Consolidation around a tunnel in a general poroelastic medium under anisotropic initial stress conditions

Computers and Geotechnics 66 (2015) 39–52

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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Consolidation around a tunnel in a general poroelastic medium under anisotropic initial stress conditions Guangjing Chen a,⇑, Li Yu b a b

European Underground Research Infrastructure For Disposal In Clay Environment (EURIDICE), 2400 Mol, Belgium Belgian Nuclear Research Center (SCKCEN), 2400 Mol, Belgium

a r t i c l e

i n f o

Article history: Received 11 September 2014 Received in revised form 13 January 2015 Accepted 14 January 2015

Keywords: Tunnel Anisotropic stress Biot coefficient Combined compressibility Liner Permeable Impermeable

a b s t r a c t In this study, a solution for the response of a poroelastic medium around a deeply excavated circular tunnel is analytically formulated based on Biot’s consolidation theory. The proposed solution considers the initial anisotropic stress in the medium around lined or unlined tunnels and is an improvement over previous solutions (Carter and Booker, 1982, 1984) because it considers Biot’s coefficient, the combined compressibility of liquid and solid phases in the porous medium, and a thick-walled liner. The solution is expressed using the Laplace transform domain, and numerical inversion techniques are used to obtain real time domain results. The new solution is verified by comparing it with previous analytical solutions and numerical results obtained using the commercial finite element software COMSOL Multiphysics (COMSOL AB, Stockholm, Sweden). Parametric studies are performed to determine the influences of Biot’s coefficient and the combined compressibility and liner properties during the consolidation process. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction During the past several decades, disposal in deep geological formations has become a promising option in many countries for managing high-level radioactive waste. Feasibility studies for radioactive waste repositories sometimes require the construction of access and experimental tunnels that reach several hundred metres below the ground for in-situ research. Proper stress, displacement and pore pressure variation estimates during and after tunnel excavation are important when designing such repositories. Although numerical methods have been extensively applied to solve tunnel problems, analytical approaches remain useful and may be advantageous because they can provide direct qualitative insights and verify numerical tools. Many analytical solutions have been developed for circular tunnels that have been excavated in isotropic and homogeneous media. Previously, the media surrounding tunnel liners was considered to behave as a single-phase material; thus, the solutions were only relevant to short- (or undrained) and long-term (or drained) conditions [17,19]. Carter and Booker [3], Carter and Booker [4] improved the solution by considering a poroelastic saturated medium and derived the time-dependent consolidation for unlined tunnels and tunnels supported by thin-walled liners. In their solutions, the solid grains ⇑ Corresponding author. Tel.: +32 14 332779; fax: +32 14 32 37 09. E-mail address: [email protected] (G. Chen). http://dx.doi.org/10.1016/j.compgeo.2015.01.007 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.

and the liquid phase were treated as incompressible, which is a widely accepted assumption for soft and fully saturated soils but may not be suitable for rock substrates or nearly saturated soils. In this study, a hydro-mechanical (HM) coupled analysis based on Biot’s consolidation theory [1] is presented for a deeply excavated circular tunnel in an isotropic, poroelastic medium of infinite extent. Within the framework of Biot’s theory, the compressibility of solid particles and the liquid phase is addressed in terms of Biot’s coefficient, n, and the combined compressibility, b. Initially anisotropic stresses are included in the solution because they often occur in situ and play very important roles. Moreover, the thinwalled liner governed by cylindrical shell theory in the solution proposed by Carter and Booker [4] is extended to include a more general, thick-walled, permeable or impermeable liner. Rice and Cleary [20] recast Biot’s work using contemporary terminology and showed that the governing equations for the general case of arbitrarily compressible constituents are not significantly more complicated. However, the effective stress coefficient and poroelastic parameters must be modified. The relevance of Biot’s and Terzaghi’s effective stress for failure phenomena around a deep borehole was discussed by Cornet and Fairhurst [9]. In addition, Detournay and Cheng [11] comprehensively analysed the poroelastic effects around deep tunnels and boreholes. This analysis has been extended to elasto-plastic media [5], a heated poroelastic medium [16], large strain elasto-plastic soil [25], cross-anisotropic rock [14,15], and other materials.

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Notation Biot’s coefficient of porous medium combined compressibility of porous medium bulk modulus of the porous medium bulk modulus of the solid grains of the porous medium shear modulus of the porous medium drained Poisson’s ratio total normal stress effective normal stress pore liquid pressure bulk modulus of the liquid phase porosity of the porous medium sometimes called Biot’s modulus an accounts for the compressibilities of solid and liquid constituents radius of the lining intrados radius of the lining extrados radius rotation angle pore liquid saturation degree total vertical stress total horizontal stress total mean stress deviatoric stress

n b Kb Ks G

m r r0 p Kl n M a b r h Sl

rV rH rm rd

1.1. Biot’s coefficient

Kb ; Ks

ð1Þ

where n is Biot’s coefficient; Ks is the bulk modulus of the solid grain; and Kb is the drained bulk modulus of the porous medium mÞ related to the stiffness of the porous medium by K b ¼ 2Gð1þ . Here, 3ð12mÞ

G and m are the drained shear modulus and Poisson’s ratio, respectively. For soft soil (Kb  Ks), the compressibility of the solid phase is negligible relative to that of the drained bulk material and n  1. Biot’s coefficient can be used to calculate the effective stress as follows:

r0 ¼ r þ np;

u,

v

ð2Þ 0

where r and r are the total stress and effective stress, respectively, with positive tension, and p is the pore liquid pressure with positive compression. Because Biot’s coefficient always satisfies n 6 1, this equation implies that the pore pressure is not totally effective for counteracting the effects of confining pressure for changing the bulk volume. Consequently, Biot’s coefficient is also known as an effective stress coefficient. Table 1 lists several poroelastic constants for different rocks. 1.2. Combined compressibility of the pore fluid and solid phase The combined compressibility of the pore fluid and solid phase, b, also known as the storage coefficient, is defined as the volume change of the pore fluid per unit volume of porous medium as a result of a unit of increasing pore pressure. For an ideal porous material characterised by a fully connected pore space (porosity n) and a microscopically homogeneous and isotropic matrix, the combined compressibility can be written as follows:

radial and circumferential displacements

ev, er, eh, crh volumetric, radial, circumferential and shear strains k k

cl B

mu C Gl

ml El El U(r, t), P(r, t) Sr(r, t), Trh(r, t) Ev(r, t),

b¼

Classical quasi-static Biot’s theory [1] describes coupled elastic deformations and diffusive flow in fully connected porous media Biot’s coefficient is defined as the ratio of the fluid volume gained (or lost) in a material element to the volume change of the porous medium and is expressed conventionally as follows [2,18]:

n¼1

rr, rh, srh total radial, total circumferential and shear stresses rr0, rh0, srh0, p0 initial stresses and pore pressure Lame parameter Darcy’s hydraulic conductivity of the porous medium unit weight of the liquid phase Skempton’s pore pressure coefficient undrained Poisson’s ratio consolidation or diffusivity coefficient shear modulus of liner Poisson’s ratio of liner Young’s modulus of liner ‘‘effective’’ modulus of the liner El ¼ El =ð1  m2l Þ V(r, t) Fourier coefficients of radial and circumferential displacement Fourier coefficients of pore water pressure Sh(r, t) Fourier coefficients of radial and circumferential stress Fourier coefficients of shear stress Xr(r, t) Fourier coefficients of volumetric and deviatoric strain

