The influence of spatial variability on 2D reactive transport simulations

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The influence of spatial variability on 2D reactive transport simulations

Marco de Lucia, Vincent Lagneau, Chantal de Fouquet, Roberto Bruno

To cite this version:

Marco de Lucia, Vincent Lagneau, Chantal de Fouquet, Roberto Bruno. The influence of spatial

variability on 2D reactive transport simulations. Comptes Rendus Géoscience, Elsevier, 2011, 343 (6),

pp.406. �10.1016/j.crte.2011.04.003�. �hal-00613170�

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on 2D reative transport simulations

Inuene de la variabilité spatiale

sur des simulations 2D de transport réatif

Maro De Luia

1,2

∗

^, ^Vinent ^Lagneau¹^, ^Chantal ^de ^F^ouquet¹^, ^Roberto ^Bruno²

1

MINESParisTeh

2

DICMA,UniversitàdiBologna

∗

orrespondingauthor's urrentaddress:HelmholtzCentrePotsdam

GFZGermanResearhCentreforGeosienes

Telegrafenberg,14473Potsdam- Germany

Tel:+49(0)3312882829,Fax:+49(0)3312881529

deluiagfz-potsdam.de

Abstrat

Inreativetransport simulations, theeets of thespatial variability of geologialmediaare gene-

rallynegleted. Theimpat ofthis variabilityis systematiallyexamined herein 2Dsimulations, with

a simple geometry and hemistry with a positive feedbak : inrease of porosity and of permeability

duringalitedissolution.Theresultshighlighttheleadingroleintheseonditionsof(i)theorrelation

length of porosity and of permeability and (ii) the kinematidispersivity, whose eets are dominant

omparedtothoseofvarianeandreationkinetis.Theimpatofstohastivariability(betweenseveral

randomdraws)isalsosigniant,asitisofthesameorderofmagnitudeastheimpatoftherangeand

dispersivity.

Keywords:spatialvariability,reativetransport,feedbak,range,dispersivity,kinetis

Résumé

La desription de la variabilité spatiale des milieux géologiques dans les simulations de transport

réatifestgénéralementnégligée.Dansetteétude,nousmettonsenévidenel'impatdeettevariabilité

de manière systématique sur les résultats de simulations 2D, sur une géométrie élémentaire, dans un

as de réation himique simple ave rétroationpositive : augmentation de laporosité et don de la

perméabilité lorsdela dissolutiondela alite.Les résultatsmettent enavant,dans es onditions,le

rle prépondérantde laportéeet de ladispersivité inématique,suivis par lavariane et lainétique.

L'eet de la variabilité stohastique (entre diérents tiragesaléatoire) n'est pas négligeable non plus,

puisqu'ilest demêmeordredegrandeurqueeluidelaportéeet deladispersivité.

Mots-lés:variabilitéspatiale,transportréatif, rétroation,portée,dispersivité, inétique

1 Introdution

Reativetransportmodellingin porous mediais apowerfultoolforstudyingwater-rokinterations;it

isused toimprovetheunderstandingof groundwaterpollution,undergroundwasteandarbondioxide

storage,metallogenesis,theenvironmentalimpat ofminingplants,et. Inthelasttwodeades, great

eortshavebeenmadetoenhanetheapabilitiesofmodellingtoolstokeeppaewiththeproblemsthat

theyansuessfullyhandle(unsaturatedmedia,multiphaseow,non-isothermalow,redoxreations,

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modelling hasoftenbeendisregarded: desriptionoftheintrinsispatialvariabilityof geologialmedia

thatispresentateverysale. Themainreasonsisalakofrealgeohemialdataneededtoevaluatethe

spatialvariabilityofthegeologialformationsattherelevantsale(i.e. betweentheREV,Representative

ElementaryVolume,andthesizeofthesimulationdomain),andtheinsuientCPU-powerneededtorun

themodels, whih limitsthe spatialdisretizations to afew thousandgrid nodesto keep theruntimes

aeptable. Very reently, mainly due to inreasing CPU apaity and a growing onern about the

heterogeneityof thegeologialmedium, aninreasing numberof authors tendto set initial onditions

inludingspatialvariabilityonsomeparametersintheirsimulations([8,12,6,15℄). However,theeets

ofthisvariabilityhavenotbeensystematiallystudiedsofar.

