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The essence of carbonate
petrophysics is (often) pore system
heterogeneity, as compared to clay
conductivity issues for clastics
The foundation of theLucia
petrophysical classification is the
concept thatpore-size distribution
controls permeability and
saturation
Thefocus of this classification is
on petrophysical properties and
not genesis
Petrophysical classifications focus on contemporary rock fabrics that include
depositional and diagenetic textures
To determine the relationships between rock fabric and petrophysical parameters, one
mustdefine and classify pore space as it exists today in terms of petrophysical
properties
Addition of vuggy pore space alters the manner in which the pore space is connected
Courtesy of Jerry Lucia
Figure 1
a :Rock Types 1 and 2 correspond to
intergranular grainstone
Limestone and dolostone
b :Rock Type 3 issucrosic dolostone with
intercrystalline porosity
c :Rock Types 4 and 5 correspond tomoldic limestone and dolostone
d :Rock Type 6 ismudstones and chalks
e :Rock Type 7 is thecombination of
matrix and moldic / vuggyporosity, ie vuggy
packstone / wackestone
f :Rock Type 8 isfracture / fissure
porosity
Cementation Exponents in ME Carbonate Reservoirs. J W Focke and D Munn, SPE Formation Evaluat ion, June 1987
Illustrative, generic thin sections
Porosity is black
Figure 2
ThemExponentinCarbonatePetrophysics
RE(Gene)Ballay,PhD
www.GeoNeurale.com
In1952ArchiestatedIndiscussingthepetrophysicsoflimestones,itisnecessarytofirst
classifytheminamannertoportrayasmuchaspossibletheessentialporecharacteristicsofa
reservoir.Theapplicationofpetrophysicalrelationshipsinlimestonescanbemuchmore
difficultthanforsandstonesbecauseoftheheterogeneity.Thisisduemainlytothevariation
ofporesizedistribution.
TheattributewithinArchiesequation
whichrepresentstheporesystem
tortuosityisthemexponent.
Complicationsincarbonateformation
evaluationcanariseduetomixed
mineralogys(whichaffectsthe
estimationoftotalporosity)and
wettability(Sweeny&Jennings)/surface
roughness(Diederix)whichdrivethe
nexponent,butfromoneperspective
themexponentcanbesaidtooften
representtheessenceofcarbonate
petrophysics.
Ascomparedtoclasticpetrophysics,
whichistypicallycompromisedbyclayconductivityissues,carbonatepetrophysicsissueswillin
manycasesrevolvearoundpropercharacterizationoftheporesystem,whichreflectsboth
depositionalanddiageneticproperties.JerryLuciaadvisesustofocusonpetrophysical
properties,notgenesis,andwethusrealizethatpetrophysicalzonesmaybegenuinely
differentthangeologicalzones:Figure1.
Whilethereareseveralcarbonateclassificationsystemsinthepublicdomain(Lucia,Lny,
etc),FockeandMunnsworkis
particularlycompleteinthatit
illustrateseachofthe
following:Figure2.
Poregeometryperthin
sections.
Laboratorym
measurements
correlatedwiththin
sectiondescriptions.
Wirelinemethodology
fordeterminingmin
thewellbore,across
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Interparticle vs moldic porosity and Sw
J W Focke and D Munn: Cementation Exponents in Middle Eastern Carbonate Reservoirs, SPE Formation Evaluation 2
(1987) : 155-167
See Also A M Borai: A New Correlation for the Cementation Factor in Low-Porosity Carbonates, SPE Formation Evaluation
2 (1987): 495-499
Schlumberger Technical Review, Volume 36 Number 3
Saturation Variations
0.00
0.20
0.40
0.60
0.80
1.00
1.5 1.7 1.9 2.1 2.3 2.5
Saturation Exponent
WaterSaturation
Boxing in the uncertainty for specific criteria
= 0.2, Rw@FT = 0.1 ohm-m, R = 50 ohm-mAt each value of 'n',a range (1.5 - 4.0) of 'm'values are displayed in steps of 0.25
m = 3.50
m = 3.00
m = 2.50
m = 2.00
m = 1.75
Pore Geometry Effects on Sw
Figure 3
Dots, ambient
pressure
Open circles,
reservoir pressure
1.9 < n < 2.1
Middle East Carbonate
Sw calculated with both constant and variable exponent
Variable (high m) exponent evaluationconsistent with water test in lower zone
Schlumberger Technical Review
Seeking the Saturation Solution M.
Watfa: Middle East Well Evaluation
Review No. 3 (1987)
Where m ~ 2.0 2.5,
calculations and test agree
Across the water test m
approaches 3.5
Failure to recognize this
yields Sw(Actual) ~Sw(m=2) / 0.25 or
Sw(Actual) ~ 4 * Sw(m=2)
Sw calculates goodbut Test is water!!
Figure 4
boththewaterlegandhydrocarboncolumn.
Comparisonofwirelinedeterminedmandlabdeterminedm.
FockeandMunnfindthat
thedifferencebetween
aninter
particle
and
vuggyporesystemcanbe
aslargeasanmof~2
versusanmof~4,
correspondingtoanSw
uncertaintyof~20%
versus~75%:Figure3.
Theconsequencescanbe
devastating:aninterval
ofhighresistivity
(apparentlypay)
may
be
nothingmorethan
relativelytortuous,
waterfilledporesystem.
Oneapproachtothis
uncertaintyistocombine
anonArchietool(suchasthedielectric)withaconventional(shallowreading,sincethe
dielectricisalsoshallow)resistivitymeasurement,andwiththismultiplicityofmeasurements
deducewhatthefootbyfootmisinthewellbore:Figure4.
