致密多孔介质中气体视渗透率的有效联系外文文献翻译.docx
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致密多孔介质中气体视渗透率的有效联系外文文献翻译.docx
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致密多孔介质中气体视渗透率的有效联系外文文献翻译
EffectiveCorrelationofApparentGasPermeability
inTightPorousMedia
FarukCivan
Abstract:
GaseousflowregimesthroughtightporousmediaaredescribedbyrigorousapplicationofaunifiedHagen–Poiseuille-typeequation.Properimplementationisaccom-plishedbasedontherealizationofthepreferentialflowpathsinporousmediaasabundleoftortuouscapillarytubes.Improvedformulationsandmethodologypresentedhereareshowntoprovideaccurateandmeaningfulcorrelationsofdataconsideringtheeffectofthecharac-teristicparametersofporousmediaincludingintrinsicpermeability,porosity,andtortuosityontheapparentgaspermeability,rarefactioncoefficient,andKlinkenberggasslippagefactor.
Keywords:
Tightporousmedia·Apparentgaspermeability·Rarefactioncoefficient·Klinkenberggasslippagefactor·Tortuosity
1Introduction
Descriptionofvariousgaseousflowregimesthroughtightporousmediahasdrawncon-siderableattentionbecausetheconvetionalDarcy’slawcannotrealisticallydescribethevarietyoftherelevantflowregimesotherthantheviscousflowregime.Forexample,Javadpouretal.(2007)havedeterminedthatgasflowinshalesdeviatesfrombehaviordescribedbytheconventionalFick’sandDarcy’slaws.Therefore,manyattemptshavebeenmadeindescribingthetransferofgasthroughtightporousmediaundervariousregimes.Sucheffortsareofutmostpracticalimportancewhendealingwithextractionofhydro-carbongasesfromunconventionalgasreservoirs,suchasshale-gasandcoal-bedmethanereservoirs.
SkjetneandGudmundsson(1995),andSkjetneandAuriault(1999)theoreticallyinvestigatedthewall-slipgasflowphenomenoninporousmediabasedontheNavier-Stokesequation,butdidnotofferanycorrelationfortheKlinkenbergeffect.Wuetal.(1998)developedanalyticalproceduresfordeterminationoftheKlinkenbergcoefficientfromlaboratoryandwelltests,butdidnotprovideanycorrelation.Havingreviewedthevari-ouscorrelationsavailable,SampathandKeighin(1982)proposedanimprovedcorrelationfortheKlinkenbergcoefficientoftheN2gasinthepresenceofwaterinporousmedia,expressedhereintheconsistentSIunitsas
wherebkisinPa,K∞isinm2,andφisinfraction.Thesignificanceofthiscorrelationisthatitsexponentisveryclosetothe−0.50exponentvalueobtainedbytheoreticalanalysisinthisarticle.
BeskokandKarniadakis(1999)developedaunifiedHagen–Poiseuille-typeequationcoveringthefundamentalflowregimesintightporousmedia,includingcontinuumfluidflow,slipflow,transitionflow,andfreemolecularflowconditions.Abilitytodescribeallfourflowregimesinoneequationaloneisanoutstandingaccomplishment.However,theempiricalcorrelationoftheavailabledataofthedimensionlessrarefactioncoefficientisamathe-maticallycomplicatedtrigonometricfunction.Asdemonstratedinthisarticle,muchaccu-ratecorrelationofthesamedatacanbeaccomplishedusingasimpleinverse-power-lawfunction.
Florenceetal.(2007)madeanattemptatutilizingtheHagen–Poiseuille-typeequationofBeskokandKarniadakis(1999)toderiveageneralexpressionfortheapparentgasperme-abilityoftightporousmediaandcorrelatedsomeessentialparametersbymeansofexper-imentaldata,includingtheKlinkenberggasslippagefactorandtheKnudsennumberbyignoringtheeffectoftortuosity,althoughitisanimportantfactorespeciallyintightporousmedia.Therefore,althoughtheiroverallmethodologyisreasonable,theirformulationanddataanalysisprocedurerequiresomecriticalimprovementsaspointedoutinthisarticlewhenattemptingtoapplytheHagen–Poiseuille-typeequation,originallyderivedforpipeflowtotight-porousmediaflow.Theirtreatmentneglectsanumberofimportantissues.TheHagen–Poiseuille-typeequationofBeskokandKarniadakis(1999)hasbeenderivedforasingle-pipeflow.Whenthebundleoftortuoustubesrealizationofthepreferentialflowpathsintight-porousmedia(Carman1956)isconsidered,thenumberandtortuosityofthepref-erentialflowpathsformedinporousmediashouldbetakenintoaccountastheimportantparameters.
Further,theapproachtakenbyFlorenceetal.(2007)forcorrelationoftheKlinkenberggasslippagefactorisnotcorrectandconsequentlytheircorrelationcannotrepresentthedataoverthefullrangeofthegasmolecularmass(commonlycalledweight).Theseerrorsarecorrectedinthisarticlebyarigorousapproachwhichleadstoaveryaccuratecorrelationoftheirdatawithacoefficientofregressionalmostequalto1.0.Inaddition,thepresentanalysislendsitselftoapracticalmethodbywhichtortuosityoftightporousmediacanbedeterminedusingtheflowdataobtainedbyconventionalgasflowtests.Totheauthor’sknowledge,suchamethoddoesnotpresentlyexistintheliterature.
