Transport of water and ions in partially water-saturated porous media. Part 3. Electrical conductivity

HAL Id: hal-02324271

https://hal.archives-ouvertes.fr/hal-02324271

Submitted on 23 Nov 2020

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Transport of water and ions in partially water-saturated porous media. Part 3. Electrical conductivity

André Revil, Abdellahi Soueid Ahmed, Stephan Matthai

To cite this version:

André Revil, Abdellahi Soueid Ahmed, Stephan Matthai. Transport of water and ions in partially

water-saturated porous media. Part 3. Electrical conductivity. Advances in Water Resources, Elsevier,

2018, 121, pp.97-111. �10.1016/j.advwatres.2018.08.007�. �hal-02324271�

ContentslistsavailableatScienceDirect

Advances in Water Resources

journalhomepage:www.elsevier.com/locate/advwatres

Transport of water and ions in partially water-saturated porous media. Part 3. Electrical conductivity

André Revil^a,∗,AbdellahiSoueidAhmed^a,StephanMatthai^b

aUniversité Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France

bDepartment of Infrastructure Engineering, University of Melbourne at Parkville, VIC 3010, Australia

a b s t r a c t

Inhydrogeophysics,weneedareliablepetrophysicalmodelconnecting(non-linearly)theconductivityofaporousmaterial(likeasoil)totheconductivityofthe porewaterandthewatersaturation.Classicalmodelsaretoosimplisticespeciallyatlowsalinities.Theconvexityoftheelectricalconductivityofaporousmaterial asafunctionoftheporewaterconductivityisduetobothatexturaleffectandthedependenceofthespecificsurfaceconductivityonsalinity.Thetexturaleffect arisesbecauseofachangeinthedistributionofporenetworkconductanceswithsalinity.Fromvolumeaveragingarguments,itispossibletoprovideageneral equationfortheconductivityofporousmaterials.ThisapproximationisbasedonaPadé approximant,connectinglowandhighsalinityasymptoticlimitsforwhich arigorousanalysiscanbemadebasedonfourfundamentaltexturalparameters.Wediscusstheconnectionbetweenthisvolumeaveragingmodelandempirical modelsaswellaswiththedifferentialeffectivemedium(DEM)solutionforgranularmedia.TheDEMcapturesthenon-linearbehavioroftheconductivitycurve withonlytwoparametersbutitisstrictlyvalidforgranularmaterialsonly.Wecomparethemodelswithfiniteelementcomputationsusingtwothree-dimensional poregeometrieswithcontinuousanddiscontinuoussolidsurfaces,respectively.Finallythemodelsarecomparedtoexperimentaldata.

1. Introduction

Seventysixyearsago,Archie(1942)developedasimpleequation tointerpretresistivitywelllogsinclean(clay-free)formations.Inhis model,theelectricalconductivityofaporousmedium𝜎(inSm⁻¹)is writtenas

𝜎= 1

𝐹𝜎_𝑤, (1)

whereF=𝜙^−m(dimensionless)describesthe(resistivity)electricalfor- mationfactor,𝜙denotestheconnectedporosity,and𝜎w(inSm⁻¹)de- notestheconductivityoftheporewater,whichdependsinturnonthe salinityoftheporewaterandtemperature.Theexponentm(dimension- less)iscalledthecementationexponentinthepetroleumengineering community.Forasetofsphericalparticles,mincreaseswithcementa- tionfrom1.1–1.3forcolloidalsuspensionsandunconsolidated sands to1.7–2.1forconsolidatedsandstoneswherethegrainsarebondedto- getherbycements(seeFriedman,2005).However,thevalueofmalso varieswithgrainshape(e.g.,Jacksonetal.,1978)andthisterminol- ogyis therefore confusing. Theterm“Archie porosity exponent” (or firstArchieexponent) shouldbe preferred. Notethatthepower law relationshipbetweenelectricalconductivityandporositywasalready knownmuchbeforeArchie(1942),seeforinstanceBruggeman(1935). ThisversionofArchie’slawF=𝜙^−mshouldalsonotbeconfusedwith Archie’slawwrittenasF=a𝜙^−m(developedlaterinthe50s)andusu- allyusedtofitaformationfactor/porositydatasetinwhicheachpoint

☆IntendedforpublicationinAdvancesinWaterResources.

