A Statistical Evaluation of Methods UsedTo Predict Pressure Losses for MultiphaseFlow in Vertical Oilwell TubingJ. David Lawson, SPE-AIME, Amoco Production Co.James P. Brill, SPE-AIME, U. of TulsaIntroductionSeveral correlations havebeen published that can beused to predict pressure losses in vertical oilwelltubing for the simultaneous, upward, concurrent, con-tinuousflowof oil, water, andgas. Becauseof theextreme complexity of multiphase flow, the proposedcorrelations are by necessity highly empirical. Thevalidityof thecorrelationsis thensomewhatlimitedto the quality and scopeof the data upon which theyare based. Therefore, some correlations performquite well for cases inthe range of the data usedin developing the correlation but fail for otherapplications.The purpose of this study is to ascertain theaccuracyof several pressure-losspredictionmethodsintermsofflowvariablesfamiliar tothepracticingengineer. Theresults of the studyshouldassist thepetroleumproductionengineerinselectingthemostaccurate method for his problemand also shouldindicate the general accuracyto be expectedfromthe methods.The correlations included in the study are those ofPoettmann andCarpenter,!Baxendell andThomas,2Dunsand Ros,3 Fancher and Brown,4Hagedorn andBrown,5and Orkiszewski.6Each of these correlationswas proposed specifically for predicting pressure lossesin vertical oilwell tubing for the upward flow of multi-phase well fluids.The pressure-loss prediction methods were pro-grammed for the IBM360 computer and testedagainst 726 well tests fromfield and experimentalwells. A statistical analysis was made to find the mostacceptablemethodfor different ranges offlowvari-ables andalsothemethodhavingthebest over-allperformance for predicting the measured pressurelosses.Pressure-Loss Prediction CorrelationsThe six correlations chosen for study will be discussedbriefly. Areview of the historyand content of multi-phaseflowliteratureandamoredetaileddiscussionof individual correlations may be found in Brown.1Thecorrelationof PoettmannandCarpenter1isarelativelysimple andpractical onetopredict pres-surelossesinflowingwells: However, its verysim-plicitylimitsitsaccuracy. Themethodwas includedinthis studyfor comparison, andbecauseit servedas a starting point for later correlations..PoettmannandCarpenter correlatedthe irrever-sible energy losses of 49 well tests with a Fanning-typefrictionterm. Thefriction termwas related to thenumerator of the Reynolds number forthe well fluidmixture. Noattempt wasmadetoaccount forliquidholdup (volume fraction liquid in a pipe section), butrather an average density of produced fluids correctedfor down-holeconditionswasused. Thecorrelationreproduced the measured pressure gradients to anaverage deviation of 1.8 percent and a standard devi-ationfromtheaverageof 8.3percent. Itwaslaterdiscovered that this excellent performance did notapply forthe wideranges of valuesof flowvariablesSeveralmethodsforpredictingpressurelosses for upward, concurrent, continuous,multiphaseflowinvertical. ai/well tubingweretestedagainst measuredpressurelossesfrom726welltests. Thiscomparisonof well knownpressure-losscorrelations revealstherelativestrengths andweaknesses of eachcorrelationandshouldbeuseful inselectingsatisfactorymethods for various applications.AUGUST. 1974903found in oil production problems. Other investigators,therefore, attemptedtomodifythecorrelationtofitbroader ranges.Baxendell andThomas2extendedthe PoettmannandCarpenter correlationtohigher flowrates andreported-+- 5 to-+- 10 percent accuracy for the higherrates.Fancher and Brown4,8applied thePoettmann andCarpenter approach to 94 tests from an experimentalwell. Produced gas/liquid ratio (GLR) was introducedasan additional parameter in thefriction factor cor-relation. TheFancherandBrownmethodpredictedthe measured pressure losses within about -+-10percent.The above correlations made no attempt to includegeometric flow configuration or "flowregime" tocharacterizepressurelosses. Also, liquidholdup, orrelatedslipvelocity(velocitydifferencebetweengasandliquid phases), forthevariousflowregimeswasnotconsidered.Ros9andalso Dunsand Ros3 gathered laboratorydataonpressurelossesfor two-phaseflowintrans-parenttubes. Theyobservedthedependencyof flowregimes on dimensionless parameters. Correlations forslip velocityalso werederived. Thecorrelation fittedthe measuredpressurelosses inthe laboratorytestsection to an average of-+- 3 to-+-10 percent, depend-inguponflowregime. Datawere takenover quitebroad ranges of flow variables and the correlation wasthusexpectedtoperformsatisfactorily formostwellconditions.HagedornandBrown5,10developedacorrelationfrom475testsina1,500-ft experimentalwell usingfluids havingviscositiesupto110cpoTheFancherdata also were used. An average mixture density cor-rectedfordown-holeconditionswasusedforcalcu-lating estimatesof pressurelossescausedbyfrictionandacceleration. Liquidholdupwasthencalculatedfromthetotal measured pressurelossandthecalcu-lated values for friction and acceleration losses.These holdup values were correlated with various flowvariables and fluid properties. Since liquid holdup wasnot measured directly, valuesof holdupgiven bythecorrelation are not always physically significant.CalculatedpressuresfromtheHagedornandBrowncorrelation matched measured pressures fromtheHagedornstudytoanaverageof 1.5percent, witha stamlard deviation from the average of 5.5percent.Orkiszewski6combinedtheworkofGriffithllforbubble flow, that of Griffith and Wallis12for slugflow,the Duns and Ros correlation for mist flow, andnewfriction anddensitycorrelations for slugflowbased ona parameter named the"liquid distributioncoefficient."WiththedataofHagedorn, thiscoeffi-cient was correlated with pipe size, superficial mixturevelocity, and liquid viscosity. The composite correla-tion was tested against 148 well tests and was reportedto predict the measured pressure losses to 0.8 averagepercent error and 10.8 percent standard deviationfromthe average error. The proposed correlationwas reported to be superior to the Duns and Ros andthe Hagedorn correlations when all methods weretested against the same well data.Other correlations have beenpublishedthat are904recommended for predicting pressure losses for multi-phaseflow, but for various reasons they were notincluded in this study. The work of Griffith andWallis12wasnot considered separately becauseof itsapplicationtoonlythe slugflowregime. However,it is partially included in the Orkiszewski methodalong withthe work of Griffithllforthebubble flowregime.A correlation by Yocum13was not includedbecause it centered on the prediction of pressurelosses inhorizontal flowlines. BakerandKeep14didnot recommendtheirmethodfor cases wheretherewas a likelihood of slip between gasand liquid.The Tek15correlation was not included in thestatistical studybecauseit predictedunusuallyhighpressure lossesfor many of the well tests. Acorrela-tion by Hughmark and Pressburg16was based ondata obtained using a relatively short tube. It was notincludedinthe studybecauseit has beenreportedthat the correlation gives values for pressure loss thataretoohighwhenappliedtodatatakenfromlongtubing strings.7TheworkofGaither et alYwasnot includedinthe statistical study, althoughit may performwellfor the small-diameter pipes fromwhichthecorre-lated data were taken.An attempt was made to include those correlationsthat have been considered most acceptable by oilindustryusers for broadranges of flowconditions.Correlations that have not attained this degree ofgeneral usagemaybequite adequatefor predictingpressurelossesinspecificapplications.Well Test DataMost ofthewell testdatawereobtained frompub-licationsonmultiphaseflowpressure-losspredictionmethods, but some field data were obtained fromunpublishedsources. Onlyoilfieldtests or tests onexperimental wells were considered. No"short tube"data were included. Every effort was madeto ensurethat the information was accurately transcribed tocomputer