Anotheroptionmightbe
topartition
the
pore
systemwithmoderntools
(Gomaaetal,
Ramamoorthy,etal,etc)
andtothen
independentlyestimate
thecorrespondingfoot
byfootmwithsome
kindofelectricalcircuit
model(Aguilera,Wang&
Lucia,etc).
Anattractionofa
mathematical
representationofthem
exponentis thatwhat
ifcalculationscanbe
done,toascertainthe
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Comparison of Empirical Models for Calculating the Vuggy Porosity and
Cementation Exponent of Carbonates from Log Responses. Fred P. Wang and F.
Jerry Lucia
In the graphic at right, cementation exponents calculated
by theArchie equation (thick black line - mres ) are
compared with those from SPI/Nugent, Nurmi
/Asquith, Lucia, modified Myers, and Dual Porosity.
Cementation Exponent Models vs Wireline m
m(Dual Porosity) ~ m(Archie)
Figure 5
0.00
0.10
0.20
0.30
0.40
0.50
0 0.1 0.2 0.3 0.4
RelativeUncertainty
Porosity
RelativeContribution ToSwUncertaintya
Rw
Phi
m
n
Rt
Below: Illustrative (Chen & Fang)Best
Estimate of each parameter, with
corresponding individual uncertainty,
and associated relative uncertainty on
Sw(Archie).
Right:Relative impact on Sw(Archie)
uncertainty of m & n,across a range
of porosity values, for afixed Phi
uncertainty.
Attribute Uncertainties Specified Individually
Light Green Cells require User Specification
Light Blue Cells are calculated resultsIndividual Best Relative Uncertainty
Attribute Uncertainty Estimate On Sw(Archie)
a 0.0% 1.00 0.0000
Rw 4.4% 0.02 0.0019
Phi 15.0% 0.20 0.0900
m 10.0% 2.00 0.1036
n 5.0% 2.00 0.0480
Rt 1.0% 40.00 0.0001
Sw 11%
Sw^n 1%
Sw^n=0.367 is an inflection point
After C. Chen and J. H. Fang.
Sensitivity Analysis of the Parameters in Archies Water Saturation Equation. The Log Analyst. Sept Oct 1986
Identifying theBiggest Bang for the Buck, in
Improved Sw Estimation
Figure 6
rangeofestimates
correspondingtoarange
ofinputuncertainties
(uncertaintyin,for
example,theporosity
partition).
Regardlessofwhat
approachisusedto
estimatethewellbore
m,duediligence
requiresthatatevery
opportunitytheresultbe
comparedagainstthe
inferredArchieestimate
inthewaterleg[where
Sw=1and
m
=
log(Rw/Ro)/Log(Phi)]:
Figure5.
Weshouldfurthermorenotlosesightofthefactthattheremaybeintervals,particularlyinthe
vicinityofthetransitionzone,acrosswhichtheArchiecalculationissimplynotvalid(Griffithset
al).
WheretoFocusAlthoughthemexponentmay(logically)bethefirstissuethatcomestomindincarbonate
evaluation,itdoesnotalwaysdominatetheuncertaintyinSwestimates.Therearetwobasic
waysofidentifyingwherethefocusshouldbe:Figure6.
Takethevarious
partialderivatives
ofArchies
equation,and
compare
magnitudesfor
locallyspecific
valuesand
uncertainties.
MonteCarlo
simulation.
Thetwooptions
complementoneanother
inthatthederivativesare
easytocodeintoa
spreadsheetor
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0.00
0.10
0.20
0.30
0.40
0.50
0 0.1 0.2 0.3 0.4
RelativeUncertainty
Porosity
RelativeContribution ToSwUncertaintya
Rw
Phi
m
n
Rt
At 20 pu, formation evaluation should focus on
improved porosity and m estimates, with n of
relatively less importance.
If porosity rises to 30 pu, however, improved
porosity estimates become more important with m
and n having similar, and less, impact.
As porosity drops to 10 pu, it is the pore
connectivity (m) that begins to dominate the
accuracy.
The Porosity Dependence
The relative importance of m and n depend not only upon their specific
uncertainty, but also upon the porosity of the interval in question; there is a link
Figure 7
After C. Chen and J. H. Fang.
Sensitivity Analysis of the Parameters in Archies Water Saturation Equation. The Log Analyst. Sept Oct 1986
0.00
0.10
0.20
0.30
0.40
0.50
0 0.1 0.2 0.3 0.4
RelativeUncertainty
Porosity
RelativeContribution ToSwUncertaintya
Rw
Phi
m
n
Rt
Light Green Cells require User Specification
Light Blue Cells are calculated results
Individual Best Relative Un
Attribute Uncertainty Estimate On Sw(Arc
a 0.0% 1.00 0.0000Rw 4.4% 0.02 0.0019
Phi 15.0% 0.20 0.0900
m 10.0% 2.00 0.1036
n 5.0% 2.00 0.0480
Rt 1.0% 40.00 0.0001
Sw 11%
Sw^n 1%
0.367 is a logarithmic inflection point
0.00
0.10
0.20
0.30
0.40
0.50
0 0.1 0.2 0.3 0.4
RelativeUncertain
ty
Porosity
RelativeContribution ToSwUncertaintya
Rw
Phi
m
n
RtLight Green Cells require User Specification
Light Blue Cells are calculated results
Individual Best Relative Un
Attribute Uncertainty Estimate On Sw(Arc
a 0.0% 1.00 0.0000
Rw 4.4% 0.20 0.0019
Phi 15.0% 0.20 0.0900
m 10.0% 2.00 0.1036
n 5.0% 2.00 0.0108
Rt 1.0% 40.00 0.0001
Sw 35%
Sw^n 13%
0.367 is a logarithmic inflection point
If the water were fresher, sayRw = 0.2
instead of 0.02, n diminishes in
importance as compared to both the
amount of porosity, and its connectivity
(m).