1 n nn ¼ þ ; M Kl Ks

ð3Þ

where the constant M is sometimes called Biot’s modulus, and Kl is the bulk modulus of the liquid phase (Kl0 = 2.2 GPa for gas-free water at 25 °C). For a highly compressible fluid (Kl  Ks), the approximated expression for b is b = n/Kl. It is important to consider the compressibility of the pore liquid during consolidation in the following two types of porous media [6]. (i) Saturated porous rock: For water-saturated rock, the pore water is not effectively incompressible. In many cases, the modulus of the porous rock (K) and the modulus of the air-free liquid (Kl0) are similar in magnitude. Therefore, the compressibility of the pore fluid should be considered [21]. (ii) Nearly saturated soil: When the degree of soil saturation is nearly 100%, the liquid phase becomes continuous, and the gas phase becomes discontinuous and occluded in the liquid phase in the form of bubbles. The pore water containing bubbles behaves as a ‘‘homogeneous compressible fluid’’. A simple analysis by Verruijt [22] indicates the following upper boundary for the compressibility of the pore fluid mixture:

1 1 1  Sl ¼ þ ; K l K l0 Pl0

1  Sl  1;

ð4Þ

where Sl is the degree of pore liquid saturation, and Pl0 is the absolute pore liquid pressure. This condition prevails at a degree of saturation with 1  Sr  1. Eq. (4) indicates that even very small amounts of gas in soils will dramatically reduce the bulk modulus of the pore fluid. 1.3. Thick-walled liner In an earlier investigation conducted by Carter and Booker [4], the liner is assumed as a thin elastic tube in intimate contact with

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G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52 Table 1 Poroelastic constants for rocks [20,13,23,24].

a

Parameter

Ruhr sandstone

Tennessee marble

Charcoal granite

Berea sandstone

Westerly granite

Weber sandstone

Ohio sandstone

Pecos sandstone

Boise sandstone

Kb Ks na n

13 36 0.02 0.65

40 50 0.02 0.19

35 45 0.02 0.27

80 36 0.19 0.79

25 45 0.01 0.47

13 36 0.06 0.64

84 31 0.19 0.74

67 39 0.20 0.83

46 42 0.26 0.85

GPa GPa – –

n represents the porosity of the porous medium.

the surrounding soil. Because its radius is much greater than its thickness, the liner is assumed to obey cylindrical shell theory. To improve the solution and consider a general liner of any thickness, this paper treats the liner as a thick-walled hollow cylinder with permeable or impermeable hydraulic conditions. 1.4. Problem description Consider a deeply excavated circular tunnel (with the tunnel depth much larger than the radius) in a saturated and poroelastic medium. The domain of the porous medium occupies the region b 6 r 6 1, and the ring-shaped liner occupies the region a 6 r 6 b (see Fig. 1a). The liner used to support the host medium is considered as a completely permeable solid (e.g., concrete) or as an impermeable solid (e.g., steel). The circular tunnel is so deep that local increases in stress due to gravity can be omitted. Hence, an initially uniform stress field occurs in the poroelastic medium before excavation, with a total vertical stress of rV and a total horizontal stress of rH. For the convenience of solution derivation, the initial stress field is treated as the sum of two parts. In part I, the total mean stress rm is considered (Fig. 1b), and in part II, the deviatoric stress rd is considered as follows (Fig. 1c):

1 rm ¼ ðrV þ rH Þ; 2 1 rd ¼ ðrV  rH Þ: 2

ð5aÞ ð5bÞ

The problem is analysed under plane strain conditions, the initial pore pressure in the porous medium is uniform, the liner is assumed to be installed before any deformation of the porous medium, and a perfectly rough interface is assumed to occur between the porous medium and liner, indicating that the radial and circumferential displacements and stresses across the interface are

continuous. The liner is assumed to be a non-porous and linear elastic medium, and the host rock (soil) is assumed to be a porous, saturated and linearly elastic medium. In the solution derivation, it is assumed that the normal and shear stresses at the inside face of the liner are only removed after the liner is in place. Thus, the present solution cannot account for the initial stress relief that occurs before installing the liner due to ground convergence. This assumption is believed to be reasonable for the case in which a liner is jacked horizontally into the host medium ahead of the excavation face [4]. The nature and geometry of the problem defined in Fig. 1 indicate that a polar coordinate system (r, h) is appropriate. The classic sign convention for poroelasticity is used in this study, in which the tensile stresses and strains are considered positive. Fig. 2 presents a definition of the coordinate system and the polar stresses and displacements.

2. Governing equations 2.1. Static equilibrium equations In the absence of body forces, the static equilibrium equations for an infinitesimal element described in polar coordinates take the following form:

@ rr 1 @ srh rr  rh þ þ ¼ 0; r @h @r r 1 @ rh @ srh 2srh þ þ ¼ 0; r @h @r r

ð6aÞ ð6bÞ

where rr, rh, srh are the total radial, total circumferential and shear stresses, respectively.

Fig. 1. Lined tunnel subject to anisotropic stresses.

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G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

sure coefficient, B, and the undrained Poisson’s ratio (mu) as follows:

n¼

3ðmu  mÞ ; Bð1  2mÞð1 þ mu Þ

M¼

2GB2 ð1 þ mÞð1 þ mu Þ : 9ðmu  mÞ

ð9Þ

ð10Þ

Skempton’s pore pressure coefficient, B, is defined as the ratio of the induced pore pressure to the change in applied stress in undrained conditions. Thus, no liquid is allowed to move into or out of the control volume. Furthermore, B is related to the porosity and several bulk moduli as follows:

B¼

mu ¼

2.2. Constitutive equations For an elastic and isotropic fluid-filled porous medium, Biot’s formulations of the linear constitutive equations are expressed using the incremental total stresses and pore water pressure as follows:

  @u u 1 @ v @u þ 2G ; þ þ @r r r @h @r

Drh þ nDp ¼ kev þ 2Geh     @u u 1 @ v u 1 @v þ 2G ; ¼k þ þ þ @r r r @h r r @h   1 @u @ v v ; Dsrh ¼ Gcrh ¼ G þ  r @h @r r

ð11Þ

By using the undrained condition, the undrained Poisson’s ratio (mu) can be derived as follows:

Fig. 2. Definitions of the coordination system, stresses and displacements.

