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THE_111THE,the,The

    THE

60

    ;JournalofHydrodynamics,Ser.B,5(2003),6064

    ;ChinaOceanPress,BeijingPrintedinChina

    ;THEPRODUCTIVITYOFWELLPATTERNWITHHORIZONTALWELLS ;ANDVERTICALWELLS

    ;ZhaoChun.sen,CuiHaiqing,SongWen?ling

    ;DepartmentPetroleumEngineering.DaqingPetroleumInstitute,Daqing163318,China,e.mail:zha-

    ;ochs@163.com

    ;ZhangZhen

    ;TheFirstOilProductionFactoryDaqingOilfieldCorporationLtd.,Daqing163001,China

    ;(ReceivedApr.10,2002)

    ;ABSTRACT:Thecalculationofproductivityandthechoice ;ofwellpatternofmixedhorizontalandverticalwellsweredis

    ;cussedinthispaper.Becausethenaturalldrainageareasof ;horizontalwellsareovalratherthancircular,rectangular ;drainagewellpatternsshouldbeadopted.Usingpseudo3-di

    ;mensionalmethodandconformaltransformation,thepotential ;functionofarowofhorizontalwellswasdeduced.Usingthis ;functionandthesuperpositionprinciple,ananalyticalsolution ;oftheproductivityfortherectangularmixedwellpatternwas ;given.Quantitativemeansfordefiningtheshapesoftheopti

    ;mumpatternweredevelopedinthispaperbyintroducingtwo ;dimensionlessparameters,theshapefactoranddimensionless ;lengthofhorizontalwel1.Theformulaforproductvitywas ;usedtooptimizethemixedwellpatterns.Thecurvesofthe ;optimizedshapefactoragainstdimensionlesslengthofhori

    ;zontalwellweredrawn.Theresultsshowthatthepatternar

    ;eaandformationthicknesshavenoeffectontheoptimalshape ;factorsandcanbeappliedtothedesignofdevelopmentforoil ;fields.

    ;KEYWORDS:horizontalwell,verticalwell,rectangular ;staggeredwellpattern,productivity,optimumpattern ;1.INTRoDUCTIoN

    ;Theuseofhorizontalwellsarethemostinter.

    ;stingissueintheoilindustryrecently.Theyhave ;largercontactareaswiththereservoircompared ;withverticalwells.Themosttheoreticalreports ;focusedonthesinglehorizontalwellproduction. ;andsomeauthorsconsideredthehorizontalwellas

    ;infinitelineinflux,whichisnotproper,because ;eachhorizontalwellhasafinitelength~.’.People

    ;thenadoptedapseudo3.dimensionalmethod,and ;consideredahorizontalwellasaverticalfracture. ;forwhichtheheightisequaltothethicknessof ;reservoirformationandthelengthisequaltothe ;horizontalwelllength.Onthisbasis,aneffective ;methodofpredictingtheproductivityofhorizontal ;wellwasthusdevelopedE.?.

    ;Assumingthatthewellpatternsonlyconsistof ;producinghorizontalwells,Duprunowlczhasuseda ;two.dimensionalapproximationofhorizontalwells ;todeterminethepatternsizeandshapetogivethe ;highestproductivityinpseudosteadyflow. ;Theresearchofthemixedwellpatternin

    ;steadyflowisanewresearchingissueinoilfield ;development.Inthispaper,thesuperposition ;principleofpotentialisusedtodeducetheproduc

    ;tivityequationofstaggeredpattern.Themethod ;ofoptimizingwellpatternsisalsopresentedinthis ;paper?

    ;AsshowninFig.1,thestaggeredpatterncon- ;sistsofinfiniterowsofhorizontalwellsandverti. ;calwells.Inthepattern,thehorizontalwellsare ;producingwellsandtheverticalwellsareinjecting ;wells.Thewellspacingis2aandtherowspacingis ;2b,sotheareaofthepatternunitis2a×2b.The ;lengthofhorizintalwellsisassumedtobe2land ;representedbyathickenedsegmentofline.Injec- ;ting?-productionbalancemustbekeptduringapro?- ;ducingperiod.

