m exponent

WWW.GeoNeurale.Com January2012 2012RobertEBallay,LLC

The essence of carbonate petrophysics is (often) pore system heterogeneity, as compared to clay conductivity issues for clastics

The foundation of the Lucia petrophysical classification is the concept that pore-size distribution controls permeability and saturation

The focus 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 must define 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 dolostoneb : Rock Type 3 is sucrosic dolostone with intercrystalline porosityc : Rock Types 4 and 5 correspond to moldic limestone and dolostoned : Rock Type 6 is mudstones and chalkse : Rock Type 7 is the combination of matrix and moldic / vuggy porosity, ie vuggy packstone / wackestone f : Rock Type 8 is fracture / fissure porosity

Cementation Exponents in ME Carbonate Reservoirs. J W Focke and D Munn, SPE Formation Evaluation, June 1987

Illustrative, generic thin sections

Porosity is black

Figure 2

ThemExponentinCarbonatePetrophysicsRE(Gene)Ballay,PhDwww.GeoNeurale.com

In1952ArchiestatedIndiscussingthepetrophysicsoflimestones,itisnecessarytofirstclassifytheminamannertoportrayasmuchaspossibletheessentialporecharacteristicsofareservoir.Theapplicationofpetrophysicalrelationshipsinlimestonescanbemuchmoredifficultthanforsandstonesbecauseoftheheterogeneity.Thisisduemainlytothevariationofporesizedistribution.

TheattributewithinArchiesequationwhichrepresentstheporesystemtortuosityisthemexponent.Complicationsincarbonateformationevaluationcanariseduetomixedmineralogys(whichaffectstheestimationoftotalporosity)andwettability(Sweeny&Jennings)/surfaceroughness(Diederix)whichdrivethenexponent,butfromoneperspectivethemexponentcanbesaidtooftenrepresenttheessenceofcarbonatepetrophysics.

Ascomparedtoclasticpetrophysics,whichistypicallycompromisedbyclayconductivityissues,carbonatepetrophysicsissueswillinmanycasesrevolvearoundpropercharacterizationoftheporesystem,whichreflectsbothdepositionalanddiageneticproperties.JerryLuciaadvisesustofocusonpetrophysicalproperties,notgenesis,andwethusrealizethatpetrophysicalzonesmaybegenuinelydifferentthangeologicalzones:Figure1.

Whilethereareseveralcarbonateclassificationsystemsinthepublicdomain(Lucia,Lny,etc),FockeandMunnsworkisparticularlycompleteinthatitillustrateseachofthefollowing:Figure2.

Poregeometryperthinsections.

Laboratorymmeasurementscorrelatedwiththinsectiondescriptions.

Wirelinemethodologyfordeterminingminthewellbore,across


Interparticle vs moldic porosity and Sw

J W Focke and D Munn: Cementation Exponents in Middle Eastern Carbonate Reservoirs, SPE Formation Evaluation 2 (1987) : 155-167See Also A M Borai: A New Correlation for the Cementation Factor in Low-Porosity Carbonates, SPE Formation Evaluation 2 (1987): 495-499Schlumberger Technical Review, Volume 36 Number 3

Saturation Variations

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Saturation Exponent

Wat

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Boxing in the uncertainty for specific criteria

= 0.2, Rw@FT = 0.1 ohm-m, R = 50 ohm-m At 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 pressureOpen circles, reservoir pressure

1.9 < n < 2.1

Middle East Carbonate Sw calculated with both constant and variable exponent

Variable (high m) exponent evaluation consistent with water test in lower zone

Schlumberger Technical ReviewSeeking 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 orSw(Actual) ~ 4 * Sw(m=2)

Sw calculates good but Test is water!!

Figure 4

boththewaterlegandhydrocarboncolumn.

Comparisonofwirelinedeterminedmandlabdeterminedm.FockeandMunnfindthatthedifferencebetweenaninterparticleandvuggyporesystemcanbeaslargeasanmof~2versusanmof~4,correspondingtoanSwuncertaintyof~20%versus~75%:Figure3.

Theconsequencescanbedevastating:anintervalofhighresistivity(apparentlypay)maybenothingmorethanrelativelytortuous,waterfilledporesystem.

Oneapproachtothisuncertaintyistocombine

anonArchietool(suchasthedielectric)withaconventional(shallowreading,sincethedielectricisalsoshallow)resistivitymeasurement,andwiththismultiplicityofmeasurementsdeducewhatthefootbyfootmisinthewellbore:Figure4.

