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1 IntroductionOverviewIntroduction Formationevaluationwithwirelinelogs,forthepurposeofestimatinghydrocarbonporevolume

(HPV), requires methods to accurately determine net-pay thickness, porosity, and hydrocarbon

saturation.Traditionalpetrophysicalalgorithms,suchasArchie’sequation,needestimatesoftrue

formationresistivity(Rt)tocalculatesaturationandHPV,butmanyfactorsinterferewiththeability

todetermineRt.ResistivitymodelingallowsforaccurateestimationofRtfrommeasuredlogdata.

Thischapterintroducestheconceptsofresistivitymodelingforformationevaluation.Thehistory

of resistivity logs and modeling is reviewed briefly, and some fundamental concepts and terms,

suchasdimensionalityof earthmodels and forwardand inverse logmodeling, aredefined.The

rationaleforwhyandwhenresistivitymodelingshouldbeappliedisintroducedbyadiscussionof

toolresolutionandaccuracy.

Contents

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TheQuestforRt

Historyofresistivitylogsandmodeling

EarthModels

Definition

Applicationsof1-D,2-D,and3-Dmodeling

ForwardandInverseModeling

Forwardmodeling

Inversemodeling

Acceptableerror

Non-uniquenessininversion

WhyModelResistivityforFormation

Evaluation?

RtforHPV

Toolresolutionandaccuracy

Chartbookvs.computermodeling

WhenShouldResistivityModelingbe

Considered?

Practicalconsiderations

“Strange”resistivityloginterpretation

Whendoesresistivitymodelinghave

highrisk?

Flowchartforapplicabilityofresistivity

modeling

Copyright©2011byTheAmericanAssociationofPetroleumGeologists.

DOI:10.1306/13211293A23414

Yin, H. 2011, Introduction, in H. Yin, Application of resis-tivity-tool-response modeling for formation evalua-tion: AAPG Archie Series, No. 2, p. 1–10.

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Ar c hI e Se rI e S, no. 2

TheQuestforRtHistoryofresistivity Perhaps the most appropriate way to summarize the history of resistivity logs is to describe the

logsandmodeling subjectasthequestforRt,thetrueformationresistivityuncorruptedbyboreholeinvadingfluids.

Such an endeavor implies the need for a resistivity tool that measures beyond the influence of

boreholeconditions,mudcake,caves,shoulderbeds,andanyotherzonesthatgiverisetoanomalous

signalsleadinginsteadtoameasurementofRa,theapparentformationresistivity.Thequestfor

RthasbeenongoingintheloggingindustryeversinceConradSchlumbergertippedhissurface-

electrical-prospectingarrayverticallytologdownawellbore,reasoningthatthecurrentfromthe

electrodesspreadoutintotheformationaroundtheborehole(Schlumbergeretal.,1934).Anearth

modelwasthenassumed,andthelawsofphysicswereusedtopredictanalyticallytheresponsefor

agivenelectrodeconfiguration(i.e.,resistivity-tool-responsemodeling).

Historically, resistivity-tool-response modeling started with resistivity-logging-tool design. The

primaryeffortinmodelingresistivitytoolswastodesigntoolsthatmeasuredRaascloselyaspos-

sibletoformationRt.Earlyexperimentalmodelingusedsmallelectrodesininfinite-conductivity

saltwaterbaths.Later,toolresponseswerestudiedusingmock-upsondesinmorerealisticenviron-

ments,createdbyusingthinimpermeablemembranestoseparatewatersofdifferentsalinity.

Starting in theearly1950s, scalemodelsandanalogcomputers (resistornetworks)werebuilt to

characterizetoolresponsesincaseswherecomplicatedmathematicalmodelsdonotyieldanalytical

solutions.

H. Guyod (1955) discussed an approximate solution of Laplace’s equation,∇2V = 0, whereV =

electricalpotential,withproperboundaryconditions,andaresistornetworkwasdesignedtosimu-

latetoolresponseforagivenformationandelectrodearrangement. Sponsoredbyseveralmajor

oilcompanies,Guyoddesignedaresistornetworkcontainingapproximately300,000resistors.The

innovativenetworkconsistedofabarrelwith2000smallmercurycups,arrangedinahelicalpat-

ternonitsinsidesurface,andacenterrotor,bearing1000brasspinsthatcanfitinsidethecupsin

ordertomovethemudandformationnetworksintoanewpositionrelativetoeachother.

