Veri f i cation of a laboratory-based dilation model for in situ conditions using continuum models

G.Wlton,M.S.Diederichs,L.R.Alejno,J.Arzú

aQueen’s University,Kingston,Canada

bUniversity of Vigo,Vigo,Spain

Veri f i cation of a laboratory-based dilation model for in situ conditions using continuum models

G.Waltona,*,M.S.Diederichsa,L.R.Alejanob,J.Arzúab

aQueen’s University,Kingston,Canada

bUniversity of Vigo,Vigo,Spain

A R T I C L E I N F O

Article history:

Received 25 June 2014

Received in revised form

18 September 2014

Accepted 26 September 2014

Available online 1 November 2014

Dilation

With respect to constitutive models for continuum modeling applications,the post-yield domain remains the area of greatest uncertainty.Recent studies based on laboratory testing have led to the development of a number of models for brittle rock dilation,which account for both the plastic shear strain and con f i ning stress dependencies of this phenomenon.Although these models are useful in providing an improved understanding of how dilatancy evolves during a compression test,there has been relatively little work performed examining their validity for modeling brittle rock yield in situ.In this study,different constitutive models for rock dilation are reviewed and then tested,in the context of a number of case studies,using a continuum f i nite-difference approach(FLAC).The uncertainty associated with the modeling of brittle fracture localization is addressed,and the overall ability of mobilized dilation models to replicate in situ deformation measurements and yield patterns is evaluated.

?2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

1.Introduction

Recently,numerical methods have become increasingly popular tools to analyze rock mass behavior.Computer programs which represent rock masses as continua and discontinua can be used to predict loads and displacements in rock structures and support or reinforcement systems or to verify hypotheses about observed behavior(back analysis).Although these tools are no longer restricted to research applications,models used in the study of civil and mining geotechnical structures are often limited in their complexity(i.e.elastic models for stress prediction).This is largely due to the questions about the validity of more complex models.In fact,the use of inadequate material models is one of the largest limiting factors in numerical analyses(Lade,1993;Carter et al., 2008).

Continuum models are more commonly used than discontinuum models in rock engineering(even when they are not necessarily appropriate).The existing experience base in the geotechnical community with respect to modeling rock masses as continua is a major driver of this phenomenon(Bobet,2010). Although rapidly evolving discontinuum and hybrid continuum/ discontinuum modeling tools provide a valuable alternative to continuum models for some applications(see Jing(2003)and Bobet(2010)),it is important to continue to improve constitutive models for use in continuum models given their relative accessibility and ease of use.

One area of particular historical de f i ciency in terms of constitutive models for rocks and rock masses is their post-yield volumetric response to continued deformation.Correspondingly, the tendency of rock masses to dilate following yield has been a topic of increased research recently.Understanding this phenomenon may be integral in allowing for the accurate prediction of yield and ground movement;this is particularly true of more brittle rocks,which tend to dilate most signi f i cantly(Hoek and Brown,1997).

In this study,different approaches for modeling dilative behavior are reviewed,and then used in a back analysis of extensometer data obtained from the Donkin-Morien Tunnel(Nova Scotia,Canada).One dilation model in particular is then applied to further case studies to illustrate its ability to successfully replicate displacements measured in situ.

2.Models for rock dilation

The tendency of rocks to expand under compression was f i rst shown to be a true material property(rather than an in f l uence of the testing system)by Cook(1970).Although the underlying mechanisms for this phenomenon are fundamentally brittle(see Brace et al.(1966)and Jaeger and Cook(1969)),different formulations based on plasticity theory have been developed over theyears in an attempt to properly capture the macroscopic stressstrain behavior of rocks.

For a Mohr-Coulomb solid,the ratios of plastic strain components are controlled by the dilation angle,ψ.This parameter uniquely de f i nes the stress gradient of the plastic potential function,which is in turn directly proportional to the plastic strain tensor for a material at yield.The connection to volumetric strain can be seen through the general de f i nition of the dilation angle in terms of plastic strain increments(Vermeer and de Borst,1984):

Early work on the post-yield deformation of plastic solids led to the concept of an associated f l ow,which requires the plastic potential surface to be coincident with the yield surface in stress space (in this case,the friction angle,φ,is equal toψ).In this case,the plastic dissipation(energy loss)associated with post-yield deformation is zero.As the study of soil and rock plasticity progressed,it was noted by many that the adoption of an associated f l ow rule was inappropriate for granular materials which dissipate energy through frictional mechanisms(Roscoe,1970;Price and Farmer, 1979;Vermeer and de Borst,1984;Chandler,1985).More recently,a number of authors have noted that for those materials,it is necessary not only to use a non-associated f l ow rule,but also to use a dilation angle which depends on con f i ning stress and is mobilized as damage accumulates in rock;note that“damage”is commonly quanti f i ed in terms of the maximum plastic shear strain, γp,taken as the difference between the major and minor principal plastic strain components.

