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SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/ doi:10.5194/se-7-1157-2016 ©Author(s)2016.CCAttribution3.0License. Archie’s law – a reappraisal PaulW.J.Glover SchoolofEarthandEnvironment,UniversityofLeeds,Leeds,UK Correspondenceto:PaulW.J.Glover([email protected]) Received:7March2016–PublishedinSolidEarthDiscuss.:5April2016 Revised:31May2016–Accepted:15June2016–Published:29July2016 Abstract.WhenscientistsapplyArchie’sfirstlawtheyoften umeofhydrocarbonsintherocks,andhencereservescanbe include an extra parameter a, which was introduced about calculated.Archie’slawisgivenbytheequation: 10yearsaftertheequation’sfirstpublicationbyWinsaueret ρ =ρ φ−m, (1) al.(1952),andwhichissometimescalledthe“tortuosity”or o f “lithology” parameter. This parameter is not, however, the- where ρ is the resistivity of the fully water-saturated rock o oreticallyjustified.Paradoxically,theWinsaueretal.(1952) sample,ρ istheresistivityofthewatersaturatingthepores, f form of Archie’s law often performs better than the origi- φ is the porosity of the rock, m is the cementation expo- nal,moretheoreticallycorrectversion.Thedifferenceinthe nent (Glover, 2009), and the ratio ρ /ρ is called the for- o f cementation exponent calculated from these two forms of mationfactor.Archie’slawinthisformwasinitiallyempiri- Archie’slawisimportant,andcanleadtoamisestimationof cal,althoughitwasrecognisedthatcertainvaluesofthece- reservesbyatleast20%fortypicalreservoirparameterval- mentation exponent were associated with special cases that ues.Wehaveexaminedtheapparentparadox,andconclude could be theoretically proven. Glover (2015) provides a re- thatwhilethetheoreticalformofthelawiscorrect,thedata view.Later,thisformofArchie’slawwouldbegivenabet- thatwehavebeenanalysingwithArchie’slawhavebeenin tertheoreticalgroundingbybeingderivedfromothermixing error. There are at least three types of systematic error that models. arepresentinmostmeasurements:(i)aporosityerror,(ii)a However, at least nine out of ten reservoir engineers and porefluidsalinityerror,and(iii)atemperatureerror.Eachof petrophysicistsdonotuseArchie’sfirstlawinthisform.In- thesesystematicerrorsissufficienttoensurethatanon-unity stead,theyuseaslightlymodifiedversionwhichwasintro- value of the parameter a is required in order to fit the elec- duced10yearslaterbyWinsaueretal.(1952),andwhichhas trical data well. Fortunately, the inclusion of this parameter theform: inthefithascompensatedforthepresenceofthesystematic errors in the electrical and porosity data, leading to a value ρ =aρ φ−m, (2) o f of cementation exponent that is correct. The exceptions are whereaisanempiricalconstantthatissometimescalledthe those cementation exponents that have been calculated for “tortuosity constant” or the “lithology constant”. In reality, individual core plugs. We make a number of recommenda- theadditionalparameterhasnocorrelationtoeitherrocktor- tionsforreducingthesystematicerrorsthatcontributetothe tuosityorlithologyandwewillrefertoitastheaparameter problem and suggest that the value of the parameter a may (Glover,2015). nowbeusedasanindicationofdataquality. A problem arises, however, when we consider the result thattheWinsaueretal.(1952)modificationtoArchie’sequa- tiongiveswhenφ→1.Thisisthelimitwherethe“rock”has 1 Introduction no matrix and is composed only of pore fluid. The resistiv- ityofsucharockmust,bydefinition,beequaltothatofthe In petroleum engineering, Archie’s first law (Archie, 1942) porefluid(i.e.ρ =ρ ).However,Eq.(2)givesρ =a×ρ . o f o f isusedasatooltoobtainthecementationexponentofrock Theapparentparadoximpliesthateithera=1always,orthat units. This exponent can then be used to calculate the vol- theWinsaueretal.(1952)modificationtoArchie’sequation PublishedbyCopernicusPublicationsonbehalfoftheEuropeanGeosciencesUnion. 1158 P.W.J.Glover:Archie’slaw–areappraisal is not valid for rocks with porosities approaching the limit Table1.Typicalrangesofcementationexponentandtheaparam- φ→1. This incompatibility that Eq. (2) has with its lim- eterfromtheliterature(Worthington,1993). iting value leads to the idea that Eq. (1) is a better qual- ity model than Eq. (2), which has some intrinsic problems. Lithology m a References While the fact that Eq. (2) breaks down when approaching Sandstone 1.64–2.23 0.47–1.8 HillandMilburn(1956) the limit φ→1 would not necessarily cause a petrophysi- 1.3–2.15 0.62–1.65 Carothers(1968) cisttobeconcerned,thequestionoughttoarisewhetherthe 0.57–1.85 1.0–4.0 PorterandCarothers(1970) 1.2–2.21 0.48–4.31 Timuretal.(1972) Winsauer et al. (1952) modification to Archie’s first law is 0.02–5.67 0.004–17.7 Gomez-Rivero(1976) valid within the range in which it is usually used. Since the Carbonates 1.64–2.10 0.73–2.3 HillandMilburn(1956) Winsaueretal.(1952)modificationtoArchie’sfirstlawusu- 1.78–2.38 0.45–1.25 Carothers(1968) allyproducesbetterfitstotheexperimentaldata,itsvalidity 0.39–2.63 0.33–78.0 Gomez-Rivero(1977) isnotquestionedfurtherandthepracticeofapplyingEq.