RESEARCH ON BEHAVIOR OF GROUNDWATER AND ITS APPLICATION TO FOUNDATION ENGINEERING

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Title RESEARCH ON BEHAVIOR OF GROUNDWATER ANDITS APPLICATION TO FOUNDATION ENGINEERING( Dissertation_全文 )

Author(s) Nishigaki, Makoto

Citation 京都大学

Issue Date 1980-01-23

URL https://doi.org/10.14989/doctor.k2315

Right

Type Thesis or Dissertation

Textversion author

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TABLEOFCONTENTS Title Page

SnmRY•-•-•••••.••••••"...m"..."""

. -e-"---e--e--e- i ACKNOWLEDG),IENT•••••••••••••••••••••••••••••••••••••••• . -e---e--ee--e-et ii CHAPTER1INTRODUCTION••••••.•••••••••••••••••.••••••• . ----e--e---e"-e--e 1 1.1Introduction...••..•.•...••..••...••.•.•.. . ---e-e"--e-e--- 1 1.2ReviewofPreviousStudies'.'.'''... .

•----•----

4 1.2.1Previousstudiesondrawdowntest... .

----•----•

6 1.2.2Previousstudiesonnumericalapproach . "t.•••••••••••••• 16 1.3ScopeofStudy••...••..••...•...••.••....••. . •••••-••••-••-•19 References'.''.''.'...'..'..'..-'..'...'..'..'.' . •••••••-•••••-•• 22 CHAPTER2PHYSICSOFSATURATED-UNSATURATEDGROUNDWATERMOTION •••••••••••• 27 2.1Tntroduction..•....•..•...•...•..•.•..•• . •-•-••••••••••••27 2.2EquationofMotion.''..'•...••.•'..'•..•...• . . •••-••••••--•- 28 2•3EquationofContinuity•••.e•...• .

''---••-••

37

2.4GoverningEquationofFlowinPorousMedia''

. ••--•••••••••••• 40 2.5InitialandBoundaryConditions...•...•....• .

".".".""..-.43

2.5•1Initialcondition'•...•...•e••.•. . •-••••e-'""''' 43 2.5.2Boundaryconditions...•.•....•....•... .

•--•-••••-•-•

44 2.5.3Sourcesandsinks....•..•...•..••..•.. . -•••-e•••-'"'' 47 References'... . •-•••---•••••• 48 CHAPTER3F!NITEELEMENTANALYSISOFFLOWINSATURATED--UNSATURATEDSOILS••• ,50 3.1lntroduction.•..•...•...•...•... . ••••••••••••••••••50 3.2Two-DimensionalFiniteElementAnalysis.e... . -•••••---•-•••• 53 3.2.1ApplicationofGalerkinmethod''''''.' . ''''''"-•••••••• 53 3.2.2Formulationbyweightedresidualprocedure ••••••••••••••• 55 3.2.3Tntegrationovertime... . ••••••••••••••••••59 3.2.4Treatmentofseepagefaces'''''''.''e. .

'"'"••-••-"••

62

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3.3AxisymmetricFiniteElementAnalysis...•... 63 3.3.1Finiteelementdiscretization.•...e... 63 3.3.2Axisymetricflowtoawell...•.. 65 3.4Three-DimensionalFiniteElementAnalysis''..'''''''''''''''' 67 3.4.1Isoparametricelements...e... 67 3.4.2NumericalintegrationbyGaussquadrature'''''''''''''' 73 References...e...e... 75 CHAPTER4EXPERIMENTALSTUDYONDETERMININGUNSATURATEDPROPERTYOFSOIL••77 4.1Introduction...•...ee... 77 4.2InstantaneousProfileAnalysis...e... 80 4.3ExperimentalApparatusandProcedure... 86 4.3.1Measurementofmoisturecontent bygammarayattenuation'''''.'....'... 86 4.3.2Measurementofporewaterpressure....;...•.... 89 4.3.3Experimentalprocedure..•...e...•... 91 4.4ExperimentalResultsandDiscussions'...e... 95 4.4.1Relationshipsbetweenhydraulicconductivity(K), volumetricmoisturecontent(e)andpressurehead(tp)'''' 95 4.4.2Relationshipsbetweenpressurehead(tp)andvolumetric moisturecontent(e)•...•...•.. 102 4.4.3EvaluationanddiscussiononGreenandAmptmodeland' pressuredistributionatequilibriumcondition''''''''' 106 4.5Conclusions...e....,... 113 References...e...e...e...e... 115 CHA[PTER5COM]?ARISONOFEXPERIIY[ENTALANDNUMERICALRESULTS•'''..'••e'''121 5.1Introduction''''''''''''''''''''''''''''''''''''''''''''''''' 121 5.2ExperimentalStudyonFlowthroughSandModel..e... 121 5.2.1Experimentalapparatusanditsprocedure

fortwo--dimensionalsandmodel...

121

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5.2.2Experimentalapparatusandits procedure

forthree-dimensionalsandmodel ---{-eeee-ete-e--e- .

ee-e 124

5.3 MaterialPropertiesandInitial-Boundary Conditions'''''' . -e-e 126

5.3.1lhaterialproperties...

.

-e-eee--e---ee-e-e--- .

ee-e 126

5.3.2Initialandboundaryconditions --e-e---t--etee-eeee . eee- 131

5.4 ComparisonsofExperimentalDatawith NumericalResults'' . ee-- 137

5.4.1Two-dimensionalsandmodel'''' ---eeeee-e--e---ee-ee . -te- 137 5.4.2Three-dimensionalsandmodel.. e-ee-ee--e-eee-e-ee-e . ee-- 145 5.4.3Effectsofinitialconditionsand hysteresis ofreteningcurve....e... e-eeeet--eee-e-eeee-- . e-ee 150 5.5 Conclusions''''''''''''''''''''''''' eee----ee-ee-e-e-e-ee . --"e 157 'References...e...e... ee-e--eeeee-e---eeeee . e-ie 158

CHAPTER6DRAIeJDOWNTESTFORDETERMININGAQUIFERCHARACTERISTICS••••.

ee-e 160

6.1 Zntroduction'''''''''''''''''''''''' ----e----ee---ee-- .

eeee 160

6.2 AnalysisofDrawdownTestDatafor PartiallyPenetratingWells -ee 162

6.2.1Introduction...•. -eee-e-e-e--e---e--e- . e--e 162 6.2.2Analyticalsolutionforpartially penetratingwell inaconfinedaquifer... --t---ee---e-ee-ee . --ee 163 A.Basicequationandsolution.e... eee--ee-ee---e-e--t-- . e--- 164 --et--eee--e--eee-e . ee-- 168

C.Methodsofanalyzingfielddata'

eeee--e-e-e-ee-eee--t . e-e- 171

a.Log-LogMethod... --e-ee-e-t-ee-eeeeee- . e--- 172 b.Log-LogDistanceDrawdownMethod ---ee-ee-e--eee---e- . -e-t 173 C.Jacob'sMethodAdjustedforPartial Penetration... . -e-e 175

d.ModifiedJacob'sMethodAdjusted forPartialPenetration e-ee 176

e.TrialandErrorMethodforUnknown AquiferThickness . -e-e 179

D.Analysisofdrawdowntestdata..

---e---ee-eeeee--et . -e-- 180 a.Log-LogMethodandTrialandError Method''''''''''' . etee 181

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. b.Jacob'sMethodAdjustedforPartialPenetration' . -e--- 183 c.ModifiedJacob'sMethodAdjustedforPartialPenetration -- 185 E.Discussionofanalysisofdrawdowntestdata.'''.' . eeeee-- 189 6.2.3Analyticalsolutionforpartiallypenetratingwell inan unconfinedaquifer...''..'...'....'''.' . "e-eeee 191 -A.Basicequationandsolution.e...•... . e---eee 192

B•Effectsofpartialpenetration''''''''.''..'

. --e-e--C.Methodsofanalyzingfielddata'''''''''''''''''' . ee-e--e 198 a.Log-LogMethod....•...e...•.•. . ee----" 198 b.Log-LogDistanceDrawdownMethod'''''''''''''''' . e-ee-e- 199 D.Analysisofdrawdowntestdata''''''''''''''''''', . teee-e- 200 a.Log-LogMethod•••...•...•..•...••...•e.. . e-eee-e 200 b.Log-LogDistanceDrawdownMethod'''''''''''''''' . t---e-- 203 E.Discussionofanalysisofdrawdowntestdata'''' . --eee-- 207 6.3 TransientFlowinGroundwatertoWellsinIslandModel Aquifer.• 208 6.3.1Introduction.•...e••...•... . -eeeee- 208 6.3.2AnalyticalsolutionforIslandModeldrawdowntest . inaconfinedaquifer'''''''''''''''''''''''''' . ee--e-e 209 A.Basicequationandsolution'''''''''''''''''''''' . eeeeeee 210

B.Effectsofconstantheadatouterboundary...

. ee-bet- 214 C.Methodofanalyzingfielddata...•...'.•.' . -e--te- 215 D.Analysisofdrawdowntestdata.'...e... t . ---e-e- 218 6.3.3AnalyticalsolutionforIslandModeldrawdowntest inanunconfinedaquifer... . ----eee 227 A.Basicequationandsolution...e... .--e"--- 227

B.Effectsofconstantheadatouterboundary.'''

. -eee-ee 233 C.Methodofanalyzingfielddata... . eee---e 234 D.Analysisofdrawdownteetdata''''''''''''''''''' . eeeeeee 237

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6.4Conclusions...e.. eeee -eee .

•••••• 243

References''''''''''''''''.''''''.''.''...'... eee- e-ee .

•••••• 244

CHAPTER7APPLICATZONTOFIELDPROBLE)vlS••••••••••••••eee" eeee.

•••••• 245

7•1Introduction...e.. ---e e--e .

•••••• 245

7.2HydraulicCharacteristicsofSoilinField'' e-e- ---- .

•••••• 245

7.3FlowthroughSandBankatFloodWaterLevel'

e-ee e--- .

•••••• 249

7.3.1Introduction'''''''''''''''''''''''''' e--- ee-e .

•••••• 249

7.3.2Selectionofboundarycondition...•..e e"ee -eee .

e••''' 252

7.3.3Determinationofhydraulicproperties

inunsaturatedregion''''''''''''•'''' eee- --e- .