After C. Chen and J. H.
Fang. Sensitivity Analysis
of the Parameters in
Archies Water Saturation
Equation. The Log Analyst.
Sept Oct 1986
The Rw Dependence
The Rw Dependence
Figure 8
petrophysicals/wpackage(forfootbyfootdisplay),whiletheMonteCarlogivesinsightinto
theup anddownsides(with95%confidence,theuncertaintywillbelessthanhighlow
calculations).
UsingChenandFangsparameterstoillustratethedifferentialapproach(Figure6)wenote
thatasporosityvaries,therelativeimportanceofmandnalsovaries.Thatis,the
importanceof
asingle
attribute,
m
for
example,
is
linked
to
the
magnitude
of
other
attributes.
Inthecaseathand,a20
puformationevaluation
shouldfocuson
improvedporosityand
mestimates,withnof
relativelyless
importance.Asporosity
dropsto10pu,thepore
connectivity(m)
begins
todominatethe
accuracy:Figure7.
Thementalimagethat
emergesisthatasporositydecreases,itsconnectivity(orefficiency)becomesincreasingly
important.
Weretheformationwaterresistivitytochange,itsquitepossiblethatthefocusshouldalso
change:Figure8.
Insummary,each
situationshould
be
evaluatedwithitslocally
specificparametersand
uncertainties,andour
focus(andbudget)applied
accordingly.
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Generalized slope-intercept straight line equation
y = m * x + b
Log(FF) = - m * Log( ) + b
At = 100 % , Log () = Log(1) = 0LeavingLog(FF) = Log(1) = 0 b = 0
to give Log(FF) = - m * Log( )
At = 10 % , FF = 100
Log(100) = 2.0 = - m * Log (0.1) = m
to givem = 2.0
1
Schlumberger Technical Review, Volume 36 Number 3
G E Archie: The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics. Petroleum Transactions
of the AIME 146 (1942): 54-62.
The Key to working with the Log-Log displays is to
think in terms of decadesand take logarithms
when working with numerical values
Figure 9
mandCarbonatePoreSystemsArchiemeasuredtheporosity,permeabilityandresistivityofbrinesaturatedcarbonateand
nonshalysandsamples,acrossarangeofbrinesalinities,toobservealinearrelation
betweenRo(brinesaturatedsampleresistivity)andRw(brineresistivity).
ThisresistivityratioisknownastheFormationFactor,where
FF=R(sample)/R(brine)=1/m
Archiecommentedthatmwasabout1.3inunconsolidatedrockandincreasedasthe
cementationincreased.Atypicalformationstartingpointvalueformis2.0.
ItwasHubertGuyodwhogaveusthetermcementationexponent,andwhoincidentally,also
suggestedtheoriginalnameoftheSPWLAJournal(TheLogAnalyst).
Themexponenttypicallyinvolvesaloglogdisplayofthebasicdata,andassuchisless
intuitivethanasimplelineardisplay.Semilogandloglogdisplaysarecommoninmany
petrophysicalendeavors(phiperm,etc),however,anditsusefultobeabletodrawourown
linethrough
the
data
and
deduce
the
corresponding
exponent.
The
Key
to
working
with
the
LogLogdisplaysisto
thinkintermsof
decadesandtake
logarithmswhen
workingwith
numericalvalues.
Sketchingourline
throughArchies1942
dataanddrawingupon
theslope
intercept
formulationofalinear
relation,revealsthat
m~2.0isindeeda
reasonablestarting
pointwiththatdata
set:Figure9.
WhileitwasMrGuyod
whogaveusthe
terminology
cementationexponent,
it
appears
that
we
derive
the
m
nomenclature
from
the
mathematiciansslopeinterceptrepresentationofastraightline,whereintheslopeisdenoted
bym:y=m*x+b.
Lookingaheadjustabit,totheResistivityIndex(incontrasttotheFormationFactor),itwould
furtherseemthatournnomenclaturemayhavearisenalphabetically,sincenfollowsm.
Interestinglyfromahistoricalperspective,thiskindofrelationshiphadbeenpostulatedearlier
bySundberg,butwithoutthesupportingdata.
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Archies 1947 Data - Sandstone and Limestone
G E Archie: Electrical Resistivity as an Aid in Core Analysis Interpretation, AAPG Bulletin 31 (1947): 350-366
Schlumberger Technical Review, Volume 36 Number 3
Suggested to Archie thatpermeability (molecular fluid
flow) and resistivity (ionic movement) were different
Formation Factor (Rformation / Rbrine )
Figure 10 Conceptually,giventhat
bothrepresentaflow,
onemightexpecta
strongerrelationship
betweenresistivityand
permeability,as
comparedtoresistivity
andporosity.Archie
addressedthisby
plottinghis
measurementsboth
ways:Figure10.
Perhapssurprisingly,the
variousFormation
Factormeasurements
convergemuch
better
whendisplayedagainst
porosity,thanagainst
permeability,promptingArchietosuggestthatmolecularfluidflow(permeability)andionic
motion(current)weredifferent.ThosewhohaveworkedbothIG/IXandchalksystemsare
alreadyawareofthis,havingnoticedthatthemforthetwoverydifferentpermeabilitiescan
bothbeabout2.
Verweretalhaveperformedadigitalimageanalysisofthinsectionsrepresentingcarbonate
plugsuponwhichresistivitymeasurementsweremade.Threeattributeswererecognizedas
playinganimportantrole.