(2) 1.7–2.3 0.35–0.8 Schön(2004) andobtaininganon-unityvalueforthea parameterremains commonpracticewithinthehydrocarbonexplorationindus- try. WhilemostscientistsfitEq.(2)tomeasurementsmadeon oilcompany,havingcommissionedoneormoreservicecom- a group of data from core plugs from the same geological panies to make the actual measurements. The company has unit or facies type on a log formation factor vs. log poros- beenaskedtoallowustoprovidetheprovenanceofthedata, ityplot,somepetrophysicistsprefertocalculatecementation but have demanded that their source remains unattributable exponents for individual core plugs than calculate a mean as a condition of their use due to the sensitivity of some of and standard deviation for a given group of measurements. themeasurements.Whilethisisnotanidealsituation,itdoes Thisapproachhasbeenconsideredjustified(e.g.Worthing- allowrealnumericaldatatobeavailableinthepublicdomain ton,1993),butrunstheriskofincludingsamplesfrommore when they would otherwise remain secret, and it shows the than one facies type by accident or oversight, whereas the typicalqualityofdatausedbytheoilindustryatpresent. useofaplotallowstheuniformityandrelevanceofthedata Inapapersuchasthis,thedatasetisveryimportant.The from all of the samples to be judged during the derivation inferencesmadeattheendofthispaperhaveabearingonthe ofthecementationexponent.Moreover,plug-by-plugcalcu- quality of data measurement. First of all, the dataset should lation of the cementation exponent is carried out with the be typical of its type within the oil industry, and prefer- equation: ably represent the best or close to the best practice within the industry. Generally service companies have very well- m=−log F/log φ, (3) developed protocols for making the best and most reliable which includes no a parameter, being derived from Eq. (1). aswellasthemostrepeatablemeasurementspossiblewithin Consequently,plug-by-plugcalculationofmeancementation tight financial constraints. Consequently, the data are often exponentandthatderivedfromgraphicalmethodsareoften ofhighquality,butnotashighasitmightbeifthemeasure- disparate. ment were carried out in an academic environment with no Therestofthispaperexaminestheapparentparadoxthat pressuresoftimeorfunding. whereas Eq. (1) has a longer and theoretically better pedi- Allofthedataanalysedinthispaperwereprovidedbyser- gree,Eq.(2)istheversionthatisoverwhelminglymorecom- vice companies, and it is understood that the great majority monly applied because it fits experimental data better. We come from a single service company. Routine core analysis show that, while the original Archie’s law is the most cor- within the service company would normally follow a very rectphysicaldescriptionofelectricalflowinacleanporous clearprotocol.Inthiscase,thesampleswouldhavebeenre- rockthatisfullysaturatedwithasinglebrine,theWinsauer ceivedaspreservedcoreorcoreplugs,andwouldhavebeen et al. (1952) variant is the most practical to apply because subsampled if required. The core plugs would have been it compensates to some extent for systematic errors that are cleaned, commonly with a Soxhlet approach, then dried in presentintheexperimentaldata. a humidity-controlled oven at a temperature low enough to Table1showstypicalrangesofvaluesforthecementation ensure the preservation of most clay structures. The poros- exponentandthea valueobtainedfromtheliterature(Wor- itymeasurementshereareallmadeusingheliumpycnome- thington, 1993). Clearly the a parameter may vary greatly. try, and are likely to have been made on an automated ba- However, some of the more extreme values given in the ta- sis. Such porosity measurements do not have the high ac- ble are probably affected by artefacts. A quick look at the curacyofthosemadeinacademicpetrophysicslaboratories, age of some of these data indicates another problem: while buthaveaverygoodrepeatability,andareusuallyaccurateto there is a huge amount of existing Archie’s law data, most ±0.005(0.5%).Theheliummeasurementisameasurement are proprietorial, and the few datasets that have been pub- ofconnectedporosityratherthantotalporosity,butthesize lishedarerelativelyold.Wehaveconductedouranalyseson of the helium molecule ensures that almost all of the pore recentdata.Thedataareallownedbyasinglemultinational space is probed by the invading helium gas. Measurements SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/ P.W.J.Glover:Archie’slaw–areappraisal 1159 ofporefluidsalinityandporefluidconductivityaregenerally A first qualitative comparison of the fits in Fig. 1 shows made on the stock solution that is used to saturate the sam- that fitted lines from both equations seem to describe the ples. The degree of saturation may not be complete depen- data very well and it would be tempting to assume that ei- dentuponthemethodused,andwhethervacuumsaturation ther would be sufficient to use for reservoir evaluation. The iscombinedwiththeadditionofahighpressureafterwards. adjustedR2 coefficientsofthefitsofEqs.