continuum and discontinuum modelling in tunnel engineering

Rudarsko-geolaqko-naftni zbornikVol. 12str. 45-57Zagreb, 2000.1CONTINUUM AND DISCONTINUUM MODELLINGIN TUNNEL ENGINEERINGGiovanni BARLA, Marco BARLADepf. of'Struclurul und Geotechnical Engineering Politecnico di Torino, Corso Ducu deglr Abruur 24, Tonno, ItalyE-mail: gburlu @polroc.poli~o. itKey-words: Tunnel engineering, Continuum and discontinuum modelmg,TBM tunnelThc paper discusses continuum and discontinuum modelling intunnel engineering. A bricf rcview of fundamentals is presented inconnection with the use of closed - formsolutions and computer bascdmethods. A few remarks are derived on the choice of eithercontinuum or discontinuum modelling of rock mass behaviour at thedesign analysis stage. Consideration is given to the validation of discontinuummodelling in connection with rock mass classification methodsand cxpected tunnel response to excavation. A wse study of aTBM tunnel (4.75 m diameter) in quartzitic micaschists is discussed indetail by paying attention to a comparison of modelling methods -including continuum and discontinuum modeling - applied at a faultzone.1. IntroductionThe prediction of the rock mass response to the excavationof a tunnel is a complex engineering problem. Theinterest of the Rock Engineer at the design stage is to assessthe stability conditions of the excavation in the "intrinsicstate" (i.e. when no support/stabilization measures are installed)and following the adoption of suitable methods oftunnel excavation/construction and support. The key to thesuccess of such a process is the level of understandingachieved in describing the rock mass conditions (in termsof geological, geotechnical, in situ stress, and hydrogeologicalparameters) and the ability to account for the fundamentalcomponents of rock mass behaviour, by usingappropriate methods for the analysis of stresses and displacementsin the rock mass around the tunnel and in thestructural components (pre-supportlpre-stabilizationmeasures; primary and final support, etc.).Anumber of methods are available for the stress analysisof tunnels, from the earliest closed-form solutions to themost recent numerical modelling methods. With the computationalpower today available at a reasonable cost, it ispossible to solve increasingly sophisticated problems. Inparticular, with the advent of numerical methods, we haveassisted to the development of techniques which have beenconceived to model realistically the rock mass behaviour.This is a quite different condition from the early start ofRock Mechanics, when the methods of analysis and thesolutions used were mostly taken from other engineeringdisciplines. This is certainly the case of the 1898 Kirschsolution for the stresses and displacements around a circularhole in a biaxially loaded, homogeneous, isotropic andlinearlv elastic date.Closed-for6 solutions are still of great value for aconceptual understanding of the response of tunnels toexcavation. One could mention in this regard the solutlonswhich are presently available for the analysis of theprogressive development of failure around a circulartunnel in a hydrostatic stress field see for example:Brown et al., 1983; Panet, 1 4 95), and for theanalysis of the interaction between the rock mass and thesupport. However, the development of modern tech-* Scientific Sym osium Rock Mechanic; and Tunnelling, 30.9.-2.10. 1999. Zagreb, RdatiaKljufne rijeti: Tunelsko inicnjerstvo, Modelirane kontinuteta idiskontinuitcta, TBM tunelU Elanku je raspravljeno modeliranjc kontinuiteta i diskontinuitetau tunelskom inienjerstvu. Saieto su opisane osnove zajedno s uporabommatematizkih rjeienja i raEunalnih numeriEkih metoda. Raspravljenje izbor modela kontinuurna i diskontinuuma stjenske maseu odnosu na analizu konstrukcije podgradivanja. U obzir je uzetopohrdivanje modela diskontinuiteta povezanog s metodama Masifikacijestjenskih masa i dkivane reakcije tunela pri iskopu. U detaljimaje raspravljen primjer studije TBM tunela (promjera 4,75 m) u kvarcititnimmikaSistirna, s posebnom pozornosti na usporedivanje metodamodelirania - ukliutuiuCi modeliranie kontinuuma i diskontinuurna- uporabfienih u iasjednoj zoni.niques of tunnel excavation and construction (i.e. thecase of pre-supportlpre-stabilization measures, the frequentadoption of pre-treatment ahead of the headingin weak rock masses, the excavation sequences, typicalof large tunnels), ... and even the complexity of rockmassconditions and behaviour, which are better describedtoday than in the past, given the modern investigationtools available, etc. make these solutions of limited valuefor design purposes.If we restrict our attention to modern numerical modellingmethods and rigorous analysis of tunnelling problems,the decision that the Rock Engineer need to makeat the design analysis stage is the choice between thefollowing two approaches:- Equivalent continuum approach: the rock mass istreated as a continuum with equal in all directionsinput data for the strength and deformability properties,which define a given constitutive relation for themedium: elastic, elasto-plastic, etc.- Discontinuum approach: the rock mass is representedas a discontinuum and most of the attention at thedesign stage is devoted to the characterization of therock elements, the rock joints and discontinuities. Themodelling approach consists in considering the blockynature of the system being analysed. Each block mayinteract with the neighbouring blocks through thejoints. The interest lies in the fact that the fundamentalpatterns of rock mass behaviour can be considered,as arbitrarily large relative displacements may takeplace at the contacts.This lecture is intended to discuss a comparison ofmodelling methods - including continuum and discontinuummodelling - applied at a fault zone in a TBMtunnel. Reference is made to a paper recently presentedat the 9'h International Congress on Rock Mechanics inParis(Bar1a & et al., 1999).2. Fundamentals2.1. Continuum versus discontinuum modelling2.1.1. Continuum modellingThe use of continuum modelling in tunnel engineeringmakes it essential to simulate the rock mass response to

