Mathematical modeling of water-rock chemistry during water injection and its impact on the composition and porosity of chalk

's

Uri*r.,t"t"i i Stavanger

DET TEKNISK-NATURVITENSKAPELIGE FAKULTET

MASTEROPPGAVE

Studieprogram/spesialisering : Petroleum Engineering /

Reservoir Technolory

Våx...semesteret. ..20 1 0...

Åpen / Iknsdensiell Forfatter: Pål Østebø Andersen

P å,1 ,A n^kn r.,

(sienatur forfatter)

Fagansvarlig: Steinar Evje Veileder(e): Steinar Evje

Tittel på masteroppgaven :

Matematisk modellering av vann-mineral-kjemi under

vanninjeksjon og dens påvirkning på komposisjon og porøsitet i kalk Engelsk tittel:

Mathematical modeling of water-rock chemistry during water injection and its impact on the composition and porosity of chalk

Studiepoeng:30 Emneord:

Water weakening Chalk Convection

Diffusion

Water-rock chemistry Strang-splitting Water injection

Sidetall: ...72.

+ vedlegg/annet: ... ...18... ..

Stavanger, ...1416, 2010..

dato/år

1 Summary 1

2 Reservoirroks and geology 2

2.1 Thegeologialaspet. . . 2

2.2 Reservoirroks . . . 3

2.2.1 Quantiation . . . 3

2.2.2 Carbonates . . . 3

2.2.3 Sandstones . . . 4

2.3 Chemialrok-uidequilibrium . . . 4

2.4 Referenes . . . 4

3 Water weakening 5 3.1 Waterweakening . . . 5

3.2 Stressandstrain . . . 5

3.3 Tests inatriaxialell . . . 6

3.4 Rokfailure . . . 6

3.5 Labtestobservations. . . 7

3.5.1 Simultaneouswaterinjetionand loading . . . 7

3.5.2 Responsetowaterinjetion inaloaded state . . . 8

3.5.3 Potentialandidatesformagnesium preipitates . . . 8

3.6 Fieldobservations . . . 9

3.6.1 Valhall. . . 9

3.6.2 Ekosk . . . 9

4 Relevantmineralsin halk replaement: Volumetrionsiderations 10 4.1 Inludingmoremineralsandvolumetrionsiderations . . . 10

4.2 Magnesium-bearingminerals . . . 10

4.2.1 Magnesite . . . 10

4.2.2 Dolomite . . . 11

4.2.3 Huntite . . . 11

4.3 Sulphate-bearingminerals . . . 11

4.3.1 Anhydrite . . . 11

4.4 Iron-bearingminerals: ankeriteandsiderite . . . 12

5 Transport-reationmodel 13 5.1 Components. . . 13

5.1.1 Solidstate: minerals . . . 13

5.1.2 Aqueousstate: ions . . . 13

5.1.3 Dissolvedgas . . . 14

5.1.4 Liquidstate . . . 14

5.2 Reations . . . 14

5.2.1 Dissolutionandpreipitationofminerals. . . 14

5.3 Porosityandvolumebalane . . . 14

5.4 Permeabilityandpossiblehysteresis . . . 15

5.5 Molarbalane . . . 17

5.6 Reationrates. . . 18

5.6.1 Chemialativity. . . 19

5.6.2 Reationratesforthemodel . . . 20

5.6.3 Aqueousreationsandhargebalane . . . 20

5.7 Transportequations . . . 21

5.7.1 Componentveloities . . . 21

5.7.2 Volume onservation . . . 22

5.7.3 Updatedequationsystem . . . 23

6 Case denitions 24 6.1 CaseI:Constantorepropertiesandinompressibleuid . . . 24

6.2 CaseII:Variableporosityandpermeability . . . 25

6.3 Reformulatingtheproblem . . . 26

6.4 Unitsanddimensioning . . . 26

7 Solution proedure 28 7.1 Operatorsplitting . . . 28

7.2 Thereationsolver . . . 29

7.2.1 Atestofthereationsolver . . . 31

7.3 Theonvetion/diusionsolver . . . 32

7.3.1 Numerialsolution . . . 33

7.3.2 Simpliation: Constantporosity . . . 35

7.3.3 TVD-analysisforstability . . . 36

7.4 Consequenesofoperatorsplitting . . . 38

7.4.1 Toohigh

∆T

^{: Washout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38}

7.4.2 Toohigh

∆T

^{: Chemial}equilibrium . . . 39

7.4.3 Toolow

∆T

^{: Left sideboundaryondition . . . . . . . . . . . . . . . . . . 39}

7.4.4 Corretionattheboundary . . . 40

7.4.5 Choieof

∆T

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42}

8 Experimentaldata 43 8.1 Experimental setting . . . 43

8.2 Fluidompositions . . . 44

8.3 Ativityoeientsandionistrength . . . 44

8.4 Reationequilibrium onstants . . . 45

8.5 Referenevalues . . . 45

9 Case I: Constantore propertiesand inompressibleuid 46 9.1 Assumptionsandgoals . . . 46

9.2 Simplepressureanalysis . . . 46

9.3 Determinationof

D

^and

α

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47}

9.4 Test ofassumption: uniform

V

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48}

9.5 Determinationofrateparameters . . . 49

9.5.1 Magnesitemodel . . . 49

9.5.2 Dolomitemodel. . . 54

9.5.3 Comparisonofmodels . . . 57

9.5.4 Inlusionofbothminerals . . . 57

10.1 Assumptions . . . 64

10.2 Theoretialpermeabilityalulations . . . 64

10.3 Test ofassumption: UniformV . . . 65

10.4 Thereationsolver . . . 65

10.5 Theonvetion/diusionsolver . . . 66

10.6 Fullsalesimulation . . . 67

11Disussion 72 A General modelin 3D 75 B Basis for

k − φ

-orrelations 76 B.1 Correlationsbasedondiretestimation . . . 76

B.2 Correlationsbasedonhangesin struture. . . 76

B.3 Comparison . . . 77

B.4 Correlationsbetweenloal permeabilityandloalporosity . . . 77

B.4.1 SuggestionI:

f = ax ^b + c

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77}

B.4.2 SuggestionII:

f = ae ^bx + c

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . 78}

B.4.3 SuggestionIII:Stepwisesmooth

f

^{. . . . . . . . . . . . . . . . . . . . . . . 78}

B.4.4 Seletedorrelation . . . 79

B.4.5 Suggestedexperimental investigationofrelationbetween

k

^and

φ

^{. . . . . . 79}

C The eetive diusionoeient

D

⁸¹ C.1 Denitionof

D

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81}

C.2 Experimental determinationof

D

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81}

C.3 CorrelationsforD . . . 82

D TVD-analysis 84 D.1 Theonvetion/diusionsolverforonstantporosity . . . 84

D.2 Theonvetion/diusionsolverforvariableporosity . . . 85

Inreentyearsmoreattentionhasbeenpaidtothehemialsideofinjeted waterusedinhalk

formationsto help produe hydroarbons. It seems the brine has atendeny to reat with the

formationitself ifit ontainstherightsubstanes, evensea waterhas thiseet. Espeially the

halk experienes the phenomenon alled water weakening whih aets the roks response to

externalloading,butalso itswettability.

Experimentshave been performed in the laboratory at theUniversityof Stavanger onhalk

oreplugs. Essentiallytheoreshavebeenexposedtoabrineunderhighpressureandtemperature

(representativereservoironditions)alongtimeto reahequilibrium. Thendierentbrineshave

beeninjetedthroughtheoreatthegivenonditionsatxedratestypiallyaround1PV(pore

volume)perday byvarying the inlet/outletpressure. Responses suh as oredeformationsand

outlet onentrationshave been measured. SEM images were used to study omposition of the

ores.

