overview - american association of petroleum...
TRANSCRIPT
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In t r o d u c tI o n
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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In t r o d u c tI o n
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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In t r o d u c tI o n
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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In t r o d u c tI o n
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