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Advances in Water Resources
journalhomepage:www.elsevier.com/locate/advwatres
The impact of pore-throat shape evolution during dissolution on carbonate rock permeability: Pore network modeling and experiments
Priyanka Agrawal
a, Arjen Mascini
b, Tom Bultreys
b, Hamed Aslannejad
a, Mariëtte Wolthers
a, Veerle Cnudde
a,b, Ian B. Butler
c, Amir Raoof
a,∗aDepartment of Earth Sciences, Utrecht University, Utrecht, the Netherlands,
bPProGRess/UGCT, Department of Geology, Ghent University, Ghent, Belgium
cSchool of Geosciences, University of Edinburgh, Edinburgh
a r t i c le i n f o
Keywords:
Carbonate dissolution Reactive transport Micro-CT Pore network model Porosity-permeability relation
a b s t r a ct
Porenetworkmodelsimulation(PNM)isanimportantmethodtosimulatereactivetransportprocessesinporous mediaandtoinvestigateconstitutiverelationshipsbetweenpermeabilityandporositythatcanbeimplemented incontinuum-scalereactive-transportmodeling.Theexistingreactivetransportporenetworkmodels(rtPNMs) assumethattheinitiallycylindricalporethroatsmaintaintheirshapeandporethroatconductanceisupdatedus- ingaformofHagen-Poiseuillerelation.However,inthecontextofcalcitedissolution,earlierstudieshaveshown thatduringdissolution,porethroatscanattainaspectrumofshapes,dependingupontheimposedreactive-flow conditions(Agrawaletal.,2020).Inthecurrentstudy,wederivednewconstitutiverelationsforthecalculation ofconductanceasafunctionofporethroatvolumeandshapeevolutionforarangeofimposedflowandreaction conditions.Theserelationswereusedtobuildanimprovednewreactiveporenetworkmodel(nrtPNM).Usingthe newmodel,theporosity-permeabilitychangesweresimulatedandcomparedagainsttheexistingporenetwork models.
Inordertovalidatethereactivetransportporenetworkmodel,weconductedtwosetsofflow-throughexper- imentsontwoKettonlimestonesamples.Acidicsolutions(pH3.0)wereinjectedattwoDarcyvelocitiesi.e., 7.3×10−4and1.5×10−4m.s−1whileperformingX-raymicro-CTscanning.Experimentalvaluesofthechanges insamplepermeabilitywereestimatedintwoindependentways:throughPNMflowsimulationandthrough DirectNumericalSimulation.Bothapproachesusedimagesofthesamplesfromthebeginningandtheendof experiments.Extractedporenetworks,obtainedfromthemicro-CTimagesofthesamplefromthebeginningof theexperiment,wereusedforreactivetransportPNMs(rtPNMandnrtPNM).
Weobservedthatfortheexperimentalconditions,mostoftheporethroatsmaintainedtheinitiallyprescribed cylindricalshapesuchthatbothrtPNMandnrtPNMshowedasimilarevolutionofporosityandpermeability.
Thiswasfoundtobeinreasonableagreementwiththeporosityandpermeabilitychangesobservedintheexper- iment.Next,wehaveappliedarangeofflowandreactionregimestocomparepermeabilityevolutionsbetween rtPNMandnrtPNM.Wefoundthatforcertaindissolutionregimes,neglectingtheevolutionoftheporethroat shapeintheporenetworkcanleadtoanoverestimationofupto27%inthepredictedpermeabilityvaluesand anoverestimationofover50%inthefittedexponentfortheporosity-permeabilityrelations.Insummary,this studyshowedthatwhileunderhighflowrateconditionsthertPNMmodelisaccurateenough,itoverestimates permeabilityunderlowerflowrates.
1. Introduction
Numerousapplications requirean understandingof fluidinterac- tionswithporoussolidphases. Examplesincludeoptimizationofthe recoveryfromhydrocarbonreservoirsandpredictionofthelong-term consequencesofCO2injectioninthesubsurface.Calcitedissolutionis
∗Correspondingauthor.
E-mailaddress:a.raoof@uu.nl(A.Raoof).
animportantreactioninthesubsurfacereservoirs.Severalstudieshave demonstrated thatduringcalcite dissolution,amultitudeof physical andchemicalprocessesoperateattheporescalewhichgoverntheevo- lutionoftheporousmedium.Suchmodificationoftheporousmedium andtherelatedporosity-permeabilityrelationsisafunctionoftheini- tialheterogeneityoftherockandofthedissolutionconditionslikethe injectionrateandchemistryoftheinjectedfluid(e.g.,Agrawaletal., 2020; Golfieretal.,2002;HoefnerandFogler,1988;Lietal.,2008; Luquotetal.,2014;LuquotandGouze,2009;MeileandTuncay,2006;
https://doi.org/10.1016/j.advwatres.2021.103991
Received10March2021;Receivedinrevisedform31May2021;Accepted28June2021 Availableonline3July2021
0309-1708/© 2021TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Molinsetal.,2014).Oneapproachtocomprehendthecomplexity of calcitedissolutionprocessisthroughusingpore-scalemodelssuchas porenetworkmodels(PNMs).
Severalporenetworkformulationshavebeendevelopedtopresent porousmedia topologyandangularporegeometries(Acharyaet al., 2005; Al-GharbiandBlunt,2005; GhanbarianB.,HuntA. G., 2016; Raoof et al., 2013; Raoof andHassanizadeh, 2012; Raoof andHas- sanizadeh,2009>). Some studieshave used porethroats with vary- ingshapes, such assinusoidal or other converging-diverging shapes (Acharyaetal.,2005;Al-GharbiandBlunt,2005),forwhichanalytical expressionscanbeappliedtocalculateporeconductance(Acharyaetal., 2005;GhanbarianB.,HuntA.G.,2016;RaoofandHassanizadeh,2012).
Correctestimationof the hydraulicconductance values of thepores throatsisrequiredtoobtainanaccuratepermeabilityforthesample.
Inthecontextofcalcitedissolution,theexistingporenetworkmodels useanetworkofsphericalporesbodiesandcylindricalporethroatsto simulatetherelatedreactivetransportprocesses(Mehmanietal.,2012; Noguesetal.,2013;Raoofetal.,2012).Theseporenetworkmodels provideporosity-permeabilityrelationsasafunctionofthespecificdis- solutioncondition,whichisakeyinputforcontinuumscalemodels.An importantconsiderationintheexistingporenetworkmodelsisrelated tothewayconductanceoftheporethroatisupdatedduringthecalcite dissolution.
Mostporenetworkmodelsassumethatporethroats,havinginitially auniformcircularcrosssectionalongtheirlength,preservetheirform throughoutdissolution,irrespectiveoftheflowandreactionregimes.