(1)and(2)tothe Onceagain,thesameprotocolwouldhavebeenusedforall data are also shown in Fig. 1 and are also summarised in samples.Thatwillleadtoagoodrepeatability,butonlyifthe Table 2. They show that Eq. (2) is a better fit in all cases, samplesareallothersimilarporosity.Ifsomesampleshave with slightly higher adjusted R2 coefficients, but the differ- amuchlowerporositythanothers,thenitispossibleforthe ence is extremely small. One might be tempted to attribute high porosity samples to be, say, 95% saturated, while the theslightlybetterfitofEq.(2)tothefactthatithasonemore lowerporositysamplesmayonlybe50%saturated.Itisnot fittingparameter. commonforfluidstobeflowedthroughtherockinorderfor Thereis,however,animportantdifferenceinthevaluesof therockandporefluidstoattainchemicalequilibrium.Con- cementationexponentthatthetwomethodsoffittingprovide. sequently, the real conductivity of the pore fluid will not be The cementation exponents that are derived from each fit the same as that of the stock solution and there is potential areshownintheregressionequationsgivenineachpanelof for error. This error might be variable, depending upon the Fig.1andaresummarisedinTable2.Itisclearthatthereis degreetowhicheachsamplecontainsmatrixmaterialinfine asignificantdifferenceinthecementationexponentsderived powderyformthatmightdissolveintheporefluidmoreeas- from the two different equations in almost every case. The ily.Protocolsareusuallysufficientlyrobusttoensurethatall extentofthedifferencesisclearinFig.2,wherethecemen- measurementsaremadeatthesametemperature,orthatcor- tationexponentscalculatedfromEq.(1)andfromEq.(2)are rections for temperature are put in place. However, there is plottedasafunctionofthemeanoftheindividualexponents thepotentialforhumanerror. calculated using Eq. (3), with the dashed line representing Inthisworkallofthedataarefromrelativelycleanclastic a 1:1 relationship. There is no significance in the almost reservoirs whose dominant mineralogy is quartz, exhibiting perfect agreement between Eq. (1) and the mean of the in- alowdegreeofsurfaceconduction.However,thereisnorea- dividualcoreplugdeterminationsasbothmeasurementsare sonwhytheargumentsmadeinthispapershouldnotapply basedonthesameunderlyingequation,thatofArchie’sorig- equallywelltocarbonates(e.g.Rashidetal.,2015a,b)orin- inal law. What is surprising is that the difference between deedanyreservoirrockforwhichArchie’sparametersmight the cementation exponents derived from using Eq. (2) dif- beusefulindeterminingtheirpermeability(e.g.Gloveretal., ferssignificantlyfromthedifferencebetweenthosethatused 2006;WalkerandGlover,2010). Eq.(1). The small, but apparently significant differences in ad- justedR2fittingstatistichavepromptedustoanalysethefits 2 Modelcomparison in greater depth in Fig. 3. In this figure the right-hand ver- tical axis shows the percentage difference between the ad- The question why the practice of using an equation that is justed R2 value from fitting Eq. (2) with respect to Eq. (1) not theoretically correct remains commonly applied in in- as a function of the parameter a from Eq. (2). In all the dustry is worth asking. The answer is that the variant form casesexceptone,thepercentagedifferenceislessthan0.5%, of Archie’s law (Eq. 2) generally fits the experimental data which is very small. The points do, however fall on a well- muchbetterthantheoriginalform(Eq.1). fitted quadratic curve that is centred on, and falls to zero at We have carried out analysis of a large dataset using the a=1.Thisshowsthatthepercentagedifferencebetweenus- twoequationsandbycalculatingthecementationexponents ingthesetwomodelsbehavespredictably,andthetwomod- for individual core plugs. Figure 1 shows formation factors elsareequivalentata=1asexpected. (bluesymbols)andcementationexponents(redsymbols)of However,thecalculatedpercentagedifferencebetweenthe thefullysaturatedrocksampleasafunctionofporosityfor cementation exponents that have been derived from fitting 3562 core plugs drawn from the producing intervals of 11 Eq.(2)withrespecttoEq.(1)asafunctionoftheaparameter unattributablecleansandstoneandcarbonatereservoirs.The (Fig.3;left-handverticalaxis)showsalinearbehaviourthat formation factor data have been linearised by plotting the passes close to zero at a=1. This time the percentage dif- data on a log axis against the porosity, also on a log axis. ferenceisnotnegligible,reachingapproximately±11%for BestfitsweremadebylinearregressionfromboththeWin- these 11 reservoirs. Such an error in the cementation expo- sauer et al. (1952) variant of the first Archie’s law (Eq. 2, nentcancauseasignificanterrorincalculatedreserves.The solid lines) and the theoretically correct first Archie’s law linear fit shows that the percentage difference between the (Eq.1,dashedlines).Inaddition,theindividuallycalculated twoapproachesisabout20%perunitchangeintheaparam- cementation exponents were calculated by inverting Eq. (3) eter.Ifsomeofthelargerandsmallervaluesoftheaparam- (redsymbols). eterthathavebeenobservedaretrue(Table1),therewould www.solid-earth.net/7/1157/2016/ SolidEarth,7,1157–1169,2016 1160 P.W.J.Glover:Archie’slaw–areappraisal Figure1.Formationfactorandcementationexponentofthefullysaturatedrocksampleasafunctionofporosityfor3562coreplugsdrawn fromtheproducingintervalsof11cleansandstoneandcarbonatereservoirs.Bluesymbolsrepresenttheformationfactorforindividualcore plugscalculatedasF =ρo/ρf andredsymbolsrepresentcementationexponentsforindividualcoreplugscalculatedwithEq.