Characterization and upscaling of hydrodynamic transport in heterogeneous dual porosity media

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Characterization and upscaling of hydrodynamic

transport in heterogeneous dual porosity media

Philippe Gouze, Alexandre Puyguiraud, Delphine Roubinet, Marco Dentz

To cite this version:

Philippe Gouze, Alexandre Puyguiraud, Delphine Roubinet, Marco Dentz. Characterization and

upscaling of hydrodynamic transport in heterogeneous dual porosity media. Advances in Water

Re-sources, Elsevier, 2020, 146, �10.1016/j.advwatres.2020.103781�. �hal-02999509�

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AdvancesinWaterResources146(2020)103781

ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

Characterization

and

upscaling

of

hydrodynamic

transport

in

heterogeneous

dual

porosity

media

Philippe

Gouze

a,∗

_,

_Alexandre

_Puyguiraud

b

_,

_Delphine

_Roubinet

a

_,

_Marco

_Dentz

b a_{Géosciences, Université de Montpellier, CNRS, Montpellier, France}

b Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Non-Fickian dispersion Heterogeneous porous media Upscaling

Time domain random walk Continuous time random walk Dual multirate mass transfer model

a

b

s

t

r

a

c

t

Westudytheupscalingofpore-scaletransportofpassivesoluteinacarbonaterocksample.Itischaracterizedby microporousregionsdisplayingheterogeneousporositydistributionthatareaccessibleduetodiffusiononly,and astronglyheterogeneousmobileporespace,characterizedbyabroaddistributionofflowvelocities.Weobserve breakthroughcurvesthatarecharacterizedbystrongtailing,whichcanbeattributedtovelocityvariabilityin theflowingmediumportion,andsoluteretentioninthemicroporousspace.Usingaccuratenumericalflowand transportsimulations,weseparatethesetwomechanismsbyanalyzingthestatisticsofresidencetimesinthe mobilephase,andthetrappingandresidencetimestatisticsinthemmobilephase.Weemployacontinuoustime randomwalkframeworkinordertoupscaletransportusingaparticlebasedimplementationofmobile-immobile masstransfer,andheterogeneousadvection.Thisapproachisbasedonthestatisticsofthecharacteristicmobile andimmobileresidencetimes,andmasstransferratesbetweenthetwocontinua.Whileclassicalmobile-immobile approachesmodelmasstransferasaconstantrateprocess,wefindthatthetrappingrateincreaseswithincreasing mobileresidencetimesuntilitreachesaconstantasymptoticvalue.Basedonthesefindingsandthestatistical characteristicsoftravelandretentiontimes,wederiveanupscaledLagrangiantransportmodelthatseparatesthe processesofheterogeneousadvectionanddiffusionintheimmobilemicroporousspace,andprovidesaccurate descriptionsoftheobservednon-Fickianbreakthroughcurves.Theseresultsshedlightontransportupscalingin highlycomplexdual-porosityrocksforwhichmobile-immobilemasstransferarecontrolledbyadualmultirate processcontrolledbytheheterogeneityofboththeflowfieldintheconnectedporosityandthediffusioninthe no-flowregions.

1. Introduction

Solutetransport in thelaminarflow throughthevoid spaceof a porousmediumisduetomoleculardiffusionandadvection.Despitethe simplicityofthesefundamentalprocesses,observedtransportis char-acterizedbycomplexfeaturessuchasstrongbreakthroughcurve tail-ing, non-Gaussianconcentration distributions,anomalous dispersion, incompletemixing,andintermittentLagrangianflowproperties(Cortis andBerkowitz,2004;Seymouretal.,2004;Bijeljicetal.,2011;DeAnna etal.,2013;Bijeljicetal.,2013;Kangetal.,2014;Holzneretal.,2015; Moralesetal.,2017).Thesebehaviorsareduetotheintricatestructure oftheporespace,andthemulti-scaleheterogeneitydistribution,which causebroaddistributionsofadvectiveanddiffusivemasstransfertime scalesandtransportpathways(Bijeljicetal.,2011;Portaetal.,2015; Puyguiraudetal., 2019a).The understandingofthese heterogeneity mechanisms,andtheirquantificationinupscaledtransportmodelsare keyissuesin manyacademicandengineeringapplicationsconcerned

∗_{Corresponding}_author.

E-mailaddress:Philippe.Gouze@UMontpellier.fr(P.Gouze).

withthelargescale(macroscopic)predictionofthefateof conserva-tiveandreactivesolutesingeologicalandengineeredmedia,suchas theassessmentofgroundwatercontaminationandremediation, geolog-icalstorageofnuclearwaste,geothermalenergyproduction,and un-dergroundstorageofcarbondioxide(DomenicoandSchwartz,1997; PoinssotandGeckeis,2012;Niemietal.,2017).

Classical upscaling approaches quantify Darcy scale transport in terms of hydrodynamic dispersion coefficients (Bear, 1972), which incorporatethe large scale dispersive effect of pore-scale ve-locity fluctuations. The key issue is evidently the determination of the macroscopic dispersion coefficient. For instance, de Jos-selindeJong (1958)andSaffman(1959)usedLagrangianstochastic modelsforpore-scaleparticlemotioninordertoderiveexpressionsfor thehydrodynamicdispersioncoefficients.Theirapproachesarebased onthefactthatvelocitiesvaryontypicallengthscales,theporelengths. Thus,particlesspendmoretimeinlowthaninhighflowvelocity re-gions.Thisbehavior,whichliesattheoriginofpore-scaleLagrangian intermittency(Dentzetal.,2016),ismodeledbyadistributionoftravel

https://doi.org/10.1016/j.advwatres.2020.103781

Received22April2020;Receivedinrevisedform10September2020;Accepted5October2020 Availableonline8October2020

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times.Thespatialtransitionsandthetransitiontimesdependonboththe porevelocitiesandmoleculardiﬀusion.Sincethesepioneeringworks, hydrodynamicdispersionanditsdependenceonthelocalPécletnumber (theratioofthecharacteristicdiﬀusiontimetotheadvectiontime)was thesubjectofnumerousexperimental,numericalandtheoretical investi-gations(Scheven,2013;Swansonetal.,2015;Pfannkuch,1963;Rashidi etal.,1996;Jouraketal.,2013;BijeljicandBlunt,2006).Systematic upscalingapproacheshavebeenbasedongeneralizedTaylordispersion theory(Brenner,1980;Sallesetal.,1993),volumeaveraging(Quintard andWhitaker,1994;Davitetal.,2012;2013)andcontinuoustime ran-domwalk(CTRW) (BijeljicandBlunt,2006).

CTRWmethodssimilartotheapproachesinvolvedintheworksof

deJosselindeJong (1958)andSaffman(1959)wereusedtomodel non-Fickian pore-scale transport features such as anomalous disper-sion,breakthroughcurvestailingandintermittentLagrangianparticle velocities(Bijeljic etal., 2011; De Annaet al., 2013; Gjetvajet al., 2015;Kangetal.,2014; Puyguiraudetal.,2019b). The implementa-tionofCTRWisoftenhandledbymodelingparticletransportthrough transitionsoverfixedspatialscalescharacterizedbyrandomtransition times(Berkowitzetal.,2006).Thetimedomainrandomwalk(TDRW) methodthatwillbeusedinthisstudy,isalsobasedonparticles mo-tionoverfixeddistance.Transitiontimesaredeterminedkinematically fromtheEulerianflowfieldandthespatiallydistributedproperties,for instancetheporosityanddiffusivity,thatcanbemappedeitherfrom tomographicimagingorfrom(statistical)models.Thus,thesemethods provideameanstorelatepore-scaleflowpropertiestoDarcyscale trans-portbehavior(Bijeljicetal.,2011;Puyguiraudetal.,2019,2020).

Thepresenceofimmobilemediumregionsthatconsistofdeadend pores,regionsoflowflowinthewakeofsolidgrainsandmicroporosity wherediffusiondominatesarecommonfeaturesofporousmediasuch asreservoirrocks.ThelargedifferenceintermsofPécletnumber be-tweenthesezonesoftheporousmediaandtheconnectednetworkof poresthatformstheusualmacroporosity(theflowingporosity)supports theuseofdualcontinuummobile-immobilemasstransferapproaches firstproposedby vanGenuchtenandWierenga(1976).Thisapproach hasbeenwidelyusedtotakeintoaccounttheoften-encountered con-trolofspatiallydistributeddiffusivezonesontheoverallhydrodynamic transportandspecificallyontheoccurrenceofoftenhighlynon-Fickian breakthroughcurvesobservedexperimentallyfromlaboratorytofield scales.Theheterogeneousmediumismodeledbyoverlappingmobile andimmobilecontinua.Ateachpointinspace,thesystemstateis de-finedbyamobileandseriesofimmobileconcentrations.Themobileand immobilecontinuacommunicatethroughlinearmasstransferin two-equationmodels(Ahmadietal.,1998;Cherblancetal.,2007),which canbe formulatedin awaythat allowstheimmobileconcentrations tobewrittenaslinearfunctionalsofthemobileconcentration,which arecharacterizedbyamemorykernel(HaggertyandGorelick,1995; Carreraetal.,1998)thataccountsforthemicroscalemasstransfer pro-cesses.Assuch,itisamethodtoupscalepore-scaletransport.Many im-plementationsofthisapproachconsideraconstantaveragevelocityin themobilemediumportion(LiuandKitanidis,2012;Portaetal.,2013; 2015).However,advectiveheterogeneity,thismeansvelocity variabil-ityin theflowingmedium portion,by itselfgivesrisetoanomalous transport(Bijeljicetal.,2011;DeAnnaetal.,2013;Kangetal.,2014; Puyguiraudetal.,2019).Thisiswhytheimportanceofpore-scale ve-locitystatisticsandtheirrelationtothecomplexmediumand hetero-geneitystructurehavebeenstudiedinaseriesofrecentexperimental andnumericalworks(Sienaetal.,2014;Matykaetal.,2016;Holzner etal.,2015;DeAnnaetal.,2017;Alimetal.,2017;Dentzetal.,2018; Aramidehetal.,2018).

