Upscaling of Single- and Two-Phase Flow in Reservoir Engineering

239 Pages·2021·20.492 MB·English

by  Hans Bruining

Upscaling of Single- and Two-Phase
Flow in Reservoir Engineering

Upscaling of Single- and Two-Phase
Flow in Reservoir Engineering
HansBruining
CivilEngineeringandGeosciences,Geoscience&Engineering,SectionReservoir
Engineering,DelftUniversityofTechnology(TUDelft),Delft,TheNetherlands
CRCPress/BalkemaisanimprintoftheTaylor&FrancisGroup,aninformabusiness
©2022HansBruining
Allrightsreserved.Nopartofthispublicationortheinformationcontainedhereinmaybereproduced,storedinaretrieval
system,ortransmittedinanyformorbyanymeans,electronic,mechanical,byphotocopying,recording,orotherwise,
withoutwrittenpriorpermissionfromthepublisher.
Although all care is taken to ensure the integrity and the quality of this publication and the information herein, no
responsibilityisassumedbythepublishersnortheauthorforanydamagetothepropertyorpersonsasaresultofthe
operationoruseofthispublicationand/ortheinformationcontainedherein.
LibraryofCongressCataloging-in-PublicationData
Acatalogrecordhasbeenrequestedforthisbook
Publishedby:CRCPress/Balkema
Schipholweg107C,2316XCLeiden,TheNetherlands
e-mail:[email protected]
www.routledge.com–www.taylorandfrancis.com
ISBN:978-0-367-76743-3(hbk)
ISBN:978-0-367-76744-0(pbk)
ISBN:978-1-003-16838-6(ebk)
DOI:10.1201/9781003168386
TypesetinTimesNewRoman
bycodeMantra
Contents
Preface.........................................................................................................................................xi
Abouttheauthor.......................................................................................................................xv
Chapter1 DutchandWorldwideEnergyRecovery;ExergyReturnonExergyInvested...1
Introduction.........................................................................................................1
1.1 FractionFossilinCurrentEnergyMix......................................................2
1.2 PossibleNewDevelopments......................................................................4
1.3 Exergy........................................................................................................7
1.4 ExergyReturnonExergyInvested(ERoEI)Analysis...............................7
1.4.1 ExerciseERoEI...........................................................................12
1.4.2 AnthropogenicEmissionsversusNaturalSequestration............12
1.4.3 Exercise:TreestoCompensateforIntercontinentalFlights.......13
Chapter2 One-PhaseFlow................................................................................................15
Introduction.......................................................................................................15
2.1 MassConservation...................................................................................16
2.2 Darcy’sLawofFlowinPorousMedia....................................................17
2.2.1 DefinitionsUsedinHydrologyandPetroleumEngineering......17
2.2.2 Exercise,EXCELNaming..........................................................19
2.2.3 Empirical Relations for Permeability (Carman–Kozeny
Equation).....................................................................................19
2.3 ExamplesthatHaveanAnalyticalSolution............................................21
2.3.1 OneDimensionalFlowinaTube...............................................21
2.3.2 Exercise,TwoLayerSandPack..................................................23
2.3.3 Exercise,NumericalModel........................................................23
2.3.4 Exercise,EXCELNumerical1-DSimulation............................24
2.3.5 RadialInflowEquation...............................................................25
2.3.6 BoundaryConditionsforRadialDiffusivityEquation...............26
2.3.7 Exercise,RadialDiffusivityEquation.........................................27
2.4 ModificationsofDarcy’sLaw.................................................................27
2.4.1 RepresentativeElementaryVolume............................................27
2.4.2 Exercise,SlipFactor...................................................................29
2.4.3 SpaceDependentDensity...........................................................29
2.4.4 WhyistheFlowResistanceProportionaltotheShear
Viscosity?....................................................................................32
2.4.5 ForchheimerEquationMustBeUsedforHighValuesofthe
ReynoldsNumber.......................................................................32
2.4.6 Exercise,InertiaFactor...............................................................34
2.4.7 AdaptationofCarman-KozenyforHigherFlowRates..............34
2.4.8 Exercise,CarmanKozeny...........................................................35
2.4.9 AnisotropicPermeabilities..........................................................35
2.4.10 Exercise,MatrixMultiplication..................................................36
2.4.11 SubstitutionofDarcy’sLawintheMassBalanceEquation.......36
v
vi Contents
2.5 StatisticalMethodstoGenerateHeterogeneousPorousMedia...............37
2.5.1 TheImportanceofHeterogeneity...............................................37
2.5.2 GenerationofRandomNumbersDistributedAccordingtoa
GivenDistributionFunction.......................................................37
2.5.3 Log-NormalDistributionsandtheDykstra-Parson’s
Coefficient...................................................................................38
2.5.4 Exercise,LognormalDistributionFunctions..............................39
2.5.5 GenerationofaRandomField....................................................40
2.5.6 Exercise,Log-NormalPermeabilityField..................................41
2.5.7 Exercise,AveragePermeabilityField.........................................41
2.6 UpscalingofDarcy’sLawinHeterogeneousMedia...............................41
2.6.1 Arithmetic,GeometricandHarmonicAverages.........................41
2.6.2 TheAveragedProbleminTwoSpaceDimensions.....................45
2.6.3 EffectiveMediumApproximation..............................................46