(3).Thesolid lineisthebestfittotheWinsaueretal.(1952)variantofthefirstArchie’slaw(Eq.2),whilethedashedlineisthebestfittotheoriginalfirst Archie’slaw(Eq.1),eachwithadjustedR2coefficients. beverysignificantdifferencesinthecementationexponents havingalowertheoretical/mathematicalqualitymodelofthe obtainedusingthetwodifferentArchie’sequations. process.Thispaperwillexaminetheimplicationsofthisob- Consequently, statistical analysis of the 3562 data points servation,examinepossiblecausesfortheapparentparadox, analysedinthisworkshowsthatEq.(2)providesabetterfit andthenmakeanumberofrecommendations. thanEq.(1),confirmingtheexperienceofmanypetrophysi- cists. Equation (2) provides a better physical quality of fit to real data despite the data being theoretically flawed, and SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/ P.W.J.Glover:Archie’slaw–areappraisal 1161 Table2.Summarydatafromthe11testreservoirs. ApplicationofEq.(2) ApplicationofEq.(1) Winsaueretal.(1952) Archie(1942) Reservoir N m a R2 m R2 m Total 3562 fromfit fromfit fromfit fromfit fromfit meanofindividual standarddeviation coreplugs ofcoreplugs A 288 1.781 0.8750 0.824 1.715 0.8229 1.713 0.102 B 365 2.135 0.8080 0.8723 2.051 0.8709 2.050 0.083 C 350 2.204 0.8835 0.8853 1.974 0.8847 1.973 0.053 D 359 1.599 1.1869 0.8684 1.666 0.8668 1.669 0.075 E 374 2.504 1.7242 0.8270 2.818 0.8136 2.831 0.177 F 379 2.417 1.2592 0.6299 2.552 0.6279 2.556 0.249 G 377 1.741 0.8720 0.7213 1.691 0.7207 1.690 0.129 H 88 1.657 1.1290 0.7598 1.669 0.7593 1.700 0.098 I 188 2.875 0.8396 0.8584 2.766 0.8572 2.759 0.230 J 396 1.916 1.0382 0.9166 1.932 0.9165 1.933 0.109 K 398 1.855 1.2791 0.3972 1.954 0.3960 1.957 0.336 Mean 2.0621 1.0813 0.7782 2.0718 0.7760 2.076 0.149 Standarddeviation 0.4041 0.2753 0.1514 0.4373 0.1512 0.436 0.088 Figure 2. Cementation exponent derived from fitting Archie’s (1942) law (Eq. 1, solid blue symbols) and the Win- Figure 3. Percentage difference between cementation exponents sauer et al. (1952) variant of Archie’s law (Eq. 2, solid orange derived from Eq. (2) with respect to that derived from the use symbols) as a function of the cementation exponent derived as of Eq. (1) (i.e. (cid:0)(cid:0)mEq.2−mEq.1(cid:1)/mEq.1(cid:1)×100) as a function themeanofthecementationexponentscalculatedfromdatafrom of the a parameter (blue symbols), with a linear least-squares individual core plugs using Eq. (3), which is based on Archie’s regression (R2=0.8005), together with the percentage differ- original law. The dashed line shows a 1:1 relationship. Each ence between the adjusted R2 fitting coefficients for fitting with symbol represents data from one of the 11 reservoirs analysed in Eq. (2) with respect to that derived from the use of Eq. (1) (cid:16)(cid:16) (cid:17) (cid:17) Fig.1. (i.e. R2 −R2 /R2 ×100)asafunctionofthea pa- Eq.2 Eq.1 Eq.1 rameter (red symbols), with a quadratic least-squares regression (R2=0.9954). 3 Implicationsforreservescalculations Wehavecomparedtheresultsofthecalculatedcementation fittingEq.(1)tothewholedatasetism=2.072±0.437,and exponents from each of the equations using the 11 reser- the mean of the cementation exponents calculated individu- voirs that are summarised in Table 2. The mean cementa- allyism=2.076±0.436.Whileformalstatisticaltestscan- tion exponent from fitting Eq. (2) to the whole dataset is not separate the use of these two equations, the cross-plot m=2.062±0.404(onestandarddeviation),whilethatfrom thatisshowninFig.4indicatesthatthereisadifferencebe- www.solid-earth.net/7/1157/2016/ SolidEarth,7,1157–1169,2016 1162 P.W.J.Glover:Archie’slaw–areappraisal Archie’s equation which contains the theoretically unjusti- fied a parameter, which seems to produce a better fit than the classical Archie’s law. However, it is not known which approachisbetteratthisstage.Theremainderofthispaper attempts to find reasons for the disparity between the two equationssothatthebestapproachcanbechosen. Therefore, there is an apparent paradox: Eq. (2) is theo- reticallyincorrectbutfitsthedatabetterthanatheoretically correctform.Therearetwopossiblereasons. 1. The theoretically correct form of the first Archie’s law iswrong. 2. Alloftheexperimentaldataareincorrect. Moreover, it is incredibly important to find out the reason fortheapparentparadox,giventheimplicationsforreserves calculationsthatwehavedescribedabove. Furthermore, Table 1 and our analysis of 11 reservoirs Figure4.Cross-plotofthecementationexponentscalculatedusing showsthatthea parametercantakevaluesbothgreaterthan Eqs.(1)and(2)foradatabaseof3562coreplugsdrawnfromthe and less than unity, indicating that there may be more than producing intervals of 11 unattributable clean sandstone and car- onecontributoryeffect. bonatereservoirs.Thesolidlineshowstheleast-squaresregression andthedashedlineshowsthe1:1ideal. 