Anotheroptionmightbetopartitiontheporesystemwithmoderntools(Gomaaetal,Ramamoorthy,etal,etc)andtothenindependentlyestimatethecorrespondingfootbyfootmwithsomekindofelectricalcircuitmodel(Aguilera,Wang&Lucia,etc).

Anattractionofamathematicalrepresentationofthemexponentisthatwhatifcalculationscanbedone,toascertainthe


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 the Archie 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 mm(Dual Porosity) ~ m(Archie)

Figure 5

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RelativeUncertainty

Porosity

RelativeContribution ToSwUncertainty

a

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 a fixed Phi uncertainty.

Attribute Uncertainties Specified IndividuallyLight Green Cells require User SpecificationLight Blue Cells are calculated results

Individual Best Relative UncertaintyAttribute Uncertainty Estimate On Sw(Archie)a 0.0% 1.00 0.0000Rw 4.4% 0.02 0.0019Phi 15.0% 0.20 0.0900m 10.0% 2.00 0.1036n 5.0% 2.00 0.0480Rt 1.0% 40.00 0.0001Sw 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 the Biggest Bang for the Buck, in Improved Sw Estimation

Figure 6

rangeofestimatescorrespondingtoarangeofinputuncertainties(uncertaintyin,forexample,theporositypartition).

Regardlessofwhatapproachisusedtoestimatethewellborem,duediligencerequiresthatateveryopportunitytheresultbecomparedagainsttheinferredArchieestimateinthewaterleg[whereSw=1andm=log(Rw/Ro)/Log(Phi)]:Figure5.

Weshouldfurthermorenotlosesightofthefactthattheremaybeintervals,particularlyinthevicinityofthetransitionzone,acrosswhichtheArchiecalculationissimplynotvalid(Griffithsetal).

WheretoFocusAlthoughthemexponentmay(logically)bethefirstissuethatcomestomindincarbonateevaluation,itdoesnotalwaysdominatetheuncertaintyinSwestimates.Therearetwobasicwaysofidentifyingwherethefocusshouldbe:Figure6.

TakethevariouspartialderivativesofArchiesequation,andcomparemagnitudesforlocallyspecificvaluesanduncertainties.

MonteCarlosimulation.

Thetwooptionscomplementoneanotherinthatthederivativesareeasytocodeintoaspreadsheetor


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RelativeUncertainty

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a

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m

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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


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RelativeUncertainty

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a

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Light Green Cells require User SpecificationLight Blue Cells are calculated results

Individual Best Relative UnAttribute Uncertainty Estimate On Sw(Archa 0.0% 1.00 0.0000Rw 4.4% 0.02 0.0019Phi 15.0% 0.20 0.0900m 10.0% 2.00 0.1036n 5.0% 2.00 0.0480Rt 1.0% 40.00 0.0001Sw 11%Sw^n 1%0.367 is a logarithmic inflection point

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a

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m

n

RtLight Green Cells require User SpecificationLight Blue Cells are calculated results

Individual Best Relative UnAttribute Uncertainty Estimate On Sw(Archa 0.0% 1.00 0.0000Rw 4.4% 0.20 0.0019Phi 15.0% 0.20 0.0900m 10.0% 2.00 0.1036n 5.0% 2.00 0.0108Rt 1.0% 40.00 0.0001Sw 35%Sw^n 13%0.367 is a logarithmic inflection point

If the water were fresher, say Rw = 0.2 instead of 0.02, n diminishes in importance as compared to both the amount of porosity, and its connectivity (m).


The Rw Dependence

The Rw Dependence

Figure 8

petrophysicals/wpackage(forfootbyfootdisplay),whiletheMonteCarlogivesinsightintotheupanddownsides(with95%confidence,theuncertaintywillbelessthanhighlowcalculations).

UsingChenandFangsparameterstoillustratethedifferentialapproach(Figure6)wenotethatasporosityvaries,therelativeimportanceofmandnalsovaries.Thatis,theimportanceofasingleattribute,mforexample,islinkedtothemagnitudeofother

attributes.

Inthecaseathand,a20puformationevaluationshouldfocusonimprovedporosityandmestimates,withnofrelativelylessimportance.Asporositydropsto10pu,theporeconnectivity(m)beginstodominatetheaccuracy:Figure7.