2.1.Mobilized dilation models

In the study of soil mechanics,there were early attempts to tie the mobilization of the dilation angle to the mobilization of friction over the course of deformation(see Rowe(1971)).Detournay (1986)extended this mobilized dilation concept to rock masses based on theoretical considerations,although his model for the dilation angle was independent of any change in the friction angle. Work by Ofoegbu and Curran(1992)represents one of the f i rst mobilized dilation models which was developed based on the study of laboratory test data and accounts for both the con f i ning stress and accumulated strain dependencies of rock dilatancy.Cundall et al.(2003)also proposed a model for post-yield dilatancy, although theirs was based solely on theoretical considerations.

The model proposed by Alejano and Alonso(2005)represented a major advancement in the study of rock dilatancy,both in that it is shown to f i t data from a wide number of lithologies,and in that it requires only one parameter to de f i ne the dilation angle for all(σ3, γp)conditions(σ3is the minor principal stress).In this model,the initial dilation angle following yield is taken to be the peak dilation angle,which is a function of the con f i ning stress.As deformation continues,the dilation angle gradually decays from its peak value. Typical volumetric strain-axial strain plots obtained from laboratory compression tests are shown in Fig.1,both for a material following the Alejano and Alonso(2005)model for dilation(AA), and for a material with a constant dilation angle.

Fig.1.Volumetric strain-axial strain curves for the Alejano and Alonso(2005)dilation angle model(top)and a constant dilation angle(bottom)(after Walton and Diederichs (2013)).

Based on a statistical analysis of in situ displacements predicted using the AA model for dilation and a variety of strength and stiffness parameters,Walton and Diederichs(2014)concluded that in many cases(particularly for near hydrostatic stresses),results obtained using the AA model can be approximated using a constant dilation angle.For preliminary models,they suggested a constant dilation angle value of

whereσcrmis the rock mass strength at uncon f i ned conditions,and σe_tis the elastic tangential wall stress,which,for a circular tunnel, has a maximum value of

whereσ1is the major principal stress.

The AA model has two major limitations.The f i rst is that it was developed based solely on a selection of sedimentary rock data,and it has since been shown that the con f i nement-dependency of the peak dilation angle as predicted by their model is too large for crystalline rocks(Zhao and Cai,2010;Arzua and Alejano,2013; Walton and Diederichs,submitted for publication).The second is that the model is based on the assumption that yield in situ is coincident with peak strength as observed in laboratory tests. Although this assumption may be true for certain weaker rock masses,for rock masses which deform through brittle fracturing processes,a different de f i nition of yield must be used(Martin, 1997;Diederichs,1999;Diederichs and Martin,2010).

In contrast to that of Alejano and Alonso(2005),the dilation angle model of Zhao and Cai(2010)de f i nes the onset of unstable cracking(CD)as yield(which is consistent with the conclusions of Diederichs and Martin(2010)for brittle rocks).The model of Walton and Diederichs(submitted for publication)(WD)uses this same de f i nition for yield,and obtains similar model f i t qualities using a lower overall number of parameters.

Like the Zhao and Cai(2010)model,the WD model begins with a dilation angle of 0?,then mobilizes dilation to a peak value before initiating a gradual decay as predicted by the AA model.Although some dilatancy caused by crack opening can be observed,it is the dilatancy which mobilizes due to shear deformation of cracks thatcontrols the major volume changes observed in laboratory tests.By fi tting this model to data obtained from laboratory compression testing,the parameters necessary to de fi ne the model can be obtained for a given rock type.Fig.2 illustrates the different phases of dilation as well as the con fi nement dependency incorporated into the WD model.The mathematical description of the WD model is represented by

The terms in Eq.(5)are listed in Table 1.

The pre-mobilization parameter(α)controls the curvature of the model up to the peak dilation angle.Avalue of 1 corresponds to a linear increase,whereas a value of 0 corresponds to an immediate rise to the peak dilation angle which remains constant for the premobilization phase.This parameter increases linearly with con f i nement,and can be broken down into its slope(α＇)and its intercept(α0):

The plastic shear strain to peak dilation mobilization(γm)has not been shown to have any consistent dependency on con f i ning stress,but the peak dilation angle(ψPeak)itself can be de f i ned for all con f i nements as a function of two parameters,in general: that under triaxial test conditions.Although the in f l uence of con f i nement on this parameter is likely a smooth function of con f i ning stress,a lack of testing data at very low con f i nements (below 0.5 MPa)precluded the de f i nition of a continuous function to capture this change.As such,they de f i ned two unique decay parameters:γ0for uncon f i ned conditions andγ＇for con f i ned conditions(Eq.(9)),whether this distinction is necessary for modeling in situ dilatancy is uncertain:

With respect to typical values,Walton and Diederichs (submitted for publication)found that more brittle rocks tended to have lower values ofα,γmandγ*,and higher values ofψPeakat all con f i nements(due primarily to higherβ0values).Fig.3 shows a comparison of dilation models f i t to quartzite and mudstone data from the literature,for comparison.The parameters associated with these models are provided in Table 2.Note that the dilation angle model parameters only control the dilation angle mobilization relative to its peak value at uncon f i ned conditions;the absolute value of this peak is equal to the peak friction angle, which in this case is 73?for the quartzite data and 46?for the mudstone data.