∗Correspondingauthorat:Université SavoieMontBlanc,ISTerre,Technolac,Bat.Belledonnes,ruedulacd’Annecy,73370LeBourget-du-Lac,France.

E-mailaddresses:andre.revil@univ-smb.fr(A.Revil),abdellahi.soueid-ahmed@univ-smb.fr(A.SoueidAhmed),stephan.matthai@unimelb.edu.au(S.Matthai).

correspondstoadistinctsample,allthesamplebeingfromthesamefa- cies(e.g.,Winsauer,1952).

Electricalconductivitytomographyisakeytechniqueinhydrogeo- physics(Binleyetal.,2015).Unfortunately,Eq.(1)remainsthebasis fortheinterpretationofresistivitydatainalargenumberofpublica- tionsinhydrogeophysics.Backinthe50sand60s(e.g.,Patnodeand Wyllie,1950,andWyllieandSouthwick,1954),petroleumengineers recognizedthatasecondcontribution,calledsurfaceconductivity,was atplay,especiallyinshalyformations.WaxmanandSmits(1968)de- velopedasimpleequationwrittenatthispointas(seealsoCremersand Laudelout, 1965,Rhoadesetal.,1976, MualemandFriedman 1991, Revil,2017a,b),

𝜎= 1

𝐹𝜎_𝑤+𝜎_𝑆. (2)

InEq.(2),𝜎S(Sm⁻¹)denotesthisextraconductivitytermcalledsur- faceconductivityorsurfaceconductanceoftheclay(e.g.,Cremersetal., 1966).Theform ofEq. (2)can be tracedbacktoPatnodeandWyl- lie(1950)andWyllieandSouthwick(1954).Intheseminalmodelof WaxmanandSmits(1968),thesurfaceconductivityiswrittenas𝜎_𝑆= (̂𝐵𝑄_𝑉)∕𝐹 where ̂𝐵(intherange2.0to4.8×10⁻⁸m²V⁻¹s⁻¹,Na⁺,at 25°C)denotestheapparentmobilityforthechargecarriers(calledcoun- terions)responsibleforthesurfaceconductivity(seealsodiscussionsin Reviletal.,1998).ThetermQ_V(inCm⁻³)denotesanexcessofcharge (ofthecounterions)perunitporevolumeduetothecationexchange capacityofthesurfaceofminerals.Claymineralsarecharacterizedby

https://doi.org/10.1016/j.advwatres.2018.08.007

Received11February2018;Receivedinrevisedform13August2018;Accepted14August2018 Availableonline18August2018

0309-1708/© 2018ElsevierLtd.Allrightsreserved.

Fig.1.Anatomyoftheelectricalconductivitycurveofaporousbodyasafunc- tionoftheporewaterconductivity.Thisshowsanexampleofnon-linearbehav- iorbetweentheconductivitydataoftheporousmaterialandtheconductivity oftheporewater.DatafromShainbergetal.(1980).

strongCECvalues,butpuresilicaischaracterizedbyareactivemineral surfaceresponsibleforsurfaceconductivity(seeReviletal.,2014).The surfaceconductivityisduetoconductionintheelectricaldoublelayer (Sternanddiffuselayers)coatingthesurfaceoftheseminerals.

AlthoughEq. (2)suggestsa linearrelationship between thecon- ductivity ofthe porousbody 𝜎 andthepore waterconductivity 𝜎w, Waxman and Smits (1968) recognized that, at low salinity, theac- tualbehaviorof𝜎isnon-linear.Theyobservedthatthisnon-linearbe- haviorisespeciallypronouncedformaterialscharacterizedbyahigh cationexchangecapacity(CEC).Toaccountforthisobservation,Wax- manandSmitsintroducedasalinitydependentcationmobility ̂𝐵(𝜎_𝑤)=

̂𝐵[1−0.6exp(−𝜎_𝑤∕0.013)](with𝜎w expressedinSm⁻¹).However,this explanationisunphysical sincethemobilityofthecationsis notex- pectedtodecreasewithdecreasingsalinity.