    ;Theanalyticalsolutionoftheproductionpo- ;tentialin3-dimensionaltobeobtaineddirectlydue ;tothecomplexityofhorizontalwellflows,so ;pseudo3dimensionalanalysisisconductedtosub- ;dividethethree.dimensionalsolutionintotwo2.di. ;mensionalsolutionsrespectivelyinXYandinYZ ;ProjectsupportedbyChinaNationalPetroleumCorporationInnovationFundation.(GrantNo:2002CX

    17)

    ;]_

    ;

    ;

    ;?Y

    ;(a.6)

;?????????????????

    ;,0.,’,

    ;g--

    ;?___

    ;Fig.1Sketchofwellpattern

    ;axialplane.

    ;IntheXYplane,thepotentialdistributionofa ;lineofhorizontalwellsisdeducedbyconformal ;transformation,thenthepotentialofanyspotin ;thewellpatterncanbedeterminedbyusingtheSU

    ;perpositionprinciple.Theproductivitycanbede

    ;terminedbythepotentialdifferencebetweeninjec

    ;tingandproducingwellsaccordingtothepercola

    ;tionflowresistanceofhorizontalwellsintheYZ ;plane.Optimizationisdonewithasuitableparam

    ;etergroupbycompilingacomputerprogram. ;

    ;1.

    ;

    ;

    ;

    ;61

    ;Flg.3~ketcholcontormaltransformation ;1sin7l”z(1a)1n

    ;叫一(1b)叫一

    ;sin2n

    ;canbedeterminedwiththetransformation ;ofWl—sin7I”z

    ;,thusthefollowingformulaecan

    ;a

    ;bededuced:

    ;】一sin7l”zch7l”v(2a)

    ;lnz(2)

    ;TERN17cozsh(2b)

    ;Consideraninfinitenumberhorizontalwells ;arrangedasinFig.1,andtakeamiddlepointofSubstitutingy0and.z1intoEq.2willresultin

    ;anyastheoriginalpoint.Assumethatraxisisa

    ;l0ngthewellrowwhileaxiiVeticaltothe ;l一一slni..

    ;rcl

    ;,17l0

    ;row.Fig.2showstherow.Zn

    ;?.r

    ;II

;??

    ;II

    ;??

    ;0:H

    ;_.J2J

    ;12a’

    ;Fig.2Sketchofarowofinfinitehorizontalwells ;InFig.2,accordingtothesymmetryofthe ;distributionofthehorizontalwellsandthefields. ;theformationcanbedividedintoaseriesofunits ;andeveryunithasimpermeableboundary.Thus ;onlyoneunitiSneededtobetakenintoaccount. ;NOWwetaketheunitenvelopedbetweenthetwo ;dashedlines.

    ;Inordertostudytheflowintheunit.two ;conformaltransformationsareusedtoturnthe ;complexflowinthezplaneintolinearflowinthe ;Waxialplane.Theformulaearegivenasfollows ;andtheresultiSshowninFig.3.

    ;1.sinz

    ;a

    ;(3)

    ;TherelationshipbetweenW1planeandtheWplane ;canbeobtained:

    ;一一sin~:chr/sinsin~:chr/

    ;171zocosh17sinzcos~:chr/

    ;a

    ;(4a)

    ;(4b)

    ;Thepotentialoftherowofhorizontalwellsinthe ;Wplane

    ;

    ;+c(5)

    ;

    ;62

    ;Therelationshipbetweenand(1,1)can

    ;gottenfromEq.C5)byeliminating

    ;(/%).(/l.

    ;1

    ;?叩十sh叩一

    ;Byusingch叩一sh叩一1,aboveequationbe.

    ;COmeS

    ;ch叩一E(Zo~++)]+

    ;,

;(6)

    ;SubstitutingEqs.(3),(4)and(6)intoEq.(5) ;yieldsthepotentialdistributionatanypoint(z,) ;ofarowofhorizontalwells

    ;

    ;h’1{sin+ch

    ;COSzz+[(sin+ch--COS2z)

    ;4sinchsin22a}+c

    ;AccordingtoMusket’S

    ;verticalinjectingwellsalong

    ;regularspacingofEq.(2a),

    ;tionoftherowis

    ;arch

    ;1__

    ;2

    ;(2n+1)

    ;2a

    ;.

    ;m

    ;

    ;2a+

    ;sh

    ;_2arch+

    ;ch

    ;.