SomeauthorshavecoupledCTRWmodelsofadvective heterogene-itywithtrappinginimmobileregions(Gjetvajetal.,2015;Dentzetal., 2018).Keyitemsforthesemodelingapproachesaretheidentiﬁcationof thedominantpore-scaletransportandmasstransferprocesses,their re-lationtothepore-scalemedium andtheﬂow properties.Theseaims have been pursued by experiments (Swanson et al., 2015),

numeri-cal simulations(deVriesetal., 2017;Ceriotti etal., 2019) and for-malupscaling usingvolume averaging(Davit etal., 2012;Orgogozo et al.,2013; Portaet al.,2015),andLagrangianCTRWbased meth-ods(Gjetvajetal.,2015;Dentzetal.,2018).Volumeaveraging delin-eatesamobileregion,theflowingporosity,andimmobileregionssuch asbiofilms(Orgogozoetal.,2013)basedonavelocitycutoff determined from aPécletcriterion (Portaetal.,2015).Thelargescaletransport modelisthenobtainedbyaveragingthemicroscaletransportequations overaunitcellthatisstatisticallyrepresentativeofthepropertiesof themediumandtheflow,andcontainstwodistinctmediumportions, which,asoutlinedabove,areconnectedthroughmasstransferacross domain boundaries. Lagrangian stochastic models (Margolin et al., 2003;BensonandMeerschaert,2009;Dentzetal.,2012;Comollietal., 2016)formulatemasstransferbetweenmobileandimmobilemedium regionsthroughcompoundstochasticPoissonprocesses(Feller,1968). Thismeansthatmasstransfereventsoccuratconstantrate,quantified bythePoissonprocess,whichrenderstheresidencetimeinimmobile regions asthesumoverindividualtrappingtimes acompound Pois-son process.Asshown byMargolinet al.(2003), Bensonand Meer-schaert(2009)anddiscussedfurtherinthispaper,thisformulationis equivalenttoEulerianmobile-immobilemasstransferformulations.

Thisstudyaimsattestingourcapabilityofcharacterizingand upscal-inghydrodynamictransportinheterogeneousnaturalreservoirswhere both velocitydistributionandimmobile domainheterogeneitycause anomalous transport,startingfrom themodelassuming that mobile-immobilemasstransfersarecontrolledbyaPoissonprocess.Forthat weuseasanexampleofhighlyheterogeneousmedia,acarbonate sam-pleimagedusingX-Raymicrotomographythatdisplaysmarkedbimodal structuralheterogeneitycausedbythepresenceof connected macro-porosityandmicroporousmaterialthatresultsfromgrain sedimenta-tionanddiagenesisevents.Theimageisprocessedinordertomapthe mobileandtheimmobiledomainanddirectnumericalsimulationsof ﬂowandtransportareperformed. Weinvestigate indetailthe statis-ticsofmobileandimmobileparticlemotionintermsoftherespective residencetimes,thetrappingratesandthemobileandimmobiletimes betweentrappingevents.Usingthedetailedstatisticalanalysis,we dis-cussthesalientfeaturesoftransportattheporescale,andquantifythem inanupscaledtransportmodelbasedonaLagrangianformulationthat implementsinasimpleformthespeciﬁcprocessthatcharacterizesthe spatialdistributionofthemobile-immobilemasstransfersin heteroge-neousmedia.

Thepaperisorganizedasfollows.Section2detailsthe methodol-ogy.Itdescribesthenumericalsolutionofthedirectﬂowandtransport problem,thesimulationsetup,theboundaryconditionsandthemodel outputswhichallowinvestigatingindetailthestatisticsofmobileand immobileparticlemotionintermsoftherespectiveresidencetimes,the trappingrates andthemobileandimmobiletimes betweentrapping events.InSection3weconsidertransportinasimplefracturematrix setupinordertopresenttheconceptofthebasicLagrangian methodol-ogyforasimplemobile-immobilesystemforwhichthesingletrapping rateupscalingCTRWformulationisdetailedandthenvalidatedusing thedirectsimulationresults.Then,followingthesameapproacheswe investigate inSection4thetransportbehaviorinthecarbonaterock sampleusingdirectnumericalsimulationsaswellastheupscalingof transportintheimmobiledomainusingastatisticalmulti-trapping ap-proach.Then,inSection5,wederiveanewupscaledLagrangianmodel andvalidateitbycomparisonwiththeresultsofthedirectnumerical simulations.ConclusionsarepresentedinSection6.

2. Materialandmethods

2.1. SampleMC10properties

TheMC10carbonatesampleisaporousandpermeablerockmade ofquasipurecalcite.Thestructurethatresultsfromcomplex sedimen-tationanddiageneticeventsismadeofimperviousgrainsofvariable

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Fig.1.Porositymapofaslicenormaltothemain ﬂowdirectionzpositionedat𝑧=450.Theimage ontheleftdisplaystheporositymapbefore thresh-old.Orangeandcyancolorsdenotethesolidphase (𝜙 =0)andthemobiledomain(𝜙 =1)respectively, whilegrayscalefromblack(lowporosity)towhite (highporosity)denotestheimmobiledomain poros-ity.Theimageontherightshowsinyellowcolor thefractionoftheimmobiledomainremoved(i.e. transformedintosolid)whenapplyingathreshold

𝜁 =0.1(deﬁnedinSection2.1).(Forinterpretation ofthereferencestocolourinthisﬁgurelegend,the readerisreferredtothewebversionofthisarticle.)

characteristiclengthrangingfromfewtenstoabout150µmand mi-croporousmaterial thatensure thecohesion of the rock. When per-formingX-Raymicrotomographyof suchsinglesolidphasematerial, theX-Rayenergyattenuationintegratedovereachvoxeloftheﬁnal 3-dimensionalimagedenotestheporosity.Forthisstudy,weuseacropped sub-volumemadeof900×900×900cubicvoxelsofsidedimension

𝑑𝑥=1.6867× 10−6_m._MC10_is_{characterized}_by_connected_macropores

ofmeansize70×10−6_m._The_microporous_material_(diagenetic_cement)

isconsideredimmobileregardingfluidflowandonlyaccessibletosolute bydiffusion.Itdisplaysvariableporositymadebyporesthataresmaller thantheimagingresolution.Accordingly,connectedmacro-porosity de-limitsthemobiledomainwhilethemicroporousmaterialdelimitsthe immobiledomain.Detailsonthesamplecharacteristicsandthe method-ologiesappliedtoprocesstheX-ray tomographicimagearegivenin

AppendixA.

TheeﬀectivediﬀusionDeineachlocationoftheimmobiledomain

(i.e.ineachvoxel)istheproductofthemoleculardiﬀusionD0times

theeﬀectiveporosity𝜙e,

𝐷𝑒(x)=𝐷0𝜙𝑒(x)=𝐷0𝜙(x)∕𝜅(x), (1)

where𝜅 denotestheimmobiledomaintortuositythatcanbeconsidered

asaconstantorafunctionoftheimmobiledomainporosity𝜙 suchas theformulationderivedfromtheelectrictortuositybyArchie(1942),

𝜅(x)=𝜙(x)1−𝑚_._Accordingly,₍₁₎_can_be_rewritten

𝐷𝑒(x)=𝐷0𝜙(x)𝑚, (2)

withmrangingfrom1(assumingthatthereisnotortuosityeffect)to about4.5inmicroporouslimestones(Gouzeetal.,2008).Thediffusion coefficientD0isconstantforallsimulationsandsetto10−9m2s−1.Note

thattheporosityoftheimmobiledomainisdefinedastheporosity ac-cessiblebyasolutediffusingfromthemobiledomain,andthuscanbe differentfromthetotalporosityoftheimmobiledomain,forinstance ifporouszonesareembeddedinzonesconsideredasnon-diffusiveas explainedbelow.

Foreachvoxeloftheimmobiledomaintransportbydiffusionis im-possiblebelowagivenporosityvalue.Differentapproachesusing,for instance,percolationtheory,critical-pathanalysisoreffectivemedium approximationtheory canbe usedtoevaluatetheporositythreshold

𝜁 belowwhichthesystemis non-percolatingfordiﬀusion(seeHunt andSahimi,2017;Hommeletal.,2018andreferencesherein).Forthe sampleconsideredhere,applyingaporositythresholdconsistsin trans-formingthefractionoftheimmobiledomainwhere𝜙 <𝜁 intosolid: 𝐷𝑒(𝐱)=

{

𝐷0𝜙𝑒 for𝜙 ≥𝜁

0 for𝜙 <𝜁 (3)

Theporosityvalueoftheimmobiledomainresultingfromthe im-ageprocessingrangesfrom0.045to0.193withmeanporosity0.108

(Hebertetal.,2015).Fig.1displaysacrosssection(normaltothemain ﬂow)inthesegmentedimageofthe9003_-voxel_sample_where_the

frac-tionofimmobiledomaincorrespondingtoporositybelowthreshold val-uesof10%areenlighten.Forinstance,applyingaporositythresholdof

𝜁 =0.1actsasremoving39%oftheimmobiledomain.Themean

poros-ity oftheremaining fractionof theimmobiledomainis then0.175. However,applyingthisthresholddoesnotchangenoticeablythearea ofthemobile-immobileinterfacewhichis1.22 × 105_m2_per_m3_of

mobiledomainwhen𝜁 =0.0and1.20 × 105_m−1_when_{𝜁 =}₀_.₁_,_i.e._a

decreaseof1.64%.