Manyerrorscanbefoundintheliteratureregardingtheimproper useofelectricalconductivityequations.Notably,whenEq.(1)isused whileEq.(2)shouldbeappliedinstead,thisresultsinunphysicalvalues anddependenciesofArchie’sporosityexponent.Forinstance,Salemand Chilingarian(1999)observed(erroneously)thatthisexponentissalin- itydependentalthoughitispurelyatexturalparameter.Thelistofsuch errorsis,unfortunately,verylongbothinpetroleumengineeringandin hydrogeophysics.Thatsaid,whileEq.(2)hasincreasinglybeenusedin hydrogeophysics,wecanwonderhowwellthisequationmodelsfresh- waterenvironmentsbecauseitdoesnotcapturethelow-salinitynon- linearbehavior(e.g.Shainbergetal.,1980,forsoils)-atleastnotina physicallymeaningfulway.

Intheprevioustwopapersofthisseries(Revil,2017a,b),wede- velopatheoryofionictransportinporousmediainunsaturatedcondi- tions.Wekeptthetheorytothehighsalinityasymptoticbehaviorfor whichsurfaceeffectsaretreatedasaperturbationofthetransportinthe connectedporespace.Inthissituation,thetransportofionsismostly affectedbythetortuosityofthebulkporespace.Inthepresentpaper, weshowthatthetortuosityaffectingthetransportoftheionschanges withthesalinity.Themaingoalofthepresentpaperistodemonstrateto thehydrogeophysicalcommunitythatbeyondEq.(2),thereisanother realmworthexploringforfreshwaterenvironments.Itmaybeimpor- tantforsoils(Rhoadesetal.,1976;Nadler,1982;1991;Binleyetal., 2015),geosyntheticclayliners(Abuel-NagaandBouazza,2016),clay suspensions(vanOlphen,1957;vanOlphenandWaxman,1958),the geophysicalmonitoringofCO₂sequestration(AlHagrey,2012;Börner etal.,2013)andreactivetransportmodeling(Day-Lewisetal.,2017), andthestudyofgashydratesandpermafrost(Spangenberg,2001;Prieg- nitzetal.,2015)tociteafewexamples.Wereviewknowledgeabout thenon-linearbehaviorofelectricalconductivityatlowsalinities(see

forinstanceBruggeman,1935,WyllieandSouthwick,1954,Bussian, 1983,LimaandSharma,1990,Schwartzetal.,1989a,b),startingwitha volumeaveragingapproachtoanon-linearconductivityequationvalid foranytypeofporousmaterials.Ouranalysiswillstartwiththeworks of Johnsonetal.(1986)andJohnsonandSen(1988).Highandlow salinityasymptoticlimitscanberigorouslyderivedandtiedtogether usingaPadé approximant(i.e.,aratioofpolynomials).Then,wewill explorehowthisequationcanbeusedtoexplaintheWyllieandSouth- wick(1954),WaxmanandSmits(1968),anddualwater(Clavieretal., 1984) models.Wewillalsocomparethemodeltothedifferentialef- fectivemediumsolutionforapackofspherescoatedbyanelectrical doublelayerandimmersedin“background” (pore)water(seeBussian, 1983,LimaandSharma,1990).Finally,thesemodelswillbecompared withnumericalsimulationsattheporescaleatsaturatedandunsatu- ratedconditions.Wewillalsoderivenewexpressionsfortheconduc- tivityandnewrelationshipsbetweenthemodelsandcheckhowthese modelscomparewithexperimentaldata.

2. Volumeaveragingapproach

Fig.1showsoneexampleofanon-linearrelationshipbetweenthe totalelectricalandtheporewaterconductivity.Weseethattheconduc- tivitycurveischaracterizedbyanisoconductivitypointforwhichthe conductivityoftheporousmediumisequaltothatoftheporewater, apropertybroadlyanalyzedinthecolloidalscienceofclaysuspensions (e.g.,Street,1963,CremersandLaudelout,1965,ShainbergandLevy, 1975)andporousmedia(Bussian,1983;LimaandSharma,1990;Revil etal.,1998).Inthissection,wesummarizethefindingsregardingthe electricalconductivityofporousmediaandespeciallythosefromthe originalworksofJohnsonetal.(1986)andJohnsonandSen(1988)in anattempttoexplainthenon-linearbehaviorshowninFig.1.Westart byconsideringtwoasymptoticlimitsfortheconductivityequationcor- respondingathighsalinities(i.e.,higherthanthesalinitycorrespond- ingtotheisoconductivitypoint)andatlowsalinities(i.e.,forsalini- tiesmuchsmallerthanthesalinitycorrespondingtotheisoconductivity point).