Rud.-geol.-naft. zb., Vol. 12, Zagreb, 2000.46 Uurfu, G., Burfu M.: Continuum and discontinuum modellingexcavation by introducing an equivalent continuum. Themost common way to solve this problem, which seems toINTACT ROCKhave gained wide acceptance, is to scale the intact rockproperties down to the rock mass properties by usingempirically defined relationships such as those given byHoek and Brown (1997).StrengthIf use is made of the Hoek-Brown criterion for describingthe rock mass behaviour, the starting point ofFs = stressthe scaling process is the definition of the intact rockmaterial parameters such as oci (uniaxial compressivestrength) and mi (material constant which depends uponthe properties of the rock), which can be obtained based,upon the results of uniaxial and triaxial laboratory testing.Then, by using well known correlations (which dependon the degree of disturbance to the rock which willvary according to rock type and excavation method) withIthe most frequently adopted rock mass indices (i.e. trun-Icated Q or RMR values, or GSI values), the rock mass >parameters such as mb and sb (rock mass constants accordingto the Hoek-Brown criterion) or c and $ (rock 't G3mass cohesion and friction angle respectively) can beestimated (Figure 1).The next ste in the continuum modelling approach isthe adoption o i' the appropriate constitutive relations for Figm I. Hoek-Brown failure criterionr for the ihtacr rock material andthe rock mass. Linear elasticity is commonly used, al-the rock mass. Ifhrtration of the scalingprocess and dejkittionof Fuctor of Safetythounh nonlinear elasfic constitutive eauations can alsobe adYopted. If the attention is posed or; the progressivefailure of the rock mass, the elasto-plastic models needinterior of the rock mass is represented mathematicallyto be utilized in order to describe the "post-peak" reasan infinite continuum. The domain methods, whichsponse. A yield function which is often chosen coincidesinclude the finite element (FEM) and finite differencewith the Hoek-Brown failure criterion: (i) the rock massmethods (FDM , imply that a physical problem is modresponseis elastic, if the state of stress is within theelled numerical 1' y by discretizing (i.e. dividing into zonesbounds defined by the yield function; (ii) the rock massor elements) the problem region, i.e. the rock mass inresponse is plastic, once the state of stress is such as towhich the excavation is to be created.reach the yield function. A number of "post-peak responsesmay be postulated depending upon rock massquality; the characteristics shown in Figure 2 are amongthe most frequently used for the solution of tunnellingproblems(Hoek and Brown, 1997):(a) Very poor quality rock mass: the rock mass behaviouris adequatelyrepresented by assuming that it behavesperfectly plastic, which means that deformation continuesat a constant stress level and that no volumechange is associated with the ongoing failure.(b) Average quality rock mass: it is reasonable to assumethat a strain-softening behaviour occurs as the rockmass strength is reduced from the in situ to thebroken state; then, once this final, "residual" state isreached, deformation will occur at a constant stresslevel.(c) Very good quality hard rock mass: in such a case(hard rock masses, such as massive granites andgneiss) it is assumed that, when the rock massstrength is exceeded, a sudden strength dro occurs.This is associated with significant dilation o ? the rockmass, which is considered to behave as amedium withzero cohesive strength and finite friction angle.As summarized by Ho e k e t a 1. (1991) - also seeStartfield and Cundal (1988) - a number ofcomputer-based numerical methods have been developedover the past few decades and these provide themeans for obtaining appropriate solutons to tunnel engineeringproblems in the framework of the equivalentcontinuum approach. These numerical methods can bedivided into two classes: boundary and domain methods.The boundary methods comprise several types of boundaryelement methods (BEM) and imply the subdivisionof the boundary of the excavation into elements, as the2.1.2. Discontinuurn modellingWith the understanding that rock joints and discontinuitiesin a rock mass play a key role in the response ofa tunnel to excavation, i.e. joints can create loose blocksnear the tunnel profile and cause local instability; jointsweaken the rock and enlarge the displacement zonecaused by excavation; joints change the water flow systemin the vicinity of the excavation (see, for example:(Shen and Barton, 1997), theuseofdiscontinuummodelling has been gaining progressive attention in tunnelengineering mainly through the use of the UDECand 3DEC codes, for 2D and 3D discontinuum modellingrespectively. However discontinuum modelling isnot being used as extensively as continuum methods andis considered to be a relatively new and "not-yet-proven"numerical technique to ap ly for analysis and design inrock engineering projects art , 1993).In the distinct element method, the rock mass is representedas an assemblage of discrete blocks which maybe considered either "not deformable" or "deformable".Joints and discontinuities are viewed as interfaces betweendistinct bodies. The important features of thismethod which make it appropriate in order to capturethe important mechanisms characterising a discontinuousmediumare(Cundal1and Hart, 1993): (i) themethod allows finite displacements detachment; (ii) themethod recognizes new contacts automatically as thecalculation progresses.In order to apply the distinct element method to thesolution of tunnel problems, there are two crucial issueswhich include the joint geometry data and the materialproperties assigned to the joints. The first issue relatesto the introduction in the model of those joints which are