Untilthelate1970s,asimilarresistornetworkhadbeenusedattheSchlumberger-DollResearch

Laboratory in Ridgefield, Connecticut (Figure 1.1). The network, consisting of nearly 500,000

resistors,ledtopredictingtoolresponsesrelativetotrueformationresistivityandtocreatingmany

oftheearlylog-correctioncharts.

Despitethedrawbackstothiskindofmodeling,suchasthedifficultyinexchangingorreconfigur-

ing thousandsof resistors tobuildeachnewcase, thescalemodelingwassuccessfullyapplied in

studyinggalvanictools.However,scalemodelingwasnotreadilyadaptabletotoolsthathavelarge

depthsofinvestigation,suchasinductiontools.

Large improvements in computing capability have introduced qualitative changes in the role of

modelingsincethe1980s.DirectnumericalsolutionsofMaxwell’sequationcanbeusedtocom-

puteresistivitytoolresponseinearthmodelswithcomplexgeometries.Inthese,thegeometrical

restrictionsaremuchlessseverethaninanalyticalandscalemodels.Realisticproblemscanbefor-

mulated,andthesolutionsappearassimpleandunthreateningnumbers,ratherthanasarcaneand

difficult-to-compute mathematical functions. Computerized modeling has reduced from weeks

tominutesthetimerequiredtocalculatemanyboreholeenvironmentaleffectsonresistivity-tool

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responses. A systematic evaluation of the effects of such environmental conditions as borehole

rugosity,caves,mudcake,invasion,dip,shoulderbeds,andformationanisotropyonresistivity-tool

responseisnowpossible.

Thesetheoreticalsimulationsoftoolresponsestolayeredandinvadedformationgeneratedbooks

ofdeparturecurves. In fact,mostcurrent resistivity-correctionchartsprovidedby servicecom-

panies are the result of numerical computer simulation, rather than physical experimental scale

modeling.

EarthModelsDefinition Resistivity-tool-response modeling may be categorized to the class of boundary-value problems.

Theearthmodelspecifiesthegeometryandboundariesofearthregionsofdifferingresistivity,and

theresistivitywithineachregion. Aresistivity toolwith itselectric source(s)anddetector(s),or

transmitter(s)andreceiver(s)locatedatspecifiedpointsisintroducedintotheearthmodel.The

source(s)ortransmitter(s)excitetheearthmediuminsomespecificmanner.Withsomeassumptions

madeaboutthespatialdistributionofresistivity,thesolutionsforresistivitytoolresponseundera

givenearthmodelcantaketheformofsimpleformulasderivedfromMaxwell’sequations,giving

Figure 1.1. The resistor network as an analog computer for two-dimensional (2-D) resistivity-tool-response modeling. This lastresistor network consisted of two racks, about 6 ft (1.8 m) tall and 12 ft (3.7 m) long, with nearly half a million resistors(Anderson et al., 1997, used with permission of Schlumberger).

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Ar c hI e Se rI e S, no. 2

theelectromagnetic(EM)fieldforanydesiredpointinspace.Formorecomplicatedcases,theEM

fieldscannotbeexpressedintermsofanalytical functions,butmustbe laboriouslycomputedat

manypointsinspacesimultaneously.Thedegreeofearth-modelcomplexitymaybesummarized

byreferringtogeometriesofdifferingdimensionalities:one-dimensional(1-D),two-dimensional

(2-D),orthree-dimensional(3-D).

ConsidertheCartesiancoordinatesystemshowninFigure1.2,specifiedbythecoordinatesx,y,

andz.

1.One-dimensional:Iftheresistivity(R)variesinonlyonecoordinatedirection,thenthegeom-

etryisreferredtoasa1-Dearthmodel.Inaparallel-layeredgeometryofsedimentaryrockwith

noborehole,theresistivityoftheformationvariesonlyinthez-coordinatedirection;R=R(z)

(i.e.,1-DVertical,whenboreholeisperpendiculartobeddingplane.Therelativedipisdefined

astheanglebetweentheboreholeandthenormalofbeddingplane).Or,inaninfinitelythick

homogeneous formation penetrated by a borehole with mud-filtrate invasion, the resistivity

variesonlyintheradialdirectionaroundtheborehole:R=R(x)=R(y)=R(r)(i.e.,1-DRadial,

whenboreholeisperpendiculartobeddingplane).

2.Two-dimensional:Ifresistivityvariesintheverticalandradialdirectionsduetobedding,bore-

holeconditions,andmud-filtrateinvasionbutisreasonablyaxisymmetricaroundtheborehole,

thentheearthmodelcanbeadequatelydefinedas2-D,R=R(x,z)=R(y,z)=R(r,z).