2.1. Theformationfactor

Westartouranalysisbyconsideringthelocalconductivityproblem intheabsenceofanelectricaldoublelayeraroundthesolidphase.The solidphaseisinsulating.Theporespaceisfilledbyanelectrolyteofcon- ductivity𝜎w.Theconstitutive(localOhm’slaw)andcontinuityequation forthecurrentdensityaregivenbyj=𝜎we_b inV_p,(i.e.,inthepore space)and∇·j=0onS(i.e.,atthemineralwaterinterface)where e_b=−∇𝜓b denotesthelocalelectricalfield(Vm⁻¹),and𝜓bthelocal electricalpotential(inV),jdenotesthelocalcurrentdensity(Am⁻²).

Thesubscript“b” referstothefactthatinthissituation,theelectrical fieldisgovernedbythedistributionofthebulkconductances(Bernabé andRevil,1995).Intheabsenceofsurfaceconductivity,theconduction problemreducesto:

∇²𝜓_𝑏=0in𝑉_𝑝 (3)

̂𝐧⋅𝐞_𝑏=0on𝑆 (4)

𝜓_𝑏=

{ΔΨ at𝑧=𝐿

0at𝑧=0 (5)

whereV_p andSdenotetheporevolumeand(specific)interfacearea betweenthesolidandthefluidrespectively;L(inm)denotesthelength ofthecylindricalrepresentativevolumeinthedirectionoftheapplied macroscopicelectrical field𝐄=−(ΔΨ∕𝐿)̃𝐳(inVm⁻¹), ̃𝐳denotesthe unitvectorinthedirectionoftheelectricalfield,ΔΨcorrespondstothe differenceofelectricalpotentialbetweentheend-facesoftherepresen- tativevolume,and̂𝐧denotestheunitvectornormaltothesurfaceof

thegrains.Thisboundaryvalueproblemcanbewrittenintermsofa normalizedelectricalpotentialΓ_b foracylindricalrepresentativeele- mentaryvolumeofporousmaterial(Pride,1994),asfollows:

∇²Γ_𝑏=0in𝑉_𝑝 (6)

̂𝐧⋅∇Γ_𝑏=0on𝑆 (7)

Γ_𝑏≡(ΔΨ 𝐿

)−1

𝜓_𝑏=

{𝐿at𝑧=𝐿

0at𝑧=0 (8)

Inabsenceofsurfaceconductivity,theformationfactorF=𝜎w/𝜎(see Eq.(1))isobtainedbysummingupthelocalJouledissipationofenergy (JohnsonandSen(1988);RevilandCathles,1999).Themacroscopic dissipationofenergycanbewrittenasD≡JEwhilethelocaldissipation ofenergyisd≡je_b whereJandjdenotesthemacroscopicandlocal currentdensities,respectively.Thereforewehave

𝐷= 1

𝑉 ∫_𝑉𝑝𝑑(𝐞_𝑏)𝑑𝑉_𝑝, (9)

𝜎(ΔΨ 𝐿

)2

= 1

𝑉 ∫_𝑉𝑝𝜎_𝑤||𝐞_𝑏||²𝑑𝑉_𝑝, (10) 1

𝐹 = (ΔΨ

𝐿 )−21

𝑉 ∫_𝑉𝑝||𝐞_𝑏||²𝑑𝑉_𝑝, (11) 1

𝐹 = 1

𝑉 ∫_𝑉𝑝||∇Γ_𝑏||²𝑑𝑉𝑝, (12)

whereVisthetotalvolumeoftheconsideredrepresentativeelementary volumeandwhere𝜎wisconsideredconstantovertheporespace.This principleisalwaysvalidsincethemacroscopicdissipationofenergyat thescaleofarepresentativeelementaryvolumeisthesumofthelocal dissipations(herelimitedtotheporespace).FromEq.(12),theinverse oftheformationfactorappearstocorrespondtoaneffectiveporosity (theconnectedporosity,apurelygeometricalquantity,is𝜙=V_p/V).

