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A New Model for Laminar, Transitional, And Turbulent Flow of Drilling Muds

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  Society of PetroleumEngineers SPE25456ANewModelforLaminar Transitional andTurbulentFlowofDrillingMuds T.O. Reed, ConocoInc. and   Pilehvari U. ofTulsa SPE Members Copyright 1993, Societyof Pf1.\roleum Engineers Inc. Thispeper was preparedforpresentation at theProductionOperationsSymposiumheld In OklahomaCity OK, U.S.A. March21 23 1993. Thispaper was selectedforpresentation by an SPEProgram Commilleefollowingreviewofinformationcontained In an abstractsubmitted by the author s . Contents of thepaper as presented have not beenreviewed bytheSocietyofPetroleumEngineers and are SUbject tocorrection by the author s .The material as presented doesnotnecessarilyreflectany pos~lon oftheSocietyofPetroleumEngineers Its officers ormembers. Papers presented atSPE meetings are subject to publicationreviewbyEditorialCommitteesoftheSocietyof PetroleumEngineers.Permission to copy Is restricted to an abstractofnot morethan300words. illustrations may not be copied. The ebstrect shouldcontainconspicuousacknowledgment of w r~ andbywhom thepaper is presented.WriteLibrarian SPE, P.O. Box833836, Richardson TX75083-3838, U.S.A.Telex 163245SPEUT.  STR CT Theconcept of an  Effective diameter is introducedforthe flow of drillingmudsthroughannuli. This new diameter accountsforbothannulargeometryandtheeffects of anon-Newtonianfluid. It providesthelinkbetweenNewtonianpipe flow andnon-Newtonian flow throughconcentricannuli.Themethod is validin any flow regimeandcan be usedtodeterminewhethera DOn-Newtonianflow is laminar,transitional, or turbulent.  n analyticalprocedure is developedforcomputingfrictionalpressuregradientsin all three flow regimes. The analysis also quantifiesbow flow transition is delayed by increasingtheyieldstress of afluid. In addition, it is shownthattransitioninanannulus is delayed to higherpumpratesastheratio of innertoouterdiameterincreases.Furthermore,themethodaccountsfor wall roughnessanditsaffectsontransition81 and turbulentpressuregradientsfornon-Newtonian flow throughpipesandconcentricannuli. Finally themethodrunsona386 P inonlya few seconds. INTRODUCTION ThestandardAPImethodsfordrillinghydraulicsassumeeitheraPowerLaw or aBinghamPlasticrbeologymodeL In reality,mostdrillingmudscorrespondmucbmoreclosely to theModifiedPowerLaw or Herschel-Bulkleyrbeologicalmodel. This distinction is particularlyimportantfortheannulargeometriestypical of normaldrillingconditionswhereshearratesareusually low andthePowerLawunderestimatesandBinghamPlasticmodeloverestimatesfrictionalpressuredrops. This paper also showsthatthese two classicalmodels,respectively,underestimateandoverestimatepumpratesrequiredfortransitionfromlaminat to turbulent flow. Althoughthere have beenanumber of paperspUblishedonthelaminar flow of yield-pseudoplasticsthroughannuli/-Jnoneclaim to beuniformlyvalidforlaminar,transitional,andturbulent flow. Thiscanbeaccomplished by introducingtbeconcept of an Effective diameterwhichaccounts.forbothannulargeometryandtheeffects of anon-Newtonianfluid. This diameter also enablestheinclusion of theeffects of wall roughnessonfrictionalpressuregradientsandtheprocess of flow transition.Furthermore,theresultingmodelcanbeusedtodeterminewhether flow of anon-NewtonianfluidthroughReferencesandillustrationsatend of paper.apipe or concentricannulus is laminar,transitional, or fully turbulent.Aderivation of themodel is presentedintheAppendix.Someadditionalbackgroundandresultsfromthemodelaregiveninthefonowingdiscussions. NEWTONIAN FLOW RelatioDShipBetween Pipe and  nnular Flows. A