2.2.Application of laboratory-based models to modeling in situ behavior

As is the case in many rock mechanics studies,the greatest challenge with respect to understanding rock dilation is determining to what degree the behaviors observed in the laboratory

whereβ0controls the con f i nement dependency at low con f i nements(σ3＜2-3 MPa)andβ＇controls the con f i nement dependency at higher con f i nements(σ3＞2-3 MPa).

In the WD model,Eq.(7)is applied to crystalline rocks.For sedimentary rocks,however,the simpler formulation proposed by Alejano and Alonso(2005)can provide an accurate representation of the peak dilation angle:

whereσcis the uncon f i ned strength of the material.Note that for sedimentary rocks,β0andβ＇need not be de f i ned,as the AA model for peak dilation can be used.

The post-mobilization decay parameter（γ*）de f i nes the amount of straining pastγmrequired to reduce the dilation angle to 1/e (37%)of its initial value.This parameter tends to decrease slightly with increased con f i ning stress,although both Alejano and Alonso (2005)and Walton and Diederichs(submitted for publication) suggested that for practical purposes,the post-mobilization decay rate can be considered similar regardless of con f i ning pressure. Under uniaxial test conditions,however,there was a tendency for the decay parameter to be signi f i cantly higher(less decay)than truly re f l ect the mechanisms which control in situ damage and deformation.Some have pointed out the dif f i culties associated with modeling the dilation of spalling fractures around underground openings(Kaiser et al.,2010).The main issue in this case is that when cracks f i rst begin to form and propagate,almost all of the irrecoverable strain is towards the excavation opening,corresponding to a dilation angle of 90?.The authors suggest,however, that similarly to the initial yield observed in laboratory tests,the initial post-yield fracture opening in situ may be insigni f i cant,and that cases where highly dilatant spalling is observed involve a notable shear component of deformation along macroscopic fractures.The examples of non-dilatant spalling shown in Fig.4 are consistent with an in situ dilation angle that rises from 0?to a peak dilation angle after some small degree of shear movement along fracture planes.The results of Zhao et al.(2010)further support the extension of laboratory-based dilatancy models to use for in situ brittle behavior;by modeling a mine-by experiment in a massive granitic rock mass using a mobilized dilation model,they were able to accurately reproduce displacements as observed in situ.To further demonstrate the applicability of laboratory-based dilation angle models for the purposes of modeling in situ brittle deformation,back analyses have been performed based on data available from the literature.

Fig.2.Different phases of post-yield dilatancy as seen in triaxial test data for coal(top) and con f i nement dependency of the peak dilation angle(ψPeak)for Carrara Marble (bottom)(after Walton and Diederichs(submitted for publication)).

3.Comparison of modeling approaches

An access tunnel for the Donkin-Morien coal mine in Cape Breton Island,Nova Scotia,Canada was driven by a shielded LOVAT M-300 TBM from January,1984 to December,1984.The maximum depth of the tunnel was 200 m below the seabed.Monitoring data for this tunnel originally analyzed by Pelli et al.(1991)using elastic models have been re-analyzed to demonstrate the ability of different modeling approaches to replicate observed displacements in situ.The extensometer data collected at chainage 2996 were selected for analysis given the quality of the data and the lack of any geological interfaces near the apparent boundary of the yield zone. At this location,the tunnel was excavated in an interbedded siltstone-mudstone unit.A tunnel cross-section is shown in Fig.5, with the principal stress directions and magnitudes interpreted by Pelli et al.(1991)illustrated.

Table 1Summaries of the terms presented in Eq.(5).

Fig.3.WD dilation angle model results for Witwatersrand quartzite(top)and mudstone(bottom)with con f i ning stresses and parameter values shown;f i t parameters from Walton and Diederichs(submitted for publication)were determined using quartzite data from Crouch(1970)and mudstone data from Farmer(1983).

Table 2Parameters used to generate the dilation angle model curves shown in Fig.3.

Fig.4.Examples of non-dilatant spalling observed in situ.(a)Excavation-parallel fracturing in a TBM tunnel;(b)A sequence of fractures in a highly stressed mine drift.

Fig.5.Donkin-Morien tunnel at study location(chainage 2996)with f i nite-difference mesh near excavation shown.

Laboratory testing results showed that uniaxial compressive strength(UCS)values ranged between 15 MPa and 63 MPa with a mean value of 36 MPa for the interbedded siltstone-mudstone unit and the values between 14 MPa and 69 MPa with a mean of 54 MPa for the siltstone unit.The Young’s modulus values varied between 4 GPa and 15 GPa with a mean of 9 GPa for the interbedded siltstone-mudstone unit and between 4.5 GPa and 25 GPa with a mean of 11.3 GPa for the siltstone unit(Yuen et al.,1987).The siltstone and interbedded siltstone-mudstone units appear to behave almost identically in situ,as the extensometer results recorded in interbedded siltstone-sandstone and siltstone units at chainage 3205 were almost identical to those recorded at chainage 2996 in the interbedded siltstone-mudstone unit.

