Remarks on axisymmetric modeling of deep tunnels in argillaceous formations

Tunnelling and Underground Space Technology 28 (2012) 70–79

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Remarks on axisymmetric modeling of deep tunnels in argillaceous formations. I: Plastic clays Alessandro Graziani a,1, Daniela Boldini b,⇑ a b

University Roma Tre, Department of Civil Engineering Sciences, via Vito Volterra 62, 00146 Rome, Italy University of Bologna, Department of Civil, Environmental and Material Engineering, via Terracini 28, 40131 Bologna, Italy

a r t i c l e

i n f o

Article history: Received 15 February 2011 Received in revised form 13 September 2011 Accepted 26 September 2011 Available online 30 November 2011 Keywords: Deep tunnels Clayey soils and rocks Hydro-mechanical coupling Analytical solutions Numerical modeling

a b s t r a c t The paper analyses the response of argillaceous formations to the excavation of deep tunnels, focusing on the role of hydro-mechanical behavior. First, two different types of geologic formations are identified: plastic, soil-like, and stiff, rock-like, argillaceous formations. This paper is specifically concerned with deep tunnels in plastic clays. In particular, the influence of pore pressure changes on the stability of the excavation and on the loading of the support systems is investigated. The state of stress and deformation is analyzed in the short- and long-term, by applying analytical and numerical solutions to the idealized situation of an axisymmetric tunnel. General remarks on the influence of hydro-mechanical coupling, artificial boundary conditions and lining permeability are presented. Theoretical predictions are compared with measurements made in some tunnels excavated in the Boom clay formation at Mol (Belgium). The suggestions provided in the paper may contribute to refining practical design approaches, hence filling the gap between the application of advanced hydro-mechanical models and conventional uncoupled elasto-plastic models. In a companion paper, the behavior of rock-like materials is considered. In particular, the effect of scaly structure, typical of many deep argillaceous formations, and fissure opening around the tunnel walls is examined. Results of conventional and novel models, purposely formulated to analyze such situations, are compared. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The design of tunnels below the water table should carefully consider the influence of water pressure and seepage forces on the load conditions of the primary and final support systems. In this study the focus is on modeling the hydro-mechanical response of clayey soils and rocks during the construction of deep tunnels. The mechanical behavior of clayey soils and rocks can vary notably as a function of percentage and type of argillaceous minerals, porosity and water content. The two latter factors depend primarily on the lithostatic pressure and past stress history. In addition, the presence of significant proportions of carbonates and quartz may increase the stiffness and brittleness of the solid skeleton and, as a consequence, strongly modify the hydromechanical response of the ground to tunneling. In this respect, the Comité Français de Méchanique des Roches (2000) proposed a crude classification, in which ‘‘plastic’’ and ‘‘rigid’’ argillaceous formations are distinguished:

⇑ Corresponding author. Tel.: +39 051 2090233; fax: +39 051 2090247. E-mail addresses: [email protected] (A. Graziani), [email protected] (D. Boldini). 1 Tel.: +39 06 57333444; fax: +39 06 57333441. 0886-7798/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2011.09.006

 ‘‘plastic’’, soil-like, clay formations generally are found at depths lower than 350 m and are characterized by a Young’s modulus E’ < 500 MPa and by a water content w > 10 %;  ‘‘stiff’’, rock-like, low-porosity argillaceous formations (indicated with several geologic and petrographic terms, such as marls, argillites, claystone, and shales) are those characterized by a Young’s modulus E’ > 2000 MPa and by a water content w < 10 %. The inter-relationship between Young’s modulus and water content is supported by the database of Fig. 1, which mainly presents data of French sites selected for deep underground repositories of nuclear wastes, generally located at depths of several hundred meters. The general link between water content and both stiffness and strength has been confirmed for different types of clay-bearing rocks (e.g., Erguler and Ulusay, 2009). The aforementioned classification has also been used as a first guidance to the choice of appropriate constitutive laws to describe material behavior and fluid–solid interaction (e.g. Einstein, 2000; Kharkhour and Jabbouri, 2001). For the geomechanical modeling of underground excavations, the main alternative is between the two-phase and the one-phase approach (i.e., the model of saturated porous medium or dry medium). The first is typically to be preferred for plastic clays, the second for low-porosity clay-bearing

A. Graziani, D. Boldini / Tunnelling and Underground Space Technology 28 (2012) 70–79

71

Nomenclature a c0 cu k n p p0 pa pR q r r0 rmax s s0 t u ua uh uv

tunnel radius cohesion undrained shear strength permeability porosity pore pressure in situ pore pressure pore pressure at the tunnel wall pore pressure at the elasto-plastic boundary support pressure radial distance from the tunnel axis external radius of the zone influenced by seepage external radius of the grid mean stress s = (r1 + r3)/2 mean effective stress s0 ¼ ðr01 þ r03 =2 time; deviatoric stress t = (r1  r3)/2 radial displacement radial displacement of the tunnel wall horizontal displacement vertical displacement

rocks. Indeed, it seems reasonable to assume that the large absorption capacity of many clay minerals, combined with the slow development of hydration reactions, can completely use up the small amount of free water available in deep formations of lowporosity (Lord et al., 1998). The same factors may also help to explain the seemingly amazing observation that deep low-porosity rocks cannot be fully saturated, even if they are, at their boundary, subjected to high pore pressures for millions of years, as has long since been observed in the field of reservoir engineering (Serafim, 1968). Concerning the choice of the solid skeleton constitutive law, the one-phase approach is often associated with the use of visco-plasticity (e.g. Boidy et al., 2002) or damage mechanics laws (e.g. Slimane et al., 2003), while more conventional soil mechanics

Fig. 1. Correlation between the natural water content w and the Young’s modulus E0 in deep clayey formations (Comité Français de Méchanique des Roches, 2000, modified); the two investigated case-histories are indicated with a larger symbols (Boom clay at Mol and Raticosa clay).

