Time dependent behavior of tunnels excavated in porous mass

Time dependent behavior of tunnels excavated in porous mass

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1453-1459, 1993 0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd Printed in Great Brita...

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Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1453-1459, 1993

0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd

Printed in Great Britain

Time Dependent Behavior of Tunnels Excavated in Porous Mass A. GIRAUD't J.M. PICARDY G. ROUSSET~"

The time-dependent behavior of a deep circular tunnel excavated in a saturated porous material is related both to the effects of hydraulic diffusion and material viscosity. The hydraulic diffusion itself is strongly affected by mechanical behavior. This paper presents a theoretical and numerical investigation of the effects of pore pressure diffusion resulting from a tunnel excavation in a non viscous pore-plastic medium. An exhaustive study is presented for the Tresca and Mohr-Coulomb plastic criteria with various hydraulic boundary conditions. Finite element calculations show that the time-dependent convergence of the tunnel wall is significant, and that even a full collapse can occur.

Then, after the mechanical problem has been defined, we use a numerical code to study the effect of the excavation and lining on the evolution of pore water pressure, and on the total stress and strain fields inside the rock mass.

INTRODUCTION

Tunnels excavated in porous rock masses, such as deep clays or marls, exhibit usually large time-dependent effects. The case of the underground laboratory at Mol Perimeter variations (mm) (Belgium) where many experimental galleries have been I I , y _ built 230 meters deep in a plastic clay is very significant. 4 00 For example, very accurate measurements performed around a gallery lined with sliding ribs, show that 300 (Bernaud et al [1]), five years after construction, the timedependent convergence of the lining represents more than 60 % of the total convergence (see Fig. 1). For this test, 200 Ribsa5 due to the specific behavior of sliding ribs, convergence of E Excavati~nRibs 15 the wall occurs for a given value of the lining pressure E 1 00 which remains constant with time (as with a "creep" loading path). These delayed phenomena are of greater importance close to the wall of the excavation, but can 0 also be observed within the rock-mass. In Fig. 2 we have plotted the evolution of the pore -100 I I I I pressure inside the rock-mass around another gallery -500 0 500 1000 1500 2000 located in the same site, lined with concrete blocks (see Time (days) Neerdael et al [10]). Before the construction of the gallery, Fig.1. In situ measurements o f sliding.Sliding ribs gallery - M o l the pore pressure field is quite stable. The pore pressure (Belgium). reponse to the construction is very sensitive • as the front progresses, pore pressure inside the rock mass decreases to a value which depends on the distance to the wall of the excavation. Then after completion of the tunnel, the pore ~ .~0q pressure field evolves very slowly due to a diffusion '/," .-.. "\ ~ process. ~ 111 For the purpose of design, it is important to have a good understanding of this time-dependent behavior. We / ~ ~ - ,, ~ . . ~ - . . . . . . . . . . . . .- ..................... can find two different causes of these phenomena. The ~ t" \ \ '~. ', ~ . " , - ' - ' " ...................................~. . . . . . I \ \ ,...-f'~'~ ----- 95mincay first cause, viscosity or viscoplasticity of the rock-mass, is ~ 0eq r \ , ..... .... ,o,.,°c,oy intrinsic and can be classically observed in laboratory by g, 1 ~ ....... 13.5 m in clay performing undrained creep tests. The second cause, the 0~-t - - - - 10.5 m incby diffusion of pore water pressure inside the rock mass, is 7.5 m in clay extrinsic ; it depends on the geometry and hydraulic 0 ,..., 100 200 300 ~00 boundaries conditions, o Time (days) The purpose of this paper is to study how the diffusion of pore water pressure can explain the time-dependent Fig. 2. Evolutionof pore waterpressure : piezometerscreen behavior of an underground cylindrical excavation. (after Neerdael et al). In the following, we first present the general poroplastic behavior of porous materials necessary for the GOVERNING EQUATIONS FOR study of the long term behavior of the gallery. POROELASTICITY

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t

"-

.....

, i , , , i

,,.

t

. . . .