Thepurposeofthisstudy istoexamineandquantifytheimpatofspatialvariabilityontheresults

of2Dreativetransportsimulations,withapartiularfousontheroleofhemistryfeedbakonhydro-

dynamis,whihistriggeredbythedissolutionofprimarymineralsresultingininreasedporosityand,

possibly,permeability. Asimplehemistryandaverysimplegeometrywerehoseninordertoproperly

handletheimpatof theinitialspatial variabilityaswellasthatofthekinetiand hydrodynamiow

onditionsontheresultsof thenumerialexperiments.

Inthisstudy,permeabilityandporosityeldswithspatialvariabilityareonsidered.Variablemineral

onentrationswere also investigated, but will bedisussed onlybriey. Non-onditional geostatistial

simulationsofpermeabilityandporositywereperformed,overingarangeofvaluesforthemostrelevant

parameters(seedisussionin setion3.1),inordertoestablishahierarhyamongthem.

The paper is organizedas follows: rst a brief overview of the relevant onepts and methods of

geostatistis and reative transport modelling, the spatial models and the hemial systemhosenfor

thisstudyaredisussed;thentheresultsforhomogeneousmediaareshown,tolarifytherepresentation

of thesystemandgiveareferenefor theheterogeneousmediasimulations. The observablequantities

hosento representthe resultsareexplained anddisussed; nally, wedisussthe resultsofnumerial

experimentspointingouttheeetsofeahparameter.

2 The tools

2.1 Spatial models and geostatistial simulations

Geostatistisis awell establishedbranh ofstatistis and probabilitytheory dealingwith spatial pro-

esses,i.e. withvariablesdenedoveraspatialdomainandreferredtoaspeivolume,alledsupport.

ItwastheoretiallyfoundedbyG.Matheroninthe1960's,andfounditsmainappliationinminingand

oilexploration. It is urrentlyapplied to virtuallyeverydisiplineof thegeosienes. Athorough and

insightfuloverviewof thetheory isgivenin [2℄, to whih thereaderanreferto formoredetails. The

fundamentalnotionofgeostatistisisthespatialautoorrelationfuntion;moregenerally,thesemivar-

iogram,orjust variogram,depitsthe spatialvariabilityoftheproessZ ^between^two^distint ^points

ofthedomain,asafuntiononlyoftheirdistane:

γ(~h) =1 2^E

h

Z(x)−Z(x+~h)i2

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Theassumption of seond-order stationarity 1

and ergodiity is suient to establishthe existene

of avariogramfuntion, whih is non-negativeand usually monotoni, and whih anbe modelled by

means of simple analyti expressions. Throughout this study, a unique variogrammodel is used, the

isotropispherial model,whihisatrunatedpolynomialfuntion:

γ(~h) =









 σZ²



 3 2

|~h| a −1

2

|~h| a

!3

 ^for 0≤ |~h| ≤a σZ² ^for a <|~h|

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The parameter a îs ^the ôrrelation ^length, âlso âlled ^range. Ît ^represents ^the ^maximum ^distane ât

whih twopoints in thedomain show aorrelation. For distanes largerthan this orrelation length,

1

seondorderstationaritymeansthatalltherstandseondmomentsoftheRandomFuntionareonstantoverthe

domain.

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thevariogramfuntion isonstantandequalstothesill. ThesillorrespondstoσZ²^,^the^variane^of^the

variable.