~ud.-geohaft. zb., Vol. 12, Zagreb, 2000.Barfa, G., Barla M.: Continuum and discontinuurn modelling 47dc - 0'sAo't - o'r/EUS~K:AQ'I - dr/'ELASTICW N SOFTENIMPERFECTLY BR~LE> > >9 €a E4( a J w t y p 0 0 r ~ l W ~ d ~ 0)- -&mass (c) wty good quality had rod^ massFigure 2. Suggested streass-strain laws for different quulily rock musses, note that the stress scules ure different (Hoek and Brown, 1997, modified)most critical to the response of the rock mass. Thesecond issue is closely connected with the need to assignto the joints in the model the stiffness and strengthproperties of the real joints in situ. Although publisheddata are readily available in the specialised literaturewhich may provide a way to guide into the choice ofappropriate parameters (Bandis, 1993), this aspect ofmodelling still remains difficult. However, it is to beremarked that a significant help in deriving the necessaryinput data for 2D and 3D discontinuum modelling maybe obtained from index testing of joints in drill cores andfrom field mapping (B a r t o n , 1998 and 1999).2.1.3. DiscussionWhen investigating a parficular problem at the designanalysis stage the decision is to choose between continuumand discontinuum modelling of rock mass behaviour.This decision may be based on the analysis of thelikely mechanism (sliding along joints, opening of joints,Q - 0.10:twIFEMlFDY 1 DEM FEWBEMFigure 3. Schemulic diugam ,sugge.~ling the runge of application of discontinuummodelling in relation to the Q-value (Bart on,1998, modified)block rotation and movement, etc.) which may influencetunnel stability and the joint spacing relative to the sizeof the excavation. Consideration may be given to a suggestedrange of Q-values for which discontinuum modellingwill be more appropriate than continuum modelling(Q = 0.1-100) as depicted in Figure 3 (Barton, 1998).Based upon experience gained so far and a closescrutiny of design analyses which are sometimes used in*Thc senior author dcrivcs thcsc observations through cxpcricncegained in reviewing a nurnbcr of dcsign analyses carricd out in Italyand abroad, in conncction with hydroclcctric, highways and high-spccdrailways projects.tunnel engineering (*), the following observations can bederived:- Isotropic continuum modelling is being used morefrequently than discontinuum modelling, even incases where the block size is significant, compared tothe size of excavation or the rock mass exhibits astrong anisotropic behaviour.- Numerical modelling is being used too often, even incases where problems are characterised by low levelsof both input data and understanding (it is appropriateto recall the classification of modelling problemsreportedby Startfield and Cundall, 1988,whorecognize this to be the case of most rock mechanicsproblems), or the size of the planned excavation issuch as not to justify this exercise.- It appears that the easy access to computer codes,which make sophisticated methods of analysispromptly available, may result in misuse of thesemethods. Sometimes, the tendency is to be happywithhaving run a model, even if the results obtained are inopen contradiction with empirical design rules andcngineering judgement.2.2. Validation of discontinuum modellingThe use of numerical modelling in engineering practicc,connected with the need to adopt modellingschcmes (continuum versus discontinuum modelling)which are the most appropriate in order to analyse agivcn problem (provided that sufficient data are availahlc),points out that "the modelling of the components,rock, rock joints and discontinuities is far more logicaland relevant than present "black box" continuum models"(B art o n, 1999). The comparison shown in Figure4 well demonstrates this uoint of view, exueciallv if criticalmechanisms of the p6ysical problem Cnder study areto be included in the analysis.There is a relevant question which arises with referenceto the use of discontinuum modelling in tunnelengineering and rock engineering, in general. This is inrelation to the complexity of the discontinuum model tobe used in order to be certain that the critical structuralfeatures have not been left out of the analysis (Hart,1993). From one side, the difficulty is in providing sufficientgeological data; from the other side, the computerhardware requirements may exceed that available.Based on the experience gained so far, a possibleapproach to this problem consists in using discontinuum