w E0 G K0 Kl Ku Kw S0 S00

m0 u0 w0 r1

r01

r3

r03 rr

r0rðr¼aÞ rh

Dp

natural water content Young’s modulus shear elastic modulus drained bulk modulus lining stiffness undrained bulk modulus for the fully-saturated medium water bulk modulus in situ total stress in situ effective stress Poisson’s ratio friction angle dilatancy angle maximum principal total stress maximum principal effective stress minimum principal total stress minimum principal effective stress total radial stress effective radial stress at the soil-lining interface total circumferential stress excess pore pressure

models based on elasto-plasticity (e.g. Gäber and Labiouse, 2003; Carranza-Torres and Zhao, 2009) or critical state constitutive laws (e.g. Wongsaroj et al., 2007) are generally preferred in the twophases approach. Indeed, a typical feature of the response of deep argillaceous formations to tunnel excavation is the importance of time-dependent deformations, which can simultaneously occur as a result of pore pressure equalization as well as of time dependent behavior of the solid skeleton. Yet, it is still common practice in the modeling of underground openings, apart from research work, to consider only one of the aforementioned time-dependent effects or analyze them separately. In this paper, attention is focused on the behavior of deep tunnels in ‘‘plastic clays’’. The case history of the tunnels excavated in the Boom clay formation at Mol (Mair et al., 1992; Bernier et al., 2007) is utilized as a benchmark to compare the results of different models and convey some general remarks. Deep argillaceous formations, such as the Boom clay, have often been considered as potential host for underground repositories of radioactive wastes. Therefore, they were the subject of extensive experimental and theoretical studies aimed at investigating the thermo–hydro-mechanical coupled behavior (e.g. Gens et al., 2007; Cui et al., 2009). In these cases, the level of geotechnical characterization of such materials is much higher than in civil engineering projects. In this paper, interest is limited to ordinary tunnels for transportation systems and the modeling approaches herein analyzed belong to practice-oriented engineering methods. In particular, the twophases porous medium with ideal elasto-plastic behavior is considered. Fully saturated conditions are generally assumed. Effects of desaturation, often related to fissure opening around the tunnel walls and to the scaly structure of many argillaceous formations are dealt with in a companion paper (Boldini and Graziani, 2011). The analytical solutions and numerical models utilized in this study refer to the plane-strain axisymmetric problem of a deep circular tunnel in an isotropic stress field. They are based on the simplified assumptions typical of the well known Ground Reaction Curve (GRC) method (AFTES, 1983). Nevertheless, the results of simple elasto-plastic axisymmetric analyses may highlight some

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essential features of the hydro-mechanical response of deep tunnels in low permeability ground. After the classical studies by Rice and Cleary (1976), Carter and Booker (1982), Detournay and Cheng (1988), the influence of pore pressure on the GRC was recently investigated by, e.g., Bobet (2003), Shin et al. (2005), CarranzaTorres and Zhao (2009) and Shin et al. (2010). Some of the main questions this paper may likely contribute to answering are the following. When does the conventional uncoupled approach (i.e., separate flow-only analysis of long-term steady-state seepage – ‘‘hydraulic problem’’ – followed by stress analysis – ‘‘mechanical equilibrium problem’’) give correct predictions of long-term deformations and support loads? In other words, when is the coupled approach (i.e., a ‘‘consolidation’’ type analysis, in which flow and mechanical equations are simultaneously solved) really necessary? What is the impact of lining permeability and of the artificial boundary conditions, which are inevitably required by the numerical modeling of deep tunnels? The problem of how to assess the relative contributions of pore pressure equalization and solid skeleton time-dependent behavior to the overall deformation is addressed in a companion paper (Boldini and Graziani, 2011). In fact, the direct interpretation of convergence measurements, as a practical tool for distinguishing the hydro-mechanical and creep components, presents serious difficulties because the trend of the time–convergence curve is usually similar whatever the driving process. 2. The case history of the Boom clay tunnels The tunnels excavated in the Boom clay, at an average depth of 200–250 m (Fig. 2), in the context of a research program for nuclear waste storage, represent a valuable source of measurements performed around deep tunnels in clayey soils. Such geotechnical situation has been therefore chosen as a reference case for the application of different analytical and numerical models and, thus, to discuss some relevant issues. The main sources of geotechnical and monitoring data (extensometres, inclinometres and piezometres installed near the tunnel walls, load cells in the lining) concerning the experimental excavations are Mair et al. (1992) and Bernier et al. (2007). A summary of the main physical and mechanical properties of the Boom clay is provided in Table 1. In the following calculations, a tunnel radius a = 2.45 m is assumed, which corresponds to the radius of the ‘‘connecting gallery’’

Table 1 Geotechnical properties of the Boom clay at the depth of 225 m (Bernier et al., 2007; Delage et al., 2007). Porosity n (%) Natural water content w (%) Undrained shear strength cu (MPa) Young’s modulus E0 (MPa) Poisson’s ratio m0 (–) Friction angle u0 (°) Effective cohesion c0 (kPa) Dilatancy angle w0 (°) Permeability k (m/s)

39 25 0.98 300 0.125 18 300 0 410-12

excavated from 2001 to 2002 and, approximately, also to the radius of the ‘‘test drift’’ (radius 2.35 m) completed in 1987. In particular, a tunnel at a depth of 225 m is considered. The total stress and the pore pressure at the tunnel axis depth are estimated to be S0 = 4.50 MPa and p0 = 2.25 MPa, respectively. The lining adopted for such tunnels can be reasonably assumed to be pervious, especially that installed in the test drift which consists of a series of concrete rings (elastic modulus 35 GPa, thickness 0.6 m), each formed by 64 blocks, separated by wood packing. On the basis of the measured load–displacement curves, the overall stiffness of the lining can be assumed equal to Kl = 90 MPa/m (i.e., Kl is the ratio of radial load q to tunnel wall deformation ua/ a). The tunnel wall displacement at the time of lining installation, estimated from the measurements given by a horizontal inclinometer installed parallel to the test drift, is ua = 90 mm.