~" G r o u p e m e n t pour l'~tucle des Structures Souterraines de Stockage

(G.3S), Ecole Polytechnique,91128 Palaiseau, France 1453

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We consider the quasistatic, infinitesimal deformations of macroscopically homogeneous and isotropic fully

1454

ROCK MECHANICS IN THE 1990s

saturated rocks with a connected pore structure.If solid and fluid phases are chemically ~ , and inertial forces and temperature changes negligible, the complete set of differential equations for linear reversible behavior (i. e. linear poroelasticity) (see Rice & Cleafy [12], Coussy [6], Kfimpel [9]), can be written : G 0~ _ bop (1) G V2Ui + 1.2v° 0x i 0x i & + b

= k V2p

(2)

with xi = coordinates in an orthogonal spatially fixed system, V2 =~-~xl~2+ ~x2x202÷ 02

L gran m r r,

t = time, ui = displacements of matrix particles, £ = 0Ul 0u2 0u3 ~ l + ~2x2+ 0x3 = volume dilatation, p = excess pore pressure of the fuid, G = shear modulus, v o ffi drained Poisson coefficient, b = Bier dimensionless coefficient of effective stress, M = Biot c o m i t y c~ent, k= Darcy conductivity (dimension length3 x time x mass -1). The unknown variables in equations (1) and (2) are the displacement components ui and the excess pore pressure p with respect to an initially equilibrium state. The Darcy conductivity k (named "permeability" in Rice & Cleary [12]), denotes the ratio between the intrinsic permeability k and the dynamic viscosity of the fluid ~. The four independent poroelastic par',aneters G, v o, b, M describe the linear elastic behavior of porous media within this theory, b and M were introduced by Biot [3]. Notations differ in Rice & Cleary [12], who use the undrained Poisson coefficient Vu and the Skempton ratio B (or pore pressure parameter) instead of b and M : 3(Vu - re) b = (I- 2Vo)(1 + vu) (3) M = 2 (1 - 2vÜX1 + re) 2 G B2 (1 - 2Vu)(Vu - Vo) (4) Drained and undrained values refer to drained and undrained conditions in jacketed laboratory tests. Equation (1) is obtained by combining the equilibrium equations : j__~ ~xi =0

(5)

with the generalized Hooke's law, extended for poroelastic media (Biot [2], [3]) :

(6) oij + bpSij = ( 1 -2vo 2 r e ) oesij + 20eij where oij is the total s~ress tensor (tensile stress is

2 G ~1,2VO V2£ = b V2p (7) Applying the Laplacian operator on equation (2) then substituting V2£ in the left hand side leads to : 1+

I' - ~2v0 l - V2o~

(V2p)

k V2(V2p)

(8)

Equation (8) has the standard form of a diffusion equation for the quantity V2p. The hydraulic diffusivity Dh is then expressed as follows (Biot [2], Rice & Cleary [12],Coussy [6],Ktimpel [9]): Dh = k 2(I- v 0) M G (I- 2re) b2 M + 2(I-re) G (9) This value of hydraulic diffusivity is valid for poroelasticbehavior. Biot [2] derived this expression by considering the consolidation of an infinite horizontal layer subjected to a uniform load. SIMPLIFIED ELASTOPLASTIC EXTENSION

The theory of plastic behavior of porous media is presented in d e ~ in C o ~ y [6]or [5]. In ~ paper, only a simplified elastoplastic extension is considered, assuming that "the solid particles that ~ the solid phase are incompressible". From fluid mass conservation, we can derive (see Coussy [6] for details) : b= 1 (10) Kfl M= ~ (11) where Kfl is the fluid bulk modulus and ~ the porosity. From (10), it appears that equation (6) agrees with the formulation of the "Terzaghi effective stress prinO.pl¢". It is worth noting that this principle is independent of an often assumed incompressibifity of the fluid. Equation (10) is commonly accepted for sattmRed soils: For rocks, parameter b takes values between the porosity ~ and one. Rice & Cleary [12] give typical values of ~ c parameters for some porous rocks. Plastic behavior is charactefi~ by permanent ~ s , which remain after complete t m l ~ to the ~ state of stressand initialpore ~ . To describe~ c behavior, the generalized Hooke's linear law (6) is modified by replacing the total strain tensor ~ij by the elasticstraintensor £ij- ~ii

:

eff =oij + PSij = (1 -~QVo) G °ij

3 5" &,,.,(ekk-~)_.j ~Y. + k=l

n

considered as positive), eij 4 (-~j+ axial) is the strain

2 G (gij - ~ )

tensor and 8ij is the KrOnecker symbol. This law

Equation (1) does not hold for poroptasfic behavior, it is therefore ~ by equations (5) ~ (12), Coussy [6] established that equation (2) is still valid when the

inlroduces the effective stress tensor o ; n = oij + b p 6ij • Unlike static deformations of purely elastic or elastoplastic media, pocoelastic deformations are timedependent phenomena because of the hydraulic diffusion. It is straight f o r w ~ to derive from (1) arid (2)a diffusion equation and t h e ~ a ~ ofthe ~ l i c ~vity that governs the characteristic time for a poroelastic evolution. Summing up ~ ~ v e s of eq~_mfion(1) yields :

of ~

(12)

~dclesho~.