Rud.-geol.-naft. zb., Vol. 12, Zagrcb, 2000.48 Bark, G., Barla M.: Continuum and discontinuum modellingFigwe 5. Estimated supporf cuiegork~ hed yon the tunnelling qutalhtyindex Q (Grimstad and Barton, 199 )- Tunnel deformation (B a r t o n , 1998)Vertical:Horizontal: Ah =where: %and ah (the vertical and horizontal in situ stresscomponents), crc [the uniaxial compressive strength ofthe intact rock material are given in consistent units (i.e.MPa); SPAN and HE1 b HT (the width and height of thetunnel) are given in mm.4. Comm~aon of la) continuum ubiuuitous iodc case and A)di~cdntinuum ;n'delling results wheh analysing typical instabiiitymechanisms around a TBMexcavated lunnel in a weak rockmassmodelIing in connection with rock mass classificationmethods in a framework of crossvalidation of the expectedtunnel response to excavation Once the specificmodel has been proven to be awe table in a given rockmass condition, it may be improve a and furthervalidatedduring tunnel excavation. A set of guidelines, given as afunction of Q values, can be used as a form of preliminaryverification of a numerical model (B art o n , 1999).-Support categoriesBased upon analyses of case records, Grim st a d andBar ton (1993) give the diagram of Figure 5 whichallows one to relate the value of the index Q to thestability and support requirements of underground excavation,once the parameter which they called EquivalentDimension is obtained. This parameter is the span,diameter or wall height of the same excavation dividedby a numerical coefficient which is intended to accountfor its use and the degree of security which is demandedof the support system installed to maintain the stabilityof the excavation.Additional guidelines are available based on the Qsystem which allow one to assess a number of additionalparameters dealing with tunnel stability and supportrequirements (Bart o n et al., 1974): a) the mmmumunsupported span, b) the permanent roof support pressure;c) the bolt length.2.2.1. Case ExamplesWith the case history in mind, which is to be discussedin the following, a 4.75 m diameter tunnel was taken asa typical problem to be used for validation of the discon-Figure 6. Four UDEC d e b wirh difierent Qvalucr: model 1-Q=8.5;made1 2 - Q=4.l; madel 3 - Q=1.9; model 4 - Q=0.67

~~d.-pol.-naft. zb., Vol. 12, Zagreb, WOO.Barla, G., Bark M.: Continuum and discontinuurn modellingtinuurn modelling by the 2D Distinct Element codeUDEC (I t a s c a, 1996). The intent has been to investigatethe influence of rock mass conditions by the Qindex, on the disturbed zone around the tunnel.Four models have been used as shown in Figure 6.With dimensions 20x20 m, three sets of persistent discontinuities(S, = schistosity, J1, J2 joints) have beenintroduced with the parameters assigned so as to obtaina range of Q values from 8.5 for model 1 to 0.67for model4. For all the models the initial state of stress is assumedto be given by: ov (vertical stress) =16.2 MPa, oh (horizontalstress) = 4 MPa, which represent a gravity inducedstress state at a depth of about 600 m, with anassumed stress ratio (ado,) of 0.25. The joint spacing foreach system shown in Figure 6 is as follows:ModelSchistosityJOINT SPACING (m)For all the models the rock blocks are deformableblocks with the assumption that the intact rock (gneiss)is considered as an elasto-plastic material which followsthe Mohr-Coulomb yield criterion. The properties areassigned as follows:Young's modulus E =60 GPaPoisson's ratio v=0.25Cohesionc= 30 MPaFriction angle $=33"The discontinuities are assumed to be Mohr-Coulombjoints, i.e. elasto-perfectly plastic joints, with normal K,,and shear K, stiffness given as follows: K,, = 40 GPaIm,K, = 4 GPa/m. The cohesion is always considered to bezero, with the friction angle $ taken as 58", 38", 22' and11' respectively for models 1, 2, 3 and 4, for both theschistosity and joints.- Unsupported tunnelA first series of analyses was carried out in intrinsicconditions, i.e. the tunnel is always left unsupported,with the disturbed zone defined as follows (S he n a n dBarton, 1997):- failure zone, where loose rock blocks are falling intothe tunnel;- open zone, where joints open up;- shear zone, where joints experience a certain sheardisplacement (3 mm).The results obtained for the 4 models, are illustratedin Figures 7 to 9, where the failure, open and shear zonesaround the tunnel are depicted. The following remarksare made: (1) the failure zones computed around thetunnel show that model 1 is stable, whereas models 2 to4 exhibit progressively critical conditions; (2) the openand shear zones appear to grow as the rock mass qualitydecreases from model 1 to 4, in line with the behaviourin terms of failure zones, (3) the results obtained seemto agree with the estimated support categories shown inFigure 5, where only the tunnel simulated with model 1would require no support, and models 2 to 4 would needdifferent and progressively more severe support measures.Figure 8. Oprm zones around the tunnel fbr models I to 4Figure 9. Shear zones around the lunnel for model9 I to 4Figure 7. Yielded and loose blocks around fhe tunnel fw modelv 1 lo 4- Supported tunnelAs a direct consequence of the analyses describedabove and based upon the guidelines of Figure 5 in termsof support requirements, the following stabilizationmeasures were introduced in models 2 to 4.

Rud.-geol.-naft. zb., Vol. 12, Zagreb, 2000.50 Burlu, G., Burlu M.: Continuum and discontinuum modellingElFigure 10. Pont Ventoux - Clarea F2 tunnelModel Bolts n. Bolt lenght Bolt spacing(') Schortcrete thickness(')~olts are supposed to be installed according to a square patternThe bolts were simulated by using the CABLE optionavailable within the UDEC code, which allows one toconsider a bolt fully bonded to the rock mass. The shot-Crete was introduced in the model by the STRUCToption which consists in simulating it as a series of beamsconnected to the rock mass.As a first estimate of the predictive capability of theUDEC discontinuum modelling of the supported tunnelthe attention is posed on comparing the verfical andhorizontal displacements Av and Ah given by the empiricalQ based formulae reported above and the results ofUDEC computations:Model233. Case studyAv [mm]Ah [mm]estimated-computed estimated-computed4.5- 5.0 2.6- 2.69.2-10.2 4.7- 5.33.1. Background informationThe case study considers the problems occurred on 11September 1997, at ch 2360 m, when crossing a fault zonewhich caused the TBM to become temporarily stuckduring