    ;-

    ;4sh”‘..

    ;2.’a——

    ;(10)

    ;whereRoisthedimensionlesspotentialdifference ;betweentheinjectingwellandtheproducingwell ;intheXYplane.

    ;InordertotestifytheaccuracyofEq.(9)in ;theXYaxialplane,letab42d,=3andthe

    ;horizontalwelllength2l2r.Underthesecon-

    ;ditions.thehorizontalwellcanbeconsideredas ;theverticalwellandthestaggeredwellpatternis ;turnedintothe5-spotverticalwellpattern.Then ;(7)Eq?(9)canbesimplifieda

    ;theory,rorarow.fQrrkhAp

    ;thez.axis,whichhas…r

    ;thepotentialdistribu-

    ;

    ;ln(chcos.rrx)+c

;(11)

    ;wheredisthedistancebetweentheinjectingwell ;andtheproducingwel1.

    ;(8)Eq?(11)isveryclosetoMusket’sproductivity

    ;formulaof5-spotverticalwellpattern: ;Fromthesuperpositionprinciple,wecangetrckhA ;p

    ;me…ofhemixedwellpatternatanypoint

    ;ofYandthepotentialdifferencebetween.6l9)( ;z,)”,…r……

    ;theinjectingwellatthepoint(0,0)andtheprodu- ;(12)

    ;cingwellatthepoint(a,b)intheXYaxialSowecansayEq.(9)iscorrect.

    ;plane.Bytakingtheinternalresistanceofhorizon-Forconvenience,thedimensionlessproductiv-

    ;talwellinYZaxialplane,theproductivityoftheityisdefinedas

    ;horizontalwellcanbedeterminedbyEq.(9). ;Q

    ;arch

    ;

    ;]?l

    ;(arch

    ;(2n1)

    ;2a

    ;.f

    ;m

    ;+

    ;(9)

    ;Q一一2rckhA

    ;p

    ;Q(13)

    ;3.OPTIMIZATIONOFWELLPATTERN

    ;FromEq.(13),thedimensionlessproductivity ;isinfluencedbythehorizontallength2l,thewell ;patternfactors2aand2b,andthereservoirthick

    ;?

    ;?

    ;+

    ;

    ;ness.Sotwodimensionlessparametersarede- ;fined:theshapefactorFa/b,andthedimen-

    ;sionlesslengthofhorizontalwelllDL/(2a×2b)

    ;.Thedimensionlessproductivityofhorizintalwell ;isdifferentwhentheshapefactorchanges,while ;theareaofthepatternisafixedvalue.Therefore,

    ;theremustbeanoptimumshapefactortogetthe ;largestproductivity.

    ;Inordertofindtherelationshipbetweenthe ;optimumfactorandthedimensionlesslength,the ;equationsinthispaperareusedtocompilecomput

    ;erprogams.Underthepracticalconditions,calcu

    ;lationwasconductedwhenthepatternunitareais ;takenas22500m.90000mand360000mandthe ;reservoirthicknessas10m,30mand50mrespec

    ;tively.

    ;Thecalculatedresultsindicatethat

    ;(1)Therereallyexistsanoptimumpattern ;shapefactorFoptogetthegreatestproductivity, ;whenthedimensionlesslengthlDisgiven.Thedi

    ;mensionlessproductivityiscalledtheoptimumdi

    ;mensionlessproductivity.

    ;(2)Thewellpatternunitareaandthickness ;havelittlerelationtoFop.Sotherelationshipbe. ;tweenFopandIDcanberegardedasstablewhenar

    ;eaandthicknesschanged,whichisshowninFig. ;4.Thereforethedefinitionofthedimensionless ;lengthofhorizintalwellinthispaperhasgreat ;theoreticalvalueandoffersatheoreticalbasisfor ;theoptimizationofvariouspatternswithdifferent ;thicknessandwellpatternunitarea.