Elastic back analyses by Pelli et al.(1991)estimated a lower bound rock mass modulus of 1.65 GPa for the interbedded siltstonemudstone based on the data obtained from chainage 2996 and a rock mass modulus for the siltstone of 5.6 GPa based on the data obtained from chainage 3205.Based on the mean siltstonemudstone laboratory stiffness(9 GPa),a geological strength index (GSI)of 80(Corkum et al.,2012),and the empirical relationship of Hoek and Diederichs(2006),a rock mass modulus of 7.9 GPa would be predicted.Based on this,it appears that 1.65 GPa is far too low. For this study,a moderate estimate for the rock mass modulus of 5.6 GPa has been used.This value was selected based on the similarity of the observed deformations in siltstone and siltstonemudstone units.

Extensometers were installed immediately behind the tunnel face.Pelli et al.(1991)concluded that at chainage 2996,only elastic deformation had occurred ahead of the face.Preliminary elastic models run by the authors predicted a total elastic crown deformation of 1.75 mm.It is reasonable to assume that～30%of this deformation(～0.5 mm)would have occurred prior to the extensometer installation(Steindorfer,1998;Vlachopoulous and Diederichs,2009).For the purposes of this study,any elastic deformation which occurred prior to the installation is ignored, given its negligible magnitude relative to the measured total displacements.

With respect to support,a wire mesh was used to retain loosened rock fragments immediately behind the TBM shield.The remainder of the support was away from the tunnel face,and was found to have no signi f i cant effect on the deformational behavior of the ground(Pelli et al.,1991).

At chainage 2996,the depth of yield was estimated to be 1.9 m (Corkum et al.,2012).The additional observation of a 60?arc of loosening and spalling in the crown allows for a reasonable constraint on the size and shape of the yield zone to be established. In the absence of additional displacement measurements at this chainage,this information is critical in establishing a physically realistic back analysis result.

Two sets of material models were tested-strain-weakening (simultaneous loss of cohesion and friction after yield)with the AA model for dilation,and cohesion-weakening-frictionstrengthening(CWFS)with the WD model for dilation.Constant dilation angle models were also run for the purposes of comparison,and to validate the parameter selection methodology proposed by Walton and Diederichs(2014).Modeling was performed using the f i nite difference program,FLAC 7.0(Itasca,2011).The mesh used consisted of a radial square mesh,with 16 cm sides(4.2%of the tunnel radius)at the excavation boundary.All models were run using a Young’s modulus of 5.6 GPa,and a Poisson’s ratio of 0.25 based on the extensive back analyses of Pelli et al.(1991)described above.

3.1.Strain weakening

Initial Mohr-Coulomb strength parameters were estimated based on the mean UCS value of 36 MPa(Yuen et al.,1987),the rock mass GSI of 80(Corkum et al.,2012),an estimated mivalue of 8 (Gomez-Hernandez,2001),and the proposed rock mass strength estimation method of Hoek et al.(2002).Although the authors acknowledge that the rock mass strength estimation method of Hoek et al.(2002)has not been thoroughly validated,in the absence of further information,it can serve as a starting point for back analyses.To achieve the desired depth and width of yield,these parameters had to be adjusted,such that the peak cohesion was slightly lower and the peak friction angle was slightly higher than those predicted according to Hoek et al.(2002).Given the apparent brittleness of the in situ failure observed,the majority of the drop in strength was constrained to occur in the cohesion component of strength.The dilation decay parameter required to de f i ne the AA dilation model for the siltstone-mudstone unit was estimated based on the data for mudstone and silty sandstone provided by Alejano and Alonso(2005)and Walton and Diederichs(submitted for publication).The f i nal back analyzed material parameters are shown in Table 3.Note that the plastic shear de f i nition used by FLAC(eps)is approximately equal toγp/2 for practical purposes (Alejano and Alonso,2005;Itasca,2011).

Table 3Back analyzed strain-weakening material parameters for the interbedded siltstone-mudstone unit at chainage 2996 of the Donkin-Morien tunnel.

Given the peak rock mass strength represented by the parameters in Table 3(uncon f i ned rock mass strength of 10.7 MPa)and the predicted elastic tangential wall stress at the tunnel crown (3σHσV=25 MPa),the preliminary best f i t constant dilation angle predicted by the methodology of Walton and Diederichs (2014)isψConstant=40??(10.7/25-0.1)=13.2?.The modeling results obtained using the above parameters both with the AA and constant dilation angle models are shown in Fig.6.

Fig.6 clearly shows that the strain-weakening model is unable to fully capture the behavior of the in situ rock mass,either with a constant or mobilized dilation angle model.Indeed,the issue is not the dilation angle model used,but the de f i nition of yield.As has been noted by many authors,where deformation occurs through brittle spalling,peak friction and peak cohesion must be mobilized at different stages of yield to achieve reasonable predictions of in situ failure(Hajiabdolmajid et al.,2002;Diederichs,2007;Edelbro, 2009;Barton and Pandey,2011).Another interesting result shown in Fig.6 is that,for preliminary modeling purposes,the results obtained using the constant dilation angle selection methodology of Walton and Diederichs(2014)do provide a reasonable approximation to those obtained using the AA model.In particular,the constant dilation angle model tends to overpredict the slope of the displacement pro f i le near the edge of the yield zone(where the mobilized dilation angle is lower)and underpredict the slope of the displacement pro f i le near the excavation wall(where the mobilized dilation angle is higher).