3. General assumptions The general hypotheses assumed in the following hydromechanical analyses are those typical of the well-known convergence-confinement method, which essentially consists in the axisymmetric analysis of ground-support interaction and thus implies the assumption of uniform and isotropic virgin stress and pore pressure at the tunnel depth. The ‘‘Ground Reaction Curve’’ (GRC) of the tunnel (tunnel wall displacement ua versus support pressure q) represents the ground response to tunnel excavation (e.g. AFTES, 1983; Brown et al., 1983), i.e., to the ideal process of reduction in internal pressure q. As known, the intersection of the GRC and of the ‘‘Support Reaction Curve’’ (SRC) (displacement versus radial load at the lining extrados) provides a simple

Fig. 2. Geological section at Mol (Mair et al., 1992, modified).

A. Graziani, D. Boldini / Tunnelling and Underground Space Technology 28 (2012) 70–79

procedure, when applicable, to solve ground-support interaction and calculate the equilibrium load. In the following analyses, it is assumed that the ground behaves as a fully saturated porous elastic ideal plastic medium according to the Mohr-Coulomb strength criterion; the plastic parameters are: cohesion c0 , friction angle u0 and dilatancy angle w0 . The static analysis of geotechnical structures in low-permeability grounds is frequently simplified by considering only two idealized limit situations: respectively, the short-term conditions (t = 0), characterized by the undrained behavior of the porous medium (i.e. absence of volumetric deformation and groundwater flow), and the long-term drained conditions (t = 1), in which a steadystate pore pressure distribution is reached. In the next paragraph, two analytical solutions relative to the aforementioned limit conditions are reported. For low-permeability clay formations it is generally realistic to assume that ideal undrained conditions hold during the excavation and support installation phases. Around the tunnel, the reduction in mean total stress due to excavation causes a reduction in pore pressure, Dp, which, together with the new boundary conditions imposed at the tunnel wall (i.e., by a pervious or impervious lining), initiates a time-dependent flow process. Pore pressure variations during the transient flow cause, in turn, increasing deformations around the tunnel, i.e., a consolidation process occurs, which generally leads to an increase in the long-term load applied on the lining. An interesting property of the stress state around a circular tunnel in axisymmetric conditions, demonstrated by Vardoulakis and Detournay (1985) in general and then applied to various problems (Graziani and Ribacchi, 1993; Mitaim and Detournay, 2004), is that all points around the tunnel (i.e., at different radial distances) follow the same stress-path during the ideal process of monotonic decrease in internal pressure. Yet, the validity of the aforementioned property in the analysis of ground-support interaction (i.e., after the installation of a lining) is questionable and will be discussed. 4. Analytical solutions The closed-form solutions herein summarized refer to a cylindrical borehole of radius a in an infinite elasto-plastic medium with uniform and isotropic in situ total stress S0 and uniform pore water pressure p0 (Fig. 3). The following solutions specifically refer to the case in which a plastic annulus forms around the tunnel wall, because it represents the most typical situation for deep tunnels. 4.1. Short-term conditions

73

of the outer boundary of the plastic zone from the tunnel axis) and the radial displacement u are given by the following expressions:

R OF  1 ¼ exp ; a 2 with OF ¼

ð1aÞ

S0  q cu

ð1bÞ

uðrÞ 1 cu a ¼ ua  ¼ expðOF  1Þ  a r 2G r

ð2Þ

The meaning of the symbols is: r, radial distance from the tunnel center, ua, radial displacement of the tunnel wall, G, shear elastic modulus, q, tunnel support pressure and, OF, the so-called Overload Factor. In the plastic annulus around the tunnel, rr and rh, the total radial and circumferential stresses, are obtained by combining equilibrium and plasticity conditions:

r a

rr ¼ q þ 2cu ln ; 

rh ¼ q þ 2cu 1 þ ln

ð3aÞ r a

ð3bÞ

Finally, the excess pore pressure Dp must be equal to the variation of the total mean stress; hence, Dp around the opening is given by:

 r for r 6 R Dp ¼ p  p0 ¼ S0 þ q þ cu 1 þ 2 ln a Dp ¼ 0 for r > R

ð4Þ

Fig. 4 shows that the theoretical prediction of radial displacements given by Eq. (2) agrees very well with the in situ measurements around the tunnels in Boom clay (Neerdael et al., 1987). In fact, in this case, a nearly axisymmetric deformation of the crosssection was observed. In contrast, in the case of tunnels excavated in the London clay (Mair and Taylor, 1992), also shown in Fig. 4, the agreement is less satisfactory; in fact, the displacements along the vertical axis of the tunnel are significantly larger than those along the horizontal axis, seemingly, because of the vicinity of the ground surface and the anisotropy of initial stress and soil stiffness. Eq. (4) indicates that no excess pore pressure should be generated outside the plastic annulus (i.e., for r > R), but some in situ observations have shown a rather different pore pressure profile. The question will be further discussed in Section 6, in which analytical and numerical predictions are compared to the observed behavior. In the framework of the GRC method, the excavation process can be simulated by the progressive reduction in support

The short-term response of the tunnel, provided by the classical solution of Salençon (1969), already utilized by many Authors (e.g., Mair and Taylor, 1992), is hereafter reported for the Reader’s convenience. The undrained behavior of the saturated porous medium is obtained by considering a null volumetric deformation and a purely cohesive shear strength, equal to cu. The plastic radius R (distance

S0

q 0

a

R

r0

r max

r

Fig. 3. Detail of the calculation model with relevant radial distances.

Fig. 4. Radial displacements measured in the experimental Boom tunnel and in tunnels excavated in the London clay (data from Mair and Taylor, 1992).