Consequently,inresponsetodissolution,poreconductancevaluescan becalculatedusingtheHagen-Poiseuillerelation.However,ithasbeen shownthatinitiallyuniformporethroatstructurescandevelopintoa spectrumofporethroatshapeswithcircularcrosssectionsofvariable diameteralongtheirlengthdependingontheflowandreactionregimes (e.g.,Agrawaletal.,2020).Forexample,underdiffusionandreaction dominateddissolutionregimes,theporethroatstructurechangesintoa half-hyperboloid(i.e.,conical)shape.Underthiscondition,theHagen- Poiseuilleequation wouldresultin anoverestimation ofpore throat conductance.Agrawaletal.(2020)showedthattheconductanceevo- lutionintheporenetworkmodelsshouldincludeinformationonpore throatshapeevolutionthattakesthereactionregimeintoaccount.As mentioned earlier,some PNMs have beendeveloped with originally non-uniformshapesofporethroats,however,noporenetwork-based studyhasincorporatedthedissolutioninducedevolutionofporethroat shapes.
Inthisstudy,welinktheevolvingshapeofporethroats,underawide rangeofconditions,totheresultingporethroatconductancevaluesus- ingreactivetransportsimulations.Weimplementedthecorrelationre- lationsfortheevolutionof conductanceasafunctionof porethroat shapeandvolume(Agrawaletal.,2020)inaporenetworkextracted fromexperimentalmicro-CTdata,tosimulatereactivetransportacross therocksampleinamodifiedPNM.Theoutcomewascomparedtothe Hagen-PoiseuillePNMapproachandtothepermeabilityevolutionde- terminedexperimentallyusingmicro-CT.Finally,theimpactofthenew conductanceevolutionrelationshiponporosity-permeabilityrelations wastestedforarangeofflowandreactionregimes.
2. Experimentalmethod
Inthisstudy,weusedanooliticlimestonesample,KettonLimestone, quarriedfromtheUpperLincolnshireLimestoneFormation(villageof KettoninRutland,England).Twoplugswithadiameterof6mmand anapproximatelengthof12mmwereextractedfromthehomogeneous ooliticblock.Hereafter,thesesamplesarereferredasK1andK2.Two setsofreactivetransportexperimentswereconductedonthesesamples.
TheinjectingsolutioncontainedHClacidwithapHvalueof3.0applied undertwoflowratesprovidedinTable1.Theenvironmentalmicro-CT (EMCT)scanner(Bultreysetal.,2016)atthecenterforX-rayTomog- raphy(www.ugct.ugent.be)wasusedtoimagetheprogressofcalcite
dissolutionwithareconstructedvoxelsizeof6μm.Eachdynamicscan consistedof2200projectionsrecordedinatotalscantimeofaround 15min.Thesamplewasscannedatthebeginningandattheendofthe experiment.Datafromraw projectionswerereconstructedintoa3D stackvolumeusingthefilteredback-projectionmethodimplemented inOctopusSoftware(Vlassenbroecketal.,2007).
Next,3Ddatasetswerefilteredusinganon-localmeansfilterand segmented with the watershed segmentation method (Avizo 9.5.0).
Table1providestheporosityvaluescomputedfromthesegmentedim- agevolumesofthesamplefromthebeginningoftheexperiments.We observedthatthecomputedporosityvaluesinthisstudyfallwithinthe rangeofearlierreportedimage-basedsegmentationporosityvaluesfor theKettonrock(e.g.,Table3inMenkeetal.,2017).Thesevaluesdiffer fromthemeasuredbulkporosity(e.g.,Table1inMenkeetal.,2017) indicatingthatimagesegmentationincludesome(unavoidable)devi- ationsfromthemeasuredbulkporosityvalues.Forbothexperiments, thesizeofthesegmented3Dvolumewas1318×1316×2494voxels fromwhichsmallersub-volumeswereusedinthisstudyasdiscussedin Section3.2.1.
3. Numericalmethods
Weperformedpore-scalemodelingatthreedifferentlevels:i)single porescale(wherewesimulatedreactivetransportforasingleevolving porespaceunderdifferentconditions,cfAgrawaletal.,2020),ii)pore networkscale(whererockdissolutionissimulatedusingPNM),andiii) DirectNumericalSimulations(DNS)using3Dimagedporestructures (wheredetailedflowfieldisobtainedbysimulatingincompressibleflow in therockimages).Thesinglepore-scalemodelprovidedinsighton theevolutionoftheconductanceofaporethroatforarangeofphysio- chemicalconditionsandwaspreviouslyvalidatedagainstexperimental microfluidicexperiments(Agrawaletal.,2020).Inthecurrentwork,we developconstitutiverelationshipsfordissolution-inducedconductance basedonresultsfromthissystematicsingleporemodel.Theevolution ofporosityandpermeabilityobservedintheporenetworkmodelwas validatedagainstaDirectNumericalSolutionofflowacrossthesample.
Next,forarangeofflowandreactionregimes,porenetworkmodels wererunwithandwithoutthenewimprovedconductancerelationships andthepermeabilityresultsof bothmodelswerecompared.Table 2 summarizeskeydetailsofvarioussimulationsusedinthisstudy.
3.1. Singlepore-scalemodel
Pore-scale reactive transport utilizing a single capillary geome- try were simulated by solving the Stokes equations for fluid flow and advection-diffusion-reaction equations for reactive transport of aqueous species (COMSOL Multiphysics®). Readers are referred to Agrawaletal.(2020)fordetailsofthesimulationsandtheirvalidation againstmicroscopicexperimentsofreactivetransportincalcitecrystals.
Themodeledrangeofboundaryconditions(i.e.,injectionrate,compo- sitionofinjectingsolution,andporethroatgeometry)correspondcon- ditionsrelevant toinjection ofCO2 oracidic solutionintocarbonate rocks.Detailsofthesimulatedflowregimes,reactionregimesandpore throatgeometriesareprovidedinTableS1.
Toinvestigatethecontrolofthereactionandflowregimesoverthe evolutionofporethroatconductance,wecalculatedthreedimensionless parametersincludingthePécletnumber,theDamköhlernumber,and theGeometryFactoratthebeginningofeachsimulation. ThePéclet number(Pe)wasdefinedas:
𝑃𝑒=𝑣𝑟
𝐷 (1)
where,vistheaveragevelocityattheinletboundary,ristheradiusof theporethroat(m)andDisthediffusioncoefficienti.e.,3.36×10−9 m2.s−1.
TheDamköhlernumber(Da)wasdefinedas:
𝐷𝑎= 𝑟𝑘
𝑣𝐶𝐶𝑎𝑙𝑐𝑖𝑡𝑒 (2)
Table1
Parametersappliedintwomicro-CTexperiments.
Experiment Porosity Flow rate (m 3.s −1) Darcy velocity (m. s −1) pH of injected solution Injection duration (min)
E1-HQ 0.18 2.05 ×10 −8 7.3 ×10 −4 3.0 102
E2-LQ 0.13 4.33 ×10 −9 1.5 ×10 −4 3.0 330
Table2
Overviewofthevarioussimulationsperformedinthisstudy.1followingmethodofAslannejadetal.(2018);2followingmethodfromRaoofet.al.(2012).
