Behaviour of Rock Fractures under Grout Pressure Loadings: Basic Mechanisms and Special Cases

Loadings

Basi Me hanismsandSpe ialCases

RIKARD GOTHÄLL

Do toral Thesis

DivisionofSoiland Ro k Me hani s

RoyalInstitute ofTe hnology

Sto kholm,Sweden, 2009

ISSN 1650-9501 SWEDEN

AkademiskavhandlingsommedtillståndavKunglTekniskahögskolanframlägges

till offentliggranskningföravläggandeavDoktorsexamen2009-01-22iSalF3.

^{RikardGothäll,November6,2009} Try k: Universitetsservi eUS AB

Sealingofundergroundex avationshasbeenadauntingengineering hal-

lengesin emanrstventuredbelowthesurfa eoftheearth. Inre enttimes,

ithasre ievedin reasedattentionasthegeneralpubli hasbe omeawareof

thepossibleissuesrelatedtounsu essfulsealingattempts. With omplexin-

frastru tureproje tsinthe entreoflarge itiesandthe hallengesfa edwhen

buildinganu learrepository,theneedforadeeperlevelofunderstandinghas

ne essitatedresear hintothe omplexme hanismsinvolved.

Inthisthesis,theme hani soffra turessubje tedtoaninternaloverpres-

sureis investigated; a problemthat hasbe ome in reasingly relevant. The

load transfer me hanismsof fra ture interfa esare investigatedin orderto

understandthe intera tionof fra turesand pressurised liquids. This me h-

anism determines the me hani alresponse of the fra tures up to a ertain

pressure. Above thatpressure, alledthe riti alpressure, the deformation

is governedbythe properties ofthe adja ent ro k mass. Two possiblero k

mass ongurations are investigated; one with a solitary fra ture and one

withtwo,simultaneouslygrouted,intera tingfra tures. Thefra turedilation

is in ea h ase shown to be proportional to the penetrated length and the

ex esspressure. Theimpli ationoffra turedilationandfra ture-fra turein-

tera tionduringgroutingisdis ussed aswellasthelimitations andpossible

advantagesofusingpressuresex eedingthe riti al pressure.

9

Hehathstret hedforthhishand

totheint,hehathoverturned

mountainsfromtheroots.

10

Inthero kshehath utout

rivers,andhiseyehathseen

everypre iousthing.

11

Thedepthsalsoofrivershe

hathsear hed, andhiddenthings

hehathbroughtforthtolight.

12

Butwhereiswisdomtobe

found,andwhereisthepla eof

understanding?

Job28:9-12

Wel ome!

This thesis is the fruit of many years of labour at the department of Soil and

Ro k Me hani s at KTH. The material presented in this thesis is a fra tion of

theentireproje t,but that doesnotmeanthat theeortput elsewherehasbeen

in vain. During the span of the proje t, many dierent leads and angles have

beenexplored,withvaryingout ome. Therehavebeeninterestingadvan esmade

in several neighbouring elds, su h as drawdown al ulations and network ow

modelling, allthoughnotallofthose resultshavemadeitto thepubli domainas

peer reviewedpapers. Thepathofs ien eisasmeanderingasthepathdes ribed

bywaterstreamingthroughthero kandastortuousasthepathtakenbythegrout

asitspreadsthroughtightly ompressedfra tures. Astheworkhasprogressed,the

goalhas nevertheless remained lear asa guiding star; thework should result in

somethingthat hasadire t appli ation forthosethat arvethebarrenro kfora

living and not onlyfor the inhabitants of Laputa. The oming years will be the

judgeon howwell this goalhas been a hieved, but hope remainsstrong that the

rulingwillbefavourable fortheplainti.

if, thegrouting pressureex eedsthe riti alpressure,thefra turein questionwill

dilate. Thedilationwill fa ilitategroutpenetrationin thatfra ture. Thedilation

mayimpedegrout penetration in adja entfra tures but that is lessof aproblem

if the fra ture spa ing is more than a few metres. Dilation in ombination with

largegrouttakeis are ipefordisaster. Fra turedilationisalsolesslikelytobea

problemin deepthaninshallowex avations.

Main Papers

This thesis in orporates the results of two paperson the subje t of fra ture be-

haviourduringgrouting:

•

^{R.Gothäll andH.Stille, Fra turedilationduring grouting in Tunnel. Un-}

dergr. Spa e Te hnol., Volume 24, Issue 2, Mar h 2009, Pages 126-135 ,

doi:10.1016/j.tust.2008.05.004

•

^{R. Gothäll and H. Stille,} Fra ture-fra tureintera tion during grouting in Tunnel. Undergr. Spa eTe hnol.,A eptedforpubli ation

Additional Publi ations

Inaddition to thetwo aforementionedpapers, theauthor has also presentedthis

resear h at the Ameri an Ro k Me hani s Asso iation's annual symposium AR-

MAro ks2008,in SanFran is o,USA.

•

^{R.GothällandH.Stille, Fra turedilationduring}high-pressuregrouting in ARMAro ks08.

Theauthor hasalso o-authoredapaperona ompletely dierenttopi that has

beenpublishedin thejournalAnalyti al Bio hemistry.

•

^{M. Wiklund, O. Nord, R. Gothäll, A.V.} Chernyshev, P-Å. Nygren, and H.

M.Hertz. 2005. Fluores en e-mi ros opy-basedimageanalysis foranalyte-

dependent parti le doublet dete tion in a single-step immunoagglutination

assay. inAnal. Bio hem.,338:90-101.

Contents 11

1 Introdu tion 13

1.1 History andBa kground . . . 13

1.2 Obje tives . . . 16

1.3 Extentand Limitations . . . 16

1.4 Denitions. . . 17

1.5 Reading Instru tions . . . 18

2 A Brief Outlineof the Thesis 19 2.1 Introdu tion. . . 19

2.2 GroutingforBeginners . . . 19

2.3 Con lusions . . . 22

3 Basi Assumptions 23 3.1 Introdu tion. . . 23

3.2 Fra tureGeometry . . . 23

3.3 GroutSpread . . . 28

3.4 Pressure Distributionduring GroutSpread . . . 31

3.5 TheMaterialPropertiesofRo k . . . 32

3.6 Stressesin Ro k . . . 32

3.7 Con lusions . . . 33

4 Fra ture Stiness 35 4.1 Introdu tion. . . 35

4.2 CompressionofaSingleFra ture . . . 36

4.3 Elementarymodelsforfra turestiness . . . 42

5 Fra ture Dilation DuringGrouting 49 5.1 Introdu tion. . . 49

5.2 Ee tiveStress . . . 49

5.3 Loadingandunloadingofafra tureduring grouting . . . 50

5.4 Fra ture-Fra tureIntera tion . . . 53

5.5 Transitionals ales . . . 64

5.6 AlternativeInterpretations . . . 65

5.7 Con lusions . . . 66

6 Measurementsof Dilation 69 6.1 Introdu tion. . . 69

6.2 Experimentalveri ation . . . 69

6.3 Prin ipalbehaviourofpre-loadedfra tures . . . 70

6.4 Summary . . . 75

7 Con lusions and Dis ussion 77 7.1 Introdu tion. . . 77

7.2 Summary . . . 77

7.3 Con lusions . . . 80

7.4 Dis ussion . . . 81

7.5 FutureResear h . . . 82

Referen es 83 A Supplementalresults from FEM-simulations 87 A.1 Dee tion duringlongrangepenetrations . . . 87

A.2 Simulationswithnon-linearstiness . . . 89

A.3 Othersimulationswithlinearstiness . . . 93

B The Kelvinfun tion;

kei

⁹⁵

II Fra ture Dilation duringGrouting 97

III Fra ture-Fra ture Intera tion duringGrouting 109

Introdu tion

Howgreatseemshumanprogress

whenwe onsiderwhereitbegan,

andhowinsigni ant, whenwe

ontemplatethegoalsforwhi h

itstrives.

FranzGrillparzer,1820

1.1 History and Ba kground

Inowtounderground onstru tionshasbeenalargeprobleminundergroundengi-

neeringsin emanrstventuredbelowthesurfa eoftheground. Inthemid1600s,

overhalfthestaattheFalun opperminewereo upiedwiththedrainageofthe

mine. Theadventsofme hanisedpumps wasabig stepforwardforunderground

engineering,butitwasnotuntilmu hlaterthateortsweremadetoa tuallystop

theinowofwaterintoundergroundfa ilities.