The plastic criterion is e ~

in term of the

effective stress o~in. Only Tresca and Mohr-Coulomb c ~

will be ~ in this study : eAT elf eft ~ . m+ oi

+o mf ,-2ccos,_
ROCK M E C H A N I C S IN THE 1990s _

eft

eft

where 4 ff _ Oli > OlI I = the three principal effective stresses, C = cohesion, ¢ = internal friction angle (for the Tresca criterion, ¢ =0). The evolution of plastic strains is governed by the plastic flow rule. In this paper, only associated flow rules are considered : If F¢(oij + p ~ij) = 0 and r e = ~ oij t~ij + 0 p p = 0 then : 0F¢

0F

= ~ °---F-q- with plastic multiplier ~ > 0 (14) 0 o~f f ~ oij' s N o hardening effects are considered, the value of the plastic multiplier is obtained by solving the complete set of equations (2) (5) (12) (13) (14) with respect to prescribed boundary conditions. N U M E R I C A L CALCULATIONS Equations (2), (5), (12), (13) and (14) form the basic set of equations solved by the numerical computer code THYME (Giraud [7]). Equation (2) is integrated with respect to time by using an implicit Euler scheme, and the plastic behavior is solved with linear poroelastic iterations. THYME is a one dimensional code for porous media, using the finite element method. Hermite polynomial basis functions are used, so the nodal unknowns are the displacements, pore pressure and their first spatial derivatives.

1455

permeable lining and undrained hydraulic boundary condition (0 p(ri,t)/0 r = 0) corresponds to an impervious lining. Apart from the hydraulic boundary condition, the timedependent behavior is also closely linked to the mechanical behavior of the lining. For the gallery lined with sliding ribs previously described, the pressure acting on the tunnel wall may be considered as constant, while convergence develops. On the other hand, a rigid lining does not allow any convergence, but is subjected to an increase of mechanical pressure. These kinds of lining are modelled further by the two following loading path cases : the "creep" case for which oi remains constant during the entire process and the "relaxation" case for which the convergence Ui = - u(ri, t)/ri remains constant during this second phase (Figs. 4 and 5). Calculation parameters

The calculations given in this paper have been done using parameters values proposed for the benchmark Interclay II [4] corresponding to a low permeability clay : M = 7 5 0 0 M P a , b = l , Vo---0.25 , G = 8 0 . 0 5 M P a , kh = 4 10-12 m s"1. Two plastic models are considered : a Tresca model (~ = 0 °, C = 1 MPa) and an associated Mohr-Coulomb model (~ = 23 °, C = 0).

ri PROBLEM

DEFINITION

We are interested in the axisymmetrical problem of a deep circular tunnel (radius ri) excavated in an isotropic homogeneous medium, initially in an equilibrium state •

.

oo

characterized by an umform stress o i j =

Fig. 3. Excavation.

ooo ~ij and //~_C~i ~

constant pore pressure p = poo. Disregarding the effects of / ~ _ ~ / " a progressive excavation, a plane strain assumption allows a one-dimensional study. ~ i ~ Stage one • excavation phase

The excavation of the circular tunnel of radius ri i s simulated by an instantaneous variation of the inner radial stress at the tunnel wall from ooo to a final prescribed value oi (see Fig. 3), corresponding to the pressure acting on the lining at the end of the tunnel construction. As we will see later, the assumption of instantaneous loading for this phase is valid provided that the time needed for the excavation is smaller than the characteristic time constant Xh of the hydraulic diffusion.

p(r i , t) = 0

~

or

~OP(ri, t)/Or

p

= 0 r

v] r i

oo

I"

'CLY Fig. 4. "Creep" case.

p(r i , t) = 0

'~ ~....-P

(u(ri,t4.)='-- 0X 0 P(ri

/0r=0 oo

l"

Stage two : time-dependent behavior

The structure is now subjected to the effect of hydraulic diffusion within the rock mass. In particular, the ultimate pore pressure distribution depends on the hydraulic behavior of the lining. We have studied two different evolutions corresponding to two different hydraulic boundary conditions which represent two opposite cases encountered in practice : fully drained hydraulic boundary condition (P(ri,t) = 0) corresponds to a

Fig. 5. "Relaxation" case.