Model Label Program Obtains
Single-pore simulation of reactive transport
COMSOL COMSOL Multiphysics® Relationship describing impact of pore shape evolution on conductance evolution 3D pore space simulation of flow only
DNS OpenFOAM Permeability change from experimental images 1
foPNM PoreFlow Permeability change from the pore network extracted from experimental images 2 3D pore space simulation of reactive transport
rtPNM PoreFlow Simulation of porosity and permeability evolution 2
nrtPNM PoreFlow Determine impact of pore shape evolution on simulated permeability evolution
where, CCalcite is the density of surface sites of calcite i.e., 0.8×10−5 mol.m− 2 andk(mol.m− 2.s − 1)is obtainedbased on theforwardrateconstantinthekineticratelawforcalcitedissolution (i.e.,thereactionrateconstantk1inEq.(11)inSection3.3.2)andthe activityofH+ions(i.e.,aH+inEq.(11)inSection3.3.2)intheinjected solution.
TheGeometryFactor,GF,wasdefinedas:
𝐺𝐹= 𝑙
𝑟 (3)
where,listhelengthoftheporethroat.
Theconductanceg(m5N−1s−1)oftheporethroatwascalculated fromtheimposedvolumetricflux,F(m3s−1),andthepressuredrop acrosstheporethroat,ΔPsp(Nm−2),as:
𝐹=𝑔Δ𝑃𝑠𝑝 (4)
Inthecaseofacylindricalporethroatgeometryandlaminarflow, Eq.(4)isequivalenttothefollowingdefinitionofconductance,g: 𝑔=𝜋𝑟4
8𝜇𝑙 (5)
where,𝜇istheviscosityofwater,0.001Pasatabout25°C.
Section4.2providesthemodifiedrelationsforthecalculationofthe porethroatconductanceforarangeofflowandreactionregimes.
3.2. Porenetworkgeneration
3.2.1. Porenetworksextractedfromthedigitalrockvolumes
A total of five pore network structures were generated from the micro-CT images of Ketton using the maximal ball algorithm (Raeinietal.,2017).Twosubsetsofsize700×700×700voxels,K1S1 andK1S2,wereobtainedfromthedigitalvolumesofthesampleK1at twodifferenttimesi.e.,t=0andt=102min.Similarsizedsubsets, K2S1andK2S2,wereextractedfromthedigitalvolumesofthesample K2atthetwoexperimenttimesi.e.,t=0andt=330min,respectively.
Thesesubsetswereutilizedforthepurposeof validationofthepore networkmodelsuchthatnetworks K1S1andK2S1werethestarting porenetworkstoperformreactivetransportsimulation.Notethatthe selectedtimedurationforbothexperimentsleadstoinjectionofcom- parableporevolumesi.e.,2442inexperimentE1-HQand1996inthe experimentE2-LQ(Table1).
Anothersubsetofsize300×300×300voxel,namely,K1S3,was extractedfromthedigitalvolumeof thesampleK1atanexperiment timestept=0min.Thissubsetwasutilizedforthesensitivitystudyof themodifiedconductancemodelpresentedinSection4.3.
Thepositionofthesesubsetswithreferencetothefullvolumesofthe KettonsampleisshowninFigs.1andS1.Fig.2showsthefrequency distributionofpropertiesoftheextractedporenetworkssuchaspore
body radius,porethroatradius,porethroat lengthandcoordination number.Table3providesthestatisticsofeachextractedporenetwork.
3.3. Porenetworkmodel
Theporenetworkmodelutilizedinthisstudyisanadaptationofthe previouslydevelopedpore-scalemodelbyRaoofetal.(2012).Inthe following,thismodelisdiscussedbriefly.
3.3.1. Flowandsolutetransport
Forthis study,single phaseflow was establishedacrossthe pore network.Thiswasachievedbyassuminglaminarflowalongeachpore throatofthenetwork(i.e.,Reynoldsnumberislessthan1).Fluxthrough aporethroatwasdescribedbytheHagen-Poiseuilleequationas:
𝑞𝑖𝑗= 𝑔𝑖𝑗( 𝑝𝑗−𝑝𝑖)
(6) whereqij istotalvolumetricfluxthroughtheporethroatij,gijis the conductanceoftheporethroatij,andpiandpjarethepressuresinpore bodiesiandj,respectively.Thesimulatedporenetworkmodelsinthis studyutilizedtheHagen-Poiseuillerelation(i.e.,Eq.(5))andmodified conductancerelations(i.e.,Eqs.(18))forcalculationoftheporethroat conductance.
Forincompressible,saturatedflowconditions,thesumoffluxesof all porethroatsconnectedtoa porebody wasequatedtozero.This impliesthattheamountoffluxgoingtoaporebodyshouldbeequalto theamountoffluxcomingoutofthatporebody:
𝑁𝑘
∑
𝑗=1
𝑞𝑖𝑗=0; 𝑗=1,2,…𝑁𝑘 (7)
whereNkisthecoordinationnumberofporebodyk.
Eq.(7)isvalidforallporebodiesofthenetworkexceptthosepresent attheinletandoutletfaceofthenetwork.CombinationofEqs.(6)and 7resultedinalinearsystemofequationstobesolvedforthepressure of theporebodiesandthevolumetricfluxthrough theporethroats.
WehaveusedaconstantDarcyvelocityboundaryconditionattheinlet face,aconstantpressure(i.e.,atmosphericpressure)boundarycondition attheoutletface,andno-slipboundaryconditionswasappliedatthe lateralfacesofthenetwork.
FromtheimposedDarcyvelocity(uD)andpressuredifferencebe- tweentheinletandoutletfaces(ΔPPNM),theabsolutepermeabilityof theporenetworkiscalculatedfrom:
𝑘𝑃 𝑁𝑀= 𝜇 𝑢𝐷𝐿
Δ𝑃𝑃 𝑁𝑀 (8)
where,Listhelengthoftheporenetworkalongtheflowdirection.
Allthechemicalcomponentsaretransportedthroughtheporenet- workbyadvectionanddiffusionprocesses.Foragivenporebodyiand
Fig. 1.pore structures and sample sub- volumes.(a)averticalsliceobtainedusingthe micro-CTvolumeofthesampleK1showingthe positionandsizesoftheextractedsub-volumes K1S1(t=0),K1S2(t= 102min)(magenta colorlines)andK1S3(t=0)(greencolorline).
Thewhitearrowshowstheflowdirection(im- ageisrotated180° comparedtoexperimental position,whereflowwasupward).Panels(b), (d)and(f)indicatethedigitalsubsets,K1S1, K1S2andK1S3,respectively.Panels(c),(e) and(g)aretheextractedporenetworksfrom thecorrespondingsub-volumes.
Fig.2.sampleporesizes.Poresizedistribu- tionfor(a)porebodyradius(b)porethroat radius,and(c)porethroatlength.(d)thedis- tributionofporecoordinationnumbers.
Table3
Statisticsofallfiveporenetworksextractedfromtheimagedrocksamples.1Porosityoftheextractedporenetwork;2Permeabilityfromsinglephaseflowsimulation ontheextractedporenetworks.