•••••• 254

7.3.4Simulationsofearthembankmentsubjectto sudden .ralse

inariverlevel... ee-- eeee.

•••••• 259

7.4OpenCutExcavation)Codel..e...t.•... e-te e-e- .

•••••• 266

7.4.1Introductione... ee-e t--e .

•••••• 266

7.4.2Simulationofseepagethroughthree-dimensional aquifer..269

7.5Conclusionse...•...•... ---e -eet .

•••••• 278

References... -e-•- eeee.

e•••'' 279

CHAPTER8Conclusions''....•.••... -ee- -eee.

•••••• 281

8.1Conclusions...•...e•... -eee ee-- .

•••••• 281

8.2RecommendationsforFutureResearch'''''''' ee-e e-ee .

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SUMMARY Thepurposesofthisthesisareprimarilytoresearchonbehaviorof groundwaterflowinsaturatedandunsaturatedzone,topresentthefundamentals ofthetheoryofgroundwaterflow,andtodevelopthemosteffectivemethodsfor so!vinggroundwaterflowproblemsrelatedtocivilengineeringpractice.The mathematicalmodelprovidesafiniteelementsolutiontotwo--orthree-dimen-sionalproblemsinvolvingtransientflowinthesaturatedandunsagurateddomains ofnonhomog'eneous,anisotropicporousmedia.Inordertodeterminerelationships' betweenvolumetricmoisturecontent(e)andhydraulicconductivity(K),and betweenpressurehead(tp)andvoiumetricmoisturecontent(e)inalaboratory, anapparatuswasconstructedandtestproceduresweredevelopedtomeasurepre--ssureheadandvolumetricmoisturecontentbyusingpressuretransducersand low-energygammarayattenuation.Thevalidityandtheaccuracyofthetwo-or three-dimensionalfiniteelementapproachhavebeeninvestigatedwithcomparing thenumericalresultswiththelaboratoryexperimentaldata.Therelationships, K-eandip-e,whichwereobtainedbythenewapparatuswereusedasinputdata'N fornumericalanalyses.Goodagreementsbetweencomputedandraeasuredpressure headprofileshavebeenobtained.Toestimatehydraulicpropertiesofaquifers, newmethodsofanalyzingdrawdowntestdataweredevelopedandillustratedwith. `someexamples.Namely,analysesofdrawdowntestdataforpartiallypenetrating wellinaconfinedoranunconfinedaquiferhavebeenshowntodetermineaniso- tropichydraulicconductivitiesandstoragecoefficients,andanalysesofdraw-downtestdatawhichareobtainedinthemuchgroundwatersuppliedaquiferwere developedwithaconceptionof"IslandModel".Todemonstratetheflexibility ofthefiniteelementapproachanditscapabilityintreatingcomplexsituations whichareoftenencounteredinthefield,thegroundwaterflowthroughsandbank atfloodwaterlevelsandtheflowthroughaqutferduetoanexcavationwereana-lyzed.

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-i'- ACKNOWLEDG),IENT-TheauthorwishestoexpresshÅ}sgratitudetoProfessorKoichiAkaiof KyotoUniversityforhisstimulatingdiscussions,sustainedguidanceand supportthroughoutthecourseofthisresearch.Heisalsomostgratefulto AssociateProfessorYuzoOhnishiformanystimulatingdiscussionsonfinite elementanalyses,forvaluablesuggestionsduringthecourseoftheresearch workandforhisfinalcritiqueofthemanuscript. Throughmanyhoursofthoughtprovokingdiscussionsonthefundamentalsof flowthroughporousmedia,ProfessorIichiroKonoofOkayamaUniversityhas significantlycontributedtothepresentworkandgivensupportforlaboratory tests.Theauthorisindebtedtohimfortheseandforhisfinalcritical evaluationofthemanuscript. HissincerethanksareduetoAssociateProfessorToshihisaAdachiofthe DisasterPrevensionResearchInstituteofKyotoUniversityforconstructive suggestionsandduetoAssociateProfessorTakaoUnoofGifuUniversityformany helpfulsuggestionsontheflowthroughunsaturatedporousmedia. ThanksarealsoextendedtoMr.MituoTeramotowhohasofferedconveni-encetotheauthorusingcomputerandtoMissKazumiOkabewhotypedthe manuscript. Finally,theauthorexpresseshisutmostappreciationsforthehelpof manymembersintheFoundationEngineeringLaboratoryinKyotoUniversityand membersinDepartmentofCivilEngineeringinOkayamaUniversity.

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CHAPTER1 INTRODUCTrON

1.1Introduction.

Waterisanessentialcommoditytomankind,andthelargestavailable sourceoffreshwaterliesunderground.Whilethroughoutrecordedhistory thereisevidencethatmankindhasfearedandrespectedthedestructivepower

ofwater.Intheformoftidesandfloods,itisoneofthemostpowerful

forcesofnature.Hiddeninrockcrevicesandsoilpores,itexertsunbeliev-ableforcesthatteardownmountainsidesanddestroyengineeringworks.Dam designersandbuilders,highwaypeople,railroadengineers,andmanyothers havelongknownofthegreatimportanceofcontrollingwaterinporesandcracks inearthandrockformations.Whengroundwaterandseepageareuncontrolled, theycancauseseriouseconomiclossesandtakemanyhumanlives. Inbrief,mostfailurescausedbygroundwaterandseepagecanbeclassi-fiedinoneoftwocategories. (1)Thosethattakeplacewhensoilparcticlesmigratetoanescapeexitand causepipingorerosionalfailures. (2)Thosethatarecausedbyuncontrolledseepagepatternsthatleadtosatu-ration,internalflooding,excessiveuplift,orexcessiveseepageforces. Category1includesfailuresofdams,levees,reservoirs,andthelike,eaused bythemigrationofsoilparticlesinducedbyavarietyofdefects.Category2 includesfailuresofdams,drydocks,andretainingwallscausedbyexcessive saturation,seepageforces,andupliftpressures.Inthiscategoryarelisted 'thedeteriorationandfailureofpavementsfrominternalflooding,theuplift-ingofcanalliningsafterdrawdown,failuresoffillsandfoundationscaused' byseepageforces,andupliftpressuresintrappedwater,landslides,and

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-1-similarcases. Thespecificproblemswhicharetobedealtwithcanbedividedinto threeparts: (1)Estimationofthequantityofseepage (2)Definitionoftheflowdomain (3)Stabilityanalysis rngeneral,theseproblemistodeterminethevelocityandthepre' waterintheinteriorofasoilmasswithgivenboundaries,undercertainim-posedconditionsalongtheseboundaries.Mathematicallyspeaking,theproblem isintheclassofboundary-valueproblems.Duringthelastfewdecadesgreat progresshasbeenmadeinthernathematicalanalysisandthesimulationtech-niquesoftheseproblems. Inthefirstprecedingworks,theLaplaceequationwascommonlyusedas thegoverningequationofsteadystateflowinporousmedia.Namelythereis nochangeinconditionswithrespecttotime,andregardingwaterasanincom-pressibleflowmakesthedensityofwateraconstant.Indealingwiththe steadystateseepageofwaterthrough,forexample,earthdamsandembankments, civilengineershavetraditionallyreliedonthegraphicalmethodofflownets, theoryofcomplexvariables,andconforma1mapping. However,asagreatmanykindsofcivilengineeringworksareperfomed withexpeditionbyusingmanykindsofconstructionmachineries,itbecomes necessarytoobtainusefulsolutionstoseepageconditionsduringthenon- steadyperiod.Formanyyears,thehydrologisthasbeenaccustomedtoanaly-zingtheflowbehaviorwithapproachofanalyticalmathematics.Engineersand hydrogeologistswhohaveattemptedtouseanalyticalmethodsarewellaware thattheseapproachisveryusefulinsomecases,butunfortunately,thereare fartoomanysituationswheretherealgeologicsystemunderinvestigationdoes notmatchthesimplifiedversionadoptedbyearlyresearchers.Inotherwords,

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ifoneattemptstoanalyzealmostanygivenhydrogeologicprobleminthefield, hesoonfindsthatthenaturalvariationinthehydraulicpropertiesofthe flowregirnepresentsmanyproblernsifheiimitshisapproachtoanalyticmathe-matics.' Withtheadventofthedigitalcomputer,however,variousnumericalmeth-odshavebeenperfectedinrecentyearsthathavemadeitpossibletosolve rathercomplexproblemsintheflowofgroundwater.Suchmethodshaveenabled engineersandhydrogeologiststodevelopamuchbetterunderstandingoftheway inwhichcomplexgeologicalconditionseontrolwatermovement.Themethods havebeenconcernedwithbothregionalgroundwaterflowproblemsandtheques-tionofthelocaldistributionofseepagearoundwellsandstructuressuchas damsandcanals. Inalloftheprecedingwork,theeffectofflowintheunsaturatedzone onthepositionofthefreesurfacehasbeenconsistentlyneglected.While therearemanysteadystatesituationswhereonecansafelymakethisassump-tion,thecaseoftransient,freesurfacemovementisanentirelydifferent matter.Thereareprobablyanurnberofsituationswheredrainageorirnbibition israpidenoughthatonedoesnoteommitaseriouserrorinusingthenonsteady numericalmethods,butitisbecomingclearthattheroleoftheunsaturated zoneincontributingtogroundwaterseepageneedsmoreattentionthanithas receivedandwithconsideringtheeffectofunsaturatedzoneinanalysisit willbepossibletosolvetheproblems,whicharealmostimpossibletobe solvedbytheconventionalmethod,suchasbehaviorsofflowthroughanearth damwhenwaterlevelraisedattheupstreamside,flowthroughanembanknent whenthewaterlevelofariverchangedandgroundwaterlevelfractuationdue towaterinjectiontoawellorinfiltrationofrain. Thereforethepurposeofthisthesisisprimarilytoresearchonbehavior ofgroundwaterflowinthesaturated-•unsaturatedzoneanditsapplicationto

