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ModellingofFlowandTransport inFractalPorousMedia ModellingofFlowand TransportinFractal PorousMedia Editedby
JianchaoCai
StateKeyLaboratoryofPetroleumResourcesandProspecting, ChinaUniversityofPetroleum,Beijing;InstituteofGeophysicsandGeomatics, ChinaUniversityofGeosciences,Wuhan,China
LiehuiZhang
StateKeyLaboratoryofOilandGasReservoirGeologyandExploitation, SouthwestPetroleumUniversity,Chengdu,China
WeiWei InstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences, Wuhan,China
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CONTRIBUTORS.........................................................................................xi
CHAPTER1 Abriefintroductiontoflowandtransportinfractal porousmedia........................................................................1
JianchaoCai,LiehuiZhang,andWeiWei
1Introduction................................................................................1
2Fractalstructuralcharacteristicsofporousmedia..................4
3Transportmodelbasedonfractalgeometryand othertheories.............................................................................5
4Modellingoftransportcharacteristicsand itsapplication.............................................................................6
5Conclusion..................................................................................8
CHAPTER2 Fractalstructuralparametersfromimages:Fractal dimension,lacunarity,andsuccolarity...............................11
YuxuanXia,JianchaoCai,andWeiWei
1Introduction..............................................................................11
2Definitionandphysicalmeaning..............................................12
3Calculatedmethod...................................................................12
4Applicationsinfractalporousmedia.......................................15
4.1Characterizationofcomplexity,heterogeneity, andanisotropy..................................................................15
4.2Fractalmodelofreservoirpermeability.........................17
4.3Fracturedistributioncharacterization............................19
4.4Permeabilityprediction....................................................21
5Conclusions..............................................................................72
Acknowledgments..........................................................................73 References.....................................................................................73
CHAPTER5 Modellingflowandtransportinvariablysaturatedporous media:Applicationsfrompercolationtheoryand effective-mediumapproximation........................................79 BehzadGhanbarianandAllenG.Hunt
1Introduction..............................................................................79
1.1Percolationtheory..... ..............80
1.2Effective-mediumapproximation....................................87
2Combininguniversalscalinglawsfrompercolation theoryandtheeffective-mediumapproximation.... ................90
3Diffusion....................................................................................92
4Electricalconductivity..............................................................93
5Permeability.............................................................................96
5.1Single-phasepermeability...............................................97
5.2Waterrelativepermeability...........................................102
6Conclusion..............................................................................111
Acknowledgment..........................................................................112 References...................................................................................112
CHAPTER6 Fractalanalysisonconductiveheattransfer inporousmedia................................................................119
XuanQin,JianchaoCai,andPengXu
1Introduction............................................................................119
2Exactlyself-similarfractalmodel...... ........121
3Statisticallyself-similarfractalmodel.......... ........................ 125
4Statisticallyself-similarfractalmodelwiththeeffect ofroughsurfaces...................................................................131
5Conclusions............................................................................136
Acknowledgment..........................................................................136 References...................................................................................136
CHAPTER7 Transportpropertyandapplicationoftree-shaped network..............................................................................141
PengXu,BomingYu,ArunS.Mujumdar,and JianchaoCai
1Introductionandbackground.................................................141
2Applicationoftree-shapednetwork... .....143
3Optimizationprinciplefortree-shapednetwork...................144
3.1TheoriginofMurray’slaw.............................................144
3.2Optimizationoftree-shapedstructure..........................147
3.3Fractaltree-shapednetwork.........................................150
CHAPTER8
4Fluidflowintree-shapednetwork.............