4 ErrorintheformulationofArchie’slaw tween the two methods that is represented by the scatter on One of the possibilities for the observed behaviour is that thisgraph,butwhichcouldeasilybeassumednottobesys- the original Archie’s law is incorrect. If that is the case we tematic. It is only when the percentage difference between canhypothesisethatthereisanunknownmechanismX oc- thetwoderivedcementationexponentsisplottedagainstthe curring in the rock which either (i) scales linearly with the parametera,(Fig.3)thatthesystematicnatureofthediffer- porefluidresistivity,orwhich(ii)scaleswiththeporosityto encebecomesapparent. thepowerofthecementationexponent(ratherthantheneg- Hence, even though Eq. (2) provides only a marginally ative of the cementation exponent). In other words, an im- better fit than Eq. (1), its application can give cementation provedArchie’slawshouldlooklikeeitherofthefollowing exponents that are as much as ±11% different from those twoequations: obtainedwithEq.(1)forthedatafromour11reservoirs,but may be even larger if the literature values are reliable (Ta- ρ =Xρ φ−m, or (4) o f ble1). ρ =ρ (cid:0)Xmφ(cid:1)−m. (5) o f For example, if one assumes arbitrarily that the true ce- mentationexponentism=2.072,andthenacceptsthatsys- Both of these equations are formally the same as Eq. (2), tematicerrorintheuseofArchie’slawis±11%,thecalcula- but are rewritten here in generic form so that they may be tionofthestocktankhydrocarboninplaceshowsanerrorof comparedwithequationslaterinthepaperthatexaminethe +20.13/−16.76%inreservescalculations.Inthislastcal- effects of errors in porosity and fluid salinity. However, we culation we have used typical reservoir values (a saturation havenotidentifiedthelinearprocessthatXcouldrepresent. exponent,n=2;porosity,φ=0.2;reservoirfluidresistivity The process cannot be that of surface conduction mediated in situ, ρφ =1(cid:127)m; effective rock resistivity, ρt =500(cid:127)m). byclaymineralsbecauseofthefollowingreasons. The error in the reserves calculation is independent of the reservoir’s areal extent, its thickness, its mean porosity, or 1. The effect occurs in clean rocks – Fig. 1 shows it op- itsformationvolumefactors.Thiserrorindicatesclearlythat eratingin11reservoirscomposedofcleansedimentary theaccuracyofourcalculationsofthecementationexponent rocks. shouldbeofprimeimportance,especiallywithreservoirsbe- 2. Surface conduction can only decrease the resistivity of coming smaller, more heterogeneous, and more difficult to the saturated rock, whereas the mechanism for which produce. we search must have the capability of both increasing In summary, apparent small differences in fit can cause anddecreasingtheresistivityofthefullysaturatedrock. significant differences in the derived cementation exponent whichwillhaveimportantimplicationsforreservescalcula- 3. Surfaceconductiondoesnotscalelinearlywiththepore tions. Moreover, it is the Winsauer et al. (1953) variant of fluidresistivityandiswelldescribedbymoderntheory SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/ P.W.J.Glover:Archie’slaw–areappraisal 1163 (Ruffet et al., 1995; Revil and Glover, 1997; Glover et al.,2000;Glover,2010). 4. It is not possible to generate the second scenario from any of the previous theoretical approaches to electri- calconductioninrocks(Pride,1994;RevilandGlover, 1997,1998). Finally, it is worth remembering that, although initially an empirical equation, Archie’s first law now has a theoret- ical pedigree since its proof (e.g. Ewing and Hunt, 2006). It seems unlikely, therefore, that the theoretical equation is wronginitself. 5 Errorintheexperimentaldata Itisworthtakingalittletimetoimaginetheimplicationsof Figure5.Thecalculatedvalueoftheparametera asafunctionof this question. It implies that the majority or even all of the thepercentageerrorinporosityforvariousvaluesofcementation electrical measurements made in petrophysical laboratories exponent(giveninthelegend).Thea parameterisindependentof around the world since 1942 have included significant sys- theactualvalueoftheporosity. tematicerrors(randommeasurementerrorsarenottheissue here).Giventheimportanceofthecalculationofthecemen- tationexponentforreservescalculations,thisstatementwill have seemincredibleandwillhavefar-reachingimplications. It is hypothesised in this paper that there have been sys- ρo=ρfφ−m=aρf(φ+δφ)−m, (6) tematicerrorsinthemeasurementoftheelectricalproperties whichallowsustocalculatetheparametera: that contribute to the first Archie’s law. The result of these errorshasbeentomaketheversionofthefirstArchie’slaw (cid:18) φ (cid:19)−m given in Eq. (2) a better model for the erroneous data than a= . (7) φ+δφ the theoretically correct model (Eq. 1), and implies that the theoretically correct model would be a better fit to accurate Itisworthnotingthatthevalueofa dependsonthecemen- experimental data. If correct, it would also imply that most tation exponent, with Eq. (7) expressed as a function of the ofthecementationexponentsthathavebeencalculatedhis- fractionalsystematicerrorintheporositymeasurementN : φ torically are correct because the errors in the