2.6.4 Pitfall: A Correctly Averaged Permeability Can Still Lead
toErroneousProductionForecasts.............................................47
2.7 NumericalUpscaling...............................................................................48
2.7.1 FiniteVolumeMethodin2–D;thePressureFormulation..........48
2.7.2 TheFiniteAreaMethod;TheStreamFunctionFormulation.....50
2.7.3 FiniteElementMethod(AfterF.Vermolen)...............................53
2.7.4 FlowCalculation.........................................................................55
2.A FiniteVolumeMethodinEXCEL...........................................................56
2.A.1 TheDataSheet............................................................................56
2.A.2 TheSheetforCalculationoftheX-dipAveraged
Permeability................................................................................58
2.A.3 The Harmonically Averaged Grid Size Corrected Mobility
inthex-Direction........................................................................58
2.A.4 TheGeometricallyAveragedGridSizeCorrectedMobility
inthey-DirectionbetweentheCentralPandtheCellS............59
2.A.5 TheSheetfortheWellFlowPotential........................................59
2.A.6 TheSheetforProductivity/InjectivityIndexes...........................59
2.A.7 TheSheetfortheWells...............................................................60
2.A.8 TheSheetforFlowCalculations.................................................60
2.B FiniteElementCalculations.....................................................................62
2.C SketchofProofoftheEffectiveMediumApproximationFormula........62
2.D Homogenization.......................................................................................64
Chapter3 TimeDependentProblemsinPorousMediaFlow...........................................71
3.1 TransientPressureEquation ...................................................................71
3.1.1 BoundaryConditions..................................................................75
3.1.2 TheAveragedProbleminTwoSpaceDimensions.....................76
3.1.3 TheProbleminRadialSymmetry...............................................77
3.1.4 BoundaryConditionsforRadialDiffusivityEquation...............78
3.1.5 Dimensional Analysis for the Radial Pressure Equation;
AdaptedfromLectureNotesofLarryLake...............................79
3.1.6 SolutionoftheRadialDiffusivityEquationwiththeHelpof
LaplaceTransformation..............................................................81
3.1.7 LaplaceTransformation..............................................................82
Contents vii
3.1.8 SelfSimilarSolution...................................................................84
3.1.9 TheDimensionalDraw-DownPressure.....................................85
3.2 PressureBuildUp....................................................................................86
3.2.1 Superposition..............................................................................86
3.2.2 TimeDerivativesofPressureResponse......................................88
3.2.3 PracticalLimitationsofPressureBuildUpTesting....................88
3.3 FormulationinaBoundedReservoir.......................................................88
3.4 Non-DarcyFlow......................................................................................90
3.A AboutBoundaryConditionatr=r ....................................................92
eD
3.A.1 Exercise,StehfestAlgorithm......................................................92
3.B RockCompressibility..............................................................................93
3.B.1 PhysicalModel............................................................................93
3.B.2 MassBalanceinConstantControlVolume................................94
3.C EquationsDisregardingtheGrainVelocityinDarcy’sLaw....................96
3.D SuperpositionPrinciple............................................................................96
3.E LaplaceInversionwiththeStehfestAlgorithm.......................................97
3.F EXCELNumericalLaplaceInversionProgramme.................................98
3.F.1 AlternativeInversionTechniques................................................99
Chapter4 Two-PhaseFlow.............................................................................................103
Introduction.....................................................................................................103
4.1 CapillaryPressureFunction...................................................................107
4.1.1 InterfacialTensionandCapillaryRise......................................108
4.1.2 Exercise,LaplaceFormula........................................................109
4.1.3 Exercise,Young’sLaw..............................................................110
4.1.4 Application to Conical Tube; Relation between Capillary
PressureandSaturation.............................................................111
4.1.5 Relation between the Pore Radius and the Square Root of
thePermeabilityDividedbythePorosity.................................112
4.1.6 Non-dimensionalizingtheCapillaryPressure..........................113
4.1.7 Exercise,RatioGrainDiameter/PoreThroatDiameter............114
4.1.8 Three-PhaseCapillaryPressures...............................................115
4.1.9 ExperimentalSetUpandMeasurementsofCapillary
Pressure.....................................................................................115
4.1.10 Cross-DipCapillaryEquilibrium..............................................116
4.1.11 Exercise,CapillaryDesaturationCurve....................................118
4.2 RelativePermeabilities..........................................................................118
4.2.1 Exercise,Brooks-CoreyRel-perms...........................................121
4.2.2 LETRelativePermeabilityModel............................................121
4.2.3 EstimateoftheLETParameters...............................................123
4.2.4 Exercise,ResidualOilandRel-perm........................................123