considerablenumber of  equivalent diameters have beenproposedoverthe years for flow throughconduitsother than circular pipes.1O-1z Thepurpose of definingsuchadiameter is to introducedefinitions of friction factor andReynoldsnumberthat will enableapplication of thewenknownrelations for pipe flow toothergeometries. In particular,theobjectiveisto be able to calculatewhatthefrictionalpressuredrop will beforagivenfluidataparticular flow rate. h shownintheAppendix,thefrictionfactorforaconcentricannulusshouldbe based ontheHydraulicDiameter, Dhy> which is simply the differencebetweentheinnerandouterdiameters.TheReynoldsnumbershouldbe based onan ~iv l nt Diameterequal to thesquare of Lamb sDiameter,D L  divided by Dhyt seeAppendixEqs. A.12, 13 and 14. Whenthefrictionfactorand.Reynoldsnumberaredefinedin this manner,theclassicalrelationsforNewtonianpipe flow e.g.,theMoodyDiagram,canbeapplied directly to concentricannuli. In fact,JonesandLeung 13 haveproventhat these definitions also apply to fully-turbulent flow throughanannulus. This was demonstrated by showingthatavariety of experimentaldataforconcentricannuliagreewiththeColebrookEquationforturbulentpipe flow which is equivalenttothe fully turbulentand hydraulica1ly roughportions of theMoodyDiagram.FlowTransition in Annuli. In additiontobeingapplicableinthelaminarandturbulentregimes,thesesamedefinitionscanalso be usedtopredictthe critical Reynoldsnumberatwhichlaminar flow endsandtransitiontoturbulencebegins.Prof.Hanksand his students have publishedexperimentaldata on flow transitioninconcentricannuli. I 4-15 Some of theirdata is shownin Fi 1. Thedashedcurve is theresults of predictionsfromatransitiontheory 469  2 A NEW MODEL FOR LAMINAR, TRANSmONAL, AND ruRBULENT FLOW OF DRILLING MUDS SPE2S456 developed by Hanks in the early1960s. 7,1 17 The solidcurvecomesfromsimplysetting the Reynoldsnumber, based on the EquivalentDiameter of Jones and Leung,equal to 2100.AReynolds number based only on Lamb's Diameter is used for the ordinatein order to displayhow the criticalvalues vary withdiameterratio. It is readily seen thatthemuch simplertransitioncriterionprovides better agreementwith Hanks experimentaldata. The newcriterionshowsthattransitionfromlaminarflowisdelayed to progressivelyhighervelocities or pump rates as diameterratioincreases. NON-NEWTONIAN FLOWPipe Flow. Prof.Metzner and hisstudentsreported some ~ion rin workin the 19508 on non-Newtonianflowthrough pipes. l-Z1 TheirexperimentswithPower-Lawfluidsinpipesprovidedthefirstcleardefinitions of how the frictionfactorvarieswithReynolds number inthelaminar,transitional, and turbulentregimes.Theyfoundthatadecreasingpower-lawexponentdelaystransition to higherReynoldsnumbersandshifts the turbulentfrictionfactorsdownward. This isadirectresult of increasingdegrees of the phenomenoncalled wshear thinning. In addition to the testdata, Metznerand hisstudents also developedanovelanalysis that provideda way to generalizetheirresultsforPower-Lawfluids to all time-independent,non-Newtonianfluids.Theysimplydefined: Wall Shear Stress = K x(Newtonian Shear Rate)N(1a) or usingstandard symbols ,N T w = K x  8 v/ D ã 1b) From thisequation,itfollows thatthe definition of the exponent WNW is: N = d(ln Twl /d(ln(8 v/ D)](1e) Theyshowedhow the twoparameters  K'w and WNW can be applied to Power-Law  PL) fluids and BinghamPlastics(BP). The Appendixprovides the details on how this approach can also be applied to aHerschel-Bulkley (BB) fluid,which is essentiallyacombination of the PL and BP rheologicalmodels. The Metzner group alsodeveloped the followingrelationforwallshearrateinapipeforageneralfluidwithtime-independentproperties. Generalized PipeShearRate:  Yw = (3N + 1)8v/  4N D)  ãã (2) This expressionledusto the followingconclusion.  