experimental datahavebeencompensatedforbytheparametera.Hence, (cid:18) 1 (cid:19)−m a= . (8) despite appearing as an empirical parameter, it would have 1±Nφ an incredibly important role in ensuring that the calculated If there is a ±10% systematic error in the measurement of cementation exponent is accurate, even with erroneous ex- the porosity of a rock, and we take m=2, we can generate perimentaldata.Afurtherimplicationisthatcementationex- values for a=1.21 and a=0.81 for the positive and nega- ponentscalculatedusingindividualcoreplugsorameanof tivecases,respectively.Figure5showsthesamecalculation individual core plug measurements are only accurate if the as a function of percentage systematic error in the porosity measurements contained none of the systematic errors that measurement. It is clear that possible systematic errors can aredescribedbelow. produce values of the a parameter that fall in the observed Thereareatleastthreepossiblesourcesofsystematicer- range. rorintherelevantexperimentalparametersusedinArchie’s Weshouldexaminethepossiblesourcesofsystematicer- laws, and others may be realised in time. Each has the po- ror in the porosity. The question should be what the correct tentialforensuringthattheWinsaueretal.(1952)variantof porosity to use in the first Archie’s law is. There is some Archie’slawwillfitthedatabetterthantheclassicalArchie’s doubtwhetherthisquestionispossibletoansweratthemo- law. These errors are associated with the measurement of ment.IfEq.(1)isfoundedongoodtheoreticalgroundsasit porosity, fluid resistivity, and temperature, and will each be seems to be, then the porosity required should be the mea- reviewedinthefollowingsubsections. sured porosity that best approaches the true total porosity 5.1 Porosity whichisfullysaturatedwiththeconductingfluid.However, ifthesample’stotalporosityisonlypartiallysaturatedwith Letusassumethatifinsteadofmeasuringthecorrectporos- conductingfluid,thewatersaturatedporositywouldlikelybe ityφ,wemeasureanerroneousporositygivenbyφ+δφ,we abettermeasure. www.solid-earth.net/7/1157/2016/ SolidEarth,7,1157–1169,2016 1164 P.W.J.Glover:Archie’slaw–areappraisal Therearemanywaysofmeasuringporosity,anditiswell solution reactions occur until the pore fluid is in physico- knownthattheygivesystematicallydifferentresults.Without chemicalequilibriumwiththerocksample. beingcomprehensive,weshouldconsideratleastthreetypes WehavecarriedouttestsonthreesamplesofBoisesand- ofporositymeasurementsthatarecommonlyusedasinputs stone,andwefindthatthefluidinequilibriumwiththerock tothefirstArchie’slawforthecalculationofthecementation can have a resistivity up to 100% less than the bulk fluid exponent. (andapHthatisupto±1pHpointsdifferent).Thesesam- pleshavealargeporosity;quartzcontentisbetween80and Heliumporosimetry iswellknowntogiveeffectiveporosi- 90%,feldsparandmicacontentisbetween10and20%,and ties that are systematically higher than other methods there is a very little clay fraction. The surface conduction becausethesmallheliummoleculescanaccessporesin was assessed as being between 13.6 and 32×10−4Sm−1 which other molecules cannot fit. Hence, it is a good (Walker et al., 2014), which is lower than most of the pe- measure of the combined effective micro-, meso-, and riodsandstoneandFontainebleausandstonesampleswehave macro-porosityofarock. recentlymeasured,andconsequentlytheBoisesandstoneis consideredtobeareasonableanaloguefortheclayfreeclas- Mercuryporosimetry Again,thismethodiswellknownto tic reservoir data used in this work. In these tests a bulk give effective porosities that are systematically lower fluid was made by dissolving pure NaCl in deaerated and than other methods because it takes extremely high deioinised water. The fluid was deaerated once again and pressures to force the non-wetting mercury into the broughttoastandardtemperature(25±0.1◦C).Thebulkre- smallestpores.Consequently,themicro-porosityisnot sistivityofthesolutionwasthenmeasuredusingabenchtop commonlymeasured,evenwithinstrumentswhichcan resistivitymeterthathadbeencalibratedusingahigh-quality generateveryhighpressures. impedancespectrometer.Twolitresofthefluidwasplacedin acontainerandpumpedthrougharocksamplethathadbeen Saturationporosimetry This method relies on measuring saturatedwiththesamefluid,andarrangedsothattheemerg- thedryandsaturatedweightsofasample,andthenus- ingfluidwasreturnedtotheinputreservoirandmixedwith ingeithercalipermeasurementsorArchimedes’method it. The circulation of fluids was continued until either 1400 forobtainingthebulkvolume,fromwhichtheporosity porevolumeshadbeenpassedthroughthesampleorthere- maybecalculated.Measurementsmadeinthiswaygen- sistivityoftheemergingfluidshadreachedequilibrium.The erallyfallbetweenthosemadeonthesamesampleus- resistivityoftheemergingfluidswasmeasuredwiththesame ingtheheliumandmercurymethods.Theproblemhere resistivitymeterinthesamewayasthebulkfluidandatthe isoneofsaturation.Ifthesampleisnotfullysaturated, sametemperature.Furtherexperimentaldetailscanbefound theporositywillbeunderestimated.Sincesaturationin in Walker et al. (2014). Figure 6 shows the