Perimeterover
area:
twodimensional
equivalent
to
the
pore
surface
/pore
volume
ratio.
Dominantporesize:theupperboundaryofporesizeswithwhich50%oftheporosityon
thethinsectioniscomposed.
Microporosity:calculatedasthedifferencebetweentheobservedmacroporosityinDIA
andthemeasuredporosityfromtheplugsample.
Theirmeasurements,nicelyillustratedwithaccompanyingthinsections,demonstratethatin
additiontoourintuitiveexpectations,
sampleswith
high
resistivity
can
have
high
permeability,
sampleswithlowpermeabilitycanhavea(relatively)lowmexponent.
AsArchiepointedoutin1952,carbonateporesystemsmaybequitevariable,causinganm=
2calculationtobenonrepresentative.WyllieandGregoryboundedthemexponentfora
rangeofbeadpacksconsistingofunconsolidatedandcementedspheres,viachemicalflushes,
andfoundtherelationbetweenformationfactorandporositycouldinvolveanadditional
parameter,C,whichdifferedfromunity.
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Wyllie and Gregory constructedbeadpacks with varying degrees of
cementation
The baseline represents a formation of
unconsolidated spheres
This dataprompted them to propose the
relation
FF = C / m
C was aformation dependent constant
m ~ 1 for unconsolidated spheres
m ~ 4 for cemented bead pack
M R Wyllie and A R Gregory: Formation Factors of
Unconsolidated Porous Media: Influence of Particle Shape
and Effect of Cementation, Petroleum Trans of the AIME
(1953): 103-110.
Schlumberger Technical Review, Volume 36 Number 3
Figure 11Archie commented that m was about 1.3 in
unconsolidated sandandbecame higher as
cementation increased
Cementation Exponents in Middle Eastern Carbonate Reservoir.
J W Focke and D Munn, SPE Form Evaluation, June 1987
Geological descriptions of hundreds of
samples reveal asystematic relation between
Rock Type and Archies m exponent
Rock with a more tortuous and/or poorly
interconnected porosity (moldic) display well-
defined trends of increasing m with
increasing porosity
The additionalmoldic porosity is less
effective at electrical conduction
In some rock m is found to rise from 2
@ 5 pu, to 5.4 @ 35 pu
m variations, within a specific Rock Type,
can bereduced by segregating the samples of a
specific Rock Type into Permeability Classes
Figure 12
Rock Type 4, themoldic limestone
grainstone, represents a diagenetic inversion
whereby the original porosity (between the
grains) was filled with cement, and the
original grains were dissolved to form the
current porosity
Symbols refer to different wells
FF=R(sample)/R(brine)=C/m
Cisformationdependent,andtherangeofmexponentswasfoundtobem~1for
unconsolidatedbeads
m~4forcemented
beads:Figure11.
FockeandMunnthen
conductedadetailed
studyofm
measurementson
actualcarbonate
samples,supported
withthinsection
descriptionstofind
systematicrelations
betweentheporosity
typeand
m.
Whiletheintergranular
/intercrystallinepore
systemhadm~2(so
longas>5pu),
consistentwithArchies
earlierlabwork,vuggyporesystemscouldexhibitanmthatreached5:Figure12.
TheirRockType4,representingadiageneticinversioninwhichOriginalPorosityRockand
OriginalRock
Porosityis
an
example
ofwhereanincreasein
porositycorrespondsto
anincreaseinvuggy
porositycontent.
Becausethevuggy
porosityisless
efficientatelectrical
conduction,them
exponent(counter
intuitively)risesas
porosityincreases:
Figure13.
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Rock Type 4, the moldic limestone
grainstone, represents adiagenetic
inversion whereby theoriginal porosity
(between the grains) wasfilled with
cement, and theoriginal grains were
dissolved to form the current porosity
Perm Class 1:Perm < 0.1 md
Perm Class 2: 0.1 md < Perm < 1 md
Perm Class 3: 1.0 md < Perm < 100 md
Perm Class 4: 100 md < Perm
Moldic Rock
1
2
3
4
5
0 5 10 15 20 25 30 35Porosity
CementExp
onent
RT4/PC1
RT4/PC2
RT4/PC3
RT4/PC4
m vs Porosity
Rock Type 4: Classes 1, 2, 3, 4
Figure 13
Cementation Exponents in Middle Eastern Carbonate Reservoir. J W Focke and D Munn, SPE Form Evaluation, June 1987
Fracture porosity has an effect opposite to vuggy porosity
Provides a conductive conduit => lowers m
m can be estimated with Charts
R Aguilera: Analysis of Naturally
Fractured Reservoirs from Sonic and
Resistivity Logs, SPE 4398, Journal of
Petroleum Technology 26 (1974)
R Aguilera: Analysis of Naturally
Fractured Reservoirs from ConventionalWell Logs, SPE 5342, Journal of
Petroleum Technology 28 (1976)
Schlumberger Technical Review, Volume
36 Number 3
Illustrative calculation
is the fraction of porositythat is fracture
Phi(Total) ~ 20 pu
~ 50 %m ~ 1.35
Figure 14
Althoughextreme,
diageneticinversionis
reportedinotherstudies,
forexampleEberlietal,
whoillustrateitwiththin
sectionsand
describe
the
processastheoriginal
grainsaredissolvedto
produceporesasthe
originalporespaceis
filledwithcementto
formtherock
FockeandMunnnext
noticedthatthe
correlationbetweenm
andporosity
could
be
improvedbybreakingthatRockTypeintopermeabilityclasses,eachofwhichcorrespondedto
diageneticallyinvertedrock,butwithdifferentpermeabilities.Weshallreturntothedifferent
mvstrendsassociatedwiththedifferentpermeabilityclasses,withinthecontextofa
digitalmexponentmodel,shortly.