Sin e then a great deal has happened, but the problem with inow of water

remainsalarge on ernin tunnel buildingproje ts. Thoughthefo ushasshifted

fromtheproblemofdrainingundergroundex avationstosealingthem,the ostof

doingsostillremainsalargepartofthetotal onstru tion ost. Theenvironmental

impa tofsealingor failingto sealunderground onstru tionshasalsomovedinto

fo us in re ent time. The inow ofwater anlowerthegroundwaterlevel whi h

willresult in damageto both thee osystemandthe lo al e onomy. Unsu essful

eortstoseal the inow may also resultin un ontrolledspread ofsealant. These

twoproblemsarenotmutuallyex lusive.

GroutingTradition in Sweden

InSwedensealingunderground onstru tionshasmainlybeendoneby ementitious

grouting. This is partlydue to high availabilityof aordable ementgrout. The

geology in Sweden onsists for the most part of hard jointed graniti ro k with

frequentwaterbearingfra tures. Thistypeofgeologyofteneliminatestheneedfor

a on reteliningforsupportpurposesbutrequiressubstantialeortstobesealed.

Groutingis usuallyintegrated asapartof thenormalex avation y le. Holes

are drilledaroundthetunnel fa e in theshapeofafan and groutis inje tedinto

these holes,onebyoneorafewsimultaneously,atmoderatetohigh pressure.

Thispro essmaybegovernedbya ontrolpro esswhereinoworpermeability

ismeasuredbeforeorafterthegroutingisperformedandtheresultwillindi ateif

anothergroutingroundistobeperformed.

Thehistoryofearlygroutingresear hinSwedenissummarisedinStille(1997).

Sin e thentherehavebeenongoingeorts on erningthe hara terisationof ro k

fromagroutingperspe tive(Fransson,2001),the hara terisationofgrout(Eklund,

2005)and thepredi tionof groutingresult(Eriksson,2002; Dalmalm,2004). The

resear h on engineering methods for sealing underground onstru tionsis losely

relatedtothatofuidowinfra turesthatis urrentlybeingperformedinSweden.

Why and When toGrout

Themostrudimentaryreasonforsealinganunderground avityistoavoidooding.

Iftheinowislowerthanthepumping apa itythisisnota on ern. Insu ha ase

itmaybetheworkingenvironmentorinstallationsthatrequiredry onditions,thus

ne essitating sealingof thero kmass. Fulllingsu hrequirementsisnotamajor

on ernin Swedish pra tiseanditmaybea omplishedwith routinepro edures.

Intheabsen eoftheunderground onstru tion,thewaterthatispupmedaway

from it would have been a part of another hydrologi al pro ess. That pro ess

may be held at a valuegreat enoughto motivate more ostly sealing pro edures

in order to leave itundisturbed. The disturban eof other hydrologi al pro esses

in thevi inityofthe onstru tionsiteisoftentheprimaryandlimiting on ernin

ex avations losetourbanorothersensitiveareas.

As previously stated, grouting is often in luded in the drill-blast-mu k y le.

Then it is often alled pregrouting. It an also be performed after the ex ava-

tion y le at greater expense and often with greater un ertainty. It is therefore

oftenstressedthatthepregroutingshouldyieldsatisfa toryresultsbeforethe y le

ontinues,preferablywithoutdelayingthepropagationofthefront.

Sometimes,oftenwhentunnellingatshallowdepth,sealing anbeperformedby

drillingfromthesurfa eintothevolumethatistobeex avatedandinje tsealant

from there. Thisform ofgroutingis onsideredalast resortfortroublesome ro k

masses orwhenanunshieldedTunnelBoringMa hine, TBM,istobeused.

Environmental on erns

Inanystablee osystemthehydrologi al y leisanimportantpartofthee ology.

Any hanges in hydrogeologi al onditionswill havean impa ton thee osystem.

Dependingonthesensitivityofthise osystem,thelargesta eptableperturbation

of the hydrogeologi al y le may be very small. All water that drains from the

e osystemwould haveplayedapartsomewhereelse. Sin ethewaterpumpedout

ofanundergroundsiteisnotusuallyreinltrateddire tlyabovethedrainagesite,a

lo aldrainageofwaterwillo urinthevi inity,eitherdire tlyabovetheex avation

ordownstream. Thismayleadtoaloweredground-waterlevelandaredu tionor

hangeinthevegetationin theae tedarea.

A loweredground-waterlevel anhaveother severe onsequen es in anurban

areaasfoundationson layorwooden pilesaredependentonaminimumground-

water level. If thedrawdown due to drainage brings the ground-water level to a

levelbelowthisminimumlevel,settlementsin layandrottingwoodenpileswould

ensue, ausing permanentdamagetothesupportedstru tures.

The la kof sealing may ause a drawdown of the ground water levelbut the

a t of sealing itself may ause environmental damage. This is not surprising for

hemi al grouts but ementitious grout has very high pH and may ause lo al

damage if it spreads outside the ro k mass. Additives and byprodu ts from the

groutmayalsoleakintotheground-waterandrendernearbywellsunusable.

Pressure-related questions in GroutingResear h

Thegoalofthegroutingpro edureistoqui klysealthero karoundthetunnelby

lettingthegroutllallvoidssurroundingtheex avationperimeter. Thiswillforma

tightsealaroundthero kmassthatistoberemoved. Thegroutisinje tedthrough

bore holes and should spread su iently to ll all a essible voids between two

adja entboreholes. Itshould not, however,spreadin amannerthat would ause

thegrouttomoveawayfromtheex avation,llingvoidsthatarenotne essaryto

sealfroma onstru tionstandpoint. Thisisnotonlyawasteofgroutbutmayalso

leadtopubli relationsproblemsfortheentrepreneurifgroutwastoleakthrough

fra turestothesurfa eashashappenedonsomeo asions. Controllingthespread

ofgroutinsidethero kmassisofhighimportan eifapredi tableand ost-ee tive

groutingresultistobea hieved.

Whentryingto a hieveoptimalgrout spread,the parametersavailable to the

designerareusuallythe hoi eofgroutmix,boreholespa ing,pressureappliedand

timeorvolumespent pumpingin ea h borehole. The hoi e ofgrout typeis also

afa tor as dierent varieties have varying ability to seal ne fra tures. In what

is often alled "the Norwegian method", the pressure is hosen to be as high as

possiblein orderto a hievethedesiredgroutingresult. Other strategiesareoften

morefo usedonthevolumeofgroutinje tedorthetimespentgrouting.

Higherpressureswillleadtohigherowratesandthusshorterpumpingtimesin

ordertoa hievetargetvolumes. Higherpressuresalsoin reasetheriskofa hieving

undesirablegroutspread.Therehasbeenseveralin identswheregrouthasemerged

atthesurfa eafterhavingtravelledlongdistan esinopenfra tures. Thattypeof

behaviourisbelievedtobelinkedtotheusageofunsuitablyhighpressures. There

arealso no indi ations that high pressuresand high grout takesare bene ial to

thesealingperforman e.

Therefore,oneofthemost urrentproblemsishowthepressurisedgroutdeforms

the surrounding ro k mass and how this will ae t the sealing result. In this

thesis the me hani s of grouting are modelled in order to better understand the

fundamental behaviouroftheseme hanisms.

1.2 Obje tives

The general obje tive of this thesis is to investigate the last problem from the

previous se tion; how the pressurised grout deforms the surrounding ro k mass.

This isdonebyfullling thefollowingsub-obje tives:

•

^{Tounderstandtheloadtransferme hanismsa rossafra tureduringgrouting.}

•

^{Tomodelhowoneormorefra turesdeform duringhighpressuregrouting}

•

^{Toputtheknowledgegainedinto awider}perspe tive

Thekeyissueisto gainunderstandingregardingaphenomenonthatis illusive

and di ultto observedire tly. Thisunderstanding is gained throughreasoning,

basedonfundamentalme hani al on epts. Ifallassumptionshold,thesystem an

beexpe tedtobehaveina ertainway,notbe auseit an,but be auseitmust.