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-3- fundationengineering.Namelyinthisthesisfollowingfourstepsofinvesti--'gationwillbetakensystematically: (1)Evaluationsanddiscussionsofthegoverningequationoffiowinthesatu-, rated-unsaturatedporousmedia (2)Developmentofnumericalmethodstoanalysesthebehaviorofgroundwater flowinbothzonesusingtwo-andthree-dimensionalfiniteelementmethod (3)Improvementofsomemethodstodeterminehydraulicpropertiesofbothzones inthelaboratoryandthefield (4)Applicationofdevelopedmethodstorealgeologicsystems

1.2ReviewofPreviousStudies•

Thestudyofphysicsofflowthroughporousmediahasbecornebasicto rnanyscientificandengineeringfields,quiteapartfromtheinterestit holdsforpurelyscientificreasons.Suchdiversifiedfieldassoilmecha- nics,groundwaterhydrology,petroleumengineering,waterpurification,indus-trialfiltration,ceramicengineering,powdermetallurgy,andthestudyofgas masksallrelyheavilyuponitasfundamentaltotheirindividualproblems. Allthesebranchesofscienceandengineeringhavecontributedavastamountof literaturesonthesubject. Whenthepreviousstudiesontheflowthroughporousmediaarereviewed, itisconvenienttoclassifythemintothefollowinggroups; (2)Studiesonmethodsfordetermininghydraulicpropertiesinthefield (3)Studiesonmethodsfordetermininghydraulicpropertiesinthelaboratory (4)Studiesonmethodsforestimatinggroundwaterbehaviors. Theobjectsofthestudiesin(l)aretoinve'stigate'thegoverningequ-ationofflowthroughporousmedia.Althoughthefundamentalsofgroundwater flowwereestablishedmorethanacenturyago,itisonlywithinrecentyears

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thatthesubjecthasmetwithscientifictreatment.Informationonflowthrough porousmediaisscatteredinamultitudeofjournalsandbooks.Amongthebooks thatdealgenerallywiththistopic2onethatmustbementionedisacomprehen-1) sivetextbookpublishedbyMuskat,whiehinspiteofitsage,stillcontainsmany usefulinforrnations.Morespecializedaspectsofthephysicsofflowthrough

porousmediaarecontainedinbooksforspecificapplications.Thus,onground-2)3)

waterhydrologyonemustmentionaclassicbooksbyTodd,Harr,andPolubarinova-Kochina.Onsoilphysics,therearebooksbyChilds,Scheidegger,andHillel. DeWiesteditedacompilationofrecentdevelopmentonthesubject.Bearpre-sented,inanorderedmanner,thetheoryofdynamicsoffluidsinporousmedia, asapplicabletomanydisciplinesofscienceandengineering. Theobjectsofthestudiesin(2)areprimarilytodevelopmethodsof determiningthehydrauliccharacteristicsofaquifersorwaterbearinglayers withadrawdowntest.Thepreviousstudiesindealingwithadrawdowntestare summarizedinsection1.2.1. Thestudiesin(3)aretodevelopexperimentalmethodsofdeterminingthe hydraulicpropertiesofunsaturatedsoils.Thissubjecthasbeenmainlystudied inagriculturalengineering.Recently,itisrecognizedthatinthemovementof thegroundwaterthewatercontentandpermeabilityoftheunsaturatedregion usuallyplayveryimportantrolesaswellasthehydraulicpropertiesofsatu-ratedsoils,sothatmanyresearcheshavebeendoneonthissubject.Someof thesepreviousworkswillbereviewedinsection4.1. Theobjectsofthestudiesin(4)aretodeveloprationaltechniquesof theseepageanaiysesandtopredictthedistributionofpressureheadandthe velocityinsoilstakingintoaccountthegeologicalformationandthehydraulic boundary.Indealingwithsteadystateseepagetraditionallythegraphical methodofflownetshasbeenreliedbycivilengineers.Forunsteadystate seepagetheearliestinvestigatorsattemptedtosolveunsteadygoverningequation

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-5- byfindinganalyticalsolutions,usingsuchtechniquesasseparationofvaria- bles,Laplacetransformation,Green'sfunctionsandconformalmapping.Adis-tinctadvantageofsuchanalyticalmethodsisthatthesolutionsarederivedin algebraicformsanditiseasytostudythebehaviorofthesystemsintermsof convenientdimensionlessvariables.However,•theprocessoffindingananaly-• ticalsolutionbecomescumbersomeanddifficultexceptinthecaseofsimple initialandboundaryconditionsandmaterialcompositionoftheflowregion !nrnanypracticalproblems,however,thedegreeofheterogeneityandani- sotropythattheengineerencountersinthefieldmaybesuchthatthesetradi-' tionalmethodsareextremelydifficulttoapplyunlesscertainsimplifyingas-sumptionsaremade.Suchdifficultieshaveledtothedeveloprnentofnumerical methods.Thepreviousstudiesonthenumericalapproachwillbepresentedin section1.2.2. 1.2.1Previousstudiesondrawdowntest Thefollowingisasurnmaryoftheoreticalandexperimentalworkthathas beendoneintheareaofflowtowells.Severalmajorareasoutsidethescope ofthepresentstudyhavebeenexcludedfromthiscompilation.Theseinclude multi-phaseflow,anisotropicandheterogeneousporou$media,compressiblefluid flow,stratifiedandmultiple-aquif.ersituations,multiple-wellproblems,and !eaky-aquiferboundaries.Withtheaboveexclusions,thefieldofflowtoward wellsinporousmediabreaksdownintothefollowingclassifications. (1)Steady-state,confinedflowinanaquiferoffinitethickness;totally penetratingwell(Fig.1.1) Forthepurposesofthisdiscussion,confinedflowmeansthattheaquifer isboundedofaboveandbelowbyhorizontal,impermeablesurface.Anaquifer consideredisfiniteinlateralextent.Confinedflowtowardatotallypene-tratingwellispurelyradialflowandisoftenmodeledbyflowbetweentwo

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.concentrlc

circles of constantpiezometric head.The .governmg

differential .equat1on .IS 1

aah

=o

. r ) (1. 1) 11) 12)

where h=lpt-X3• Muskat andPolubarinova-Kochina havederived the solution, which

canbe found .In

anybasic textonground waterflow.The outer circle is the

"radiuS of influence lt

of thewell. This conceptisnotwell difined, .

sincethe 11)

piezometric head .varles

logarithmically withtheradius. However,Muskat

ob-served that any reasonable assumption for the"radiusof influence tt'glves

suf-ficiently .preclse

values forthe discharge flowrate.

Åë x Ground level l

-

--.---.-. .-...--.

N

N"N

Nx

l .-..- --m-p Piezometric head 'Aquifer 1l1 1 lll t11 I

1impermeable

ll strata

Fig.1.1 Totally penetrating well

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(2) Steady-state,confinedflowin an aquiferoffinitethickness; partially

penetratingwell(Fig.1.2)

Nowthereisanadditional velocity direction.

Thus,thedifferentialequationis

lg.(rgh.)+Qggl},,h-o (1.2)

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Muskat distributedsinksalongthe well axisandadjustedtheirintensity to

approximatetheboundarycondition of constantpotentialalongthewell bore.

A theoreticalsolutionforthecase of afinitewellradiuswasproposed by

13)

Kirkham.Kirkhamusednumerical techniques toevaluatecoefficientsof a

Fourier-Besselseries. Åë Groundlevel , .-b---e--...-

"--

-"-'

•=. N.,..,.

NÅ~

x

l /.Åqk

--. Piezometrichead tl t, 11II L J rmpermeablestrata Fig.1.2Partially penetratingwell

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(3)Steady-state,unconfinedflowinanaquiferoffinitethickness;totally penetratingwell(Fig.1.3) Unconfinedflowmeansthatpartoftheboundaryoftheflowdomainisa freesurfacewhoselocationisinitiallyunknown.Thismakesthemathematical solutionmuchmoredifficulttoobtain.Themostcommonwaytohandlethefree surfacetheoreticallyhasbeentousetheDupuit-Forchheimertheory.This theoryisbasedontheassumptions,aspresentedbyDupuit,that1)forsmall inclinationsofthefreesurface,thestreamlinesareessentiallyhorizontal; and2)velocitiesareproportionaltotheslopeofthefreesurfaceandinde- pendentofdepth.Actually,onedoesnotneedtheassumptionofdepth-indepen-dentvelocitiestoformulatethetheory.FlowthatobeysDarcy'slawandthat hasnegligibleverticalvelocity(impliedbyassumption1)enablestheequations offlowtobeaveragedintheverticaldirectionyieldingtheDupuitequation

1d

i9ii;(Bh2 Asmentionedpreviously,surfaeesofseepageareneglectedinthistheory. Flowratesareexact,butthefree-surfaceprofilesareinerror,especially wherecurvaturesofthefreesurfacearelarge;e.g.,intheneighbourhoodofa wellbore.Themathematicaldifficultiesoffree-surfaceflowmotivatedmany researcherstoturntoexperimentalandnumericalmethods.Finitedifference 14) techniqueswereusedbyBoultontoobtainflowratesandfree-surfaceprofiles whichincludedthesurfaceofseepage.Boultonalsoworkedwithasandbedmodel i5) forexperimentalverification.Kirkhamgaveananalyticaisolutionbasedupon afictitiousflowregionabovethefreesurfaceandemployinganinteration, procedure.•Analternateapproach,usingforceanalysisandtrialnglesoffilt-16) rationrequiringnoiteration,wasdevelopedbyKashef.

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-9-Åë 'Groundlevel

lII.I1Freesurface

'

11 ll Nsurfaceofseepage

Aquiferl

s,1l111lrmpermeablestratum Fig.1.3Totallypenetrating well

(4) 'Steady-state,unconfinedflowin anaquiferoffinitethickness; partially

penetratingwell(Fig.1.4)

Dupuittheorycannotbeapplied tothiscasebecausethevertical

velo-city

nearthebottomofthewellisnot

negligible.Thefinitedifference

tech-17)

.mques,similaronebyBoultonwasalso employedbyBorelifortheflow

toa

18)

partially,penetratingwell.Taylorand Brownobtainednumericalsolutions from

19)

afinite elementanalysis.Daganused matchedasymptoticexpansionsto derive

an analyticalsolution.Thewellwas modeledbyalinesinkofuniform strength,

and thesurfaceofseepagewasneglected. Aninnerexpansion,validfor

dis-tances .farfromawellbore,wasobtained bythemethodofmatchedasymptotic

ex-.panslons. Thezerothordertermofthe

outersolutionistheDupuit solution

fora totallypenetratingwell.This techniquegivesanapproximate analytical

(21)

10)

and Muskat'sconfinedflowsolution for apartiallypenetratingwell.

