m exponent ok plan

8/13/2019 m Exponent Ok Plan

1/18

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 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


2/18


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


3/18


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


4/18


0.00

0.10

0.20

0.30

0.40

0.50

0 0.1 0.2 0.3 0.4

RelativeUncertainty

Porosity


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


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


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.


5/18


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


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.


6/18


Archies 1947 Data - Sandstone and Limestone

G E Archie: Electrical Resistivity as an Aid in Core Analysis Interpretation, AAPG Bulletin 31 (1947): 350-366


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.


7/18


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.


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.


8/18


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.


9/18


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.


10/18


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


11/18


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 ]


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 ]


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.


12/18


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


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.


13/18


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.


14/18


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.


15/18


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.

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JournalofPetroleumTechnology26(1974):12331238

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JournalofPetroleumTechnology28(1976):764772


16/18


AlGhamdi,AliandBoChen,HamidBehmanesh,FarhadQanbari&RobertoAguilera.An

ImprovedTriplePorosityModelforEvaluationofNaturallyFracturedReservoirs.Trinidadand

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www.GeoNeurale.Com.

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Ballay,Gene.

Coffee

Or

Tea.

October

2009.

www.GeoNeurale.com.

Ballay,Gene.SplitPersonality.December2009.www.GeoNeurale.com.

Ballay,Gene.StatisticsArePliable.February2010.www.GeoNeurale.com.

Ballay,Gene.TheBiggestBangfortheBuck.April2011.www.GeoNeurale.com

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Equation.TheLogAnalyst.SeptOct1986

Diederix,K.M.AnomalousRelationshipsBetweenResistivityIndexandWaterSaturationsinthe

RotliegendSandstone(TheNetherlands),TransactionsoftheSPWLA23rdAnnualLoggingSymposium,CorpusChristi,Texas,July69,1982,PaperX

Eberli,GregorP.andGregorTBaechle,FlavioSAnselmetti&MichalLIncze.Factorscontrolling

elasticpropertiesincarbonatesedimentsandrocks.THELEADINGEDGEJULY2003

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17/18


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18/18


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

m exponent ok plan

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