excavation of the F2 tunnel, one of the majorelement of the Pont Ventoux- Susa Hydropower Projectin the Susa Valley, near Torino. This tunnel (4.75 mdiameter) is being excavated by an open TBM configurationthrough uartzitic micaschists under a coverwhich is to reach 'B 00 m maximum (Figure 10).Following the first 1800 m approximately where therock mass conditions where generally good, with RMRvalues ranging from 65 to 75 (Figure ll), TBM tunnellingin the F2 tunnel became progressively more severewith the approaching of the fault zone. At present, thetunnel is experiencing very severe dificulties with consid-Figure 1 I. A view of the F2 tunnel in the section where the gneiss rockmuss was generally gooderable delays with respect to the expected advance rate.Figure 12 shows two photographs taken following chainage2360 m, where the fault of interest in this papercrossed the tunnel, affecting progress significantly.3.2. Description of the overstress problem at ch 2360 mThe sketch shown in Figure 13 gives a simplified illustrationof the conditions at the tunnel face where theTBM became temporarily stuck as a consequence ofoverstressing and a 25 cm block movement of the rightsidewall (Figure 14). The quartzitic micaschist is chara~terized by the presence of three to four joint systemsincluding foliation. Based on geologic mapping, whichwas carried out by the contractor's geologist (Pont Ventoux,1997) once the TBM could drill a few meters aheadof the section where it jammed (Figure 15), at least twosub-parallel discontinuities could be evidenced, the secondof which (a fault with stnke N66E and dip 83 to theS, which intercepts the tunnel axis) has a clay filling andgouge with aperture ranging from a few centimeters tomore than a decimeter.The rock mass conditions were estimated on a 7 rntunnel length, from ch 2349 to ch 2356 m, with RMRindex equal to 31. According to a more complete Q-loggingestimate due to Barton (1997), from ch 2350 to ch

Rud.-Seol.-naft. zb., Vol. 12, Zagreb, 2000.G., Barlu M.: Continuum and discontinuum modelling 51-ure 12. A view of flu F2 tunnel: (a) lef! (b) right walls, where stability conditions became very difiull so us to require continuos placement of liner" plalerFigure 13. Skelch of the rock corulitwns a1 rltr JitceFigure 15. Rock mass conditions on the tunnel (a) left (b) rig& walls. Thepholograph was taken when the TBM was moved aheadFigure 14. Phologruph showing the heau OJ TBM jammed wilh rockoverstressing and block movement of the right .stdewaII2360 m, an extreme range of Q-values of about 0.007("exceptionally poor" - locally) to 0.3 ("poor") showed aweighted mean of all recordings of about 0.05. Water isflowing through the joints at a temperature of 20'. Noquantitative data have been recorded of the water pressure,which seems unlikely to be exceeding, with anoverburden of 650 m, a maximum of 6 to 7 MPa (outsidethe tunnel drainage area).3.3. Rock mass strengthThe rock mass strength in the section of interest islargely controlled by the generally "poor" conditions andthe limited shear strength along discontinuities. Typicalstrength properties derived from triaxial testing of rocksampIes are as follows:

Rud.-gco1.-naft. zb., Vol. 12, Zagreb, 2000.52 Rarlu, G., Barla M.: Continuum and discontinuum modellingm., r[-I6.1where o~,~ and a o,, are the uniaxial compressivestrength values (peak and residual) and mipp and mi,, arethe corresponding Hoek-Brown empirical constants.For very good quality rock masses, which is the casefor a significant tunnel length (Figure ll), typical rockmass properties can be defined on the basis of the GeologicalStrength Index (GSI) as follows:Intact rock strengthoCi = 144 MPaHoek-Brown constant m, =7.0Geological Strength Index GSI = 70Hoek-Brown constant mb =2.4Hoek-Brown constant s =0.02Rock mass compressive strength o,,= 27.2 MPaDeformation modulus E, =35 GPa3.4. In situ stress conditionsIn view of the high overburden and closer valley side,associated with evidence of deformation and thin slabbingin isolated sections along the tunnel length, it wasrecommended that stress measurements by means of flat-jack tests and hydraulic minifracture tests be carriedout. At present only the flat-jack test data are available Gas follows, derived from measurements around the sec- -r=02 /tion of the tunnel at ch 950 m, where the overburden isapproximately 400 m:Figure 16. (a) Mmum princrpul .!tre.r\ contours. (6) Computed andmeusured tangential strrsse.s Fht-jack slot tests, wtth overbur-Angle with respect toJack presyureden of 400 m approximatelyFlat-jack slot horizontal axisPI[MPalvalues of oCi and s = 1. The stability of the tunnel wasMI 0 34.8 assessed by computation of the strength factor contoursas shown in Figure 17. The implication is that the rockM2 +50 3.3mass inside the unit strength factor contour (SF(m,ol = 1)M3 -45 23.5 will be unstable, unless well retained.Based on a stress concentration study in linearly elas-the above simplifiedtic conditions around the tunnel using the finite element linearly elastic costitutive model, no influence of hy-method and the phase2 code cur aband c ku *, drostatic head considered, etc.) the tunnel experi-1997), as shown in ~i~~~~ 1 , the initial stresses in the ences localized failure at the sidewalls which deepensplane perpendicular to the tunnel axis were evaluated to inside the rock mass as the micaschist intact rockbe as follows: uniaxial compressive strength decreases from 100MPa to 75 MPa. This would signify that in the highlyol = maximum principal stress =13 MPaanisotropic stress regime of the tunnel, as evidencedo3 = minimum principal stress = 2.6 MPawith a I?, value equal to 0.2, localized slabbing insta-O1 = angle of ol with respect