Åë Groundlevel l 1 Freesurface v Aquifer t

NSurfaceofseepage

l l t

"a

e !mpermeablestratum . L:ig.1.4Partially penetratingwell

(5) Steady-state}partallypenetrating wellinaninfinitelythick aquifer

with animpermeabieroof(Fig.1.5)

20)

Anapproximatesolutionwas first presentedbyHarr.Thewellwas

re-presentedbyaline-sinkofconstant intensityperunitlength,andthe we11

12)

wall becameanellipsoidofconstant potential.Polubarinova-Kochina gavethe

samesolutipnandshowedthatitwas validforfree-surfaceflowwith

thelinea-rized ,free--surfaceboundarycondition .Shementionedthatimproved accuracy

can

tt

beobtainedbyvaryingthestrength ofthesinkdistributionalog thewell.

.axls

.Mathemetically,thelinearized ' free-surfaceboundarycondition saysthat

the free-surfaceforunconfinedflow .IS atthesameelevationasthe piezometric

head forconfinedflow.

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11-Åë ' Groundlevel , ---- -.-k..'--b.-.. N.---..

N

'

.-.-e Piezometrichead

--

--

--Aquifer 1 ' ,L J 1 Fig.1.5Confinedflow'

m

aninfinitelythickaquifer (6) Unsteady,confinedflowinan aquifer 'offinitethickness;totally pene-trating well(Fig.1.6) '

Theintroductionoftimecomplicates themathematicalsolution

consider-ably. Hereoneisconcernedwith time-dependent,radialflow.Theaquifer

con--sidered isinfiniteinlateralextent . Thedifferentialequation,which

.m-21)

cludes waterandaquifercompressibility effects,waslucidlyderivedby Nornitsu

inthe formasfollow

laa.(rg:)-\ge

(1.4)

where histhepiezometrichead,Sis the istioragecoefficient[S=yb(ct+nB)] ,and

(23)

Åë Groundlevel .--b.-.. '--b.-.

'

--

'

N

'..Nb 1

t.-'

'N

'

N

N..

-

'

NN

N

-/x Piezometrichead 1 l Aquifer !mpermeablestrata

Fig.1.6 Totally .penetratlng

well

dependencehasbeenintegrated outof the .equatlon

.

Theis,known asthe fatherof unsteady we11solutions,gave the first,

widely popularized, unsteady solution . His"well function"solution to theabove

equation hasbeenextensively tabulated . The differentialequation strictly

describes aconfined, .arteslan

aquifer ' butTheis' solutionhasalso beenused

asan .approxlrnate solution forfree-surface flow withalinearized, free-surface

boundary condition.

22)

Jacoband Lohmansolved thecase of the time-varyingdischarge

undercon-23)

ditions ofconstant headandcomparedtheirresults withfieldtests . Hantush

24)'

dealtwith we11sinsandsof non-uniform thickness. Boultonintroduced elastic

and delayed-yield effectsto theTheis modelbyaddingatime-dependent integrai

termto the differential equation. Dischargeflow ratesthatwere certain

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- 25)26)cifiedfunctionsoftimeweretreatedbyAbu-ZiedandScottandHantush.Peri-•27)28)odicpumpingrateswereinvestigatedbyLennoxandBerg.Kriz,etal.derived theTheissolutionbyreducingthepartialdifferentialequationtoanordinary differentialequationthroughasimilaritytransformofdimensionZessvariables. Theydevelopedanimprovedprocedurefordeterminationofthestoragecoeffici-entandtransmissivityusinggraphical"typecurves". (7)Unsteady,unconfinedflowinanaquiferoffinitethiekness;totally penetratingwell(Fig.1.7) ThebasisformanyanalysesinthisareaistheBoussinesqequationuti-lizingtheDupuitassumptions:

iaah-BL-ah

wherehisthepiezometrichead.Thisnon-linearpartialdifferentialequation muststillbelinearizedinordertoobtainananalyticalsolution.Asmentioned previously,theTheistsolutionhasbeenappliedasanapproximationtothecase ofunconfinedflow.Thus,thefree-surfacedrawdownnearthewellisquitein-accuratebecausenoaccountistakenoftheverticalvelocity.Thisvertical velocityissignificantnearthewellbore. 29) ThepotentialtheoryformulationwassolvedbyBoultonusingalinearized, free-surface,boundaryconditionwhichwasvalidforrelativelysma11drawdowns. Hemodeledthewellwithaline-sinkofconstantdischargeperunitlength.The surfaceofseepagewasneglected,butacorrectionwasappliedtothesolution tocompensateforthisassumption.Thesolutionforthefree-surfacedrawdown asafunctionoftimewasobtainedforconstantdischargeconditionsbyusinga Fourier-Besselanalysis.TheTheiS7solutionwasshowntobevalidforflowin

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anunconfinedwellprovidedthedurationofpumpingwassuffieientlylong. Throughnumericalintegration,correctioncurveswereobtainedfortimesnear thestartofpumping. 30) GloverandBittengerimprovedtheTheis7solutionbyemployinganiteration proceduretotakeintoaccountthereductioninsaturatedthicknessoftheaqui- . ferasthefreesurfacefalls.AlinearizedBoussinesqequationwasderivedby 31) Hantushforwellsinslopingsandsunderconditionsofconstantdischarge. 32) McNeary,etal.solvedthecaseofcbnstant-headoperationwithvariablestora-tivityandtransmÅ}ssivitybynumericalintegrationoftheparabolicBoussinesq equation. 33) Boultonextendedhisearliersolutioninthecaseofconstant-headpro-ductionforawelloffiniteradiusbyFourier-Besselanalysisandcontour integration.Forthistypeofpumpingoperation,thesurfaceofseepagecanbe retainedinthemathematicalformulation.Thesolutionforthewelldischarge asafunctionoftimeisnearlyconstantatfirst,butthenapproachestheJacob 22) andLohmanresultforanartesianaquiferatlongtimesafterpumpstartup. 34) AnanalogmodelusinganRCnetworkwasbuiltbyStallman. Numeriealintegrationofthenon--linearBoussinesqequationwasperformed 35) byKriz,et' al.forconstantdischargeflowratesandlargedrawdowns.The partialdifferentialequationwastransformedintoandordinarydifferential equationbyadimensionalsimilaritytransformation.Anapproachusingthe 36) hydraulictheoryoffree-surfaceflowwastakenbyMahdaviani,whonumerically integratedthecharacteristicequationsobtainedfromtheenergyandcontinuity equationsforthesystem.Hydrostatieequilibriumwasassumed,whichrnadethis 37) approachsimilartotheBoussinesqtheory.FoxandAliduplicatedthiswork. EsrnaliandScottextendedtheresultsofKriz,etal.toconstant-headopera-tion. Boulton'sanalyticalsolutionwasverfiedbyHerbertwitharesistance

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-15-networkanalog. Thissolutionis particularlyusefulforshorttimes afterthe 40)

startofconstant-head welldrawdowntest.TaylorandLuthinhavetreated a

finite-radiuswell withasurfaceof seepageandpartiallysaturated flowabove

41)

thefree-surface usingafinite--difference method.Boultonanalyzed

somedraw-downtestsfor totallypenetratingwells inanunconfinedaquifer.He teokinto

accounttheslow yieldofwaterfrom abovethefreesurfaceasitfalls

concludedthat drawdownversusdistance curveswerelesssensitiveto delayed

yieldeffects thandrawdownversustimecurves.

Åë Groundlevel 1 l

IIfIFreesurface

'et tt

ItImpermeablestratum

Fig.1.7Totally penetratingwell

1.2.2Previous studiesonnumerical approach

Theexplosive developmentofthe numericalapproachinrecenttimes has

beenmainlydue totherapidevolution ofhighspeeddigitalcomputers inthe

(27)

scienceandtechnology,complexproblemshavenowcomewithinthereachofcri-' ticalanalysis.FinnemoreandPerryhaveadaptedarelaxationtechniqueinana-lyzingsteadyseepagethroughanearthdam.Anotherfinitedifferenceapproach 43),44) tothesteadystateseepageproblemhasbeendescribedbyJeppson.!nrecent yearsanincreasingnumberofattemptshavebeenmadetosimulatenonsteadyflow 45) withafreesurfacebymeansoffinitedifferencemodelbyDesai,Szaboand 46) McCraig. Finitedifferencemodelshavecertainrestrictionsastothekindofgeo-logicalsituationsthatcanbehandled.Thefiniteelementmethod,ontheother hand,hasbeenfoundtobeeasilyadaptedtoproblemsofseepagethroughcomplex systemswhereafreesurfaceexists.Thefiniteelementmethod,whichwasfirst devisedasaprocedureforstructuralanalysis,hascometoberecognizedasan effectiveanalysistoolforawiderangeofphysicalproblems.Amongtheseare-. problemsinthefieldofflowanalysis,asubjectofwhichishereininterpretedto encompassnotonlytheflowoffluidsbutalsoheatflow. Theapplicationofthefiniteelementmethodtoflowanalysisproblemswas 47),48)

developedbyZienkiewiczinrelativelyrecentyears,but,nevertheless,asig-49)50)

nificantliteratureonthetopichasalreadyemerged.TaylorandBrownandFinn werethefirsttoapplythismethodtosteadystate,freesurfaceproblems.Finn's approachhasbeenextendedbyVolkertpincludenonlinearflow.Neumanwas abletodevelopaniterativemethodofconvergingtoasolutionthateliminates

49)53)

theambiguityeffectreportedbyTaylorandBrown.Neumanhasdevelopeda finiteelementmethodforanalyzingnonsteadyflowwithafreesurfacethatis anextensionofthetechniqueusedinsolvingsteadystateproblems[Neuman andWitherspoon,1970].Themethodisbasedontheoriginalformofthenon-lineargoverningequationsandthereforeenablesonetoinvestigatenonsteady freesurfac.eproblemsforawidevarietyofconditions.Largeverticalgradi-entsareeasilyhandledinsystemswithcomplexboundariesandarbitrarydegrees