4.3 TheoryofBuckley–Leverett..................................................................124
4.3.1 Exercise,VerticalUpscalingRelativePermeability..................125
4.4 MaterialBalance....................................................................................126
4.4.1 SolutionsoftheTheoryofBuckley–Leverett...........................129
4.4.2 Equation of Motion (Darcy’s Law) and the Fractional
FlowFunction...........................................................................129
4.4.3 AnalyticalSolutionoftheEquations........................................130
viii Contents
4.4.4 ConstructionoftheAnalyticalSolution;Requirementofthe
EntropyCondition.....................................................................132
4.4.5 Exercise,BuckleyLeverettProfilewithEXCEL......................134
4.4.6 DerivationoftheShockCondition...........................................135
4.4.7 AnalyticalCalculationoftheProductionBehavior .................136
4.4.8 Exercise,BuckleyLeverettProductionFile..............................137
4.4.9 Exercise,AnalyticalBuckleyLeverettProductionCurve........137
4.4.10 Determination of Relative Permeabilities from Production
DataandPressureMeasurements.............................................138
4.4.11 DeterminationoftheRelativePermeabilitiesbyAdditional
MeasurementofthePressureDrop...........................................139
4.5 FiniteVolumeApproachtoObtaintheFiniteDifferenceEquations
fortheBuckleyLeverettProblem..........................................................140
4.5.1 Exercise,NumericalSolutionofBuckleyLeverettProblem....143
4.6 Vertical Equilibrium as a Basis for Upscaling of Relative
PermeabilitiesandFractionalFlowFunctions.......................................143
4.6.1 Dake’sUpscalingProcedureforRelativePermeabilities.........144
4.6.2 Exercise,SortingFactorDependence.......................................147
4.6.3 Hopmans’sFormulation............................................................148
4.7 PhysicalTheoryofInterfaceModels.....................................................149
4.7.1 DerivationofInterfaceEquationofMotionandProductions
forSegregatedFlow..................................................................149
4.7.2 Stationary Interface (Mobility Number
<GravityNumber+1)..............................................................150
4.7.3 Exercise,InterfaceAngleCalculations.....................................153
4.7.4 ProductionBehaviorforStationarySolution,i.e.,
M<G+1.................................................................................154
4.8 Non-stationaryInterface........................................................................154
4.8.1 TheVolumeBalanceintheFormofanInterfaceEquation......155
4.8.2 Dietz-Dupuit-Approximation....................................................157
4.8.3 ApproximateEquilibriumEquation..........................................157
4.8.4 DerivationofFlowRateQ fromDarcy’sLaw......................158
wx
4.8.5 Quasi Stationary Solution of the Dietz-Dupuit Equation
forM<G+1...........................................................................159
4.8.6 Exercise,ShockSolutionversusInterfaceAngleSolution.......160
4.8.7 AnalyticalSolutions..................................................................160
4.8.8 AnalyticalExpressionsfortheInterfaceasaFunctionof
PositionintheReservoir...........................................................162
4.8.9 AnalyticalExpressionsfortheProductionBehavior................162
4.8.10 SummaryofAnalyticalProcedureforInterfaceModels..........165
4.8.11 Exercise,AdvantageofM≤G+1...........................................166
4.A NumericalApproachforInterfaceModels............................................166
4.A.1 Exercise.BehaviorforM>G+1............................................167
4.B Numerical Approaches for Buckley Leverett and Interface Models
ImplementedwithEXCEL....................................................................168
4.B.1 SimpleSheetforBuckley–LeverettModel...............................168
4.C NumericalDiffusionforFirstOrderUpstreamWeightingScheme......169
Contents ix
Chapter5 DispersioninPorousMedia...........................................................................173
5.1 Introduction............................................................................................173
5.2 MolecularDiffusionOnly......................................................................178
5.3 SolutionsoftheConvection-DiffusionEquation...................................181
5.3.1 InjectioninaLinearCore.........................................................181
5.3.2 Taylor’sProbleminaCylindricalTube....................................184
5.4 DerivationoftheDispersionEquation..................................................184
5.5 StatisticsandDispersion........................................................................186
5.5.1 RandomWalkModels...............................................................186
5.6 VarianceofConcentrationProfileandDispersion.................................188
5.7 DispersivityandtheVelocityAutocorrelationFunction.......................189
5.8 Exercise,Numerical/Analytical1DDispersion.....................................191
5.9 Exercise,GelharRelation......................................................................193
5.10 NumericalAspects.................................................................................195
5.A Higher-OrderFluxFunctionsforHigher-OrderSchemes.....................196
5.B NumericalModelwiththeFiniteVolumeMethod................................198
Glossary...................................................................................................................................201
ListofSymbols........................................................................................................................203
References................................................................................................................................207
Index.........................................................................................................................................219