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t = (σ1-σ3)/2 0

4.2. Long-term conditions =

Lembo-Fazio and Ribacchi (1984) proposed to evaluate the state of stress and deformation for the long-term drained conditions (t = 1) assuming that a stationary radial flow is established inside an annular zone spanning from the tunnel wall to a radial distance r0, which represents the limit of the zone influenced by seepage (i.e., p = p0 for r P r0) (Fig. 3). The corresponding pore pressure distribution has a logarithmic shape:

σ'

3

STRESS PATH: effective

h e ng t r str s he a ' ϕ in 's ϕ'+s 'cos t=c

total

limit

B

A A'=B'

pðrÞ ¼ pa þ ðp0  pa Þ ln

S'0

S0

s = (σ1+σ3)/2 s' = (σ'1+σ'3)/2

Fig. 5. Effective and total stress paths of a point around the tunnel in undrained conditions.

pressure q from the initial value, equal to S0 ðwith S0 ¼ S00 þ p0 Þ, to the final equilibrium value, q = 0 if any, if no support system is applied to the tunnel wall. Fig. 5 shows the total and effective (indicated with the apex 0 ) stress-paths of a point located at the tunnel wall. During the initial phase of stress relief, the ground response is elastic (segment S0A), the effective stress path is vertical and it reaches a maximum value of shear stress given by the point A0 ; in the subsequent plastic phase, the total stress path moves from A to point B, while the effective stress remains fixed at point A0 . As already mentioned, points located at increasing distances from the tunnel wall follow the same stress-path, but for a length that becomes progressively shorter (i.e., they do not reach point B). The shear stress in point A0 = B0 corresponds to the undrained cohesion cu, which can therefore be easily expressed as a function of the effective strength parameters c0 and u0 and initial effective stress S00 :

cu ¼ ðS0  p0 Þ sin u0 þ c0 cos u0

ð5Þ

The pore pressure p is constant during the elastic phase S0A, while it decreases during the plastic phase, eventually reaching also negative values, if B lies to the left of B0 as in the case of the Boom clay tunnel (Fig. 5). For the same case, the GRC in undrained conditions (t = 0), obtained by Eq. (2), is represented by the bold line in Fig. 6.

undrained conditions

drained conditions, impervious tunnel

3

  ua 1 1  2m0 þ sin u0 ðS0  p0 Þ sin u0 þ ðp0  pR Þ ¼ ðR=aÞKþ1 0 2G a 2ð1  m Þ h i þ ðR=aÞKþ11  ½ðS0  p0 þ c Þð1  2m0 Þ  1 þ NK  m0 ðK þ 1ÞðN þ 1Þ ð7Þ ðq  p0 þ c0 Þ K þ1 where pR is the value of pore water pressure at the boundary between the elastic and elasto-plastic domains (i.e., at a distance from the tunnel center equal to the plastic radius R) and m0 is the Poisson’s ratio. The parameter h takes the influence of the pore pressure gradient into account; N and K are, respectively, the triaxial and dilatancy factors:

h¼

p0  pa ; ðN  1Þ  lnðr 0 =aÞ

ð8aÞ

N¼

1 þ sin u0 ; 1  sin u0

ð8bÞ

K¼

1 þ sin w0 1  sin w0

ð8cÞ

ð9Þ t=∞

2

σ'r (r =a) < 0 1

t=∞ t=0 0

150

300

ð6Þ

where pa is the pore pressure imposed at the tunnel wall. Comparison of Eq. (6) with more realistic solutions, where a plane steady flow with undisturbed or partially lowered water tables are considered, shows that this can be considered a satisfactory approximation of the real distribution, at least in a zone close to the tunnel boundary (Ribacchi et al., 2002). The general solution for stresses and displacements around the tunnel is described in Lembo-Fazio and Ribacchi (1984); afterward, it was specialized by Graziani and Ribacchi (2001) to the case of uncompressible fluid and of an elasto-plastic solid skeleton. For an ideal plastic medium with the Mohr-Coulomb strength criterion, the long-term GRC is given by (Graziani and Ribacchi, 2001):

σ'r(r=a) = 0, q = p0

0

r  0 a

1 h i 9N1 8 ð12m0 Þ 0 0 0 R <ð1  sin u Þ  S0  pR  2ð1m0 Þ ðp0  pR Þ þ c  cot g u  h= ¼ ; a : q  pa þ c0  cot g u0  h

drained conditions, pervious tunnel

support pressure q (MPa)

a

ln

The external boundary of the plastic zone around the tunnel in the steady state situation is given by the following plastic radius R:

ANALYTICAL SOLUTIONS:

4

 r .

450

600

tunnel wall displacement ua (mm) Fig. 6. GRCs of the tunnel in undrained (t = 0) and drained conditions (t = 1).

Eq. (9) highlights that a minimum support pressure is generally required for tunnel stability. Such a critical value of the total support pressure q can be easily obtained as the value that makes the ratio on the right side of Eq. (9) tend to infinity:

qcrit ¼ pa þ h  c0  cot g u0

ð10Þ

Fig. 6 shows the long-term GRCs for the two limit cases of perfectly pervious and impervious boundary, which are respectively obtained by imposing the pore pressure at the tunnel boundary equal to pa = 0 and pa = p0 in Eqs. (7)–(9). For the calculations of the curves in Fig. 6, the limit radius r0 was assumed equal 20a.