Fractureporosityhasaneffectoppositetovuggyporosity,conceptuallyformingakindof
shortcircuit,whichisrepresentedmathematicallybyadecreaseinvalue:Figure14.
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TheDual Porosity Cementation Exponent model follows from a simple two
component (intergranular and vuggy porosity) electrical model
1/R(equivalent) = 1/R1 + 1/R2 = C0 = C1 + C2
Each component satisfies Archies relation between resistivity and formation factor
Resistivity = Rw * FF Conductivity = Cw / FF
Formation Factor is related to porosity as FF = a / m
Thetwo component parallel circuit equation is
then
C0 = Cw [1/FF1 + 1/FF2]
C0 = Cw [ (1)m(1)/ a(1) + (2)m(2)/ a(2)]Now take a(1) as 1.0, m(2) as 1 and allow a(2) to
vary from 1 to infinity [av(2) represents the
tortuosity of the vuggy partition]
intergranular vuggy
Figure 15
Comparison of Empirical Models for Calculating the Vuggy Porosity and
Cementation Exponent of Carbonates from Log Responses. Fred P. Wang
and F. Jerry Lucia
Type Iformula:Assuming that mv is 1 and that av varies from 1 to infinity, we can
write the parallel circuit equation as below
The deduction of the net effective m from the conductivity equation follows from
(relative to the simple Archie relation)
Ro = Rw / ( m) => Co = Cw * ( m)
Take logarithm of Co - Cw equation
log(Co) = log(Cw) + m log()
Following exhibit
Details
following
exhibit
Figure 16
Comparison of Empirical Models for Calculating the Vuggy Porosity and Cementation Exponent of Carbonates from Log
Responses. Fred P. Wang and F. Jerry Lucia
DigitalmExponentModel
WithintheLuciasystem,
vuggyporosityis
everythingthatisnot
interparticle,and
includesboth
touching
vugs(fractures,etc)
andseparatevugs
(moldic,intraparticle,
etc).Whilethereare
chartsavailableto
estimatem,asa
functionofvuggy
porositycontent,forthe
variousscenarios,it
would
be
advantageous
tohaveadigitalmodel.
Thereareseveral
digitalmodelsavailable,andhereweuseWang&Luciaforillustrativepurposes,because:
Itisbaseduponasimplecircuitmodelthatiseasytofollowforthosenotparticularly
comfortablewithelectricalcircuittheory,
Itallowsforbothtouchingandseparatevugs,andeverythinginbetween,
WangandLuciaincludedindependentQCchecksontheviabilityoftheirmodel,
WangandLuciarepresent
vuggyrockasatwocomponent,parallelcircuit:
Figure15.
Eachcomponent
(independently)satisfies
Archiesequation,andthey
thensimplifytheresulting
netconductivity
expressionbysettingthe
mofthevuggyfractionto
be1andrepresentingthetortuosityofthevugs(be
theytouching,separateor
somethinginbetween)
withaparameterav:Figure
16.
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Type Iformula:Assuming that mv is 1 and that av varies from 1 to infinity, we can write
the parallel circuit equation as below (details in exhibits following)
The deduction of m from the conductivity equation follows from (relative to the
simple Archie relation)
Ro = Rw / ( m) => Co = Cw * ( m)
log(Co) = log(Cw) + m log()
m log() = log(Co) - log(Cw) = log(Co / Cw) = log [ ]
m = log(Co / Cw) / log() = log [ ] / log()
Figure 17
Comparison of Empirical Models for Calculating the Vuggy Porosity and Cementation Exponent of Carbonates from Log
Responses. Fred P. Wang and F. Jerry Lucia
This indicates
that equation 30
can be used to
model reservoirs
with separate
vugs, touching
vugs, and
fractures using
appropriate
values of av.
av representsthe connectivity, or lack thereof, of the vuggy portion of the pore syste
Figure 18
Thisleavesthemwithan
expressionforthenet
cementationexponent
asafunctionofthetotal
porosity,theporosity
partition,the
exponent
oftheinterparticle
fraction(mip)andthe
connectivityofthe
vuggyfraction(av):
Figure17.
Withthisdigitalmodel
inhand,duediligence
requiresthatitbe
tested,withonesuch
testbeing
acomparison
ofthecalculatedmto
thatinferredfrom
Archiesequationinthewaterleg,Figure5,wheretheyfindthatanappropriatechoiceofav
resultsinamatch.
Whilethereisatendencytothinkofnonfracturevugsasseparate,withoutcontributionto
permeability,aliteraturesearchwillrevealinstancesofthepermeabilityactuallyincreasing,
withthepresenceofvuggycontent.Thisbehaviorbringsforwardtheimportanceofparameter
av,whichisavailabletorepresentsuchasituation.
WangandLuciaalsotestedtheirmodelinsuchasituation,drawinguponMeyersdata,where
theyagain
found
that
a
locallyappropriatechoiceof
avbroughtmeasurements
andestimatesinto
agreement:Figure18.
Theattractionofadigital
modelisthatitallowswhat
ifcalculations.Suppose,for
example,avisual
examinationofcore
indicatesthat
about
1pu
of
vuggyporosityispresent,
dispersedamongstthe
matrixporosity.Ifthatvuggy
porosityisnotwell
connected(ieav>>1)and
thetotalporosityis~5puor
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So long as ~ 5 pu and greater,only at av ~ 1
does a small ( v ~ 1 pu) cause the CementationExponent to differ from mip = 2.0
Dual-porosity Model / What If Characterizations
0
1
2
3
4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
CementExponent
Total Porosity [ Phi(v)=0.01 ]
Dual Porosity / Type 1
av=1
av=10
av=100
av=1000
ip + v = 2 pu
0
1
2
3
4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
CementExponent
Total Porosity [ Phi(v)=0.05 ]
Dual Porosity / Type 1
av=1
av=10
av=100
av=1000
ip + v = 6 puSo long as ~ 10 pu (or more)and the vugsare not present as fractures (or connected vugs),
an m of ~ 2.5 would be a reasonable starting
point.