Theexperimentsshowedthatresultsaresensitivetoinjetiononentrationoftheions

Ca ²⁺

^,

M g ²⁺

^and

SO ² ₄ ⁻

^{. Rokomposition hangedafterooding. Espeiallyinjeting}

M gCl 2

^-solution

gavepreipitationofamagnesium-basedmineral,andoodingwithseawatergavepreipitationof

asulphate basedmineral. The resultsare believed relatedto dissolution/preipitationreations

inaninterplaywithonvetion,diusionandaqueoushemialreations. Amathematialmodel

[22℄hasbeendevelopedthat isabletorepliatetheoutletmeasurementswith goodauray. It

was developedbyS. Evje, A.Hiorth, M.MadlandandR. Korsnes. Thesameauthors presented

supportiveexperimentaldataandsomealterationsin [23℄.

The fous of this thesisis to expand the original model. Espeially we inlude the mineral

dolomiteasapreipitateandweletrokpropertiessuhasporosityandpermeabilityhangewith

rok omposition. Some relevant experiments are also suggestedto better estimate parameters

usedinthemodel.

The waterweakeningeet hasimpat on areas suh as porosity, permeability (pluggingor

openingof pores), ompressibility(higherrokexpansionmeansmoreproduedporeuid), ten-

silestrength(an aet fraturepressure),wettability,residualsaturations,waterbreakthrough,

reoveryandsubsidene.

Summary

Inthis thesisthe model developed in [22, 23℄ hasbeen investigated and further developed. The

mineral dolomite was inluded to the minerals alite, magnesite and anhydrite. Porosity was

inludedas afuntion ofthemineral omposition. Some suggestionsare givento exploreeets

onpermeabilityand pressure,but underthe assumptionsofthe modeltheyare botheliminated

fromthesystemandnorelevantdatawas availablefortesting.

Computersimulationsshowthatdolomitebyitselfandmagnesitebyitselfastheonlymagnesium-

bearingmineralpreipitatingintheoreanexplaintheeuentsmeasuredattheoutlet. However,

to explain SEM observations thepresene of bothis required. Severalombinationsof rate pa-

rametersarepossibletottheexperimentaleuentdataineahmodel(dolomiteonly,magnesite

only, dolomiteand magnesite),but the magnesitemodelgavemoreoptionsto determineabest

tthanthedolomitemodel.

The simulations predited a steady dissolution of alite and preipitation of the minerals

magnesite,dolomiteandanhydritewhentheenvironmentsuggestedso. Theneteetwasavery

lowvariationin porosity(from 0.48to0.47),bothloally andonaverage,evenafter aperiodof

20days. Thereasonisthatthedissolvedmineralsarereplaedbypreipitatingmineralsandthe

ompositionhanges. Thisonlusionissupportedbythemassbalaneofionswhereexess

Ca ²⁺

isprodued while

M g ²⁺

^-and

SO ² ₄ ⁻

^{-ionsare retainedin theoreomparedto asimulationwith}

noreations.

2 models were tested: one with onstantporosity in the equations, with porosity only as a

funtionofthesolutionofmineralomposition. Theotherwhereporosityvariedintheequations

aswellbeingoupledwiththerest ofthesystem. Thelowvariationin porositymadetheresults

fromthe2modelsundistinguishable.

Themodeldoesnotaountforavailablesurfaeareainthereationsandthatwouldprobably

improvethetwithexperimentaldataatearlytimestoagreatextent.

Reservoir roks and geology

2.1 The geologial aspet

Whenmineralsaredeposited,buriedandompatedtheybeomepartofasedimentaryrok,per

denition. Thedepositionanourbytransportofgrains,hemialsanpreipitatefromsolution

orsmallorganismsanleaveshellsandskeletonsof mineralomposition. Duringtheompation

the spaebetween thegrains is redued sinethe aumulatingoverburden fores will fore the

grainstopakintotighterongurations. Inthisproessthevolumeoupiedbyuidsisredued

either beause they esape or beause they are ompated more easily until the pore pressure

fraturesanopening. Weakmineralsan begroundintosmallerpiees leavingadenserpaking.

However, most sedimentary roksretain a relatively large fration of pore volume, porosity, of

manytensperentandthat iswhysedimentaryroksaregood forstoringhydroarbons.

Theburialproessisalsokeytotheformationofpetroleum. Whenorganimaterialisburied

in a manner that preserves it from oxidation then it will be exposed to a gradual inrease in

temperature and pressure. Smaller organimoleules transform into largeromplexsubstanes.

The organi material is by denition divided into kerogen and bitumen. Kerogen is the part

insolublein organisolvents,whilebitumen(oilinsolidstate)isthesolublepart. Suhproesses

beginshallowomparedtotheformationofpetroleum. Whenkerogenisexposedtohighpressure

andtemperatureoverlongtimeitturnsintopetroleum. Theoilwindowisarangeoftemperatures

where oil generation is possible. Oil beginsto form at 60

o C

^{with optimal onditions between}

100-120

o C

^{. At} temperatures higherthan 180

o C

^{a proess alled rakingbreaks down heavy}

moleules intosmalleromponents. Gasformationis stillpossibleabovethese temperaturesbut

approahing225

o C

^{mostoftheseproesseshavealreadyhappened.}

One petroleum(oil orgas)beomesmobile itwill tryto esapetowardsthe surfaesineit

has lower density than water. If it does notesapefrom the soure rok (where the kerogenis

beingtransformed)itwillbedestroyedasexplainedpreviously. Thehydroarbonswillthenfollow

amigration routealongporehannelsin therokuntilitreahesthesurfaeandisdestroyedby

bateriaor until it reahes aboundary that doesnot allow ow in theupward diretion. This

requiresthatapermeableandporousformation,whihweallareservoir,intersetsthemigration

routeandthataaprok/trapoverlaysutsotheroute. Also,thesealmustbeinplaebefore

the oil an esape. The seal must keep the hydroarbons trapped for maybe millions of years

until present. Geologi ativityin the rustan disturbthis, but also reate new possibletrap

ongurations. Thegoldenzoneisthetemperaturerangewhereoilreservoirsareatuallyfound.

Itpeaksaround90

o C

^{butrangesfromabout60to150}

^o C

^.

2.2.1 Quantiation

Although everyrok is in somesense unique,wean quantify a rokspropertiesby performing

labtestsonoresandevaluatelogsandthinuttings.

•

^Porosity

φ

^{isthevolumefrationofarokthatislledwithuidssuhasbrine,gasandoil.}

Highporosityindiates ahighstorageapaityandisgivenasafrationbetween0and1.

•

Permeability

k

^{measurestheabilityarokhastoletauidowasasinglephasethroughthe}

rokin agiven diretion. Permeability generallyis anisotropi(varies with diretion)and

isoftenlowerin thevertialdiretion. Itismeasuredin dary. Highpermeabilityindiates

arokwithlittle owrestritionin the givendiretion, whilealowpermeabilityindiates

narrowporethroatsoromplexporehannels.

•

Wettability indiates theinterplay betweenthe rokand thepore uids. When two uids are plaedontheroktheywill bedividedbyaninterfae. Oneuidstendeny to spread

ontherokwillbegivenbytheangletheuidinterfaemakeswiththeroksurfae. Ifitis

muhlessthan90 degreestheuid iswetting, iftheangleismuhmorethan90theother

uid is wetting. If theangle islose to90 degreesthe rokis notpreferentially wetted by

either uid. Neutralwettabilityispreferableforhigh reovery.