Fig.6.Strain-weakening results compared to in situ extensometer data when using the AA dilation angle model(top)and a best-estimate constant dilation angle(bottom).

3.2.Cohesion-weakening-friction-strengthening(CWFS)

In contrast to the strain-weakening strength model,the CWFS model begins with cohesion at its peak value,but with the friction angle at a low value(usually between 0?and 15?)(Diederichs, 2007);as cohesion drops with continued deformation,the friction angle is eventually mobilized to its peak value(Hajiabdolmajid et al.,2002).

To start,the initial friction angle was set to 10?to be within the acceptable range de f i ned by Diederichs(2003).Next,the peak cohesion was set such that the uncon f i ned crack initiation strength would be equal to 15.1 MPa(=0.42UCS,best f i t relationship for sedimentary rocks obtained from Perras and Diederichs(2014)). Ultimately,this crack initiation strength was found to be too high, and was subsequently reduced in further modeling.With respect to the peak(f i nal)friction angle,it is reasonable to assume that it is equal to the peak dilation angle under uncon f i ned conditions (Alejano and Alonso,2005;Zhao et al.,2010;Walton and Diederichs,submitted for publication).Based on the data for silty sandstone and mudstone data obtained from Farmer(1983),a value of 45?was deemed reasonable for the siltstone-mudstone unit.This corresponds to a lower bound estimate of the spalling limit for residual strength-σ1/σ3≈6 versus the range of 10-20 suggested for crystalline rocks(Kaiser et al.,2000;Diederichs, 2003).Residual cohesion was initially set as 0.1cPeak,and ultimately lowered to achieve the desired yield zone size.With respect to the parameters used for the WD dilation model, reasonable values were tested based on available data for siltysandstone and mudstone.Because the peak dilation angle for sedimentary rocks can be accurately predicted using the peak yield strength based on the f i ndings of Alejano and Alonso(2005),no values ofβ0orβ＇were required.The f i nal back analyzed material parameters are shown in Table 4;note that the parametersγm,γ0, andγ＇have been replaced with epsm,eps0,and eps＇in this table to re f l ect the difference between the plastic shear strain de f i nition adopted by Walton and Diederichs(submitted for publication)and that used in FLAC(γp/2≈eps)(Alejano and Alonso,2005;Itasca, 2011).

For the purpose of comparison,a constant dilation angle model was also run,withψConstant=15.5?(as based on the method of Walton and Diederichs(2014)).The results of the models run with both the mobilized and constant dilation angles are shown in Fig.7.

The results obtained using the mobilized dilation angle model show a good agreement to the in situ deformation measurements. There is a relatively high degree of error in the f i t between 0.8 m and 1.7 m from the excavation,because the model is unable to capture the non-increasing nature of the displacement pro f i le slope in this region(this is discussed further in Section 5).Even these errors,however,are relatively small(＜15%of the measured displacement).The constant dilation angle model again provides a decent result,although in this case the mobilized dilation angle model is clearly preferable.For the purposes of comparison,the yield zones obtained using the strain-weakening and CWFS strength models are shown in Fig.8.Note that in the CWFS case,the observations on the depth and extent of yield are accurately reproduced(as well as the displacement measurements).

To illustrate the in f l uence of each of dilation model parameters on the model displacements,each parameter was individually varied from the best f i t CWFS model to an extreme value(or extreme high/low values).The results of this sensitivity analysis can be seen in Fig.9.No results are shown for variation ofα＇,since varying this parameter was found to have minimal effect on the model displacements.This is likely because of a combined low sensitivity of the model toα,since the high plastic strains at equilibrium mean the majority of the deformation occurred post-mobilization,and also due to low range of con f i ning stresses in the yield zone(0-5 MPa).

Although the model has an overall low sensitivity to the value of α,signi f i cant changes to the parameter do have an effect on the resulting displacements,particularly away from the excavation wall,where less total deformation occurs,and therefore the premobilization dilation phase has a relatively signi f i cant in f l uence on displacements.The values tested forα0(0 and 0.25)appear to represent practical lower and upper bounds for this parameter based on the data presented by Walton and Diederichs(submitted for publication).

The AA peak dilation predictions for the siltstone-mudstone layer were found to correspond approximately to WD peak dilation parameter values ofβ0=0.5 andβ＇=0.2.Since these appear to be nearly lower and upper bound values for these parameters, respectively,the opposite extreme for each parameter was tested. Using aβ0of 1(typical for a crystalline rock)resulted in an extremely large increase in model displacements,corresponding to a signi f i cant increase in the peak dilation angle at the low con f i nement levels presented in the yield zone.This result is not physically meaningful,however,as a rock with a high value ofβ0would tend to be much stronger,and therefore experience less yielding for the same stress conditions.The change toβ＇also increased model displacements,although its in f l uence was much less signi f i cant.