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-17-ofanistropyandheterogeneity.' wriendealing withradialflpwtoawelloperat-ingataprescribedrate,onemusttakeinto accountbothstorageinthewell andtheactualdistributionofvelocitiesalong thewellbore.Themethodcan alsohandletheeffectsofelasticstorage,an importantconsiderationinana-lyzingmultipleaquifersystems.Inaddition,

watercanbeaddedtoortaken

awayfromthefreesurfaceatprescribedrates tosimulateinfiltrationand evaportranspiration.Toobtainvariationswith time,themethodusesanimpli-cit,timecenteredschemethatisaccurateand unconditionallystable.Asa resultonlyasmallnumberoftimestepsare requiredtoreachthefinalsteady state. Inalloftheprecedingwork,theunsaturated regionabovethefreesur-faceisnotconsideredinthecalculation.It standstoreasonthatinorder tomodelaflowregionwherebothsaturatedand unsaturatedregimescoexist, oneshouldcombinethephysicalfeaturesgoverning

'thenatureofeachofthese

importantphenomena.Inrecentyears,there havebeenagrowingnumberof attemptstoconsiderflowinthesaturatedand unsaturatedzonessimultaneously. Finitedifferenceschemesforsimulatingboth ofthesezoneshavebeenemployed

56)57),58)58)

byRubin,TaylorandLutin,VermaandBrutsaert ,andFreeze.Freezehasbeen abletosimulatetransientsubsurfaceflowin athree-dimensional.Withthis 58) three-dimensionalmodel,Freezehasbeenable tosimulatenaturalflowsystems inhypotheticalbasinsanddevelopaninsight intothemechanismsinvolvedin thedevelopmentofperchedwatertablesandin thearealvariationofwater tablefluctuations.Theseproblemsarealmost impossibletobesolvedbythe precedingwork.Thefiniteelementmethodwas firstappliedtoproblemsin-59) volvingsaturated-unsaturatedflowinporous mediabyNeuman,butnotmany researcheshavebeendoneyet.

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1•3ScopeofThisStudy

Thepresentworkisessentiallyanattempttodevelopthefiniteelement methodforanalyzingtransient,multi-dimensionalfluidflowinsaturated--un-saturated,heterogeneousporousmedia.Thisnewmethodmakesitpossibleto identifythosefieldsituationswhereflowintheunsaturatedregionisan importantfactorthatcannotbeneglectedandinthemultiple-aquiferwithfree surface.Unfortunately,considerationeftheunsaturatedzoneinvolvesadded rnathernaticalcomplexityandrequiresdataonunsaturatedsoilproperties. Then,thepurposesofthisthesisaretobreaktheiceintreatingbothzones simultaneously. Chapter2discussesthephysicsofthesaturated-unsaturatedgroundwater motion.Thegoverningequationofsaturated-unsaturatedflowisderivedfrom thelawofmassconservationandfromtheDarcy'slawandRichard'sequationof motionandiscomparedwiththeKlute'sdiffusionequationwhichhasbeenwide-lyusedintheanalysisofunsaturatedflow.Typicalboundaryconditionsare enumerated. rnChapter3,thegoverningequationofflowthroughsaturated-unsaturated zonesisformulatedintothefiniteelementdiscretizationswhichareevolved intothestudyofeithertwo--dimensionalmodelsorthree-dimensionalmodels withradialsymmetry.Thesemodelstakeintoaccounttheeffectsofhysteresis inthevolumetricmoisturecontent-pressureheadrelationshipsinunsaturated zone.Inconjunctionwiththefiniteelementdiscretizationweightedresidual Chapter4dealswiththeexperimentalstudyofhydraulicpropertiesof unsaturatedSoil.rntreatingunsaturatedzone,agreatdealmoredataare requiredthanarerequiredforthesaturatedzone,butthesepropertiesof soilsmustbeknowntoapplythefiniteelementapproachtoactualflow problems.,Themainpurposesofthischapteraretoproposearationalbasisof gettingexperimentalrelationshtpsbetweenpressurehead(tp)andhydraulic -19.b

(30)

conductivity(K)andbetweenpressurehead(th)andvolumetricmoisturecontent(e) with"theinstantaneousprofilemethod"byusingthesourceoflow-energygamna rayattenuationandpressuretransducer,andtoutilizetheseexperimentaidata inChapter5tocheckthenumericalsolutions. InChapter5)inordertocheckthevalidityofthefiniteelementap•-proachdevelopedinChapter3,laboratoryexperimentalstudyoninfiltration anddrainageintwo-andthree-dimensionalsandmodelboxiscarriedoutrespec-, tively.Thenumericalmodelisthenusedtosolvethesetypicalproblemsand theresultsareeomparedwithexperimentalresultsforconsistency.Therela-' tionshipsofthepressurehead-moisturecontentandthehydraulicconductivity -moisturecontentwhichareobtainedinChapter4areusedasinputdata. InChapter6,newmethodsofanalyzingdrawdowntestsaredescribedandil-lustratedwithsomeexamplestodeterminehydraulicpropertiesthatarerequired tosimulateapracticalflowprobleminthefield.Firstly,somemethodsof analyzingdrawdowntestswithpartiallypenetratingwellisdevelopedtodeter-mineanisotropichydraulicconductivitieswhicharetakenintoaccountquite easiiyinthefiniteelementmethod.Methodsofhandlingtheeffectofpartial penetrationaredescribedinsection6.2,wheretheresultsofsuchmethodsare successful,onecannotonlydeterminetheanisotropicpermeabilitiesand storagecoefficientbutalsoobtainsomeideaofthethicknessoftheaquifer beingtested.Secondarily,theanalyticalsolutionsofunsteadyflowdueto drawdowntestarederivedintheconceptionof"lslandModel"thattheshapeof groundwaterlevelisfixedbytheeircularwatersupplywhichisequilibrium withthepumpingrate.Byusingthesesolutions,newmethodsofanalyzingdraw-downtestswhichareperformedinaconfinedaquiferandanunconfinedaquifer aregivenrespectivelyandtheeffectofinfluenceregionisevaluated. InChapter7,havinglookedintothereasonablenessandvalidityofthis finiteelementmodelinChapter5,thepossibleapplicationofthismodelis

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finallydescribed.Before progressinginto

thevariouslevelsofapplica-tionstheinputdataand boundaryconditions arediscussedandevaluated.The

applicationsofmodelsto fieldsituation aretheflowsthroughsandbankat

floodwaterlevelsandthe flowthrough aquiferduetoexcavation.

Finally,inChapter 8,thesummaryof thisthesisispresentedandthe

.varioussuggestionsfor futurestudiesare

'

.gzven.

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,--21•-References 1)Muskat,M.:Theflowofhomogeneousfluidsthroughporousmedia,lsted. McGrawHill,1937. 2)Todd,D.K.:Groundwaterhydrology,Wiley,NewYork,1959. .3)Harr,M.E.:Groundwaterandseepage,McGraw--Hill,NewYork,1962. 4)Polubarinova-Kochina,P.Ya:Theoryofgroundwatermovement,Princeton UniversityPress,1962. 5)Childs,E.C.:Anintroductiontothephysicalbasisof-soilwaterphenomena, Wiley,NewYork,1969. 6)Scheidegger,A.E.:Tehphysicsofflowthroughporousmedia,Thirdedition, UniversityofTorontoPress,1974. 7)Hillel,D.:SoilandWater,physicalprinciplesandprocess,AcademicPress, NewYork,1971. 8)Dewiest,R.J.M.:Flowthroughporousmedia,AcademicPress,NewYork,l969. 9)Bear,J.:Dynamiesoffluidsinporousmedia,Americanelsevier,NewYork, 1972. 10)Cedergren,H.R.:Seepage,drainage,andflownets,JohnWiley,NewYork,1967. 11)Muskat,M.:TheFlowofhomogeneousfluidsthroughporousmedia;McGrauHill, N.Y.,1937,pp.15-276. 12)Polubarinova-Kochina,P.Ya.:Theoryofgroundwatermovement,(Trans.by -R.J.M.SeWiest),PrincetonUniv.Press,Princeton,N.J.,1962,pp.360-362. Geophys.Res.,64,9,1959,pp.1317-1327. 14)Boulton,N.S.:Theflowpatternnearagravitywellinauniformwaterbearing medium,Jour.Inst.CivilEng.,36,1951,pp.534-550. 15)Kirkham,D.:Exacttheoryfortheshapeofthefreewatersurfaceabouta wellinasemi-confinedaquifer,Jour.Geophys.Res.,69,12,1964,pp.2537-2549.

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16) Kashef,A.1.: Exactfreesurfaceofgravitywells,Proc.AmeF.Soc.Civil Eng.,91,HY4,1965,pp.167-184. 17) Boreli,M.: Free-surfaceflowtowardpartiallypenetratingwells,Trans.Am. Geophys.Union, 36,4,1955,pp.664--672. 18) Taylor,R.L. &C.B.Brown:Darcyflowsolutionswithafreesurface,Proc. Amer.Soc. CivilEng.,93,HY2,1967,pp.25-33. 19) Dagan,G.:A derivationofdupuitsolutionofsteadyflowtowardwellsby matchedasynptoticexpansions,WaterResourcesRes.,4,2,1968,pp.403-412. 20) Harr,M.E.: Groundwaterandseepage,McGrawHIII,N.Y.1962,pp.259-262. 21) Momotsu,T.&K.Yama$hita:Anadvanceinthetheoryofwells,II,Chikylibutsuri (Geophsics) 7,1,pp.21-40,(1943) 22) Jacob,C.E.& S.W.Lohman,Non-steadyflowtoawellofconstantdrawdownin anextensive aquifer,Trans.Amer.Geophys.Union,33,4,19S2,pp.559-569. 23) Hantush,M.S. :Flowofgroundwaterinsandsofnon--uniformthickness-3,flow towell,Jour .Geophys.Res.,67,4,1962C,pp.1527-1534. 24) Boulton,N.A.: Analysisofdatafromnon-equilibriumpumpingtestsallowing fordelayed .yieldfromstorage,Proc.Inst.CivilEng.,26,pt.3,1963. N pp.469-482. 25) Abu-Zied,M. &V•H•Scott:Non-steadyflowforwellswithdecreasingdischarge, Proc.Amer. ,Soc.CivUEng.,89,HY3,1963,pp.119-i32. 26) Hantush,M.S.:Drawdownaroundwellsofvariabledischarge,Jour.Geophys. Res.,69,20 ,(1964B).pp.4221-4235. 27) Lennox,D.H. &A.V.Berg:Drawdownsduetocyclicpumping,Proc.Amer.Soc. CivilEng., 93,HY6,1967,pp.35-51. 28) Kriz,G.J.,V .H.Scott&R.H.Burgy:Graphicaldeterminationofconfinedaquifer parameters, Proc.Amer.Soc.CivilEng.,92,HY5,1966A,pp.39-48. 29) Boulton,N.S. :Thedrawdownofthewatertableundernon-steadyconditions nearapumpedwellinanunconfinedforrnation,Proc.Znst.CivilEng.,3.