•

^{Mehanialpropertiesexplainhowtherokdeformstodierentloadings.Testsanquantify}

drivemehanismssuhas rokexpansionbyporepressuredepletionandboreholestability.

•

^{Chemial omposition and the} distribution of the grains an be important if the rok is hemiallyreative. It is wellknown that laysare espeially reativedue to high surfae

area omparedto volume. They an work as atalystsfor hemial reations,an expand

orompressduetoionexhangeandbindwater. Theavailablesurfaeareaoftheommon

grains isalso ofimportanetotherateofreations.

•

Temperatureand pressure at reservoironditions is a ritial fator sinethe behaviorof rok,uidandhemistryanhangedramatially.

2.2.2 Carbonates

Carbonates are minerals ontaining the

CO ² ₃ ⁻

^{-anion in ombination with dierent ations. In}

reservoirengineeringespeiallythearbonatemineralsalite,

CaCO 3

^{,anddolomite,}

CaM g(CO 3 ) 2

^,

areofimportanesinelimestoneformationsanddolomiteformationsrespetivelyhavethesemin-

eralsasthemajoringredient. Lessknownarbonatesarearagonite

CaCO 3

^{(otherstruturethan}

alite), siderite

F eCO 3

^,magnesite

M gCO 3

^andankerite

Ca(F e, M g, M n)(CO 3 ) 2

^.

Carbonatereservoirsareamongtheworldslargest. Theyarefoundworldwideandabout

40%

oftheworldhydroarbonprodutionisfromarbonates.

Forpetroleumstorageonlymarinearbonatesmatter. Thesearbonatesedimentsareproduts

from living organisms (suh as pellets), dead organisms (shellsand skeletons)and preipitation

of salts. The depositional environment is mostly shallow: ramps and platforms (the limestone

reservoirGhawar in SaudiArabiais agood example),reefsor evaporites. Howeverwealso nd

reservoirsaftergreatdepthdepositionbyarbonateturbiditesandasremainsofpelagireatures.

Pelagiarbonates(made from anient oolithospheres) givesoriginto halk. The NorthSea

ontainsthegiantEkoskoileldwhihmainlyonsists ofhalkrok.

Chalkformationsareharaterizedbyhigh porosity(anapproah

70%

^{,but ismostlyin the}

area

15 − 50%

^)andverylowpermeability(a fewmD).Natural fraturingimprovestheeetive largesale permeability to therange of 100md. Chalksare mostlyoilwetting andhave alarge

reativesurfaearea.

ismostsevereintheseroks. Howeverthesimilarhemialompositionoflimestonesinpartiular

suggeststhat waterweakeninganplayarolealsointhese formations.

Dolomites are often assoiated with evaporiti environments. This mineral is not formed

diretly,but requiresthepreseneof

CaCO3

^{(eitherasaliteorpreferablyaragonite)andmag-}

nesium ions. The transformation of a limestoneinto dolomiteis alled dolomitization and this

proessisbelievedto haveformedmostdolomitereservoirs. Basially

Ca ²⁺

^{ispartlyreplaedby}

M g ²⁺

^{intherokstruture.}

2CaCO 3 + M g ²⁺ ⇋ CaM g(CO 3 ) 2 + Ca ²⁺

^(2.1)

The onditions for this proess to move to the right is that

CaCO 3

^{is unstable, the uid is}

oversaturatedondolomiteand

M g ²⁺

^issuppliedadequately.

2.2.3 Sandstones

Sandstonesare lasti (madeof grains from pre-existing roks). Wesort lasti roksby grain

size andsandstone ison theoarsesideof thesale(as opposed to laystonewith muh smaller

grains). Sandstonesontainmostlyquartz,

SiO 2

^{, and feldspars} (tetosiliates ontainingSi, O, Na,K,Al,Ca). However,mineralpreipitationfromuidsanontributetolltheporespaein

aproess alled ementation. Suh minerals arealiteand other arbonates,quartz, laysand

zeolites.

2.3 Chemial rok-uid equilibrium

A rok an under normal irumstanes be assumed to be in equilibrium with its pore uids,

meaningthatanyhemialreationratesarenegligible. Thesystemisharaterizedbytheloal

pressureandtemperatureonsite andtheloalompositionoftherokanduids.

When introduing, letus say, sea waterto the system it may have a low temperature, if it

is injeted there will beapressure gradientand the omposition of the seawatermaybe quite

dierent from the one in equilibrium with the rok. A front will move from the injetion site

haraterized by that in front the uid is in equilibrium with the rok, while behind the front

the stateis dierent. Moving a uid from one PT state to another an inuene the solubility

of its salts. Salt preipitation an redue owarea in pores and pipes and should generallybe

avoided. A highertemperaturewillinreasesolubilityinmostases,butanimportantexeption

is

CaCO 3

^{whih behavesexatlyopposite. This behaviourisalled retrogade}solubility. So even iftheompositionsarethesameahangeinthermodynamialstatean imposereations.

Given2unequaluids thatan betreatedas asinglephasethe ionswillspreadbydiusion

(drivenbyonentrationgradients),onvetion(uidowduetopressuregradients)andhemial

reations (workingto establishanewrok-uidequilibrium). These proessesaregenerallyvery

oupledsinethereations depend onloal onentrationsand state,theonvetiondepends on

pressuredrop,rokpermeabilityanduidvisosity. Changesinuidompositionandstateanal-

tervisosity,hangesintherokmehanialpropertiesandgraindistributionhangepermeability

andporosity. Diusion depends onomponentdistribution,owonditionsandpore struture.

A model desribing how the distribution of hemial substanes progresses during injetion

was developedin [22℄ and[23℄. Thistransportmodelwill beexplainedstartingin hapter 5and

reformulatedduringthis thesis.

2.4 Referenes

[1,2,3,4, 5,6,22℄

Water weakening

3.1 Water weakening

Inshortwords,waterweakeningmeansaroklosessomeofitsabilityto resistdeformationfrom

thesurroundingfores. Thishangeisrelatedtoreations withareativebrine.

To understand water weakening one should have a basi understanding of rok mehanial

theory. The setions 3.2, 3.3 and 3.4 give asummary of important onepts, relationsand test

methods. Theyare mostlybasedon [6℄, abook reommendedif amorethoroughdesriptionis

needed.

Inthelastsetionswewillpresentsomeobservationsmadeoneldsaleandinthelaboratory

thatillustrate theeets.

3.2 Stress and strain

Theoneptofstressis denedasforedividedbyarea.

σ ≡ dF

dA

^(3.1)

Stressisnormaliftheforeworksperpendiulartothesurfaeandshearifitatsparalleltothe

surfae. Foranisotropimaterialstressisatensorsineaforeanatin3diretionsonsurfaes

normalto 3axis. Assuming fore andmoment equilibrium thistensor is symmetri. Thestress

tensoran bedividedintoahydrostatipart(with onlynormalstressesnonzeroandhavingthe

value of the mean normal stress) and a deviatoripart (whih is simply the remaining partof

the matrix). The hydrostati partindiates a level of ompressive or expansive loadwhile the

deviatoripartindiateshowtheunequalstressdistributionompares.

Given a stress tensor we an nd 3 perpendiular axis orresponding to zero shear stresses

and thus all stresses are direted along the oordinate axis. These normal stresses are alled

prinipalstressesanddenethestressstatealongwiththeirdiretion.Inanydiretionthatisnot

exatlyonone oftheaxistherewill alsobeashearstress, whih an beexpressed asafuntion

ofthe prinipalstress values. Note that iftheprinipal stressesareidential theloadingwillbe

hydrostatialseenfrom anyangle. If 2prinipalstressesareequaltheplane thatontainsthem

ontainsnoshearstress.