Thementalimagethatemergesisthatasporositydecreases,itsconnectivity(orefficiency)becomesincreasinglyimportant.

Weretheformationwaterresistivitytochange,itsquitepossiblethatthefocusshouldalsochange:Figure8.

Insummary,eachsituationshouldbeevaluatedwithitslocallyspecificparametersanduncertainties,andourfocus(andbudget)appliedaccordingly.


Generalized slope-intercept straight line equationy = m * x + b

Log(FF) = - m * Log( ) + bAt = 100 % , Log ( ) = Log(1) = 0

Leaving Log(FF) = Log(1) = 0 b = 0 to give Log(FF) = - m * Log( )

At = 10 % , FF = 100Log(100) = 2.0 = - m * Log (0.1) = m

to give m = 2.0

1

Schlumberger Technical Review, Volume 36 Number 3G 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 decades and take logarithms

when working with numerical values

Figure 9

mandCarbonatePoreSystemsArchiemeasuredtheporosity,permeabilityandresistivityofbrinesaturatedcarbonateandnonshalysandsamples,acrossarangeofbrinesalinities,toobservealinearrelationbetweenRo(brinesaturatedsampleresistivity)andRw(brineresistivity).

ThisresistivityratioisknownastheFormationFactor,where

FF=R(sample)/R(brine)=1/mArchiecommentedthatmwasabout1.3inunconsolidatedrockandincreasedasthecementationincreased.Atypicalformationstartingpointvalueformis2.0.

ItwasHubertGuyodwhogaveusthetermcementationexponent,andwhoincidentally,alsosuggestedtheoriginalnameoftheSPWLAJournal(TheLogAnalyst).

Themexponenttypicallyinvolvesaloglogdisplayofthebasicdata,andassuchislessintuitivethanasimplelineardisplay.Semilogandloglogdisplaysarecommoninmanypetrophysicalendeavors(phiperm,etc),however,anditsusefultobeabletodrawourownlinethroughthedataanddeducethecorrespondingexponent.TheKeytoworkingwiththe

LogLogdisplaysistothinkintermsofdecadesandtakelogarithmswhenworkingwithnumericalvalues.

SketchingourlinethroughArchies1942dataanddrawingupontheslopeinterceptformulationofalinearrelation,revealsthatm~2.0isindeedareasonablestartingpointwiththatdataset:Figure9.

WhileitwasMrGuyodwhogaveustheterminology

cementationexponent,itappearsthatwederivethemnomenclaturefromthemathematiciansslopeinterceptrepresentationofastraightline,whereintheslopeisdenotedbym:y=m*x+b.

Lookingaheadjustabit,totheResistivityIndex(incontrasttotheFormationFactor),itwouldfurtherseemthatournnomenclaturemayhavearisenalphabetically,sincenfollowsm.

Interestinglyfromahistoricalperspective,thiskindofrelationshiphadbeenpostulatedearlierbySundberg,butwithoutthesupportingdata.


Archies 1947 Data - Sandstone and Limestone

G E Archie: Electrical Resistivity as an Aid in Core Analysis Interpretation, AAPG Bulletin 31 (1947): 350-366Schlumberger Technical Review, Volume 36 Number 3

Suggested to Archie that permeability (molecular fluid flow) and resistivity (ionic movement) were different

Formation Factor (Rformation / Rbrine )

Figure 10 Conceptually,giventhatbothrepresentaflow,onemightexpectastrongerrelationshipbetweenresistivityandpermeability,ascomparedtoresistivityandporosity.Archieaddressedthisbyplottinghismeasurementsbothways:Figure10.

Perhapssurprisingly,thevariousFormationFactormeasurementsconvergemuchbetterwhendisplayedagainstporosity,thanagainst

permeability,promptingArchietosuggestthatmolecularfluidflow(permeability)andionicmotion(current)weredifferent.ThosewhohaveworkedbothIG/IXandchalksystemsarealreadyawareofthis,havingnoticedthatthemforthetwoverydifferentpermeabilitiescanbothbeabout2.

Verweretalhaveperformedadigitalimageanalysisofthinsectionsrepresentingcarbonateplugsuponwhichresistivitymeasurementsweremade.Threeattributeswererecognizedasplayinganimportantrole.

Perimeteroverarea:twodimensionalequivalenttotheporesurface/porevolumeratio.

Dominantporesize:theupperboundaryofporesizeswithwhich50%oftheporosityonthethinsectioniscomposed.