Drr þ nDp ¼ kev þ 2Ger ¼ k

1=K b  1=K s : 1=K b  1=K s þ n=K f  n=K s

ð7aÞ

3m þ Bð1  2mÞð1  K b =K s Þ ; 3  Bð1  2mÞð1  K b =K s Þ

ð12Þ

with a range of 0:5 P mu P m. 2.4. Initial and boundary conditions 2.4.1. Part I: release of the total mean stress rm and drainage of the pore water pressure p0 (Fig. 1b) (1) Hydro-mechanical boundary conditions at r = 1

ð7bÞ

ð7cÞ

where Drr = rr  rr0, Drh = rh  rh0, Dsrh = srh  srh0, Dp = p  p0, and rr0, rh0, srh0, p0 are the initial values of the stresses and pore pressure, and u and v are radial and circumferential displacements, respectively (see Fig. 2 for the definition of stresses and displacement). In Eqs. (7a) and (7b), Drr + nDp and Drh + nDp are known as effective stress increments (Dr0r and Dr0h , respectively). Thus, Biot’s coefficient, n (as defined in Section 1.1), is sometimes interpreted as an effective stress coefficient. In Eqs. (7a)–(7c), the shear modu2Gm lus, G, and Lamé parameter, k, are introduced (k ¼ 12 m, m is Poisson’s ratio) with the assumption of linearity to relate the effective stress increments of Dr0r and Dr0h and the shear stress increment of Dsrh with the volumetric strain, ev, radial strain, er, circumferential strain, eh, and shear strain, crh. The strains are related to the radial and circumferential displacements u and v according to a compatibility expression, see Eqs. (7a)–(7c).

The variations of the stresses and pore liquid pressure induced by the tunnel are assumed to be negligible in the far field of the porous medium.

pð1; tÞ ¼ p0

ð13aÞ

rr ð1; tÞ ¼ rm

ð13bÞ

(2) Hydraulic boundary conditions at r = b

pðb; tÞ ¼ 0 for permeable liner

ð14aÞ

 @pðr; tÞ @r 

ð14bÞ

¼ 0 for impermeable liner

r¼b

(3) Mechanical continuity conditions at r = b The continuity between the poroelastic medium and the liner results in the following conditions:

uðbþ ; tÞ ¼ uðb ; tÞ

ð15aÞ

rr ðbþ ; tÞ ¼ rr ðb ; tÞ:

ð15bÞ

2.3. Continuity equation for the liquid phase Mass conservation of a compressible fluid yields the following continuity equation:

k

cl

r2 p ¼ n

@ ev 1 @p þ ; M @t @t

ð8Þ

where k is the conventional Darcy’s hydraulic conductivity of the porous medium; cl is the unit weight of the liquid phase; the 2

2

Laplace operator is defined as r ¼ @r@ 2 þ 1r nates; and

1 M

¼

n Kl

þ

nn , Ks

@ @r

þ r12

2

@ @h2

in polar coordi-

where n is the porosity of the porous med-

ium and Kl is the bulk modulus of the liquid phase. The Biot’s coefficient, n, and the so-called Biot’s modulus, M, in Eqs. 7a, 7b and 8 can also be expressed by Skempton’s pore pres-

(4) Mechanical boundary conditions at r = a As shown below, the inside face of the liner is free of stress.

rr ða; tÞ ¼ 0

ð16Þ

(5) Initial conditions The initial stresses in a poroelastic medium include the following:

rr ðr; 0Þ ¼ rm

ð17aÞ

rh ðr; 0Þ ¼ rm :

ð17bÞ

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G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

The initial pore liquid pressure in the porous medium is defined as follows:

pðr; 0Þ ¼ p0 :

ð18Þ

2.4.2. Part II: release of deviatoric stress rd (Fig. 1c)

where x ¼  1r @u þ @@rv þ vr . @h The overall solution for the problem defined in Fig. 1a can be found by superimposing the two individual solutions established for parts I (Fig. 1b) and II (Fig. 1c). 3.1. Part I: Solutions due to the release of the total mean stress rm and pore pressure drainage p0 (Fig. 1b)

(1) Hydro-mechanical boundary conditions at r = 1 In the far field of a host rock, the following boundary conditions exist:

By applying the Laplace transformation and considering all variables as independent of coordinate ‘‘h’’, Eq. (8) is reduced to

pð1; h; tÞ ¼ 0

ð19aÞ

k

rr ð1; h; tÞ ¼ rd cosð2hÞ

ð19bÞ

cl

srh ð1; h; tÞ ¼ rd sinð2hÞ:

ð19cÞ

(2) Hydraulic boundary conditions at r = b

pðb; h; tÞ ¼ 0 for permeable liner  @pðr; h; tÞ  ¼ 0 for impermeable liner @r r¼b

ð20aÞ ð20bÞ

ð27Þ

R1 where ~f ðsÞ ¼ 0 f ðtÞest dt is the Laplace transform of a given variable. By using Eq. (25) and by considering the solutions as bounded, we obtained the following equation:

~ ¼ A1 ðk þ 2GÞ~ev  np

ð28Þ

Substituting the above expression of ~ev into Eq. (27) results in the following differential equation relative to the pore pressure in the Laplace transform domain:

! ! ~ 1 @p ~ @2p cl n2 c n c ~þ l sA1  l bp0 þ þ b sp ¼ @r 2 r @r k k þ 2G k k þ 2G k

(3) Mechanical continuity conditions at r = b For a perfectly rough interface between porous medium and a liner, compatibility across the interface gives the following conditions:

uðbþ ; tÞ ¼ uðb ; tÞ

! ~ 1 @p ~ @2p ~  p0 Þ; þ ¼ ns~ev þ bðsp @r 2 r @r

The pore pressure solution should be bounded, therefore the solution of Eq. (29) takes the following form:

ð21aÞ

v ðbþ ; tÞ ¼ v ðb ; tÞ

ð21bÞ

rr ðbþ ; tÞ ¼ rr ðb ; tÞ

ð21cÞ

srh ðbþ ; tÞ ¼ srh ðb ; tÞ:

ð21dÞ

~ ¼ B1 K 0 ðqrÞ   p

The inside face of the liner is free of stress.

n kþ2G 2

n kþ2G

bp  A1 þ  2 0  n þb þb s kþ2G

ð30Þ

where A1 and B1 are constants that can be determined from the boundary conditions, and K0(qr) is the modified Bessel function with an order 0. In addition,

q2 ¼

(4) Mechanical boundary conditions at r = a

ð29Þ

! n2 s þb s¼ ; C k þ 2G

cl k

ð31Þ

rr ða; h; tÞ ¼ 0

ð22aÞ

where C is the consolidation coefficient or diffusivity coefficient and takes the following form:

srh ða; h; tÞ ¼ 0

ð22bÞ

C¼

(5) Initial conditions The initial total stresses in the poroelastic medium are defined as follows:

rr ðr; h; 0Þ ¼ rd cosð2hÞ

ð23aÞ

rh ðr; h; 0Þ ¼ rd cosð2hÞ

ð23bÞ

srh ðr; h; 0Þ ¼ rd sinð2hÞ:

ð23cÞ

(6) The initial pore liquid pressure in a porous medium is defined as follows:

pðr; h; 0Þ ¼ 0:

k

k þ 2G

cl n þ bðk þ 2GÞ 2

¼

k 2GB2 ð1  mÞð1 þ mu Þ2 : cl 9ð1  mu Þðmu  mÞ

ð32Þ

By considering boundary conditions (13a) and (14a) for a permeable liner, the solution for the pore water pressure in a porous medium is obtained as follows:

~ðr; sÞ ¼  p

p0 K 0 ðqrÞ p0 þ ; r P b ðfor permeable linerÞ: s K 0 ðqbÞ s

ð33aÞ

By substituting Eq. (33a) into Eq. (28), the radial displacement ~ ðr; sÞ in a porous medium is obtained, which is then substituted u ~ r ðr; sÞ and r ~ h ðr; sÞ. into Eqs. (7a) and (7b) to obtain r

~ ðr; sÞ ¼ u

np0 K 1 ðqrÞ C 1 þ ; r P b ðfor permeable linerÞ qsðk þ 2GÞ K 0 ðqbÞ r

ð24Þ

ð33bÞ 2np0 G K 1 ðqrÞ C 1 rm ;  2G 2 þ sqðk þ 2GÞ rK 0 ðqbÞ r s r P b ðfor permeable linerÞ

r~ r ðr; sÞ ¼ 

3. Solution formulation By using Eqs. (7a)–(7c), the equilibrium Eqs. (6a), (6b) for a porous medium can be rewritten as

ðk þ 2GÞr2 ev ¼ nr2 p

ð25Þ

r2 x ¼ 0;

ð26Þ

r~ h ðr; sÞ ¼ 

ð33cÞ

2np0 G K 01 ðqrÞ C 3 rm ; þ 2G 2 þ sqðk þ 2GÞ K 0 ðqbÞ r s

r P b ðfor permeable linerÞ

ð33dÞ

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G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

Eq. (33) lists the general solutions for pore pressure, radial displacement and stresses in porous medium for a case with a permeable liner. The constants C1 and C3 will be determined based on the continuity conditions (Eqs. (21a) and (21c)) between the liner and porous medium. The classical solution for an elastic hollow cylinder subjected to an axisymmetric load is used for the liner. Because the liner is considered a non-porous solid, no hydraulic coupling is involved, and the solutions of stresses and displacements for the liner are expressed as follows: 2 ðb;sÞ ~ ðr; sÞ ¼ 2b 2 r~ r2G u b a



a2 r

þ ð1  2ml Þr   2 r~ r ðr; sÞ ¼ b2ba2 r~ r ðb; sÞ 1  ar22   2 r~ h ðr; sÞ ¼ b2ba2 r~ r ðb; sÞ 1 þ ar22 l

9 > > > > = a 6 r 6 b ðFig: 1bÞ; > > > > ;

3.2. Part II: solutions due to the release of deviatoric stress rd (Fig. 1c)

ð34aÞ ~ r ðb; sÞ is the radial contact stress at the interface between where r the porous medium and the liner, and can be solved as follows based on the radial continuity conditions shown in Eqs. (15a) and (15b):

r~ r ðb; sÞ ¼

2

rm

ðb  a2 ÞGl : s ðb2  a2 ÞGl þ ða2 þ ð1  2ml Þb2 ÞG

latter part is induced by the release of rm, which is not time dependent. For the case with an impermeable liner, no drainage occurs in the porous medium. Thus, the solution (Eq. (36)) is purely determined by the total mean stress, rm. The drainage does not contribute to the stresses and displacements in porous medium. Furthermore, the solutions of a porous medium for an unlined tunnel can be easily obtained by taking Gl = 0 in Eqs. (35), (36), as presented in Eqs. (A1), (A2) in Appendix A for permeable and impermeable tunnels.

ð34bÞ

Therefore, solutions for the radial displacement and stresses in a porous medium are obtained by substituting Eq. (34b) into Eqs. (33b)–(33d). The solutions for the case with an impermeable liner can be obtained in a similar way. In summary, solutions for the pore pressure, radial displacement and stresses in a porous medium are expressed using the Laplace transform domain as follows. With a permeable liner:

Following Carter and Booker [3], the solutions for pore pressure, displacement and stress in a porous medium due to the release of deviatoric stress, rd, with boundary conditions listed in Section 2.4.2 are sought using the following forms:

uðr; h; tÞ ¼ Uðr; tÞ cosð2hÞ

ð37aÞ

v ðr; h; tÞ ¼ Vðr; tÞ sinð2hÞ

ð37bÞ

pðr; h; tÞ ¼ Pðr; tÞ cosð2hÞ

ð37cÞ

rr ðr; h; tÞ ¼ Sr ðr; tÞ cosð2hÞ rh ðr; h; tÞ ¼ Sh ðr; tÞ cosð2hÞ srh ðr; h; tÞ ¼ T rh ðr; tÞ sinð2hÞ ev ðr; h; tÞ ¼ Ev ðr; tÞ cosð2hÞ xðr; h; tÞ ¼ Xðr; tÞ sinð2hÞ;

ð37dÞ ð37eÞ ð37fÞ ð37gÞ ð37hÞ

where U(r, t), V(r, t), P(r, t), Sr(r, t), Sh(r, t), Trh(r, t), Ev(r, t), and Xr(r, t) are the Fourier coefficients of the displacements, pore water pressure, stresses, and strains [3].

9 p0 K 0 ðqrÞ p0 > > þ > > > s K 0 ðqbÞ s > > >   2 2 > > np0 K 1 ðqrÞ b K 1 ðqbÞ 1 rm b ða2 þ ð1  2ml Þb Þ > > ~ ðr; sÞ ¼ > þ u  > > r 2s ðb2  a2 ÞGl þ ða2 þ ð1  2ml Þb2 ÞG sqðk þ 2GÞ K 0 ðqbÞ r K 0 ðqbÞ = rPb ( )   2 2 2 2Gnp0 1 K 1 ðqrÞ b K 1 ðqbÞ r b Gða þ ð1  2 m Þb Þ > ðFig: 1bÞ m l >  r~ r ðr; sÞ ¼  1 >  > > sqðk þ 2GÞ r K 0 ðqbÞ r K 0 ðqbÞ s r2 ðb2  a2 ÞGl þ ða2 þ ð1  2ml Þb2 ÞG > > > > ( ) >  0  2 2 > 2 > 2Gnp0 K 1 ðqrÞ b K 1 ðqbÞ r b G ða þ ð1  2 m Þb Þ > m 1 l > þ r~ h ðr; sÞ ¼  þ 1 þ 2 > ; sqðk þ 2GÞ K 0 ðqbÞ r K 0 ðqbÞ s r2 ðb2  a2 ÞGl þ ða2 þ ð1  2ml Þb2 ÞG