Inthispaper,diﬀerenttortuositymodelsareinvestigated.Assuming aporosity thresholdof 𝜁 =0.1andtortuositydeﬁnedby𝜅(𝜙)=𝜙1−𝑚

withm=2.5isthemostrealisticmodel(Garingetal.,2014),butmodels withm=1.5and4.5aswellaswithaconstanttortuositymodel𝜅 =1.8 and𝜁 =0areinvestigatedinordertoexplorethefeedbackcontrolofthe

immobiledomaindiﬀusivityontheoverallsolutetransportandonthe upscalingfeasibility.Acomprehensivecharacterizationofthediﬀusion propertiesaccordingtotheassumptionmadeontortuosityaregivenin

AppendixB.

2.2. Mobiledomainﬂow

WeconsidertheﬂowinthesampleatlowReynoldssothatthe pore-scaleﬂowvelocityv(x)issolutionoftheStokesequation

∇2_𝐯₍_𝐱₎₌ 1

𝜇∇𝑝(𝐱), (4)

where p(x) is the ﬂuidpressure. The 9003 _cubic _voxels _mesh_is

di-rectly usedasthemeshed domainforOpenFOAMcalculationsusing apermeameter-likeconﬁguration:(i)constantpressureisappliedatthe inlet(z=0)andtheoutlet(𝑧=𝐿𝑧)boundarieswhere20-pixelslayersof

unitaryporosityareaddedinordertoobtainanaccuratedetermination ofthevelocitycomponentsattheinletandoutletofthedomain,(ii)the domainisboundedbysolidatx=0,𝑥=𝐿𝑥,y=0and𝑦=𝐿𝑦,(iii)

no-slipconditionsareappliedatthemobile-solidandtheimmobile-solid domainboundaries.Lx,LyandLzdenotethedomainlengthsinthex,y

andzdirections,respectively.

The flow equations are solved via a finite volume scheme implemented in the SIMPLE algorithm of OpenFOAM (https://cfd.direct/openfoam/user-guide/v7-fvsolution/). This al-gorithm solves the steady state Stokes equation (4) and continuity equation∇⋅𝐯(𝐱)=0followinganiterativeprocedure.Convergenceis reached whenthedifferencein termsof pressure andvelocity com-ponentsbetween thecurrentandthepreviousstepsissmallerthana threshold.Onceconvergencehasbeenreached,weextractthevelocity fieldcomponentsthatarecomputedateachofthevoxelinterface.

Theﬂuidvelocityinthedirectionalongthez-axis(themainﬂow direction) displays an asymmetric shape with some negative values

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Fig.2. Left:normalizedfluidEulerianvelocityPDFinthedirectionofthemainflow𝑣𝑧 ∕𝑣𝑧 (circles)andinoneofthedirectionperpendiculartothemainflow𝑣𝑥 ∕𝑣𝑧 (solidline).Right:Eulerianvelocitynorm|V|PDF(plainline)andfluxweightedEulerianvelocitynormPDF(dashedline).

thatemphasizethehighcomplexityoftheflowfieldtriggeredbythe highheterogeneityofthemobiledomain.Thefluidvelocity perpendic-ulartothemainflowdirection isquasi-symmetric(Fig.2).Theflux weightedvelocitynormPDF𝑃◦

𝐸(|𝑉|)=|𝑉|∕⟨|𝑉|⟩𝑃𝐸(|𝑉|),wherePE(|V|)

denotesthePDFoftheEulerianvelocitynorm,isdisplayedinFig.2.

Puyguiraudetal.,2019showedthatforstationarysystem𝑃◦

𝐸isequalto

theLagrangianvelocityPDFwhichisthecoreinformationrequiredfor upscalingadvectivetransportinthemobiledomain(Puyguiraudetal., 2019).Upscalingoftheadvectivetransportusing𝑃◦

𝐸forthishighly

het-erogeneoussampleisbeyondthescopeofthepresentworkthatfocuses onupscalingtheimmobiledomaintransport,andwillbepresentedin afuturededicatedpaper.Nevertheless,wenotethatthePDF𝑃◦

𝐸

pre-sentedinFig.2isquitesimilartothatofthesandstonesamplepresented inFig.2inPuyguiraudetal.,2019forwhichupscalingmethodswere proposedbytheauthors.

2.3. Transportsimulations

Transportinthemobile-immobiledomainisdescribedbythegeneric advection-diﬀusionequationwhichisconsideredtoapplyatthescale ofeachvoxel:

𝜕𝑐(𝐱,𝑡)

𝜕𝑡 +𝐯(𝐱)⋅∇𝑐(𝐱,𝑡)−𝐷𝑒∇

2_𝑐(_𝐱,_𝑡₎₌₀_, ₍₅₎

whereDeistheeffectivediffusioncoefficientandv(x)istheflow

veloc-ity.Inthemobiledomain,Dereducestothemoleculardiﬀusion

coeﬃ-cientD0whereastheﬂowvelocityiszerointheimmobiledomain.

Equation (5) is solved numerically using a time-domain random walk (TDRW) method (Russianet al., 2016),which is basedon the formulationofEq.(5)asamasterequationusingaﬁnitevolume dis-cretizationofthespatialoperators.AcompletedescriptionoftheTDRW method,a demonstrationof itsequivalencewithEq.(5),andits im-plementationusingvoxelizedimagesofporousmediacanbefoundin

Dentzetal.(2012)andRussianetal.(2016).Themainfeaturesofthe methodaregivenbelow.Thedomaindiscretizationusedfortransportis thesameastheoneusedforcomputingtheflow.TheTDRWapproach modelsthedisplacementofparticlesinspaceandtime,theirensemble averagegivingthesolutionofthetransportequationfortheconsidered media.Foreachparticle,eachmotioneventisdenotedbyasinglejump fromonevoxeltooneofthe6face-neighboringvoxels.Assuch,the jumpdistance𝜉 isconstantandequaltothevoxelsizedx.The direc-tionandthejumpdurationarecontrolledbythelocalpropertiesofthe voxels,i.e.thefluidvelocityandtheeffectivediffusioncoefficient.The

recursiverelationsthatdescribetherandomwalkfrompositionxj to

positionxiofagivenparticleatjump𝑛is

𝐱𝑖(𝑛+1)=𝐱𝑗(𝑛)+𝝃, 𝑡(𝑛+1)=𝑡(𝑛)+𝜏𝑗, (6)

with|𝝃| =𝜉 denotingthetransitionlength.Theprobabilitywijfora

tran-sitionoflength𝜉 frompixeljtopixeli,andthetransitiontime𝜏j

asso-ciatedtopixeljaregivenby

𝑤𝑖𝑗= 𝑏𝑖𝑗 ∑ [𝑗𝑘]𝑏𝑘𝑗, 𝜏𝑗 =∑ 1 [𝑗𝑘]𝑏𝑘𝑗, (7)

wherethenotationΣ_[_jk_]indicatesthesummationoverthenearest neigh-borsofpixelj.Thebijaregivenby

𝑏𝑖𝑗= ̂ 𝐷𝑒𝑖𝑗 𝜉2 + |𝑣𝑖𝑗| 2𝜉 ( _𝑣 𝑖𝑗 |𝑣𝑖𝑗| +1), (8)

where𝐷̂𝑒𝑖𝑗 denotestheharmonicmeanofthediﬀusioncoeﬃcientsof

pixelsiandj,andvijdenotesthevelocitycomponentofvjinthedirection

ofpixeli,𝑣𝑖𝑗=𝐯𝑗⋅𝝃𝑖𝑗.Asaconvention,voxeliisdownstreamfrompixel jifvij>0.

NotethattheTDRWcanbeseenasacontinuoustimerandomwalk (CTRW)becauseittreatstimeasanexponentiallydistributed contin-uousrandomvariablewhosemeanmayvary betweenvoxels.Inthis paper,weusethetermTDRWforthenumericalrandomwalkmethod usedtosolvethedirectproblem,andthetermCTRWfortheupscaled randomwalkframework.

2.4. TDRWsimulationssetup

Theappliedboundaryconditionatthesampleinlet(𝑧=0)isapulse

ofconstantconcentrationinthemobiledomainonly.Thisisperformed byapplyingaﬂuxweightedinjectionoftheparticlesat𝑡=0.By con-structionthepulseisformallyanexponentialconcentrationfunctionof characteristictime𝜏𝑗|𝑧=𝑑𝑥∕2 (Russianetal.,2016).Themainresultis

givenbytheﬁrstpassagetimeattheoutletofthemobiledomainwhich denotestheinerttracerbreakthroughcurve (BTC).No-ﬂuxboundary conditionissetat𝑥=0,𝑥=𝐿𝑥,𝑦=0,𝑦=𝐿𝑦,aswellas𝑧=0and𝑧=𝐿𝑧

intheimmobiledomain.