In rok mehanis it is usual to use positive stress for ompression and negative stress for

tension,andtheprinipalstressesarelabeledin desendingorderas

σ 1

^,

σ 2

^,

σ 3

^.

Normalstrainisdenedashangein lengthdividedbytheoriginallength

L 0

^{oftheunloaded}

material:

ε ≡ L 0 − L L 0

(3.2)

arelinearlyrelated.

Givenaporoussamplesomeoftheloadisarriedbytheporeuid,givenbytheporepressure,

p f

^{timesBiot'soeient,}

α

^{. Theeetivestress}

σ ^′

^{thatisarriedbytherokgrainsisthen}

σ ^′ ≡ σ − αp f

^(3.3)

Thedeformationresultsfromloadingtherokand relatestoeetivestressbyYoungsmodulus

E

^:

σ ^′ = Eε

^(3.4)

Aloadinoneaxialdiretion

z

^ausesdeformationofoppositesignalongtheotheraxes

x, y

^related

byPoissons ratio

ν

ν ≡ − ε x

ε z

(3.5)

Volumetrideformationisgivenby

ε V = V 0 − V

V 0 = ε x + ε y + ε z

^(3.6)

If avolume is hydrostatiallyloaded (all prinipal stressesequal) bythe load

σ _c ^′

^{the volumetri}

deformationisgivenby

σ ^′ _c = Kε V

^(3.7)

where

K

^{isthebulkmodulus.}

3.3 Tests in a triaxial ell

A ylindrial ore sample is plaed vertially between two axial bolts and sealed from the sur-

roundingsbyathinsleeve. Aonningpressure

σ c = σ r = σ θ

^{(foraylindrialgeometryweuse}

theoordinates

r, θ, z

^{)inthehorizontalplaneisprovidedbyaonninguid. Axialstress}

σ z

^is

provided byinreasingthepressurein auid hamberabovetheupperaxial bolt that pushes it

down against theore sample. Wemust orret forfrition, but in prinipleweknow theaxial

load. Smallopeningsinthebolts allowirulationofuidandthusaporepressureweanvary.

Axialstrainismeasuredbydisplaementofthebolt(afterorretingitsowndeformation)and

radialstrainismeasuredbysensorspointedtowardstheoresurfae.

Indrainedtestsuidanesapeandtheuidarriesaonstantload

p f

^{. Inastandardtriaxial}

ompression test the load is inreased hydrostatially (

σ _c ^′ = σ ^′ _z

^{) and the bulk modulus of the}

framework

K f r

(representing the porous roks abilityto resist deformation) is measuredas the slope

K f r = ∆σ ^′ _z

∆ε V = ∆σ _z ^′ 3∆ε z

(3.8)

Afterthishydrostatiphasehasreahedaertain

σ c

^{, theonningloadiskeptonstantandthe}

axialloadis inreasedfurther. TheYoungsmodulusoftheframeworkisthendeterminedas

E f r = ∆σ _z ^′

∆ε z

(3.9)

inthis deviatoriphase.

3.4 Rok failure

Materialsandroksoflowporositydonotfailhydrostatiallyuntilatveryhighpressures.However

halkis veryporous andunder enoughpressurethepores an ollapsebyloal shear failure. In

thedeviatoriphasewedenetheyieldpointastheeetivestressthatisfollowedbyanonlinear

onethisstresshasbeenreahedarelaxationofthestressallowsfurtherdisplaementevenatlower

stressbeforethesamplenallyruptures. Thisexplainswhyaproessofinrementaldisplaement

ispreferredoverinremental loading,toobservethelast phase.

Chalk an also experiene reep. It is a timedependent deformation that ours under on-

stant stressand temperature. Note that the applied stressan be lessthanwhat ausesplasti

deformation(permanentstrain). Weandividethereepintoatransientstate(dereasingstrain

rate),steadystate(onstantstrainrate)andaeleratingstate(inreasingstrainrate)eventually

leadingto rupture.

3.5 Lab test observations

3.5.1 Simultaneous water injetion and loading

In [21℄ several lab experiment results are presented. In one of them halk ores at 130

o C

^are

oodedwithdierentbrineswhilebeingloadedhydrostatially. Theresultingstress-straindiagram

isrepeatedleftin Fig 3.1. Itwas observedthat theoresgotaloweryieldstress(averageof6.5

MPa)whentheywereoodedwiththesulphateontainingbrinesthanwiththesulphate-deient

ones (average of 8.5MPa). Thesulphateexposedores alsogota muh higherompation (2.5

times the strain than those not exposed to sulphate at high stress). Note also that the bulk

modulus (given by one third of the initial linear slope, as in eq. (3.8)) is lessfor the weakened

samples(byafatorofa2/3).

Figure 3.1: Left: Stress-strain diagram for hydrostati loading of halk ores at 130

o C

^while

ooding brine at onstant rate. Right: The following reep diagram at 10 MPa ompressive

stress.

Whenreahing10MPastressthisloadwaskeptonstantandtheresultingreepwasobserved.

The reep phase resultsare given right in Fig 3.1. Again the sulphate-exposed oresshowed a

muhhigherdegreeofompationthantheothers. Flooding withahighonentrationsulphate

brine(doubleofseawater)ledtopluggingoftheore,probablydue topreipitationofanhydrite

CaSO 4

^.

An importantonlusionin thepaperwas thattheions

M g ²⁺

^,

Ca ²⁺

^and

SO ₄ ^{2 −}

^(inamounts

omparable to that found in seawater) an impat the mehanial behavior and wettability of

halk.

In[13℄sandstoneoreswereleanedusingmethanolandtoluene,thendried. Theoreswerethen

saturatedwithdeaneandloadedin atriaxialellsuhthat

∆σ ^′ _c

∆σ ^′ _z = 0.25

^{. Theoreswerekeptat}

axed stress state several days and noreep strain was observed. Slow injetion with

3%

^KCl

solutionintheoresresultedinimmediateresponseeitherbyshearfailureorquitenotieableaxial

and/orradialstrain. Creep(ontinuingdeformation) was alsoobserved. Thisdemonstratesthat

waterweakeninganberelevantalsoforsandstones,butthatothermehanismsmaybeinvolved.

NorthSeahalkwassaturatedwithmineraloilandloadeduniaxiallywithaonstantloading

rate. Thestraininreasewasapproximatelylinear withtime. After 290hoursNorthSea water

was injetedintotheoreandarapidinreasein axialstrainwasobservedfollowedbyreep.

3.5.3 Potential andidates for magnesium preipitates

Floodinghalkoreswith

M gCl 2

^{-brineresultinwaterweakening,aordingto[16℄. Theooding}

showed a lower outlet onentration of

M g ²⁺

^{than ould be explained by adsorption and ion}

substitution. It was onluded that amagnesium based mineral preipitating in the ore ould

explaintheobservations. Forthegivenexperiment(0.219M

M gCl 2

^,T=130

o C

^,P=8bar,

P CO2 = 10 ^{− 3.5}

⁾ simulations using EQAlt showed that several magnesium minerals were supersaturated givenbythevalueofionprodutratio

Q

^{oversolubilityonstant}

K

^{beinggreaterthan1. Espeially}

huntite(

CaM g 3 (CO 3 ) 4

^)andhydro-magnesitehadlargesuhnumbers,butsimplermineralssuh as dolomite and magnesite were also supersaturated (see Fig 3.2). Note that the large

Q/K

Figure3.2: Supersaturatedmagnesiumminerals, tablefrom[16℄

ratioofhuntiteanbeexplainedbyitsdependeneon

M g ²⁺

^and

CO ² ₃ ⁻

onentrations. Assume bothdolomiteandhuntiteareexatlysaturatedatagivenstate(

Q/K = 1

^{)inseparatesolutions.}

Doublingtheonentrationof

Ca ²⁺

^,

M g ²⁺

^andof

CO ₃ ^{2 −}

^wouldmake

(Q/K) dolomite = 2 ¹ · 2 ¹ · 2 ² = 16

^while

(Q/K) huntite = 2 ¹ · 2 ³ · 2 ⁴ = 256

^{. If} preipitation leads to the initial equilibrium onentrationsthesamenumberofmolesarepreipitatedin eah solution.