Fractalcharacterizationoffracturenetworksand productionpredictionformultiplefracturedhorizontal wellsinunconventionalgasreservoirs............................165 QiZhangandGuanglongSheng 1Introduction............................................................................165
2Fractalfracturepropertydistribution....................................167
2.1Fractaldimensionsofinducedfractures......................167
2.2Fractalfractureporosity,permeability, andcompressibilitydistribution.. 168
2.3Resultsanddiscussion..................................................170 3DMFDEconstruction..............................................................174
3.1Diffusivityequationsofdual-mediasystems...... ..........174
3.2Modelvalidationandapplication...................................182
CHAPTER9 Applicationoffractaltheoryintransientpressure propertiesofhydrocarbonreservoir................................193 LiehuiZhang
1Introduction............................................................................195
2Fractalwelltestingmodelforaverticalwellina homogeneousoilandgasreservoir........................ ............ ..196
2.1Physicalmodeldescription............................................197
2.2Mathematicalmodelanditssolution............................197
2.3Pressureresponseanalysis..... ..........201
3Fractalnonlinearseepageflowmodelfordeformable dualmediareservoir..............................................................204
3.1Background....................................................................204
3.2Problemsstatement......................................................206
3.3Solutionanalysis............................................................211
3.4Applicationtopressureanalysis........................ ...........220
4Transientpressurefractalanalysisofaverticalwellin acompositereservoir............................................................221
4.1Physicalmodel...............................................................221
4.2Mathematicalmodel......................................................222
4.3Solutiontothemodel....... ..........224
4.4Resultsanalysis.............................................................227
5Fractaltheoryinshalegas reservoir.... ..........233
5.1Background....................................................................233
5.2Fractalmodelforshale.................................................233
5.3Multilayerfractaladsorptionmodel..............................236
5.4Experimentalresultsanddiscussion.. ..........................239
6Conclusions............................................................................246
References...................................................................................246
INDEX...................................................................................................................251
Contributors JianchaoCai StateKeyLaboratoryofPetroleumResourcesandProspecting,ChinaUniversityof Petroleum,Beijing;InstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences,Wuhan, China
BehzadGhanbarian PorousMediaResearchLab,DepartmentofGeology,KansasStateUniversity, Manhattan,KS,UnitedStates
AllenG.Hunt DepartmentofPhysicsandDepartmentofEarthandEnvironmentalSciences,Wright StateUniversity,Dayton,OH,UnitedStates
ArunS.Mujumdar DepartmentofMiningandMaterialsEngineering,McGillUniversity,Montreal, QC,Canada
XuanQin InstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences,Wuhan,China
GuanglongSheng SchoolofPetroleumEngineering,YangtzeUniversity,Wuhan,China
ShifengTian StateKeyLaboratoryofCoalMineDisasterDynamicsandControl;SchoolofResources andSafetyEngineering,ChongqingUniversity,Chongqing,China
WeiWei InstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences,Wuhan,China
YuxuanXia InstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences,Wuhan,China
XuefuXian StateKeyLaboratoryofCoalMineDisasterDynamicsandControl;SchoolofResources andSafetyEngineering,ChongqingUniversity,Chongqing,China
PengXu CollegeofScience,ChinaJiliangUniversity,Hangzhou,China
KangYang StateKeyLaboratoryofCoalMineDisasterDynamicsandControl;SchoolofResources andSafetyEngineering,ChongqingUniversity,Chongqing,China
BomingYu SchoolofPhysics,HuazhongUniversityofScienceandTechnology,Wuhan,China
LiehuiZhang StateKeyLaboratoryofOilandGasReservoirGeologyandExploitation,Southwest PetroleumUniversity,Chengdu,China
QiZhang SchoolofEarthResources,ChinaUniversityofGeosciences,Wuhan,China
JunpingZhou StateKeyLaboratoryofCoalMineDisasterDynamicsandControl;Schoolof ResourcesandSafetyEngineering,ChongqingUniversity,Chongqing,China
Abouttheeditors JianchaoCai isaprofessorattheStateKeyLaboratoryofPetroleumResources andProspecting,ChinaUniversityofPetroleum(Beijing),andtheInstituteof GeophysicsandGeomatics,ChinaUniversityofGeosciences(Wuhan).Hehas publishedmorethan140peer-reviewedjournalarticles,threebooks,and numerousbookchapters.Heisthefounderandtheeditor-in-chiefof Advances inGeo-EnergyResearch andservesasanassociateeditororeditorialmemberfor severaljournals.HereceivedtheProjectoftheNationalScienceFoundationof ChinaforOutstandingYouthFoundationin2017.