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-23-pt.3,1954, pp•564-579. 30) Glover,R.E.& M.W.Bittinger:Drawdownduetopumpingfromanunconfined aquifer,Proc. Amer.Soc.CivilEng.,86,IR3,1960,pp.63--70. 31) Hantush,M.S.: Hydraulicsofgravitywellsinslopingsands,Proc.Amer.Soc. CivilEng.,88, HY4,1962,pp.1-15. 32) McNeary,S.S.1.Remson&S.C.Chen:Hydraulicsofwellsinunconfinedaquifers, Proc.Amer.Soc.CivilEng.,88.HY6,1962,pp.115--123. 33) Boulton,N.S.: Thedischargetoawellinanextensive,unconfinedaquifer withconstant pumpinglevel,J.Hydrology,3,2,1965?pp.124-130. 34) Stallman,R.W.: Effectsofwatertableconditionsofwaterlevelchangesnear pumpingwells, WaterResourcesRes.,1,2,1965,pp.295-31!. 35) Kriz,G.J.V.H. Scott&R.H.Burgy:Analysisofparaminationofanunconfined aquifer,Proc. Amer.Soc.CivilEng.,92,HY5,1966,pp.49-56. 36) Mahdaviani,M.A. :Steadyandunsteadyflowtowardsgravitywells,Proc.Amer. Soc.CivilEng. ,93,HY6,1967,pp.135-146. 37) FoKJ.A.&r.Ali: Unsteady,unconfinedflowtogravitywells,Proc.Inst. CivilEng.,40, 1968,pp.451-469. 38) Esmali,H.&V.H .Scott:Unconfinedaquifercharacteristicsandwellflow, Proc.Amer.Soc.CivilEng.,94,TRI,1968,pp.115-136. 39) Herbert,R.:Time variantgroundwaterflowbyresistancenetworkanalogues, .Jour,Hydrology, 6,3,1968,pp.237•-264. 40) Taylor,G.S.&J .N.Luthin:Computermethodsfortransientanalysisofwater tableaquifers, WaterResourcesRes.,5,1,1969,pp.144-152. 41) Boulton,N.S.: Analysisofdatafrompumpingtestsinunconfinedaquifers, J.Hydrology,10, 4,1970,pp.369-378. 42) Finnemore,E.J. &B.Perry:Seepagethroughanearthdamcomputedbythe relaxation technique,WaterReso.Res.,Vol.4(5),1968,pp.1059-1064.

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-43) Jeppson,R.W.:Seepagefromditchessolutionbyfinitedifferences,Proc. ASCE,94(HYI),1968,pp.259--267. 44) Jeppson,R.W.:Seepagethroughdamsinthecomplexpotentialplane,Proc. ASCE,94(IRI),1968,pp.23-30. 45) Desai,E.S.&Sherrnan,W.C.:Unconfinedtransientseepageinslopingbanks, Proc.ASCE,(SM2),1971,pp.357-373. 46) Szabo,B.A.&McCraig,I.W.:Amathematicalmodelfortransientfreesurface flowinnon-homogeneousoranisotropicporousmedia,WaterReso.Bull. Vol.4(3),l968,pp.5--23. 47) Zienkiewicz,O.C.&Cheung,Y.K.:Finiteelementsinthesolutionoffield problems,TheEngineer,1965,pp.501-510. 48) Zienkiewicz,O.C.&Cheung,Y.K.:Solutionofanisotropicseepageproblemsby finiteelements,Proc.ASCE,Vol.92,(EMI),1966,pp.111-120. 49) Taylor,R.L.&Brown,C.B.:Darcyflowsolutionsandafreesurfaces,Proc. AsCE,Vol.93,(Hy2),1967,pp.25-33. 50) Finn,W.D.:Finiteelementanalysisofseepagethroughdams,Proc.ASCE, 51) Volker.R.E.:Nonlinearflowinporousmediabyfiniteelements,Proc.ASCE, Vol.95,(HY6),1969,pp.2093-2101. 52) ,Neuman,S.Pe&Witherspoon,P.A.:Finiteelementmethodofanalyzingsteady seepagewithafreesurface,WaterReso.Res.,Vol.6,No.3,1970,pp.889-897. 53) Neuman,S.P.&Witherspoon,P.A.:Analysisofnonsteadyflowwithafreesurface usingthefiniteelementmethod,waterReso.Res.,Vol.7,1971,pp.611-623. 54) Rubin,J.:Theoreticalanalysisoftwo-dimensionaltransientflowofwater inunsaturatedandpartlyunsaturatedsoils,SoilSci.Soc.Amer.Proc.,32, 1968,pp.607--615. 55) Taylor,G.S.&Luthin,J.N.:Computermethodsfortransientanalysisofwater-tableaquifers,WaterReso.Res.,Vol.5,No.1,1969,pp.144-152.

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-2S-56) Verma,R.D.&W.Brutsaert:Unconfined aquiferseepagebycapillaryflow theory,Proc.ASCE,Vol.96(HY6),1970, pp.1331-1344. . 57) Freeze,R.A.:Themechanismofnatural ground-waterrechargeanddischarge, 1,one-dimensional,vertical,unsteady, unsaturatedflowabovearecharging ordischargingground--waterflowsystern, WaterReso.Res.,Vol.5,No.1 ,1969, pp•153-171.

58) Freeze,R.A.:Three-dimensional,transient ,saturated-unsaturatedflow .ma

groundwaterbasin,WaterReso.Res.,Vol .7,No.2,1971,pp.347--365.

59) Neuman,S.P.:Saturated-unsaturatedseepage byfinite•elements,Proc., ASCE,

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CHAPTER2 PHYSICSOFSATURATED-UINSATURATEDGROUI-IDWATERMOTION

2.1Introduction

lnsolvingaspecificphysicalproblem,suchastheflowofaliquidthrough aspecifiedporousmediumdornain,itisnecessarytodevelopthefundamental equationsdescribingthetransportoffluidinaporousmedium. Inthischapter,firstly,Darcy'slawisdiscussedastheequationofmo-tion.TheexperimentofDarcywillnotbeuniquelydefined,therefore,thereis considerableambiguityinpostulatingadifferentialequationwhichwouldbe equivalenttotheresultsoftheexperiments.Infact,thedifferentialequa- tionwhichisnowcommonlycalled"Darcy'slaw"isnotanequivalentexpres-sionforDarcy'sfinding,althoughthesedofollowfromit.However,theywould equallywellfollowfromothertypesofdifferentialequations• ThisisespeciallytrueifgeneralizationsofDarcy'slawtoanisotropic compressibleandunsaturatedporousmediaareattempted.Somediscussionwillbe devotedtothissubjectlaterinthischapter. ItistobeexpectedthatDarcy'slawwillhavelimitations.Indeed,sueh 'limitationsoccurgenerallyashighandlowflowrates,aswellasinrelation tovariousothereffects.TherangeofvalidityofDarcy'slawanditslimita--tionswillalsobediscussedinthischapter. Secondly,Richard'sequationisdiscussedasthelawofcontinuityorcon-servationofmass.Coupledtheequationofmotionwiththelawofcontinuity, thegoverningequationsforsaturated-unsaturatedflowwillbederivedandcom-paredwiththeKlute'sdiffusionequationwhichwaswidelyusedintheanalysis ofunsaturatedflow.

WhenaproblemisdescribedsimultaneouslybyanumberQfdependentvaria-

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-27-bles,thesamenumberofequatiorisisneededforacompletesolution.Beinga generaldescriptionofanactualphenomena,itisobviousthatthepartialdif- ferentialequationitselfdoesnotcontainanyinformationconcerningthespe-cificva!uesofquantitiescharacterizingaspecificcaseofaphenomenon. Therefore,anypartialdifferentialequationhasaninfinitenumberofpossible solutions,eachofwhichcorrespondstoaparticularcaseofthephenomenon. Toobtainfromthismultitudeofpossiblesolutionscorrespondingtoacertain specificproblemofinterest,itisnecessarytoprovidesupplementarytoacer- tainspecificproblemofinterest,itisnecessarytoprovidesupplementaryin-formationthatisnotcontainedinthepartialdifferentialequation.Finally .section2.5willincludeadiscussionoftheinitialandboundaryeonditions offlowoffluidsthroughporousmedia.