Figure3.3: Comparisonofweightdistributionofanalysiswithweightdistribution ofknownmin-

erals

In[17℄apresentationofompositionanalysisusingSEM(sanningeletronmirosope)showed

aweightdistributionofthemoleulesinpreipitatedmineralgrainsthatlookedsimilartohuntite.

magnesite and huntite in Fig 3.3. It is seen that the analysis results an be explained as the

preipitation of huntite, but a ombination of the minerals magnesiteand dolomite (taking the

average of their distributions) gives almost exatly the same distribution as huntite (a better

weighedaveragewouldtevenbettertotheanalysis).

In the model [22, 23℄ magnesite is the only magnesium basedmineral inluded. We expand

this by inluding dolomitealso. It should be onsidered thoughthat huntite is just as relevant

andperhapsan evenberepresentativefortheentire magnesiummineralpreipitation.

3.6 Field observations

3.6.1 Valhall

Inapaper[15℄ from 1989rokompressibilitywas onluded to be animportantparameter for

thehighporosityhalkeld Valhallausingporosityredution,ompationof reservoirintervals

andseabedsubsidene.

3.6.2 Ekosk

Aasestudyof thehalkeldEkoskin theNorthSeaispresentedin[14℄ from1999. Theeld

startedproduingin1971,waterinjetionbeganin1987. Seaoorsubsidene(seeleftinFig. 3.4)

inreasedin the90'sand theseaoor droppedatarateof 25to42 mperyear. Overtheyears

thisresultedinseveralmeters. In94theinjetionwasinreasedtoreplaetheproduedreservoir

uid volume, but the subsidenedid not derease signiantlyand kepta steady rateabove 35

m/ymostofthe90s. Themodelsusedso far(mathinghistorialoilrate,waterinjetion,GOR

and water ut proles)ould not explain the observedompation after 93, when the pressure

delinewas beginning to stopby inreasedsupport. Inluding awaterweakeningmehanismto

the model gavejust as good predition of the previousparameters, but theompation volume

was betterestimated(rightin Fig. 3.4).

Figure3.4: Observedsubsidenerate(left)andhistorymathingofompationvolume(right)at

Ekosk

Relevant minerals in halk

replaement: Volumetri

onsiderations

4.1 Inluding more minerals and volumetri onsiderations

We want to onsider what happens if alite

CaCO 3

^{dissolves and is replaed by another pre-}

ipitating mineral. Ifthe newmineral takeslessspaethere should beinreased porosity,while

minerals taking more spae would redue porosity. For simpliity we assumethat the molesof

ionsinsolutionarenegligibletothosethathavepreipitated. Inthiswayweanquiklyestimate

whether an inreaseor redutionin porosityis likelyfor theinjeted brine and whihions that

should beprodued. Fromanotherpointofview,giventhebrineand outletomposition wean

makeaqualiedguessofwhihreationsaretakingplaein theore. Forthealulationsweuse

thatalitehasdensity

2.71 g/cm ³

^{andmolarweight}

100.087 g/mol

^{so1molaliteorresponds}

to

1mol ∗ 100.087g/mol

2.71(g/cm3) = 36.93cm3

^(4.1)

Intheoriginalmodel[22,23℄onlyalite,magnesiteandanhydritemineralswereonsidered. We

evaluatesomedierentmineralsandtheirpossiblerelevanetowaterweakening.

4.2 Magnesium-bearing minerals

4.2.1 Magnesite

Magnesite

M gCO 3

^{reatedfromaliteanbedesribedas}

CaCO 3 + M g ²⁺ ⇋ M gCO 3 + Ca ²⁺

^(4.2)

Magnesitehasadensityofa.

3.1 g/cm ³

^{(atuallybetween3.0and3.2)andmolarweight}

84.314 g/mol

^{. 1molofalitewouldhaveavolumeof}

36.93 cm ³

^andifitwastransformedintomagnesite thesolidvolumewouldbe

1mol ∗ 84.314g/mol

3.1(g/cm3) = 27.20cm3

^(4.3)

avolume redutionof

26.3%

^{. With uid allowedto esapeit is easyto see how suh aproess}

ouldberelevanttowaterweakening. Foronethingitwouldselfontratthematrixandenhane

ompation,despiteiftheuidheldthesamepressure.Seond,loadarryinggrainmirostrutures

would bedestabilizedandthestrengthoftherokshouldderease.

Asmentioneddolomite,

CaM g(CO 3 ) 2

^{,isloselylinkedwithaliteinitsgeologialformationand}

itisreasonabletothink theyouldtransforminto eahother hemiallyundertherightirum-

stanes. Espeiallythesupplyofmagnesiumionsisneessary,buttherateofthistransformation

is also important (whether the reations happen fast enoughto matter). We an onsider the

transformationasanetreationoftheform

2CaCO 3 (s) + M g ²⁺ (aq) ⇋ CaM g(CO 3 ) 2 (s) + Ca ²⁺ (aq)

^(4.4)

Dolomitehasdensity

2.85 g/cm ³

^{andmolarweights184.401g/mol}respetively. 2molesofalite hasavolumeof

2 ∗ 36.93 = 73, 86cm3

^(4.5)

whileifthesemolesweretransformedto1moldolomitethevolumeofsolidwouldbe

1mol ∗ 184.401g/mol

2.85(g/cm3) = 64, 70cm3

^(4.6)

A omplete transformation ofalite into dolomitewould meanalmost

12.5%

^{redutionin rok}

volume.

4.2.3 Huntite

As mentioned huntite

CaM g 3 (CO 3 ) 4

^{an bea very relevant mineral for water weakeninggiven}

resultsfromSEMmeasurements. Calite-huntitetransformationouldgoas

4CaCO 3 (s) + 3M g ²⁺ (aq) ⇋ CaM g 3 (CO 3 ) 4 (s) + 3Ca ²⁺

^(4.7)

4molesalites hasavolumeof

4 ∗ 36.93 = 147.72 cm ³

^{. Huntite has density 2.87}

g/cm ³

^(from

[26℄)andmolarweight353.029

g/mol

^{so1molhuntitehasvolume}

1mol ∗ 353.029g/mol

2.87(g/cm3) = 123.01cm3

^(4.8)

leadingto arokvolumeredutionof

16.73%

^.

The transformation of aliteinto magnesium-bearing minerals seems to redue the matrix

volume.