Simulations are performed for diﬀerent values of Pécletnumber whichis deﬁnedby𝑃𝑒=⟨|𝑉|⟩𝑙∕𝐷0 wherelisacharacteristiclength

whichistakenhereastheaverageporelength.Eachsimulationinvolves atleast107_particles._The_statistics_concerning_the_{characteristics}_of_the

trappingevents,suchasthetrappingtimeintheimmobiledomainand thesurvivaltimeinthemobiledomainbetweentwotrappingevents,are

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obtainedbysamplingmorethan109_events_(the_deﬁnition_of_a_trapping

eventisprovidedinSection2.5).

2.5. Modeloutput

Themasstransfersoccurringinthesampleareprobedbyasetof statisticaldistributionswhicharegivenasprobabilitydensityfunctions (PDFs),denoteď𝜓.(.),wheretheoverlyingreversed-hatsymbolindicates

thattheyarederivedfromtheresultsofthedirectTDRWsimulations. ThesePDFsdescribetheadvection-diﬀusiontransportinthemobile do-main,theexchangebetweenthemobileandtheimmobiledomains,and thediﬀusivetransportintheimmobiledomain.Intermsofrandomwalk process,wewillnameeachintrusionofaparticleintotheimmobile do-maina”trappingevent”.

TrappingtimePDFdenoted ̌𝜓𝜏𝑖𝑚 isthePDFofthetime𝜏im(p,n(p))

spentbytheparticlesp(𝑝=1,.,𝑃)intheimmobiledomain dur-ingthetrappingeventsn(p)(𝑛(𝑝)=1,… ,𝑁(𝑝)),wherePisthe totalnumberofparticlesexitingatthesample’soutletandN(p) isthetotalnumberoftrappingeventsencounteredbyparticlep.

Immobile time PDF denoted ̌𝜓𝑡𝑖𝑚 is the PDF of the time tim(p)

spentbytheparticlesintheimmobiledomaintocrossthe en-tiredomain(i.e.from𝑧=0to𝑧=𝐿𝑧).Foragivenparticlep, 𝑡𝑖𝑚=∫𝑛(𝑝)𝜏𝑖𝑚𝑑𝑛.

SurvivaltimePDFdenoted ̌𝜓𝜏𝑠 isthePDFofthetimes𝜏s(p,n(p))

spentbytheparticlespinthemobiledomainbetweentrapping events𝑛−1andn.

MobiletimePDFdenoted ̌𝜓𝑡𝑚 isthePDFofthetimetm(p)spentby

theparticlesinthemobiledomaintocrosstheentiredomain.For agivenparticlep,𝑡𝑚=∫𝑛𝜏𝑠𝑑𝑛+𝜖,where𝜖 isthesumofthetime

spenttomovefromtheinlettothelocationoftheﬁrsttrapping eventandof thetimespenttomovefrom theexitlocationof trappingeventN(p)totheoutlet.

TrappingratePDFdenoted ̌𝜓𝛾isthePDFof𝛾(𝑝)=𝑛(𝑝)∕𝑡𝑚(𝑝).

FirstpassagetimePDFdenoted ̌𝜓𝑡𝑡 isthePDFoftheﬁrstpassage

timett(p)spentbytheparticlestocrossthedomainandis

equiv-alenttothebreakthroughcurve(BTC).Bydeﬁnition,foreach particlep

𝑡𝑡(𝑝)=𝑡𝑚(𝑝)+𝑡𝑖𝑚(𝑝)=𝑡𝑚(𝑝)+ 𝑁(_∑𝑝)

𝑖=1𝜏𝑖𝑚

(𝑝,𝑖). (9)

3. Modelingtransportinasinglefracturewithmobile-immobile masstransfer

Inthissection,weinvestigatethecaseofthetransportofapassive tracerin a simplemobile-immobiledomain systemthat canbe ade-quatelyrepresentedasasinglelinearfracture(themobiledomain) cross-ingacontinuousporousmatrix(theimmobiledomain).Thediﬀerent PDFscharacterizingthetransportprocess(describedinSection2.5)will becomputedusingdirectTDRWsimulationsandwilllaterbecompared tothoseresultingfromTDRWsimulationsperformedfortheMC10 car-bonatesample.Furthermore,wepresenta1DCTRWmodelthatupscales transportinthefracturematrixsystemandintroducesthemainfeatures andconceptsusedfortheupscalingoftransportintheMC10carbonate sample.Thedetailedfracture-matrixtransportmodelcanbeformulated inthemostgeneralformas

𝜙(𝑦)𝜕𝑐(x,𝑡)

𝜕𝑡 +𝑢(𝑦) 𝜕𝑐(x,𝑡)

𝜕𝑥 −∇⋅[𝐷(𝑦)∇𝑐(x,𝑡)]=0, (10)

where𝜙(y)isporosity, whichisequalto𝜙m withinthefractureand 𝜙iminthematrix;u(y)istheDarcyvelocity,whichisequaltouinthe

fractureand0inthematrix;similarly,D(y)isthediﬀusioncoeﬃcient, whichisequaltoDm𝜙minthefractureandequaltoDim𝜙iminthematrix,

whereDmandDimdenotethediﬀusioncoeﬃcientinthemobileandthe

immobiledomainrespectively.

Inthefollowing,weconsidertwoequivalentupscaledtransport ap-proaches.

3.1. Upscalingbyverticalaveraging

Upscaledtransportinthisfracture-matrixsystemcanbedescribedby amultiratemasstransfermodel(HaggertyandGorelick,1995;Carrera etal., 1998).Inthefollowing,we brieﬂyoutline thestepsthatlead tosuchadescriptioninordertohighlighttheunderlyingassumptions. Theupscaledmultiratemasstransferdescriptionforthefracture-matrix system isobtained byverticalaveraging. Tothisend,we deﬁnethe averagesconcentrationoverthefractureandmatrixcross-sectionsas

𝑐𝑚(𝑥,𝑡)=_𝑑1 𝑚∫ 𝑑𝑚 0 𝑑𝑦𝑐(𝐱,𝑡), 𝑐𝑖𝑚(𝑥,𝑡)= 1 𝑑𝑖𝑚∫ 𝑑𝑖𝑚 0 𝑑𝑦𝑐(𝐱,𝑡), (11)

wheredmisthewidthofthefractureanddimofthematrix.Averaging

(10)overthefracturecross-sectiongives

𝜙𝑚𝜕𝑐𝑚_𝜕𝑡(𝑥,𝑡)+𝑢𝜕𝑐𝑚_𝜕𝑥(𝑥,𝑡)−𝐷𝑚𝜙𝑚𝜕 2_𝑐 𝑚(𝑥,𝑡) 𝜕𝑥2 =−_𝑑1_𝑚𝜙𝑚𝐷𝑚𝜕𝑐𝑚_𝜕𝑦(𝐱,𝑡)|||| 𝑦=0, (12) whiletheequationforpurelydiﬀusivetransportinthematrixdomain is 𝜙𝑖𝑚𝜕𝑐𝑖𝑚 (𝐱,𝑡) 𝜕𝑡 −𝐷𝑖𝑚𝜙𝑖𝑚 𝜕2𝑐𝑖𝑚(𝐱,𝑡) 𝜕𝑦2 =0. (13)

Theboundaryconditionare𝑐𝑖𝑚(𝑥,𝑦=0,𝑡)=𝑐𝑚(𝑥,𝑦=0,𝑡)asan

ex-pressionofconcentrationcontinuity.Weapproximate𝑐𝑚(x,𝑡)≈ 𝑐𝑚(𝑥,𝑡),

which assumes fastequilibration over the fracturecross-section. Us-ingﬂux continuityacrossthefracturematrixinterface,weobtainin

AppendixC 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑡 +𝑣 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑥 −𝐷𝑚 𝜕2𝑐𝑚(𝑥,𝑡) 𝜕𝑥2 =−𝛽 𝜕𝑐𝑖𝑚(𝑥,𝑡) 𝜕𝑡 , (14)

wherewedeﬁnedthecapacitycoeﬃcient𝛽 =𝑑𝑖𝑚𝜙𝑖𝑚∕𝑑𝑚𝜙𝑚andthepore

velocity𝑣=𝑢∕𝜙𝑚.Theaveragematrixconcentrationcanbeexpressedas

alinearfunctionaloftheaveragefractureconcentration(AppendixC)

𝑐𝑖𝑚(𝑥,𝑡)=_∫ 𝑡

0 𝑑𝑡 ′_𝜑₍_𝑡₋_𝑡′₎_𝑐

𝑚(𝑥,𝑡). (15)

Thememoryfunctioniswellknown(Carreraetal.,1998),andcan beexpressedinLaplacespaceas

𝜑∗(𝜆)=tanh( √ 𝜆𝜏𝐷) √ 𝜆𝜏𝐷 , (16)

where we deﬁne the characteristic diﬀusion time 𝜏𝐷=𝑑𝑖𝑚2∕𝐷𝑖𝑚 in

the matrix.Combining (14) and(15), weobtain the integro-partial-diﬀerentialequation 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑡 +𝑣 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑥 −𝐷𝑚 𝜕2𝑐𝑚(𝑥,𝑡) 𝜕𝑥2 =−𝛽 𝜕𝜕𝑡∫ 𝑡 0 𝑑𝑡 ′_𝜑₍_𝑡₋_𝑡′₎_𝑐 𝑚(𝑥,𝑡), (17)

whichisequivalenttothemultiratemasstransfermodelofHaggertyand Gorelick(1995);Carreraetal.(1998).Inthefollowing,wedescribethe formulationofthisupscaledmodelinaLagrangianframework.