storage by double-checking to eliminateclerical errors.Each of the well tests was examined for unreason-able values by comparing the measured pressurelosses for a well test with the pressure losses predictedby all the multiphase flowcorrelations. However,because of the rather wide range of answers given bydifferent correlations,none of the available well testscouldberuledout bythistechnique. Therefore, allthe data usedinthe statistical study are probablyreasonably correct,although the absolute accuracy ofthe well data remains unknown.The following information was included in thedata storage foreach well test:1. Assigned well test number,2. Data source code number,3. Inside tubing diameter, in.,4. Oil production rate, STB/D,5. Water production rate, STB/D,6. Produced GLR, standard cubic feet (60F,14.7 psia) per STB of liquid,7. Depth of pressure measurement,ft(tubingheaddepth =0),JOURNALOFPETROLEUMTECHNOLOGY8. Measured flowing pressure at depth, psia,9. Measuredflowingpressure at thetubinghead,psia,10. Tubinghead temperature, of,11. Temperature at depth, P,12. Specific gravity of stock-tank oil, API,13. Specific gravity of stock-tank water (pure waterat 600P= 1),14. Specific gravity of total produced gas(Air = 1at 600Pand 14.7 psia),15. Viscosityofstock-tankoil at 1000Pand14.7psia, cp,and16. Viscosityofstock-tankoil at 2100Pand14.7psia,cpoSome data were not available for well tests, sothemissingvalueswereestimated. Tubingheadanddown-hole temperatures were estimated for manytests, and. temperatureprofileswerealwaysassumedto be linear withdepth. Theroughness of thetubingwall always was assumed to be 5X10-5ft. Oilviscositiesat100 and 2100Pwere usually not avail-able, so estimates were made using any availableviscositydataandthe Beal correlation. IS The lackof measuredvalues for fluid properties andtubingroughnessis adrawback, but theincreasednumberof complete well tests available after the approximatevalues were included was considered to be a valuableaddition to the well data.Porsomewell tests, moreinformationwasavail-ablethanWas used; e.g., formationvolumefactors,solutionGaR's, andtemperatureandpressurepro-files. Ratherthanusetheextra. informationonfluidproperties, thesevalueswereestimated fromvariousphysical property correlations that were programmedalongwitheachpressure-losspredictioncorrelation.When pressure profiles were available, only thedeepest measurement was used. The exclusion ofsome available i,nformation was a necessarycompro-mise between exactnessand standardizationof data-handling and programming procedures.Therangesofflowvariablescoveredbythewelldata and the arithmetic average of some of the valuesforthe welldataaregiven inTable 1.Eighty percent of the well tests were taken on wellspredictedtobeintheslugflowregimebytheflowregime boundary definitions of Griffith and Wallisfor bubble flowand thoseof Dunsand Rosformistflow. Pivepercent of the well tests were predictedto be in the bubble flowregimeand10 percent wereindicated tobe inacombination of bubbleflowandslugflow. None of the tests ever enteredthe mistflowregime, but 5 percent were predicted to bein a combination of slug flowand the transitionregionbetweenslugandmistflow.Brief descriptionsof thegroupsofdataandtheirsources are given in Table 2. A substantial partof the well test data usedinthis study was fromHagedorn,lO which may give the Hagedorn andBrown correlation a more favorable comparison.However, the Hagedorn data were taken under care-fully controlled conditions and therefore are con-sideredaccurate data that should be included in thatdata bank.Because of the length of the data list, it is notAUGUST. 