4.3 Sulphate-bearing minerals

4.3.1 Anhydrite

The last mineral used in the original model was anhydrite:

CaSO 4

^{. It should be noted that}

anhydritean bond withwaterto form gypsum

CaSO 4 · 2H 2 O

^{. A net}transformationof alite intoanhydriteanbedesribedby

CaCO 3 + SO ² ₄ ⁻ ⇋ CaSO 4 + CO ² ₃ ⁻

^(4.9)

Anhydrite has density

2.97 g/cm ³

^{and molar weight}

136.139 g/mol

^{. 1mol alite} transformed intoanhydritewouldgofrom36.93m3solidvolumeto

1mol ∗ 136.139g/mol

2.97(g/cm3) = 45.84cm3

^(4.10)

an inrease of

24.1%

^{suggesting that if this reationis dominant we should observe a redued}

permeability and perhaps even plugging. It an be mentioned that gypsum has lower density

(

2.31 − 2.33 g/cm ³

^{)andhighermolarweightsuggestingthatapartialonvertionofanhydriteinto}

gypsumwouldfurtherlltheporesbyinreasingthesolidvolume. Gypsumishowevermoderate

solublewhileanhydriteislesssolubleandthusmorerelevant.

Thisisjust formentioning. Ironionshavenotbeeninludedinthemodel sofar,but an playa

role. Espeially intheaseofdrilling,partilesfrom pipesorequipmentan bearried withthe

oweither as grains ordissolvedand aetaloal region(ironhasanegligibleonentrationin

seawater). Ifthisissigniantaskinan developlosetothewell.

Siderite

F eCO 3

^{and ankerite}

CaF e(CO 3 ) 2

^{havedensities}

3.5

^and

2.9 − 3.1 g/cm ³

^andmolar

weights

115.854

^and

215.941 g/mol

^{. Followingthe}transformationsofaliteas

CaCO 3 + F e ²⁺ ⇋ F eCO 3 + Ca ²⁺

^(4.11)

2CaCO 3 + F e ²⁺ ⇋ CaF e(CO 3 ) 2 + Ca ²⁺

^(4.12)

wegetforsideriteavolumeredutionof

10.4%

^{,whileforankeriteweangetsomewherebetween}

.81%

^expansionand

9.4%

^{redution. Both aseslean toward a redutionin} matrix-volume. In otherwordsit seemsironionswillnotausehemialdamagetolimestoneandhalkreservoirs.

Nearholedamageis likelymoreaetedbymudpartilespluggingtheporethroats.

Transport-reation model

Thetransportmodelsuggestedin [22℄onsiderstheproessofintroduingabrineinto aporous

rok ontaining an original brine in hemial equilibrium. The solution an be desribed by

indiating the onentration of eah hemial at a given loation, whether it be rok minerals,

waterordissolvedsubstanes. Speiallytheunknownswesolveforaretheporeonentrations

C i

^{of omponentsin uid phase, thetotal volume} onentrationsof minerals

ρ i

^{and pressure}

p

^.

All thesevariablesare funtionsof position andtime

(x, t)

^. Temperatureis onsideredonstant, asisthepartialpressureofdissolvedgasinwater.

Tosolvetheequationsweusemolarbalaneequations,equationsforinstantwaterequilibrium

and a hargebalane. Inorporatedinto these equations are rate expressionsfor the rok/uid

reationsandtheuidomponentveloities.

Theappliationispartiularlyrelevantforhalkreservoirsormoregenerallyarbonatereser-

voirsandthisisreetedintheonsideredhemialreations.

5.1 Components

Wedivideallhemialomponentsinto4groups. Theyarepresentedbyname,hemialompo-

sitionandprimaryunknownwithindexusedforrefereneinequations. Dolomitehasbeenadded

tothemodelseeifitmakesabettertthanmagnesiteorifbothmineralsshouldbeinluded.

5.1.1 Solid state: minerals

•

^Calite,

CaCO 3

^,

ρ c

•

^Anhydrite,

CaSO 4

^,

ρ g

•

^Magnesite,

M gCO 3

^,

ρ m

•

^Dolomite,

CaM g(CO 3 ) 2

^,

ρ d

5.1.2 Aqueous state: ions

•

^Calium,

Ca ²⁺

^,

C ca

•

^Magnesium,

M g ²⁺

^,

C mg

•

^Sulphate,

SO ² ₄ ⁻

^,

C so

•

^Sodium,

N a ⁺

^,

C na

•

^Cloride,

Cl ⁻

^,

C cl

•

^Hydron,

H ⁺

^,

C h

•

^Hydroxide,

OH ⁻

^,

C oh

•

Biarbonate,

HCO ₃ ⁻

^,

C hco

•

^Carbonate,

CO ² ₃ ⁻

^,

C co

5.1.3 Dissolved gas

•

^{Carbondioxide,}

CO 2

^,

P CO2

^{(assumedgivenby}temperature)

5.1.4 Liquid state

•

^Water,

H 2 O

^,

C l

Note that themineralsare assumedto exist onlyin solid phasewhile theother omponents are

assumedtobepartofthewaterphase,either asions,dissolvedgasorwater.

5.2 Reations

5.2.1 Dissolution and preipitation of minerals

•

^Calite:

CaCO 3 + H ⁺ ⇋ Ca ²⁺ + HCO ⁻ ₃

•

^Anhydrite:

CaSO 4 ⇋ Ca ²⁺ + SO ₄ ^{2 −}

^,

•

^Magnesite:

M gCO 3 + H ⁺ ⇋ M g ²⁺ + HCO ⁻ ₃

^,

•

^Dolomite:

CaM g(CO 3 ) 2 + 2H ⁺ ⇋ Ca ²⁺ + M g ²⁺ + 2HCO ⁻ ₃

These reations between uid and rok ourwith a nite rate dened in setion 5.6. We use

themto denetheratetermsinthedierentialequations.

5.2.2 Aqueous reations

• CO 2 + H 2 O ⇋ HCO ₃ ⁻ + H ⁺

• HCO ₃ ⁻ ⇋ CO ² ₃ ⁻ + H ⁺

• H 2 O ⇋ H ⁺ + OH ⁻

The reations in the uid phaseour at high ratesompared to themineral reations and are

assumed to be in equilibrium. Theyare used as onstraints,that is3 equationsto determine 3

unknowns.

5.3 Porosity and volume balane

Intheformermodels[22, 23℄avariableporosityhasnotbeenfullyonsidered. Thissetionwill

attempttomakeaphysiallymeaningfuldenitionofporosityasafuntionoftheloalvariables.

Givenall theomponentswean separate them into thoseexisting in solid phase(minerals)

and those in the uid phase(water, dissolved ions and gas). Considerasmall partof the ore

sample with volume

V

^{. At agiven time all omponents havedened their total} onentration

ρ i

^{, where}

i

^{representsthe givenomponent. If wealso know themolarmasses,}

M i =

^mass

mol , and

eetive densities,

ω =

^mass

eetivevolume

(by eetivevolume we mean the volume the omponent

numberofmoles

n i

^,mass

m i

^andvolume

V i

^:

n i = ρ i V

^(5.1)

m i = M i ρ i V

^(5.2)

V i = M i ρ i V ω i

(5.3)

V i

V = M i ρ i

ω i

(5.4)

Note thatthelast equationis thevolumefration ofomponent

i

^{. Sinethetotal volumeis the}

sumofeetivevolumes

V = A∆x = X

i

V i = X

i

M i ρ i V ω i

(5.5)

X

i

M i ρ i

ω i = 1

^(5.6)

Thevolumefrationofsolidphaseisthen

V minerals

V = X

i:minerals

M i ρ i

ω i

(5.7)

andtheporosityisperdenitiontheremainingvolumefration

φ = 1 − X

i:minerals

M i ρ i

ω i

(5.8)