n  Effectivewdiametercan be defined so that the generalizedshear rate can beput into thesame form as theshearrate forNewtonianpipe flow, viz., (8v/D).Using this logic, the Effective Diameter forgeneralizednonNewtonianpipeflow is: WElTective w PipeDiameter = D eff = 4N D/ 3N + 1)....(3) The wEffective w diameter for non Newtonian pipe flow is the diameter   a circular pipethat would havetheidentical pres- sure drop for flow   a Newtonianfluid with viscosityequalto the apparent viscosity and the same average velocityas thenon Newtonian flow. Note that this diameterisless than the physicaldiameterforpseudoplastics  N < 1) and isgreater thanthe physicaldiameterfordi1atantfluids  N > 1). As will be shown,theEffective Diameter makes iteasier to relatenon-NewtonianandNewtonianflows. The advantage of suchaconnectionisthatthewell-establishedfrictionfactorrelationsforNewtonianflowscanthen be applied to non-Newtonianflows. This greatlysimplifies the task of computingfrictionalpressure drops forsuchflows. The remainingvariablethatisneeded to define the GeneralizeReynolds Number is the  apparent Newtonianviscosity. This parameterisdefined by: App.Viscosity = ShearStress/Shear Rate @ Wall(4a) 1J.w,app = K (8 v/ D)N/  Yw .......................ã........... u (4b) Withthesedefinitions, the GeneralizedREynoldsnumber  GRE) isdefinedas: NRe,O =   vDeffl 1J.w,app ããããããããããããããããããããããããããããããããããããããããããã  ãã (5) This expresses the Reynolds number fornon-Newtonianpipeflowin thesame algebraicform as foraNewtonian fluid Metzner & Reed gshowed thatthe GRE isrelated to laminarfrictionfactor by thesame classicalequationforNewtonianflow, viz,   c = 16/ NRe,o ,LaminarNon-Newtonian Flow...... (6) These authorsshowedthatexperimentalpipeflow data foravariety of non-Newtonianfluidsfollowedthisrelationship. Later experi.mentswithPower-LawfluidsinturbulentpipeflowledDodge   Metzner3l to modify the classicalColebrookequationforturbulentfrictionfactorsinNewtonianpipeflow, see Eq. A 45 This was necessaryin order to correlate thedata because the shear-rate exp0- nent was found to be asignificantparameter. In particular,frictionfactors decrease withdecreasingvalues of the Power-Lawexponent.Figure Z shows the correlations Dodge   Metznerdevelopedfor laminar and turbulentflow. The transitionzoneswere not correlated by theseauthors, but theirexperimental data follows the indicatedcurvesbetween theend of laminarflowand the beginning of fully- turbulentlow. As describedin the Appendix,a new correlationfor the transitionzones bas been developed based onthe experimental data of Dodge & Metzner. The curvesin the transitionzones are from this newcorrelation. The generalizedshear-rateexponent  Nil can be used to extendthe data of Dodge   MetznerforPower-Lawfluids toothernon· Newtonianfluids. In partiCUlar, Dodge   Metznerderivedtheappropriateequationsfor the GRE, shearrate,andapparentviscosityforflow of a BP throughpipes.TheirworkisextendedtoHerschel-Bulkleyfluidsandconcentricannuli.in the Appendix.However,beforediscussingannularflow,itisappropriate to firstdiscuss some resultsfrom the newtransitioncriterionforfluidswithayieldstress. Hanks and Pratt J proposedamodelforflow of BinghamPlasticsthroughpipes.Their paper also includesatransitioncriterion.Theychose to base theircriterion on the so-calledBinghamPlasticReynolds numberandtheHedstrom number.Thesetwodimensionlessparameters are definedin the Appendix, see Eqs.A-50and A 51 Unfortunately, the BP Reynolds number is not alegitimatemeasure of Reynolds number effectssimplybecausethePlastic Vis· cosity (PV) isaconstant and does not vary withshearrateastheapparentviscosity