A. Graziani, D. Boldini / Tunnelling and Underground Space Technology 28 (2012) 70–79

Obviously, according to Eq. (6), the second case means that pore pressure is constant and equal to the undisturbed value p0 in the whole calculation domain. The two curves overlap perfectly up to the point of yielding initiation at the tunnel wall. Then, they separate and exhibit a very different trend. The pervious tunnel requires a much lower support pressure for the same value of convergence. It can also be observed that, paradoxically, there is a range of q values for which the long-term convergence ua is even lower than that obtained for the short-term undrained conditions (Fig. 6). This outcome raises doubts about the adequacy of the uncoupled solution. For the impervious tunnel, the total pressure q decreases until it reaches the value q = p0: from this point on, the ground detaches from the lining if the tensile strength of the interface is null, otherwise, the effective radial stress at the interface r0rðr¼aÞ will become negative. In this case, the load on the lining derives only from water pressure, and a gap forms at the ground-lining interface which then widens progressively. The GRC for an impervious tunnel could also be obtained by considering the effective in situ stress S00 ¼ S0  p0 , carrying out the calculation in terms of effective stress, with the same equations valid for a tunnel in a dry medium, and adding, only at the end, the undisturbed pore pressures p0 at the previously obtained support pressure q0 . This procedure is not applicable for a pervious tunnel, because in this case the pore pressure varies with radial distance and seepage forces must be taken into account. Fig. 7a and b shows, respectively, the stress-paths for the pervious and the impervious tunnel. For the pervious case, pore pressure gradients around the tunnel modify the state of stress even

(a) PERVIOUS TUNNEL

q

= 3

=

σ'

3

σ'

STRESS PATH: undrained

0

t = (σ1-σ3)/2

drained

C'

A'=B'

h e ng t r str s he a ' ϕ in limit 's ϕ'+s 'cos t=c

C

B

A

D=D' S'0

S0

s = (σ1+σ3)/2 s' = (σ'1+σ'3)/2

(b) IMPERVIOUS TUNNEL

75

during the initial phase of stress relief, when ground response is still elastic: such effect is visible as a rightwards bending of the effective stress path (curve S00 C). It is also important to remember that the uncoupled solution of the long-term porous-elasto-plastic problem cannot take possible phenomena of elastic rebounds into account. In fact, the effective stress of all points inside the plastic annulus are forced to stay on the yield surface.

5. Numerical models To verify the influence of hydro-mechanical coupling on the tunnel convergence and support load, it is necessary to utilize numerical models. In fact, coupled analytical solutions of consolidation problems around circular boreholes are only available for elastic porous media (e.g. Rice and Cleary, 1976; Carter and Booker, 1982; Detournay and Cheng, 1988). Linear elastic solutions are of little interest for deep tunnels in weak soils and rocks; furthermore, it can be demonstrated (Lembo-Fazio and Ribacchi, 1984) that, for purely elastic behavior, the time-dependent process of pore pressure equalization does not affect the displacement of the tunnel wall nor the radial load acting on the tunnel support. Plane-strain numerical analyses were performed by the finite difference code FLAC (ITASCA, 2005), adopting a coupled solution scheme. A full axisymmetric model, which implies that gravity is disregarded, has been preferred in order to equate the numerical solutions as much as possible to the analytical solutions. The grid used represents a quarter of a tunnel, with the interval (a, rmax) of the calculation domain subdivided into 60 elements (Fig. 3). The main differences between the analytical and numerical approaches are to be found in the limited extent of the grid (rmax) and, as a consequence, in the type of boundary conditions assumed for r = rmax (in terms of imposed stress or displacement). All the numerical analyses are arranged into two main stages, corresponding respectively to the phase of undrained excavation and installation of the support system (short-term equilibrium) and to the subsequent time-dependent process of consolidation. The ground-support interaction has been represented by two different approaches: in the first one, the support acts as a rigid restraint applied to the gridpoints on the tunnel walls, in the second one, a deformable lining composed of beam elements is introduced. The ground is modeled as a two-phase medium, i.e., an elastoplastic porous skeleton saturated with a compressible fluid. Indicating with K0 and Kw, the drained bulk modulus of the solid skeleton and the bulk modulus of water respectively, the undrained bulk modulus for the fully-saturated medium is given by Ku = K0 + Kw/n, where n represents porosity. In the numerical analyses Kw = 2 GPa and n = 0.35 were assumed.

t = (σ1-σ3)/2 q =

σ'

3

σ'

3

=

0

6. Results

sh limit

B

t=c

A'=B'

st ear

'cos

re ng

ϕ'+s

th

'sinϕ

'

A

D

D' S'0

S0

s = (σ1+σ3)/2 s' = (σ'1+σ'3)/2

Fig. 7. Effective and total stress paths of a point around the tunnel in drained and undrained conditions for pervious (a) and impervious (b) lining.

6.1. Coupled versus uncoupled solutions The grid utilized for the first set of analyses extends up to a distance rmax equal to 50a (i.e., 122.5 m), where a constant radial pressure of 4.50 MPa is applied. The limit radius of the seepage zone r0 is equal to 20a, a distance which corresponds approximately to the maximum length of the drainage path in the Boom clay formation. In fact, the clay layer, confined between two more permeable sandy layers, is characterized by a thickness of 100–120 m and the tunnels are located almost in the middle (Fig. 2). Fig. 8 plots the GRCs calculated for undrained (t = 0) and drained conditions (t = 1), assuming a pervious as well as an impervious boundary at the tunnel wall. The short-term tunnel

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A. Graziani, D. Boldini / Tunnelling and Underground Space Technology 28 (2012) 70–79

1.5

(a) PERVIOUS TUNNEL

drained conditions ( t =∞) NUMERICAL SOLUTIONS:

(a) t = c'cosϕ'+s'sinϕ'

undrained conditions ( t =0)

4)

3)

2)

0.5

1)

drained conditions ( t =∞)

0 E'

2 E

0 D'

1

2

2.5

(b)

B'

1

t = c'cosϕ'+s'sinϕ'

C

A' t=∞

B t=0

0 0

1.5

impervious lining

C'

D

0.5

1.5

150

A 300

450

600

t (MPa)

support pressure q (MPa)

undrained conditions ( t =0)

3

1

t (MPa)

ANALYTICAL SOLUTIONS:

4

pervious lining r /a: 1) 1.00 2) 1.23 3) 1.65 4) 3.38

1

0.5

4)

tunnel wall displacement ua (mm)

1)

3)

2)

0

(b) IMPERVIOUS TUNNEL

0

0.5

1

1.5

2

2.5

s' (MPa) Fig. 9. Stress paths of some points at varying radial distance r/a around the tunnel calculated through coupled numerical analyses.