Number of pore bodies K1S1 K1S2 K2S1 K2S2 K1S3
2088 1787 2015 1960 184
Number of pore throats 3794 3108 3324 3293 316
Mean coordination number 3.63 3.48 3.30 3.36 3.43
Mean pore body radius(m) 48.38 ×10 −6 53.55 ×10 −6 46.51 ×10 −6 48.67 ×10 −6 48.22 ×10 −6 Mean pore throat radius(m) 26.23 ×10 −6 31.22 ×10 −6 24.80 ×10 −6 26.52 ×10 −6 25.94 ×10 −6 Mean pore throat length(m) 112.41 ×10 −6 118.40 ×10 −6 107.29 ×10 −6 110.14 ×10 −6 117.78 ×10 −6
1Porosity 0.15 0.16 0.12 0.13 0.19
2Permeability ( k foPNM) (mD) 2809.6 6511.6 1226.1 2865.5 5515.6
Table4
EquilibriumratelawsofCO2–H2Osystemandrespectiveequilibriumconstants.
Reactions
Equilibrium rate constant from Plummer and Busenberg, 1982 CO 2(aq) = H ++ HCO 3− a4.5 ×10 −7
HCO 3−= H ++ CO 32− a4.78 ×10 −11 H 2O = H ++ OH − 1 ×10 −14
fortheflowconditionsunderwhichfluidflowingfromporebodyjto- wardsporebodyi,throughporethroatij,wecandefinethetransport ofthekthchemicalcomponentas:
𝑉𝑖𝑑 𝑑𝑡
(𝑐𝑘,𝑖)
=
𝑁∑𝑡ℎ𝑟𝑜𝑎𝑡𝑖𝑛
𝑗=1 𝑞𝑖𝑗𝑐𝑘,𝑖𝑗− 𝑄𝑖𝑐𝑘,𝑖+
𝑧𝑖
∑
𝑗=1𝐷𝐴𝑖𝑗
(𝑐𝑘,𝑖𝑗−𝑐𝑘,𝑖)
𝑙𝑖𝑗 −𝑅𝑘,𝑖 (9) where,Vi,Qiandck,iarethevolume,volumetricflowrateandconcen- trationofkthcomponentinporebodyi,respectively.Theleft-handside (L.H.S)oftheEq.(9)providestherateofchangeofconcentrationfor thekthcomponentintheporebodyiovertime.Thisrateisacombined outputoftheadvective,diffusiveandreactiverates.Thefirsttermon theright-handside(R.H.S)oftheEq.(9)calculatesthetotalamount oftheincomingfluxofeachcomponentthrough𝑁𝑖𝑛𝑡ℎ𝑟𝑜𝑎𝑡upstreampore throats(i.e.,porethroatsflowingintoporebodyi).Thiscalculationuti- lizesthevolumetricflowrate(qij)andconcentration(ck,ij)inporethroat ij.ThesecondtermontheR.H.Scalculatesthetotaloutfluxofthekth component.ThethirdtermontheR.H.Scalculatesthediffusionrateof thekth componentthroughziconnectedporethroats.Thiscalculation utilizesthemoleculardiffusioncoefficient(D)andthecross-sectional areaofthethroat(Aij).ThelastR.H.Sterm,Rk,i,defineschangesdue tochemicalreactionsofthekthcomponenttakingplaceinsidethepore body.
Similarly,themassbalanceequationforporethroatijandforthe casewhenthefluidisflowingfromporebodyjtoporethroatij,canbe writtenas:
𝑉𝑖𝑗𝑑 𝑑𝑡
(𝑐𝑘,𝑖𝑗)
=𝑞𝑖𝑗𝑐𝑘,𝑗−𝑞𝑖𝑗𝑐𝑖𝑗+𝐷𝐴𝑖𝑗
(𝑐𝑘,𝑗−𝑐𝑘,𝑖𝑗) 𝑙𝑖𝑗 +𝐷𝐴𝑖𝑗
(𝑐𝑘,𝑖−𝑐𝑘,𝑖𝑗)
𝑙𝑖𝑗 −𝑅𝑘,𝑖𝑗 (10)
whereVij,qij andck,ijarethevolume,volumetricflowrateandcon- centrationofkthcomponentinporethroatij,respectively.Rk,ijprovides feedbackfromreactionsofthekth componenttakingplaceinsidethe porethroat.
3.3.2. Reactionsandratelaws
Todescribethecalcitedissolutionsystem,wehaveincorporateda totaloffourreactions.Outofthosefourreactions,threereactionsin- volvingonlyaqueousspeciesaredescribedusingequilibriumrelation- ships,whereasthedissolutionofcalciteisdescribedbyakineticrate law.DetailsofequilibriumreactionsareprovidedinTable4.
(Plummer et al., 1978) has identified three main mechanisms throughwhichcalcitedissolutiontakesplace.Accordingly,wehavede- finedthecalcitedissolutionrateas:
𝑅𝑎𝑡𝑒= 𝐴(
𝑘1𝑎𝐻++ 𝑘2𝑎𝐻2𝐶𝑂∗3+ 𝑘3
)(
1−𝑎𝐶𝑎2+𝑎𝐶𝑂32−
𝐾𝑒𝑞 )
(11)
where,Ais thereactivesurfaceareaofcalcite andai istheactivity ofspeciesi.k1=8.64×10−5(m.s −1),k2 =4.78×10−7 (m.s− 1) andk3 =2.34×10−9 (m.s− 1) arereaction rateconstants at25 °C (PlummerandBusenberg,1982).Keq=10−8.48istheequilibriumcon- stantforcalcitedissolution(PlummerandBusenberg,1982).
3.3.3. Modelingparameters
Wehaveutilizedtwosetsofflowandchemicalconditionsforpore networkmodeling.Thefirstsetisderivedfromtheexperimentalparam- eters(Table1)andthesecondsetcoverstherangeofreactionandflow
regimeswhicharerelevanttoacidstimulationexperimentsincarbonate reservoirs.
Intheexperiments,weinjectedthesolutionsatafixedflowrate.
Table5providesthemagnitudeofthecalculatedDarcyvelocityforthe experimentsE1-HQandE2-LQ.Modelscorrespondingtotheseexperi- mentsutilizedtherespectivevalueoftheDarcyvelocityforinjectinga solutionwiththepHvalueof3.0.AdditionalsetsofutilizedDarcyveloc- itiesandpHoftheinjectedsolutionsareprovidedinTable5.LowerpH valuesoftheinjectedacidarerelevanttoacidstimulationapplications incarbonatereservoirs(Al-AmeriandGamadi,2020;Janbumrungand Trisarn,2017;LeongandBenMahmud,2019).Thecompositionofthese solutionswascalculatedusingPhreeqC(ParkhurstandAppelo,2013) andphreeqc.datdatabase(TableS2).
Inordertocategorizethesimulatedflowandreactionregimes,we havedefinedthemacroscopicPecletnumber(𝑃̃𝑒)andDamköhlernum- bers(𝐷̃𝑎𝐼and𝐷̃𝑎𝐼𝐼)as:
𝑃̃𝑒= 𝑢𝐷𝐿
𝐷 (12)
𝐷̃𝑎𝐼 =𝐿2𝑘1𝑎𝐻+
𝐷𝐶𝐶𝑎𝑙𝑐𝑖𝑡𝑒 (13)
𝐷̃𝑎𝐼𝐼 = 𝐿𝑘1𝑎𝐻+
𝑈𝐷𝐶𝐶𝑎𝑙𝑐𝑖𝑡𝑒 (14)
where,uDistheDarcyvelocity(m.s−1)andListhelengthofthepore networkalongtheflowdirection.Table5provides𝑃̃𝑒,𝐷̃𝑎𝐼 and𝐷̃𝑎𝐼𝐼 numbersforallthesimulatedmodels.