Microporosity:calculatedasthedifferencebetweentheobservedmacroporosityinDIAandthemeasuredporosityfromtheplugsample.

Theirmeasurements,nicelyillustratedwithaccompanyingthinsections,demonstratethatinadditiontoourintuitiveexpectations,

sampleswithhighresistivitycanhavehighpermeability, sampleswithlowpermeabilitycanhavea(relatively)lowmexponent.

AsArchiepointedoutin1952,carbonateporesystemsmaybequitevariable,causinganm=2calculationtobenonrepresentative.WyllieandGregoryboundedthemexponentforarangeofbeadpacksconsistingofunconsolidatedandcementedspheres,viachemicalflushes,andfoundtherelationbetweenformationfactorandporositycouldinvolveanadditionalparameter,C,whichdifferedfromunity.


Wyllie and Gregory constructed bead packs with varying degrees of cementation

The baseline represents a formation of unconsolidated spheres

This data prompted them to propose the relation

FF = C / mC was a formation 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 sand and became 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 a systematic 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 additional moldic 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 be reduced by segregating the samples of a specific Rock Type into Permeability Classes

Figure 12

Rock Type 4, the moldic 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/mCisformationdependent,andtherangeofmexponentswasfoundtobem~1for

unconsolidatedbeadsm~4forcementedbeads:Figure11.

FockeandMunnthenconductedadetailedstudyofmmeasurementsonactualcarbonatesamples,supportedwiththinsectiondescriptionstofindsystematicrelationsbetweentheporositytypeandm.

Whiletheintergranular/intercrystallineporesystemhadm~2(solongas>5pu),consistentwithArchies

earlierlabwork,vuggyporesystemscouldexhibitanmthatreached5:Figure12.

TheirRockType4,representingadiageneticinversioninwhichOriginalPorosityRockandOriginalRockPorosityisanexampleofwhereanincreaseinporositycorrespondstoanincreaseinvuggyporositycontent.Becausethevuggyporosityislessefficientatelectricalconduction,themexponent(counterintuitively)risesasporosityincreases:Figure13.


Rock Type 4, the moldic 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

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

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RT4/PC1RT4/PC2RT4/PC3RT4/PC4

m vs PorosityRock 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 porosityProvides 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 Conventional Well Logs, SPE 5342, Journal of Petroleum Technology 28 (1976)Schlumberger Technical Review, Volume 36 Number 3

Illustrative calculation

is the fraction of porosity that is fracture

Phi(Total) ~ 20 pu ~ 50 %m ~ 1.35

Figure 14

Althoughextreme,diageneticinversionisreportedinotherstudies,forexampleEberlietal,whoillustrateitwiththinsectionsanddescribetheprocessastheoriginalgrainsaredissolvedtoproduceporesastheoriginalporespaceisfilledwithcementtoformtherock

FockeandMunnnextnoticedthatthecorrelationbetweenmandporositycouldbe

improvedbybreakingthatRockTypeintopermeabilityclasses,eachofwhichcorrespondedtodiageneticallyinvertedrock,butwithdifferentpermeabilities.Weshallreturntothedifferentmvstrendsassociatedwiththedifferentpermeabilityclasses,withinthecontextofadigitalmexponentmodel,shortly.

Fractureporosityhasaneffectoppositetovuggyporosity,conceptuallyformingakindofshortcircuit,whichisrepresentedmathematicallybyadecreaseinvalue:Figure14.


The Dual 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 / FFFormation Factor is related to porosity as FF = a / m

The two 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


Type I formula: 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


DigitalmExponentModelWithintheLuciasystem,vuggyporosityiseverythingthatisnotinterparticle,andincludesbothtouchingvugs(fractures,etc)andseparatevugs(moldic,intraparticle,etc).Whiletherearechartsavailabletoestimatem,asafunctionofvuggyporositycontent,forthevariousscenarios,itwouldbeadvantageoustohaveadigitalmodel.

Thereareseveraldigitalmodelsavailable,andhereweuseWang&Luciaforillustrativepurposes,because:

Itisbaseduponasimplecircuitmodelthatiseasytofollowforthosenotparticularlycomfortablewithelectricalcircuittheory,

Itallowsforbothtouchingandseparatevugs,andeverythinginbetween, WangandLuciaincludedindependentQCchecksontheviabilityoftheirmodel,

WangandLuciarepresentvuggyrockasatwocomponent,parallelcircuit:Figure15.