to the vertical axis =15O bility cannot be ruled out, if the rock strength is in therange 100-75 MPa.3.5. Continuum modelling3.5.1. The m = 0 approachIn order to determine to what extent the adoption ofa continuum modelling approach can provide a reasonableinterpretation of tunnel instability, numerical analyseswere first carried out to assess the stability conditionsby using a constant deviatoric stress criterion as proposedby (M a r t i n et a 1. (1997).The initial stresses o, and 03 at the cross section ofinterest (ch 2360 m) were assumed to be proportionallyhigher in relation to the higher overburden of 650 m.Also, the stress ratio (q,) of the minimum stress tomaximum stress (03/01) in the plane of the tunnel crosssection was considered equal to 0.2, as in the section ofthe flat-jack measurements.The m = 0 analyses were performed by using thephases2 code and the Hoek-Brown criterion for different3.5.2. The elastic-brittle-plastic modelThe analyses above considered overstressing of therockmass around the tunnel periphery, without accountingfor the presence of discontinuities. Therefore, it wasdecided to analyse the same instability problem by introducingthese features in the numerical model.As shown in Figure 18, the model comprises twoparallel discontinuities near the right sidewall, dippingtoward the tunnel. In order to account for the rock massdisturbance due to jointing in the proximity of the discontinuitiesand the apparently more massive rock onthe left sidewall, where relatively little slabbing and overstressingwas present, it was decided to introduce threedifferent regions in the model with the following materialproperties, according to the Hoek-Brown failurecriterion:

~ud.-geohaft. zb., Vol. 12, Zagreb, 2000.Barla, G., Barla M.: Continuum and discontinuurn modelling 53Material properties Region (Fibwrc 18) Joint propcrtics Joint (Figurc 19)Em [GPalvm [-Ioci [MPalm, 1-12 17.5 12500.35 125150 0.080.74 01.00for: Gci = intact rock strength; mb , mbr, sb, Sr = Hoek-Brown constants; Em = rock mass cleformation modulus;Vm = Poisson's ratio.The rock mass surrounding the tunnel was specifiedas plastic and the elastic-brittle-plastic option of phase2was activated b specifying the constants for peakstrength (mb , S$ different than residual strength (mb,sr) only for tie more massive rock mass conditions. Thevalues were chosen on the basis of personal judgementwith the purpose to account for the better rock massconditions on the sidewall. The discontinuities were introducedin the model by using the joint option of phase2,with the following joint properties:for: K,, K, = normal and shear stiffness; h = joint0.001 aperture; c = joint cohesion; + = joint friction angle.Figure 18. Defuil ofthe FEM model near fhe funnel showing regions wifhdifferent material propetties ad joidv (refer to vulues in previousfables)Figure 19 shows the deepening of the failure zones inthe right and left walls as predicted by the elastic-brittleplasticmodel. The results of such an analysis is the moresignificant deepening of breakout on the right wall inagreement with observations in the tunnel where stressfailure and block movement actually occurred duringface advance, resulting in jamming of the TBM.It is conduded that the structural discontinuities nearthe tunnel govern to a significant extent the stabilityconditions. This is a very relevant aspect combined withthe high overburden and highly anisotropic stress regimein the resent situation, and eftect of water, which wasnot inc 7 uded in the numerical analysis.o,, = 75 MPaFigure 17. Strength factor conlours determined with the m = 0 upproachfor the funnel at ch 2360 m (on.=IOO, 75 MPu).3.5.3. Lessons learnedThe future geological and hydrogeological conditionsalong the tunnel alignment anticipate the presence of clayfilled discontinuities with unfavorable orientation as experiencedin the actual situation, with high hydrostatic head.Then, the lessons learned so far need to be taken intoaccount in order to carefully assess the interactive behaviourbetween the rock surround and TBM. In fact, thetunnelling conditions could become more problematic,as they involve already highly stressed rock masses, withthe overburden expected to rise up to 800 m maximum.With this in mind, also considering that the TBM isdriven along a curved axis, it is of interest to use thepredictive capability of the elastic-brittle-plastic modelto analyse two possible scenarios:1) the discontinuities run parallel to the tunnel axis, atthe crown;2) the discontinuities are at the left wall.

Rud.-geol.-naft. zb., Vol. 12, Zagreb, 2000.54 Burla, G., Burlu M.: Continuum and discontinuurn modellingtigated by adopting discontinuous models, with the 2Ddistinct element code, UDEC (I t a s c a, 1996). The rockmass surrounding the tunnel' was represented by twodiscontinuous models as follows.Figure 19. Computed ykldedwnes around the tunnel and dejormed meshusing* elustic-brittle-phstic coastitutive model3.6.1. Deterministic modelThe rock mass was considered to be intercepted bythree sets of joints, and foliation. Different spacing,degree of persistence and shear strength properties wereintroduced in the model so as to simulate the rock massconditions away from the fault zone as illustrated inFigure 21. The joints were again assumed to be Mohr-Coulomb joints, i.e. elasto-perfectly plastic joints. Theblocks were treated as an elasto-plastic material whichfollows Mohr-Coulomb criterion. The properties of rockblocks and joints are listed below:MaterialProperties(*)Zone (Figure 8 and 10)Em .tGPa]The analyses were carried out with the same assumptionsgiven above for the material and joint properties vm [-Iand the in situ state of stress. The results obtained aregiven in Figure 20 by showing the plot of the yielded c [MPalzones around the tunnel and the deformed mesh.4 r7(*> c = cohesion: &I = friction angle: E, = rock mass\ I ? 