~ðr; sÞ ¼  p

By employing Eqs. (37c) and (37g), Eq. (26) can be rewritten as follows:

With an impermeable liner:

9 > > > > 2 2 > ml Þb2 Þ > ~ ðr; sÞ ¼ 1r rm2sb 2 2ða þð12 u > = rPb ðb a ÞGl þða2 þð12ml Þb2 ÞG n2 o 2 2 Gða þð12ml Þb Þ r b m > ðFig: 1bÞ r~ r ðr; sÞ ¼  s r2 ðb2 a2 ÞG þða2 þð12m Þb2 ÞG  1 > > l l > n2 o > > 2 2 Gða þð12ml Þb Þ rm b ; ~ rh ðr; sÞ ¼ s r2 ðb2 a2 ÞG þða2 þð12m Þb2 ÞG þ 1 >

! o @2 1 @ 4 n ~ ¼ 0: þ  2 ðk þ 2GÞ e E v  np 2 r @r r @r

~ðr; sÞ ¼ ps0 p

l

ð35Þ

ð36Þ

l

Solutions for the liner in Eq. (34) indicate that the stress and displacement in liner, and the contact radial stress rr(b, t) at the interface between the liner and porous medium are not time dependent and are proportional to the total mean stress, rm. This finding indicates that the drainage of pore water pressure in the porous medium has no effect on the liner behaviour. For the case with a permeable liner, the stresses and displacements in the porous medium (Eq. (35)) for are described in Fig. 1b in two parts. The first part is induced by the drainage of pore pressure p0 and evolves with time, and the

ð38Þ

Because the solution should be bounded, Eq. (38) is solved as follows:

~ ¼ GD1 r2 : E v  np ðk þ 2GÞ e

ð39Þ

E v with By applying Laplace transforms to Eq. (8) and replacing e the expression in Eq. (39), the differential equation with respect to pore pressure is expressed as

~ 1 @p ~ @2p þ ¼ @r 2 r @r

(

cl k

! ) n2 s 4 c nsG ~þ l D1 r2 : þ bs þ 2 p k þ 2G r k k þ 2G

ð40Þ

The bounded solution requirement causes the pore water pressure in Eq. (40) to take the following form:

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

~ðr; sÞ ¼ E1 K 2 ðqrÞ  p

cl nsG D1 r 2 ; q2 kðk þ 2GÞ

r P b:

Furthermore, the solutions for a porous medium for an unlined tunnel can be easily obtained by taking Gl = 0 in Eq. (42), as presented in Appendix C, for permeable and impermeable tunnels. So far, all solutions are given in the Laplace transform domain. To obtain the solutions in the real time domain, a numerical inversion

ð41Þ

Similar to the solutions shown in Section 3.1, the solutions for displacements, stresses and strains in porous medium and the liner are solved and expressed using the Laplace transform domain as follows:

9 > > > > > > > > > > > > > > > > > > =

n o e e e E 1 n1 cl n2 s G e E v ðr; sÞ ¼ @@rU þ Ur þ 2rV ¼ kþ2G K 2 ðqrÞ þ 1  q2 kðkþ2GÞ D r 2 kþ2G 1 e

e

e

~ ðr; sÞ ¼ 2 U þ @ V þ V ¼ D1 r 2 X r @r r

e sÞ ¼ nE1 2 K 0 ðqrÞ  1 D1 r1 þ 1 F 1 r 3 Uðr; 2 2 2 ðkþ2GÞq n o rPb e ðr; sÞ ¼  2nE1 2 K 2 ðqrÞ þ 1 2 bG V D1 r 1 þ 12 F 1 r 3 2 n þbðkþ2GÞ r ðkþ2GÞq > ðFig: 1cÞ > n 0 o > K 2 ðqrÞ rd > 4K 2 ðqrÞ n2 þbðkþGÞ 2GnE1 2 4 > e >  D r  3GF r  S r ðr; sÞ ¼  ðkþ2GÞq þ 2G 1 1 2 2 2 s > r r n þbðkþ2GÞ > > >  2 4

> rd 2GnE1 > 1 0 4 e > q þ ðqrÞ  K ðqrÞ r þ K þ 3GF S h ðr; sÞ ¼  ðkþ2GÞq 2 1 2 > s r 2 r2 > > n o > 0 2 > K 2 ðqrÞ rd K 2 ðqrÞ n þbðkþGÞ 4GnE1 ; 2 4 ~ T rh ðr; sÞ ¼ ðkþ2GÞq2  D r  3GF r  þ G 1 1 2 2 s r r n þbðkþ2GÞ

e e e Gl e E v ðr; sÞ ¼ @@rU þ Ur þ 2rV ¼ k þ2G ðD1l r 2 þ D2l r2 Þ l

e

e

l

e

~ ðr;sÞ ¼ 2 U þ @ V þ V ¼ D1l r 2  D2l r2 X r @r r kl e D1l r 3  12 D2l r 1 þ 12 F 2l r Uðr;sÞ ¼ 12 F 1l r3  16 k þ2G l

l

e ðr; sÞ ¼ 1 F 1l r 3 þ 1 2kl þ3Gl D1l r 3 þ 1 Gl D2l r 1  1 F 2l r V 2 6 k þ2G 2 k þ2G 2

9 > > > > > > > > > > > > > > > > > =a 6 r 6 b

> ðFig: 1cÞ > > > > > > l l > > > > þGl e > S h ðr;sÞ ¼ 2Gl D1l r 2 kklþ2G þ 3Gl F 1l r 4  Gl F 2l > > l l > > > k þG k þG 2 2 4 l l l l T~ rh ðr; sÞ ¼ k þ2G Gl D1l r þ k þ2G Gl D2l r  3Gl F 1l r  Gl F 2l ; l

l

l

l

l

ð42Þ

scheme using Crump´s algorithm [10], which is based on a summation in Durbin´s Fourier series approximation [12], is adopted. 4. Verification of the solution

;

l

þGl e S r ðr; sÞ ¼ 2Gl kklþ2G D2l r2  3Gl F 1l r 4 þ Gl F 2l

l

45

l

ð43Þ where K2(qr) is the modified Bessel function with an order of 2 and D1, E1, F1, D1l, D2l, F1l, F2l are constants to be determined from the boundary conditions and continuity (with a perfectly rough interface) conditions at the interface. The expressions of these constants are listed in Appendix B.

The correctness of the new analytical solutions in this paper is extensively verified as described below. 4.1. Unlined tunnel with n = 1, b = 0 Solutions for the porous medium with an unlined tunnel are presented in Appendix A (due to the release of the total mean stress rm and pore pressure drainage p0) and C (due to the release of deviatoric stress rd). When assuming that Biot’s coefficient is equal to 1 (n = 1) and that the combined compressibility is zero (b = 0), the solutions in Appendix A are reduced to the sum of the solutions in Cases I and Case II, and the solutions in Appendix C are reduced to Case III, as defined by Carter and Booker [3].