Thelastitemin thelistisperhapsthemost hallenging. Whoseperspe tiveis

themostrelevant,andisitpossibleforsomeonewhola ksrsthandknowledgeof

thepra ti aloperationsto havethesameperspe tiveasonewhohassu hknowl-

edge? Istheperspe tiveoftheentrepreneurreallyrelevant? Perhapsthedesigner,

who has to make de isions based on a minimum of information, is the one who

has the most relevant perspe tive? Although the impli ations of the results are

dis ussed, it is notthe authors intentto for e his perspe tiveon the reader, but

rather to present the knowledge gained through the re ent years work in su h a

waythatthereaderathemselves anform su haperspe tive.

1.3 Extent and Limitations

Thisthesisdoesnot overorintendto overtheproblemssurroundinggroutingin

soils, lays,orpoorqualityro k.

Thefo usoftheresear hin thisthesisisonthelongtermgoaloftheSwedish

Nu lear FuelWasteManagementCorporation,SKB.It thereforemainly on erns

theme hani sandproblemsofgroutinginhighqualityhardjointedro kwithhigh

probabilityofwaterbearingfra tures. Thenalrepositorywillbelo atedatgreat

depth but the onstru tionoftherepositorywillstartat groundlevel. Therefore,

in this thesismodellingwill notonly be onnedtosituationswith high onning

stressandhigh waterpressure.

Thisthesisdoesnotinanyway overtheproblemsrelatedtogroutproperties,

su hasstability,ltration,oradditives. Forallaspe tsofthisthesis, ementitious

groutis onsideredto beahighlyvis ous NewtonorBinghamuid,depending on

thesituation. Thismakesmostmodelsandassumptionsvalidforothersealantsas

well.

It is not possibleto overall previous resear h on fra tures and the oupling

betweenhydrologi alpropertiesand me hani alproperties. Theliterature review

inthateldhasbeenextensive,butverylittleofthepreviousresear hisappli able

togroutingproblemssu hasthosedes ribedinthisthesis.

Therearealsomanyotherwaystota kletheproblemsinvolvedandmanyother

aspe tsthat are not overedormentioned. Theworkdes ribedin this thesishas

hada learfo usonunderstandingthemostbasi andfundamentals enariosrather

thanthe omplete hartingofevery on eivable ombinationoffra tures,stresses,

andgroutingte hniques.

It should also be noted that the modelling is limited to situations where the

pressuredoesnotimposealoadonthero kmassthat ex eedsthebearing apa -

ity. Thefollowing hapters dealwith dierenttypes ofelasti deformations. The

analysis anbeextendedtosituationsthatwouldin ludethepossibilityofinelasti

failure,butthisisnotin ludedinthisthesis. Forareviewofthattypeofproblems

seee.g. Brantbergeretal.(2000)

1.4 Denitions

A hievable radius: Also knownas a hievable distan e, usually designated

R

^or

I max

^{,thisisthemaximumdistan eaBinghamuid antravelin a onduit.}

This is usually not the same thing asthe penetration distan e

r g

^{whi h is}

the a tual distan e from the inje tion pointto the rimof the grout spread

pattern.

Criti alpressure Theminimumpressureneededtoindu eja king. Ex esspres-

sureis alledpost- riti alpressure.

Hydrauli fra turing: Theformationofnewfra tures,orpropagationofexisting

fra tures,duetoanappliedhydrauli pressure. Hydrauli fra turingmaytake

pla e during ja kingand vi eversa, but there is astri t dieren e between

thetwo on epts.

Ja king: Thestatewheretheee tivestressin thefra ture,orparts ofthefra -

ture, is zero. The ja king an either be reversible, elasti ja king, or irre-

versible,plasti ja king.

Load-ae ted volume: The volume of ro k that is subje tto relevant hanges

instrainduetothegroutingoperations.

Sealinge ien y: Ameasureofthesu essofagroutingoperation. Itisusually

measuredas theper entageofinowremovedfromanunderground ex ava-

tion.

Ultimatepressure: Thepressureneededtoindu euna eptabledeformations.

Un ontrolledgrout ow: A onditionwheretheowofgroutwilldeviatefrom

thedesiredspread. Mostoftenthedesiredspreadpattern isa ylindri al2D

pattern with the borehole in the entre. An example of un ontrolled grout

owiswhenthegroutndsanunusuallylarge onduitleadingalongdistan e

awayfromthezoneintendedtobesealed.

Uplift: A deformation of thero k mass that primarily onsists of translation as

opposed toa hangeinstrain.

1.5 Reading Instru tions

The target audien efor this thesisare individuals with someba kgroundin ro k

me hani s and ursory knowledge in re ent resear h in the eld, espe ially the

resear hperformedinSweden.

Chapter2isanattemptatdes ribingthekernelofthisworkin amannerthat

islessstringentbuthopefullymorea essibletothosethatareunfamiliarwiththe

notationandformulationsofa ademi writings. Forthosethatla kinterestinthe

mathemati aldes riptionbutwanttoavailthemselvesofsomelevelofunderstand-

ing,this hapterand hapter7should su e.

Chapter3providesthene essaryba kgroundmaterialforthemodellingin the

following hapters. It anbeseenasasele tstudyoftheavailableliterature.

Chapters5and4 onstitutethebulkoftheresear h. Thepredi tionsmadein

these haptersarethenusedtointerprettheresultsfromafewsetofmeasurements

madebyotherresear hers. Thosemeasurementsandtheinterpretationofthemare

presentedin hapter6.

Themajorpartof thedis ussiontakespla ein hapter7. The haptertriesto

build abridgebetweenthetheoryanditsappli ations.

It should also benoted that the papers do not ontainany vitalinformation

above andbeyond what is presented in thethesis. They arehoweverin luded in

this thesisforthepurposeofreferen e.

A Brief Outline of the Thesis

Everygeneralisationis

dangerous,espe iallythisone.

MarkTwain

2.1 Introdu tion

Inthis hapter I will tryto des ribe themodelling and theresultsin awaythat

islesss ienti but, hopefully, a essibleand easyto read. Formulæ aregreat as

theyhavetheabilityto onveylargeamountsofmeaning,buttheydonarrowthe

audien eabit. Thiswill hopefullybeahandy hapter for thosewhoare pressed

fortimebut have onden e inmymodellingandjustwantabriefoutlineaswell

asforthosewhohavehadthesensetoprioritiseotherthingsinlifethana ademi

studies.

2.2 Grouting for Beginners

Ro kisaninterestingmaterial. Whereasmost onstru tionsarebuiltbythejoining

ofinta t buildingblo ks, anunderground ex avation is ompletely dierent. The

ro kis already broken when the onstru tion startsand atunnel is built by the

destru tion and removal of material. In S andinavia, the ro kis usually of high

quality, and the remains after an ex avation is usually self supporting with little

need for additional support. Even though the ro k is onsidered to be of high

quality,thero kislitteredwithfra tures. Mostfra turesarethinnerthanasheetof

paperandtightly ompressedbytheweightofthesurroundingro k. Thefra tures

are also oarse and rough to the tou h. Even though the fra tures are tightly

ompressed, the roughness will prevent them from being ompletely losed and

waterwillbeabletoowthroughthem.

It isthis waterthat is theproblem, asitwill keepowinginto theex avation

untiliteitherhasbeenlledorthewaterrunsout. Usually,neitherofthoseoptions

area eptableandtheonlyoptionthatremainsistotrytosealthefra tures,whi h

is done by pumping ement into the fra tures. This pro ess is alled grouting.

A su essfully grouting operation is the result of a orre t design and awless

exe ution. The design part onsists, among other things, of the hoi e of grout,

drilling geometry, pressureand stop riteria. A awless exe ution would in lude

arefulhandlingofthe omponentsforthegroutandatimelyinje tionofthegrout

into thero kfra tures.

Ea h oneofthese steps would beaworthy andidate forresear h, but in this

thesis, themain andidatefor examinationhasbeenthepressure. Howto hoose

atwhatpressuretopumpthegroutisnotobvious. Higherpressureswillmakethe

groutowfaster,whi hisgood,butalsofartherawayfromwhereitisinsertedinto

thero k,whi hisgoodonlyuptoa ertainlimit. Ifthegroutmovestoofaraway

fromwhereitisintendedtogo,itisnotonlywastedbutitmaybe omeaproblem

as itmaysurfa eandpollutethesurroundings.

The hoi e ofahigh pressureisalso asso iatedwith anotherrisk. If thepres-

surebe omeshigh enough,theweightofthero kmaynotbeenoughtokeepthe

fra turessqueezed. Thisproblemhasbeenthesubje tofspe ulationforsometime

andithasbeenthegoalofthisresear htotrytoanswerthequestionsofwhenand

howthis ould happen. Ifa fra ture isopenedby thegrout pressure,it is alled

ja king.