With respect to the plastic strain to peak dilation mobilization, changing this parameter effectively changes the weighting of how much of the deformation occurs prior to and following the mobilization of peak dilation.Because of the large strains predicted in this case,the model result was not very sensitive to this parameter, although lowering it did increase displacements away from the excavation wall(peak dilation angle attained further into the rock mass)and increasing it increased displacements near the excavation wall(peak dilation angle mobilized later,meaning less postmobilization strain to cause dilation angle decay).

Table 4Back analyzed CWFS material parameters for the interbedded siltstone-mudstone unit at chainage 2996 of the Donkin-Morien tunnel.

Fig.7.CWFS results compared to in situ extensometer data when using the WD dilation angle model(top)and a best-estimate constant dilation angle(bottom).

Fig.8.Contours of plastic shear strain(epsin 103)obtained using strain-weakening strength model with AA dilation model(left)and CWFS strength model with WD dilation model(right).

Fig.9.Sensitivity of model results to different dilation model parameters;in each case,the model parameters were kept the same as those of the best f i t model(see Table 4)with the exception of the parameter(s)speci f i ed in the legend.

4.Case studies from the Couer d’Alene mining district

Mining is the primary industry in the Couer d’Alene district in Northern Idaho.This region is the home to several signi f i cant lead, zinc,and silver deposits.Mineralization in the region typically occurs in the form of galena-sphalerite and tetrahedrite veins in a sequence of Proterozoic rocks belonging to the Belt Supergroup (Fleck et al.,2002).To further illustrate the applicability of the proposed model for brittle rock dilatancy,data from two mine shafts constructed in the region were used for the purposes of back analysis.

4.1.Lucky Friday Mine-Silver Shaft

First,the Silver Shaft from Hecla Mining Company’s Lucky Friday just East of Wallace,Idaho was considered.In particular,extensometer records originally presented by Barton and Bakhtar(1983) are analyzed.These instruments were installed at a depth of 1582 m in the shaft.According to the stress model of Whyatt et al. (1995),the major and intermediate principal stresses are thought to be 110.4 MPa and 66.4 MPa,and oriented NW-SE and NE-SW, respectively;the minimum principal stress is sub-vertical and is approximately equal to the overburden weight in magnitude (42.7 MPa,assuming a density of 2700 kg/m3)(Barton and Bakhtar, 1983;Pariseau et al.,1992;Whyatt et al.,1995).The shaft was excavated in a weakly foliated quartzite unit,with the foliation oriented NW-SE(parallel to the major principal stress)and having a near vertical dip.The shaft and the relative locations of the extensometers studied are shown in Fig.10.

Back analysis by Barton and Bakhtar(1983)based on the elastic deformation seen in the data from EXT 2 suggested a Young’s modulus on the order of 20.7-27.6 GPa.Unfortunately,this back analysis was based on assumptions about the stress f i eld which are inconsistent with the model of Whyatt et al.(1995),suggesting that their rock mass modulus range is too low.Borehole deformation tests by Patricio and Beus(1976)and USBM(1980)using a Colorado School of Mines(CSM)cell led to small scale modulus estimates for the bedded quartzite in the range of 48.3-75.8 GPa.If we consider the upper end of this range representative of the intact quartzite, we can estimate the rock mass modulus as approximately 55.5 GPa using the method of Hoek and Diederichs(2006)and the GSI value of 70 suggested by Gomez-Hernandez(2001).This value is close to the lower-bound rock mass modulus estimated based on the borehole deformation tests,so can be considered reasonable as a starting estimate.

Fig.10.Silver shaft stress and instrumentation geometry(EXT 1 and EXT 2).

To re f i ne the rock mass modulus estimate,the deformations recorded by EXT 2(deemed purely elastic)were matched using elastic FLAC models.First,however,the measurements required a correction to account for elastic displacements which occurred prior to instrument installation.As in the case of the Donkin-Morien tunnel,the instruments were installed at the face,so it can be assumed that 30%of the elastic deformation occurred prior to instrument installation.To add this missing deformation to measurements,f i rst a quadratic function was f i t to the extensometer measurements.The measurement from the anchor nearest to the excavation wall was ignored due to its anomalous nature;this anomaly could be due either to an instrumentation problem,or a locally anomalous set of rock properties as discussed by Pelli et al. (1991).The quadratic f i t to the remainder of the data was then considered to represent 70%of the elastic deformation pro f i le. Correspondingly,3/7 of the f i t value was added to the original measurement at each anchor to obtain the corrected data.Using a Poisson’s ratio of 0.25(after Barton and Bakhtar(1983)),a Young’s modulus value of 52 GPa was found to provide an optimal f i t to the EXT 2 data.The data correction process and elastic back analysis result are illustrated in Fig.11.