3.Three-dimensional:WhenR=R(x,y,z),theearthmodelissaidtobe3-D.

Applicationsof1-D, Applicationsof1-D,2-D,and3-Dresistivitymodelingarelistedbelow.Theassociatedearth-model

2-D,and3-Dmodeling geometriesareillustratedschematicallyinFigure1.2.

Table 1.1. Applications of 1-D, 2-D, and 3-D modeling

Modeling Dimension Effects addressed1-D Bedthickness,shoulderbeds,dippingbeds

2-D Boreholeandcaves,invasion

3-D HeterogeneousRt

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ForwardandInverseModelingForwardmodeling Thenumericalprocessofreconstructingaresistivitylogforagivenearthmodelisoftenreferred

toasforwardmodeling.Forwardmodelingtypicallyresultsinauniqueandrobustsolutionforthe

modeledresistivitylog.

Forward modeling entails the analytic or numerical solution of Maxwell’s equations in a math-

ematical boundary-value problem, where the earth model specifies the boundaries and shapes

of earth regionsofdifferent resistivities.The resistivity logging tool is introduced into the earth

modelbylocatingitstransmitter(s)andreceiver(s)atspecifiedpoints.Thetransmittersexcitethe

physicalmediumoftheearthmodelinamathematicallydefinedmanner.Fora1-Dearthmodel,

thesolutionsofMaxwell’sequationscantaketheformofsimpleformulas,givingtheEMfieldfor

anydesiredpoint inspace.For2-Dor3-Dcases, theEMfieldscannotbeexpressed in termsof

analytical functions, but must be laboriously computed at many points in space simultaneously.

Themathematicalandnumerical techniquesused in forwardmodelinghavebeenthesubjectof

manypublications,anddiscussionofthesetechniquesisoutsidethescopeofthisvolume.See,for

example,AndersonandGianzero(1983);Andersonetal.(1996);Chewetal(1984);Gianzeroetal.

(1985).

Figure 1.2. Schematic resistivity profiles for earth models in one dimension (1-D) if the resistivity varies in borehole axialdirection (R = R[z]); two dimensions (2-D) if resistivity varies in the vertical and radial directions (R = R[x,z] = R[y,z] =R[r,z]) axisymmetric around the borehole (where Rxo = resistivity of invasion zone, Rm = borehole mud resistivity, Di =diameter of invasion zone, and DB = diameter of borehole); and three dimensions (3-D) if resistivity varies in all major axes(R = R[x,y,z]).

R5”

3-D

R(x,r,z)y

x

z

R4”

1-D

Relative DipR(z)

z

R6

R7R8 R9

R6”

R2”

R1”

2-D

R(r,z)

z

R4’ Rxo

R2’

R1’

DB

Di

R3’

R5’

R4

R2

R1

R3

R5 Rm

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Ar c hI e Se rI e S, no. 2

Earth Model

Shallow Medium Deep

Resistivity

Rt

1-D Forward Modeling

Tool Response Functions

Aschematicrepresentationof1-DforwardmodelingisshowninFigure1.3.Ontheleftistheearth

model. Itassumesnoboreholewashoutorellipticityandnoinvasion. Simplifiedtool-response

functionswiththreedifferentdepthsof investigation(shallow,medium,anddeep)areshownin

the middle. In principle, the depth of investigation of an induction tool is proportional to the

spacingbetweenthetransmitterandreceivercoilpairs(i.e.,thelongerthespacing,thedeeperthe

toolmeasures).Theright-handsideshowsthetrueresistivityoftheformation(orange),andthe

shallow(blue),medium(green),anddeep(red)resistivitylogs.Thedeepresistivitytoolreadsthe

lowestvaluesovertheinterval,becausethethicknessesofthebedsarebelowtheverticalresolution

ofthedeeptool,whereastheshallowtoolreadsnearlythetrueRthereintheabsenceofinvasion.

Inversemodeling TheprocessofderivinganearthmodelincludinganRtprofilefromasetofgivenfieldlogswith

appropriateresistivity-tool-responsefunctionsisreferredtoasinversemodeling.Inversemodeling

entails iterativelyadjustingtheformationresistivity(Rt),bedboundaries,and(fora2-Dmodel)

invasiondepthandinvasion-zoneresistivity(Rxo)ineachlayeroftheearthmodel,andrepeating

theforward-modelcalculation,untilanacceptablereplicationoftheobservedfieldlogsisachieved.

Inversemodelingmaynotyieldauniquesolution.