LiehuiZhang isaprofessorattheCollegeofPetroleumEngineeringandthe vicepresidentatSouthwestPetroleumUniversity.Hehaspublishedmorethan 270peer-reviewedjournalarticlesandauthoredorcoeditedeightbooks.He alsoservesonnumerouseditorialboards.HeistherecipientoftheChina NationalFundsforDistinguishedYoungScientistsin2012,awardedbythe NaturalScienceFoundationofChina.HewasalsonamedtheCheungKong Scholars’distinguishedprofessorin2014bytheChineseMinistryofEducation.
WeiWei nowworksattheInstituteofGeophysicsandGeomaticsofChina UniversityofGeosciences(Wuhan).HereceivedhisBScinGeophysicsfrom theGuilinUniversityofTechnologyin2012aswellashisMscinGeological EngineeringandPhDinGeophysicsfromChinaUniversityofGeosciences (Wuhan)in2015and2018,respectively.Hisresearchfocusesonmodelling theelectricalpropertiesofporousmediaaswellasfractaltheoryanditsapplication.Hehaspublished14peer-reviewedjournalarticles,whichhavenearly 550totalcitationsfromtheWebofScience.
Preface Porespaceand/orasolidmatrixofporousmediahavebeenobservedtofollow fractalproperties,includingself-similarityandself-affinity.Featuressuchas power-lawprobabilitydensityfunctionandscaleinvariancecanhelpcapture thecomplexitytostudyflowandtransportphenomenainnaturalporoussystemssuchassoilsandrocks.Asthisisagrowingresearcharea,itisnecessaryto highlightnewcharacteristicsoffractalscalinglawsandilluminatepractical applicationsfromdifferentviewpoints.Scientistshavebeenattemptingtofill thegaptocharacterizeporousmediausingfractaltheoryforalongtime.Inthis book,wetrytoshedfurtherlightontwoconcepts:(1)howtoanalyzethestructuralcharacteristicsofporousmediabymeansoffractaltheory,and(2)howto effectivelyutilizefractalcharacteristicstoinvestigatetheflowandtransportprocess.Therefore,apracticalknowledgeoffractalporousmediawillcontributeto anaccuratecharacterizationoftheirmicrostructuresandanin-depthunderstandingoffluidtransportmechanisms.
Thebookpresentsacomprehensiveoverviewoftheflowandtransportpropertiesoffractalporousmediathatwillenhancethebasicunderstandingofthe theoreticalmodellingandapplicationoftransportphenomena.Itcoversawide rangeofapplicationsusingdifferenttheoretical,mathematical,andnumerical approacheswhilealsoprovidingananalysisoftheflowandtransportpropertiesinfractalporousmedia.