NotethatthePeandDanumberswithout~signweredefinedatthe scaleofanindividualthroatwhilePeandDanumberswith~signwere definedatthescaleofporenetwork.
3.4. Directflowsimulationofpore-scaleimages
Inorder tocomparethe permeabilityvalues obtained fromPNM results, we have performed pore-scale flow DNS.We have used the VolumeofFluid(VOF)methodimplementedinOpenFOAMComputa- tionalFluidDynamic(CFD)packagetosolveincompressibleflowusinga semi-implicitmethodforpressure(Aslannejadetal.,2018;Bedramand Moosavi,2011).
Numerical three-dimensionalsimulation of incompressible linear- viscousflowwasdonebysolvingtheNavier-Stokesequations:
𝜌(𝜕𝑢
𝜕𝑡+𝑢.∇𝑢)
=−∇𝑝+𝜇∇2𝑢 (15)
∇.𝑢=0 (16)
where,pispressure,uisvelocity,𝜇istheviscosityofwater(𝜇=0.001Pa s)and𝜌isthedensityofwater(𝜌=1000kgm−3).
Theflowequationsaresolvedbyapplyingaconstantpressurediffer- encebetweentheinletandtheoutletfaces(ΔPI)andano-slipboundary conditionattheremainingfaces.Thisyieldedthevelocityandpressure foreachvoxelinthedomain.Theflowrate,Q(m3s−1)wascalculated as𝑄= ∫𝑢𝑥𝐴𝑥,whereAx(m2)isthecross-sectionalareaoftheinlet faceperpendiculartothedirectionofflowxanduxistheinletfaceve- locitycomponentinthedirectionofflow.Fromtheflowrate(Q)and theimposedpressuredifference(ΔPDNS),absolutepermeabilityofthe 3DXMTimage,kDNS(m2)iscalculatedusingDarcyequation:
𝑘𝐷𝑁𝑆= 𝜇 𝑄𝐿𝑥
Δ𝑃𝐷𝑁𝑆𝐿𝑦𝐿𝑧 (17)
where,Lx,LyandLzarethedomainlengthsineachdirection.
Table5
Statisticsrelatedtothesimulatedporenetworks.
Sample Experiment Darcy Velocity (m. s −1) pH of injecting solution Simulation Duration (min) Injected Pore Volumes 𝑷 ̃𝒆 𝑫 ̃𝒂 𝑰 ̃𝑫 𝒂 𝑰𝑰
K1S1 E1-HQ 7.3 ×10 −4 3.0 35.5 2442 974 5.82 ×10 4 5.98 ×10 1
K2S1 E2-LQ 1.5 ×10 −4 3.0 115 1996 192 5.82 ×10 4 3.03 ×10 2
K1S3 1.5 ×10 −4 1.0 100 56 1.70 1.04 ×10 6 6.13 ×10 5
K1S3 8.3 ×10 −5 1.0 20 56 8.50 1.04 ×10 6 1.23 ×10 5
K1S3 15 ×10 −4 1.0 10 56 17.00 1.04 ×10 6 6.13 ×10 4
K1S3 30 ×10 −4 1.0 5 56 34.00 1.04 ×10 6 3.07 ×10 4
K1S3 75 ×10 −4 1.0 2 56 85.00 1.04 ×10 6 1.23 ×10 4
K1S3 1.5 ×10 −4 3.0 3000 1670 1.70 1.07 ×10 4 6.30 ×10 3
K1S3 8.3 ×10 −5 3.0 600 1670 8.50 1.07 ×10 4 1.26 ×10 3
K1S3 15 ×10 −4 3.0 300 1670 17.00 1.07 ×10 4 6.30 ×10 2
K1S3 30 ×10 −4 3.0 150 1670 34.00 1.07 ×10 4 3.15 ×10 2
K1S3 75 ×10 −4 3.0 60 1670 85.00 1.07 ×10 4 1.26 ×10 2
K1S3 1.5 ×10 −4 5.0 3000 1670 1.70 3.87 ×10 2 2.28 ×10 2
K1S3 8.3 ×10 −5 5.0 600 1670 8.50 3.87 g ×10 2 4.56 ×10 1
K1S3 15 ×10 −4 5.0 300 1670 17.00 3.87 ×10 2 2.28 ×10 1
K1S3 30 ×10 −4 5.0 150 1670 34.00 3.87 ×10 2 1.14 ×10 1
K1S3 75 ×10 −4 5.0 60 1670 85.00 3.87 ×10 2 4.56
4. Resultsanddiscussions
Section 4.1 presents a comparison of the pore network models againsttheobservationsfrommicro-CTexperiments.Next,usingsingle- poresimulations,wederivedcorrelationrelationsfortheconductance ofporethroatsinSection4.2.Theserelationswereimplementedinthe newPNM(nrtPNM).Section4.3providescomparisonofpermeability evolutionfromnrtPNMversusrtPNMfortheexperimentalconditions.
Section4.4extendsthiscomparisonforarangeoftransportandreaction regimes.InSection4.5wepresenttheimpactofflowandreactioncon- ditionsovertheporosity-permeabilityrelationsobtainedfromnrtPNM versusrtPNM.
4.1. Validationofthereactivetransportporenetworkmodel(rtPNM)with themicro-CTexperiments
Tworockporenetworksi.e.,K1S1andK2S1weresimulatedwith similarinitialandboundaryconditionstotheexperimentsE1-HQand E2-LQ,respectively.Attimet=0,sampleswerefilledwithafluidso- lutionequilibratedwithcalcite.IntheexperimentE1-HQ,injectionof asolutionwitha pHvalueof 3.0initiated calcite dissolutionwhich showsitselfasadecreasingamountofcalciteinFig.3e.Fig.3a-dshows thecorrespondingtime-basedevolutionoftheaveragechemicalspecies concentrations.ThepHofthesolutioninsidethesampledecreasedcon- tinuouslyasalarger numberofpores werepenetratedbytheacidic solution(Fig.3a-d).Theacidarrivaltimeforeachporethroatdepends onitsdistance fromtheinletboundaryanditsconnectivitywiththe dominantflowchannels.Theinletareaofthesamplereachesasteady stateinshortertimecomparedtotheoutletarea(FigureS2).Moreover, thesteadystatepHoftheinletareaislowerthanthatoftheoutletarea (FigureS2).Thisindicatesthatwithinthesimulationtime,asignificant numberofporethroatslocatedintheoutletareawerenot yetfilled withtheinflowsolutionchemistry.FromthedistributionofpHofthe solutioninsidethesample,afterinjectionofaround2442porevolumes, weobservedthatmorethan40%ofporethroatswerefilledwiththein- flowsolutionpHvalueof3.0,while~ 1%porethroatsretainedtheir calciteequilibratedsolution(FigureS3a).ComparedtoexperimentE1- HQ,thelowerinjectionrateinexperimentE2-LQfurtherdelayedthe attainmentofsteadystatefordifferentpartofthesample(FigureS4).