Figure 1.4 is a schematic representation of 2-D forward and inverse modeling. The sand layers

havebeeninvadedbymudfiltrate.Thetool-responsefunctionsfortheshallow,medium,anddeep

resistivitytoolsareshowninthemiddle.Theright-handsideshowsthetrueresistivityofthefor-

mationRt,resistivityoftheinvadedzoneRxo,andtherespectiveshallow,medium,anddeeptool

readings.Inthiscase,theshallowresistivitylogissignificantlylowerthantrueresistivity(Rt)but

closelyagreeswiththeinvadedzoneresistivityRxobecauseofinvasion(contrastwithFigure1.3).

Figure 1.3. Schematic representation of 1-D forward modeling of resistivity in ohm m of an earth model where the sandsare not invaded. Rt = true resistivity.

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Acceptableerror Inthisvolume,resistivity-tool-responsemodeling,whenappliedforformationevaluation,involves

replicatingtheobservedfieldlogbynumericallysolvingthemathematicalboundary-valueproblem

of the electromagnetic fields generated from a specific resistivity tool under a pre-defined earth

modelontrial. Tothedegreethatthefield-logandthemodeled-logresponsesare inacceptable

agreement, the underlying earth model can be considered one possible representation of the

formation’strue-resistivityprofile.Iftheassumptioninthedefinedearthmodel(1-D,2-D,or3-D)

isclosetotherealitywherethefieldlogdatawereobtained,suchanacceptableagreementcanbe

definedbythetool-sondeerrorinmanycases.

Non-uniquenessin Resistivity modeling may not yield a unique solution. Multiple acceptable replications of the

inversion observed field log may be achieved through resistivity modeling by variation of the formation

resistivities,bedboundaries,andinvasiondepths.Hence,thederivedearthmodelmaynotbethe

true formation-resistivity profile, but one possible representation that yields a similar observed

response.

Non-uniquenessisparticularlyproblematicinformationswherethebeddingfrequency(inversely

related to bed thickness and spacing) exceeds the so-called blind frequency of the logging tool.

Theblindfrequencyisthemaximumspatialfrequencybelowwhichindividualbedscanbedistin-

guishedonthelog.Forbeddingabovetheblindfrequency(andthinnerthanthecorresponding

minimumthickness and spacing), individualbeds cannotbedistinguished, andnon-uniqueness

Figure 1.4. Schematic representation of 2-D forward- and inverse-modeling process of resistivity in ohm m for an earthmodel where the sands are invaded.

Earth Model

Shallow Medium Deep

Resistivity

Rt

Forward Modeling

Inverse Modeling

Tool Response Functions

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Ar c hI e Se rI e S, no. 2

becomesamajorfactorininversion.Asanexample,theblindfrequencyofthedualinductionlog

isintherangeofonebedperfoot(0.3m).SeeChapters3and6formoredetails.

Tominimizenon-uniqueness,itisessentialtointegratelogswithdifferentdepthsofinvestigation

andverticalresolution,andtoemployconstrainingdatafromallpossiblesourcesintheapplication

ofresistivitymodelingforformationevaluation.Forexample,coreimagesorborehole-imagelogs

maybeutilizedtoidentifybedboundariesandthicknesses.Petrophysicallimitsfromporositylogs

orcoredatamaybeusedtoderivepossiblemaximumandminimumresistivityvalues.Knowledge

oftoolphysics,localgeology,andoffsetwelldatamayleadtheresistivitymodelertoamorerealistic

earthprofileofformationresistivity.

WhyModelResistivityforFormationEvaluation?RtforHPV Theaccuracyofhydrocarbonsaturationcalculatedfromwelllogsisfundamentallydependentupon

accuratedeterminationofRt. True formation resistivity is anessential input intohydrocarbon-

resourceassessmentsinpetrophysics.Iftheapparentresistivity(Ra)measuredbyaresistivitytool

isuseddirectlytoestimatehydrocarbonsaturationandhydrocarbonporevolume(HPV),errorsin

HPVwillgenerallyresult.

Toolresolutionand Because of tool physics and borehole conditions, formation resistivity measured by a logging

accuracy tool, Ra, is not equal to Rt in most logging environments. All deep-reading resistivity tools are

influencedbytheresistivitydistributionina largevolumesurroundingthesonde. Forexample,

the deep induction resistivity tools may not resolve beds less than 2 ft (0.6 m) thick even after

resolution-enhancementprocessing,because theirvertical resolution is limitedby induction-tool

physicsandthespacingofthedeep-readingcoils.