Thisbookprovidesasuitablereferenceforgraduateandupperlevelundergraduatestudents,andforpractitionersintheresearchanddevelopmentofresource andenergyinacademiaandindustry.Itconsistsofninechapterstointroduce ModellingofFlowandTransportinFractalPorousMedia. Chapter1 presentsabrief summarizationrelatedtotheflowandtransportpropertiesoffractalporous media,whichcanenhanceresearchers’basicunderstandingoffractaltheoreticalmodellingandtheapplicationoftransportphenomenainporousmedia. Chapter2 analyzeshowtoapplyfractalparameterizationtoevaluatethestructuralcharacteristicsofporousmediafromtheviewsofcomplexity,heterogeneity,andanisotropywhilealsoprovidingafeasiblerelationshipbetween
transportpropertiesandfractalfeatures. Chapter3 presentsaframeworkofcalculatingthetortuosityfortheelectricalandhydraulictransportinfractal porousmediaanddiscussesthepower-lawrelationshipbetweentortuosity andfractaldimensions.In Chapter4,theadsorptionandflowcharacteristics areanalyzedwithfractalparameters,andanoverviewisprovidedontherelationshipbetweenthemostcommonmethodsofcapturingporestructuralcharacteristicsandfractaldimensions. Chapter5 reviewstheapplicationsof percolationtheoryandeffective-mediumapproximationformodellingthe flowandtransportinunsaturatedporousmediaandfindsthemodelsof transportproperties,includinghydraulicandelectricalconductivityaswell asgas/solutediffusionrelatedtoporesizedistribution. Chapter6 explores theapplicationoffractaltheoryintheheattransferofporousmedia,and obtainsgeneralizedmodelsforeffectivethermalconductivitybasedonthe effectivemediummodel,theSierpinskimodel,andthefractalroughsurface model. Chapter7 proposesaprospectiveviewoftree-shapednetworkstostudy thetransportpropertiesoffluidflow,andpresentstheoptimalprincipleoftreeshapednetworksthatcanprovideanin-depthanalysisforpulsatilefluidflow, heatconduction,andelectricaltransfer. Chapter8 developsacoupledmethod tocorrelatethefractalcharacteristicsoffractureapertureandfracturespacing, andemploysthisapproachintheflowsimulationformultiplefracturedhorizontalwellsinunconventionalgasreservoirs. Chapter9 solvesthedetailed analyticalprocessofafractalmodelforahomogeneousoilandgasreservoir, adeformabledualmediareservoir,acompositereservoir,andashalegasreservoirbasedontheLaplaceintegraltransform.
Theeditorsacknowledgethecontributingauthors.Specialthanksaredueto AmyShapiro,LindsayLawrence,BharatwajVaratharajan,andMilesHitchen ofElsevier’sbookproductiongroupforeditingthisbook.Wealsoacknowledge theNationalNaturalScienceFoundationofChina(Nos.51534006, 41722403,42004086)forsupportingourseriesofstudiesonflowandtransportinfractalporousmedia.
Abriefintroductiontoflowandtransport infractalporousmedia JianchaoCaia,b,LiehuiZhangc,andWeiWeib aStateKeyLaboratoryofPetroleumResourcesandProspecting,ChinaUniversityofPetroleum, Beijing,China, bInstituteofGeophysicsandGeomatics,ChinaUniversityofGeosciences,Wuhan, China, cStateKeyLaboratoryofOilandGasReservoirGeologyandExploitation,Southwest PetroleumUniversity,Chengdu,Sichuan,China
1Introduction Innature,porousmediaaswellasitsmatrix,pores,andfracturesconstitutea complexmicrosystem,forexample,soils,rocks,buildingmaterials,andbiologicaltissues.Thestructuralfeaturesofthesesystems,includingpore(grain)size distribution,fracturelengthdistribution,andsurfaceroughness,usuallyshowa statisticalpowerlawwithmeasuredscale.Thesestructuresarereferredtoasselfsimilarinfractaltheory,thustheporousmediaisusuallyknownasfractal porousmedia.Thestudyoffractalporousmediahasreceivedsubstantialattentionbyresearchersinthelastdecades.Itcanbeseenin Fig.1 thatthepublished paperskeepincreasingwithtime.Italsomeansthatthemodelandtheoryof fractalporousmediaaremoreconvenienttodescribethecomplexporestructurecomparedwithregularandtraditionalapproaches.