Initial conditions correspond to a 250 m deep tunnel : ooo = -5 MPa and poo = 2.5 MPa. The tunnel radius ri is taken equal to r i = 2.5 m and the inner radial stress to oi = -2.5 MPa.

1456

R O C K M E C H A N I C S IN T H E 1990s

Results

The results are presented hereafter in a dimensionless form. First we define the characteristic hydraufic diffusion time Xh : Xh = r~ / Dh (Xh = 2.13 years, equ.(9)). Normalized radius r' and normalized time t' are respectively : r' = r]ri and t' = t/Xh Radial convergence Ui = -u(ri,t)/ri, normalized pore pressure p' = p/p**, normalized inner radial stress at the tunnel boundary tYi = o i / t ~ . , , normalized boundaries of plastic zone y' = y/ri and x' = x/ri are the other dimensionless parameters of the problem. STAGE O N E : EXCAVATION PHASE

The excavation generates a plastic zone close to the tunnel wall, in which pore pressure drops, while the outer zone remains elastic. As noted previously (see Fig.2), this phenomenon corresponds to in situ observations : piezometers located near the cavity showed an instantaneous drop of pore pressure. No pore pressure variation develops in the elastic zone, which is not surprising because in linear elasticity, excavation in an infinite medium is locally an isovolumetric process (see Fig.6, ~'i -- 0.5). --

Normalized pore pressure P' I I I l 1.2

J

1 -

0.8

-

0.6

-

0.4

-

0.2

Tresca and Mohr-Coulomb models lead to a similar behavior during the excavation phase. The Mohr-Coulomb model predicts larger drops of ~ pressure but smaller extensions of plastic zones than the Tresea model (see Fig.6). It is worth noting that for large mechanical loading, poroelastoplastic models predict negative pore pressure near the tunnel boundary during the excavation phase. For example, considering the Tresca model, we can deduce from equation (16) the particular value of ai beyond which pore pressure becomes negative on the tunnel wall after the excavation stage : p(ri , t = 0 + ) > 0 f o r a * * < a i
Just after the excavation, the pore pressure distribution is not uniform within the rock mass. Thus, for any mechanical and hydraulic boundary conditions, fhlid flow occurs inside the rock mass, causing time-depe~ent changes for the structure. The ultimate pore pressure distribution differs whether the lining is impervious or not. For undrained c o n d i ~ s , the ultimate distribution is constant whereas for drained conditions, pore pressure tends to a log type profile. In the first case the pore pressure in the plastic area increases and in the second case, it decreases (see Fig. 7 for the Tresca criterion). "Creep" case

I

o

0.5

1

I

I

I

I

1.5 2 2.5 3 3.5 Normalized radius r'

0 4

Fig. 6. Normalized pore pressure p' after the excavation phase. For this instantaneous loading, there is no fluid diffusion during the excavation and thus, no fluid mass supply throughout the rock mass. That means that the material reacts instantaneously as an undrained material. In this situation, an ideal poroplastic medium is exatly equivalent to a hardening elastoplastic monophasic medium, the hardening parameter being the plastic dilatancy. This equivalence makes it possible to find closed form solutions for the instantaneous convergence, pore pressure, plastic zone boundary and stresses (Giraud [7]). For example, in the case of the Tresca model : PlastiC zone boundary y(t = 0 +) = ri exp [ k~i "a**l - C] C Elastic zone r > y : p(r, t = 0 +) = p** Plastic zoner< y : p(r, t = 0 +) = 2vf/d 130o- (l_2vo) M + G [ o i - a** - C - 2 C Log(~-.)] 11

(15)

(16)

We first examine what happens if the mechanical pressure on the lining is constant ("creep" case). Due to the absence of evolution in the mechanical equilibrium expressed in terms of total stress, it may be noticed that total stress variations within the rock mass are limited in this particular case (constant radial stress at the boundary ~i = - 2.5MPa). Because of the change of pore water pressure field previously described, the effective stress change tends towards tension in case of an impervious lining, and towards compression in case of a permeable lining. Therefore, a Mohr-Coulomb model, which is limited in tension only, give more important plastic flow in the case of undrained boundaries conditions. The numerical results confirm this qualitative analysis. Indeed, only very small convergence develops in the drained case for the Mohr-Conlomb criterion (see Fig. 8). On the contrary, the undrained case show high timedependent convergence for the Mohr-Coulomb and even for the Tresca model. Calculations show that for the Tresca model, time-dependent convergence represents approximately 25 % of the ultimate convergence and for the Mohr-Coulomb model, it is higher and a full collapse may occur.

