Process-Based-Reconstruction-of-Sandstones-and-Pre

TransportinPorousMedia46:311–343,2002.c
2002KluwerAcademicPublishers.PrintedintheNetherlands.311ProcessBasedReconstructionofSandstonesandPredictionofTransportPropertiesP˚AL-ERICØRENandSTIGBAKKEStatoilResearchCentre,N-7005Trondheim,NorwayAbstract.Wepresentaprocessbasedmethodforreconstructingthefullthree-dimensionalmicro-structureofsandstones.Themethodutilizespetrographicalinformationobtainedfromtwo-dimensionalthinsectionstostochasticallymodeltheresultsofthemainsandstoneformingprocesses–sedimentation,compaction,anddiagenesis.WeapplythemethodtogenerateFontainebleausand-stoneandcomparequantitativelythereconstructedmicrostructurewithmicrotomographicimagesoftheactualsandstone.Thecomparisonshowsthattheprocessbasedreconstructionreproducesadequatelyimportantintrinsicpropertiesoftheactualsandstone,suchasthedegreeofconnectiv-ity,thespecificinternalsurface,andthetwo-pointcorrelationfunction.Astatisticalreconstruc-tionofFontainebleausandstonethatmatchestheporosityandtwo-pointcorrelationfunctionofthemicrotomographydatadiffersstronglyfromtheactualsandstoneinitsconnectivityproperties.Transportpropertiesofthesamplesaredeterminedbysolvingnumericallythelocalequationsgov-erningthetransport.ComputedpermeabilitiesandformationfactorsofprocessbasedreconstructionsofFontainebleausandstonecomparewellwithexperimentalmeasurementsoverawiderangeofporosity.Keywords:3Dreconstruction,microstructure,Fontainebleausandstone,percolation,permeability,formationfactor.1.IntroductionThemicrostructureofaporousmediumandthephysicalcharacteristicsofthesolidandthefluidsthatoccupytheporespacedetermineseveralmacroscopictransportpropertiesofthemedium.Understandingtherelationshipbetweenmicrostructureandtransportisthereforeofgreattheoreticalandpracticalinterestinmanyfieldsoftechnology.Importantindustrialexamplesincludetheproductionofoilandgasfrompetroleumbearingreservoirsandthetransportofcontaminantsinsoilsandaquifers(Dullien,1992).Thepredictionofaverageormacroscopictransportpropertiesfromtheirmicro-scopicoriginsrequirestwomainsteps:(i)aquantitativegeometricdescriptionofthecomplexmicrostructureoftheme-dium,and(ii)exactorapproximatesolutionsoftheequationsofmotionthatgovernthetransportphenomenaofinterest.312P˚AL-ERICØRENANDSTIGBAKKEWiththeadvanceincomputationalfluiddynamics,itisnowpossibletocomputetransportpropertiessuchaspermeabilityandelectricalconductivityonquitelargesystems.Giventheeverincreasingspeedofcomputers,thebottleneckinelucidat-ingtherelationshipbetweenmicrostructureandtransportmaybetoactuallyobtainanadequatedescriptionofthecomplexmicrostructure.Throughouttheyears,anumberofdifferentmicroscopicmodelshavebeenproposedtorepresenttheporestructureofporousmedia.AnoverviewoffrequentlyemployedmodelsisgivenbyHilfer(1996).Severaldirecttechniquesprovideadetailedthree-dimensional(3D)descriptionoftheporestructure.Asetoftwo-dimensional(2D)serialsectionsofasamplecanbeimagedandcombinedtobuilda3Dimageoftheporestructure(LinandCohen,1982;Holtetal.1996).However,serialsectioningisatediousandtime-consumingtask.AdvancedsynchrotronX-raymicrotomographystudieshavealsodirectlyimagedthe3Dporestructureofsedimentaryrocksatmicrometerresolu-tion(Dunsmoiretal.,1991;Colesetal.,1994;Spanneetal.,1994;Hazlett,1995;Colesetal.,1996;Cokeretal.,1996).Althoughmicrotomographyisextremelyuseful,itisstillnotreadilyavailable.Inpractice,informationabouttheporestruc-tureofporousmaterialsisoftenlimitedto2Dthinsectionimages.Anattractiveapproachisthereforetoreconstructa3Ddescriptionoftheporestructurefromreadilyavailable2Dimagesofthesystem.Recently,anumberofstatisticalmodelshavebeenproposedforreconstructing3Dporousmediafrom2Dthinsectionimages(Joshi,1974;Quiblier,1984;Adleretal.,1990;Adleretal.,1992;Adler1992;Hazlett,1997;Roberts,1997;YeongandTorquato,1998a,b).Historically,theprocesswasdevisedbyJoshi(1974)andQuiblier(1984).Inshort,itconsistsofdeterminingthefirsttwomoments(i.e.theporosityandthetwo-pointcorrelationfunction)ofthebinaryphasefunctionfrom2Dthinsectionimagesofthesample.Random3Dmodelsarethengenerat-edinsuchamannerthattheymatchthemeasuredstatisticalproperties.Adleretal.(l990)usedthistechniquetoreconstructFontainebleausandstone.Compu-tationsofpermeabilities(Adleretal.,1990)andelectricalconductivities(Adleretal.,1992)ofthereconstructedmodelswereinfairagreementwithexperimentalmeasurements.However,thecomputedresultswereconsistentlylowerthantheexperimentaldata.Thisisprobablycausedbythefactthatthepercolationthresholdofthesestatisticalmodelsisabout10%(Adler,1999)whilethethresholdporosityforFontainebleausandstoneissignificantlyless(2–3%).Incontrasttostatisticalmodels,processbasedmodelstrytoaccountforthefactthattheporestructureisoftentheresultofphysicalprocesses.Whilesuchmodelsareroutinelyappliedtounconsolidatedporousmedia,theyarerelativelyrareforconsolidatedmedia(Hilfer,1996).Bryantetal.(1993)developedaprocessbasedmodelforsedimentaryrocksbymodellingdifferentrock-formingprocessessuchascompactionandcementationonapackingofequalspheresconstructedbyFinney(1968).Thisapproachwasshowntoprovidevaluableunderstandingab