does. Furthermore, theHanks and Pratt criterion doesnot agreewithexperimental data at highervalues of theHedstromnumber. J This disagreementisshowninFig.3.Otherinvestigatorshave also notedthisdisagreement. Z1 22 In contrast,thesolidcurvein Fig. 3 is based on thenewtransitioncriterionwhichrequires the criticalconditionforanyfluid to occurwhen the product 470  SPE 25456 T.REEDAND   PILEHVARI 3 of theFanningfrictionfactor andthe GRE equals16.1. (As may be seenfrom Eq. 6, this product is 16 in Jaminar flow, and it exceeds this valuewbentransiqonbegins.) This generalizedtransitioncriterionisdesigned to reduce to acriticalReynolds number of 2100forNewtonianpipe flow, see lastsection of Appendix. The underprediction of thedata by the Hanks and Pratt criterion is even more apparentwhencriticalvalues of the GRE are plottedasafunction. of Hedstrom number, Fig. 4. Here itbecomesclearthattbe HanksandPratt criterionpredictsdecreasing flow rates for transitionas the YieldPoint (YP) increases. This certainlyviolatestbeexpectedtrend. The newgeneralizedcriterionshow the expectedtrend,i.e.,higher pump rates, are required to achievetransition as tbe yP of a BP increases. It is instructive to plot the frictionfactorsfor BP pipeflowthroughall three flowregimeswith Hedstromnumberas aparameter.This is sbownin Fig. 5. Thisfigureshowsbow an increasing yP delaystransition to higher pump rates and also extends tbe transitionzoneoveraprogressivelywiderrange of pump rates. For example,wben tbe Hedstrom number is500,000, the flawis not fullyturbulentuntilaReynolds number of 100,OOO This compareswithacommonvalue of about3,000forNewtonianpipeflow. It is particularlyimportant to observe that all of the curvesfor ~ ns t on eventuallymergeinto tbe turbulentcurveforNewtonian flow. This occursbecause the yP becomesprogressivelylesssignificant as sbear rate increases.Hence, the curveshave the expectedasymptotes.. There is one other point to noticefromcomparingFigs.5 and 3. A BP bas a sbear-rateexponent of 1; wbereas,a PL fluidgenerallybasanonunityexponent.This is the basicreasonwby the frictionfactorcurvesfordifferentvalues of the sbear-rateexponent do not merge and remaindistinct for all Reynoldsnumbers. In tum, this impliesthataHerschel.Bulkleyfluid will exhibitsomewbatdifferentbehaviorsinceitcombines tbeBP andtbe PL rbeologicalmodels. In particular, one caninferthatayieldstress will delayandextend the transitionzone, andtbe transitionalvalues of frictionfactor will eventuallymergewith the fully-turbulentcurvefora PL fluidwith an exponentequal to whateverappearsintbeHerscbel-Bulkleymodel, see Eq. 7below. At tbispoint,itisappropriate to introduce some viscometer data forarepresentativedrillingmud.Sometypicalrotating-cupdata are shownin Fig. 6. Dial readingsfor six: rotaryspeeds are given. The twoparametersin the BP model  YP and PV and the PL model  K and n ) are determinedaccording to standard API procedures by fittingthese two rheologicalmodels to the 300 and 600 rpm data. In contrast, the threeparameters of the HB modelallowagoodfit to all six datapoints.Sinceannularflowshearrates are normallybelowthe100 rpm (170 sec-I) viscometerdata, it is important to haveamodelwhichisvalid at low shear rates. On theother hand, the HB modelcanalso be used to estimatefrictionalpressuredrops and flowregimesforflowthrough drill pipeswhereshearrates are high. The HB parametersfor the particular data of Fig. 6are: T = YieldStress   K(Shear Rate - ::: 14   0.281  ,y o.m Ibf/l00 ft2 [0.4788 Pal (7) Theseparameters will be usedforillustrationpurposesinFigs.7 tbru 11. Aweight of 12 ppg[1,438 kglmJ] will also be assumedforthisexamplemud.Figure7presentsfrictionfactorsthrough