Foranytypeofporousmaterial,thiseffectiveporosityisbuiltbygiving someweighttothethroatsandnoweighttodead-endsofpores.This canbeshownforinstanceforthematerialsketchedinFig.2forwhich thenumericalsimulationisperformedathighsalinity.Thedistribution oftheΓ_b-fieldisshowninFig.3.This∇Γ_b-fieldisstronginporethroats andnullindead-ends(Section6belowexplainsindetailhowthesecom- putationsareperformed).Thiswouldremaintrueforanytypeofpore network,idealizedornot.Theformationfactorgoestoinfinityaspores becomedisconnected(andverylargenearthepercolationthreshold).

Thispaperdoesnotfocusonthespecificphysicsofelectricalconduc- tionclosetopercolation.

2.2. Highsalinityasymptoticlimit

Nowwediscusstheimpactofanelectricaldoublelayer,coatingthe surfaceofthegrains(Fig.4a),contributingextrasurfaceconductivity totheporousmedium.WeintroduceΣ_S(inS),thespecificsurfacecon- ductivity/conductanceoftheelectricaldoublelayer(inS)(seeFig.4, ande.g.,vanOlphen,1957,JohnsonandSen,1988),

Σ_𝑆=

∫

∞ 0

(𝜎(𝑥)−𝜎_𝑤)

𝑑𝑥, (13)

where𝜎(x) denotesthelocalconductivityin thevicinityof themin- eralsurface/porewaterinterface(seedetailsinFig.4b).Outsidethe electricaldoublelayer,𝜎(x)=𝜎wwherexdenotesthelocalcoordinates normaltotheporewater/solidinterface.Inthethindoublelayeras- sumption,thelocalelectricalconductivitydistributioniswrittenas𝜎(x)

=𝜎w+Σ_S𝛿(x)where𝛿(x)denotesthedeltafunctioncharacterizingthe positionofthemineralsurface.Toobtainthehighsalinityasymptotic behavioroftheconductivity,wereplacetheconductivityofthepore

Fig.2. Sketchofthefirst3Dporousmaterialusedtoillustratetheconductivity problem.a.Geometrywithboundaryconditions.Thematerialismadeofan insulatingmineral(nullconductivity)coatedbyaconductiveelectricaldouble layer(conductivityof1Sm⁻¹).The(connected)porespacecontainsondead- endandonethroat.TheelectrodesAandBareusedtoinjectthecurrentI (inA)whilealltheotherboundariesareinsulating.Thevectorndenotesthe normalunitvectortotheexternalboundariesoftheporousbodyand𝜓the electricalpotential.b.MeshusedforthefiniteelementcalculationswithComsol MultiphysicsoftheOhmicconductivityproblem.

water𝜎wbythelocalconductivity𝜎(x)assumingthatinthishighsalin- itylimit,theelectricalfieldremainsroughlythesame.Thisyields 𝜎(ΔΨ

𝐿 )2

= 1

𝑉 ∫_𝑉𝑝𝜎(𝑥)||𝐞_𝑏||²𝑑𝑉_𝑝. (14) Athighsalinity,theelectricalfieldiscontrolledbythedistribution ofbulkconductancesandwereplacethelocalconductivitybyitsex- pressionobtainedwiththethin doublelayerassumption,i.e.,𝜎(x)= 𝜎w+Σ_S𝛿(x).Thisyields(Johnsonetal.,1986),

𝜎(ΔΨ 𝐿

)²

= 1 𝑉 ∫_𝑉𝑝

[𝜎_𝑤+Σ_𝑆𝛿(𝑥)]||𝐞_𝑏||²𝑑𝑉_𝑝, (15)

𝜎(ΔΨ 𝐿

)2

= 𝜎𝑤

𝑉 ∫_𝑉𝑝||𝐞_𝑏||²𝑑𝑉_𝑝+Σ_𝑆

𝑉 ∫_𝑉𝑝𝛿(𝑥)||𝐞_𝑏||²𝑑𝑉_𝑝, (16)