Eqn(5.6)anintheorybeusedasaonstraintontheunknowns(justasthesumofsaturations

should be

1

^{in amultiphaseproblem). Inpratiethere areafewdiulties though. Ifwehave}

properlydenedthehemialstrutureofeahomponentthenall

M i

^{an befoundfromtables.}

(Eetive) density for roks and wateris also available in the literatureand it an be adjusted

for temperature and pressure using the minerals oeient of thermal expansion (

α = _{V dT} ^dV

^),

thepressureompressibility (

β = − V dp ^dV

^{) and thedeviations from thereferene state. Inreased}

pressureandinreasedtemperaturetendtohaveoppositeeet,and bothvaluesaretypiallyof

low order(perents) forsolid roksand liquidsandweassumethedensities remainonstantfor

simpliity. Therewillbemoreunertaintyrelatedtohowmuheetivevolumeisoupiedbythe

dissolvedionsandgas. Water ispolarandouldsometimes bepushedawaybyequalhargesto

inreasetheeetivevolume. Injust thesamewayitouldwork toshrink. Theionsthemselves

willperhapsoupymoreeetivespaeiftheyaremoreharged. Onepossibilityistoassumethe

atomsaresofarapartduetolowonentrationsthattheireetivevolumeisthesame,espeially

thesameaswater,whihisknownwithgreat auraysineitsdensityisknown. Wewillgivea

betterdenition ofvolumebalanelaterusingthewaterphaseasawhole.

Thementionedunertaintiesdonoteet(5.8)sinetheionsarenotinluded,butthevolume

balanemustbeaonstrainttodenetheporositythewaywedo.

Intheoriginalmodel[22, 23℄itwasassumedthat porositywas onstant. Lettingit varywill

inreasetheouplingofvariablesinthedierentialequations.

5.4 Permeability and possible hysteresis

Inshort wordswetreatloal permeabilityas afuntion of loalporosity. In[22,23℄ ithasbeen

assumedonstant.

Chalkhasnarrowporethroats,butlargepores,resultinginhighporosityandlowpermeability.

When grains are fored againsteah other they will tend to dissolveat theontatpointsand

Bernoulli'slawitislearthattheveloitywillbegreaterintheporethroatsandtheporepressure

less. Higherveloitywilldragonthegrainsandreduedporepressurewillinreasetheloadarried

bytherok. Thesemehanismswouldfavoranimprovementinpermeabilitybyinreasedporosity.

On theother hand, whenthe ow enters the widepores and thepressureis largerand veloity

smaller,grains shouldsettle andpossiblypreipitationwouldourmoreeasilyin these regions.

Following this reasoningdeposition should not eet the size of thepore throatsverymuh. A

redution in porosity should redue permeability less than a leaning eet would inrease it.

Thiswouldleadtoaform ofhysteresis,meaningthataporosityinrease,followedbyaporosity

redutiontothesamelevelwouldgiveabetterpermeability. Wenegletanysuhbehavior,partly

forsimpliation, andpartlybeausetheporosityshould gomainlyin one diretion. Themodel

assumesnomovementofsolidpartilesarriedbytheuid. Thatmeanspartileslargerthanthe

porethroatsdonotauseanypluggingeet.

It should be noted that the proesses desribed will depend on uid veloity, its ability to

arry grains (involves visosity), the variation in area from pore to throat, rates of dissolu-

tion/preipitation,stressesinthe rok,uidpressureandprobablyother fators. Sine itwould

bepratiallyimpossibleto makeaurate measurementsrelatingsuhpore saleeetsto per-

meabilitywhihismeasuredonoresalewesettleforamoreunertainrelationthatjustrelates

permeabilityto porosity,that is

k = k(φ)

^{. This an bejustied bythinking oflow}permeability as aregion ofloally lowporosity. The measuredpermeabilityovertheorelength will depend

onthewhole distribution,espeiallyonthesmallestvalues.

Assumewehaveaninitialdistributionofbothpermeabilityandporosity:

k(x, t = 0) = k 0

^and

φ(x, t = 0) = φ 0

^{. Weassumethereis afuntion}

f ( · )

^suhthat

k

k 0

= f ( φ φ 0

)

^(5.9)

With nohysteresisinitial

φ

^{orrespondsto initial}

k

^{, so}

f (1) = 1

^{. Improvingone should improve}

the other so

f ^′ > 0

^{. Both should bezero at thesame time so}

f (0) = 0

^{. If it is truethat the}

throatsareattakedrstthentheeetshouldbemostrapidlosetotheoriginalstatewhenthe

throatsare small ompared to thepores. Also theeet should be lesspowerfulwhen they are

omparableinsize,suggestingthat

f ^′′ < 0

^.

Somesuggestionsareevaluatedinappendix??withreferenes. Itisshownthatdependingon

thehoieofformulationof

f

^{werequireaorrelationtot}

f (x) =

( x ^a 0 < x < 1

bx ^c + 1 − b x > 1

^(5.10)

with

a > 1; b > 0; 0 < c < 1

^(5.11)

or

a > 1; b, c < 0

^(5.12)

orwean usetheorrelation

f =

( _e ax − 1

e ^a − 1 0 < x < 1 be ^cx + 1 − be ^c x > 1

(5.13)

with

a > 0; b, c < 0

^(5.14)

Thesearederivedfromtypialpermeability-porosityorrelationswhereparametersshoulddepend

onlithologyandthemehanismofthestruturalhangesinvolved.

Theassumptions are that moles aretransported in the uid phasewith aertain veloity. The

veloity depends both onuid veloity anddiusion. Solid omponentsare not transported by

theow,butaumulateordiminishloallybypreipitationordissolution.

Assumeathinuttingofaorethathasrossetionalarea

A

^{(assumedonstantwithtimeand}

position)andlength

∆x

^{. Atposition}

x

^{uidentersthevolume,andat}

x + ∆x

^{uidleaves. During}

thetime

∆t

^{there isahangein thetotalontentofmolesofthesubstaneduetotransportand}

hemialreation. Wedenote porosityas

φ

^{,omponentveloityin thepore spae}

v

^(positivein

thex-diretion),onentrationofomponentasmolespervolumeuid

C

^{andhemialprodution}

ofmolespervolumeuidpertime as

r ˙

^{. For aomponentin theuidphasewehave:}

(AφCv) x ∆t − (AφCv) x+∆x ∆t =

^{molesaddedbyow (5.15)}

A∆x r∆t ˙ =

^{molesreatedbyreationsinside thevolume (5.16)}

(A∆xφC) t+∆t − (A∆xφC) t =

^{hangeinnumberofmoles (5.17)}

Sinetheaumulationisthesumofhemialgenerationandtransportarossboundarieswehave

(A∆xφC) t+∆t − (A∆xφC) t = (AφCv) x ∆t − (AφCv) x+∆x ∆t + A∆x˙ r∆t

^(5.18)

Divide by

A∆x∆t

^andletboth

∆x, ∆t → 0 (φC) t+∆t − (φC) t

∆t = (φCv) x − (φCv) x+∆x

∆x + ˙ r

^(5.19)

∂(φC)

∂t = − ∂(φCv)

∂x + ˙ r

^(5.20)

∂(φC)

∂t + ∂(φCv)

∂x = r ˙

^(5.21)

Fluidonentrationis denedas

C =

^mol

porevolume

. Sineporosityis

φ =

^porevolume

totalvolume

weandene

totalonentrationsas

ρ =

^mol

totalvolume

=

^porevolume

totalvolume

mol

porevolume

= φC

^(5.22)

Eqn(5.21) anthenbewritten intermsoftotalonentrationsas

∂ρ

∂t + ∂(ρv)

∂x = ˙ r

^(5.23)

Forthesolidomponentsthereisonlyhemialontributiontotheaumulationsoasimilar

derivationresultsin

∂ρ

∂t = ˙ r

^(5.24)

Equations(5.21)and(5.24) arethoseoriginally used. Wewillmakeasmall alterationbynoting

thattheratetermsshouldberelatedtotheporevolumes,sinethatiswherereationstakeplae.