Figure 19
Monte Carlo Dual-porosity Model
What If Characterizations
TheMonte Carlo method relies on repeated
random sampling to model results
Advantages of Monte Carlo include
any type of distribution can be used tocharacterize the uncertainty distribution of
any parameter
normal, log normal, etc
insight is gained into the upside and
downside
0
100
200
300
400
500
600
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00
Frequency
DualPorosity"m"
MonteCarloDistribution
"m"DualPhi
0
1
2
3
4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
CementExpone
nt
Total Porosi ty [ Phi(v)=0.05 ]
Dual Porosity / Type 1
av=1
av=10
av=100
av=1000
Figure 20
more,thedualporositymodel
wouldindicatethataneffective
mofabout2isareasonable
startingpoint:Figure19.
If,however,that1puispresent
inthe
form
of
well
connected
vugs(fractures),thenav~1and
theeffectivembecomesmuch
moresensitivetotheporosity
partitionandconnectivity.
Nextconsideranintervalwith
about5puofvuggyporosity
(Figure19,again),withamatrix
cementationexponentof2.So
longasthetotalporosityis10pu
orgreater,
areasonable
initial
valueoftheeffectivemwouldbeabout2.5,solongasthevuggyportionisnotwell
connected.
Buildinguponthisconcept,werealizethatifitispossibletopartitionporosity,footbyfootin
thewellbore,thenonemayalsocalculatemfootbyfootandusethatexponentvalueto
thenestimateSw.Insuchasituation,itwouldbeadvisabletoseekoutawaterintervalatthe
firstopportunityandcomparetheresultingSw(whichishopefullycloseto100%)withthat
deducedfromaninversionofArchie(asWang&Luciadid,inFigure5),forvalidationpurposes.
Ifthecalculateddualporosity
exponentis
afair
representation
oftheinvertedArchiem,then
theevaluationofvuggyintervals
hingesupontheaccuracyofthe
porositypartition,whichmayin
factbeahurdle.Limitationsthat
willariseare,
theimagelogwillbe
compromisedifthevug
sizeislessthanthebutton
resolution,about
3
5mm,
incarbonatestheNMRT2
willlosetheporesize
relationatporebodysizes
ofabout50100um.
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Matrix has a relatively uniform amount of
interparticle porosity (the background), with
variable amounts of vuggy porosity, with
different degrees of connectivity
mip = 2.0, ip & avconstant across range of v
Moldic Rock
1
2
3
4
5
0 5 10 15 20 25 30 35Porosity
CementExponent
RT4/PC1
RT4/PC2
RT4/PC3
RT4/PC4
Focke & Munns Data
Rock Type 4, Perm Classes 1 => 4, amoldic
limestone grainstone: representing a
diagenetic inversion whereby theoriginal
porosity (between the grains) wasfilled with
cement, and theoriginal grains were
dissolved to form the current porosity
Dual Porosity / Type 1
1
2
3
4
5
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Total Poros ity [ Phi(ip)=0.01 ]
Ceme
ntExponent
av=1
av=10
av=100
av=1000
ip + v = 2 pu
TheDual Porosity Modelhas captured
the essential trends of the independent
Focke & Munn data.
The Dual Porosity Model vs Lab Data
Figure 21 Thereisyetanother
advantageofexpressing
thevuggyporesystem
exponentin
mathematicalform:it
becomespossible
to
performaMonteCarlo
simulation:Figure20.
Nowweareableto
accountforuncertainty
inthevariousinputs(the
porositypartition,asan
example),and
characterizetheBest
Estimateintermsof
probabilities.That
is,
ratherthangowitha
high/lowestimate,one
canidentify(for
example)95%oftheMCdistribution.Inmanycaseswewillfindthat,ifwecanaccepttwo
standarddeviationsofuncertainty(95%),thelikelyhigh/lowvaluesarenotsoextremeasthe
possiblehigh/low.Thisisbecause,ingeneral,itisunlikelythatallthehigh(orlow)input
valueswilloccursimultaneously.
Theprecedingillustrationsrepresentafixedvuggyportioninthepresenceofmoreorless
matrixporosity.Itisalsopossiblethatmostoftheporosityisintheformofvugs,withsome
smallamount
of
(background)
matrix
porosity:
Figure
21.
Thatis,insteadofaconstant1pu(or5pu)ofvuggyporosity,asthetotalporositygoesupand
down(Figure19),onemighthaveasmallamountofinterparticleporosity,withmoreorless
vugspresent.Thedualporositymodelexponenttrendsarenowverydifferent,becausethe
bulkoftheporosityisnotefficient.
Solongasthevugsarenotconnected,Figure21revealsatrendofincreasingm,with
increasingporosity.Asavvariesfromslightlyconnected(av~10)toseparate(av~1000),the
differenttrendsseparate,andexhibitapatternverysimilartowhatFockeandMunnfound
experimentally.
Fockeand
Munns
Rock
Type
4(diagenetically
inverted),
with
different
permeability
classes,
can
berepresentedwiththedualporositymodel.Atasimplelevel,theavvaluescorrespondtothe
chokingoffoftheporethroats,whichbothreducespermeabilityandincreasesm.