difference be- any laboratory is generally governed by its protocols, tweentheresistivityofthebulkfluidandtheresistivityofthe attainment of an only partially saturated sample would actualporefluidsforarangeoffluidswithdifferentstarting besystematic. salinities. The figure shows clearly that low resistivity bulk There is scope for a study to discover which method for fluidsbecomesignificantlylessresistiveastheyequilibrated measuringporosityisthebestforusewithArchie’slaw.Such with the rock, and this has been associated with dissolution astudy,however,wouldneedtoremoveallothersourcesof of rock matrix in the fluid. The effect is sufficiently large systematic error in order to find the best porosity measure- at low salinities to preclude the possibility of having a very mentmethodreliably. low-salinityfluidequilibratedwiththerock,andcanleadto increases in fluid conductivity of up to 100% if the initial 5.2 Porefluidsalinity bulkfluidhasaconductivityoflessthan10−3Sm−1.How- ever, the effect is significant, even at greater salinities with Itisimportanttodistinguishbetween(i)thebulkporefluid bulkfluidswithaninitialconductivityof0.1Sm−1undergo- resistivityand(ii)theresistivityofthefluidinthepores.The inganincreaseofupto16%.Thereiseventheintimationof bulk pore fluid resistivity is that fluid which has been made very high initial salinity bulk solutions decreasing in salin- inordertosaturatetherock.IthasagivenpHandresistivity, ityandconductivityslightlyuponequilibrationwiththerock which may be measured in the laboratory, but is sometimes sample,aneffectthatweassociatewithaslighttendencyto calculated from charts, using software, or empirical models precipitatesaltwithintherockortoreactwithit. suchasthatofSenandGoode(1992a,b).Itistheresistivity Theapparentcleardifferencebetweentheresistivityofthe of this fluid that petrophysicists have most commonly used bulkfluid,whichisusedasaninputtoArchie’sfirstlaw,and intheiranalysisofdatausingthefirstArchie’slaw. the resistivity of the fluid, which should be used, is clearly However,thefirstArchie’slawisnotinterestedinthebulk the source of an invisible systematic error to which many fluid resistivity, but the actual resistivity of the fluids in the petrophysicallaboratorieshavesuccumbed. pores.Whenanaqueousporefluidisflowedthrougharock Let us assume that instead of using the resistivity of the sample, it changes. Precipitation and, more commonly, dis- fluidintheporesρ ,wehaveusedtheresistivityofthebulk f SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/ P.W.J.Glover:Archie’slaw–areappraisal 1165 Figure 6. Percentage difference between the conductivity of the fluidintheporesandthatofthebulkfluidoriginallyusedtosat- uratetherockasafunctionoftheresistivityofthefluidinthepores forthreesamplesofBoisesandstone. Figure7.ResistivityofanaqueoussolutionofNaClasafunctionof temperatureforanumberofdifferentporefluidsalinitiesusingthe fluidgivenbyρ +δρ ,whereδρ willbepositiveforlow- methodofSenandGoode(1992a,b).Dashedlinesshowthechange f f f inconductivityresultingfromadifferenceintemperaturebetween and medium-salinity fluids due to dissolution and negative ◦ 20and25 C.Notethatthenormalisedcurvesfromthewholerange forhigh-salinityfluidswheretheremaybeprecipitation.We ofsalinitiesincludinginthefigurearealmostcoincident. thenhave ρ =ρ φ−m=a(cid:0)ρ +δρ (cid:1)φ−m, (9) o f f f aqueous solution of NaCl as a function of temperature and whichallowsustocalculatetheparameteraas salinity up to 100◦C. This model has been implemented in Fig. 7 for conductivity and for a range of fluid salinities. ρ a= f (10) Inthisfigurewehavenormalisedthecurvesforeachofthe ρ +δρ f f salinities to that at 20◦C. This allows us to see that the rel- oras ativevariationofconductivityforalltheporefluidsalinities inthefigureareapproximatelythesame,aswellasenabling 1 a= , (11) thedifferenceinconductivitywithrespectto20◦Ctobecal- 1±N ρf culatedeasily. Ifwemeasuretheporefluidresistivity,orcalculateitus- where N is the fractional systematic error in the fluid re- ρf ingthemodelat25◦C,butmeasuretheresistivityofthesat- sistivitymeasurement. If there is a +10% systematic error in ρ , which is uratedrocksampleat20◦C(orviceversa),wewillintroduce f the case approximately for a fluid solution of 0.1moldm−3 asystematicerrorinthemeasurementsthatcanbelarge.Fig- (Fig.6),wecancalculatea=0.91,whichisintherangeof ure7showsthattheerrorinsuchatemperaturemismatchis approximatelythesameforallfluidsalinities,andwouldbe observed values. Hence the erroneous assumption that the between−12.03and12.34%dependingonthefluidsalinity bulk fluid resistivity represents the resistivity of the fluid in (largestforthehighestsalinities).Equation(10)canbeused the pores can easily produce the observed effect, and much tocalculatethatavalueofa=1.25wouldbeintroducedto biggervaluesofawouldbepossibleiflowerbulkfluidsalin- thefirstArchie’slawfittingwhenusingEq.