3.2. Upscaledlagrangianmodel

The upscaled Lagrangian approach models one-dimensional advective-diﬀusivetransportalongthefracture,whichisinterruptedby trappingeventsthatarePoissondistributed.Thismeansthattransitions from the fracturetothematrixoccur at constantrate𝛾, which can

be quantiﬁedbythediﬀusionrateover thefracturecross-section.At eachtrappingevent,aparticleistrappedforarandomtimedistributed accordingto𝜓im(t).Thesearetheprincipalingredientsoftheupscaled

transport model. In the following, we formulate this model in the TDRWframework.

One-dimensionaladvective-diffusionparticlemotionatconstant ve-locityvanddiffusioncoefficientDmisdescribedbyparticletransitions

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overtheﬁxeddistance𝓁byarandomtimetm.Theprobabilitywufor

upstreamparticlemotionis

𝑤𝑢= 𝐷_𝓁𝑚2𝜏𝑣. (18)

Theprobabilityfordownstreammotionisaccordingly𝑤𝑑=1−𝑤𝑢.The

time𝜏visdeﬁnedby 𝜏𝑣=₁₊𝓁∕_2∕𝑣_𝑃_𝑒, 𝑃𝑒= _𝐷𝑣𝓁

𝑚. (19)

Thetransitiontimetmisexponentiallydistributed

𝜓𝑖𝑚(𝑡)=𝜏𝑣exp(−𝑡∕𝜏𝑣). (20)

These rules represent mobile transport as a TDRW model for advection-diffusionwith constantvelocity vanddiffusioncoefficient

Dm(Russianetal.,2016).Thismotioniscombinedwiththetrapping

rulesoutlinedinthefollowing.During atransitionof durationtm,nt

trappingeventsoccur,suchthatthetotaltransitiontimeisgivenby

𝑡𝑡=𝑡𝑚+ 𝑛𝑡

∑

𝑖=1𝜏𝑖𝑚. (21)

ThenumberntoftrappingeventsisdistributedaccordingtothePoisson

distribution

𝑃(𝑛|𝑡)=(𝛾𝑡)𝑛𝑡 exp(−𝛾𝑡)

𝑛𝑡! , (22)

withmean⟨𝑛𝑡⟩ =𝛾𝑡.Thus,thetotaltransitiontimettdescribesa

com-poundPoissonprocess.ItsPDF𝜓(t)canbeexpressedinLaplacespace

as(Margolinetal.,2003;Dentzetal.,2012)

𝜓∗(𝜆)=₁₊ 1

𝜆𝜏𝑣+𝛾𝜏𝑣[1−𝜓𝑖𝑚∗(𝜆)]. (23)

Laplacetransformedquantitiesaremarkedbyanasterisk,theLaplace variableisdenotedby𝜆.

InordertoshowtheequivalenceofthisLagrangianformulationwith theMRMTmodel(17),wederiveinAppendixDfortheconcentration

c_m(x,t)inthefracture 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑡 +𝑣 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑥 −𝐷𝑚 𝜕2𝑐𝑚(𝑥,𝑡) 𝜕𝑥2 =− 𝜕𝑐𝑖𝑚(𝑥,𝑡) 𝜕𝑡 . (24)

Theconcentrationcim(x,t)inthematrixisgivenby 𝑐𝑖𝑚(𝑥,𝑡)=𝛾 ∫

𝑡

0 𝑑𝑡 ′_𝜗₍_𝑡₋_𝑡′₎_𝑐

𝑚(𝑥,𝑡′), (25)

wherethememorykernelϑ(t)isdeﬁnedby

𝜗(𝑡)=_∫ ∞

𝑡 𝑑𝑡

′_𝜓

𝑖𝑚(𝑡′). (26)

Itdenotestheprobabilitythatthetrappingtimeislargerthant.Using

(25)in(24),weobtainforcm(x,t)thegoverningequation 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑡 +𝑣 𝜕𝑐𝑚(𝑥,𝑡) 𝜕𝑥 −𝐷𝑚 𝜕2𝑐𝑚(𝑥,𝑡) 𝜕𝑥2 =−𝛾 𝜕𝜕𝑡∫ 𝑡 0 𝑑𝑡 ′_𝜗₍_𝑡₋_𝑡′₎_𝑐 𝑚(𝑥,𝑡′). (27)

Thisequationandequation(17)areequivalentif

𝛾𝜗(𝑡)≡ 𝛽𝜑(𝑡). (28)

Weﬁrstrecallthat𝜑(t) isnormalizedto1,which canbeseenby

takingthelimit𝜆 → 0in(16),whiletheintegraloverϑ(t)isequalto ⟨𝜏im⟩,themeantrappingtime.Thus,weobtain

𝛾⟨𝜏𝑖𝑚⟩ =𝛽. (29)

Thisequivalenceidentiﬁesthetrappingrate𝛾 andtrappingtime

dis-tribution𝜓im(t)asthekeyquantitiesintheupscaledmodel.Both

quan-titiescanbeaccessedbyrandomwalkparticletrackingsimulationsas outlinedinSection(2.4).Inthefollowing,weusethisgeneral frame-workfortheupscalingoftransportintheMC10carbonatesample.

Fig.3. PDFsofthesurvivaltimě𝜓𝜏𝑠computedbyTDRW(circles)andthe expo-nentialtrend(Equation(30))correspondingtothesamemeanvalue⟨𝛾⟩ =1∕⟨𝜏𝑠 ⟩ =0.697s−1_(dashed_line)_for_diﬀerent_Péclet_numbers._The_colored_lines_denote

thetrappingratePDFš𝜓𝛾wheretheaverage⟨𝛾⟩ =⟨𝑛∕𝑡𝑚 ⟩(verticaldashedline) is0.697s−1_.

3.3. CTRWupscaledmodelversusTDRWmodelresults

We tested the CTRW model by comparing the results with di-rect TDRW simulationsforthesimplest idealized2-dimensional rep-resentation of a single fracture system. The computational domain is a porousmedium (the immobiledomain) of dimension Lz=20000

× Ly=1001pixelsembeddingafracture(themobiledomain)of

aper-ture1pixel,locatedaty=500,sothattheimmobiledomaindepthon eachsideofthefractureis𝓁_im=500pixels.Thepixelsizeisdenoted𝜉

asinSection2.3.

Theﬂowvelocityvinthefractureisconstant,theinletislocatedat

𝑧=0whereapulseinjectionisapplied(seeSection2.4)andthe out-letislocatedat𝑧=20000wherethePDFoftheﬁrstpassagetime(or breakthroughcurve) ̌𝜓𝑡𝑡 ismonitored.Withafractureaperture𝜉,the

problemissimplycharacterizedbythePécletnumber𝑃𝑒=𝑉𝜉∕𝐷0.We

performedsimulationsforconstantdiﬀusivityintheimmobiledomain (𝐷𝑒(𝑥,𝑦)=𝐷)andforrandomlognormaldistributionwithDe(x,y)taken

asthespatialgeometricmeanofthepixeldiﬀusion.Simulationsare per-formedwith𝜉 =10−5_m,_𝐷

0=10−9m2· s−1,𝐷𝑒=1.774× 10−11m2· s−1

and10−12_≤_𝐷

𝑒(𝑥,𝑡)≤10−9m2· s−1forthelognormaldistributed

diﬀu-sionmodelinthematrix.

A mainattribute of the compound Poisson process described in

Section3.2isthatthedistributionof thesurvivaltimein themobile domain𝜏sisexponentiallydistributed:

𝜓𝜏𝑠 (𝑡)=𝛾 exp(−𝛾𝑡), (30)

where𝛾 =1∕⟨𝜏𝑠⟩.

Fig.3displaysthesurvivaltimedistribution ̌𝜓𝜏𝑠 computedfromthe

TDRWwhichiswellﬁttedbyanexponentialdistributionofmean⟨𝛾⟩ = ⟨n/tm⟩ =1/⟨𝜏s⟩.

Fig. 4 shows the perfect agreement between the breakthrough curves,orﬁrstpassagetimePDFs ̌𝜓𝑡𝑡 ,resultingfromtheupscaledCTRW

simulationsandthoseobtainedfromtheTDRWsimulations,forPe val-uesrangingfrom1to100.Theresultsaresimilarforthehomogeneous immobiledomainandfortherandomlognormaldistributionwiththe samegeometricmeandiﬀusion,asexpected(NœtingerandEstebenet, 2000;Russianetal.,2016).

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Fig.4.Green,redandbluecirclesdenotě𝜓𝑡 𝑡,thePDFsoftheﬁrstpassagetime

tt computedbyTDRW,forconstantDe =1.774×10−11m2· s−1inthe immo-biledomainforPe=1,10and100,respectively.Theblackcircles,thatare almostcompletelyoverlappedbytheredcircles(𝑃𝑒=10),denote ̌𝜓𝑡 𝑡for ran-domlognormalporositydistributionwithgeometricmeandiﬀusionequalto 1.774×10−11_m2_{· s}−1_._The_green,_red_and_blue_dashed_lines_denote_the

equiva-lent𝜓𝑡 𝑡computedbytheCTRW.Thecurveplottedasacontinuousblacklineis thememoryfunction𝜑(t)ofslope−1∕2thatcharacterizestheimmobiledomain whiletheverticalblackdashedlineindicatesthediﬀusioncharacteristictime

𝑡𝑑 =𝓁_𝑖𝑚2∕(2𝐷𝑒 )=7.044×105s.(Forinterpretationofthereferencestocolour inthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

4. TDRWmodelingoftransportinthecarbonatesample

ThissectionconcernsdirectsimulationsperformedwiththeTDRW model,i.e.simulationsofthe3-dimensionaldomain.Simulationsare performedaccordingtothealgorithmandtheboundaryconditions de-scribedinSections2.2and2.3,respectively.Theresultspresentedin thissectionfocuson4distinctmodelsthatcharacterizetheimmobile domaindiﬀusivity distributionin termsof tortuosity 𝜅 and porosity

threshold𝜁 (seeTableB.1).Thesimplestmodelassumesconstant tor-tuosity𝜅=1.8andnoporositythreshold,whilethethreeotherassume

porosity-dependenttortuosity𝜅 =𝜙𝑚_,_with_m₌_1.5,_2.5_or_4.5_and_a

porositythreshold𝜁 =0.1.