    ;(3)FromFig.4aconclusioncanbedrawn

    ;thatwiththeincreaseofthedimensionlesslength, ;theoptimumshapefactorincreases,thatistosay, ;thepatterndeviatesfromaquadratemore.Re. ;gressionindicatesthattherelationshipbetween ;themcanbeexpressedbythefollowingequation: ;Fop0.024/z+0.2612/D+1.4478(14)

    ;Eqs.(9)and(14)offeratheoreticalbasisfor ;thedesignofdevelopmentanddynamicanalysis, ;especiallyforthedesignofthepattern. ;Whentheoptimumshapefactorandthelayer ;thicknesschange,therelationshipbetweenopti. ;mumdimensionlessproductivityanddimensionless ;lengthofhorizintalwellisgiveninFig.5. ;Fromabovefiguresweknowthat

    ;(1)Thedimensionlessproductivityincreases ;withtheincreaseofthedimensionlesslength. ;The

    ;63

;Fig.4Therelationbetweenoptimumshapefactoranddi

    ;mensionlesslength

    ;0.12

    ;0.08

    ;0.04

    ;O.OO

    ;0l2

    ;

    ;0.12

    ;0.08

    ;0.04

    ;O.O0

    ;0l2

    ;0.12

    ;0.08

    ;0.04

    ;O.0o

    ;0l2

    ;Fig.5Therelationofoptimumdimensionlessproductivity

    ;anddimensionlesslengthofhorizonta1wel1 ;incrementbecomesverylittleafterdimensionless ;lengthreachesacertainvalue.

    ;(2)Thedimensionlessproductivitydecreases ;withtheincreaseofthereservoir,buttendsto ;withincreasingthepatternunitarea.Forexam. ;pie,theinfluenceofthethicknessisslightwhen ;theunitareais36000m.

    ;(3)Thedimensionlessprouctivitydecreases ;whenthepatternunitareaincreases.

    ;8

    ;6

    ;2

    ;()

    ;i,i

    ;Fig.6Theincreasedpercentofdimensionlessproduction ;oftheoptimumrectangularwellpattern ;Inordertoexplaintheadvantagesofrectan. ;gularpattern,curvesaredrawntocomparetheop. ;timumdimensionlessproductivityoftheoptimum ;rectangularpatternwiththatofthesquarepat. ;tern?TheresultisshowninFig.6.Thelargestper. ;l/.:

    ;

    ;64

    ;centageoftheincrementisabout7%,anditis ;moreobviouswhenthereservoirthicknessiSsma1. ;1erandthewellpatternunitareaissmaller. ;4.CoNCLUSIoNS

    ;(1)Thepotentialofarowhorizontalwellin ;XYaxialplaneiSdeduced,whichiSimportantfor ;theresearchofhorizontalwellpattern.

    ;(2)Thepercolationtheoryofthemixedwell ;patternofhorizontalandverticalwellsiSpresen. ;ted.Theaccurateproductivityofthepatternin ;pseudo3.dimensionaliSgiven.

    ;(3)Therelationshipbetweentheoptimum

    ;shapefactorofthewellpatternandthedimension. ;1esslengthofhorizontalwelliSpresented.Thein. ;fluencefactorstotheproductivityarediscussed. ;REFERENCES

    ;[1]MUSKATM.Flowofhomogeneousfluidsthroughpor- ;OIlSmedia[M3.NewYork:McGrawHill,1937,273

    ;275.

    ;[2]SUPRUNOWICZR.Verticalconfinedwaterdriveto ;[3]

    ;[43

    ;[5]

    ;[6]

    ;horizontalwell,PartI:waterandoilofequaldensities ;rA].The3rdTechnicalMeetingoftheSouthSaskatche- ;wanSection[C].ThePertroleumSocietyofCIM.Regi

    ;na,1989,123134.

    ;JOSHIS.D.Productionforecastingmethodsforhori

    ;zontalwells[AJ.SPE17580[C3.usA,1988,303316.

    ;KARCHERM.J.Giger:somepracticaltormulasto ;predicthorizontalwellbehavior[A3.SPE15430[c3. ;USA,1988,142156.

    ;SUPRUN0WICZR.Thechoiceofpatternsizeand ;shapeforregulararraysofhorizontalwellsEJ].Journal ;ofCanadianPetroleumTechnology,1992,31(1):3944.

    ;SUPRUN0WICZR.Theproductivityandoptimum ;patternshapehorizontalwellsarrangedinstaggeredrec- ;tangulararrays[J].JournalofCanadianPetroleumTech- ;nology,1992,31(6):4146.

    ;

    ;

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