Therelationofthetopic“porousmedia”mainlyconnectedwithflow/transport fromthedatabaseoftheWebofScienceCoreCollectionisshownin Fig.2.It alsoindicatesthatthesolutionfortheflow/transportprobleminporousmedia isalong-termresearchtheme.Theflowmechanismofporousmediaisaffected byfluidproperties,porestructure,andmatrixconstituent.Forfractalporous media,thetransportprocessadditionallyneedstoconsidertheinfluenceof theself-similarporestructureandthemultiordersofmagnitudeoftheporesize. Fromsearchresultsonthetopic“fractalporousmedia,”asshownin Fig.3,we findthatthefocuspointisthefractaldimensionofthemicrostructure,and obtainingtheflowortransportbehaviorofporousmedia.Severaltheories andmethodsareusedtoanalyzetherelationshipbetweentransportcharacteristicsandpore/fracturegeometryparameters,suchasthepercolationtheory, theeffectivemediumtheory,thefractaltheory,andtheporenetworkmodel.
CONTENTS 1.Introduction..... 1
2.Fractalstructural characteristicsof porousmedia... 4
3.Transportmodel basedonfractal geometryand othertheories.. 5
4.Modellingof transport characteristics andits application....... 6
5.Conclusion....... 8
Acknowledgments 8 References........... 8
FIG.1
Thenumberofpublishedpapersforthetopic“fractalporousmedia”from2010to2019.Thesearch criteriaare"fractal"and(“porousmedia”or“soil”or“rock”or“fiber”or“building”or“biological”or “reservoir”or“material”);thisdataistakenfromtheWebofScienceCoreCollection.
FIG.2
Morethan14,000paperswerepublishedfrom2015to2019relateddirectlyorindirectlytovarious aspectsofflow/transportinporousmedia.ThisdataistakenfromtheWebofScienceCoreCollection.
FIG.3
About5000paperswerepublishedfrom2010to2019,relateddirectlyorindirectlytovariousaspectsof thetopic“fractalporousmedia.”Thesearchcriteriaare“fractal”and(“porousmedia”or“soil”or“rock”or “fiber”or“building”or“biological”or“reservoir”or“material”);thisdataistakenfromtheWebofScience CoreCollection.
Thesetheories/methodscouldeffectivelydescribethefluidflowprocessin porousmediafromdifferentviews.
Fractalgeometryisasuccessfulmethodtocharacterizethestatisticalrelation betweenthegeometrystructureversuslengthscale.Thescalingexponentand fractaldimensioncouldsimplifythemathematicalexpressionfordepicting complexstructuralfeaturesofporousmedia.Besides,theanalysisoftheheterogeneityandconnectivityofthepore-throatstructurealsohasgoodperformance basedonotherfractalparameters,suchaslacunarityandsuccolarity.Combiningwithotheranalyticalmethods,fractalgeometryisnowconsideredtobean integratedtooltomodeltheflowandtransportinporousmedia.Asthistoolhas obtainedmanyachievementsinseveralfieldsofproduction(e.g.,reservoirengineering,geologicalsequestrationofcarbondioxideandgeothermalenergy),systematicallyunderstandingtheapplicationoffractaltheoryforflowand transportinporousmediaisofmoreconcern.
Inthischapter,wepresentabriefsummarizationrelatedtotheflowandtransportpropertiesoffractalporousmediathatwillenhanceourbasicunderstandingofthetheoreticalmodellingandapplicationoftransportphenomenain differentfields.Thischapterincludesthreeparts:thefractalstructuralcharacteristicsofporousmedia,thetransportmodelbasedonfractalgeometryand
othertheories,andmodellingthetransportcharacteristicsanditsapplication. Itcoversawiderangewithdifferenttheoretical,mathematical,andnumerical approachesaswellastheanalysisofflowandtransportphenomenainfractal porousmedia.