Hence,theprimaryobjectivesofthisarticlearethreefold:
(1)Correlationoftherarefactioncoefficient
(2)Derivationoftheapparentgaspermeabilityequation
(3)CorrelationoftheKlinkenberggasslippagefactor
Theseissuesareresolvedandverifiedinthefollowingsectionsbytheoreticalmeansandrigorouslyanalyzingexperimentaldata.
2CorrelationoftheRarefactionCoefficient
BeskokandKarniadakis(1999)derivedaunifiedHagen–Poiseuille-typeequationforvolu-metricgasflowqhthroughasinglepipe,givenbelow:
wheretheflowconditionfunctionf(Kn)isgivenby
whereKnistheKnudsennumbergivenby
whereRhandLhdenotethehydraulicradiusandlengthofflowtube,andλdenotesthemean-free-pathofmoleculesgivenby(Loeb1934)
wherepistheabsolutegaspressureinPa,TistheabsolutetemperatureinK,Misthemolecularmassinkg/kmol,Rg=8314J/kmol/Kistheuniversalgasconstant,andµistheviscosityofgasinPa.s.
Equation2describesthefundamentalflowregimes,namelytheconditionsofcontinuumfluidflow(Kn≤0.001),slipflow(0.001 TheparameterαappearinginEq.3isadimensionlessrarefactioncoefficientwhichvariesintherangeof0<α<αoover0≤Kn<∞.BeskokandKarniadakis(1999)provideanempiricalcorrelationas: whereα1=4.0,α2=0.4,andαoisanasymptoticupperlimitvalueofαasKn→∞(representingfreemolecularflowcondition),calculatedby: Here,bdenotesaslipcoefficient.Theyindicatethatα=0andb=−1intheslipflowcondition,andthereforeEq.7becomes: TheexpressionofEq.6ismathematicallycomplicated.Inthefollowingexercise,itisdemonstratedthatasimpleinversepower-lawexpressionasgivenbelowprovidesamuchmoreaccurateandpracticalalternativetoEq.6fortherangeofdataanalyzedbyBeskokandKarniadakis(1999): whereAandBareempiricalfittingconstants.NotethatEq.(9)honorsthelimitingconditionsof0<α<αoover0≤Kn<∞.Infact,itcanbeshownthat Fig.1PresentapproachusingEq.9accuratelycorrelatesthedataofbothLoyalkaandHamoodi(1990)usingthetheoreticallypredictedupperlimitvalueofαo=1.358andTisonandTilford(1993)usinganadjustedupperlimitvalueofαo=1.205.Thepresentapproachyieldsaccuratefitofdatawithcoefficientsofregressionsverycloseto1.0 AsillustratedinFig.1,thepresentapproachusingEq.9accuratelycorrelatesthedataofbothLoyalkaandHamoodi(1990)usingthetheoreticallypredictedupperlimitvalueofαo=1.358andTisonandTilford(1993)usinganadjustedupperlimitvalueofαo=1.205.Consequently,thedataofLoyalkaandHamoodi(1990)iscorrelatedas Thus,A=0.1780andB=0.4348.Ontheotherhand,thedataofTisonandTilford(1993)iscorrelatedas Figure2showsthatthedataofLoyalkaandHamoodi(1990)canbecorrelatedaccuratelybyboththepresentcorrelationapproachusingEq.9withacoefficientofregressionofR2=0.9871andtheempiricalequationgivenbyBeskokandKarniadakis(1999)withacoefficientofregressionofR2=0.9697usingthetheoreticallypredictedvalueofαo=1.358.However,asindicatedbythecomparisonofthecoefficientsofregressions,thepresentapproachyieldsamoreaccuratecorrelationthanthatofBeskokandKarniadakis(1999). Fig.2DataofLoyalkaandHamoodi(1990)canbecorrelatedaccuratelybyboththepresentcorrelationapproachusingEq.9withacoefficientofregressionofR2=0.9871andtheempiricalequationgivenbyBeskokandKarniadakis(1999)withacoefficientofregressionofR2=0.9697usingthetheoreticallypredictedvalueofαo=1.358.However,asindicatedbythecomparisonofthecoefficientsofregressions,thepresentapproachyieldsamoreaccuratecorrelationthanthatofBeskokandKarniadakis(1999) Figure3showsthatthepresentcorrelationwithEq.9usingtheadjustedvalueofαo=1.205representsthedataofTisonandTilford(1993)accuratelywithacoefficientofregressionofR2=0.9486,closeto1.0.Incontrast,theempiricalequationgivenbyBeskokandKarniadakis(1999)usingtheadjustedvalueofαo=1.19leadstoalowerqualitycorrelationwithacoefficientofregressionofR2=0.7925,lessthan1.0.Asindicatedbythecomparisonofthecoefficientsofregressions,thepresentapproachyieldsamuchmoreaccuratecorrelationthanthatofBeskokandKarniadakis(1999) Fig.3PresentcorrelationwithEq.9usingtheadjustedvalueofαo=1.205representsthedataofTisonand
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