1974included here. Copies of the data bank and calculatedresults areavailablefromtheauthorsuponrequest.Programming of MethodsA separate main program was written for eachpressure-losspredictionmethod, withfluidpropertycorrelations handled as subroutines. Subroutines werewrittenfor calculatingvalues of formation volumefactor and solution GaR, oiland water viscosity, oiland water surface tension, gas viscosity, and gascompressibility.Calculations were made for changes in tubingdepth corresponding to assigned pressure changesstarting with the measured tubinghead pressure. Whenvalues of fluid properties representative of the averageconditions in the tubing section were needed, the sub-routines wereenteredwitharithmeticaveragepres-sure and temperature. When temperature varied withdepth, the calculation was by trial and error to matchthe giventemperature gradient. The gravityof thefreegas and that of the gas dissolved in the oil phaseat higherpressureswerebothsetequal tothetotalproduced gasgravity.The values of pressure, temperature, and otherfluid conditions calculatedfor the exit of the firsttubingsection usedasinlet valuesforthenextlength.The calculation proceeded in this incrementalmanner until the total depth was reached or exceeded.The bottom-hole pressure was then calculated bylinear interpolation of the last two pressure/depthcoordinates. It should be noted that calculations start-ing at the measured bottom-hole condition and endingat the tubinghead would not necessarily give the samepressure drop.Thepressureincrement sizeswereselectedtobebetween 5 and10 percent of the absolute pressure tominimize the inaccuracies of averaged physical prop-erties and at the same time to maintain areasonablyshort computationtime. The pressure-loss methodsstudiedrequiredonlyafractionof asecondontheIBM 360/85to calculate a pressuretraverse.Certain extensions were made to some correlationstocoverextremesencounteredinanalyzingthewelltest data. When a dependent variable in a correlationTABLEI-MINIMUM, MAXIMUM, ANDAVERAGEVALUES OF WELL DATAArithmeticMinimum Maximum AverageInternal diameter, in.1.049 8.760 1.878Producing liquidrate, STB/D 1.0 5,082.0 395.2ProducingGLR, scf/STB21. 788,000.Well depth, ft918. 12,458. 3,993.Measuredbottom-hole 104. 5,140. 1,252.pressure, psiaMeasured wellhead pressure, psi a 20. 2,856. 518.Wellhead temperature, 0 F 50.0 180.0 93.9Bottom-hole temperature, 0 F 80.0 330.0 126.0Producedoil API gravity 9.5 56.2 34.4Producedwater gravity 1.00 1.15 1.03Producedgas gravity 0.600 1.400 0.850Viscosity of produced oil 0.90 20,000.00at 100OF, cpViscosity of produced oil 0.26 140.00at 210OF, cp905. was defined in only a certain range of the independentvariable, therange of defini!ionof the dependentvariable was extended to include these extremevalues of independent variable. For example, thefrictionfactorcurvesofthePoettmann andCarpen-ter, Baxendell and Thomas,and Fanher and Browncorrebltions Were extrapolatedtothelowandhighReynolds-nllmber numerator ranges to cover all welltest conditions, The' extrapolationofthesecurvesis. consideredtobeof relativelyminor importance tothe analysis of the different correlations since the greatmajorityofwell test datawerewithintherangeoftheoriginal curves. Theextensionsareexplainedint!J.e Appendix.No part of the Duns and Ros correlation was extra-polatedbecause of the unpredictable shape of theregressioncoefficient curves. Onlythewelltest datawithin the range of the Duns and Ros correlation wereusedinthestatistical analysis.It is emphasized that such extrapolations may notalways be advisable and no recommendation is givenfor suchextensions. For this study, the extensionswerenecessarytoreducethehandlingof dataandto simplify the presentation of results.Fluid Physical Property CorrelationsAll thepressure-loss predictioncorrelations requirevalues for fluidphysical properties that areusuallynot known andmust therefore be estimated usingvariousempirical correlations. The physical propertycorrelations used in this study are presented inTable 3.Theaccuracyofthecorrelationsis usuallyquitegood fortheparticular fluidsandconditionsusedinderivingthem. However, theiraccuracyisunknownwhentheyareappliedtovariousother hydrocarbonmixtures. Additional uncertainties result fromthesometimes necessary extrapolation of the methodsbeyondtheiroriginallydefinedranges.Proper