Matrix Heterogeneity Effects on Gas Transport and

TranspPorousMed(2009)80:281–304DOI10.1007/s11242-009-9359-4MatrixHeterogeneityEffectsonGasTransportandAdsorptioninCoalbedandShaleGasReservoirsEbrahimFathi·I.YücelAkkutluReceived:28May2008/Accepted:9February2009/Publishedonline:3March2009©SpringerScience+BusinessMediaB.V.2009AbstractIncoalbedsandshales,gastransportandstorageareimportantforaccuratepredictionofproductionratesandfortheconsiderationofsubsurfacegreenhousegasseques-tration.Theyinvolvecoupledfluidphenomenainporousmediumincludingviscousflow,diffusivetransport,andadsorption.Standardapproachtodescribegas–matrixinteractionsisdeterministicandneglectstheeffectsoflocalspatialheterogeneitiesinporosityandmaterialcontentofthematrix.Inthisstudy,adoptingweak-noiseandmean-fieldapproximationsandusingastatisticalapproachinspectraldomain,matrixheterogeneityeffectsareinvestigatedinthepresenceofnon-equilibriumadsorptionwithrandompartitioncoefficient.Itisfoundthatthelocalheterogeneitiescangeneratenon-trivialtransportandkineticeffectswhichretardgasreleasefromthematrixandinfluencetheultimategasrecoveryadversely.Macro-transportshows1/[1+NPe/(1+NPe)]dependenceontheP´ecletnumber,andpersistsatthediffusiveultra-lowpermeabilitylimit.Macro-kineticsisdirectlyrelatedtoThielemodu-lusbythefollowingexpression:NTh/(1+2NPe).Itleadstotrappingofgasintheadsorbedphaseduringitsreleasefromthematrix,andtoanadsorptionthresholdduringthegasuptakebythematrix.Botheffectsareproportionaltotheinitiallyavailableadsorbedgasamountandbecomesmorepronouncedwiththeincreasingvarianceoftheporosityfield.Consequently,anewupscaleddeterministicgasmassbalanceisproposedforpracticalpurposes.Numericalresultsarepresentedshowingfreeandadsorbedgasdistributionsandfractionalgassorp-tioncurvesforuniporecoalmatrixexhibitingGaussianporositydistribution.Thisstudyisauniqueapproachforourfurtherunderstandingofthecoalbedsandgasshales,anditisimportantforthedevelopmentofsoundnumericalgasproductionandsequestrationmodels.KeywordsHeterogeneity·Upscaling·Gasadsorptionkinetics·Macro-transport·Macro-kineticsE.Fathi·I.Y.AkkutluMewbourneCollegeofEarthandEnergy,UniversityofOklahoma,Norman,USAI.Y.Akkutlu(B)SchoolofPetroleumandGeologicalEngineering,SarkeysEnergyCenter,UniversityofOklahoma,RoomT-311,100EastBoyd,Norman,OK73079-1003,USAe-mail:[e-mail],I.Y.AkkutluListofSymbolsB0Absolutecoalpermeability(cm2)CFreegasconcentration(mol/ccpore)CµAdsorbedgasconcentration(mol/ccsolid)CµsMaximumadsorbedgasconcentration(mol/ccsolid)DMoleculardiffusioncoefficient(cm2/s)DApparentdiffusioncoefficient(cm2/s)EAdsorbateadsorbentinteractionenergy(J/mol)gAveragefreegasconcentration(mol/cc)KPartition(distribution)coefficient(fraction)kfGasadsorptionratecoefficient(1/s)krGasdesorptionratecoefficient(1/s)kr∞Gasdesorptionrateconstantatzeroenergylevel(1/s)RUniversalgasconstant(JK−1mol−1)rPorehalfwidth(cm)tTimecoordinate(s)TTemperature(K)xSpacecoordinate(cm)GreekSymbolsαEffectivedriftvelocity(m/s)φPorosity(fraction)Solid-to-bulkvolumeratio(fraction)σ2fVarianceofporosityfluctuationsµGasviscosity(kg/cms)λPorositycorrelationlength(cm)1IntroductionProductionfromcoalbedsandshalegasreservoirsmakeupnearly15%ofthetotalannualnat-uralgassupplyintheUnitedStates.Manyothercountriescurrentlyinvestigatethepotentialoftheseunconventionalresources.Australia,Canada,China,andIndiahavecommercialpro-jectsoncoalgasproduction,whileothersidentifynewshalegasresourcesastheyconsidertheincrementalshalegasproductionintheexistingreservoir.Thesenaturalgasresourcesareestimatedtoexceed25,000Tscfgloballyand,undertheprojectedenergyportfolio,theyarepredictedtoplayanimportantroleonenergysupply(JenkinsandBoyer2008).Despitetheircommercialimportance,exploitationoftheseresourcesraisestechnologicalchallenges.Gaswellproductivitiesareinfluencedprimarilybylow-permeablenatureofthereservoirformation.Porosityandabsolutepermeabilityofthematricesaresignificantlylessthanthosebelongtotheconventionalgasreservoirs,takingvaluestypicallyintherangeofamillidarcyintheproducingareas.Inaddition,duetolargeinternalsurfaceareasofthematrices,coalbedsandgasshaleformationscontainingsignificantamountsoforganicmatter(e.g.,Devonianshales)retainalargeportionofnaturalgasatanadsorbedstate.Thelatterisaphysicalmechanismwhichplaysanimportantroleduringtheestimationofgas-in-placeandthefuturereservoirpredictions(King1990).123MatrixHeterogeneityEffectsonGasTransportandAdsorptioninCoalbedandShaleGasReservoirs283Tightnatureofthematricesandtheirabilitytoretainthegasatanadsorbedstatealsomaketheseenvironmentsimportantgastrappingandstoragelocations.Coalbedsareconsid-eredtobeoneofthetargetedsubsurfaceenvironmentsforthegreenhousegassequestration.Althoughnolarge-scaleprojectscurrentlyexist,fieldtestsarebeingperformedforCO2injectionandenhancedcoalbedmethaneproduction,i.e.,CO2–ECBM.Gasshalesarealsolikelytoplayaroleinsequestrationsimilarinmagnitudetocoalsinnearfuture(Nuttall2005).Avibrantandfast-growingliteratureexistsrelatedtovariousaspectsofcoalsandgasshales,includingoperational(e.g.,drilling,completion,andproduction)andtechnologicalchallenges.Thela