Eachcomponent(independently)satisfiesArchiesequation,andtheythensimplifytheresultingnetconductivityexpressionbysettingthemofthevuggyfractiontobe1andrepresentingthetortuosityofthevugs(betheytouching,separateorsomethinginbetween)withaparameterav:Figure16.


Type I formula: 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


This indicates that equation 30 can be used to model reservoirs with separate vugs, touching vugs, and fractures using appropriate values of av.

av represents the connectivity, or lack thereof, of the vuggy portion of the pore system

Figure 18

Thisleavesthemwithanexpressionforthenetcementationexponentasafunctionofthetotalporosity,theporositypartition,theexponentoftheinterparticlefraction(mip)andtheconnectivityofthevuggyfraction(av):Figure17.

Withthisdigitalmodelinhand,duediligencerequiresthatitbetested,withonesuchtestbeingacomparisonofthecalculatedmtothatinferredfrom

Archiesequationinthewaterleg,Figure5,wheretheyfindthatanappropriatechoiceofavresultsinamatch.

Whilethereisatendencytothinkofnonfracturevugsasseparate,withoutcontributiontopermeability,aliteraturesearchwillrevealinstancesofthepermeabilityactuallyincreasing,withthepresenceofvuggycontent.Thisbehaviorbringsforwardtheimportanceofparameterav,whichisavailabletorepresentsuchasituation.

WangandLuciaalsotestedtheirmodelinsuchasituation,drawinguponMeyersdata,wheretheyagainfoundthatalocallyappropriatechoiceofavbroughtmeasurementsandestimatesintoagreement:Figure18.

Theattractionofadigitalmodelisthatitallowswhatifcalculations.Suppose,forexample,avisualexaminationofcoreindicatesthatabout1puofvuggyporosityispresent,dispersedamongstthematrixporosity.Ifthatvuggyporosityisnotwellconnected(ieav>>1)andthetotalporosityis~5puor


So long as ~ 5 pu and greater, only at av ~ 1 does a small (v ~ 1 pu) cause the Cementation Exponent to differ from mip = 2.0

Dual-porosity Model / What If Characterizations

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ip + v = 6 puSo long as ~ 10 pu (or more) and the vugs are 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

The Monte Carlo method relies on repeated random sampling to model results

Advantages of Monte Carlo include

any type of distribution can be used to characterize the uncertainty distribution of any parameter

normal, log normal, etc

insight is gained into the upside and downside

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av=1

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Figure 20

more,thedualporositymodelwouldindicatethataneffectivemofabout2isareasonablestartingpoint:Figure19.

If,however,that1puispresentintheformofwellconnectedvugs(fractures),thenav~1andtheeffectivembecomesmuchmoresensitivetotheporositypartitionandconnectivity.

Nextconsideranintervalwithabout5puofvuggyporosity(Figure19,again),withamatrixcementationexponentof2.Solongasthetotalporosityis10puorgreater,areasonableinitial

valueoftheeffectivemwouldbeabout2.5,solongasthevuggyportionisnotwellconnected.

Buildinguponthisconcept,werealizethatifitispossibletopartitionporosity,footbyfootinthewellbore,thenonemayalsocalculatemfootbyfootandusethatexponentvaluetothenestimateSw.Insuchasituation,itwouldbeadvisabletoseekoutawaterintervalatthefirstopportunityandcomparetheresultingSw(whichishopefullycloseto100%)withthatdeducedfromaninversionofArchie(asWang&Luciadid,inFigure5),forvalidationpurposes.

IfthecalculateddualporosityexponentisafairrepresentationoftheinvertedArchiem,thentheevaluationofvuggyintervalshingesupontheaccuracyoftheporositypartition,whichmayinfactbeahurdle.Limitationsthatwillariseare,

theimagelogwillbecompromisedifthevugsizeislessthanthebuttonresolution,about35mm,

incarbonatestheNMRT2willlosetheporesizerelationatporebodysizesofabout50100um.


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 & av constant across range of v

Moldic Rock

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Focke & Munns DataRock Type 4, Perm Classes 1 => 4, a moldic limestone grainstone: representing 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


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av=1

av=10

av=100

av=1000

ip + v = 2 pu

The Dual Porosity Model has captured the essential trends of the independent Focke & Munn data.

The Dual Porosity Model vs Lab Data

Figure 21 Thereisyetanotheradvantageofexpressingthevuggyporesystemexponentinmathematicalform:itbecomespossibletoperformaMonteCarlosimulation:Figure20.