0 " , .adeformation modulus; v, = Poisson's ratio.JointProperties(*)1K,, [GPa/ml 40Ks [GPa/ml 4c [MPa] 0.1+ r"l 33JointZone (Figure 8 and 10) (Figure 8and 10)(*) K,,, K, = normal and shear stiffness; c = jointcohesion; $ = joint friction angle.Figure 20. Compufedyieldedzone.s uround the tunnel and deformed meshusing the elastic-hriftle-plasli con.$titutive model for two po,~siblegeologicul condilionv along the tunnel aris3.6. Discontinuum modellingWith the understanding that discontinuities (majordiscontinuities and jointing) play a key role in the developmentof tunnel instability, the problem was also inves-3.6.2. Discrete Feature Nefwork (DFN) modelA Discrete Feature Network model was also createdby using the procedures implemented in the FracMancode (Dershowitz et al.. 1995). The results ofgeologic mapping on the right wall of ihe tunnel formedthe basis for defining; the input data in terms of orientation,size and degree-of fracturing for the ioint sets J1.52and ~3 a, b. The& were superim~osed wiih foliation anddiscrete features represented by the two sub-paralleldiscontinuities described above. The 3D volume realizedin this way is shown in Figure 22, where also shown is theplot of the joint sets given with the FracMan code.The UDEC model was obtained by taking a crosssection orthogonal to the tunnel axis through the 3DD m volume of Figure 22. This network of 2D fracturesis not directly amenable to UDEC analysis as the fratturesneed to be well connected and no isolated fracturescan be handled by the code. At the same time, it wasnecessary to cross-validate the model against the rockmass conditions around the tunnel. This was carried outby increasing the block size in the original 2D networkmodel both at the crown and on the left wall, to account

~"d.-geol.-naft. zb., Vol. 12, Zagreb, 2000.Barla, G., Barla M.: Continuum and discontinuum modelling 55Figure 21. Detail of UDEC &termhistic model showing regi~n,~ withdifferent material and joint properiiesfor the generally better rock mass conditions in thesezones.A random process which cancels the fracture tracesexceeding a given limit-length fixed was used in conjunctionwith the Set edge command which is available withthe UDEC code. A typical model finally developed isshown in Figure 23. As for the deterministic modeldescribed above, the joints were assumed to behave asMohr-Coulomb joints and the blocks were treated as anelasto-plastic material. The input data are the same aslisted above and shown in Figure 21.TUNNEL FACEFigure 22.30 Dkcrete Feature Network model created by the Frac-Mancode with plot ofjoint sets. This is typical for rock conditions onfhe rigk wall3.6.3. ResultsFor both the deterministic and DFN models the samestress conditions as assumed for the continuum modelsWere applied.Figure 23. Detail of UDEC-DFN model showing regions with differentmaterid properlies and jointsIn order to investigate whether the continuum approachcan yield similar results to the discontinuum approach, thecalculations were carried to simulate tunnel excavation asfor the finite element models by using stress boundaryconditions. Also, no account was taken for the water pressuredistributed around the tunnel. The main interest wasposed on the yield zones, as shown by blocks which arefailing, and on shear displacements induced along joints(Figure 24). Attention was also paid to the pattern of blockmovements at the right wall (Figure 25).The results obtained show that both the deterministicand DFN models capture well the overstressing conditionsand block movements of right wall. As for theelastic-brittle-plastic model which accounts for the presenceof the two sub- arallel discontinuities, the deepeningof breakout on t l!e right wall is well reproduced withthe discontinuum approach (compare Figure 19 withFigure 24). However, in the latter case (Figure 24) nosignificant instability is seen to develop at the left wall,unlike the continuum approach (Figure 19). The similarityis evident with the actual conditions in the tunnel,where the TBM jammed because of instability and blockmovement at the right wall (Figures 14 and 15).The continuum ap roach appears to capture some ofthe adverse factors w E ich were significant in the instabilityconditions that developed during TBM excavation:- the elastic-plastic-brittle behaviour of rock material,- the highly anisotropic stress regime with a very lowstress ratio,- the presence of unfavourable dominant discontinuitiesat the right wall.In comparison, the discontinuum approach shows in aremarkable manner the block movement developing onthe right wall, whereas the tunnel remains stable alongthe periphery away from the fault zone at the crown andright wall.4. ConclusionsModern numerical modelling methods by the use ofeither the equivalent continuum or discontinuum approachhave been discussed by underlying some of the fundamentalconcepts and guidelines, when investigating tunnelproblems at the design analysis stage. With reference tomodelling in engineering practice, the interest has been