Fig. 3. Pore pressure and radial displacement due to drainage around an unlined tunnel. The results were calculated using the present analytical solution and the FE code COMSOL with n = 0.667, b = 0.0002 MPa1.

46

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

Fig. 4. Radial displacement and radial stress due to release of deviatoric stress rd around an unlined tunnel. The results are calculated by using the present analytical solution and the FE code COMSOL with n = 0.667, b = 0.0002 MPa1.

4.2. Unlined tunnel with n 6 1; b P 0 The commercial software COMSOL Multiphysics [8] is a powerful and well-validated finite element code for solving thermohydro-mechanical coupled problems in geomechanics. The values of Biot’s coefficient, and the compressibility of the fluid and solid in porous medium can be defined by the user in a code. This code is used to numerically model a quarter of the tunnel, including the liner and porous medium, and the size of the modelled region is 1000 m tall and 1000 m wide with a geometry meshed by 860 triangular elements. The pore pressure profiles and radial displacement around an unlined tunnel due to drainage that were obtained using the new analytical solutions and COMSOL are compared in Fig. 3a and b for a case with n = 0.667, b = 0.0002 MPa1 (Kb = 10 GPa, Ks = 30 GPa, Kf = 2 GPa, n = 0.39, which are typical values for stiff sandstone, see Table 1). The pore pressure ratio, p/p0, radial displacement, u⁄ = (k + 2G)  u/p0/b, and time T ¼ bCt2 (C is the consolidation coefficient given in Eq. (32)) are used in the figures. In Fig. 4, the profiles of radial displacement and radial stress ratio (Sr/rd) due

to the release of deviatoric stress (rd) around an unlined tunnel are compared. The comparisons are conducted at various times. Excellent agreement between the results using analytical solutions and the finite element code COMSOL can be observed in Figs. 3 and 4.

4.3. Lined tunnel with (b  a)/b = 0.01 As previously mentioned, the analytical solution presented by Carter and Booker [4] for the consolidation around a lined tunnel considers a porous medium with n = 1, b = 0, and considers the liner as a thin-walled tube that obeys cylindrical shell theory. In addition, the present analytical solution allows n 6 1; b P 0 for the porous medium, and considers the liner as a thick-walled cylinder of any thickness. When the ratio of the liner thickness to the radius, (b  a)/b is small enough, the results obtained by these two methods should become very close, which is verified in Fig. 13 when using n = 1, b = 0 and (b  a)/b = 0.01 in the new solution.

Fig. 5. Dimensionless pore water pressure and radial displacement in the porous medium around an unlined tunnel with a permeable boundary due to pore pressure drainage p0.

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

47

Fig. 6. Radial displacement coefficients due to the release of deviatoric stress, rd.

Fig. 7. Radial stress coefficients due to release of deviatoric stress, rd.

5.1. Effects of n and b on consolidation around an unlined tunnel

5.1.1. Due to pore pressure drainage p0 In this section, the effects of pore pressure drainage p0 on the hydro-mechanical responses in the porous medium are studied. Based on Eqs. (33a), (A1) and (A2), the study could be conducted in dimensionless form. Fig. 5 presents the dimensionless pore pressure, p/p0, and the dimensionless radial displacement, (k + 2G)  u/p0/b/n, at various dimensionless times, T ¼ bCt2 . Poisson’s ratio of the porous medium is selected as zero. In addition, the results presented in Fig. 5 are compared with those obtained by Carter and Booker [3] using n = 1, b = 0 with excellent agreement.

Solutions for an unlined tunnel with a permeable boundary can be found in Eq. (33a) for pore pressure and in Eq. (A1) for radial displacement, radial stress and circumferential stress. Eq. (A1) indicates that the solution is the sum of two individual parts. The first part is induced by the drainage of pore pressure p0 and evolves with time, and the second part is induced by the release of rm and is constant with time. Thus, the total mean stress, rm, is decoupled from the fluid behaviour and independent of Biot’s coefficient, n, and the combined compressibility, b.

5.1.2. Due to the release of deviatoric stress rd Solutions due to the release of deviatoric stress rd for an unlined tunnel with permeable boundary are presented in Appendix C and indicate that the pore pressure, displacement and stress responses are time-dependent. Three cases, with n = 1, b = 2  104 MPa1 (Kb = 0.1 GPa, Ks = 30 GPa, Kf = 2 GPa, n = 0.39, typical values for saturated Boom clay [7]) for case a, with n = 0.667, b = 2  104 MPa1 (Kb = 10 GPa, Ks = 30 GPa, Kf = 2 GPa, n = 0.39, typical values for stiff

5. Numerical results and parametric study This paper extends the previous solutions by Carter and Booker [3], Carter and Booker [4] by considering the Biot’s coefficient n 6 1, the combined compressibility b P 0, and a liner of any thickness. In this section, the effects of these factors on consolidation around unlined and lined tunnels are investigated.

48

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

Fig. 8. Pore pressure coefficients due to the release of deviatoric stress, rd.

Fig. 9. Circumferential displacement coefficients due to the release of deviatoric stress, rd.

Fig. 10. Circumferential stress coefficients due to release of deviatoric stress, rd.

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

49

Fig. 11. Effects of liner stiffness and liner thickness on the radial displacement at the interface due to the release of total mean stress, rm.

Fig. 12. Effects of liner thickness on the responses at the interface due to the release of deviatoric stress, rd, with n = 1, b = 2  104 MPa1, El =G ¼ 100 and m = 0 (permeable liner).

Fig. 13. Effects of liner thickness on the responses at the interface due to the release of deviatoric stress, rd, with n = 1, b = 2  104 MPa1, El =G ¼ 100 and m = 0 (impermeable liner).

50

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

Fig. 14. Effects of liner stiffness on the responses at the interface due to the release of deviator stress, rd, with n = 1, b = 2  104 MPa1, (b  a)/b = 0.1 and m = 0 (permeable liner).

sandstone, see Table 1)) for case b, and with n = 1, b = 2  103 MPa1 (Kb = 0.1 GPa, Ks = 30 GPa, Kf = 0.2 GPa, n = 0.39, nearly saturated Boom clay with Sl = 99% and a pore water pressure of 2 MPa) for case c, are compared. Poisson’s ratio of the porous medium is selected as zero. The radial displacement coefficient profiles (2G  U/rd/b), radial stress coefficient (Sr/rd), pore pressure coefficient (P/2rd), circumferential displacement coefficient (2G  V/rd/b), and circumferential stress coefficient (Sh/rd) at different times (T ¼ bCt2 ) are presented in Figs. 6–10, respectively. Comparisons between case a and case b reveal that Biot’s coefficient n significantly influences the time evolution of radial and circumferential displacement, whilst negligible differences between case a and case c indicate rather limited influences of the compressibility b. For case b, the stress and displacement quickly reach steady states, almost independent of time. Carter and Booker [3] used n = 1, b = 0 (also presented in Figs. 6–10), and observed results that agreed well with the results obtained using n = 1, b = 2  104 MPa1 (as in case a). These corresponding results indicated that liquid with a compressibility of b = 2  104 MPa1 could be considered as incompressible.