Consequently,afteraround1996porevolumesofacidinjectioninthe experimentE2-LQ,asignificantfractionofporethroatswerefilledwith solutionshavingpHvalueslargerthan7(FigureS3b).
Fig.4comparestheevolutionofporosityandpermeabilityinthere- activetransportmodel(rtPNM)againsttheexperimentalporosityand permeabilitydata,obtainedfromthetime-lapsemicro-CTimages.Ex- perimentaldataofchangesinporositywerecalculatedfromtheporosi-
tiesoftheextractedporenetworksthatwerederivedfromthedigital imagesofthesamplebelongingtothestartandendoftheexperiment (Table3).Themodelshowedasmalleramountofdissolvedcalcitecom- paredtothemeasuredexperimentalvalues(Fig.4a).Apossiblecause couldbethedeviationoftheimplementedratesinmodelsfromthose operatingintheexperiments.Suchdeviationcan cause,forexample, slowerpenetrationrateoftheacidinthesimulationscomparedtothe experiment.ForthemodelrelatedtotheexperimentE1-HQ,2442pore volumesofacidinjectionwerenotsufficienttoattainacompletesteady statepHinthesystem(Fig.3e).
Forbothexperiments,weobservedafairlygoodagreementbetween the permeability evolution in the reactive transport model (krtPNM) and in the dissolutionexperiments (kfoPNM and kDNS) (Fig. 4b). Ad- ditionally,weobservedthatfortheexperimentE2-LQ,kfoPNMagreed well withkDNS,while fortheexperimentE1-HQ,thetwovaluesdif- ferredsignificantly(Fig.4b).Thedifferencebetweenpermeabilitiesob- tained throughfoPNMandDNSislikely duetothesimplificationof thepore-scalegeometricalfeaturesduringextractionofporenetwork (Boeveretal.,2016).
4.2. Derivationoftheconstitutiverelationsforcalculationofporethroat conductance
Agrawalet.al.(2020)haveshownthattheimposedflowandreac- tionregimesdeterminetheevolutionoftheporethroatshapeandsub- sequentevolutionofconductanceduringcalcitedissolution.Figs.5and S5presentthedependencyoftheconductanceofaporethroatonthe volumeofthatporethroatforthesimulatedflowandreactionregimes whilethelengthandinitialradiusoftheporethroatarefixedatavalue of500×10−6and40×10−6m,respectively.
Weobservethat,atanytimeduringcalcitedissolution,theupdated poreconductancecan berelatedtothemodifiedvolume ofthepore throatusinganexponent,b:
𝑔𝑡
𝑔𝑜 = (𝑉𝑡
𝑉𝑜 )𝑏
(18)
wheregtandgoaretheconductanceoftheporethroat,calculatedfrom Eq.(4),goattimet=0sandgtattimetduringthecourseofthedisso- lution;similarly,VtandVoarethevolumeofporethroatattimet=0s andattimet.Exponentbrepresentsthedistinctivenessandtopology oftheevolvedporeshape.Sincetheevolutionoftheporethroatshape wasguidedbyanumberofparameterssuchasinitialgeometryofthe porethroatandimposedflowandreactionregimes,bisafunctionof theseparameters(Fig.6a).
Fig.6ashowsthatunderadvectiondominatedregimes,indicatedby Pe>1,andreaction-controlledregimes,indicatedbyDa<1,thebcoef-
Fig.3. concentrationfields.(a-d)show3DdistributionofpHwithinthereactivetransportporenetworkmodel(rtPNM)correspondingtotheexperimentE1-HQ attimescorrespondingtoporevolumesof(a)0,(b)35,(c)1032,and(d)2442.(e)evolutionoftheconcentrationofaqueousspecieswiththeinjectednumberof porevolumes.Note:toobtainthe1Dtimeplot,the3Dconcentrationfieldsareaveragedwithintheentiresampleatdifferenttimes.
Fig.4. (a).Porosityevolutionand(b)perme- abilityevolutionagainsttheinjectedporevol- umesforexperimentsE1-HQ(highflowrate) andE2-LQ(lowflowrate).Diamondmarkers correspondtortPNMwhichusetheinitialsam- pleporestructureastheirinitialconditionand performreactivetransporttocalculateandpre- dictevolutionofsamplesovertime.Asterisks andsquaresshowresultsofflowonlysimula- tionsperformedonthe3Dimageofthesample fromthebeginningandendoftheexperiments:
asterisksshowresultsfromDNSandsquares showresultsfromfoPNMsimulations.
Fig.5.poreconductancerelations.Normalizedvaluesofporethroatconduc- tanceasafunctionofnormalizedvolumeofporethroatforsomeofthesimu- latedflowandreactionregimes(porethroatlengthandinitialradiusarefixed at500.0×10−6mand40.0×10−6m,respectively).Foreachdataset,markers representthedatapointsandthesolidlinesrepresentthefittedcurves(Eq.(18)).
ficientapproachedavalueof2.0.Undertheseregimes,theshape of theporethroatremainedcylindricalthroughoutthedissolutionperiod (Agrawal etal., 2020: Fig.7b). Underthis condition (i.e.,b =2.0), ourrelation(Eq.(19))becameequaltotheHagenPoiseuillerelation (Eq.(5)).Fordiffusiondominatedregimes,whenPe<1,andreaction dominatedregimes,whenDa>1,thecylindricalshapeevolvedintohalf- hyperboloidshapes(Agrawaletal.,2020:Fig.7a).Forthesecases,the valueofbbecamesmallerthan2.0andtheaccurateporeconductance canbecalculatedfromEq.(19).Itshouldbenotedthathere,shapeevo- lutionimpliedthattheprofileoftheshapealongtheflowpathchanged fromcylindertohalf-hyperboloidlongitudinalwhilecross-sectionofthe porethroat,i.e.,perpendiculartotheflowpath,remainedcircular.
Therelationforcoefficientbcanbeexpressedas:
𝑏=𝐴1+ 2−𝐴1
1 + 𝐷𝑃𝑎𝑒𝑝𝑝34
(19)
𝐴1=𝑝1𝐺𝐹−𝑝2 (20)
Valuesoffittingparameters(p1,p2,p3andp4)wereobtainedusing nonlinearleastsquaresmethod (SISectionS1,FigureS6a).Wehave obtaineda R-squarevalueof0.94indicatingthat thefittedrelation- shipanditsparameterswereabletodescribedatapoints(FigureS6b).
Fig.6bshowsthe3Dplotofthefittedrelationandthecorresponding datapoints.Ingeneral,forreactionregimescorrespondingtoDa>1and Da<1,thefittedrelationshipcapturedthedependencyofboverPenum- ber,DanumberandGF(Fig.6b).FigureS7showsthesensitivityofthe derivedmodeloverthetestingdataset.Formostofthetestdatapoints,
the%differenceinthevalueof bcalculatedfromthefittedmodelvs thatobtainedfromnumericalsimulationwaswithin10%(FigureS7b).
Thisassuresthepredictivecapacityoftheproposedmodel.