4.1. TrappingpropertiesoftheMC10sample

Here,thetrappingcharacteristicsoftheMC10sampleareanalyzed andcomparedtotheCTRWmodeldiscussedinSection3.Werecallthat thismodelischaracterizedbythefollowingfeature:theconditionalPDF

Pn(n|tm)thatmeasuresthenumberoftrappingeventsconditionedtothe

timespentinthemobiledomainisaPoissondistributionwithaconstant trappingrate𝛾.Thismeansthatthetimespentinthemobiledomain

betweentwotrappingevents,orsurvivaltime𝜏s,ischaracterizedbyan

exponentialdistribution.ThePDF ̌𝜓𝜏𝑠 computedfortheMC10sample

andtheonecorrespondingtoEquation(30)withthesameaverage val-ues⟨𝜏s⟩aredisplayedinFig.5,whilethePDFPn(n|tm)computedfor

theMC10sampleandtheonecorrespondingtoEquation(22),where theconstanttrappingrate𝛾 =⟨𝛾⟩,aredisplayedin Fig.6. Thelatter isobtainedbycomputingthePDFof𝑛𝑡𝑚 fromEquation(22)foreach

rangeoftm.ThesurvivaltimePDF ̌𝜓𝜏𝑠 doesnotdependontheaverage

fluidvelocityinthesample,i.e.doesnotdependonthePevalue,andis controlledbythetransportpropertiesatthemobile-immobiledomains interfaceandthusiscontrolledbytheeffectivediffusionofthe immo-biledomaininthevicinityofthemobile-immobileinterface,thatisto saybythelocalporosity.Fig.5showsthatthe ̌𝜓𝜏𝑠 PDFs arevisually

identicalwhenapplyingaporositythreshold𝜁 ≤0.1ornot,

emphasiz-Fig.5. PDFsofthesurvivaltime𝜏s ,̌𝜓𝜏𝑠,fordiﬀerentvaluesof𝜅 oralternatively

mand𝜁.Forcomparison,thecontinuouslinesdenotetheexponentialtrend

(Equation(30))correspondingtothesameaveragevalues⟨𝜏s ⟩.

ingthatthevalueoftheporositythresholddoesnotchangenoticeably thepropertiesoftheimmobiledomainatthemobile-immobileinterface noritstopology,asitisshownalsoinAppendixB.Asexpected, ̌𝜓𝜏𝑠 is

stronglyshiftedtowardlargertimevalueswhentheimmobile diﬀusiv-ityatthemobile-immobileinterfacedecreases.Theimportantpointis that ̌𝜓𝜏𝑠 curvesare,asageneralrule,notexponentialdistributionsand

displayanover-representationoftheshortsurvivaltimes.Weseealso largermaximumvaluescomparedtowhatispredictedbythe exponen-tialdistribution,butthisfeaturedecreaseswhenmincreases.

TheconditionalPDFPn(n|tm)resultingfromtheMC10simulationsis

comparedtotheonecomputedassumingaPoissondistribution follow-ingEq.22withaconstanttrappingrate⟨𝛾⟩ inFig.6.Theconditional PDFPn(n|tm)resultingfromtheMC10simulationsisnoticeablydiﬀerent

fromtheonecomputedassumingaPoissondistributionwithaconstant trappingrate.Foragivenmobiletime,thetheoreticalPoissonmodel predictslesstrappingeventsthanwhatismeasuredfortheMC10 sam-ple.Thisdiscrepancyincreaseswiththevalueoftm.

Thetrappingratedistributionencompassestheinformationabout thetrappingprocesswhichis controlledbythecomplex interactions betweenthemobileandimmobiletransportprocess.Assuch,onecan expectthatthetrappingratedistributionisamacroscopicobservable thatcharacterizesthemobile-immobilemasstransfer,andinasimilar mannerthatthememoryfunctionisthemacroscopicobservablethat encipherstheentirepropertiesofimmobiledomaindiﬀusivetransport properties.Fig.7displaysthePDFof𝛾, ̌𝜓𝛾andreportsthepercentage

oftheparticlesthatdonotencountertrappingwhentravelingfromthe inlettotheoutletfordiﬀerentpropertiesoftheimmobiledomain.This percentagedependsevidentlyonthevalueofthePenumberbut also onthepropertiesoftheimmobiledomain.Forinstance,itrangesfrom 1.8%to95.6%forPe=100dependingonthevalueofm.Itfollowsthat theaveragetrappingrate⟨𝛾⟩ cannotbeinferredfrom1/⟨𝜏s⟩becausethe

statisticsof𝜏sconcernonlyparticlesthatencountertrappingwhereasa

certainnumberofparticlesnevervisittheimmobiledomain.Yet, inter-estingly,theresultspresentedinFig.7showthatallthePDFs ̌𝜓𝛾have

almostthesameaveragevalue,⟨𝛾⟩ =1.51 ± 0.18s−1_{independently}

oftheimmobiledomainproperties,whichmeansthatthisvalueisan intrinsicpropertyoftheMC10sample,probablyrelatedtothe geome-tryofthemobiledomainanditsinterfacewiththeimmobiledomain, despitethedistributions ̌𝜓𝛾beingstronglydissimilar.

Altogether,theseresultssuggestthattheassumptionssupportingthe CTRWmodelofSection3.2arenotstrictlymetforthecarbonatesample

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Fig.6. Left:conditionalprobabilitylog10(Pn (n|tm )) ob-tainedfortheMC10sample.Right:conditional prob-abilitylog10(Pn (n|tm ))correspondingtothetheoretical Poissondistributioncomputedusingasinglerate⟨𝛾⟩. Blackcolorindicatesvaluesoflog10(Pn (n|tm ))smaller than–5whilecolorscalefromdarkredtowhitedenotes valuesoflog10(Pn (n|tm ))rangingfrom–5to–0.68.The greendashedlineisavisuallandmark.Resultsaregiven for𝑃𝑒=100,𝜁 =0.1and𝜅(x)=𝜙(x)1−𝑚 _with_m₌_2.5. (Forinterpretationofthereferencestocolourinthis ﬁg-urelegend,thereaderisreferredtothewebversionof thisarticle.)

Fig.7. PDFsofthetrappingratě𝜓𝛾,where𝛾(𝑝)=𝑛(𝑝)∕𝑡𝑚 (𝑝)fordifferent proper-tiesoftheimmobiledomainanddifferentvaluesofPe.Thevaluesinparenthesis denotethepercentageofparticlesptravelingtheentiredomaininthemobile domainwithoutbeingtrappedintheimmobiledomain.Theverticalred discon-tinuouslineindicatesthemeantrappingrate⟨𝛾⟩ whichisequalto1.51 ± 0.18 forallthecurves.(Forinterpretationofthereferencestocolourinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)

consideredhere.Takingintoaccounttheseresults,theissuethatwill beinvestigatednextistoevaluatetowhichextenttheCTRWmodelis robustenoughtomodeltransportinheterogeneousmediasuchasthe MC10sample,oralternativelywhatadditionalrelationshipbetweenthe trappingratepropertiesandthemobiledomainpropertiesarerequired toderiveareliableupscaledmodelfor complexsystemssuchasthe MC10sample.

4.2. Upscalingtheimpactofdiﬀusionintheimmobiledomain

Foreachtrappingevent,theparticlesthatentertheimmobile do-main at a given location can exit at another location. In the case of thesinglefracturemodelwithhomogeneousequivalentimmobile domain, therelocation distance along the linearcontinuous mobile-immobileinterfaceisasharpdistribution(welldescribedbyitsmean value0).Conversely,therelocationoftheparticlesintheMC10 sam-pleismuchlesspredictableduetothestrongheterogeneityofthe sys-teminwhich theimmobiledomainis formedof heterogeneous

clus-tersspatiallydistributed.Thisistriggeredprincipallybynon-continuous mobile-immobile interfaces (lacunar interface)and thepossibility of particlestoutilizetheimmobiledomaintotakeashortcutfromagiven ﬂowpathtoanother.Conversely,the1-dimensionalCTRWmodel im-posesbyconstructionthatparticlesenterandexittheimmobiledomain atthesamelocationforeachtrappingevent.