2.2EquationofMotion

rn1865,HenriDarcypublishedinanappendixtohisbook"LesFontaines PubliquesdelaVilledeDijon"theresultsofhisexperimentsontheflowof waterthroughgranularmaterial.Usingaeylindricalsample,thedirgctionof flowbeingalongthecylinder,hefoundthedischargeperunitareaofcross--sectiontobeproportionaltothegradientofpiezometricheadinthedirection offlow,i.e. Here,Qisthedischargethroughthesample,Aisthegrossareaofthesample's cross-section,AhistheheadlostinalengthAÅíandKisconstantforagiven sample.Thisexpression,andvariousrearrangementsofit,havebeennamed

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Darcy'slaw anditisthebasicrelationshipinquantitativestudyoftheflow offluids throughporousmedia. Darcy's laworiginallywaslimitedtoone-dimensionalflowinasteady statefor ahomogeneousincompressiblefluid.WhenDarcy'slawisextendedto aformal generalizationoftheequationofmotion,someproblemsariseasfollows: (1)to three-dimensionalflow (2)tothe flowinanisotropicmedium (3)tothe flowinananisotropicmedium (4)to unsteadystateflow Several researchershavederivedDarcy'slawfromthegeneralNavier-'Stokesequa-tionsfor viscousflowtoextendDarcy'slawtoaboveproblems.Underunsteady state conditionstheNavier-StokesequationforlowReynoldsnumbers(neglecting thehigher orderinertialterm)becomes av.

p--Ka--t+p-gtilli-:-=vv2vi(2.2)

HereÅëis aforcepotentialdefinedas

tp=-xigi+-gi=(O,o,-g) istheaccelerationduetogravity,p,vandparerespectiveiythe pressure, 1)*viscosityanddensityofthefluid.Themassaveragevelocityviisin-troduced intoEq.(2.2). av.*

Inabove equationpV2v*irepresentsthedensityoftheforceduetothefluid"s

viscosity, whichresiststhemotion.Itisaviscousforceperunitvoiumeof

fluid.These resistanceforcesdependuponthefrictionofflutdparticleswith

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-29-soilparticles.Theforcesofinternalfrictionfluidparticleswithfluid particlesarenegligiblysmallincomparisonwiththeforcesofexternal friction.Theresistanceforcesofferedbyasinglesphereforthespecialcase ofslowviscousflowofaNewtonianfluidisgivenbyStokeg?'ia)w.Thislawcan beexpressedinageneralizedformas inwhichf pistheresistanceordragofasingleparticle,urepresentsthedy-namicviscosityofthefluid,distheparticlediameter;forirregular-shape nonuniformparticles,dwouldbethecharacteristiclengthoftheaverage-size velocity),andXrepresentsacoefficientthattakesintoaccounttheeffects- ofneighboringparticles.ThecoefficientXwilldependuponthelocalstream-lineconfigurationaroundtheparticleand,hence,mustbesomefunctionofthe geometryoftheporesystem.MorespecificallyXwilldependupontheporosity, theshapeoftheparticles,andthedistributionofthesizesoftheparticles. ThetotalresistanceFofferedbyalloftheparticlesintheelementwillR thusbe inwhichNrepresentsthenumberofparticlesintheelement.FRisrewrittenin theanotherform ConsideringthatthecoefficientBretainsthesamevaluefortheunsteadymove-mentasforthesteady.Eq.(2.4)withthehelpofEq.(2.7)becomes *p"/i+pgO..=--(vlB)v*i(2•8)t'

(41)

andthisequation canbeshownin.thenextform *av.*B)agtll'(B)p'-Ka'r]-'t'(2'9)

v.=-(IUIV

4) Averagingover thecross-sectionweobtain

*k.-BÅë"'BaV"i•(2.lo)

Vi'-

.pPexi"(V)at

-

e-wherevisthe

averagekinematicviscosity(=u/p).Thus,theequationofmotion derivedinthe formofEq.(2.10)isvalidforaninhomogeneousfluidinlaminar flow throughan ' permeability willbediscussedindetailinotherchapter.Eq.(2.10)isthege-neralizedform ofDarcy'slawfornonsteadystateflowandtheformoftheexten--sionofDarcy's lawtothree-dimensionalflowinanisotropicmedia. Another derivationsofDarcy'slawfromNavier-Stokeseq.arepresentedby 4),5),6) Whitaker(1966 7),8),9),1967,1969)andbySlatteryÅq1967,1969,1972).Theyused theSlattery-Whitaker averagingtheoremwhichdiscoveredsimultaneouslyandinde-pendentlyby Slattery(1967)andbyWhitaker(1967).Thistheoremenables onetoexpress thevolumeaveragesofspacederivativesintermsofthespacede--rivativesof volumeaverages,therebymakingitpossibletoproceedwiththein-tegrationof differentialequationsfromonescaleofmeasurementtoanotherin amathematically reigorousfashion. Thesecond termontheright-handsideofthemotionequationEq.(2.10)ex-pressesthe averageacceleration.SinceinpracticeDarey'slawisalwaysex-pressedinthe formnegl,ectingthelastterminEq.(2.10),itisimportantto knowunderwhat conditionscanonejustifyneglectingthisterm.Forflowsin whichtheloeal inertialforcescanbeneglectedwithrespecttotheviscous( resistance) .forces,polubarinova-Kochik9)(lg62)hasindeedshownthatthe.

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-31--accelerationtermintheequationofmotiontends veryrapidly(e.g.,withina

fractionofasecond)tozeroaftertheonsetof

flow.Hence,oneisjustified indeletingthistermfromtheequation. Thesameinterestingwayoflookingatthis problemhasbeensuggestedby 6) Whitaker.Tousetheauthor'sownwords(Whitaker ,1969,pp.25)"lfatubefill-edwithfluidissubjectedtoasuddenchangeis thepressuredrop,essentially steadyflowoccursfortimesgreaterthantowhere

to=d2!4v.Heredisthetube

diameterandvisthekinernaticviscosity.Forthe purposeofestimatingmicro-scopictransienttimesinporousmedia,apractical lowerboundonvis10-'2cm21

sec,andanupperboundondmightbeontheorder

of10-icm.Thisgivesamicro-• scopictransienttimeontheorderof1sec,and ifthetransienttimeforthe macroscopicprocessismuchlarger(sayonthe orderofminutes),weshouldbe treattheflowasquasi-stead..."khenthefluid isincompressibleandthepor-ousmediumisrigid,allthetransienteffectsare causedentirelybytemporal ehangeintheexternalboundaryconditions.In practicethecharacteristictime ofsuchchangesisusuallyatleastontheorder ofminutes,indicatingthatthe timederivativeinEq.(2.10)canprobablybe neglectedinmanysituations. Thenforahomogeneousincompressiblefluid, p=const,u=constandthemotion .Eq.(2.10)maybewrittenintermsofthepiezometric headfi=z+p'/V•

*---efi

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vi='(KijYIV)o.j

. ThisequationisDarcy'sexperimentallaw. Althoughupperevaluationsdonotcontribute totheformulationofanew law,theseconfirmtheearlybeliefthatDarcy's lawisofthenatureofstatis-ticalresultgivingtheempiricalequivalentof Navier-Stokesequations. Itisimportanttodefinearangeofvalidity ofDarcy'slaw,becauseDarcy's

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lawisnotuniversallyvalidforallconditionsofliquidflowinporousmedia. InderivationofDarcy'slawfromNavier-Stokesequation,itisassumedthatthe flowislaminar(i.e.,nonturbulentslippageofparallellayersofthefluid oneatopanother,andinertialforcesarenegligiblecomparedtoviscosforces). Laminarflowprevailsinsiltsandfinermaterialsforanycommonlyoccurring hydraulicgradientsfoundinnature.Incoarsesandsandgravels,however,hy- draulicgradientsmuchinexcessofunitymayresultinnonlaminarflowcondi-tions,andDarcy'slawmaynotalwaysbeapplicable.Inflowthroughconduits, theReynoldsnumber(Re),adimensionlessnumberexpressingtheratioofiner-tialtoviscousforces,isusedasacriteriontodistinguishbetweenlaminar flowandturbulentflow.Byanalogy,aReynoldsnumberisdefinedalsoforflow throughporousmedia: R=v*dlv(2.12)e wherev*isthemeanflowvelocity,dtheeffectiveporediameter,andvisthe kinematicviscosityofthefluid. Thelimitcanbefoundbyplottingthedimensionlessfanningfrictionfac--torf,usedinhydraul.i'cs,againstRe.Thefactorfisdefinedby (2.13) 2pLv*2 whereApisthepressuredifferenceoveralengthofporousrnedia,Lmeasured alongthelineofflowandotherquantitiesareasdefinedabove.Datafromsev- eralinvestigationsareplottedinFig.2.1.Departuresfromalinearrelation-12) shipappearwhenRereachestherangebetweenabout1and10,thusindicating ,anupperlimitforthevalidityofDarcy'slaw. Tfwetakev=O.O18cm21sec(forwater),then

v*dÅqO.O18•.O.18(2.14)

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-33-wherev*is expressed

incmlsee,d-in

cmnext

swe taked=O.1cm,then

'v*Åq

O.18-"1.8(cmlsec) (2.15) lo5 A rnvestigators -"lobo e Bakhmeteff andFeodoroff "ves

eBurke and Plmmer

-g z AMavisand Wilsey

ori

e xRose

" e

.Vlo3

"Saunders andFord

p e lp 5 oo : e H: $nx :di eeett in lo2 o f.-yt2gg-oo nxeu D Re aN!.. e NNÅq5' . 10

tceeeoeeN

o.Ol 10

io2io3

Reynolds numder, Re

Fig.2.1 Relationoffanning friction factor toReynoldsnumberfor

12)

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thisrneansusedDarcy'slaw,then,

V*=Ke.O•18•-1•8(cmlsec)

whenwetakeK=1.0Å~10-2cmlsec

ahÅq18180

thisvaluesareverylargehydraulicgradient,andDarcy'slawcanbeemployed inthevastmajorityofcasesconcerningtheflowofwaterinsoil. Onthecontrary,forverylowvelocities,someinvestigators(Swartzen- druber(1962),MillerandLow(1963))haveclaimedthat,inclayeysoils,lowhy-draulicgradientsmaycausenofloworonlylowflowratesthatarelessthan proportionaltothegradient,whileothershavedisputedsomeofthesefindings 15) (Olsen(1965)).Apossiblereasonforthisanomalyisthatthewaterinclose proximitytotheparticlesandsubjecttotheiradsorptiveforcefieldmaybe morerigidthanordinarywater,andexhibitthepropertiesofa"Binghamliquid" (havingayieldvalue)ratherthana"Newtonianliquid."Theadsorbed,or bound,watermayhaveaquasicrystallinestructuresimilartothatofice,or evenatotallydifferentstructure.Soraesoilsrnayexhibitanapparent"thresh- 'oldgradient,"belowwhichthefluxiseitherzero(thewaterremainingappar-entlyimmobile),oratleastlowerthanpredictedbytheDarcyrelation,and onlyatgradientsexceedingthethresholdvaluedoesthefluxbecomeproportion--altothegradientFig.2.2. Finallythereseemtobeaveryimportanthydrologicalproblemthatmayre-quiretheapplicationoftheDarcy'slawtotheflowinunsaturatedsoils.For theaboveproblem,anexperimentforthedirectverificationofDarcy'slawin 16) unsaturatedrnaterialshasbeencarriedout(ChildsandCollis-George,1950). Theydevisedamethodwherebythemoisturecontentandsuctiondownalongcolumn

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-35-

x:-,k-'

o

Yield

Hydraulicgradient

Fig.2.2 Possibledeviations fromDarcy'slawatlowgradients

ofporousconductor wereuniform, thepotentialgradientbeingduesolelytothe

gravitational component.Various magnitudesofpotentialgradientwereimposed

bysuspendingthe columnatvarious anglesofinclinationtothevertical.From

theresultsitcould besafely inferredthattherateofflowforagivendegree

ofsaturationwas .proportional

,tothepotentialgradient,asinthecaseofsat-uratedmaterials. Thusifmay

bepossibletoassumethattheDarcy'slawisap-plicabletounsaturated flowin anisotropicmediaintheform:

v.