5.2. Effects of liner thickness and liner stiffness on responses at the interface 5.2.1. Due to the release of total mean stress, rm Regarding situations when the total mean stress is released and the pore pressure is drained, as described in Fig. 1b, the radial displacement solutions shown in Eqs. (34)–(36) clearly show that draining the pore pressure p0 in a porous medium does not influence the radial displacement at the interface between the liner and the porous medium (i.e., r = b, and the radial displacement at the interface is not time dependent). If the radial displacement for an unlined tunnel at r = b is noted as U0 (Eq. (A1) in Appendix A), the ratio of radial displacement at the interface for a lined tunnel, U, to an unlined tunnel, U0 can be expressed as h i1 2 =b2 ÞG =G l þ 1 , which is a function of a/b, Gl/G and U=U 0 ¼ ðað1a 2 =b2 þð12m ÞÞ l

the liner Poisson’s ratio, ml. Fig. 11 presents variations in the ratio of U/U0 for two cases with liner Poisson’s ratios of ml = 0.0 and ml = 0.4. Thus, ml = 0.0, 0.4 is selected to cover the possible range of the liner Poisson’s ratio for the parametric study. When the U/U0 ratio decreases as the Gl/G ratio increases, the liner is stiffer and less radial displacement

Fig. 15. Effects of liner stiffness on the responses at the interface due to the release of deviatoric stress, rd, with n = 1, b = 2  104 MPa1, (b  a)/b = 0.1 and m = 0 (impermeable liner).

51

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52

occurs at the interface. Decreasing the U/U0 ratio as the a/b ratio decreases results in a thicker liner and less radial displacement at the interface. Fig. 11 also shows that the liner Poisson’s ratio has some influence on the radial displacement at the interface. Fig. 11 shows that the convergence of the host medium is significantly reduced as the liner stiffness increases. It should be emphasised that the analytical solution for the lined tunnel in this paper is obtained by assuming no convergence of the porous medium before lining installation, which is not realistic for many practical engineering applications. For most tunnel excavation problems, some convergence and relief of the initial stress state occurs before the installation of the liner. Thus, the present solution overestimates the stresses acting on the lining and underestimates the convergence of the host medium before the liner installation. 5.2.2. Due to the release of deviatoric stress, rd The ratio of liner thickness to the interface radius, (b  a)/b and the ratio of the ‘‘effective’’ modulus of the liner to the shear modulus of the porous medium, El =G, with El ¼ El =ð1  m2l Þ [4] are defined here. Figs. 12 and 13 show that responses occur at the interface due to the release of deviatoric stress rd with various values of (b  a)/b for permeable and impermeable liners, with the (b  a)/b = 0 case corresponding to an unlined tunnel. Figs. 14 and 15 are used to investigate the responses at the interface due to the release of deviatoric stress, rd, with various values of El =G for both permeable and impermeable liners and El =G ¼ 0 corresponding to an unlined tunnel. Figs. 12–15 clearly demonstrate that the radial displacement and radial stress at the interface increase with the time. Increases in the liner thickness or liner stiffness efficiently reduce the displacement. Comparisons between the results with permeable liners (Figs. 12 and 14) and the results from using impermeable liner (Figs. 13 and 15) also indicated that the drainage boundary conditions influence the evolution of radial displacement and radial stress with time at the interface but do not effect their final states. As shown in Fig. 13, when (b  a)/b reaches a value as small as 0.01, the results become very similar to those obtained by Carter and Booker [4]. Thus, when the ratio of the liner thickness to the radius is small enough, the cylindrical shell theory could provide a good approximation to classic elastic theory for the hollow cylinder studied in this paper. 6. Conclusions An analytical solution for consolidation around a circular, unlined and lined, deeply excavated tunnel in a general poroelastic medium is derived using Biot’s poroelasticity theory. The new solution improved previous solutions [3,4] by considering a Biot’s coefficient of less than 1 (which occurs in many saturated porous media), the compressibility of the liquid and solid phases in the porous medium, and the liner with any thickness. In the solution, the liner is treated as a non-porous solid; thus, drainage is assumed

D2l ¼  1 4ð1ml Þ

D1l ¼ g2 D2l 1 2 F 1l ¼ 4ð1 mÞa l

rd sðD1 1ÞG



1

1 3

This work is undertaken in close co-operation with, and with the financial support of ONDRAF/NIRAS, the Belgian Agency for the Management of Radioactive Waste and Enriched Fissile Materials, as part of its programme on geological disposal of mediumlevel long-lived waste. Appendix A Solutions for unlined tunnels with permeable boundaries due to the release of total mean stress, rm, and drainage, p0, are expressed as follows:

 br

K 1 ðqbÞ K 0 ðqbÞ

o

9 b2 rm 1 > ~ ¼ 2G u > s r >  > = 2 r b m r~ r ¼ s 1  r2 :  > > > 2 ; r~ h ¼ rsm 1 þ br2 >

l

2

rd b 1 2 2 D1 ¼  GðDG1l1Þ 4ð1 ml Þ fa  b gfb D1l þ a D2l g  sðD1 1ÞG n h io 2 4 6 D3 1 1 1 2 6 F 1 ¼ 4ð1 D1l þ 4ð1 m Þ D1 fa  b gb þ 3 a þ ð3  4ml Þb mÞ 2

l

ðA1Þ

ðA2Þ

Appendix B The expression of seven constants in Eqs. (42), (43) (i.e., D1, E1, F1, D1l, D2l, F1l, F2l) can be solved using the boundary conditions shown in Eqs. (19)–(22) as follows:



2

K 1 ðqrÞ K 0 ðqbÞ

Solutions for unlined tunnels with impermeable boundaries due to the release of the total mean stress, rm, and drainage, p0, are expressed as follows:

fa2 b2 gD

D1l a4 þ D2l

n

9 2 mb > þ 1r r2Gs > > = n o 2 > 2Gnp0 1 K 1 ðqrÞ rm b b K 1 ðqbÞ ~ rr ðr; sÞ ¼  sqðkþ2GÞ r K 0 ðqbÞ  r K 0 ðqbÞ  s r2  1 : > n 0 o   > > > K 1 ðqrÞ 2Gnp0 rm b2 ; b K 1 ðqbÞ þ r~ h ðr; sÞ ¼  sqðkþ2GÞ þ þ 1 s K 0 ðqbÞ r 2 K 0 ðqbÞ r2

np0 ~ ðr; sÞ ¼ sqðkþ2GÞ u

9 > > > > > > > > > > > > > > > =

1 2 2 F 2l ¼ 4ð1 m Þ fD1l a  D2l a g

l

Acknowledgements



Gl 1 1ÞG

D1 þðD

to occur at the interface between the liner and the surrounding porous medium. Analytical solutions for pore water pressure, stresses and displacements are obtained using the Laplace transform domain, and a numerical inversion technique with high accuracy and stability is used to obtain a solution in the real time domain. The new solution for lined and unlined tunnels is analytically and numerically verified from several aspects. The new solution considers the initial anisotropic stress in the porous medium. The responses due to release of mean stress and deviatoric stress are investigated separately. Parametric studies of Biot’s coefficient and combined compressibility show that Biot’s coefficient significantly influences the consolidation process in the porous medium, whilst the combined compressibility has less effect. Studies of the liner properties indicate that the thickness and stiffness of the liner have strong effects on the radial displacement and radial stress at the interface between liner and porous medium.