Inotherwords

r ˙

^{meansmolesgeneratedpertimeperporevolume fromnowon. Toonvertthis}

intoratespertotalvolumeagainsothebalaneisorret,thetermsare multipliedbyporosity:

moles

time

·

^totalvolume

=

^porevolume

totalvolume

moles

time

·

^{porevolume (5.25)}

˙

r tot = φ · r ˙ pore

^(5.26)

Themolarbalaneequationsarenow

∂(φC)

∂t + ∂(φCv)

∂x = φ r ˙

^{fornonsolidomponents (5.27)}

∂ρ

∂t = φ r ˙

^{forsolidomponents (5.28)}

Assumeareationoftheform

aA + bB ⇋ cC + dD

^(5.29)

ofhemialreatantsAandBandprodutsCandDwhere

a, b, c, d

^arestoihiometrioeients thatpreservemolarandhargebalane. Therateofthereationisdened (seealso[10,11℄)by

˙ r = − 1

a dn A

V dt = − 1 b

dn B

V dt = 1 c

dn C

V dt = 1 d

dn D

V dt

^(5.30)

where

n

^{is molesand}

V

^{is porevolume. Therate ispositivewhen thereationis shiftedto the}

right(AandBareonsumed,CandDareprodued).

Wearereallyinterestedin thederivativesontherightsidewhihisthereationsontribution

totheomponentratesusedin theequations. For examplewean saythat for

A

˙

r A = dn A

V dt = dC A

dt = − a r ˙

^(5.31)

statingthatifthereationmovesto theright(

r ˙

^{positive)then}

A

^{isonsumedbyanamountof}

a

omparedtothereationrate.

Thereationrateisafuntionofthehemialativity,

a i

^{oftheinvolvedomponents. Ativity}

isdiretlyrelatedto uidonentration

C i

^by

a i = γ i C i

^(5.32)

where

γ i

^{istheativityoeientofomponent}

i

^{,to bedisussedlater.}

Therateofwhihtheleftandrightsideomponentstransforman begivenas

k +1 a ^a _A a ^b _B

^and

k − 1 a ^c _C a ^d _D

^where

k +1

^and

k − 1

^{are positiveonstants, but spei for thegivenreation and the}

temperatureofonsideration. Thenetrateofthereationis

˙

r = k +1 a ^a _A a ^b _B − k ⁻ 1 a ^c _C a ^d _D

^(5.33)

Suhaformulationwas madein[22℄.

In[23℄,arateexpressionoftheform

˙

r = k(1 − Ω) ⁿ

^(5.34)

was adopted from[10℄, onlyusing

n = 1

^{forsimpliity. Thismodelwill alsobeappliedhere. Its}

appliation isdissolution reations and

Ω

^{is dened as theativityprodut ratiodividedby the}

solubilityonstant.

Ω = Q/K

^(5.35)

We will show whih assumptions an lead to suh a model: The point of view is that the

reationsofonsiderationaredissolutionreationswithomponent

A

^{beingthemineral. Minerals,}

water and

CO 2

^{are here assumed to have ativity equal to}

1

^{. Gas omponents are normally}

representedby theirpartial pressurein reationrates, but it isassumed herethat all gasexists

dissolvedin thewaterphaseandthat thisamountisgivenbytheonstanttemperature.

Dissolutionhasrate

k +1 a ^a _A = k +1

^whilepreipitationhasrate

k ⁻ 1 a ^c _C a ^d _D

a ^b _B

^{. Thenetreationrate}

isthen

˙

r = k +1 − k − 1

a ^c _C a ^d _D

a ^b _B

^(5.36)

Having dened the reations the exponents are known and given the urrent state, so are the

ativities. Ifweknow

k +1

^and

k − 1

^{weanspeifythereationrateandthusthehemialprodu-}

tion/onsumptionofagivenomponentdueto thisspeireation.

k +1

^and

k ⁻ 1

^arerelatedby

thesolubilityprodut

K

^{whihanbefound}experimentallyor perhapseveninhemistrytables.

Atequilibriumthereationrateis

0

^andwedene

K ≡ k +1

k − 1

= a ^c _C a ^d _D

a ^b _B

^(5.37)

Thesamevalueof

K

^{resultsfrombothrate}formulations,butthespeivaluesof

k +1

^and

k − 1

anbedierent.

Wethenwritethereationrateas

˙

r = k +1 (1 − a ^c _C a ^d _D

a ^b _B K )

^(5.38)

andnotethatthementioned

Ω

^{istheratioofativityprodutsdividedbythe}equilibriumonstant forthereation.

Dissolutionan happenonlyaslongasthemineralexists. Eahdissolutionreationratewill

thereforebemodiedsothat iftheonentrationofthemineralis

0

^{thereationrateannotbe}

positive,but issetto

0

^.

Therateexpressioniswrittenasafuntion

F

^timesthe

k +1

^{fator. Wethenseparate}

F

^into2

termsaordingtowhenitispositiveor negative. When

F

^{ispositiveandmineral}onentration iszero,rateisset to

0

^.

F ≡ 1 − Ω = (1 − a ^c _C a ^d _D

a ^b _B K ) = F ⁺ − F ⁻

^(5.39)

F ⁺ ≡ max(0, F ), F ⁻ ≡ max(0, − F ), sgn ⁺ (x) =

( 1

^if

x ≥ 0 0

^else

(5.40)

˙

r = k +1 [sgn ⁺ (ρ)F ⁺ − F ⁻ ]

^(5.41)

5.6.1 Chemial ativity

Ion ativities

a i

^{are relatedto uid}onentrations

C i

^as

a i = γ i C i

^.

γ i

^isomponent

i

^'sativity

oeient,givenbytheDebye-Hukelformula(see[10,12℄)

− log ₁₀ (γ i ) = A(T )Z _i ² √ I 0

1 + a ⁰ _i B(T ) √ I 0

(5.42)

I 0 = 1 2

X

i

C i Z _i ²

^(5.43)

where

I 0

^{is theioni ativity and}

Z i

^{arethe ioniharges.}

a ⁰ _i

^{are omponent spei onstants}

indiating the eetive size of the hydrated ion measured on angstrom and an be found from

tablessuhasin [12℄. Theonstantsweuseare

Z ca = +2, Z mg = +2, Z so = − 2, Z na = +1, Z cl = − 1, Z h = +1, Z oh = − 1, Z hco = − 1, Z co = − 2

a ⁰ _ca = 6, a ⁰ _mg = 8, a ⁰ _so = 4, a ⁰ _na = 4, a ⁰ _cl = 3, a ⁰ _h = 9, a ⁰ _oh = 3.5, a ⁰ _hco = 4, a ⁰ _co = 4.5

A(T )

^and

B(T )

^are orrelations of the density of water, the dieletri onstant of water whih dependsontemperatureandtemperatureitself. Suhrelationsaregivenin[12℄.

Thetemperatureweonsider isaonstant

130

^{degreesCelsiusandwehave}

A(T = 130) = 0.6623 B(T = 130) = 0.3487

^(5.44)

whih were alulated in [22, 23℄ using thesimulator EQAlt.

I 0

^{, the ioniativity is evaluated}

withtheomposition oftheinjeteduid andassumedonstant. Intotalallativityoeients

arethentreatedasonstantsforagivensimulation.