is taken into account for the coefﬁcient gc of the concrete material is 2/3 1 and fracture rock gc 1. In Figs. 8–10, radial displacement, pore pressure and stress isochrones are plotted for different permeabilities j = 0.01, 1, 100 and 1000 at the time t* = 1. For any value of permeability, the displacements (Fig. 8) calculated by the analytical solution are equivalent at the boundary of the tunnel. The displacement decreases along the direction of the radius and increases with the increasing of permeability j. The effect of j on pore pressure, shown in Fig. 9, is remarkable. Combined with Fig. 10, analysis can be carried out for a certain j, i.e. j = 1. As the lining is assumed to be inﬁnitely ﬂexible and the thickness can be neglected, the solid of lining can not support the inner water pressure. So, the stress for different values of j is rr = 0.4q0 at the boundary of the tunnel when C = 0.4. The pressure supported by the pore water in the rock is rr = 0.15q0 (shown in Fig. 9) and that by the pore water in the lining is rr = 0.45q0. Further more, an extreme case, i.e. j ? 0, is discussed in the following. When the lining is approximately sealed, the tunnel is impermeable and the pore pressure in the rock approaches zero. The other

1.6 1.4

0.6 =0.01

0.5

=1

1

=100

+2G)ur/r2q0

El → 0

0.8

r / r2 = 1

κ = 100

0.6

=0

0.4

=0.4

(

( +2G)ur/r2q0

1.2

=0.7 =1.0

0.2

2

4

6

8

El → 0 t* = 1 Γ = 0.4

0.3 0.2 0.1

0 0

=1000

0.4

10

t*

0 1

2

3

4

5

Fig. 7. Histories of normalized radial displacement for different values of coefﬁcient C.

6

7

8

9

10

r / r2 Fig. 8. Isochrones of normalized radial displacement variation with radius for different values of parameter j.

5.2. Effect of the permeability parameter

0.6 =0.01 =1

0.5

=100 =1000

0.4

p/q0

iation of displacement is remarkable near the boundary of tunnel for different values of C. The radial displacement increases with the increasing of C. When C is equal to 1, there is the maximum value of displacement at the boundary of the tunnel (without considering the effect of lining thickness) and when C = 0, the displacement is equal to zero at the boundary of the tunnel and there exists deformation in the rock mass which is induced by the distribution of tension stress (shown in Fig. 4). For certain radial distance (r/r2 = 1), the displacement increases with time and tends to an asymptotic value when the time t P 1 (Fig. 7).

El → 0 t* = 1 Γ = 0.4

0.3

j 0.2

In order to assess the sealing effects, it is more appropriate to consider a partially sealed condition along the lining–rock interface and consequently to include the inﬂuence of tunnel permeability in the formulation. When the lining is intact, its permeability is much lower than that of the rock and this is the case j < 1. However, when the lining suffers from annular shrinkage cracks or pressure-induced longitudinal cracking, its permeability may be greatly increased being j > 1. So in this analysis a wide range of j is considered. As mentioned in a previous section,

0.1 0 1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 9. Isochrones of normalized pore pressure variation with radius for different values of parameter j.

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G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

Referring back to Figs. 11 and 12, it can be observed that, when the stiffness ratios of the lining elastic modulus to shear modulus

0

-0.1 0

r /q0

-0.2 El → 0

-0.1

t* = 1 Γ = 0.4

-0.3

r /q0

=1

-0.4

=100

1

2

3

4

5

6

7

8

9

κ = 100 h / r2 = 0.1

-0.3

=1000

-0.5

t* = 1 Γ = 0.4

-0.2

=0.01

RS=1

10

RS=1000

-0.4

r / r2

RS=10000 RS=100000

Fig. 10. Isochrones of normalized stress variation with radius for different values of parameter j.

-0.5 1

5.3. Effect of the stiffness El =G and thickness h=r2 Generally, the material lining has some stiffness, which affects the distribution of the applied load between solid and ﬂuid. In this section, the displacement and stress values are calculated for various values of the stiffness El/G and thickness h/r2. The results of considering the stiffness and thickness are presented in Figs. 11–14 for a wide range of stiffness ratios (deﬁned as RS) El/ G = 1, 103, 104 and 105, and for four values of thickness h/ r2 = 0.01, 0.05, 0.1 and 0.15 at time t* = 1. A value of Poisson’s ratio ll = 0.1 was used for the lining.