How GroutMoves Inside Fra tures

When grouting, a number of boreholes are drilled into the ro k and then lled

withgrout. Fra turesthatinterse tthoseboreholesarethenalsolled withgrout.

If everything goes a ording to plan, als fra tures interse ting other grout lled

fra tures will be lled with grout. The grout will travelthrough all interse ting

fra turesuntiltheentirefra turenetworkhasbeenlleduporpumpingisstopped.

Thispro esstakessometimeasthegrout anbequitethi kandthefra turesare

verythin. The grout also tends to ow along thepath of least resistan e, whi h

meansthatitwillowthroughlargeopeningsrstuntiltheyarefull, thussealing

thenestfra tureslast.

Whenthegrouthaspenetratedabitintothefra tures,itwillbe omeharderto

pump andtheowwillgraduallyde rease. To ompensate,thepressureisusually

in reased. Nowthis iswherethingsget interesting. Theweightof thero kexerts

a pressure on the fra tures, but so does the grout. When the grouting pressure

be omeslargeenoughto rivaltheweightofthero k,itwill begintoliftthe ro k

andthefra tureswillthen easetobetightlysqueezed. Atthatpoint,ja kinghas

ommen ed.

Ja king of Fra tures

Whathappensafterja kinghasbegundependsonthe ir umstan esandthesur-

rounding ro k. The groutwillowmoreeasilyinside thefra tureswhen theyare

ja ked openand itis reasonableto believethat itwill rea hfurther into aja ked

Figure 2.1: A s hemati illustration of a possible s enario. The dark grey grout

deformstheadja entro kandfra ture.

fra ture. The ro k that is being moved by the grout will most likely not have

anywhereelsetomove,sotheamountofja kingmaybelimitedratherqui kly.

It should be noted that it will be the largest fra ture that will open up the

most. Thiswillhappenat theexpenseofthesurroundingfra tures. Itistherefore

reasonabletoassumethatja kingmaymakethegroutgofurtherinlargefra tures,

butatthesametimelimitthegrout'stravelinthenefra tures.

Anotherthingthatshouldbenotedisthatwhenafra tureisja kedopen,grout

isnot theonly thing that will owmoreeasily inside thefra ture -sowill water.

Thepointoftheoperationwasto makethewaterowmoreslowly,orrathernot

atall. Ifafra tureisleft ja kedopenwhen thegrouthardens,theparts that are

notlledwith grout will a t ashighwaysfor thewater, and thus, quite possibly,

ompletelynegatetheextrasealing.

Two Spe i Cases

Therstspe i aseistheoneofasingle,verylarge,fra ture deepinside aro k

mass. It will nothave any neighbouringfra tures to squeeze losed, only ro kto

ompress. If the fra ture is assumed to be symmetri around the borehole, this

will also apply to the grout spread pattern. Let us also assume that the grout

haspenetratedinto thero kforawhile andthat thepressurehasbeenin reased

graduallyuntil it ex eededtheweightof thero k onthefra ture. Atthat point,

thegroutwill liftaregion ofro k losetotheborehole,but asthepressuredrops

awayfromthe entre,notallgroutwillliftupro k.

The load on the ro k will be the ex ess pressure and the ro k will resist the

loadbydeforming andthusspreadingtheloadoveralargerregion. Theshapeof

the deformation an be found in a number of textbooks as this is awell known

problemin lassi alme hani s. As anbeseen(ingure2.1),eventhoughtheload

is on entratedtothe entre, thefra tureopensupsomedistan e away.

The se ond rather spe i ase is probably a bit more ommon. It involves

two fra tures, still of innite extention, parallel to ea h other. The ro k that is

between them an be said to besuspended between the fra tures. If we further

assume that thefra tures areslightly dierent in size,one beinglarger than the

other, and that they are being grouted at the same time, we will get a bending

loadonthefra ture. Theshapeofthebending anbe al ulatedeitherbyhandor

with a omputerand theshapeofthedeformation anbeseeningure5.4. Even

though thepressurein bothfra tures is largeenoughto ountera t theweightof

the ro k, thelarge fra ture will pushharder on thero kslab between them and

push it into the thinner fra ture, ee tively keeping it losed and negating any

advantageexpe tedfrom the higherpressure. This typeof fra ture intera tionis

likelytoo urifthefra turespa ingislessthanafewmetres.

2.3 Con lusions

In many ases, it willbe thesealing of thethinnest fra tures that will determine

the su ess of the grouting operation. If the grout has not travelled far enough

into thosefra tures,thesealingwill notbegoodenough. Ifthe hosenpressureis

toolow,thegroutwillnottravelfastenoughtoa hievegoodpenetrationbeforeit

hardensoroperations ease.

Ontheotherhand,ifthepressureistoohigh,thepresen eoflargefra tures an

tightenthesqueezeonthinfra tures,sealingthemfromgrout,butnotfromwater.

Thepressure anthereforebebothtoohighandtoolowatthesametime. Insu h

asituation,boththepressureandthepropertiesofthegroutmust be hanged.

Basi Assumptions

One annotexpressla kof

knowledgeinarmative

language.

Mi heldeMontaigne,1588

3.1 Introdu tion

This hapter will des ribe some of the fundamental assumptions that have been

madeinordertomodelgroutingrelatedproblems. Itisbasi allyaliteraturestudy,

althoughwithafo usonprovidingne essaryba kgroundmaterialforthefollowing

hapters. Thetopi sinthis hapteraresele taswellasessential. Theinsightinto

theme hani sof fra tures omes largely from thestudy ofwhat is impossibleto

knowabouttheirgeometry. Thetopi offra turegeometryhasbeenstudiedquite

extensively, but with ala kof resultsappli ableoutside afewspe ial dis iplines.

Thetopi isimportant,but asne essityandbrevityarenotmutuallyex lusive,it

isonly overedin ursorydetail. Soisalsothepartongroutowmodels,but not

be ausethereisala kofunderstandingonthetopi ,butratherbe ausethemost

basi modelsprovidessu ienta ura yforthesubsequentreasoning.

3.2 Fra ture Geometry

The Parallel Plate Model

The most simple and ee tive model of a fra ture is the parallel plate model in

whi h a fra ture is modelled as the void between two at, smooth, and parallel

surfa es without onta t. The advantages of the parallel plate model is that it

iseasy to graspand hasanalyti alsolutionsto the owequations forbothwater

and grout (Gustafson and Stille, 2005). The downside of the model is that it

does not faithfully represent any me hani al properties of the fra ture, su h as

fra turestinessoraperturevariations. Thersteortsmadetoappendadditional

propertiestothis model, fromagroutingperspe tive,wasby Hässler(1991),who

introdu edthePie e of ake-model, wheretheparts ofthefra ture wallsthatare

in onta twithea hotherarea ountedfor. Itmodelsgrouttakemorea urately

than the parallel plate model but does not in lude any other hydrome hani al

propertiestobepredi tedmorea urately.

The parallel plate model always yields rotationally symmetri alowpatterns

for grout. Fluidsin realfra tureswith smallandvarying aperturesdonotowin

that fashion. The variations in aperture guides the owof uids in the fra ture

into " hannels" (Pyrak-Nolteet al., 1997). This hannelling ee t anbe learly

seenwhenawater- ondu tingfra tureinterse tsanundergroundex avation. The

majorityofthewaterseepingthroughthefra ture omesat afewdis retepoints

alongthefra turetra e.

Fra tal and Self-aneModels

Sin etheadventof haostheory(Mandelbroot,1984),fra talmeasureshavebe ome

in reasingly popular when hara terising irregular and omplex geometries. The

surfa eofaro kfra ture onsistsof lustersof rystals,whi hareinturn lustered

together in larger and larger formations. It is apparentfrom looking at pi tures

of ro k out rops or ro k surfa es that there are s aling phenomena at work. If

noreferen eobje tsarepla edinthepi turetoshows ale,thedimensionsof the

images an be very hallengingto determine. The very nature of thero k seems

toindi atethatfra talmathemati s ouldbeanee tivetoolintryingtodes ribe

thegeometryoffresh ro ksurfa es.

Fra tal mathemati s is howevernot as simple asit may seem at rst glan e.