useof thecorrelationsforpredicting solu-tionGORandformationvolumefactor incalculat-ingdown-holevolumes andspecificgravities of oiland gas phases requires that the gas and liquid phasesbe in equilibrium. Actually, equilibriumdoes notexist between oil and gas phases in the tubing becauseof theeffectsof gasslippage. However, theassump-tion of equilibrium is probably the best approximationthat can be made.Liquid-phasepropertiesareespeciallydifficult, ifnot impossible, topredictwhentheliquidisatwo-phasemixture of oil andwater. For this studyanarithmetic average value weighted onthe basis ofvolume fractionof oiland water intheliquid phasewas calculated from individual values of the propertyTABLE2-DESCRIPTIONOFWELLTEST DATAData SourcePoettmann andCarpenter1Fancher8Hagedorn10Baxendell andThomas2Orkiszewski6Espanol Herrera23U. of TulsaData Group 1camach024NumberofTests498334625224456101General DescriptionFielddata; mostly2%- and2'l's-in.-ODtubing; mediumflowrates; gas-oil andgas-ail-water; low tomediumpressures.One field wellwith gas injection; 2%-in.-00tubing; medium-highflowrates; 95percent watercut; lowpressures.Experimental well withair injection; 1-,11,4-, and 11h-in. nominal tubing; air-water and air-refined oil flow; oil vis-cositiesfrom10 to110 cp; broadrangeof flowrates; lowpressures.Field data from2'l's - and3% - in. - 00tubing; high flowrates; gas-oil flow; lowtomediumpressures.Fielddatafrom31h-in.-ODtubingwithtwo testson8.76-in.-ID tubing; mediumto highflowrates; heavyoils; moderatepressures.Fielddataon2%- and2'l's-in.-ODtub-ing; mostly lowflow rates; light oils;high pressures.Fielddataon2%- and2'l's-in.-ODtub-ing; mostly lowrates; lowto mediumpressures.Fielddataon2%- and2'l's-in.-OD tub-ing; gas wells making lowto mediumamounts of water; mediumpressures.Comments onEstimated QuantitiesSurface temperature assumed to be80F with 1.0F/100 ft temperaturegradient. ViscositiesfromBeal correia-tion.18Produced gasgravitywas taken to be0.6 asanaveragebetweensolutiongasgravityof0.65andinjectedgasgravityof 0.57. Measured pressure at lowestdepth above gas injection depth wasused. Measured pressures were readfromplots inAppendixto MSthesis.Temperaturewasset constant at80F.Pressuremeasurement at lowest depthabovegasinjectiondepthwasused. Airviscosity was setat 0.0185 cp; air com-pressibilityfactor at 1.0; solution GLRat O. Surfacetensionandliquid phaseviscosity were set at values given inPhDdissertation.Temperature was set constant at 180F.Oil viscosities estimated from data givenat 160and 200F. Deepest pressuremeasurement wasused.Fluidsfromall wellswereassumedtohave same properties as those given forWell 22 except for oil APIgravities givenfor each well. Well 22 temperature gradi-ent wasused for other wells.Inside diameters of1.996 and2.376 in.wereused.Viscosities were estimated using Bealcorrelation.906 JOURNALOFPETROLEUMTECHNOLOGYfor each phase. Average values of liquid-phase surfacetensionandliquid-phase densitywerecalculatedinthismanner. Averageliquid-phaseviscosity wascal-culated from values of oil and water viscosity weightedbythevolumefractionof eachphaseinthe liquidaccordingtothe Arrhenius formula19, as suggestedbyHagedornandBrown.5Theeffects of oil-wateremulsions were not considered in this study.Volumefradions of oil andwater inthe liquidphase for down-hole conditions were calculated usingoilandwater productionratesandassumingnoslipbetweenthe oil and water phases. Actually, sincetheoj!is usuallyless densethanthewater, the oilwill tend to risefaster. Thereforetheactual volumefractionof oil in the liquid phase for down-hole con-ditions will usuallybe smaller than the calculatedvolumefraction. Theinaccuracies ,of estimatingrep-resentative valuesof physical properties for oil-watermixturesmaybeaserious limitationtothe properuse of multiphase pressure-10ss correlations for three-phase oil, water,and gas flow.' ,Theinaccuraciesofthephysical propertycorrela-:-tions andthewayinwhichtheyareusedlimit