EXT 1 showed signi fi cant displacement when compared with EXT 2,indicating a relatively large yield zone on the SW side of the shaft.Support was installed after displacements measurements had stabilized,and so its effect is neglected.Unfortunately,no data were available within the 1 m of rock closest to the shaft wall,and further from the wall,there is relatively poor constraint on the exact extent of the yield zone.A starting estimate of CI(the crack initiation stress)for uncon fi ned conditions was obtained by taking 0.47UCS, where the UCS was reported as 125 MPa by Gomez-Hernandez (2001)(Perras and Diederichs,2014).Dilation parameters were estimated based on the properties of other crystalline,brittle rocks studied by Walton and Diederichs(submitted for publication).The fi nal back analyzed parameters are shown in Table 5.Although the model produced using these parameters predicts some minor yield on the NW side of the shaft which is not seen in EXT 2,it is likely that there may be some strength anisotropy due to the foliation.It is expected that the strength perpendicular to the foliation(at the position of EXT 2)might be slightly higher than that is re fl ected by the parameters shown in Table 5.It is assumed that the actual lack of yield in situ at the NW side of the shaft relative to the minor yield predicted by the model has a negligible effect on the observed displacements on the SW side of the shaft.The f i nal model displacements are compared to those measured by EXT 1 in Fig.12.

Fig.11.Correction of extensometer data to account for missed elastic deformation ahead of the shaft face(top)and elastic back analysis result for EXT 2 using E=52 GPa (bottom).

4.2.Caladay Shaft

The Caladay Shaft(at Callahan Mining Corp.’s Calladay Mine)is located less than 1 km west of the center of Wallace,Idaho,and approximately 10 km west of the Lucky Friday Mine.The shaft is rectangular in shape,and was excavated to a depth of 6300 ft(1920 m)below surface in virgin ground.Instrumentation was installed at a depth of 5950 ft(1814 m)below surface in the same weakly foliated quartzite unit(of the Revett formation)as found in the Silver Shaft of the Lucky Friday Mine(Whyatt et al.,1995).Extensometers installed on the NW and SE sides of the shaft were installed perpendicular to the strike of the quartzite bedding,and experienced signi f i cant inelastic deformation(Whyatt and Beus, 1987).Given the proximity of the Caladay Shaft to the Lucky Friday,it was assumed that the stress model developed by Whyatt et al.(1995)for the latter could be applied to the former;using this model,one obtains stress estimates ofσ1=126.9 MPa, σ2=76.1 MPa,andσ3=49.0 MPa.Timber supports were installed in the shaft approximately 25 ft(7.6 m)behind the face of the shaft; given this distance and the relative softness of the support,its effecton the measured displacements was considered negligible(Whyatt and Beus,1987).Fig.13 shows the orientation of the shaft,bedding, instruments,and stresses at the site of interest.

Because of the similarity in the geological conditions at the Caladay and Silver Shafts,the back analysis properties derived from the Silver Shaft case study were used as a starting point.These parameters resulted in a slight underestimation of the depth of yield,so the strength was gradually increased until a good f i t to the observed data was obtained.The f i nal set of parameters was equivalent to those shown in Table 5,exceptwith the peak cohesion value changed from 35 MPa to 40 MPa,and the residual cohesion value changed from 0.8 MPa to 1 MPa.This degree of minor variation in back analyzed strength is reasonable,given the level of variability expected across the region,and could be attributed to the difference in bedding characteristics,such as spacing or degree of healing.The obtained model results are compared to the extensometer records in Fig.14.The data for both EXT 1 and EXT 2 are presented together,because for the type of homogeneous model used in this study,the results for two diametrically opposed measurement lines are equivalent.The fact that the model result lies within the range of displacements recorded on both extensometers indicates that it has captured the overall behavioral trend of the rock mass,and that the back analyzed parameters used for the Caladay and Silver Shafts are reasonable.The actual discrepancies in the measurements recorded by EXT 1 and EXT 2 could be due to geologically controlled variability in any of the geotechnical model parameters used,but to capture this degree of variability in a model is impractical given the relative scarcity of data that is available for routine geomechanical investigations.

Table 5Back analyzed CWFS material parameters for the foliated quartzite present at 1582 m depth in the Silver Shaft.

5.Strain localization in situ

The progressive fracture of a brittle rock mass in situ was f i rst comprehensively documented by Martin(1993),in the case of the Lac du Bonnet Granite at Atomic Energy of Canada Ltd.’s Underground Research Laboratory in Manitoba,Canada.Part of his thesis described the formation of a notch in the roof of an excavation as being driven by the gradual formation and removal of individual rock slabs separated by spalling fractures.In the case where fractured material is retained,either by a support system or by virtue of the system geometry(i.e.slabs presented in the shaft wall not loosened by gravitational loading),the distribution of fractures and ground movement may not be completely regular.

Fig.12.Back analysis model results for the Silver Shaft EXT 1 data when using a mobilized dilation angle model.

5.1.Evidence of irregular strain localization in situ

Continuum numerical models tend to predict smooth displacement pro f i les,with the slope of the pro f i le increasing regularly towards the excavation boundary.This,however,is not always the case in displacements recorded by extensometers. Fig.15 shows three extensometer records from different case studies,all with some indication of irregular fracture dilation within the yield zone.

5.2.Modeling brittle strain localization

A potential explanation for the irregular strain distributions measured in situ was found in performing a back analysis of extensometer data from a deep mine shaft in Arizona(bottom of Fig.15).Although the use of a standard mesh size resulted in a regular displacement pro f i le,with a very f i ne mesh,strain localized into three areas:a thin skin of damage around the excavation boundary,a small notch which extends slightly deeper into the rock,and a broader arc of strain just beyond the f i rst notch.In between the notch and the arc,an elastic portion of rock is presented.The elastic portion of the rock as predicted by the f i nelymeshed f i nite-difference model corresponds well to the area of reduced displacement in the extensometer data(see Fig.16).