Formultivariateproblems,theross-variogrambetweenvariablesZ1^andZ2^represents^the^orrelation

betweentheirinrements:

γ12(~h) = 1 2^E

h

Z1(x)−Z1(x+~h)i

×h

Z2(x)−Z2(x+~h)i

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Toaddressand model theorrelationbetweenporosityandpermeability, theintrinsimodel of o-

regionalizationwashosen: all diret- and ross-variogramsare multiples of abase model [2℄. This is

ahievedbythefollowingonstrutionforthesimulationofpermeability(notedask⁾^and^porosity⁽ω^):

Y^k=ρY^ω+p

1−ρ²·R ⁽⁴⁾

where ρ îs ^the ôrrelation ôeient ât ^the ^same ^point ^between Yk ând Yω^, ^whih âre ^the ^redued

Gaussian elds respetivelyassoiated withpermeability and porosity; Y^ω ^and R ^are^twoindependent geostatistialsimulations,randomlydrawnwiththesamevariogrammodel.

Thejointspatialdistributionsofpermeabilityandporosityareobtainedbynon-onditionalgeostatis-

tialsimulationsonne,regulargridsbytheDisreteFourierTransformalgorithm[18℄. Thistehnique

washosenforitsspeedandfortheimmediatepossibilityof obtainingafamilyofGaussianelds with

dierent orrelationlengths from asingleset of randomnumbers. This feature allowsthediret om-

parison of results obtained from geostatistial simulations with dierent orrelation lengths (or other

parameters), eetivelyltering theeet of the additionalvariability that would havebeenobtained

fromseveralrealisations.

TheGaussian random variables are then transformed into log-normal elds to math thelassial

distributionusedforpermeability:

K=e^m^k⁺^σ^k^Y^k ⁽⁵⁾

andsymmetriallyfortheporosityω^. ^This transformationstepallowsusto takeintoaountamaro- sopi orrelation between porosity and permeability, as observed in natural systems. Among those

availablein theliterature,wehosetheempiriallawofBretjinski[16℄, whihreads:

ω= 0.593√⁷

K ⁽⁶⁾

inwhihK îsêxpressedâs^m·^s⁻¹^. ^Tâking^the^logarithm:

log(ω) = log(0.593) +1

7log(K) ⁽⁷⁾

makesiteasytoderivearelationshipbetweentheoeientsmω^,mk ^andσω^,σk ^to^be^usedⁱⁿ^the^log-

normaltransformation. TwolognormaldistributionsarethusobtainedwhoseGaussiantransformations

arerelatedbytheorrelationoeientρ^and^whihstohastiallyobeyBretjinski'slaw. Notethatifa

σlogk îsîmposed,^this^xes^theσlogω^, ^thus^reduing^the^degreesôf^freedomôf^the^model.

Tosummarize, when the variogrammodel and theaverages of porosityand permeability are xed

oneandforall,themultivariatespatialmodelisompletelydenedbythreeparameters:theorrelation

lengtha^of^thevariograms,thestandarddeviation σlogK ^ofpermeabilityandtheorrelationoeient

ρ^betweenlogK ^andlogω^.

2.2 Reative transport ode

Reativetransport simulationsweremade withtheoupledprogramHyte, developed attheMINES

ParisTeh[22℄. InHyte, thehydrodynami part(ow,multiomponenttransport, heat transport)is

solvedbyanite-volumesapproahonunstruturedgridsbasedonVoronoïpolygons. Reativehemistry

isevaluatedbyhess,alsodevelopedatMINESParisTeh[23℄. Itdeterminesaqueousspeiation,ioni

exhange,surfaeomplexation,mineralpreipitationanddissolution,assumingeitherloalequilibrium,

oradynamimixedstatusof equilibrium/kinetis.

An upsalingmethodforpermeabilityisneessarytolink thene-gridded geostatistialsimulations

and the oarse, non-regular grids used in the oupled transport-hemistry modelling [19℄. Suh teh-

niqueswereintroduedandthoroughlydisussedin[4,5℄: theyweredenedinthemostgeneralaseof

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surfaeoftriangles onbothsidesof theboundarybetweenelements) andof thenite volumes sheme

(the alulation isrestrited tothe omponentof ow orthogonalto theboundarybetweenelements).

Thereaderisreferredto[5℄forfurtherdetails.