theability of a multiphase flow pressuure-loss correlationto make an accurate prediction. Therefore, thepressure-loss prediction method, made up of a combi-nationof pressure losscorrelation and fluidphysicalproperty correlations, must be consideredasa pack-agewhentestedagainst measuredpressure losses.Validation of ProgrammingBecauseoftheintricacyofsomeofthecorrelationsandthecomplexities of the programminginvolved,pressurelosses calculatedbythecomputerprogramfor eachpressure-losspredictionmethodwerecom-paredwithcalculatedpressurelossesavailablefromanother source for the same well test data. Closeagreement between the calculated pressure lossesfrom this study and the independent calculationindicatesthat theprogrammingwas donecorrectly.Minordifferencesincalculatedanswersfromdiffer-ent sources are tobe expectedbecause of the useof different physical propertycorrelations andpro-gramming techniques. However, it is difficult toTABLE3-FLUIDPHYSICALPROPERTYCORRELATIONSFluidPhysical PropertyPseudocritical temperatureand pressure of hydro-carbon gasesHydrocarbon gas compress-ibi lityfactorSolution GOROil formation volum.e..factorHydrocarbon gas viscosityOil viscosityWater viscosityOil surface tensionWater surface tensionagainst airWater surface tensionagainst hydrocarbon gasAUGUST, 1974CorrelationUsedKatz25,26Standing and Katz25,27Lasater2s,29Standing30,31Carr et al.32Bea11S,29 andChewand Connally29, 33BealiSBaker34,35Data of Ref. 36Data of Hough et a/.37;also Katz25CommentsThe curve for miscellaneousgases wasused.Thecorrelation becomes morein error asthenonhydrocar-boncontent of thegasesisincreased, butmay still beaccu-ratetoabout 2%whenonlysmall amountsof nonhydrocar-bons are present. z-factors for reduced temperatures lessthari 1.05 werecalculatedfor reducedtemperaturesof 1.05.Predictedvaluesof gasmol fractiongreater than0.85weretakenas 0.85. TheAPI vseffectivemolecular weightof tankoil correlationwasextrapolatedfor oil API lower than18 andgreater than 55. This correlation predi'cts the bubble-pointpressuresof the mixtures used togeneratethecorrelationto an average of 3.8%. Accuracy for predicting solution GOR'swouldbeexpected tobeabout 5%also fortheoriginal data.Thecorrelation hasanaccuracyof 1.17%when appliedtomixturesIJsedtodevelopthecorrelation.Thecorrection factors for H2S, CO2and N2were not usedsince gas analyses were not available. Fortunately, correc-tionsaresmall fornonhhydrocarboncontent less thanabout5%. Effect of water vapor on viscositywas not considered.Beal'scorrelationwas usedtoestimate"deadoil" viscosityat 100 and210F asa functionof API gravity. Thesevaluesmay be in error several-fold, since oil compositionalso affectsoil viscosity. Temperature dependency of viscosity was as-sumedtofollowtheBeal correlation. WhentherangeoftheChew and Connally correlationwasexceeded,values at1,600Scf/bbl wereused.Dataare foraspecificgravity of 1.0. Thecurvewasextrapo-latedtohigher temperatures. Noattempt was madetoac-count fortheeffectsof salt content, dissolvedgascontent,or pressure. 'Valuesat68F wereusedfortemperaturesbelow 8 ~ F andvalues at 100Fwere used for temperatures above 100F.Thefractional changeof surfacetension with pressurewastakenas0.1for pressuresover3;900psia.Datafor purewater against air were used. Surfacetensionwas assumed to vary linearly with temperature and havevalues of 69.56 dyne/cmat 104Fand 58.9 dyne/cmat212F.Surface tension values were found by linear interpolationbetweendatafor 74 and280Finthemethane-water sys-tem. Valuesfortemperaturesbelow74Fandabove280Fwere takenas thevaluesat 74 and280F, respectively.907generalizeastoexactlyhowmuchdeviationincal-culated pressure lossescould be expected because ofthose differences.Theprograms developedfor the PoettmannandCarpenter, Baxendell and Thomas, Fancher andBrown, and Hagedorn and Brown methods werefound toreproducethepressure lossescalculated byeach method quite closely. Comments on comparisonsfor each method are given below.Poettmann and Carpenter: The computer programof this work matched the 49 calculated pressuregradientsinthePoettmannandCarpenterpapertoanaverageof 2.6 percent and weregenerallywithin10 percent.Baxendell andThomas:No independently calculatedpressure gradients were available, but comparisonsof pressure gradients predicted by the computer pro-gram.of this work against measured pressure gradientsintheBaxendell andThomas paperweregenerallythe same +5to+10 percent reported by Baxendelland Thomas.Fancher and Brown:The same general + 10 percentdifferences betweencalculatedandmeasured pres-surelosses originallyfoundbyFancherwerefoundwiththecomputerprogramofthisstudy. Thepres-sure losses calculated by Fancher were read from thegraphs in the Fancher MS thesis.Hagedorn and Brown:The calculated pressure lossesfrom the computer program of this study agreed withthe pressure losses calculated originally by Hagedornas wellascould be determined by readingthecalcu-latedpressure lossesfromgraphs inthe B:agedorndissertation.Duns and Ros: The programming of the Duns and Roscorrelation wastestedbycomparingestimatedpres-sure losses for the same input well data with theprogramoftheDunsandRoscorrelationofD. R.McCordandAssociates, Inc.20Calculatedpressurelosses for more than 100different well tests werecompared. All the comparisons were within + 10TABLE5-STATISTICALRESULTSFORALLWELLTESTSAveragePercent StandardMethod Difference DeviationPoettmann and Carpenter - 107.3 195.7Baxendell andThomas - 108.3 195.1Fancher and Brown 5.5 36.1Duns and Ros 15.4* 50.2"Hagedorn and Brown 1.3 26.1Orkiszewski 8.6 35.7"no-slip" static head + 53.5 33.0"Results for the 427 well tests in the range of the correlation.percent and most were within+5 percent.Orkiszewski: The predictedpressure losses for theentire 726 well tests by the computer programofthisstudy for theOrkiszewski methodwerecheckedagainstcalculated pressure lossesofthesamecorre-lationprogrammedbytheContinental Oil Co: Re-search Dept., Ponca City, Okla. The average percentdifference. betweencalculatedpressurelossesof thetwo different computer programs for the 726 well testswaslessthan 2 percent. Larger discrepancies in cal-culatedpressurelossesoccurred forwellsproducingwater and oil together, but the differences were tracedto the use of different techniques for calculating liquid-phase average viscosity,2lStatistical data on deviations between measuredandpredictedpressurelossesgivenbyOrkiszewski6for theOrkiszewski, Duns andRos, andHagedorncorrelations could not always be reproduced using thecomputer programs of this study for the same sets ofwell testdata.. (Table4compares statistical resultscalculatedbyOrkiszewski andstatistical resultscal-culated fromthecomputer program estimates of thisstudy. Statistical parameters are defined in Discussionof Results.) Part of the discrepancies could be causedby the use of a different definition of average percenterror. Orkiszewski didnot state his average errorformula. Althoughthestatistical resultsof this workwere different froin those reported in the Orkiszewskipaper, agreement on the calculated pressure losses forthe example problem in Appendix D of the Orkiszew-ski paperwas quitegood. Orkiszewski calculatedaTABLE4-ORKISZEWSKI'SSTATISTICAL RESULTS COMPAREDWITHTHOSEOFTHIS STUDYData GroupOrkiszewski data on heavy-oil wells(22 well tests)Average error,percentStandard deviation, percentBaxendell and Thomas(25 well tests)Average error, percentStandard deViation, percentPoettmann and Carpenter(49 well tests)Average error, percentStandard deviation,percent908PredictionMethodOrkiszewski Ros Hagedorn and BrownOrkiszewski Results of Orkiszewski Results of Orkiszewski Results ofResults This Study Results This Study Results ThisStudy-1.2 -12.0 +22.7 -42.4 +16.4 -29.710.4 13.1 18.7 19.7 41.4 42.0-2.1 -3.7 +2.3 -13.9 +8.7 +6.011.1 41.1 20.0 39.5 12.7 11.6-1.0 +3.7 +5.8 -4.8 -13.0 +22.812.0 13.7 12.4 15.1 22.2 24.7JOURNALOFPETROLEUMTECHNOLOGY1. 049 L380 1. 610 1. 