ROCK MECHANICS

IN T H E 1990s

1457 Normalized pore pressure P'

Normalized pore pressure P'

0

0.5

1

1.2

1.2

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 Normalized radius r' Normalized radius r' Fig.?. Profiles ot normalized pore pressure p' (Tresca model).

4

I

I

I

Tre,c,ca

I

I

I

20

I

J

i

Undrained condition

J

i

I

/

t

16

Undrained condition /

/

2.5

i

Mohr-Coulomb

3 12

Drained condidon

2

Instantaneous convergence

1.5 ~-

4

Radial convergence (%)

Radial convergence (%)

3.5

0 3.5

/

8

1

4

/

/

/ °°" ,°°

-

•_-.----""

Drained condition

0.5 0

I

I

I

I

I

10 .4 10.3 10 .2 10"1 100 101

I

_J

I

I

0

104

102 103

I

I

I

I

~

10 ,4 10 .3 10 .2 10"1 100 101 1 0 2 Normalized time t'

Normalized time t'

I

10 3

Fig.8. Radial convergence evolution. Plastic zones boundaries

Plastic zones boundaries I

I

I

I

Mohr-Coulgmb 2.5

2,5 =

._=

2

"la

-~ 1.5

1.5

O N

1"4

m

1

E O

z

2

--/

t

O

0.5

z

10-3

10 - 2

10 -1

10 0

0.5

101

10 -4

Plastic zones boundaries I

I

I

l

I

Tresca 2.5

-

I

10-~

10-z 10-1 100 Normalized time r

101

Plastic zones boundaries I

3

Undrained Elastic zone

Mohr-Coulomb -

.,.. 2.5

o~

:5

Residual elastic

-

Normalized time t'

°

Elastic zone Plastic zone

E

10 - 4

_

Dmine~

2

t,

-

-

/ n_~"

Elastic zone /

/

2

-

Plastic zone

astic zone ._~ 1.5

[

-o 1.5 • N

Residual elastic

idual elastic

ea

z

0.5

z

Excavated zone

-

ZOne

o.5

o 10 -4 10.3 10.2 10 "1 100 101 102 103 104 Normalized time t'

10 -4 10-3 10-2 10-1 100 101 102 103 104 Normalized time t'

Fig. 9. Plastic zones evolution.

104

1458

ROCK MECHANICS IN THE 1990s EXTENSION OF THE MODEL : PORO-ELASTOVISCOPLAS~CITY

More can be learned by studying the curves showing the evolution of the boundary of the plastic zone (see Fig. 9). In the drained case, the outer radius of this zone decreases and after a certain time, all the rock-mass behaves elastically. Consequently, plastic flow is low for the drained case, and the total amount of time-dependent convergence remains limited. The evolution is more complex for the undrained case : first, the boundary of the plastic zone diminishes, but after a small time, it extends notably. A stabilisation can be noticed for the Tresca model, whereas the progressive plastiflcation of the rock mass is greater with the Mohr-Coulomb model. Other calculations show progressive collapse of the tunnel as hydraulic diffusion proceeds. The ultimate extension of the plastic zone (for the Tresca model) shows that the mechanical behavior of the rock-mass near the tunnel is irreversibly modified by the hydraulic diffusion process. The evolution near the tunnel wall is therefore rather complicated • it consists of a succession of plastic loading, elastic unloading and sometimes plastic reloading, due to the pore pressure diffusion. So, it is important to discretise very accurately the time and to consider very short time steps compared to the hydraulic diffusion time. The ultimate state cannot be reached by a one step calculation and a complete knowledge of the loading path is necessary. More calculations have been performed with various values of t~i. On Fig. 10, we have plotted the convergenceconfinement curves corresponding to the instantaneous behavior, immediately after excavation and to the ultimate equilibrium of the tunnel. The differences between the curves show the volume of time-dependent behavior and illustrates once more the great influence of the choice of the hydraulic properties of the lining on the long term equilibrium of the gallery.