Fig.13.Caladay shaft stress and instrumentation geometry(EXT 1 and EXT 2).

In situ,the tendency of strain and fracture dilation to localize into distinct areas within the rock mass is likely to be a function of both the resistance of the rock matrix to fracturing as well as the location and orientation of anomalously weak or strong structures. In a f i nite-difference model,the strain localization depends on the interplay between different strength components(cohesion and friction)as they evolve,as well as the characteristics of the mesh used(Varas et al.,2005).Although numerical modeling results were capable of replicating the observed displacements in the case of the Arizona mine shaft illustrated in Fig.16,because of the number of factors involved and the great degree of uncertainty associated with the in f l uence of distinct structures on fracture evolution in a rock mass,it is not reasonable to expect that this type of strain localization can be accurately modeled in general.Instead, it is preferable to obtain generally representative results from back analyses,such as those shown in Sections 3 and 4.

Fig.14.Back analysis model results for the Caladay Shaft extensometer data when using a mobilized dilation angle model.

6.Conclusions

Using several case studies,the ability of an appropriate mobilized dilation model combined with a CWFS strength model to accurately replicate observed brittle deformation in situ has been demonstrated.Although there is still uncertainty associated with exact parameter values obtained from the back analyses performed due to the relative lack of in situ data available for any one case study,the applicability of the mobilized dilation angle is clear.

It appears that the range of parameter values obtained from laboratory-testing results is appropriate for modeling brittle rock masses in situ;this is consistent with the concept that for sparsely structured rock masses,the structure presented has limited in f l uence on the overall yield process(Hajiabdolmajid et al.,2002; Diederichs,2007;Carter et al.,2008).One key deviation from the laboratory results is that it appears that it may be possible to represent the dilation decay parameter(γ*or eps,*)by a single value in situ,rather than using separate values for uniaxial and triaxial conditions.This requires further veri f i cation,however,particularly for cases with high data density near the excavation wall.

In the absence of a well-de f i ned guideline for parameter selection during the user-controlled iterative back analysis process, input values were varied according to the degree of uncertainty associated with each parameter.As is shown by the sensitivity analysis presented in Section 3.2,changes in individual parameters from the back analyzed solution do not provide an improved model-data f i t.Although the parameter sets obtained have not been objectively con f i rmed to be global optima,they do represent reasonable solutions given the available data constraints.Despite the fact that the parameter solutions may be non-unique,theexistence of solutions for multiple cases con f i rms the appropriateness of the constitutive model used for rocks which deform through brittle processes.

Fig.15.Extensometer data from the Donkin-Morien tunnel(top),the Silver Shaft (middle),and a mine shaft in Arizona(bottom);areas where the displacement pro f i le slope is not regularly increasing are circled.

Although the actual brittle failure process involves irregular strain localization,it is dif f i cult to accurately capture this behavior through the use of continuum models.Even with the use of extremely f i ne meshes,the number of factors involved in both the numerical and physical bifurcation during yield makes any numerical result indicate an irregular yield zone potentially suspected.The main conclusion of this study is that even if bifurcation does occur in situ,the overall behavior of the rock mass(as recorded by extensometer measurements)can still be captured reasonably well for practical purposes using a mobilized dilation model,even if the details of strain localization are not fully resolved due to mesh size constraints.

Fig.16.Arizona mine shaft case study:extremely f i ne mesh used to model strain localization(top),contours of plastic shear strain with extensometer location indicated (middle),and comparison of model results and extensometer data(bottom).

Con f l ict of interest

The authors wish to con f i rm that there are no known con f l icts of interest associated with this publication and there has been no signi f i cant f i nancial support for this work that could have in f l uenced its outcome.

Acknowledgments

This work would not be possible without funding support from the Natural Sciences and Engineering Research Council of Canada (NSERC),the Center for Excellence in Mining Innovation(CEMI), and the Nuclear Waste Management Organization of Canada (NWMO).

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Gabriel Walton obtained a Bachelor’s degree in Geological Engineering from Queen’s University,Kingston,Canada(2011). His initial area of study was focussed on applied geophysics, but his main interest shifted to rock mechanics as he started a Ph.D.with Dr.Mark Diederichs(2014).Gabriel’s research aims to improve our understanding of the impact of brittle rock dilatancy on the stability of underground excavations and to enhance thecapabilities ofcontinuummodels forrepresenting in situ rock mass behavior.He also has an interest in applications of geophysics for tunnelling and mining,as well as rock mass characterization using LiDARand photogrammetry data.

*Corresponding author.Tel.:+1 613 893 4223.

E-mail address:7ggw@queensu.ca(G.Walton).

Peer review under responsibility of Institute of Rock and Soil Mechanics, Chinese Academy of Sciences.

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http://dx.doi.org/10.1016/j.jrmge.2014.09.004

Continuum models

Case studies

Brittle rock