Nowweareabletoaccountforuncertaintyinthevariousinputs(theporositypartition,asanexample),andcharacterizetheBestEstimateintermsofprobabilities.Thatis,ratherthangowithahigh/lowestimate,onecanidentify(for

example)95%oftheMCdistribution.Inmanycaseswewillfindthat,ifwecanaccepttwostandarddeviationsofuncertainty(95%),thelikelyhigh/lowvaluesarenotsoextremeasthepossiblehigh/low.Thisisbecause,ingeneral,itisunlikelythatallthehigh(orlow)inputvalueswilloccursimultaneously.

Theprecedingillustrationsrepresentafixedvuggyportioninthepresenceofmoreorlessmatrixporosity.Itisalsopossiblethatmostoftheporosityisintheformofvugs,withsomesmallamountof(background)matrixporosity:Figure21.

Thatis,insteadofaconstant1pu(or5pu)ofvuggyporosity,asthetotalporositygoesupanddown(Figure19),onemighthaveasmallamountofinterparticleporosity,withmoreorlessvugspresent.Thedualporositymodelexponenttrendsarenowverydifferent,becausethebulkoftheporosityisnotefficient.

Solongasthevugsarenotconnected,Figure21revealsatrendofincreasingm,withincreasingporosity.Asavvariesfromslightlyconnected(av~10)toseparate(av~1000),thedifferenttrendsseparate,andexhibitapatternverysimilartowhatFockeandMunnfoundexperimentally.

FockeandMunnsRockType4(diageneticallyinverted),withdifferentpermeabilityclasses,canberepresentedwiththedualporositymodel.Atasimplelevel,theavvaluescorrespondtothechokingoffoftheporethroats,whichbothreducespermeabilityandincreasesm.

Itisimportanttorealizethat,conceptually,someporesystemsmaynotsatisfytheparallelcircuitassumptionandthatadditionallyinsomereservoirstheporesystemisatripleporositysystem,notdual.Insuchasituation,thisparallelcircuitdualporositymodelisnotgoingtobesufficient.


Cementation Exponents in Middle Eastern Carbonate ReservoirsJ W Focke and D Munn, SPE Formation Evaluation, June 1987

m(EPT) & m(core), foot by foot, with the two m estimates compared in crossplot in later graphic

Immobile oil in black, movable oil shaded, both as a fraction of porosity, calculated with Schlumbergers Global package

MoldicIG / IX

Upper Interval

m from logs, foot-by-foot,

compared to core

Figure 22

mfromnonArchieTechniquesIftheporesystemcanbepartitioned,amathematicalmodelofthemexponentwillallowonetothenevaluatethehydrocarboncolumnwithvariablem.Analternativeapproachistoincludeanadditionaltoolinthesuite,whichwhencombinedwitharesistivitymeasurement,allowsonetodeducethelocalm,andtothenusethatmintheinterpretationofthedeepresistivity.

FockeandMunncombinedthedielectriclogwithanRxomeasurementinjustthisway,to

calculatemfootbyfootinthehydrocarboncolumn, comparethewellboremestimateswithlabm.

Althoughtheconceptisstraightforward,FockeandMunnwerecarefultopointoutkeyissuesthatmustbekeptinmind.

TheevaluationrequiresthatthedielectricbepairedwithroutineporosityandRxotools,whichhaveagreaterdepthofinvestigationandlessverticalresolution.Theyarethenreflectinga

largervolumeofreservoirthanisthedielectric.

Rxomayalsobereflectingamixofmudfiltrateandformationbrine.Ifso,aneffectivevalueRmfeshouldbeused.

Archiessaturationexponentisassumedtobe2.0andcarbonatescanassumenonwaterwetvalueswithn>2.0.

Intheirillustrativewell,twoporesystemsarerecognized,moldicandIG/IX:Figure22.

Asexpected,themoldicporosityexhibitsanincreasedm,andreaches~3.5inplaces,asseenwithbothlabdataandthedielectricbasedestimate.


Wireline m(EPT) vs Laboratory m(core) are in general agreement

m and from corem from logs vs from logsCementation Exponents in Middle Eastern Carbonate Reservoirs J W Focke & D Munn, SPE Form Evaluation, June 1987

Figure 23

Similar Trends

Labandwirelinecanalsobecomparedviacrossplot.Recognizingthedifferentvolumesofinvestigation,andattemptingtocompensate,FockeandMunncrossplotmversusporosityforcore,andforlog:Figure23.