As an extension of the previous analysis, we have studied the case of the imperviously lined tunnel in a poroelasto-viscoplastic model (Tresca criterion). The constitutive model is that of Bingham (a linear dash pot is in parallel with the sliding element). The only additional parameter of this model is the viscosity 1]; when 1] + 0 we find again the previous poroelasto-plastic model. The "creep" test is illustrated on Fig. 13 : we have plotted convergence Ui versus time for the poro-elasto-plastic model (P) (the same that in Fig.8) and the poro-elasto-viscoplastic model (VP). The differences between both evolutions is very sensitive at the beginning of the process : the response of the VP model at the first instantaneous stage is purely elastic because viscosity prevents any plastic strain. Then the evolution is made of two different parts : first, convergence evolves due to the viscous process. So, in the case of this particular VP model, we can answer the question set in the introduction : the magnitude of time dependent closure is mainly due to viscosity (range of 60 %; and 40 % due to hydraulic diffusion). CONCLUSION The two main factors responsible for time-dependent behavior of a circular tunnel excavated in poroplastic media such as clay are the nature of the hydraulic boundary condition at the tunnel wall (nature of the lining) and the nature of the plastic model. For Mohr-Coulomb or Tresca models, the choice of a permeable lining (which corresponds to a drained hydraulic boundary condition) leads to negligible timedependent convergence. An impervious lining can lead to important delayed convergence and plastic behavior depending on the mechanical loading applied and on the constitutive model. For the Tresca model, exhaustive parametric studies presented in Giraud [7] show that the time-dependent convergence is smaller than a value of (approximately) 40 % of the instantaneous convergence for the Boom clay.

"Relaxation" case

The "relaxation" case (Ui remains constant ) produces similar results. We have plotted on Fig. 11 and Fig. 12 the evolution of pore pressure in the rock mass (r' = 1.2) for the Coulomb model and pressure acting on the lining for the Tresca and the Coulomb criteria. Time dependent variations are negligible with a permeable lining and significant with an impervious lining, especially for the Coulomb model. I

,

i

i

,

Tresca

o~

t0 0.8

(undrainedcondition)

._~ 0.6 "0

N 0

1

0,8 -

0.4

0.2

-

z

convergence i ,

0

5

I

I

10 15 20 Radial c o n v e r g e n c e

I 25 (%)

"~

0.6

~

0.4

~

0.2 0

30

k

I~ ~

'

i

/

'

0

, -

'

i

'

1

-4

Ultimate convergence] ,~

I

J

1

4 8 12 Radial c o n v e r g e n c e (%)

Fig. 10, Instantaneous and ultimate convergenees.

]

Mohr-Coulomb|

Ultimate convergence

~,~o,~n~n,~ I

,

i

/ 16

ROCK MECHANICS IN THE 1990s

1459

Radial c o n v e r g e n c e (%)

So a porous plastic model such as the Tresca model cannot predict very high delayed convergence. For the Mohr-Coulomb model, an impervious lining can lead to high delayed convergence, including the full collapse of the tunnel. It is of major importance to determine the value of the angle o f internal friction o f the rock mass. If this value is reasonably high, it would be advisable to avoid an impervious lining in order to limit the convergence and the extension of the plastic zone to reasonable values. The study concerning the above two criteria is now complete. On going work concerns different criteria such as Modified Cam Clay model. The influence of the intrinsic viscosity o f the material is also under study, as well as the effects of thermal loading which are relevant to the nuclear waste disposal studies. Normalized

3.5

I

3

I

I

I

I

I

I

-

hydraulic diffusion

2

-

~ / . viscosity -'P-----~Poro-elasto-plastictty / ,,/

1.5

-

2.5-

o~

I

1 0.5 0

--~

-

Poro-elasto-vlscoplasUc~ty _

(vP)

Instantaneous convergence i (vP) ~' I

10 -7

~

I 10 -5

I

I 10 -3

I

I 10 -1

1

1 101

Normalized time t'

Fig. 13. Radial convergence evolution. Comparison between plastic and viscoplastic behaviors (Tresca model).

pore p r e s s u r e P'

Acknowledgment- This research was supported by the Agence nationale pour la gestion des d6chets radioactifs (ANDRA). 92 266 Fontenay aux roses, FRANCE.

0.8

-

-

-

0.6

REFERENCES

0.4

-

0.2

--

I

0 0

I

0.4

I

I

0.8 1.2 1.6 N o r m a l i z e d time t'

2

Fig.11. Evolution of normalized pore pressure p' at r' = 1.2 for Mohr-Coulomb model ("relaxation"case). Normalized

0.8

radial s t r e s s I

I

I

I

I

I

Unarained condition

(Mohr-Coulomb)

0.7

0.6

//

0.5

U~n~ed condition _

ffresca}

Drainedcondition

0.4

I 10 -4

I 10 -2

I

I 10 0

Normalized

I

I 10 2

I 10 4

time t'

Fig. 12. Evolution of normalized radial stress acting on lining ("relaxation" case).

104R5 30: 7-~t

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