the transitionzonefor the example mud flowingthroughpipes. The PL modelleads to transition at lower pump rates.Initiation of transitionisdelayed to higherpumprates based onthe HB model, but fully-turbulentflowoccurs at about thesame GRE as for tbe PL case. TheHB curveforfullyturbulentflowdoes not mergewitb the PL curve at higherReynoldsnumbersbecause the sbear rate exponentfor the PL modelisonly 0566. The resultsfor theBP modelis as expected i.e.,transitionisdelayed tbe most by using this model. Furthermore, the extent of. thetransitionzoneisexpandedsignificantly, and the GRE is an order-of-magnitudelargerbeforefully-turbulentflowisachieved. In order to relatetheseresults to atypicalfieldapplication,considerflowthroughastandardweight5-incb drill pipewith an ID of 4.276incbes.Fu1Iy-turbuientflow is achieved at a GREof 3800 and a pumprate Of 270 gpm [17.0 lis] using the PL model. The correspondingresultsfor theHB modeloccurs at a GRE of 4100 and a pump rate of 310 gpm [19.6 lis]. In contrast, tbe BP modelleads to aprediction of fully-turbulent flow at a GRE of 22,300anda pumprate of 670 gpm [42.3 Urn]. (Note: The analysisdoesnotaccountfor the effects of restrictions at tooljointswhicbcouldcauseflowtransition to begin at lower pump rates.) It is also of interest to compare the frictionalpressuregradientsforarealistic pump rate of 600 gpm[37.85 lis]. The resultsfor the PI.., HB, and BP models,respectively, are 0.085,0.099, and 0.104psilft[22.62kPa/m]. The relativeclosevaluesfor the HB andBP fluids are aresult of the HB beingfullyturbulent andthe BP beingintransition at this pump rate. Again if tooljointscausedtransition to occur at lower pump rates andthe BP wereinfully-turbulentflow at 600 gpm [37.85 lis], thenthe differencebetweenpressuredropsfor the HB and BP would be larger. Annular Flow. The flowmodelcan be extended to concentricannuli by altering tbe definition of the  Effective diameter to include the effects of the differentgeometry. The  Effective diameterforaconcentricannuli is equal to DJIy IG. The  Effective diluneterfor non ewtonianflow through a concentric annulus isthe dia meter   a circular pipethatwouldhavetheidentical pressuredrop forflow   a ewtonianfluid with a viscosity equalto the  effective viscosity which is basedonthe average wall shear rate in t annulus and has a velocity equaltothenon- ewtonianannularflow velocity. The  an function is based on acorrelation of the analyticalsolution by Fredrickson and Bird forflow of aPower-Lawfluidthroughaconcentricannulus. This functionisdependent on bothtbe ratio of inner to outer diameters andthe shear-rateexponent. The function is definedin the Appendix by Eq. A-34. This correlationcan be applied to other fluids by using the generalizedexponent  N , Eq. 1c, inplace of the usual PL exponent n. The details are givenintheAppendix undertbe beading Non-NewtonianAnnularFlow. The followingdiscussion will focus on some resultsforatypicalfieldsizeannulus. A 12-1/4 x 5 [31.12 emx 12.7 em] annulus is selectedforillustrativepurposes. The same mud rbeology, Eq. 7,andaweight of 12ppg[1,438 kg m J] will again be used forcomputingfrictionalpressuregradientsthrough the annulus. It is pertinent to note that the weight of the mud does not affectlaminarfrictionalgradients, but weightdoesinfluencetransitional and turbulentgradients.Figure8sbowsbow the ratio of tbe generalizedexponent  N over the shear-rateexponent Om varieswith shear rate for the example HB fluid. As may be seenfrom Eq. 7,  m bas the numericalvalue of 0.792.Figure8showsbow N approacbes m as wall sbear rateincreases.Thisis the expectedtrendbecause the influence of theYieldStress(YS) on T w becomesprogressivelylesswithincreasingsbearrate,compare Eqs. 1band 7. In the limit of infinitesbearrate, N m Figure8 also identifies tbe initiation of transition at 1910gpm [1205 lis], and fully-turbUlentflowbegins at 2010 gpm [126.8 lis]. This simplytellsus thatthe flowislaminarover the range of pump ratesthat are normally used in the field.Unfortunately, the ~o l does not tellusbow drill stringdynamics may influenceflow 471  4 ANEWMODELFORLAMINAR, lRANsmONAL, AND TURBULENTFLOWOF DRTI..LING MUDS SPE2S456 transition.   