> > > > > > > > > > > n h i o > > > 2 2 D3 2 2 2 ; fa  b g  a b þ ð3  4ml Þ b D2l > D1

ðB1Þ

52

G. Chen, L. Yu / Computers and Geotechnics 66 (2015) 39–52



where g1 ¼  D ¼ g1 þ g2 .

Gl ðD1 1ÞG

h

2

1b2



i

a

D1 þðD 1

h

þD1

i

a

1

Gl 1 1ÞG

2

ð34ml Þþb2

,

fa2 b2 gb2

g2 ¼  Gl G

3fa2 b2 g



G 1 Gl

where



ðb6 a6 Þþa6 þð34ml Þb6

,

G D1 ¼ ðkþ2GÞ

D3 ¼  12

1 D1 b K 2 ðqbÞ 2

K 1 ðqbÞ n2 n2 þbðkþ2GÞ qbK 2 ðqbÞ

n2 þbðkþGÞ n2 þbðkþ2GÞ

G þ ðkþ2GÞ

þ 12

n2 þbðkþ3GÞ n2 þbðkþ2GÞ

n2 1 n2 þbðkþ2GÞ q2 bK 2 ðqbÞ

9 > > > > = : > >  0 > > K 2 ðqbÞ  2 K 2 ðqbÞ ; b

ðB2Þ With an impermeable liner 2nG E1 ¼  n2 þbðkþ2GÞ 2G D1 ¼  ðkþ2GÞ

D3 ¼

9 > > > > =

1 D1 b3 K 02 ðqbÞ

K 1 ðqbÞ n2 n2 þbðkþ2GÞ qb2 K 02 ðqbÞ

2 þbðkþGÞ  12 nn2 þbðkþ2GÞ



þ 12

n2 þbðkþ3GÞ n2 þbðkþ2GÞ

n2 2G 1 ðkþ2GÞ n2 þbðkþ2GÞ q2 b2 K 0 ðqbÞ 2

> >  0 > > K 2 ðqbÞ  2b K 2 ðqbÞ ;

:

ðB3Þ Appendix C Solutions for unlined tunnels with permeable boundaries due to the release of deviatoric stress, rd, are expressed as follows:

  9 e ¼  2rd 1 K 2 ðqrÞ  b22 K 2 ðqbÞ > P > Gs v r > > >   > 2 2 > 0 2 r n þbðkþ2GÞ K ðqr Þb n 1 3 2 0 > e ¼ d 1 > U K ðqrÞ  dr  > 2 Gs v ðkþ2GÞq2 2 r 2nG > > n o > 2 > 2rd 1 2K 2 ðqrÞ > n 1 b K 2 ðqr0 Þb 1 3 e = V ¼ Gs v ðkþ2GÞq2 r  2 n dr  2 r n  0  o 2 2 > rK ðqrÞ4K ðqrÞ rd 1 K 2 ðqbÞb 2Gn 2 2 e S r ¼ 2Gs  2½n þbðkþGÞ þ 3Gdr 4  rsd > > > v ðkþ2GÞq2 n r2 r2 > > >  2 4

> 2rd 1 rd 2Gn 1 0 4 e > > S h ¼ ðkþ2GÞq2 Gs v q þ r2 K 2 ðqrÞ  r K 2 ðqrÞ  3Gdr þ s > > > n o > 0 2 2 b2 > rK ðqrÞK ðqrÞ 2 r r n þbðkþGÞ K ðqbÞq 2 4Gn 4 ; 2 d T~ rh ¼  Gsd v1 ðkþ2GÞ 2 q2 r2  3Gdr   n s r2

2 2n where ¼ ðkþ2GÞqb K 1 ðqbÞ  n þbðkþGÞ K 2 ðqbÞ, nG 2 4 4K 2 ðqbÞþqbK 1 ðqbÞ ½n þbðkþGÞ þ 2nG K 2 ðqbÞgb . q2 b2

v



rd 1 d0 ¼ 2Gs v0

n

n ðkþ2GÞ

0 4 n2 þbðkþGÞ bK 2 ðqbÞgb . 4nG

References

With a permeable liner nG E1 ¼ n2 þbðkþ2GÞ

qbK 1 ðqbÞþ4K 2 ðqbÞ q2 b2

2

0 2n K 1 ðqbÞ v0 ¼ kþ2G þ n þbðkþGÞ bK 2 ðqbÞ, qb 2nG

d¼

2rd 1 Gs v

n

ðC1Þ

n ðkþ2GÞ

Solutions for unlined tunnels with impermeable boundaries due to the release of deviatoric stress, rd, are expressed as follows:

9   e ¼  2rd 10 K 2 ðqrÞ þ b32 K 0 ðqbÞ > P > 2 > Gs v 2r > >  > 3 0 > 2 > 0 2r d 1 n þbðkþ2GÞ K 2 ðqr 0 Þb n 1 0 3 e > U ¼  Gs v0 ðkþ2GÞq2 K 2 ðqrÞ þ 2nG d r  > 2 r > > > n o > 3 0 > 2rd 1 2K 2 ðqrÞ > n 1 b K 2 ðqbÞb 1 0 3 e = V ¼ Gs v0 ðkþ2GÞq2 r þ 2 n r 2d r n  0  o 2 > rK 2 ðqrÞ4K 2 ðqrÞ K 02 ðqbÞb3 rd 1 2Gn e > S r ¼ 2Gs þ 2½n þbðkþGÞ þ 3Gd0 r 4  rsd > > v0 ðkþ2GÞq2 n r2 r2 > > > 

> 2rd 1 rd 0 4 2Gn > 2 4 1 0 e > S h ¼ ðkþ2GÞq2 Gs v0 q þ r2 K 2 ðqrÞ  r K 2 ðqrÞ  3Gd r þ s > > > n o > 0 0 > 2 b3 2 rK ðqrÞK ðqrÞ K ðqbÞq 2rd 1 n þbðkþGÞ 2 rd > 4Gn1 0 4 2 2 ~ þ T rh ¼  Gs v0 ðkþ2GÞ  3Gd r  s ; n r2 q2 r 2

ðC2Þ

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