3 E'' D''

C''

B''

A''

q = p0 t=∞

2 E D 1 C B

t=0 A

0 0

150

300

450

600

tunnel wall displacement ua (mm) Fig. 8. GRCs of the Mol tunnel in undrained and drained conditions: pervious (a) and impervious (b) tunnel.

characteristic curves obtained with the analytical and numerical solutions agree well with each other. In the numerical approach, the long-term characteristic curves were obtained through a series of fully-coupled analyses. Each analysis starts from a different point on the undrained curve, corresponding to a variable degree of stress relief (points A–E in Fig. 8). For each point, tunnel convergence during the consolidation phase is fixed equal to the short-term undrained value, thus simulating the installation of a perfectly rigid lining. At the end of the consolidation analysis the numerical solution for a perfectly pervious lining (points A0 –E0 ) gives a much larger value of the lining load than that obtained by the analytical uncoupled solution for drained conditions (Fig. 8a). On the contrary, the difference is negligible for the case of impervious lining (Fig. 8b). A possible reason for this discrepancy can be identified by observing the stress paths of points located at increasing distances from the tunnel axis (Fig. 9). In the case of impervious lining (Fig. 9b), the stress path of the point located at the tunnel wall reaches the limit strength curve and remains on it during the consolidation phase. Conversely, in the pervious case (Fig. 9a), the stress-path of the same point is characterized by an elastic rebound during the final steps of the consolidation phase. The occurrence of an elastic rebound is a-priori neglected in the uncoupled calcula-

tion, which, therefore, can be considered satisfactory only for the case of impervious lining. More generally, it can be observed that the case of pervious tunnel belongs to the category of elasto-plastic problems whose solution is markedly stress-path dependent. Thus, a realistic solution must resemble the effective physical process, i.e., for the problem at hand, a contemporary variation in pore pressure and ground deformation. In contrast, it was found that an uncoupled approach can lead to unrealistic, and in this case unsafe, results, unless in the very special case of fully drainage ahead of the tunnel face (Ramoni and Anagnostou, 2011). The previous finding is of general validity; in fact, it has been confirmed by various analyses performed by the Authors for different p0/S0 ratios as well as different shear strength parameters (see also, Graziani and Ribacchi, 2001). As already pointed out in Section 4.2, two possible situations may take place (Fig. 8b): namely, the extrados of the lining can remain in contact or separate from the ground if the tensile strength of the soil is reached. Only in the former case, all the points within

3.0 E''

2.5

support pressure q (MPa)

support pressure q (MPa)

4

D''

p0

A'', B'', C''

2.0

E

E'

D

1.5

D' C'

1.0 C

B' A'

B 0.5

impervious lining pervious lining

A 0.0 1

10

100

1000

10000

time t (30-day months) Fig. 10. Time histories of the lining load for impervious and pervious linings (the letters refer to the vertical paths in Fig. 8, corresponding to the consolidation phase).

A. Graziani, D. Boldini / Tunnelling and Underground Space Technology 28 (2012) 70–79

the plastic zone remain on the yield surface, otherwise, as in the example of Fig. 8b, an elastic rebound occurs also in the case of impervious lining (Fig. 9b). Therefore, for all points from A00 to D00 , the load on the lining is simply equal to the undisturbed pore pressure (q = p0). Fig. 10 shows the time evolution of the lining load for points A to E of Fig. 8. As already noticed with respect to the GRC for impervious lining (Fig. 8b), consolidation analyses starting from points A to D reach the same final load, corresponding to the in situ pore pressure p0. The time required to reach practically stationary load conditions is one order of magnitude higher in the impervious case than in the pervious case (i.e., the time is approximately equal to 100 and 10 years, respectively).

6.2. Influence of far field boundary conditions The GRCs of Fig. 8 have highlighted the influence of lining permeability and hydro-mechanical coupling on long-term convergence. In this section, the attention is turned back to the shortterm stage of tunnel response, with the aim of analyzing the influence of far field boundary conditions, whose importance is sometimes overlooked. In fact, the prediction of tunnel convergence and of the support load in the short-term turned out to be strongly affected also by a number of factors, such as the limited extent of the grid (rmax), the possible presence of an external annulus (r0 < r < rmax) in permanent drained conditions, and the type of boundary conditions applied at r = rmax. To investigate these issues, several undrained analyses were performed by varying the limit radii r0 and rmax, and type of external boundary conditions. In general, two different boundary conditions, in terms of fixed stress or displacement, can be imposed at the external artificial limit of the mesh. In the case of ‘‘stress boundary conditions’’ the radial stress is set equal to far field stress S0, while in the case of ‘‘displacement boundary conditions’’ the external grid-points are fixed. The GRCs obtained for ‘‘stress boundary conditions’’, by varying r0 and rmax in the range 20a–50a, are almost coincident and well approximate the analytical solution. For sake of simplicity, in Fig. 11 only the case with r0 = 20a and rmax = 50a, already considered in the analyses of Figs. 8–10, is represented. On the contrary, the GRCs obtained with ‘‘displacement boundary conditions’’ show a remarkably stiffer response, even if a very large extent of the calculation domain is adopted, unless an external annulus (from r0 to rmax) of the calculation domain is maintained under drained condi-

analytical solution

4

1) r0 = 20 ⋅a, rmax = 50 ⋅a

support pressure q (MPa)

2) r0 = 20 ⋅a, rmax = 20 ⋅a

stress boundary conditions

3) r0 = 30 ⋅a, rmax = 30 ⋅a 4) r0 = 30 ⋅a, rmax = 30 ⋅a

3

5) r0 = 20 ⋅a, rmax = 20 ⋅a

displacement boundary conditions

6) r0 = 20 ⋅a, rmax = 30 ⋅a

2

support reaction curve (Kl = 90 MPa/m)

1

0 0

100

200

300

400

tunnel wall displacement ua (mm) Fig. 11. Short-term GRCs for the tunnel calculated by numerical analyses with different boundary conditions and characteristic curve of the lining (SRC).