Measurements with fra tal methods require the utmost are, both in the a tual

measurementsand intheinterpretationofthedata(WalshandWatterson,1993).

Thenumberofworksthathavetriedtodes ribedierentfra turepropertieswith

fra tal methodsis overwhelming. Despiteallthese resear heortstherehasbeen

verylittleprogressinthiseld. TheearlyresultsofBrownandS holtz(1985a)have

not beenimproveddespite several eorts(see gure3.1). This is in theauthor's

opinion largely due to the inherent, but unapparent, di ulties in the required

analysis.

Fra tal Dimension

Themostdistinguishingnew on eptoffra taltheoryisthatoffra taldimensionor

Hausdor dimension. Itisas alingpropertyoffra talsthatisremotelyrelatedto

spatialdimensionalitybutthetwoshouldnotbe onfused. TheHausdordimension

and thefra taldimensionarepurely mathemati al on eptsunlikethetopologi al

dimensionthat anbe onsideredto bereal andalwaysaninteger.

Fornon-fra talentities thefra taldimensionisalwaysthe sameasthespatial

dimension, but for anobje twith fra tal properties, thefra tal dimension is not

aninteger. Aro ksurfa ewithfra talpropertieshasafra taldimensionbetween

2and3. Thevalueof thedimensionis loselyrelatedto the orrelationbehaviour

ofthesurfa e. Moreover,it anneverbelessthan2ormorethan3.

Thefra taldimensionofthefra turesurfa eis onne tedtothe orrelationbe-

haviourofthefra turesurfa e,whi hinturnis onne tedtothestinessbehaviour

ofthesurfa e(Pyrak-NolteandMorris,2000).

Ifallinherentdi ultieswithfra taltheory ouldbeover omeon eandforall,

afra talandself-anemodel ouldbeveryinterestingmodelfordes ribingseveral

oftheme hani alpropertiesoffra tures.

Ana uratemodelofthegeometryofafra turewithasmallnumberofmeasur-

ableparameterswouldbeofgreatvaluefortheadvan ementoffra tureme hani s.

Although no model that ould be used for all relevant me hani alproblems has

been found in the literature, there are somepropertiesthat anbe derived from

thefra talnature ofthegeometry that areusefulforthemodellingin this thesis,

theprimaryonbeingthat there arenoatfa esonafra talsurfa e. This makes

everypointalo alextreme,andanygradientmeasurewouldhavetobeanaverage

over someregion. This insight is afundamental assumption for the reasoningin

se tion4.2.

Fra ture Statisti s

Thetopographi aldatafromafra turemeasurementisusuallyanalysedwitheither

Fouriermethods or auto orrelation methods. Theresult fora Fourieranalysis is

usuallyaplotlookinglikegure3.1.

A power-spe trum anbe seenasameasure ofhowlargefeaturesof a ertain

s alearewithrespe ttothoseofothers ales. Apower-spe trumplotoftopologi al

datawillusually haveapowerlawresemblan ein somepartofthespe trum and

a gradual transformation into white noise at higher frequen ies. In order for it

to really be apower-lawspe trum the plotmust be linear overseveral orders of

magnitude. The problem with determining the range and proportionality for a

power law relationship from a power-law plot is that, sin e it is a

log / log

^-plot,

avast majority ofthesampleswill bein theupperend ofthespe trum,whi h is

dominatedbynoiseandsamplingartefa ts. Theproportionality onstant al ulated

fromsu haplotisthereforeextremelysensitivetothe hoi eofhighfrequen y ut

ooranynoiselteringmethods used.

The auto orrelation fun tion of a surfa eis losely related to boththe power

spe trum and the Kriegingfun tion of the surfa e. The waythe auto orrelation

fun tiontendstozero analsoberelatedtothefra taldimension.Theauto orrela-

tionfun tionalsohastheadvantagethatitaveragesoveralargenumberofpoints,

redu ingitssensitivitytonoise. Ifdonein2dimensions,boththeauto orrelation

fun tionandtheFourierpowerspe trawill revealanyanisotropyinthedata (see

gure3.2).

If thefra ture is anisotropi , it is likelyto haveundergone shear and there is

ahigher probability of a high, stress-independent, residual transmissivity. When

thefra ture surfa esmoverelative toea h other, theybe omeunmated, reating

Figure3.1: TheFourierspe trumforasetofprolometertra es(fromBrownand

S holtz(1985b)). Noti ehowthelogarithmi s ale ondensesvirtuallyallmeasure-

ments to the upperregion of thespe trum. This makes urve-tting sensitiveto

low-frequen ynoise.

largeropenori esthatwillremainopenevenduringhighnormalloads. Thiswill

result in ahigh residual owthat will be unae tedby in reases in normal load

(Pyrak-NolteandCook,1988).

Figure 3.2: The auto orrelation fun tion of a fra ture (from Pyrak-Nolte et al.

(1997))The orrelationfun tionsfordierentdire tionshavedierentslopesindi-

atinganisotropy.

Measurements on Fra ture Geometry

There are many properties of fra tures that an be harted, the most ommon

beingthelo ation,size,andorientation. Therearealsoseveralhydrauli teststhat

anbeperformedonafra tureorasetof fra tures. Afra ture's mostimportant

propertyis,inmanyaspe ts,itsaperture. Theapertureofthefra turegovernsthe

water-bearing apa ityandthegroutabilityofthefra ture. Ingeneralthereisalso

ahigh orrelation betweenthe aperture of afra ture and its spatial extension in

theotherdire tions(Gale,2004). Afra turehasanapertureineverypoint. These

aperturesformdistributionsand anbe orrelated. Iftheapertureinasinglepoint

ismeasured,it annotreally des ribethefra ture inanyotherwaythanthat the

maximumfra tureapertureisatleastthatlarge. Su hameasureisoften alledthe

me hani al aperture ofthefra ture. Itshould benoted,however,thatthe on ept

ofsu h a measure annot be uniquelydened and that statementsregarding the

me hani alaperture of fra tures are impossible to ompareunless thedenitions

are ompatible. In this thesis, this measure has beenavoided wheneverpossible.

Fortunately,a hangeinapertureislesssus eptibleto onfusionthantheaperture

itself, whi h hasthus beenthesalvation formu h of thereasoningin the oming

hapters.

Mostmeasurementsonthegeometryofafra turesurfa einvolveameasurement

onthetopologyofthesurfa e,eitherviaaprolometer(BrownandS holtz,1985a;

Poonet al.,1992),ormoreadvan ed methods(Xieet al.,1998;Montemagnoand

Pyrak-Nolte,1999;Lanaro,2000;BabadagliandDeveli,2003;Chaeet al.,2004).

A dierentandoften more ommonmethod fordetermining theaperture ofa

fra tureisthehydrauli aperture. Byassumingthattheparallelplatemodelholds

and that uid owin the fra ture anbe des ribed by Dar y'slawthe aperture

anbe al ulatedfromthefollowingformula

Q = − ρ W g µ W

W b ³

12 ∇ϕ.

^(3.1)

With thisdenitionoffra tureaperture,theapertureiswelldenedaslongas

thewidth,

W

^{,overwhi htheowismeasured,is hosenproperly. Thisdenition}

ofaperturedoesnot orrespondwellwithotherdenitionsandisalsoonlyvalidfor

al ulationsofowofwater,orotherNewtonuids. Measurementofthehydrauli

aperture also requires that the hydrauli gradient,

∇ϕ

^{, is known. If a fra ture}

is open into anongoingex avation, thehydrauli gradientis time dependentand

maybesomewhatelusive. Neverthelessthe hydrauli aperture hastheadvantage

of beingeasy tomeasure in amorerepeatableway. If onlyones alarmeasure is

to beused todes ribeafra turethehydrauli apertureistheonethatmakesthe

mostsense.

Con lusions on Geometry

Any deterministi modelregardingtheme hani albehaviouroffra turesoruids

movinginsidethemwouldrequireaprofoundknowledgeregardingthea tualgeom-

etry ofthefra ture. Inla kofsu hknowledge,simpli ityisthefavouredproperty

when hoosing a model. The parallel plate model does not apture any relevant

me hani alpropertiesof the fra ture, but is good enoughwhen it omes to uid

ow. Forthemodelling des ribedinlater hapters,theparallel platemodelyields

theworst ases enariointermsofloading.