2.3 Hydrohemial setting

The investigated hemial system depitsthe dissolution of alite(CaCO3⁾ ^following^the ^injetion ^of

hydrohloriaidHCl. Thisreationhasseveralappliations,e.g. inthework-overofoil-produingwells.

More generallyspeaking, it an be onsidered a shemati illustration of an aidattak on arbonate

roks, a proess expeted to govern, for instane, the natural development of karsti systems, or in

geologialstorageofCO2^. ^F^rom ^the^pointôf^viewôf^hemistry^feedbakâtingôn^thehydrodynamis, itrepresentsaaseofinreasingporosity.

Thereationanbewrittenforexample:

CaCO3+ 2^HCl⇀↽^Ca²⁺+^CO2^(aq)+^H2^O+ 2^Cl⁻ ⁽⁸⁾

oralternatively

CaCO3+^HCl⇀↽^Ca²⁺+^HCO⁻₃ + 2^Cl⁻ ⁽⁹⁾

ThepHofthesolutionisontrolledbyaarbonatebuertriggeredbyalitedissolution,whihonsumes

theaid;atthesametime, thereationproduesalium,hlorideandCO2 ⁱⁿ ^solution. ^F^ollowing^the

reation,themineralvolumedereases,thus produinganinreasein porosity.

Under the thermodynami equilibrium assumption, the mass ation law onstrains the hemial

speiationvia theformationonstantK^:







K = ^[^H⁺^]

[^Ca2+

]·[^HCO− 3 ]

ifaliteispresent

K < ^[^H⁺^]

[^Ca2+]·[^HCO− 3 ]

otherwise

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Optionally,thedissolutionreationanbekinetiallyontrolled. Wehoseawidelyusedrst-orderlaw

[11℄toexpress thevariationofonentrationS ^of^a^mineralⁱⁿ^time:

dS

dt =−k^h(1−Ω^calcite) ⁽¹¹⁾

In this form, the kineti onstant kh îs êxpressed ⁱⁿ ^[molal · ^s⁻¹^℄, ^therefore împliitly înluding ^the

(onstant) reative surfae of the mineral. The subsript h ^refers ^to ^the implementation in Hyte.

Ω^calciteîs^the^ratio^between^theÎonÂtivity^Produtând^the^formationônstantôfâlite;^therefore^the

non-dimensionalterm(1−Ω^calcite)^aounts^for^the^deviation ^from equilibrium. This kinetilawdoes notdepend expliitlyon theporosity, but only onthe ativitiesof the speies. Other types ofkineti

lawsanbeused,partiularlyto betterdesribetheeetofvarying reativesurfaeareas,oratalyti

eets. However,theauthorsbelievethattheuseofotherkinetilawswouldnotsigniantlyalterthe

leadingonlusions ofthestudy.

Intuitively,primarymineraldissolutionmodiestheporestrutureofthemedium, thustriggeringa

feedbakmodiationofitshydrodynamiproperties,aetingowandtransport. Themodelsworking

at the REV sale (Representative Elementary Volume) are intrinsially not able to take into aount

suh feedbak; itis therefore ruial to introduefurther information in the model. The simplest way

is to relate porosity (an extensive variable, alulable by a mineral volume balane) and permeability

by anempirial relationship. To this purpose, thesame Bretjinski's law (eq. 6) used to orrelatethe

marosopi porosity and permeability spatial variability was hosen to hange the permeability asa

funtionof porosityhanges:

K K0 =

ω ω0

^κ

withκ= 7 ⁽¹²⁾

but therelationshipis nowxedand notprobabilisti. Otherforms of theporosity/permeabilityrela-

tionship ould havebeen used, suhasthe well known Kozeny-Carmanlaw. Thelattergivesaslower

inreaseofpermeabilitythanBretjinski'sfollowingtheinreaseofporosity. Nevertheless,theinueneof

suhahoiehastobefurtherinvestigated,optimallyinludingexperimentaldatarelatedtotheatual

mediumandthemodelledreation.