7

8

9

10

Rh=0.01 Rh=0.05

0.25

Rh=0.1 Rh=0.15

0.2

t =1 Γ = 0.4 κ = 100 *

0.15

El / G = 105

0.1 0.05 0 1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 13. Isochrones of normalized radial displacement variation with radius for different thickness.

-0.04

t* = 1 Γ = 0.4 κ = 100

-0.08 r/q0

( +2G)ur/r2q0

6

0

t* = 1 Γ = 0.4 κ = 100 h / r2 = 0.1

0.3

5

0.3

RS=1 RS=1000 RS=10000 RS=100000

0.4

4

Fig. 12. Isochrones of normalized stress variation with radius for different stiffness.

0.6 0.5

3

r / r2

( +2G)ur/r2q0

pressure (rr = 0.6q0) dissipates completely in the lining. In addition, the effect of permeability on the distribution of stress shown in Fig. 10 is similar to that on displacement (Fig. 8). More speciﬁcally, Figs. 8–10 show the distribution of displacement, pore pressure and stress for four different values of permeability j. It is concluded that the special sealing boundary condition can properly model ﬂow phenomenon within the tunnel for all considered cases. The lining–rock interaction can be considered approximately. The assessment of sealing condition is relevant for calculation of stress, strain and pore pressure ﬁelds of rock around the tunnel.

2

El / G = 105

-0.12

0.2

Rh=0.01 Rh=0.05

-0.16

0.1

Rh=0.1 Rh=0.15

0

-0.2

1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 11. Isochrones of normalized radial displacement variation with radius for different stiffness.

1

2

3

4

5

6

7

8

9

10

r / r2 Fig. 14. Isochrones of normalized stress variation with radius for different thickness.

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G.-b. Liu et al. / Tunnelling and Underground Space Technology 25 (2010) 407–414

of rock increased from 1 to 105, the displacement and stress in the rock decreased, which is consistent with general mechanical concepts. It is obvious that the lining supports most of the loading. The trend of displacement and stress, from which an increase in the thickness ratios of the lining, to the radius of the tunnel, we obtain the increasing values h/r2 from 0.01 to 0.15, is given in Figs. 13 and 14. It is, therefore possible to investigate the behavior of the lining for the typical geometrical situations that one could encounter, i.e. the thin lining case, with particular attention being paid to the inﬂuence of thickness on the mechanical behavior. As far as the thickness ratios are concerned, the results of the numerical calculation has shown that the inﬂuence of the thickness ratios are signiﬁcant on the displacement and stress, and that the displacement and stress decrease with the increasing of the values h/r2. 6. Conclusion A parametric study was carried out using the semi-analytical solution presented in this paper for the evaluation of a porous tunnel embedded in an inﬁnite saturated elastic rock. A partially sealed boundary condition is used to model water ﬂow across the tunnel–rock interface. A loading coefﬁcient C is introduced to denote the value of inner water pressure supported by the pore water and the rock material, respectively. Mechanical interaction between lining and rock is discussed and the effect of lining stiffness and thickness can be assessed, as well as permeability and the parameter C by inverting Laplace transform method. The analytical solution provides a rational method for the design of pressure tunnels since all relevant parameters are considered in the formulation. However, it should be restricted to those cases that fall within the scope of the analysis e.g. limited ground yield. The solutions can also be used to examine the poroelastic process on the condition leading to formation breakdown during a high inner water pressure acting on the tunnel, and on the stability of the pressure on the tunnel by analyzing the tangential and radial stress. In summary, although the analytical solution presented in this paper is limited in scope, it can be used for a preliminary design of tunnels in porous material. References Abousleiman, Y., Cheng, A.H.D., Roegiers, J.C., 1993. A micromechanically consistent poroviscoelasticity theory for rock mechanics applications. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30 (7), 1177–1180.

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