3.3 Grout Spread

Most of thepreviouswork in theeld of grouting relatingto the spreadofgrout

inside fra tures. There are several, highly interesting, topi s in that eld, some

relatestotheuidowin thespe ialgeometry,othersto thebehaviourof emen-

titious groutin narrow hannels. Ee ts, su h asltration orseparation,are not

onsideredtohaveanyimpa tonthepressuredistributioninthefra ture,atleast

notinanywaythatisrelevantforthisworkandarethereforeomitted.

GroutSpread Models

Cementitiousgroutisa ompli atedliquidwiths aleandtime-dependentproper-

ties. Insome ases,it anbe onsideredtobehavelikeaNewtonuid,butinother

situations,theshear strength of thegroutwill ome into playand it will a tlike

aBinghamuid. Astimepassesthegroutwillhardenandtheshearstrengthwill

in reaseuntilithassolidied ompletely. TheBinghampropertiesof ementitious

groutis thereforeoftendominantduringthelatterstagesofgrouting.

Ifthelengths alesaresu ientlysmall thegrout annolongerbe onsidered

tobeauidbutratheraparti lesuspension. Ifthegroutismodelledasaparti le

suspensioninaNewtonuid,propertiessu hasplugsinsmallapertures,suspension

stabilityandlteringmaybetakenintoa ount(Eklund,2005).

Spe ial Bingham Properties

ABinghamuiddiersfromaNewtonuidsu haswaterinonerespe t: aNewton

uidhasno relevantshearstrength. Ifthestrainrate in aowingBinghamuid

islowerthantheyieldvalueoftheuid,therewillnotbeanyshearandthatpart

oftheuidwill owwith onstantvelo ityandwithnegligibledeformation.

The dieren e in rate of shear and velo ity prole between a Bingham uid

and aNewton uid an be seenin gure3.3. When the shear is at a minimum,

the velo ity prole of the Bingham uid will be onstant. This region is alled

the plug ow region. If the plug ow region is mu h smaller than the aperture

of the hannel the dieren e will be ome negligible (Bernander, 2004). For one-

dimensionalow,su hasowinawide,straight hannelwithparallelwalls,there

will be onesigni antdieren e; As thegrout propagates throug the hanel, the

shearstrengthofthegroutwill ountera tthepressurethatpropellsitforwardand

asthegroutpenetratesthe hannel, theareaoverwhi htheshearstrength ofthe

groutwilla tin reases. Thiswill leadto a ontinuousredu tioninowrateuntil

theow ompletely stops. Whentheowratede reasesthe plugowregionwill

growinsize andwhentheowstops theplugwilllltheentire hannel.

For any given set of ow geometry, grout mix and pressurethis will yield a

maximumdistan e fromtheboreholethatthegrout antravel. Insomeinstan es

thiswillbethelimitingfa torofgroutpenetration.

Cylindri al Flow

Iftheparallelplatemodelisassumed,thegroutwillspreadinaradiallysymmetri

wayfromtheborehole. Thismodelisonlyvalidforfra tureswithlarge 1

apertures

(Brown,1987)butisneverthelessinterestingsin eit anbesolvedanalyti ally. The

understandingoftheproblemthattheanalyti alsolutiontothismodelprovidesis

vitaltotheunderstandingoftheowofgroutinmore omplexgeometries.

1

Largeinthis ontextmeansthattheapertureislargerthanthelo alroughnessofthefra ture

sothatowisnotdominatedby hannelow.

Figure3.3: Thevelo ityandshearrateprolesforaNewtonuid(top)andaBing-

ham uid (bottom) in aparallel plate owmodel. Theshear rateisproportional

to the hangeinvelo ityand,wherethat hangebe omestoosmall,theBingham

uidwill a tasasolid.

When the grout spreadsradially from the bore-hole into the fra ture in this

model,itwilldrop invelo ityratherrapidly asitpropagatesawayfrom thebore-

hole. Thisisnotonlydueto thein reasingareaofthegroutfrontbutalsoto the

Binghampropertiesofthegroutmentionedin thepreviousse tion. Thedieren e

between Bingham ow and Newton ow is howeveronly relevant when the ow

velo ityislow. Themaximumdistan e thegrout anspreadfrom theboreholeis

alled thea hievableradius,

I max

^{and anbe al ulatedwithequation3.2}

I max = b∆P 2τ 0

(3.2)

Theboreholeradiusis inmost asesseveralordersof magnitudesmallerthanthe

a hievableradiusand anthereforebenegle ted.

In reality the maximum obtainable penetration distan e is smaller than the

a hievableradius. Thegroutwill propagate veryslowlytowardstheend andwill

not ome loseto

I max

^{in anyreasonableamountoftime.}

The solution to the time dependent problem is des ribed in great detail by

AmadeiandSavage(2001)andGustafsonandStille(2005)

Non-Symmetri Flow

The symmetri al grout spread pattern result from the parallel plate model is a

onsequen e of the symmetri geometry of the model. The geometry of a real

fra ture is notlikely to haveany kindof symmetri al properties. Theroughness

and natural undulations in a fra ture will reate variations in the aperture of a

fra ture. With the ubi law (equation 3.1) givinga high pre eden e for ow in

larger apertures the ow of grout will tend to ow in any dire tion where the

aperture is larger. For a rough naturalisotropi fra ture this will still lead to a

grout spread pattern that is statisti ally rotationally symmetri (Gustafson and

Stille,1996).

3.4 Pressure Distribution during Grout Spread

Theexa t pressure distributionin afra ture during grouting is ompli ated and

will inherently depend on the geometry of the fra ture more than anything else.

As the grout propagates through the fra ture the pressure will drop for several

reasons. Theprimaryreasonissimplythat thefrontareagrowswiththedistan e

awayfromtheborehole. Thiswillmakethepressuredropinverselyproportionalto

thepenetrateddistan e. CementitiousgroutisaBinghamtypeuid,whi himplies

thattheuidhastheabilitytoresistasmallamountofshear.Inthefra turethis

manifestsitselfasashearingfor ealongsidethefra turefa es. Theshearresistan e

isusuallyverysmall,ontheorderofafewPa,makingits ontributionnegligibleas

longastheowratesarehighandthepenetrateddistan eshort. Aftersometime

ofpumping it will, however,be thedominant ontribution to theowresistan e,

unless the grout has hardened. Close to the borehole, the ow velo ity will be

high and the drag will be dominated by vis ous for es. At some distan e away,

thevis ousfor es an benegle teddue to thelowowrates andthedrag willbe

dominatedbytheshear resistan eof thegrout. This willmakethepressuredrop

towardszero almost linearlywith distan e. If the grout spreadis assumed to be

rotationallysymmetri ,theresultingpressuredistribution anbeapproximatedas

oni al,withthepressuredroppinglinearlyfrom theboreholetothegroutfront.

Asmentionedinse tion1.3,themodellingassumeselasti onditions,e.g. that

theloadindu edbythegroutpressuredoesnotex eedthebearing apa ityofthe

ro kmass. Thisis mostlikely tobeanissue ifthefra turesare very loseto the

ex avation fa es orthe ground surfa e. It is, however, notpossible to formulate

thisin termsofgroutpressurealone,astheloadisafun tionofboththepressure

andthepenetrationaswellastheinsitu stress(seeequation5.9andBrantberger

et al.(2000)).

3.5 The Material Properties of Ro k

Ro k is a material that an havea range of dierent properties. In the S andi-

navian shield, the dominant ro k typesare hard and jointed. The inta tro kis

elasti andbrittlebutthejointinggivesthero kmassmoredu tileproperties. The

ompositionmayseemtrivial, butit yieldsa omplexs ale dependen y. Inorder

toobtainaverageproperties,volumeslargerthanthe oheren elengthneedstobe

averaged. On amedium s ale, the blo kinessof the material be omesdominant,

and at sub-blo klevel,the granularityof thero k be omes dominant. The latter

s ale isthemostimportantforthe modelling inthe following hapter asthe on-

ta tasperities arelikelyto beofthesames aleasthegrains inthero k. Atthat

level,thevariationin propertiesbe omespronoun ed. Grainsthat arefavourably

orientedare likelyto haveex eptionallyhigh ompressivestrength and it anbe

arguedthatitisthosegrainsthat formthe onta tasperities,astheyaretheones

most likely to have survived the genesis of the fra ture. In the following hap-

ters, theuniaxial ompressivestrengthofthe ro kis usedasaparameterand the

unusuallyhigh valuesforthatparameteraremotivatedbythes aleofthesystem.