(2)tocalculate itieswereusedtosaturatetherockifitweretheresistivityof the cementation exponent with a bulk rock resistivity mea- thosefluidsthatwasdirectlyusedinthefirstArchie’slaw. surementthatismadeatatemperature5◦Clowerthanthat 5.3 Temperature atwhichthebulkfluidhadbeenmeasured.Hence,onceagain asystematicerrorofthecorrectmagnitudeisobtainedfrom Temperaturealsoaffectstheporefluidresistivitythatweuse alackoftemperaturecontrol. in the first Archie’s law. The resistivity of an aqueous pore Thesystematicerrorcanberemovedbymeasuringthere- fluid changes by about 2.3% per ◦C at low temperatures sistivity of the fluid emerging from the rock sample at the (<100◦C).SenandGoode(1992a,b)provideanextremely same time or just after the resistivity of the bulk rock has usefulempiricalmodelforcalculatingtheconductivityofan been measured because the bulk rock and the emerging flu- www.solid-earth.net/7/1157/2016/ SolidEarth,7,1157–1169,2016 1166 P.W.J.Glover:Archie’slaw–areappraisal fluidsaturationortemperature,andthefinalresultinreality willbeamixtureofallthreeerrors. First,letusassumethatweareusingArchie’slawtocal- culatethecementationexponentofasinglecoreplug.Equa- tion(1)canbeusedinitsrearrangedform(Eq.3).Ifweas- sumethatthemeasurementsofporosity,fluidresistivity,and coreplugresistivityareallaccurate,thenEq.(3)willgivean accuratevalueofcementationexponent.Converselyifthere is an error in any of the input parameters, they will be in error in the cementation exponent. Equation (2) can also be rearranged for the calculation of cementation exponent, but sincethereisnoaprioriknowledgeofthevalueofthea pa- rameter,thecalculationcannotbecarriedout.Consequently, whencalculatingasinglevalueofcementationexponentfor asinglecoreplug,theaccuracyofcementationexponentsde- pendscriticallyupontheaccuracyoftheinputdata,whether theerrorsarerandomorsystematic. Nowlet usexamine the calculationof cementationexpo- nent from a population of core samples by the fitting of the original or Winsauer et al. variants of Archie’s law. Three possibilitieswillbeanalysed,ofwhichtheresultsoftwoare shown in Fig. 8. The three possibilities are that there is a random error (which is not shown in the figure), a system- aticerrorresultinginthemeasuredporosityofeachsample being overestimated or underestimated by a constant value (Fig. 8a), and a systematic error resulting in the measured porosity of each sample being overestimated or underesti- matedbyaconstantfractionoftherealvalue(Fig.8b).Both graphsaregivenasafunctionoftherealerror-freeporosity. Thefigureassumesthattherealfluidresistivityis10(cid:127)m, andtherealcementationexponentis2.1;however,theseare genericvalues,andanyvaluefortheseparameterscouldhave been taken with the same result. The horizontal green lines Figure 8. Modelling of the calculated cementation exponent and inbothpartsofthefigurerepresenttherealcementationex- a parameter for error-free data and data containing two types of ponent, m=2.1. This value has been used to calculate the systematicerrorinporosity,(a)measuredporosityexceedsthereal resistivity of the sample, which varies with porosity. Con- porosityby0.02,and(b)measuredporosityexceedstherealporos- ity by a factor of 1.2. Solid curves and dots refer to the left-hand sequently, we take the resistivity of the sample, that of the verticalaxis.Theorangedashedlinereferstotheright-handaxis. pore fluid, and the value m=2.1 to represent the target re- ality. The next step is to use the values of the resistivity of thesampleandthatoftheporefluidtogetherwiththemea- sured porosity in order to calculate the value of the cemen- ids should both have the same temperature. Providing the tationexponent.InFig.8aweassumethateveryinstanceof pore fluid has been equilibrated properly with the sample, measuredporosityisoverestimatedby0.02(ortwoporosity thisprocedurealsoremovesanyerrorsassociatedwithusing units).Thatrepresentsaverylargefractionalerrorforthose the resistivity of unequilibrated bulk fluids in Archie’s law samplesthatreallydohavealowporosity,andasmallfrac- calculations. tionalerrorforsampleswitharealporosityapproaching0.3. Using Eq. (1) to calculate the cementation exponent of this error-pronedata,weobtainthebluecurveinFig.8a,which 6 Discussion varies from m=3.02 at low porosities (x axis=0.005) to m=0.32 at high porosities (x axis=0.3). Clearly, the con- Equations (6) to (11) mathematically imply that the use of stantover-orunderestimationofporosity,byasmallamount, the a parameter might be compensating for any systematic hasalargeeffectuponthecalculatedcementationexponent errors. This perhaps surprising result is analysed in the fol- ifEq.(1)isused. lowingsection.Theimplicationsoferrorsinporositywillbe Now let us use Eq. (2). In practice the value of the a pa- analysed, though the argument applies equally to errors in rameter would be known from the fitting of Eq. (2) to the SolidEarth,7,1157–1169,2016 www.solid-earth.net/7/1157/2016/

Description:
a group of data from core plugs from the same geological unit or facies type on a log formation factor vs from the producing intervals of 11 clean sandstone and carbonate reservoirs. Blue symbols represent the . fluid resistivity and is well described by modern theory. Solid Earth, 7, 1157–1169,
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