Rud.-geol.-naft. zb., Vol. 12, Zagreb, 2000.56 Barla, G., Barla M.: Continuum and discontinuum modelling%I(a) iFigure 24. YeIded blocks undsheardi.splacements along the join& aroundthe tunnel; (a) determmisfic model, (b) DFN modelplaced on the adoption of discontinuum models whichgive a far more realistic and representative picture ofrock mass behaviour than equivalent continuum models.Modelling of rock mass around a 4.75 m diametertunnel excavated by an open TBM configuration throughquartzitic michaschists has been described, with attentionpaid to the instability conditions that developedwhen crossing through a fault zone. The calculationswere performed by using the finite element method andthe distinct element method, according to a two dimensionalmodelling format, three-dimensional representationbeing too limited and not justifiable in respectof the uncertainties of the input data.With the finite element method an equivalent continuumapproach was applied, including the influence oftwo major discontinuities on the right wall. With thedistinct element method a fully dicontinuum approachwas used by either a deterministic or discrete featurenetwork model. The discontinuum representation of therock mass captures in a remarkable manner the instabilityconditions that developed at the TBM head duringexcavation. In comparison, the continuum representationgives a more idealised illustration of the instability.It is concluded that for stability analysis peculiargeologic conditions such as fault zones around the tunnelrequire specific models, which reflect the given conditionsin the most realistic way possible.Received: 2000-05-24Accepted: 2000-09-21Figure 25. Block movemertls around the funnel, (a) deterministic model,(6) DFN d lREFERENCESBarla, G., M. Barla, L. Repetto (1999): Continuug anddiscontinuum modelling for design analysis of tunnels. 9 . Int.Coner. on Rock Mech. Paris. France.Band is ,"s . (1993): ~n~ineerine properties and characterization ofrock discontinuities. Com~rehenslve Rock Eneinee~e. Volume1. Fundamentals ~udsoi J.A, editor). pp. 153-183. PergamonPre~s, Oxford (U Ii ).Barton , N ., R . ' ~ e.n i , J . L u n d e (1974): Engineering classificationof jointed rock maqses for the design of tunnel supportRock Mechanics. Vol. 6, pp.189-236.Bart on, N. (1997): Pont Ventoux Hydropower Project. TunnelF2-Poet Ventoux. Assessment of geological condilwns andsuppofifor ch 2350-dWO Report to Nocon. 25 October;%gdBarton, N. (1998): Quantitative description of rock mawes for thedesign of NMT reinforcement Spec~al Lecture 1). Int. Conf. onHydro Power Development In h imalayas. Shimla, IndiaBart on, N. 61999): General report concerning some 2Dth centurylessons an 21S'century challenges in ap lied rock mechanics. 9thInternational Congress on Rock ~ecianics. August 25 -28.1999, Paris.Brown, E.T., J.W. Bray, B. Ladanyi, E. Hoek 1983):Characteristic line calculations fur rock tunnels. J. Geotec I, . Eng.Div. Am. Soc. Civ. Enp. 109. pp.15-39.Cundall, P. A., R. D. Hart (1993): Numerical modelling ofdiscontinua. Comprehensive Rock Engineering. Volume 2.Analysis and Design Methods (Hudson J.A., editor). pp. 231-261. Pergamon Press, Oxford (UK).Curran, J. H. and Corkun, B. T. (1997): phase2. 2D finiteelcment program for calculating stresses and estimating su portaround underground excavations. Reference Manual and futorialManual. Rock Engineering Group. University of Toronto.Dershowitz, W., G. Lcc, J. Geier, P. La Pointc,(1995):FracMan Interactivc Discrcte Feature Data Analysis,

Rud.-geol.-naft. zb., Vol. 12, Zagreb, 2000.~~rlrr, G, Barlu M.: Continuum and discontinuurn modelling 57Geometric Modelling and Exploration Simulation. User docu- di derivuione nel tratto Pont Ventoux - F2 (Dott. Geol. G.mentation. Version 2.50. Seattle. Golder A.s.wciates Inc.Venturini).G rims t ad , E ., N . B a r to n (1993): Updating the Q-system for She n , B ., N . Barton (1997): The disturbed zone around tunnelsNMT. Proc. of Int. Symp. on Sprayed Concrete. Modern use of in jointed rock masses. lnf. J. Rock Mech. Min. Sci. Vol. 34, pp.wet mix sprayed concrete for underground sup orf. Fagernes 117-125.me om pan, 0pS.m and erg, editors). ~onvcgian LnCretc Asso- St art f i e 1 d , A. M ., C u n d a l 1 P . A. (1988): Towards a methciation.Oslo.odology for rock mechanics modelling. Inf. J. Rock Mech. Min.art , R . D . (lF3): +n introduction to distinct etement modelling Sci. Vol. 25, pp. 99- 106.for rock engmeenng. Comprehensive Rock Engineering. Volume2. Analysis and Design Methods Hudson J.A., editor). pp. 6. Acknowledgements245-263. Pergamon Press, Oxford (U k ).Hock, E., M. W ., Grabinsky, M. S. Dicdcrichs (1991): The work described in this paper was carried out withNumerical modelling for underground excavation design: Trurw. the financial support of the [talian Ministry for Univer-Imt. Min. hfebl. VollOO. January-April1991, PP. A22-A30. sity and Technological Research (M.U.R.S.T.) as part ofHoe k, E, E., T.B ro w n (1997): Practical estimates of rock massstrength. Inl. J. Rock Mech. Mn Sci. Vol. 34, pp.1156- 1186. the R~~~~~~. programme tnnne~ing in difficult condi-Itasca Consulting Grou (1996): UDEC (Universal Distinct Ele- tions" 40%). The authors would like to acknowledge thement ~ e ) version , 5.0. volume 1: user's manual. volume 11: help o G. Giampieri and L. Repetto, former students atVerification roblems and example applications. Minneapolis, the Politecnico di Torino.Minnesota, &A.Martin, C. D., D. R. Mc Creath, M.The permission of AEM (Azienda Energetica Metro-Stochmal (1997):Btimatingsu port demand-load caused by stress-induced failure politana) Torino, Owner the Pant VentOux-Susa H ~-around tunncfh. Int. Symp. on Rock Support, Lillehammer, June dropower Project, to publish this paper on the F2 tunnel1997. is gratefully acknowledged. The authors would also likepane t , M . (1995): Le calcul des tunnels par la mCthod convergence- to thank the following companies involved in the projectcontinement. Presses de 1'Ecole Nationale des Ponts etChau.~~.described in the paper: ~ ~ ~ s.~.A., ~ l~~~l~d and i N ~ ~p o n t V e n t o u x (1997): Nota relativa al sopralluogo effettuato indata 8/10/1997 In corrispondenza del fronte di scavo della galleria