995 2.376 2.441 2.992. (311 (175) 1140l ,(228) (411 (60) (46)VALUES OF INSIDETUBINGDIAMETER, INCHES( NUMBEROFWElLTESTS I N EACH GROUP I N PARENTHESES)Fig. I-Statistical results for well data groupedby internaltubing diameter.probabilitythat themeasuredpressurelosseswouldbe predicted within+26.1percent(l standarddevi-ation) of the average percent difference and a 95percent prdbability that thepressure losses wouldbepredictedwithin+52.2percent (2standarddevia-tions) of the average percent difference. Anormaldistributionwouldbeexpectedfor thislargecollec-tionof data, andthereforetheforegoingconfidencelevelsareprobablysuitablefor drawingconclusionsastothe accuracyof predictions fromeachof themethods.Table5 also includessatistical resultsforcompar-ingmeasuredpressuredropwiththat resultingfromastaticheadofthefluidsifnogasslippageoccurs.As expected, this prediction method underpredictsthe pressure drop, since friction,acceleration, andgreater values ofliquid holdup resultingfromgasslippagearenotconsidered.In addition to the over-all statistical results in Table5, furtherinformationabout theperformanceofthepressure-losspredictionmethodsisshowninFigs. 1through' 5. Thefigures illustratetheeffects that in-ternal tubing diameter, produced water / oil ratio(WOR), producedGLR, producedoil API gravity,andsuperficial liquidvelocity at tubingheadcondi-tions exert ontheperformanceof eachof thefourpredictionmethods that performbest over all. Theflowvariables areinunitsfamiliar tothepracticingpetroleum production engineec Since these flow vari-ables have amajor effect onthepressuregradientsinproducingwells, ananalysis of predictionerrorsbasedonranges ofthesevariablesshouldpoint outsomestrengthsandweaknessesof the individ-ualcorrelations.pressureloss of approximately 850psi (Fig. 14ofRef. 6); the computer program of this study calculated870 psi; the Continental computer program calculated872psi.Discussion of ResultsThe statistical results for the prediction methodsappliedtoall 726well tests are giveninTable5.Definitions of percent difference, PD, arithmeticaverage of percent differences, APD, and standarddeviationof percentdifferencevalues fromtheaver-age percent difference, SD, are given below.PD= e:..pm- e:..pc 100 percent,e:..pmnL PDiAPD= =-.:1=---_nSD=" n- 1whereAPe=calculated pressuree:..pm =measured pressure lossn =number of well tests.AnanalysisoftheequationforPDindicatesthatthe value of the percent difference between measuredand calculated pressure losses can be a large negativenumber for overpredictionof pressure loss;' i.e"e:..pc >>e:..pm, but isboundedby+100percent forunderprediction of pressure loss. The average percentdifference values in Table 5 therefore tend to empha-sizemoretheerrorsfor methods that grosslyover-predict pressure loss. Likewise, the values of standarddeviation mayemphasizemorethescatter of grosslyoverpredictedpressurelosses.Table 5 indicates that boththe Poettmann andCarpenter andthe Baxendell andThomas methodsoverpredict the measured pressure losses on the aver-agebyafactor oftwoasindicatedbythe -107.3and -108.3 values of average percent difference.Thereisalso considerablescatterof thepercent dif-ferencevalues about their averageas shownbythehigh values of standard deviation.TheFancherandBrowncorrelationis similar totheBaxendell andThomas andthePoettmannandCarpenter correlations, but is based on a broaderrange of well conditions. The Fancher andBrownfriction factor correlation usually predicts lowervaluesof friction factorand lower pressure lossesasshown by the lower valuesof averagepercentdiffer-enceandstandarddeviation.The pressure-loss predictions of some of themethodsinTable5showsmallervalues ofaveragepercent difference andstandard deviation. For ex-ample, the averagepercent difference andstandarddeviation values forthe Hagedornand Brown corre-lationare-1.3and26.1percent, respectively. Thisimpliesthat, ifpercent differencesarenormallydis-tributed about their mean value, there is a 67 percent40'" u2015'"'" IisI-015u'"'" 0-'" '-"ffi -20>