Next,alongwithHagenPoiseuille’srelation,wehaveusedourde- velopedrelationtoperformporenetworkmodelsimulations.Hereafter, thesetwoporenetworkmodelswillbereferredasrtPNMandnrtPNM (i.e.,newmodel).InnrtPNM,ateachtimestepofthesimulation,using theupdatedPenumber,Danumber,andGF,thebvaluewascalculated foreachporethroatoftheporenetwork.Subsequently,foreachpore throatoftheporenetwork,themodificationinconductancewascalcu- lated.InSections4.3and4.4,wecomparetheimpactofthesetwopore networkmodelingapproachesontheevolutionofthepermeabilityof thesample.
4.3. nrtPNMunderflowandreactionregimesofmicro-CTexperiments
For theexperiment E1-HQ,diffusiondominatedpore throats are highlightedin redinFigureS8aafterinjectionof around2442pore volumes.TheseporethroatseitherhadPe<0.0001(i.e.,diffusionwas themajortransportmechanism)or0.0001<Pe<1(i.e.,bothdiffusion andlowvelocityconvectionwerecontrollingthetransport).Partofthe diffusiondominatedporethroats,mostlyporethroatswithPe<0.0001, hadaDanumbergreaterthan1(highlightedingreencolorinFigure S8b).Theseporethroatswilltransformintonon-uniformporeshapes, andavalueofbsmallerthan2.0isobtainedforthecalculationofcon- ductanceusingEq.(19)(highlightedinyellowcolorinFigureS8c).In theexperimentE1-HQ,thefraction ofsuchdiffusiondominated,and reactiondominatedporethroatswass~8%,whereasintheexperiment E2-LQ,thisfractionwas~11%(FigureS8d-S8f).
WeobservedthatbothrtPNMandnrtPNMprovidedporosityand permeabilityevolutionsthataresosimilarthattheresultswouldover- lapin FigureS9.Thisisbecausetheflowconditionsforbothexperi- ments,whichwereselectedtotriggervisibleporositychangeswithin thetimeframeofthemicro-CTexperiments,leadpredominantlytoad- vectiondominatedregimesasindicatedbylarge𝑃̃𝑒numbers(Table5).
Therefore,undertheseflowconditions,89–92%porethroatskepttheir cylindricalshapeduringdissolutionandnrtPNMagreedequallywellas rtPNMwiththeexperimentallydeterminedporositiesandpermeabili- ties.
4.4. nrtPNMunderdifferentflowandreactionregimes
ThissectionexplorestheinfluenceoftheinjectionvelocityandpH valueoftheinjectedsolutionoverthecalcitedissolutionprocessesand subsequentevolutionofthepetrophysicalpropertiesofthesample.Cal- citedissolutionisacombinationofthreeprocesses:i)transportofthe reactantstowardsthecalcitesurface,ii)interactionofreactantswiththe calcitesurfaceandiii)removaloftheproductsawayfromthecalcitesur- face.Thefluidvelocityinsidetheporethroatdeterminesthetransport
Fig.6. poreconductanceparameters.(a)De- pendencyofthebcoefficient(shownusingthe colorlegend)onthestrengthofreactionand flow regime (the horizontal axes) and pore throat lengths (shown as segregated colors.
i.e.,eachsurfacecorrespondstoacertainpore throatlength).Highervaluesofbcoefficient correspondtohigherPe numbersandlower Danumbers. Pore throatlength of value of 500.0×10−6mshowsahigherrangeforthe bcoefficientcomparedtoasmallerporethroat lengthwithavalueof 250.0× 10−6 m.(b) thefittedcurvedsurfacetogetherwiththedata pointsusedtoobtainthebestfit(SISectionS1;
TableS1).
Fig.7.Evolutionofporosityandpermeability.(a)Porosityvsinjectedporevolumes(b)Permeabilityvsinjectedporevolumesand(c)PermeabilityvsPorosity, fromrtPNM(solidlines)andnrtPNM(dottedlines).TheinflowsolutionhadapHvalueof1.0andwasinjectedusingfivedifferentDarcyvelocities.Notethatthe legendprovidedinpanel(a),iscolorcoordinatedwiththeplottedlinecolorsandapplicableforpanels(b)and(c),aswell.
Fig.8. theobtainedsimulationaccuracy.Relativepercentagedifferenceinthe finalpermeabilityvalueofthesamplesimulatedusingthenewconductancere- lation,nrtPNM,andtheHagenPoiseuillerelation,rtPNM,underdifferent𝐷̃𝑎𝐼𝐼 numbers.
rateforthefirstandthelaststeps.ThepHofthesolutioncontrolsthe rateof thesecondstep.Calcitedissolutionincreaseswhenasolution withalowerpHvalueisinjectedortheinjectionratesareincreased.
Fig.7showstheimpactofchangingtheinjectionrate,resultingina different𝑃̃𝑒number,ontheevolutionofporositywhilethepHofthe injectedsolutionwasfixedatavalueof1.0.Fortheinjectionofthesame numberofporevolumes,loweringoftheacidinjectionratesledtodisso- lutionofahigheramountofcalcite(Fig.7)asitincreasedtheresidence timeofacidwithinthesample.Weobservedthatthereactivetransport conditionsinsidetheporenetworkevolvedconstantlyduringdissolu- tion(FigureS10).These updatedreactivetransportconditionsofthe porethroatwereutilizedinthecalculationofnewconductancevalue usingEq.(19)and20.ComparingresultsfromnrtPNMandrtPNM,we observedsimilarityintheporosityevolutionbutdifferenceintheper- meabilityevolution(Fig.7).FiguresS11andS12providetheevolution ofporosityandpermeabilityforremainingsimulatedcases.
Differencesintheevolutionofpermeabilityinthesemodelsarepro- portionaltothenumberofporethroatstransformingintonon-uniform shapes during dissolution. As mentioned earlier, these sets of pore throatsbelongtoadiffusiondominatedtransportregimeandareac- tiondominateddissolutionregime.Foreachofthesimulatedboundary conditions,wehaveidentifiedthefractionofporethroatswhichful- filledthiscriteria(TableS3,FigureS13).Forexample,forasimulated networkS5,injectionofasolutionwithpHof1.0andaDarcyvelocity of3.17×10−6m.s−1resultedintodevelopmentofanon-uniformshape foraround60%ofporethroats(Fig.7:𝑃̃𝑒=1.70and𝐷̃𝑎𝐼𝐼=6.13×105).
Consequently,forthiscase,weobservedthatrtPNMoverestimatedthe permeabilitybyaround27%ascomparedtonrtPNM.Forallsimulated boundaryconditions,Fig.8summarizesthepercentagedifferenceinthe predictedvalueofpermeabilityfromthesetwomodels.Thisshowsthat forflowandreactionregimescorrespondingto𝐷̃𝑎𝐼𝐼 greaterthan100 thetwoporenetworkmodelsstartedtodeviateandfor𝐷̃𝑎𝐼𝐼 greater than103significantdifferencesincurredinthepermeabilityprediction ofthesample.
4.5. Implicationsforporosity-permeabilityrelationshipusedatthefield scale
Manycontinuum-scalemodelsusea powerlawrelation todeter- minetheevolutionofpermeabilityfromthechangesinporosity(e.g., Hommeletal.,2018):
𝑘𝑡=𝑘𝑜
(∅𝑡
∅𝑜 )𝛼
(21)
wherekoand∅oaretheinitialpermeabilityandporosityofasample.