Simulating such a situation while keeping the complete (3-dimensional) computation of thetransport in the mobile domain is viewedaspotentiallyinstructiveforunderstandingconjointlytheeﬀect oftheparticlesrelocationatthemobile-immobileinterfaceowingtothe strongheterogeneityoftheinterfaceandthestatistical representative-nessofthetrappingtimePDF ̌𝜓𝜏𝑖𝑚 formodelingtheimmobiledomain

transportpropertiesatthescaleofthesample.Tothisend,theTDRW solverismodiﬁedsuchthattransportinthemobiledomainandthe trap-pingprocessarekeptunchanged,butthetimespentintheimmobile domainisdrawnfromthetrappingtimePDF ̌𝜓𝜏𝑖𝑚 previouslycomputed

duringthecorrespondingTDRWsimulationinvolvingthefulldirect sim-ulationofthetransportinthemobileandtheimmobiledomain.Doing thisimposesbyconstructionthatparticlesenterandexittheimmobile domainatthesamelocationforeachtrappingeventsimilarlytothe CTRWupscaledmodel.Fromnowon,themodelinwhichthetrapping timePDFisusedtomodelthetimespentintheimmobiledomainat eachtrappingeventiscalledtheUPSCALTDRWmodelincontrastto theFULLTDRWmodel.

4.2.1. Controloftheimmobiledomaindiﬀusionpropertiesover mobile-immobilemasstransfer

Fig.8compilesthemaininformationconcerningtheresultsinterms ofﬁrstpassagetimett,mobiletimetm,immobiletimetimandsurvival

time𝜏sfortheFULLTDRWmodelandtheUPSCALTDRWmodel.These

dataareveryvaluableforunderstandingthecontroloftheimmobile domainpropertiesonthewayparticlessamplethesystem.The capac-ityoftheimmobiledomaintotriggershortcutsbetweenzonesofthe mobiledomainwithdiﬀerentﬂowpropertiesdecreaseswhenmoving fromthe𝜅 =1.8-𝜁 =0modeltothem=2.5-𝜁 =0.1modelandm= 4.5-𝜁 =0.1modelbecause1)applyingaporositythresholddecreases

theprobabilityofhavingimmobiledomainclustersconnectedtomany poresand2)increasingthevalueofmactsasincreasingthetortuosity, i.e.theeffectivediffusiontimeintheimmobiledomain(seeTableB.1). Consequently,comparingtheresultsfortheFULLmodelwiththe UP-SCALmodelforwhichparticlesareforcedtoexittheimmobiledomain wheretheyenteredforeachofthetrappingeventsallowsnotonlyto understandthefeedbackeffectoftheparticlerelocationprocessatthe mobile-immobileinterfaceontheoveralltransportthatisquantifiedby thefirstpassagetimePDF,butalsotodecomposetheoveralltransport processintermsofthetimespentinthemobiledomainandthe immo-biledomain.

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Fig.8. PDFsoftheﬁrstpassagetimestt (̌𝜓𝑡 𝑡,plaincircles),themobiletimestm (̌𝜓𝑡 𝑚,plainlines)andtheimmobiletimestim (̌𝜓𝑡 𝑖𝑚,dashedlines)fortheFULLTDRW model(black)andtheUPSCALTDRWmodel(red).Simulationswereperformedwiththefollowingparameters:Pe=100,𝜅 =1.8and𝜁 =0,and𝜅(x)=𝜙(x)1−𝑚 _with

m=2.5or4.5and𝜁 =0.1.Thegray-ﬁlledcirclesdenotethePDFsoftt forthecasewherethereisnoimmobiledomain(recallthat ̌𝜓𝑡 𝑡= ̌𝜓𝑡 𝑚inthiscase).(For interpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

Forthe𝜅 =1.8-𝜁 =0modeltheﬁrstpassagetimePDF ̌𝜓𝑡𝑡 obtained

fortheFULLandtheUPSCALmodelarenoticeablydissimilar;their re-spectiveshapebeingfullycontrolledbytheimmobiletimedistribution forintermediateandlongtimes.Conversely,themobiletimePDFare thesamebutdifferentfromthemobiletimePDFcomputedassuming noimmobiledomain,i.e.dependingonlyonthemobiledomain proper-ties.Thisindicatesthatthetransportisstronglycontrolledbythebroad spatialredistributionoftheparticlesamongmobilezonesofdistinctly differentflowrates.Asageneralrule,onecanconcludethatthe discrep-ancybetweentheUPSCALandtheFULLmodelintermsoffirstpassage timePDF ̌𝜓𝑡𝑡 (Fig.8) originatesfromthefactthatboththetrapping

rate𝛾 andthetrappingtimeintheimmobiledomain𝜏im arediﬀerent

asaresultofthedistinctparticlerelocationprocesses when encoun-teringtrappingevents.Fromtheseobservations,onecanspeculatethat forthe𝜅 =1.8-𝜁 =0modeltheupscalingof suchasystemwitha

one-dimensionalmodelwhereparticlessampletheimmobiledomain accordingtotheensembleaveragestatisticsofthemobiledisplacement willfailevenifoneconsidersanon-uniquetrappingratethatwouldbe relatedtothemobiletime.

Thetwoothermodelsofimmobiledomain(m=2.5and4.5)share thesamespatialgeometry,i.e.thesameboundaries,becausetheyshare thesameporositythreshold𝜁 =0.1,butdiﬀerfromtheeﬀective

dif-fusionspatialdistributionandmean.Increasingthevalueofmactsas decreasing1)themeandistanceofpenetrationoftheparticleintothe

immobiledomainand2)therelocationdistancebetweentheentrance andtheexitoftheparticleintheimmobiledomainduringeachtrapping event.Assuch,themodelcharacterizedbym=4.5isthemostsimilar tothesimplefracturemodelpresentedinSection3.3intermsof geom-etry.Indeed,theresultspresentedinFig.8form=4.5-𝜁 =0.1show

thattheﬁrstpassagetimePDFs ̌𝜓𝑡𝑡 arealmostsimilarfortheFULLand

theUPSCALmodels,whilethemobiletimePDFsoftm(̌𝜓𝑡𝑚 )overlapthe

PDFsoftmforthecasewherethereisnoimmobiledomain.Thismeans

thattheimmobiledomainheterogeneitydoesnotcontroltheadvective transportinthemobiledomainsimilarlytowhatoccursinthesimple fracturemodel.

Forthemodelwhereonesetsmtothevalueof2.5,whichisthemost realisticparameterization,Fig.8tellsus,followingthesame argumen-tationasforthe𝑚=4.5case,thattheimmobiledomainheterogeneity weaklycontrolsthemobiledomaintransport.

4.2.2. Onthecontrolofthemobiledomaintransportonthetrappingrate Fig.9showsthatthetrappingrate𝛾 isnotconstantbutdependson

themobiletimet_m.Thefunction𝛾(t)dependsonthepropertiesofthe

immobiledomaincontrolledby𝜅 and𝜁 butalsoonthePevaluewhich

meansthatthisfunction𝛾(tm)isnotanintrinsicpropertyofthesystem,

butdependsontheﬂowrate.Yetweobserved,forinstanceforthe im-mobiledomaincharacterizedby𝑚=2.5and𝜉 =0.1thatthetrapping rateisactuallyconstantforvalueoftmlargerthan200s(materialized

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Fig.9. 𝛾(t)versustm ,fordiﬀerenttortuositymodelfortheimmobiledomain (with𝜁 =0.1).Grayﬁlledsymbolsareresultsfor𝑃𝑒=100whileredandblue

circlesareresults(forthe𝑚=2.5model)forPe=10and1000,respectively. (Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereader isreferredtothewebversionofthisarticle.)

bytheverticaldashedlineinFig.9).Thesystembehavesasaconstant trappingrateforrangeoftmwhichincreasesasmincreases.

Fig.10compares thetheoreticalconditionalPDFPn(n|tm)

assum-ingaPoissondistributionfollowingEq.22wherethetrappingrateisa functionoftmusingthevaluesgiveninFig.9withtheconditionalPDF Pn(n|tm)resultingfromtheMC10simulations.Itcanbeseenthat

us-ingthe𝛾(tm)functionreestablishestheconsistencywiththecompound

Poissonprocessmodel(comparedtoFig.6).Thisgivesusthesound ba-sisforimplementingtheCTRWapproachpresentedinSection3.2,but implementedwiththe𝛾(tm)function,forupscalingthetransportinthe

MC10sample.

5. CTRWupscalingofMC10

Asshownabove,thedependenceofthetrappingrate𝛾 onthetime

spentbyaparticleinthemobiledomaintmisacriticalfeaturetriggered

bytheheterogeneityofthemobile-immobiledomaininterface. Accord-ingly,theproposedmodelisbasedontheimplementationofthe mobile-timedependenceofthetrappingrateintheCTRWupscalingmodelthat wasusedtomodelthetransportinthesinglefractureinSection3.2.One speculatesthatthenumberoftrappingevents𝑛𝑡𝑚 duringamobile

tran-sitionofdurationtmisPoisson-distributedinwhichthetrappingrateis

givenbythe𝛾(tm)function.Totestthisassumptionwebuildasimple

upscaledmodelinwhichthetransportprocessesismodeledbythe un-conditionaldownstreammotionofparticleswithconstantdistanceLz

andarandomtransitiontimetmdistributedaccordingtǒ𝜓𝑡𝑚 (plottedin

Fig.8).Theﬁrstpassagetimeforeachparticleinjectedatt=0issimilar toEquation(9)

𝑡𝑡=𝑡𝑚+ 𝑛𝑡

∑

𝑖=1𝜏𝑖𝑚. (31)

Herethenumberoftrappingeventsntisarandomvariabledistributed

accordingtothePoissondistribution

𝑃(𝑛|𝑡𝑚)=

(⟨𝑛𝑡⟩)𝑛exp(−⟨𝑛𝑡⟩)

