Archie's law

13 Pages·2016·3.84 MB·English

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
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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
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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
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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-
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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.
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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
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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-
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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
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