3.6 Stresses in Ro k

The stress statein ro k is often hara terised by the rst prin ipal stress or the

amount of overburden present. This view is not always fully representative as

stressisatensorpropertythat anvarylo ally. Inthemodellingperformedinthis

thesis,thestressa tingnormaltoafra turewillbeofinterestanditisimportant

tore ognisethatthisstress anassumeanyvalueuptotherstprin ipalstress. A

fra ture perpendi ulartothethird prin ipalstresswouldbeless ompressedthan

oneperpendi ular to therst prin ipal stress. Duringtheformation offra tures,

thefra tureorientationisoftenrelatedtotheorientationofthestresseld,butthis

eldislikelytohave hangedovertime,duetote toni movementsandgla ialloads,

andthefra turesmaybeveryold. Theorientationofthefra tureswithrespe tto

the prin ipalstress dire tions ould thereforebe onsidered asrandomoratleast

unknown,andthenormalstressa rossasinglefra tureshouldbe onsideredtobe

unknownbut limitedbythemajorand minorprin ipalstress.

Due to the te toni movements, the S andinavian shield is horizontally om-

pressed. Thehorizontalstressisthereforehigherthantheverti alstressatnormal

ex avationdepths. Thissituationisquite ommonunlesstheex avationis loseto

amajorfault orslope.

To further ompli ate theissue, the ex avation will indu e ase ondarystress

eld in thero kmass that is being grouted. During typi al onditions, this will

in reasethe ompression tangentialto any ex avated fa es and redu e the stress

normal to the same fa es. This will make fra tures parallel to the tunnel walls

less ompressed, and as will be shown in the following hapters, more prone to

dilation. It will alsobeshownlater on that thedilation maybe omesubstantial

and in ombination with low retaining pressures, this will impose a lear risk of

ro kbursts.

Duringnormalgroutingpro edures,thegroutingholesextendfromthetunnel

frontfa ein adivergingpattern,typi ally20m long. Mostfra turesaretherefore

likelytobeseveralmetresinto thero kmasswhengrouted,butthereisalwaysa

riskofhavingfra tures losetotheex avation.

3.7 Con lusions

This hapterhasdes ribedsomeavailable modelsandmotivatedthe hoi e ofthe

mostsimpleones asabasisforthemodels inthe omming hapters. Theparallel

plate model will be used for estimation of uid ow in the fra ture and as it is

augmentedbyasuitableapproximationfor fra turesstiness, it willalso beused

topredi tthe hangeinuidowduringdilation. Eventhoughthegroutspreadis

likelytobemorea uratelydes ribedbyanetworkmodelduringtheinitialstages

ofgrouting,dilationofthefra tureislikelytoredu etheinuen eofroughnesson

theresult (Cornet et al., 2003). The spe ial Bingham properties of ementitious

groutalso giveapressuredistributionthat is virtuallyunae tedby thewaythe

grouthasspreadthroughthesystemasthe resultingpressuredistribution willbe

approximatelylinearforboth hannelowandparallelplate ow.

Fra ture Stiness

Uttensio,si vis

RobertHooke,1678

4.1 Introdu tion

The stiness of a fra ture is a vague on ept. The fra ture is just a void spa e

andas su h, has norelevantme hani alproperties. Despitethis, the presen e of

fra tureshasalargeimpa tonthebehaviourofaro ksampleorro kmass. When

afra ture isgrouted,itwill intera t,notwith thevoidspa eit devours,butwith

theadja entro kmass,mostspe iallytheasperitiesthat bridgethevoidbetween

the fra ture fa es. This hapter tries to on eptualise that intera tion in a way

thatmakesitpossibletounderstand andmodelthebehaviourofin situ fra tures

duringgrouting.

Fromagroutingperspe tive,thegenesisofthefra tureisoflittleinterest,only

thepresen ematters. Despitethat,this hapterdes ribestheloadingofafra ture,

butonlybe auseitisamore onvenientdes ription. Duringgrouting,thepro ess

anbe onsidered to operatein reverse, withnosigni antdieren esasnonon-

linearpro essesareregarded. Thepro essisofimportan eforthe on lusionsand

interpretationsandhasthereforebeendealt withinaseparate hapter.

Dierent modelsfor Fra ture Stiness

Aspreviouslymentioned, the on eptof fra ture stiness in notentirelyobvious.

Alot of resear hhas beendone in thiseld, boththeoreti al and empiri al. The

most ited models from ea h ategory is the Goodman model (Goodman, 1976)

andtheBarton-Bandismodel(Bandisetal., 1983). Boththese models, andmost

other published variants, have drawba ks that make them umbersome to use in

thistypeofresear h. Themaindrawba kisthattheempiri almodelsarebasedon

laboratorymeasurementsofsmallsamplesandthatbothmodelsutiliseparameters

thataredi ultorimpossibletoobtain,andsometimesdene,forinsitufra tures.

Parameters likethe maximumpossible losure orthe initial stiness requirethat

a measurement with variable normal load is made. Su h a measurement is not

possibleforafra turethatissituateddeepinthethero kmass. Ifa oresampleis

removedandbroughttoalab,thestressandmatednessarenotpreserved,making

thevalidityofanylabmeasurementsun ertain. Itisalsoverydi ulttotalkabout

the me hani alapertureof afra tureof unknownsize and dire tion. Attheedge

ofthefra ture,thesizeofthevoidspa eismost ertainlylessthanawayfromthe

edge. Ifthe fra tureis large,the normalstressand the hangesin stress indu ed

by theex avation may vary,and onsequentlytheamountof ompression. It has

thereforebeenavitalpartofthisresear htondades riptionofthefra tureand

its stiness that isindependent ofanyarti ial orunobtainable parameters. The

main fo us is not the aperture of the fra ture, but the lo al hange in aperture

during grouting.

4.2 Compression of a Single Fra ture

Conta t Pressure Hypothesis

As an essentialpart of des ribing the unloading of afra ture due to grouting, a

onta tpressurehypothesishasbeendevelopedwherethepressureinea h onta t

pointis assumedto be onstantand not afun tion of loadorgeometry. This is

rst dis ussed from asingle pointof onta t perspe tive andthen later extended

to afra turewithmultiple onta tpoints.

The me hani s of a single point of onta t

Considertwohypotheti alsurfa esthat are fra taland notexa trepli asof ea h

otherthatarebroughtinto onta t. Ataninnitelysmallnormalloadthesurfa es

will losethegapbetweenthembutstopastherst onta tismade. The onta t

point will be innitely small, sin e it will onsist of two lo al extreme points on

ea hsurfa e. Ifthenormalloadisin reasedto somearbitrarilybut notinnitely

small value, thestress atthe onta tpoint will approa h innity andthe ro kat

thatregionwillfail. Thefailurewillprogressuntilthe onta tareaislargeenough

to a ommodate theloadwithoutex eedingsome riti alstress level. Thisstress

anbeassumedtobesomewherebetweentheuniaxial ompressivestrengthofthe

materialandthetransitionstressbetweenplasti andbrittlebehaviour(Steskyand

Hannan,1987). Astheloadin reases,theareaofea h onta tpointwillin reaseto

a ommodatethestress. Thedeformationwill alsobringnewpointsinto onta t.

This pro essisillustratedin gure4.1.

Two on lusions anbedrawnfrom this:

•

^{If both the maximum stress level and the load are known, so is the total}

onta tarea.

a b c

Figure4.1: a) Thetworandom, unmatedsurfa es,are broughtinto onta twith

aminimumnormalload. b)Theloadhasin reasedandthero kisbeing rushed.

Equilibriumisrea hedwhenthe onta tareaislargeenoughto a ommodatethe

loadwithout ex eeding riti al stress levels. ) The loadis in reased again and

thedeformationofthero kin reases. Thedeformationissolargethat additional

points of onta t haveformed. Thetotal onta t areais signi antly largerin

thanb.

•

^{Ifthein reaseinstressissmallenough,the}deformationwillnothavebrought anyotherpointsinto onta twithea hother. Itwillonlyin reasethe onta t

areaofthea tive onta tpoints.