77

tions (case 6 in Fig. 11). In fact, such drained annulus, as a whole, acts as a ‘‘deformable boundary’’ for the internal part of the domain and, therefore, makes the displacement of the tunnel wall much more similar to that obtained by adopting stress boundary conditions. An explanation of the previous results can be found by observing the short-term distribution of pore pressure p around the tunnel (Fig. 12a). The analyses with displacement boundary conditions and without the drained outer annulus are characterized by a significant decrease in pore pressure Dp, a consequence of the ‘‘incompressibility’’ of the porous medium in undrained conditions. The magnitude of the Dp decrease, which is uniform in the entire external elastic domain (for r > R), is strongly dependent on the extent rmax of the grid (Fig. 12a) and, therefore, can be viewed as an effect of the artificial boundaries of the model. However, a careful analysis of the in situ measurements of pore pressure suggests further considerations. Short-term measurements around the test drift at the time of installing the lining (year 1987) and 56 days later (Mair et al., 1992) are reported in Fig. 12a while Fig. 12b and c show the pore pressure recorded 3 ad 17 years after excavation by piezometres installed in horizontal and downward boreholes (Bernier et al., 2007). The sharp bend in the short-term profile of measured pore pressure is remarkably coincident with the plastic radius calculated for undrained conditions (Fig. 12a). So, for r > R, the measurements exhibit a constant threshold, as predicted by the analytical solution (Eq. (4)), but such threshold is significantly lower than the undisturbed pore pressure (p0), contrary to Eq. (4). Therefore, the measured pore pressure profile appears to be, quite surprisingly, in much better agreement with the results of numerical models with fixed displacement boundary conditions (and without the drained external annulus). At some sections of the experimental tunnel, short-term variations in pore pressure were measured up to 70 m distance from the tunnel (Mair et al., 1992). This represents further evidence that the influence zone of excavation-induced excess pore pressure can be much larger than the plastic zone. A possible explanation of the seemingly better prediction of models with fixed displacement boundary conditions can be found in the ‘‘confinement’’ effect provided by the sandy layers (Fig. 2) which bound the Boom clay deposit and may represent a kind of ‘‘rigid’’ restraint for the clay layer. However, since the mechanical properties of these sandy layers are not well known, the previous explanation is reasonable but only speculative. A similar situation can occur in fault zones, where fine-grained cataclastic rocks are confined between much stiffer rock masses (e.g., Vogelhuber et al., 2004). Finally, the calculation of short-term equilibrium (intersection point of GRC and SRC) requires, as customary in the GRC method, the partial relief of initial stress at the phase of lining installation be evaluated. A relaxation factor equal to 72% was back-calculated in order to obtain a tunnel convergence at the time of lining installation equal to ua = 90 mm, namely the value indicated by measurements. The SRC (Fig. 11) is characterized by a stiffness Kl = 90 MPa/m (see Section 3). The general finding of the previous analyses is that the shortterm equilibrium load (Fig. 11) and pore pressure distribution (Fig. 12a) can differ dramatically depending on the type of boundary conditions which can be reasonably hypothesized at the far boundary of the calculation domain. 6.3. Time history of pore pressure and lining load Fig. 12b and c show the pore pressure distribution calculated for increasing consolidation time (t = 3.2 years and t = 1). There is a reasonable agreement with in situ measurements: for the interme-

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1) r0 = 20⋅ a, rmax = 50⋅a, pa = 0 MPa 4) r0 = 30⋅a, rmax = 30⋅a, pa = 0 MPa

pore-water pressure p/p0

5) r0 = 20⋅a, rmax = 20⋅a, pa = 0 MPa plastic radius R

t =0

stress boundary conditions displacement boundary conditions

Calculated at t ≅ 3.2 years:

t =∞

1

0.5 Short-term measurements: lining installed (1987)

0

Successive measurements:

56 days later

(a)

Successive measurements:

horizontal 1990

minimum pressure measured

(b)

horizontal 2004

(c)

downward 1990

downward 2004

-0.5 1

3

5

7

9

1

3

5

7

9

distance r /a (m)

distance r /a (m)

1

3

5

7

9

distance r /a (m)

Fig. 12. Comparison between pore pressure profiles measured around the test-drift tunnel in the short- (Mair et al., 1992, modified) and long-term (Bernier et al., 2007, modified) and pore pressure profiles calculated for the pervious tunnel.

diate situation (t = 3.2 years), the calculation case with fixed displacements as external boundary conditions seems to give the pore pressure profile closest to measurements. In addition, pore pressure measurements near the tunnel wall appear to be systematically lower than the calculated pore pressure profiles. This fact can be tentatively explained as a consequence of the presence of a disturbed, and possibly fissured zone around the tunnel wall characterized by higher permeability and/or partial saturation. Experimental evidence of the presence of a higher permeability annulus around the tunnel is reported by several authors, e.g., Bauer et al. (2003) and Wongsaroj et al. (2007). Simple approaches that account for the effect of fissure opening are dealt with in a companion paper (Boldini and Graziani, 2011). Fig. 13 shows a comparison between measured and computed lining loads. Measurements were performed for 4 years with load cells cast in the lining ring. The load time-histories indicate a rapid build up, characterized by a logarithmic law, from the first months after installation. Analyses with stress boundary conditions give a better prediction of the short-term undrained load (conventionally

1) r0 = 20⋅a, rmax = 50⋅a 4) r0 = 30⋅a, rmax = 30⋅a 5) r0 = 20⋅a, rmax = 20⋅a

stress boundary conditions displacements boundary conditions

2

1.6

q (MPa)

Symbols indicate in situ measurements

1.2

0.8

0.4

0 0.1

1

10

100

t (30-day months) Fig. 13. Comparison between the lining load measured at increasing times (Mair et al., 1992, modified) and the time histories of load calculated through different numerical analyses (30 days load is conventionally assumed as the undrained load condition).