2Fractalstructuralcharacteristicsofporousmedia Infractaltheory,therearemainlythreeparameterstodescribethecomplex microstructureofporousmedia:fractaldimension,lacunarity,andsuccolarity [1–3].Bytheapplicationofthethreeparameters,thestructuralcharacteristics ofporousmediacanbeevaluatedfromtheviewofcomplexity,aggregation, andconnectivity,respectively.Thefractaldimensionrepresentshowmuchan objectoccupiesthemetricspace.Lacunaritydenotestheclusterdegreeofpores (fractures);moregenerally,itquantifiestheheterogeneity.Succolarityreflectsthe abilityofafluidflowinginaporousmedium.However,exceptingthefractal dimension,thecalculationoflacunarityandsuccolarityiscumbersomeand time-consumingfortheheterogenousstructureinthemicro-andnanometer scale.Fortunately,withthedevelopmentofimagingtechnologyandhighperformancecomputers,thebox-countingmethodforlacunarityandsuccolarity candealwiththestructureimagingofhighresolutionwell [4–6]
Althoughthecomplexity,heterogeneity,andanisotropyofporousmediacan beaccuratelycharacterizedbycombiningthethreeparameters,theemphasisof thethreefractalparametersisclearlydifferent.IntheworkofXiaetal. [7,8], theyusefractaldimension,lacunarity,andsuccolaritytoestimatethepermeability.Akeyfindingisthatthesuccolaritycouldbetterfitthepermeability ofsandstonethanotherparameters.Theyillustratethereasonthatpermeability iscloselyrelatedtoporeinterconnectioninlow-permeabilityrock,andthe valueofsuccolarityissensitivetotheinterconnectionofporousmedia.
Aswecansee,theaforementionedfractalparametershaveadvancedperformance,buttheyaredifficulttodescribetheflowlengthoftheelectricalor hydraulictransportprocessinporousmedia.Oneofthedifficultiesisthatthe tortuosityofthefluidflowchangesconstantlywiththelocaldifferenceofpore configuration.WheatcraftandTyler [9] assumedthattortuosityhasaself-similar behaviorandpresentedthecalculationmethodoftortuositybasedonfractal theory.Theyutilizedthemeasuredlengthandthefractaldimensiontoexpress thetortuosityinadifferentscale.Thetransportbehaviorrelatedtotortuosity alsocanbecharacterizedbythefractaldimension.However,theapplication offractaltheoryforcalculatingtortuosityholdsseveraluntowardpoints [10,11].Oneofthemostimportantpointsishowtoobtainthefractaldimensioninthepower-lawexponentparameterbetweentortuosityandlengthscale.
Forthesolutionofthepower-lawexponent,Weietal. [12] presentedaframeworktocalculatethetortuosityfortheelectricalflowlengthinfractalporous
media.Thisresultshowsthatthepower-lawexponentbetweentortuosityand measuredscalecanbecalculatedbytwofractaldimensions,thetortuosityfractaldimensionandthefractaldimensionoftherandomwalker.Theyfoundthat thedifferencevalueofthepower-lawexponentfromtwofractaldimensions approaches1.TheanalysisfromCaietal. [13] indicatesthatthedifferent calculatingmethodoftortuosityalsocaninfluencethepower-lawexponent, namelythatthetortuositysuchaselectricalandhydraulicflowcannotberepresentedinthesametortuositymodel.
Intheprogressoffractalparametersandtortuositycalculation,thestructural imagingandstatisticsofporousmediaalsoneedtobenoticed.Themostcommonlyusedcharacterizationmethods,suchasopticalmicroscope,scanning electronmicroscopy,nanoCT-scan,mercuryinjectioncapillarypressure, CO2 adsorption,N2 adsorption,andnuclearmagneticresonancetests,can reflectthefractaldimensionsofporousmedia [14–16].However,fractal dimensionsindifferenttestmethodsaredissimilar.Forexample,theporefractaldimensioncanbecalculatedbyscanningelectronmicroscopyorgasadsorption.Thefractaldimensionfromscanningelectronmicroscopyisbasedon imageanalysis,whilethefractaldimensioncalculatedbygasadsorptionisfrom theanalysisofadsorptioncapacityinfluidinjectionprocess.Especially,the transportcapacityofshalewithorganicmatterporesisrelatedtothemineral matrixorparticles,interparticle,intraparticle,andfracturepores.Thesecomponentscanaffectthetestresultsforthefractalcharacteristicsofporestructure obtainedfromdifferentmethods.