Chartbookvs. Traditional environmental correction charts usually do not provide satisfactory remedies for

computermodeling estimating environmental effects on resistivity logs. The charts that are provided by the service

companieshavelimitingassumptionsthatseldommatchreal-worldexamples.Computermodeling

canbedonetoconvertapparentresistivityfromlogsintoaresponseprofilethatmaybecloserto

reality.Infact,anymodernenvironmentalcorrectionchartprovidedbyaservicecompanyisthe

resultofcomputermodeling.

High-resistivityanomaliesmaynotcorrespondtoresistivebeds,butrathertotheinterfacebetween

twobeds,eachwithlowerRtthantheRaindicatedonthelog.Suchananticorrelationmayoften

bemistakenasdepthmismatch,andonlymodelingmayresolvethecorrectdepthandresistivityof

eachbed.

Withtheadventofmodelingcodes(computerprograms)tosimulateresistivity-toolresponse,and

withsignificantincreasesincomputingpower,resistivitymodelinghasbecomeafeasiblealterna-

tive tochartbooks for formationevaluation. Whenappropriatelyused, resistivitymodelingcan

reduceuncertaintyinresistivity-basedreserveestimates.

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WhenShouldRtModelingbeConsidered?Practical Although resistivity modeling is becoming practical with the continual increase in computing

considerations power,resistivity-tool-responsemodelingmaystillbequitetimeconsuming,especiallyin2-Dand

3-Dmodeling.Whetherresistivity-tool-responsemodelingispracticalforagivenwellshouldbe

objectivelyjudged,basedonprojectrequirementsandavailableinformationfromfieldlogs.

Whenaccurateresistivity-basedHPVcalculationisrequired,resistivitymodelingshouldgenerally

beappliedwhenthechart-book-specifiedenvironmentalconditionsdonotapproximatethebore-

holeconditionsinwhichthelogswerecollected.

“Strange”resistivity Wellsareoftennotidealforlogging.Resistivitylogsfromdeviatedandhorizontalwells,drilledto

loginterpretation optimizeproductivity,oftenappearcounter-intuitive,andarenotcorrectablebychartbooks.The

physicalprinciplesembodied inMaxwell’sequationsofresistivity toolsarewellunderstood,and

withmodelingcapableofhonoringenoughdetailsofthetoolphysicsandboreholeenvironment,

suchresponsesarepredictable. Toolresponseartifacts suchaspolarizationhorns,whichappear

aslargetransientovershootsintheresistivityresponseatbedboundariesindeviatedwells,canbe

readilyunderstoodthroughmodeling.

Whendoesresistivity Defining resistivity for beds thinner than the vertical resolution of deep-reading resistivity tools

modelinghave ishighlyuncertainwithoutadditionalconstraints,suchascore-plugsaturationdatatodefinebed

highrisk? resistivity and high-resolution log or core images to rigorously define beds and bed thicknesses.

Thisisduetothetools’physicallimitsinaccuracyandsensitivityforbedsthinnerthanthevertical

resolutionofthetool.Theverticalresolutionoftoday’sarrayinductiontoolsisabout2ft(0.6m),

andthatisalsoabouttheverticalresolutionofmostporositylogs.Becauseoflimitedresourcesfor

modelingporositylogs,resistivitymodeledforbedsthinnerthan2ft(0.6m)maynotcompletely

addresstheuncertaintyinHPVcalculation,exceptwhenthewellhascore-plugporosity.Moreover,

whenbedsarethinnerthantheverticalresolutionofdeep-readingresistivitytools,othertechniques

mayworkwellandbemoreeconomical,suchastheparallel-conductivitymodeland/orvolumetric

laminatedsandanalysis(VLSA)(Passeyetal.,2006).

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Ar c hI e Se rI e S, no. 2

Flowchartfor Thefollowingflowdiagramsummarizeswhenandwhereresistivitymodelingshouldbe

applicabilityof considered.

resistivitymodeling

Figure 1.5. Flow diagram summarizing when and where toapply resistivity modeling. Sw = water saturation

yes

yes

yesyes

yesyes

no

no

nono

nono

Boreholeand invasionproblems?

Start

Are sands> 2 feet?

Are sands< 20 feet?

Is dipping orwell deviation

>30°?

Significant“horn” or resistivity

contrast?

Boreholeand invasionproblems?

Will Inductionor Laterolog be used

in Sw and HydrocarbonPore Thicknesscalculations?

Rt Modeling is notnecessary for HPT

estimation

1-D Resistivitymodeling

Use modeling results and conductArchie or shaly sand FE for

Hydrocarbon Pore Thickness

2-D Resistivitymodeling