assumed as average load at 1-month after lining installation), but the calculated increase in load due to consolidation is negligible and much lower than the measured value. The assumption of a perfectly pervious lining, initially believed to be more representative of the real segmental lining, highly underestimates the lining loads. A progressive increase in pore pressure at the tunnel wall, as a consequence of the long-term reduction in lining permeability, would determine higher loads, as demonstrated by Fig. 10, but such hypothesis is not supported by the pore pressure profiles measured for longer time periods. In the end, the previous discussion suggests that other phenomena, such as viscous deformation and/or progressive softening of the soil skeleton, not accounted in the analyses described in this paper, may be responsible for most of the observed time-dependent increase in load. 7. Conclusions For sake of simplicity, a preliminary crude distinction between plastic soil-like and low-porosity rock-like clay formation was made. The questions covered in this paper deal with the first situation, typically modeled as a saturated porous medium. Fissure opening and desaturation effects, more typical of the second situation, are dealt with in a companion paper. The questions formulated at the beginning of the paper summarize some issues of common interest in the analysis and design of deep tunnels under the water table in plastic clay formations. The first question concerns the application of the conventional uncoupled approach for the analysis of long-term ground-lining interaction. The numerical analyses, performed for the tunnels excavated in the Boom clay formation, have shown that this approach is satisfactory and leads to reliable results in terms of support loads only for the particular case of impervious linings. In general, the coupled hydro-mechanical approach (i.e., ‘‘consolidation’’ analysis) is needed in order to correctly evaluate long-term loads, since the predictions afforded by the conventional analysis are dangerously unsafe. Moreover, analysis of ground-support interaction for lining subjected to external water pressures must take into account also the possibility that lining and ground separates and a gap opens. The previous findings also entail that the popular GRC method can be applied only for impervious tunnels – a rather rare situation for deep tunnels under high hydraulic heads (practically, greater than 80–100 m), apart from particular cases such as headrace tunnels for hydroelectric plants.

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The second question deals with topics of more specific interest for numerical modeling. The extent of the calculation grid, as well as the type of boundary conditions, proved to have a marked influence on the short-term response of the tunnel. In particular, boundary conditions in terms of fixed displacement at the far boundary of the calculation domain may induce a larger decrease in pore pressure not only inside the plastic annulus but also in the outer elastic region, in contrast to the prediction of models with stress boundary conditions. The larger drop in pore pressure results in a much stiffer response of the ground and, eventually, in a lower equilibrium load on the lining. A lesson learned from the previous calculations is that fixed displacement boundary conditions should be generally avoided for undrained analyses, unless an outer annulus in drained conditions is present or unless the ‘‘artificial’’ boundaries of the model match ‘‘real’’ geological limits, such as bedrock and stiffer strata. In the case of the Boom clay tunnels, a tentative explanation of the seemingly better prediction of models with displacement boundary conditions could be tentatively found in the geologic formations which bound the clay deposit and which may act as ‘‘rigid’’ restraints. References AFTES, 1983. Recommandations sur l’emploi de la méthode convergenceconfinement. Tunnels et Ouvrages Souterrains 59, 206–222. Bauer, C., Piguet, J.P., Wileveau, Y., 2003. Disturbance assessment at the wall of a vertical blasted shaft in marls. In: Proceedings of the 10th ISRM Congress, Johannesburg, South Africa, pp. 93–98. Bernier, F., Li, X.L., Bastiaens, W., 2007. Twenty-five years’ geotechnical observations and testing in the Tertiary Boom Clay formation. Géotechnique 57 (2), 229–237. Bobet, A., 2003. Effect of pore water pressure on tunnel support during static and seismic loading. Tunnelling and Underground Space Technology 18, 377–393. Boidy, E., Bouvard, A., Pellet, F., 2002. Back analysis of time-dependent behaviour of a test gallery in claystone. Tunnelling and Underground Space Technology 17, 415–424. Boldini, D., Graziani, A., 2011. Remarks on axisymmetric modeling of deep tunnels in argillaceous formations. II: Fissured argillites. Tunnelling and Underground Space Technology 28, 80–89. Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E., 1983. Ground response curve for rock tunnels. Journal of Geotechnical Engineering 109 (1), 15–39. Carranza-Torres, C., Zhao, J., 2009. Analytical and numerical study of the effect of water pressure on the mechanical response of cylindrical lined tunnels in elastic and elasto-plastic porous media. International Journal of Rock Mechanics and Mining Sciences 46, 531–547. Carter, J.P., Booker, J.R., 1982. Elastic consolidation around a deep circular tunnel. International Journal of Solids and Structures 18 (12), 1059–1074. Comité Français de Méchanique des Roches, 2000. Manuel de méchanique des roches. Tome 1: Fondaments. Roches Argileuses, Les Presses de l’Ecole des Mines, Paris, France, pp. 22–253. Cui, Y.J., Le, T.T., Tang, A.M., Delage, P., Li, X.L., 2009. Investigating the timedependent behaviour of Boom clay under thermomechanical loading. Géotechnique 59 (4), 319–329. Delage, P., Le, T.T., Tang, A.M., Cui, Y.J., Li, X.L., 2007. Suction effects in deep Boom Clay block samples. Géotechnique 57 (2), 239–244. Detournay, E., Cheng, A.H., 1988. Poroelastic response of a borehole in a nohydrostatic stress field. International Journal of Rock Mechanics and Mining Sciences 25 (3), 171–182. Einstein, H.H., 2000. Tunnels in Opalinus Calyshale. A review of case histories and new developments. Tunnelling and Underground Space Technology 15, 13–29. Erguler, Z.A., Ulusay, R., 2009. Water-induced variations in mechanical properties of clay-bearing rocks. International Journal of Rock Mechanics and Mining Sciences 46, 355–370. Gäber, R., Labiouse, V., 2003. Influence of pore water on the design of deep galleries in low permeability rocks. In: Proceedings of 10th ISRM Congress, Johannesburg, South Africa, pp. 347–350.

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