Letus ontinue to onsider asinglepointof onta tbetweentwofra turesur-

fa es, as in the previous paragraph. The ro k material in, and adja ent to, the

onta tpointwillbeinaplasti failureregime. Thevolumeofro kwithnonelasti

propertieswillbeverysmallasthestressintensitywilldiminishrapidlyawayfrom

thefra turesurfa e(PoulosandDavis,1940). Theplasti partsofthedeformation

areofnointerestasthemodellingstrivesto apturethebehaviourduring unload-

ing. Thus, if the small volume of failed ro k is ignored, the deformation of the

asperities in onta t an be approximated by the deformation of an innite half

plane.

Ifthedeformationofaninnitehalfspa eloaded byauniform ir ularloadis

approximatedbythe displa ement ofthe midpointof theloaded area, theelasti

deformation anbewrittenas(PoulosandDavis,1940):

w = 2σr l

E (1 − ν ² )

^(4.1)

where

σ

^{isthepressureundertheload,}

E

^{istheYoung'smodulusofthehalfspa e,}

r l

^{istheradiusoftheloadedzoneand}

ν

^{thePoissonratio. The}deformationofthe onta tasperities will be equalto half the hange in aperture at that lo ationas

bothfa eswilldeform.

Usingthe onta tpressurehypothesis,

r l

^{anbe repla edbyafun tion of the}

normalloadandthestrengthofthematerial,

r ² _l πσ p = r _a ² πσ n ⇒ r ^l = r a

√ σ n

√σ p

(4.2)

where

r a

^{istheaveragedistan ebetweenthe onta t}asperities,

σ n

^{thenormalload}

onthefra ture,and

σ p

^{thepeakstrengthofthero k.}

Withequation(4.2)insertedintoequation(4.1)theelasti deformationbe omes

w = 2 r a

E

√ σ n σ p (1 − ν ² ).

^(4.3)

Thisapproximatestheelasti deformationofasingle onta tpointinthefra -

ture. As theirreversible ata lasti ow and any plasti deformations are disre-

garded, the a ura y of the approximation will depend on the matedness of the

surfa es in onta t. Forunmated labsamples, thiswill resultin asubstantialde-

viation from themeasuredresult,itis, however,reasonableto assumethatunless

disturbed,thein situfra tureswill exhibitahigh levelofmatedness.

Thedistan e betweenthe asperities determinesthe totalfor e a ting on ea h

asperity. This distan e will be a measure of the matednessof the fra ture, and

perhapsthemostimportantgeometri almeasure ofthe fra ture. Sin ethestress

inthe onta tregionisapproximatedtoequalthepeakstrengthofthematerial,the

onta tareaandthusthedeformationwillbegiven. Thestinesswillbeobtained

asthederivativeofthestressasafun tionofthedeformation:

K N = dσ n

dw = E√σ n

2(1 − ν ² )r a √σ p

.

^(4.4)

The stiness of ea h asperity will in rease slowly with load as the onta t area

in reases,as anbeseeningure4.2.

Usingthe onta tpressurehypothesis,typi alvaluesoftheelasti modulus(

80

GPa) and the uniaxial ompressivestrength for inta tsamples of small size (

400

MPa),thestraininthe onta tregionwillbe

500

mi rostrain. Thedeformationof the onta tasperity anthenbe al ulatedbyestimationofthedistan e between

the onta tpointsandthetotalload. Thisiswherethes aleofthesystem omes

in. For verylow loads,the s alefor alab sample an be takenasthe size of the

sample but the initial s ale for in situ fra tures should be of the same order of

magnitudeasthelevelofmatedness ofthefra ture.

Letus illustrate this with anexample, assuminga normalloadof

3

^MPaand

a span of

10

^{m betweenthe onta t points in addition to the material parame-}

ters asin the previousparagraph. The onta tareawill then haveadiameter of

approximately5mmandthe ompressionofthear ha rosstheratherlargeopen

voidspa e will be

50 µ

^{m. Forthese valuesit is,however,likely thatnew onta t}

pointswouldhaveformedthatwouldhave arriedsomeoftheloadandredu edthe

deformation. With four onta t pointsin the sameregion, thetotal onta tarea

0 0.2 0.4 0.6 0.8 1 1.2 0

2 4 6 8 10 12 14 16 18 20

Deformation, mm

Normal Load, MPa

Figure4.2: Thedeformationalbehaviourofasingle onta tasperitya ordingto

the onta tpressurehypothesis. Inthisguretheaverageasperitydistan eis1m.

wouldbethesame,buttheaveragear hlengthwouldbehalfandthedeformation

wouldbe

25 µ

^{m. Thusanin reaseinloadwillresultinanequallylargein reasein}

onta tarea,whetherit omesfromthein reaseinsizefromasinglepointorfrom

theadditionofpoints. Theaddition ofpointswill, however,havealargerimpa t

onthestinessof thesystem. With in reasedloading,theloadwill be arriedby

in reasingly sti subsystems that will leadto aprogressivelystier system. The

rateofin reasewilldependonthematednessofthefra turesurfa es,wherehighly

matedsurfa eswillhaveaveryrapidin reaseof stiness. This isin linewith lab

measurement results. The example assumes the same 1D movement of the ro k

asin alab setup. Forin situ fra tures the example will des ribe the movement

ofasperities situated lose enoughnottoexperien eany dieren ein loaddueto

pressuregradients.

Multiple onta t pointswith 1D normal load

Using the superpositional prin iple the problem an be separated into a sum of

Goodmanfra tureswhereall thespringsarelinearandwithequalmaterialprop-

erties. By engaging the springs at dierent degrees of deformation the stiness

hara teristi s anbemadeto mimi that oflab tests. Figure 4.3illustrates this

examplewiththreelevelsofdetail. At3

M P a

^,the ompressionisapproximately17

µ

^{mwithea hleveltakingthesameamountofloadandthushavingthesametotal}

onta t area. Duringthe rst 7

µ

^{m of} ompression, onlythe large s alefeatures are a tiveandthestiness isthenlower. As theothertwolevelsof detailengage,

the stiness rampsup a ordingly. 17

µ

^{m is ahigh level of} ompression but well within therangeof whatis tobeexpe tedfrom alabtestwith a10 mdiameter

sample. (Raven andGale, 1985). Shear testsof mated naturalfra tures seemto

indi atethatthemeandistan ebetweenlarges aleasperitiesisapproximatelyone

orderofmagnitudehigherthantheapertureofthefra ture. Intheexampleabove,

thiswouldgiveamaximum ompressionofthefra turelessthan1

µ

^m.

The fra ture in gure 4.3 onsists of only three levelsof information yet it is

not di ultto imaginewhat the urveswould look likewithadditional levels. If

ontinuousspe traofdetailsare superposedto reatethesurfa e,thesurfa ewill

be ome fra tal, and the response urve be ome a smooth and progressive urve

similarto thosemeasuredin labtests.

Ifthelinearspringsareex hangedforspringsbehavinga ordingtothe onta t

pressure hypothesis, the deformation anbe al ulatedby integratingthe inverse

of the stiness as afun tion of the load. Fora set of asperities, all engagingat

dierentdeformations,itwillbe omealittlemore onvolutedbuttheprin iplewill

be the same. As the deformationin reases, new onta t points will engage ea h

otherandalterthetotalstinessofthesystem. Withthespringsaddedinparallel,

thestinesswillin reasewithea hengagingasperity. Theintegraloverthestiness

will be omeaseries. Ea htermin theserieswillbetheresultoftheintegralover

asmallin reasein stresswherethenumberof onta tshasbeen onstant. Foran

arbitrarysystem,thedeformation

w

^{willbegivenby}

w(σ) =

k

X

i=0

Z σ i

σ _i−1 i

X

j=1

K _N ⁻¹ _j dσ,

^(4.5)

where

k

^{is thenumberof subsystems engagedat thestress}

σ k

^and

K N j

^{the sti-}

ness ofthe

j

^{-thsubsystem. Inserting equation(4.3)into equation(4.5)yields the}

expression

w(σ) = 4(1 − ν ² )√σ p

E

k

X

i=1

√ σ i − √ σ _i−1
P i

j=1 1 r _aj

.

^(4.6)

Thisisundoubtedlyanawkwardwayofexpressingthedeformationofthefra ture.

It does, however,allowfornumeri al al ulationsoffra ture behaviourandorder

ofmagnitudeestimations.

+ +

=

a 3

0.5 1.25

7 12 17

MPa

mm b

Figure 4.3: a) Three dierent sets of linear springs are added together to form

dierentlevelsof detailforthis parti ularfra ture. b) Theresultingdeformation

whenthesurfa eispressedagainstaatsurfa e.