Itisimportanttorealizethat,conceptually,someporesystemsmaynotsatisfytheparallel
circuitassumptionandthatadditionallyinsomereservoirstheporesystemisatripleporosity
system,notdual.Insuchasituation,thisparallelcircuitdualporositymodelisnotgoingtobe
sufficient.
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Cementation Exponents in Middle Eastern Carbonate Reservoirs
J W Focke and D Munn, SPE Formation Evaluation, June 1987
m(EPT) &m(core),foot by foot, with the
two m estimates compared incrossplot in
later graphic
Immobile oil in black, movable oil shaded,
both as a fraction of porosity, calculated with
Schlumbergers Global package
Moldic
IG / IX
Upper Interval
m from logs,
foot-by-foot,
compared to core
Figure 22
mfromnonArchieTechniques
Iftheporesystemcanbepartitioned,amathematicalmodelofthemexponentwillallow
onetothenevaluatethehydrocarboncolumnwithvariablem.Analternativeapproachisto
includeanadditionaltoolinthesuite,whichwhencombinedwitharesistivitymeasurement,
allowsonetodeducethelocalm,andtothenusethatmintheinterpretationofthedeep
resistivity.
FockeandMunncombinedthedielectriclogwithanRxomeasurementinjustthisway,to
calculatemfootbyfootinthehydrocarboncolumn,
comparethewellboremestimateswithlabm.
Althoughtheconceptis
straightforward,Focke
andMunnwerecareful
topointoutkeyissues
thatmustbekeptin
mind.
Theevaluation
requiresthatthe
dielectricbe
pairedwith
routineporosity
andRxotools,
whichhavea
greaterdepthof
investigation
andless
vertical
resolution.They
arethen
reflectinga
largervolumeofreservoirthanisthedielectric.
Rxomayalsobereflectingamixofmudfiltrateandformationbrine.Ifso,aneffective
value Rmfe shouldbeused.
Archiessaturationexponentisassumedtobe2.0andcarbonatescanassumenon
waterwetvalueswithn>2.0.
Intheir
illustrative
well,
two
pore
systems
are
recognized,
moldic
and
IG/IX:
Figure
22.
Asexpected,themoldicporosityexhibitsanincreasedm,andreaches~3.5inplaces,as
seenwithbothlabdataandthedielectricbasedestimate.
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Wireline m(EPT) vsLaboratory m(core)are in general agreement
m and from corem from logs vs from logs
Cementation Exponents in Middle Eastern Carbonate Reservoirs J W Focke & D Munn, SPE Form Evaluation, June 1987
Figure 23
Similar Trends
Labandwirelinecanalso
becomparedviacrossplot.
Recognizingthedifferent
volumesofinvestigation,
andattemptingto
compensate,Focke
and
Munncrossplotm
versusporosityforcore,
andforlog:Figure23.
Inthemoldicintervals,the
tworesultsdisplayan
almostidenticaltrendof
increasingmas(moldic)
porosityincreases.
RasmusandKenyon
provideyet
another
illustrationofthecontributionthatadielectriclogcanmakeinthepresenceofoomoldic
porosity.
InthetimesinceFockeandRasmus,significantadvanceshavebeenmadeindielectric
measurements,andtheynowoffermoresophisticatedoptions(ZappingRocks.Romulo
Carmona,etal.OilfieldReview.Spring2011.).
SummaryIn1952Archiepointedoutthecomplicationsthatarisewithcarbonateporesystems,whichcan
resultintortuouswaterfilledporesystemshavingaresistivitysimilartoaninterparticlepore
systemthat
is
hydrocarbon
charged.
In
one
sense
then,
and
recognizing
that
other
complicationscanbepresent,thecementationexponent(whichrepresentstheporesystem
tortuosity)representstheessenceofcarbonatepetrophysics.
WyllieandGregoryboundedthemwithlaboratorybeadpackstudies,findingthatinapack
ofunconsolidatedbeadsm~1,whilem~4inachemicallycementedpack.
FockeandMunninterpretedhundredsofcarbonateformationfactormeasurements,within
thecontextofthinsectiondescriptions,tofindasystematicrelationbetweenRockTypeand
m.
Inthecaseofmoldicporosity,FockeandMunnfoundthatmsystematicallyincreasesas
total
porosity
increases:
the
additional
porosity
is
simply
not
effective
in
the
electrical
conductionsense.
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Petrophysical Characterization of Permian Shallow-Water
Dolostone. M H Holtz, R. P. Major. SPE 75214, 2002
http://www.beg.utexas.edu/mainweb/presentations/2002_presentations/holtz_spe0402ab.pdf
Pore geometries controlthe interrelationship ofpetrophysical properties.
The three most important pore-geometry characteristics are
amount andtypes of pores or shape
interconnectedness of pores (tortuosity)
size of interconnectingpore throats
Various m exponentsas a
function of vuggy porosity ratio
Note the Myers trend is relatively
flat: the vugs in this data set exhibit
touching behavior.
As the separate vug
fraction increases, so too
does the Archie m
Figure 23
Ingeneral,thepresence
ofvuggyporosity
correspondstoan
increaseinm,withthe
magnitudeofthat
increasediffering
from
onedatasettothenext:
Figure23.
Therearebothcharts
andmathematical
modelsthatallowoneto
estimatemforvarious
porositypartitions.The
advantageofthe
mathematicalmodelis
thatthey
easily
allow
bothfootbyfoot
estimatesandMonteCarlosimulation.