Forthesetestmethods,Zhouetal. [16–18] providedafindingthattheadsorptionandgasflowbehaviorsinshalearedifferentduetotherangeofporesize distribution,whichmeansthattheapplicationofreasonablefractalparameters indifferentporecharacterizationmethodsissignificant.Therefore,acombinationofaseriesofporecharacterizationmethodsisneededtoobtainthefull-scale poresizedistributionofshale.Furthermore,fractaltheoryindifferentporecharacterizationmethodsiseffectivetoreflecttheheterogeneousporesatdifferent sizeranges.
3Transportmodelbasedonfractalgeometryand othertheories Thecapacitytodescribetheself-similarbehaviorofporousmediaisoneofthe advantagesoffractaltheory.However,forsomespecialinfluencesfortransport propertiesresultingfromtheinhomogeneousdistributionofporousmedia, suchastheeffectivestatisticalinformationofporestructureandpercolation status,theapplicabilityoffractaltheoryisrelativelyweak [19,20].Applying acombinationofdifferenttheories/methodstoanalyzethetransportcharacterizationhasbeenapopularideaintheexplanationofafractalporesystem.
Amongthesetheories,theeffectivemediumtheoryandpercolationtheorycan effectivelymakeupthedisadvantagesoffractaltheory,especiallyintheprocess offractalgeometrydescribingthebundle-of-tubesstructureofporousmedia.
Thedrawbackfromthepoordescriptionforporeinterconnectivityinporous mediacanbeperfectlysolvedbycombiningthefractalbundle-of-tubesmodel withthepercolationtheoryandeffectivemediumtheory [21].Themethods havebeensuccessfullyappliedtodescribefluidflowandtransportinlattices, porenetworks,andporousmedia [22–24].Therearemoreadvantagessuchas thepresenceofapercolationthresholdbelowwhichthefluidwithinthepore spacelosesitsconnectivity,andaccordinglymacroscopictransport coefficients [25]
Thecombinationofseveraltheoriescanbettercharacterizethetransportproperties,includinghydraulicandelectricalconductivityandgas/solutediffusion. Theaccuratedescriptionofheattransferinporousmediaisavailablebasedon fractalgeometryandeffectivemediumtheory.Relativetothefluidflowprocess inporousmedia,theheattransferneedstoconsiderthethermalconductivityof eachphase,suchasgas,grain,andfluid.Oneofthecommonlyusedmodelsfor thermalconductivitycharacterizationinporousmediaistheMaxwell-Eucken model,wheresphericalporesareassumedtobewidelydispersedinacontinuousmedium [26].Theeffectivemediumtheoryisanapproximatedmethodto modelthethermalconductivityofmacroscopicallyhomogeneousandisotropicmediacontainingrandomlydistributedgrainsandpores.However,the impactsofmicrostructuressuchassolid/poregeometryandsize,tortuosity, andtheroughsurfacesofmicroporesonthethermalconductionofporous mediaremaindemanding [27]
Onthebasisofaneffectivemediumapproach,thethermalconductivityof porousmediaisstronglyinfluencedbythemicrostructuralfeaturesofporous media,whichcanbewellcharacterizedbyfractalgeometry.Inarecentwork, Qinetal. [27,28] derivedageneralizedmodelforeffectivethermalconductivity dependingontheeffectivemediummodel,theSierpinskimodel,andthefractal roughsurfacemodel.Thesemodelsconsidertheinfluenceofgrainandporesize distributions,liquidsaturation,androughsurface,whichisinagreementwith thepublishedexperimentaldataandnumericalsimulationdata.