is reasonableto expect the energy addedtothe flowby theagitation may causetransitionat lower pumprates.Anotherparameter in themodel is the ratio of Effective diameter over theEquivalentdiameter. This ratio partia1Iy removes geometric effects and highlights thenon-Newtonian effects.  ipre shows this ratioapproaches unity as  N goes toone. This is the expected limiting condition for Newtonian flow. SinceFig. 8 shows that  N increases with pumprate,the Effective diameter also increases with pumprate. Thishas theneteffectof reducing themagnitudeof the increase in frictional pressuregradientaspumprate is increased, i.e., the fluid is  shear thinning. This leads tothequestion,whathappenstothe  apparent viscosityas pumprate increases? An Effective viscosity is defined in a man neranalogousto pipe flow. The primary difference for anannulus is: the  Effective viscosity is defined by theaverage wall shearstress acting ontheinnerandouter walls and divided by the average wall shearrate,seeEq. A-43.This viscosity is presented in Fig 10 as a function ofpumprate.   continuouslydecreases with pumprateand. has anoticeabledropattransition from laminar toturbulent flow. This is caused by the rapidlyincreasing shearrate as transition occurs. Figure 11 presents friction factors as a function o theGRB for the example annulusandmudproperties.Forsmooth walls, theincrease in friction factorthroughthetransitionzone is small.  s indicated inFig.8, thetransitionzone occurs betweenpumpratesof 1910 and 2010 gpm [120.5-126.8 I.Js]. An example oftheeffectsofrough walls is also included inFig. 11.  t may be noticedthattheinitiationoftransition occurs atthesameGRB This is consistent with theexperimentaldataofNikuradse for Newtonian pipe flow, see Schlichting. 24  s indicated in the figure, transitiontoturbulent flow takes place overasmallerrangeofpump rates, 1910 to 1970 gpm [120.5 to 124.3 I.Js], andthe fully turbulentpressuredropsare approximately 70 percentlargerthanthecorre sponding turbulent friction factors for smooth walls. The  relative roughness for this illustration is approximately0.01.This is basedon dividing theabsoluteroughness height by the  Equivalent diameter,Eq. A-14. However, thecorrect definition of relative roughness fornon-Newtonian flowsis theratioofabsolute roughnessover the  Effective diameter.Hence, relativeroughness varieswith the fluid and the shearrate.Table1presenta list ofsomeoftherelevantparametersthroughthetransitionzone for the rough-wallcase.  t can beseenthatthe relative roughnessdecreases as pumprate increases. Theauthors believe thesearethe first computations for theeffectsof wall roughness onnon-Newtonianpressuregradients in thetransitional and turbulent regimes. (Note:pressuregradientsare given in unitsof inches ofwaterper foot; theseunitsareequalto 0.0361 psiIft or 0.8164kPa/m.) Finally, Fig.12 comparesthe model with testdata for a Mixed Metal Hydroxides  MMH mud system flowing througha 5.023 x 2.375 [12.76 cm x 6.03 em] annulus.Thetests were conducted in Amoco s Drilling Hydraulics Test Facility by personnel from thePetroleum Engineering Department,Tulsa University. Pressuredrops were measuredoveralengthof 50 ft[15.24 m]. Pump ratj:swerevaried from 50 to 500 gpm [3.16 to 31.55 I.Js]. Experimentaldata for a high and a low rheology MMH fluid are included inFig. 