Chartsandmodelsshouldalwaysbetested,atthefirstopportunity,bycomparingthem
estimatetothatdeducedbyinvertingArchiesequationinthewaterlegofarepresentative
well.InthecaseoftheDualPorositymodelusedforillustrationsherein,thatcomparisonis
reasonable.TheDualPorositymmodelisalsofoundtomatchtheindependentlab
measurementsofMeyers,andtocapturethepatternsreportedbyFockeandMunn.
AwellborealternativetommodelsistoaddanonArchietooltothetoolsuite,whichallows
aneffectivemtobededuced.Thatmisthenusedtointerpretthedeepresistivity
measurement.
Regardlessoftheapproachused,weshouldbearinmindthatparticularlyinthevicinityofthe
transitionzone,thereservoirmayfailtosatisfythebasicArchiecriteria.
Acknowledgement
MohamedWatfasArchiesLaw:ElectricalConductioninClean,WaterbearingRockisan
importantsinglepointhistoricaloverviewsource.FockeandMunnsCementationExponentsin
MiddleEastCarbonateReservoirsliterallysetthestandardforasystematicinvestigationof
theissue,backin1987.TothesemustreadarticlesIhaveaddedmorerecentmaterial,and
mypersonalthoughts/techniques.
Thisone
is
for
my
Mother,
who
is
always
there,
through
thick
and
thin.
ReferencesAguilera,R.AnalysisofNaturallyFracturedReservoirsfromSonicandResistivityLogs,SPE4398,
JournalofPetroleumTechnology26(1974):12331238
Aguilera,R.AnalysisofNaturallyFracturedReservoirsfromConventionalWellLogs,SPE5342,
JournalofPetroleumTechnology28(1976):764772
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AlGhamdi,AliandBoChen,HamidBehmanesh,FarhadQanbari&RobertoAguilera.An
ImprovedTriplePorosityModelforEvaluationofNaturallyFracturedReservoirs.Trinidadand
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Characteristics.PetroleumTransactionsoftheAIME146(1942):5462.
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(1947):350366
Archie,G.E.ClassificationofCarbonateReservoirRocksandPetrophysicalConsiderations.
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Ballay,GeneandRoyCox.FormationEvaluation:CarbonatevsSandstones.2005.
www.GeoNeurale.Com.
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Ballay,Gene.RollingtheDice.July2009.www.GeoNeurale.com
Ballay,Gene.
Coffee
Or
Tea.
October
2009.
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Ballay,Gene.SplitPersonality.December2009.www.GeoNeurale.com.
Ballay,Gene.StatisticsArePliable.February2010.www.GeoNeurale.com.
Ballay,Gene.TheBiggestBangfortheBuck.April2011.www.GeoNeurale.com
Carmona,Romuloetal.ZappingRocks.OilfieldReview.Spring2011
Chen,CandJ.H.Fang.SensitivityAnalysisoftheParametersinArchiesWaterSaturation
Equation.TheLogAnalyst.SeptOct1986
Diederix,K.M.AnomalousRelationshipsBetweenResistivityIndexandWaterSaturationsinthe
RotliegendSandstone(TheNetherlands),TransactionsoftheSPWLA23rdAnnualLoggingSymposium,CorpusChristi,Texas,July69,1982,PaperX
Eberli,GregorP.andGregorTBaechle,FlavioSAnselmetti&MichalLIncze.Factorscontrolling
elasticpropertiesincarbonatesedimentsandrocks.THELEADINGEDGEJULY2003
Focke,J.W.andDMunn.CementationExponentsinMECarbonateReservoirs.SPEFormation
Evaluation,June1987
Gomaa,N.andA.AlAlyak,D.Ouzzane,O.Saif,M.Okuyiga,D.Allen,D.Rose,R.Ramamoorthy
&E.Bize.CasestudyofPermeability,VugQuantificationandRockTypinginaComplex
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Hartmann,DanandEdwardBeaumont.PredictingReservoirSystemQualityandPerformance.
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Overview
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&
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Chattanooga shale
Mississippian limestoneR. E. (Gene) Ballays35 years in petrophysics include
research and operations assignments in Houston
(Shell Research), Texas; Anchorage (ARCO), Alaska;
Dallas (Arco Research), Texas; Jakarta (Huffco),
Indonesia; Bakersfield (ARCO), California; and
Dhahran, Saudi Arabia. His carbonate experience
ranges from individual Niagaran reefs in Michigan to
the Lisburne in Alaska to Ghawar, Saudi Arabia (the
largest oilfield in the world).
He holds aPhD in Theoretical Physics withdouble minors in Electrical Engineering
& Mathematics, hastaught physics in two universities,mentored Nationals in
Indonesia and Saudi Arabia, publishednumerous technical articles and been
designatedco-inventor on both American and European patents.
At retirement from the Saudi Arabian Oil Company he was the senior technical
petrophysicist in the Reservoir Description Division and had represented petrophysics
in three multi-discipline teams bringing on-line three (one clastic, two carbonate) multi-
billion barrel increments. Subsequent to retirement from Saudi Aramco he establishedRobert E Ballay LLC, which provides physics - petrophysics training & consulting.
He served in the U.S. Army as a Microwave Repairman and in the U.S. Navy as an
Electronics Technician; he is a USPA Parachutist, a PADI Nitrox certified Dive Master
and a Life Member of Disabled American Veterans.
Watfa,M.etal.ArchiesLaw:ElectricalConductioninClean,WaterbearingRock.Technical
Review,Volume36Number3
Watfa,M.SeekingtheSaturationSolution.MiddleEastWellEvaluationReviewNo.3(1987)
Wyllie,M.R.andARGregory:FormationFactorsofUnconsolidatedPorousMedia:Influenceof
ParticleShape
and
Effect
of
Cementation,
Petroleum
Trans
of
the
AIME
(1953):
103
110.
Biography
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