4Modellingoftransportcharacteristicsandits application
Thepurposeofusingfractaltheory/methodsistosolvethevariedproblemsin transportingforporousmedia.Thetree-shapedstructureusuallyisappliedto
optimizethetransportprocessinindustrialengineering.However,akeyproblemishowtoeffectivelyutilizethetree-shapedstructuretodesignthenatural transfersystems.SimilartoHagen-PoiseuillelawandFick’slaworiginating fromabiologicalcontext,Murray’slawisproposedinabiologicalsystem andisabasicphysicalprincipleforbothlivingandnonlivingtree-shapedtransfernetworks.Thepreviousresearchofthetree-shapedstructurebasedonMurray’slawhasbeenextendedtothestudyofthetransportperformanceofporous media [29,30].Theoptimizationofthetree-shapedstructureismainlyfrom fourparameters:theminimumsurfacearea,theminimumvolume,theminimumdrag,andtheminimumpower.Therefore,thebalanceofthefourparametersintree-shapednetworksneedstobefurtherinvestigated.
Fromtheviewofoptimizationofthetree-shapednetwork,Xuetal. [31–33] gaveanapplicationframeoftree-shapednetworkstostudythetransportpropertiesoffluidflow.Furthermore,theypresentedtheoptimalprincipleoftreeshapednetworksthatprovidesin-depthanalysisforpulsatilefluidflow,heat conduction,andelectricaltransfer.Theoptimalresultsshowedthattheflow featuresarerelatedtothemacroscopictransportpropertiesandmorphological parametersoftree-shapednetworksbasedonfractalgeometry.Thesefeaturesof thetree-shapednetworkscouldhelpresearcherstounderstandtheflowmechanismofreservoirrock.
Inunconventionalreservoirs,suchascoalformation,shale,andtightreservoirs,fractaltheoryalsoplaysanimportantrole.Itcanbeappliedtoanalyze theinducedfracturespacingandapertureintheprocessofconnectingthepreexistingnaturalfractures [34] ,andsolvetheanalyticalsolutionofwellbore pressureforthetransientflow [35,36].Theresultsareserviceableformatchingandpredictingproductionrateofreservoirs.Asthedevelopmentofthe calculatingmethodoftortuosityindex,fractaldimension,fractureporosity, permeability,andcompressibilityare graduallyusedtodescribethecoupled relationshipbetweentheapertureandthespacinginfractal-induced fracture [37].
Forhydrocarbonreservoirs,thestudyoftheapplicationoffractaltheoryin transientflowmodelsisgraduallygrowing [38,39].Anonlineartransientflow modelofdualfractalmediacanbeestablishedbasedonembeddingthetreeshapedfractalnetworksintoafractalporousmedia.Inthederivativecurvefor thewelltestingorexperimentaldata,thetransientflowmodelsshowthatthe flowregimesarerecognizable,whichmeansthatthefractalmodelsaresuitable foramajorityofreservoirs [38,39].Inseveralcases [40–42],theresultsshowed thatthemicrostructuralevaluationoffractalporousmediaaswellastheevaluationofoilandgasresourceabundancearepromisinginprovidingnewideas andmethodsforoilandgasexplorationanddevelopment.
5Conclusion Linkingthemodellingofflowandtransportwithfractalporousmedia,itis recognizedthatfractaltheoryhasbeenasuccessfulapproachtodescribethe characterizationofthecomplexstructureofaporoussystem.Inthisbook, wepresentacomprehensiveoverviewrelatedtotheflowandtransportpropertiesoffractalporousmedia.Thebookfurtherexploresandextendstherecent applicationoffractaltheoryinthemodellingofthefluidtransportprocess basedonutilizingthenovelfractalstructuralparameters,combiningothertraditionaltheories,andsolvingseveralnewmathematicaltransportmodels.It willhelpmoreresearcherslearnadvancedideasaboutthestudyofflowand transportinfractalporousmedia.
Acknowledgments Wewouldliketothanktheindividualchapterauthorsofthebookfortheirinspiringcontributions anddiligentworks.WespeciallyacknowledgetheElsevierpressforprovidingtheopportunityto bringthisbooktoreaders.J.CaialsoacknowledgetheNationalNaturalScienceFoundationof China(Nos.41722403,41572116)forsupportingtheseriesofstudiesonthemodellingofflow andtransportinfractalporousmedia.
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