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ler

Consideringthemarosopi(Dary)sale,thegoverningequationsareestablishedwithoutattemptingto

deriveanupsaledequationfrommassandmomentumonservationequationsdenedatthemirosopi

sale.

Theadvetion-dispersionequationforatransportedonentrationCⁱ ^reads:

ω∂Ci

∂t − ∇ ·(D^∗· ∇Cⁱ) +∇ ·(~u Cⁱ) =νⁱk^h(1−Ω^calcite) ⁽¹³⁾

whereωîs^the^porosity^,Ciîs^the^molalityôfâtransportedspeies,D^∗=De+αk~ukîs^the^loal^dispersion

tensor, sumofaneetivediusion oeientandadispersiveterm,in turn proportionalto thenorm

oftheloalDaryveloity~u^; Deîs^theêetive^diusion ôeient⁽ⁱⁿ^m2

s

−1

)-thusinorporatingthe

eetof tortuosity-and α^thedispersivity(inm).

The right hand side of this equation orresponds to a reation rate governed by the only mineral

(alite) present in the domain. kh îs ^the ^kineti ônstant âdopted ⁱⁿ ^Hyte ^(eq. ¹¹⁾ ând νi ^the

stoihiometrioeientofthe i^-th âqueous^speiesⁱⁿ ^the ^hemial^reationôf âlitedissolution(eq.

8and9). All variablesareat thispointdened overthebloksof thespatial disretizationused inthe

reativetransportode,whihissupposed tobesuiently largerthantheRepresentativeElementary

Volumefortheinvestigatedporousmedium.

Inreationkinetis,ompetitionarisesbetweentheamountofhydrodynamiallytransportedreative

substanesandthatonsumedbythereations. Thisrelationshipissummarizedbythenon-dimensional

Damkhölernumber(Da^). Da^is^generally^dened^as^the^ratio^between^theharateristitimesofkinetis andadvetion;however,itspreisedenition is tosomeextentarbitrary,andhastobeadapted tothe

partiularproblemandthespeiformulationofreationkinetis adopted(i.e. thekinetiorder-see

[7℄andfurther referenestherein,[1, 9,14, 17℄). Wehosethedenition foradomain-saleDamköhler

numberby[10,11℄:

Da= kh·L

v·c^∗ forP e >1 ⁽¹⁴⁾

Inthisformulation,kh îs^the^kinetiônstant^forâlitedissolution,whosedimensionis[molal/s℄inour geohemialmodel,assumedoftherstorderandinludingimpliitlythereativesurfaeofthemineral.

This quantityis onstantoverthedomain giventhe kinetiformulation(eq. 11). Lîs âharateristi length[m℄,arbitrarilyset asthedomainlength;vîs ^the^normôf^the^seepage^veloityâveragedôver^the

domain[m·^s⁻¹^℄^andc^∗^aharateristionentration[molal℄,hosenastheonentrationofaidinthe solutioninjetedat theinowboundary.

ThePéletnumberP e^isⁱⁿ^turn^dened ^as:

P e= vL D^∗

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where D^∗ ^is ^the domain-averageddispersion tensor, redued to a salar in Hyte, with dimensions [m

2

s

−1

℄;itsexpressionin termsofeetivediusivityandkinematidispersionwasintroduedabovein

(eq. 13). Moreover,under thehypothesisofdominantonvetiveow,theexpressionofP e^redues^to:

P e= L

α ⁽¹⁶⁾

Inpratieitisdiulttoseparatetheontributionoftheeetivediusionandthatofthekinemati

dispersion;thereforein thefollowingwewillvary thePéletnumberonlybyadjusting thedispersivity

α^, ând ^have ^xed ^the êetive ^diusion ôeient Dê ⁱⁿ ^the ^models ât â ^quite^small ^value⁽1·10⁻¹⁰

m

2

s

−1

).

TheombineduseofP eândDa^,âs^pointedôutⁱⁿ ^[3,^7℄,îsâ^means^to^lassify^the^behaviourôf^the

system with respet to the dissolution pattern (formation of wormholes,ramied hannels oruniform

dissolution).