12 Thecorrespondingrotating-cupviscometer readings atthe six standard speeds are: 327,57.3,61.5,63.0,66.5 and 25,13.5,18.3,220,28.0. These values aretheaverageof readingsfor fourdifferent samples taken duringthetimeofa tesL  s indicated inFig. 12, the high-rheologyfluid appearsto have been in the laminar-flow regimefor theentirerangeofpumprates.The model predicts transitionathigherpumpratesand friction factors that continuously decrease throughthetransition zone According to the model, the flow is not fully turbulent until aGRBof 18,000 andapumprate of 77JJ gpm [45.42 I.Js]. The predicted values ofturbulent friction factorsare nearly constant for a wall roughness of 0.00018 in [0.0046 mm]. The above valueof wall roughness gave good agreement between predictedturbulent friction factors andexperimentaldata for the low- rheologyfluid. Unfortunately, water teststodetermine wall roughness forthe annuluswerenotperformed. However, Fig. 12 shows encouraging agreement betweenpredictionsfor the low rheology fluid andthecorrespondingdatathrough  u three flow regimes. Additional tests were conducted with bentonitemuds flowing throughrough pipes.  n these cases wall roughness was determined via watertests, andpredictions from the model, using the experimental values of wall roughness, agree v ry closely with thetestdata for  u three flow regimes. This data covers aGRBrangeof 500 to 270,000 and includes relative roughnessvalues upto 0.002. Becauseof spacelimitations, these non-Newtonian pipe-flow dataarenotpresentedhere. CONCLUSIONS 1. A newanalysis has been developed for non-Newtonian flow through pipes andconcentric annuli. Themethod is based oil relatingnon-Newtonian flows to Newtonian flows. The advan tage is that well-established results  or Newtonian flows canbe applied to non-Newtonian flows. 2The  Effective diameter is a key conceptofthe method.  t accounts for bothgeometricand non-Newtonian fluid effectsonfrictional pressuregradients in pipes and annuli.3. Resultsagree with finite-difference computations for laminar flow ofa Herscbel-Bullcley fluid through concentric annuli.4. The analysis is valid for the laminar, transitional, and fully turbulent flow regimes. Themethodincorporatesa new transi- tion criteriathataccountsfora delay in flow transition with increasing ratiooftheinner-to-outerdiameters in concentric annuli. This agrees with experimentaldatareported in theliterature. 5. Model predictionsof critical Reynolds numbersagree with published data forBPpipe flows. 6. The Herschel-Bulkley rheolOgical model is used which includes Newtonian, Power-Law, and BinghamPlastics asspecial cases. This IDOre accurate rheological model is equally valid for predictions ofpressuregradients in drill pipes and the large annuli typical of full-scaJe  ri ling~ 7. Whenusing thesameviscometerdata,results fromtheanalysesshow transition from laminar toturbulent flow occurs athigherpumpratesthan for a Power-Law fluid, but signifi cantlylower than thecorresponding BinghamPlastic fluid 8. TheColebrookequationforturbulent friction factors has been extended so that itapplies to non-Newtonian flows through pipes and concentricannuli with smoothorrough walls. 9. Themethodaccounts for theeffectsof wall roughness on frictional. pressuregradients in transitional flow ã. 10. Model predictionsofpressuregradients in  u three flow regimeshave been verified by experimental data forMMHand bentonite  ~ s flowing through pipes and annuli withvarying degreesof wall roughness. 472

CB Maideen

Jul 23, 2017
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