𝑛! , (32)

with⟨𝑛𝑡⟩ =𝑡𝑚𝛾(𝑡𝑚).Thetrappingtime𝜏imisalsoarandomvariable

dis-tributedaccordingtǒ𝜓𝜏𝑖𝑚 .TheresultsoftheupscaledCTRWmodelare

ﬁrstcomparedwiththoseobtainedwiththeUPSCALTDRWsimulations intheleftplotofFig.11.TheUPSCALTDRWsimulationsintegratethe

samerelocationprocessasintheupscaledCTRWmodel.Assuch,and becauseboththemodelssharethesamemobiletimedistribution ̌𝜓𝑡𝑚

computedassumingnoimmobiledomain,thecomparisonoftheCTRW withtheTDRWUPSCALmodelisasoundvalidationoftheapproach usedtomodelthetrappingrateandthetrappingtimeintheimmobile domain,independentlyofretro-actionoftheimmobiledomainonthe mobiledomaintimedistribution.TheresultsgiveninFig.11showthat theCTRWisperfectlyreproducingtheBTCcomputedwiththeTDRW

UPSCAL modelfordiﬀerent propertiesof theimmobiledomain.The comparisonoftheBTCscomputedwiththeCTRWmodelusingthe mo-biletimedistribution ̌𝜓𝑡𝑚 resultingfromtheFULLTDRWmodelwith

thosecomputedwiththeTDRWaregiveninFig.11(rightplot)for dif-ferentimmobiledomainparameters,aswellasfordiﬀerentvaluesofPe

inFig.12.Asexpected,theCTRWmodeldoesnotreproducewellthe TDRWdataforthecasewherem=1.5and𝜁 =0forwhichthe

immo-biledomainprovidesthelargestopportunityforparticletoshortcutthe mainflowstreams.Clearlythisspecificprocesscannotbetakeninto ac-countbytheLagrangianCTRWmodel.Conversely,weobserveagoodfit oftheupscaledCTRWmodelresultswiththosecomputedbythedirect TDRWsimulationsforthereferencecasewhere𝜅(x)=𝜙(x)1−𝑚_with_m₌

2.5.Speciﬁcally,theupscaledmodelreproducesperfectlythedatafor longtimes(t≥102_s)_for_both_{𝜁 =}₀_and_0.1_while_the_ﬁt_at_intermediate

timesisbetterforthecasewheretheporositythresholdisapplied,i.e.

𝜁 =0.1.TheabilityoftheCTRWmodeltoreproducethedirect

simu-lationsfortimesrangingoversixordersofmagnitudeandfordiﬀerent valuesofthePenumberisshowninFig.12.Thisﬁgurealsodisplaysthe PDFs𝜓𝑡𝑡 computedbytheCTRWmodelassumingasingletrappingrate

value𝛾 =⟨𝛾(𝑝)⟩thusallowingustoﬁgureoutthenoticeable improve-mentofusingthetemporallyevolving𝛾(tm)forupscalingtheBTCsat

longtimes.

6. Conclusions

Withtheobjectiveofupscalingtransportinheterogeneousmedia,a Lagrangiantransportmodelthatseparatestheprocessesoftransportin mobileandimmobiledomainsispresentedalongwithitsparticle-based CTRWimplementation.Assuminganhomogeneousimmobiledomain andacontinuous mobile-immobileinterface,masstransferoccursas acompoundPoissonprocesswithconstantmobile-immobileexchanges rate.Weshowthatthismodelisequivalenttothemultiratemasstransfer modelofHaggertyandGorelick(1995)withthemeantrappingevent numberoftheLagrangianmodelbeingequaltothecapacityratioof themultiratemasstransfermodel.Inotherwords,thetransportprocess intheLagrangianCTRWmodelischaracterizedbythemeantrapping eventnumberwhichalsodenotestheratioofthesolutemassin the immobiledomaintothatinthemobiledomainatequilibrium.Weshow thatthis1-dimensionalCTRWmodelperfectlyreproducesthedirect 2-dimensional TDRW simulationsofthe transportof soluteina linear fractureembeddedintoaporousmatrix.

However,thissimpleupscaledmodelisnotabletoreproduce accu-ratelythetrans-portinthedigitalizedcarbonatesampleMC10thatis usedtoillustratehighlyheterogeneousporousmediaforwhichwe per-formeddirect3-dimensionalTDRWsimulations.Yet,thedirect simula-tionsallowthethoroughstatisticalanalysisofthemasstransfer dynam-icswithinthetwodomainsandofthemassexchangedattheirinterface whichisspatiallydiscontinuousandheterogeneousintermsoftrapping rate.Wefoundthatthisheterogeneityofthemobile-immobileinterface, togetherwiththecomplexityoftheﬂowinthemobiledomaincausesa deviationfromtheCTRWmodelpresentedinSection3.2orequivalently adeviationfromtheMRMTmodel.

Forinstance,thediscrepancybetweenthecomputedsurvivaltime distributionforMC10andtheexponentialdistributioncharacterizing thesingletrappingratemodelarisesfromthenon-uniquenessof the trappingrate;𝛾 isafunctionoftm(Fig.9).Thesurvivaltime

distribu-tiondisplaysapowerlawtrendforshortsurvivaltimeswhichdenotes thesuperpositionofexponentialdistributionsof𝜏s(tm)withdistinct

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av-Fig.10. Left: conditional probabilitylog10(Pn (n|tm )) obtainedfortheMC10sample.Right:conditional prob-abilitylog10(Pn (n|tm )) correspondingtothe theoreti-calPoissondistributioncomputedusing thefunction

𝛾(tm ) displayed in Fig. 9. Blackcolor indicates val-uesoflog10(Pn (n|tm ))smallerthan-5whilecolorscale fromdarkredtowhitedenotesvaluesoflog10(Pn (n|tm )) rangingfrom-5to-0.68.Thegreendashedlineisa vi-suallandmarkidenticaltoFig.6.Resultsaregivenfor

𝑃𝑒=100,𝜅(x)=𝜙(x)1−𝑚 withm=2.5and𝜁 =0.1.(For

interpretationofthereferencestocolourinthisﬁgure legend,thereaderisreferredtothewebversionofthis article.)

Fig.11. ThefigureontheleftdisplaysthecomparisonofPDFsofthetotaltt ̌𝜓𝑡 𝑡computedbytheTDRWfortheUPSCALmodelwith𝜓𝑡 𝑡 thetotaltimePDFs resultingfromtheCTRWusingasinputthemobiletimePDFs ̌𝜓𝑡 𝑚computedbytheTDRWmodelwithoutimmobiledomain(correspondingtotheidenticalgray filledcirclecurvedisplayedinallplotsof Fig.8).ThefigureontherightdisplaysthecomparisonofPDFsofthetotaltt ̌𝜓𝑡 𝑡computedbytheTDRWfortheFULL modelwith𝜓𝑡 𝑡thetotaltimePDFsresultingfromtheCTRWusingasinputthemobiletimePDFs ̌𝜓𝑡 𝑚computedbytheFULLTDRWmodel(correspondingtothe immobile-domain-properties-dependentblacklinecurvesdisplayedinFig.8).

erage:⟨𝜏s⟩(tm)≡ 1/𝛾(tm).Conversely,theincreaseoftheoccurrenceof

largersurvivaltimescomparedtotheexponentialdistributiondenotes thelacunarityofthemobile-immobileinterface;thedistance(andthus thetime)between twotrappingeventscanbeaugmentedduetothe absenceofavailableimmobiledomaininsomepartsofthemobile do-main.

Introducingthisfunctionaldependenceofthetrappingratetothe mobiletimeallowscomplyingwithamobiletimedependentcompound Poissonprocess.Inotherwords,themasstransfersatthescaleofthe sample can be modeled as the ensemble average of residence-time-dependent masstransfersthat canindividuallybe modeled assingle rateprocesses.

The comparison of the upscaled model against the direct 3-dimensional TDRWfordifferent assumedproperties of theimmobile domainanddifferentvaluesofthePécletnumberpermitstoprovethe efficiencyofthemodeltoreproducethecomplexmasstransfersinthe twodomainsandattheirinterface,aslongasthespreadingofsolute duetotheimmobiledomaindoesnotreachalevelwhereitproduces astrongdecorrelationof thevelocityexperiencedbytheparticlesin themobiledomain.Thissituationoccurswhenimmobiledomain

clus-tersallowshort-cutconnectionsbetweenzonesofthemobiledomain displayingdistinctlydifferentflowrates.Fortunately,thissituationis quiteunlikelyinreservoirrockssincethediffusivityintheimmobile domainisgenerallydecreasingfromthemobile-immobileinterface en-suringtogetherwiththepresenceofnon-diffusingzonesacertain insu-lationbetweenadjacentflowingporenetworks.

Thefinalconclusionofthisstudyisthattheproposedupscaled La-grangiantransportmodelprovidesanaccuratedescriptionof the ob-servednon-Fickianbreakthroughcurvesinheterogeneousdual-porosity media,evenwhentheyaredisplayingbroaddistributionsofflow veloc-ityvaluesandhighlyheterogeneousimmobilezones,suchasthe carbon-ateexamplestudiedhere.Thismodelisadualmultiratemasstransfer model(DMRMT),inwhichthemultipleratesoftrappingarisefromboth theheterogeneityofthediffusioninimmobiledomainandthe hetero-geneityoftheflowinthemobiledomain.

Acknowledgments

PG andDRacknowledge fundingfrom theCNRSIEAthrough the projectCROSSCALE(ex-PICSn\260280090).MDandAPacknowledge