Thefittingparameter,𝛼,dependsontheinitialpropertiesoftherock andontheprocessescontrollingthechangesintheporegeometry(e.g., Parvanetal.,2020).The𝛼valuecaneitherbeobtainedthroughtheo- reticalderivationorthroughfittingofexperimentsandmodels.Inthis
Fig.9. the𝛼 coefficientinthe permeability relation.Powerlawexponentcoefficient𝛼as afunctionofporositychangeforsimulations with(a)constantinjectionvelocity(i.e.,con- stant𝑃̃𝑒number=1.70)usingdifferentsolu- tionpHvalues,and(b)injectionofasolution withaconstantpHvalueof1.0usingthreedif- ferentvelocities.Notethatthesolidlinescorre- spondtortPNMandthedottedlinescorrespond tonrtPNM.
study,weobservedthatforaverysimilarrocksample,𝛼isafunction oftheinjectionparameterssuchasDarcyvelocityandacidityofthe injectedsolution(Fig.9)(seealsoNoguesetal.,2013).Moreover,we observedatemporalvariationin𝛼whichindicatesdifferentphasesof dissolution(Fig.9).
Fig.9shows𝛼asafunctionoftheporosityofoursamplesforasetof simulations.First,forsomeoftheinjectionconditions,rtPNMprovided larger𝛼values,byafactorofover50%,ascomparedtothenrtPNM (Fig.9:𝑃̃𝑒=1.70and𝐷̃𝑎𝐼𝐼 =6.13×105).
Theinitiallylarge𝛼valuesuggeststhatthefirsttargetoftheacid reactionswerethehighlyconductivepathswhereasmallincrementin theporosityresultedinlargeincrementinpermeability(Bernabé etal., 2003;Noguesetal.,2013).Weobservedthattheincrementinthein- jectionvelocityorinthepHoftheinjectedsolutionledtoaconsistent increaseof𝛼(Fig.9),irrespectiveofwhichmodelwasused(rtPNMor nrtPNM).This canbe explained bythefactthat, inadvectiondomi- natedregimesorinreaction-controlledregimes,thedissolutiontakes placemoreuniformlyinthesample.Fig.9againshowsthatatcertain conditions,rtPNMwasfinetobeused,butatotherconditionsitwill giveanoverestimation.
5. Conclusions
Inthecontextofreactiveporenetworkmodelsforcalcitedissolution, wehavepresentednewconstitutiverelationstoupdateporethroatcon- ductance,whichtakescareoftheevolvingporethroatshapeasperthe imposedflowandreactionregimes.Thiswasachievedthroughnumer- icalsimulationofreactivetransportprocessesonascaleofsinglecap- illary/porethroatandlaterextractionoftheevolutionofconductance ofthecapillary.Weobserved that, fortheadvectiondominatedand reaction-controlleddissolutionregimes,theconductancecanbeupdated using theHagen-Poiseuillerelation,sincethe initialcylindrical pore throatshapeismaintainedduringdissolution.Onthecontrary,forthe diffusiondominatedandreaction-dominateddissolutionregimes,the porethroatshapechangestothehalf-hyperboloid(i.e.,conical)shape andtheproposednewconductancerelationsprovidesanimprovedde- scriptionoftheimpactonconductanceevolutionoftheporethroat.
Next,thederivedconductancerelationswereimplementedinpore networksimulationsandthepredictionofpermeabilitywascompared withtheexistingporenetworkmodels.Weobservedthatcorresponding tothemicro-CTexperimentalconditionsi.e.,𝑃̃𝑒=974,Pe=974,Da= 59.8andPe=192,Da=303,only.
𝐷̃𝑎𝐼𝐼=59.8and𝑃̃𝑒=192,𝐷̃𝑎𝐼𝐼=303,onlyaround11%oftotalpore throatsweresubjectedtotheconditionswhichareresponsibleforthe developmentofthenon-uniformporethroatshapes.Consequently,es- timatedpermeabilityfromtheporenetworkmodelremainedindiffer- enttotherelationsusedforupdatingofconductanceofindividualpore throats.Asaresult,newandexistingPNMprovidedequallygoodagree- mentwiththepermeabilityofthereactedsampleimageobtainedafter theexperiment.
Finally,weperformedasensitivitystudyofthenewconductancere- lationsacrossarangeofflowandreactionregimes.Thisrevealedthat theinjectionofhighlyacidicsolutionswithapHvalueof1.0andatan injectionvelocityof3.17×10−6m.s−1caused60%oftotalporethroats tomodifytheirinitiallycylindricalshapesintonon-uniformshapesdue todissolution.Weobservedthat,comparedtothenewlymodifiedPNM, utilizationof existingPNM canlead toanoverestimationof 27%in thepredictedpermeabilityvalueandanoverestimationofover50%in thefittedexponent𝛼usedintheporosity-permeabilityrelationship.In thecontextofapplicationssuchastheinjectionofCO2oracidicsolu- tionintocarbonaterocks,porenetworkmodelsarethemost promis- ingtoolforthepredictionofporosity-permeabilityrelationsattherep- resentativeelementaryvolume level.Dependingon thereactive-flow regime,incorporationofthedissolutioninducedchangesofthepore throatshapemayberequiredin ordertoenhancethepredictionca- pacityofexistingporenetworkmodelsandpossiblybringthemodeled
outcomesclosertofieldobservations,forexample,realisticprediction ofporositypermeabilitychangesinthecaseofdevelopmentofdifferent dissolutionpatternswhenvariedrangeofinjectionvelocityorsolution chemistryareutilized.Insummary,PNMprovideconstitutiverelations forREV’sincontinuummodelsthatcanbeusedtosimulatefieldsites andweproposedimprovedconstitutiverelationships.
Supportinginformationavailable
Additionaldetailsonthepresentedresults,Tables1-3andFigs.1-13.
DeclarationofCompetingInterest
Theauthorsdeclarethattheyhavenoknowncompetingfinancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinfluence theworkreportedinthispaper.
Acknowledgements
TheresearchworkofP.A.,M.W.andA.RispartoftheIndustrialPart- nership Programme i32Computational Sciencesfor EnergyResearch thatiscarriedoutunderanagreementbetweenShellandtheNether- landsOrganizationforScientificResearch(NWO).M.W.hasreceived funding fromtheEuropean Research Council (ERC)underthe Euro- pean Union’s Horizon2020research andinnovationprogram (grant agreementNo.[819588]).TheXMTexperimentsinthisworkwereper- formedatGhentUniversity’scenterforX-rayTomography(UGCT),a centerof ExpertisefundedbytheGhentUniversitySpecialResearch Fund(BOF-UGent)undergrantBOF.EXP.2017.007.TB,AMandVCre- ceivedfundingfrom theResearch Foundation–Flanders(FWO)under projectgrantG051418N.TBisapostdoctoralfellowoftheResearch Foundation‐Flanders(FWO)andacknowledgesitssupportunderGrant 12×0919N.
Supplementarymaterials
Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.advwatres.2021.103991.
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