Inthemoldicintervals,thetworesultsdisplayanalmostidenticaltrendofincreasingmas(moldic)porosityincreases.

RasmusandKenyonprovideyetanother

illustrationofthecontributionthatadielectriclogcanmakeinthepresenceofoomoldicporosity.

InthetimesinceFockeandRasmus,significantadvanceshavebeenmadeindielectricmeasurements,andtheynowoffermoresophisticatedoptions(ZappingRocks.RomuloCarmona,etal.OilfieldReview.Spring2011.).

SummaryIn1952Archiepointedoutthecomplicationsthatarisewithcarbonateporesystems,whichcanresultintortuouswaterfilledporesystemshavingaresistivitysimilartoaninterparticleporesystemthatishydrocarboncharged.Inonesensethen,andrecognizingthatothercomplicationscanbepresent,thecementationexponent(whichrepresentstheporesystemtortuosity)representstheessenceofcarbonatepetrophysics.

WyllieandGregoryboundedthemwithlaboratorybeadpackstudies,findingthatinapackofunconsolidatedbeadsm~1,whilem~4inachemicallycementedpack.FockeandMunninterpretedhundredsofcarbonateformationfactormeasurements,withinthecontextofthinsectiondescriptions,tofindasystematicrelationbetweenRockTypeandm.

Inthecaseofmoldicporosity,FockeandMunnfoundthatmsystematicallyincreasesastotalporosityincreases:theadditionalporosityissimplynoteffectiveintheelectricalconductionsense.


Petrophysical Characterization of Permian Shallow-Water Dolostone. M H Holtz, R. P. Major. SPE 75214, 2002http://www.beg.utexas.edu/mainweb/presentations/2002_presentations/holtz_spe0402ab.pdf

Pore geometries control the interrelationship of petrophysical properties. The three most important pore-geometry characteristics are

amount and types of pores or shapeinterconnectedness of pores (tortuosity)size of interconnecting pore throats

Various m exponents as a function of vuggy porosity ratioNote 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,thepresenceofvuggyporositycorrespondstoanincreaseinm,withthemagnitudeofthatincreasedifferingfromonedatasettothenext:Figure23.

Therearebothchartsandmathematicalmodelsthatallowonetoestimatemforvariousporositypartitions.Theadvantageofthemathematicalmodelisthattheyeasilyallowbothfootbyfoot

estimatesandMonteCarlosimulation.

Chartsandmodelsshouldalwaysbetested,atthefirstopportunity,bycomparingthemestimatetothatdeducedbyinvertingArchiesequationinthewaterlegofarepresentativewell.InthecaseoftheDualPorositymodelusedforillustrationsherein,thatcomparisonisreasonable.TheDualPorositymmodelisalsofoundtomatchtheindependentlabmeasurementsofMeyers,andtocapturethepatternsreportedbyFockeandMunn.

AwellborealternativetommodelsistoaddanonArchietooltothetoolsuite,whichallowsaneffectivemtobededuced.Thatmisthenusedtointerpretthedeepresistivitymeasurement.

Regardlessoftheapproachused,weshouldbearinmindthatparticularlyinthevicinityofthetransitionzone,thereservoirmayfailtosatisfythebasicArchiecriteria.

AcknowledgementMohamedWatfasArchiesLaw:ElectricalConductioninClean,WaterbearingRockisanimportantsinglepointhistoricaloverviewsource.FockeandMunnsCementationExponentsinMiddleEastCarbonateReservoirsliterallysetthestandardforasystematicinvestigationoftheissue,backin1987.TothesemustreadarticlesIhaveaddedmorerecentmaterial,andmypersonalthoughts/techniques.

ThisoneisformyMother,whoisalwaysthere,throughthickandthin.

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Chattanooga shale

Mississippian limestoneR. E. (Gene) Ballays 35 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 a PhD in Theoretical Physics with double minors in Electrical Engineering & Mathematics, has taught physics in two universities, mentored Nationals in Indonesia and Saudi Arabia, published numerous technical articles and been designated co-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 established Robert 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.

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Wyllie,M.R.andARGregory:FormationFactorsofUnconsolidatedPorousMedia:InfluenceofParticleShapeandEffectofCementation,PetroleumTransoftheAIME(1953):103110.

Biography

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