1

ah=--K..(Åë)ax.IJJ

(2.18)

wherev.isthei (volumetric)flux (volumeofwaterperunitareaunittime).

ahlax.isthehydraulici

head

gradient,whichmayincludebothsuctionandgravi-tationalcornponents andK..isaIJ functionofthepressurehead(th)orafunction

ofthevolumetric watercontent (e),andnotaconstant,asinsaturatedflow.

17)

Eq.(2.18)issometimes calledthe Richard'sequation,becauseRichardshasshown

firstlythatthe modifiedDarcy' slawappliestounsaturatedsoils.Millerand

18)l9)

Miller(1956)point .outanlmportant

differencethatsincethemoisturecharac-teristicissubject tohysteresis

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thehistoryofwettinganddryingbywhichthatvolumetricmoisturecontent. isreached. However,therelationofhydraulicconductivity(sornetimescalledcapil--, laryconductivity)tovolumetricmoisturecontentK..(e)isaffectedbyhyste-IJ resistoamuchlesserdegreethanistheK..(th)function,atleastinthe.IJ20) mediathusforexamined(ToppandMiller,1966).Thus,Darcy'slawfor'unsatu-ratedsoilcanalsobewrittenas vi=-Kij(e)ax.J which,however,stillleavesonewiththeproblemofdealingwiththehysteresis betweentpande.Amorecompletereviewofthisproblemwillbebroughtata laterchapterofthisresearch. 2.3EquationofContinuity Intrasientgroundwatermotion,thedistributionofpotentialwithinthe ' flowregionundergoescontinualchangewithtime.Thenatureofthetime-dependenceofconservationofmass,subjecttotheconstraintsoftheequations ofmotionandofstate.Theconservationlawasappliedtofluidorheat transferisalsoknownastheequationofcontinuity. ConsiderafinitesubregionoftheflowregionasdepictedinFig.2.3, Undertransientconditionsofflow,waterentersandleavestheflowregionat differentratesatdifferentpartsoftheenclosingsurface(controlsurface). Theamountandidentityofmatterinthecontrolvolumemaychangewithtime, buttheshapeandpositionofthisvolumeremainfixed.Thespecialcase,when theinflowexactlyequalstheoutflowsothatthereisneitheramassexcess

-37-.

(48)

N 1

1

NN /

Fig.2.3 Fluxes crossingboundarysurfaceef

an arbitrary

.

subregion oftheflowregion

nora massdeficiency

,isthe

phenornenonof$teadystateflow .Since

'masscan

neither becreatednor destroyed, themassconditionhasto be adsorbedintoor

released fromthe particularsmall partoftheflowregion under consideration.

The equationof continuity,then, isastatement,equating the summationof

therates atwhich massentersor leavesthecontrolvolume ofthe flowregion

andthe rateatwhich .massls

absorbedbythesubregion. According toSokolnikoff

21)

and Redheffer(1966), theiawof conservationofmasscanbe stated as

ÅÄÅÄ

V'nd6=

ÅÄdivVdV (2.20)

s v

ÅÄwhereV

isthevelocity ofthe fluid

ÅÄn

istheunit normal

s isthecontrol surface

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Eq.(2.20)amountstodividingafiniteregionSintoalargenumberofsma11 voumeelementsandsumminguptherateoffluidincreaseineachelemntin ordertoobtaintheoverallfluidmassincreaseinV.Thelefthandsideof Eq(2.20)simplystatesthatthefluidexcessarisingoutoftirasientflowof fluidacrosscontrolsurface(lefthandsideofEq.(2.20))isaccommodatedby '-anequivalentincreaseinthefluidcontentwithintheelement(righthandside ofEq.(2.20)).Thisisbutarestatmentoftheconservationequation.Note alsothatthelefthandsideofEq.(2.20)isthecauseandtherighthandside istheeffect. 17) Insoilphysicsliterature,Richards(1931)wasprobablytheearliestto expresstheequationofcontinuityfortransientsoilwatermovementinthe formofaparabolic,partialdifferentialequation.Familiarlyknownasthe Richard'sequation,itcanbewrittenincartesiancoordinatesas, '

-divpV=-e.pV=a2(pe)(2.2i)

ÅÄwherepisthedensityofwater,VisspecificfluxorDarcyvelocityvector andeisthevolumetricmoisturecontent,definedasthevolumeofwaterper unitvolumeofsoil.InEq.(2.21)thedensityofwaterpisassumedtobeinde-pendentofspaceandtime,Eq.(2.21)reducedto,

•-e.•ee

-divV=V=

whichisalsoanexpressionoftheconservationofmass.

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-39-2.4GoverningEquationofFlowinPorousMedia Inpracticalproblemsoffluidflowinporous media,themosteasily measuredphysicalparameteristhefluidpotential. Also,initself,the variationinfluidpotentialisaphenomenonof considerableinterestin studyingflowthroughporousmedia.Hence,inthe practicaluseofEq.(2 . 22),itisconvenienttomakefluidpotentialthe dependentvariable.Thus ' substitutingEq•(2.19)intoEq.(2•22),assumingthat .X3ISaconstant duringthetimeinterval

divK(e)=bt(e)

(2.23) Eq.(2.23)isthegoverningequationforflowthrough porousmedia. RewritingEq.(2.23)inthecartesiancoordinate andinthetensor notation,onehas (224)

at=

ax.[D(e)a..+Ki3(e)]

. whichistheso-calleddiffusionequationderivedby D(e)=Kij(e)-

gaig-e--'(2.25)

whereD(e)isthesoil-moisturediffusivityinsoil physicsliterature. ltmaybeappreciatedthataV/aeisa'measureofthe storagepropertiesof

thegeologicalmaterialandisthefirstderivativeof thepressurehead, vJe

totheincrementofthevolumetricmoisturecontent,

e.Thefunctionalre-lationshipbetweenthande(so-ca'11edwaterretention curve)isshownin Fig.

2.4.AsseemfromFig.2.4,themoisturecontentetends tobecomeconstant

whichisequaltoporosityofsoil,n,assoonasÅëÅrtp ,i.e.,whenthe

cr

suctionisles$thantheairentryvalue(IJcr)andD becomesessentially

discontinuousasaVJ!aeÅÄco.ThereforeEq(2.24)cannot beusedforthe

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A

3i!,, (Unsaturated zone) ve v=2 =thca 2g i;,cr SslooZr o Vo1urnetricmoisture content(e) (Saturated zone)

Fig.2.4 Variationsinthemoisture content

zonemustbe distinguished fromthemovementofwaterinsaturated zone,it

isquite inconvenientto solvetheproblemsuchas advancementofwetting

frontin the saturated-unsaturated sdilmedia. AlthoughEq.(2.24) hasbeen

22)23)24) used .qulte

ofteninthe flowanalysis,solutionsare obtainedonly inthe

problems for unsaturated zone.

Meanwhile,itisknown thatthevolumetric .molsturecontent

eisex-pressed as

e=nsW

(2.26)

wherenis theporosityof thedegreeof .saturatlon.

Choosing thas onlyone dependentvariableandapplying thechain ruleof

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-41-differentiation tothetime derivativeontherighthandsideofEq.(2. 23),

Eq.(2.23)canbe transformedto thefollowingequation.

div K(tp)i(l;(.

thX3

)=(nS)atw

a

'

d-Qa31{-.

=dtp(nS.)at

dS (2.27)

[S.dtp"dv]at

Thetwotermsontherighthand sideofEq.(2.27)denotetwodistinct

physi-25)

calphenomena.Thefirstterm representsthedeformabilityofthesoil

skeletonwhichis anologousto thecosolidationproblemexpressedby Biot's

26)

equation.This termisusually ignoredintheproblemsofflowthrough porous

mediabyassumingthattheporous mediaisrigid.Hereweconsiderthat the

soilmediumis slightlycompressible andalsoincludesthelastterm which

representsthe desaturationof thepores,thatis,thecapacityofthe soil

toabsorborrelease .mo1sture duetosaturationchanges.

Assumingthat theporosity ndoesnotchangeduetothevariation of

pressuretpin unsaturatedzone, Eq.(2.27)rnayberesucedto:

div K(tp)K5(v+x3

)=(c(e,)+Bs.)"a,+s(2.

28)

where

Unsaturatedzone

B-Saturatedzone

whereSdenotess thespeeific storage(definedaSthevolumeofwater

instantane-ouslyreleased frornstorageper unitbulkvolumeofsaturatedsoilwhen this

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deldth)andSisasinkorsourceterm.InEq.(2.28),theadvantagebeing thatC(e)remainsfinitethroughouttherangeofflow. C(V)attainsamaximumvalueincoarsesand,neartheairentryvalue (tpcr),wherelargechangesineoccuratsmallchangesoftp.Itvanishesat suctionsmallerthantheairentryvalueorforQÅrtpcr(looselycalled,"at saturatedsoil"). ClearlyEq.(2.28)hastheadvantageoverEq.(2.24)thatisappliedfor thewholeflowregion,includingsaturatedandunsaturatedflow.SoEq.(2.28) iscalled"thegoverningequationforsaturated-unsaturatedflowthrough porousmedia." 2.5InitialandBoundaryConditions Thesupplementaryinformationthat,togetherwiththepartialdifferen-tialequation,definesanindividualproblemshouldincludespecificationsof: (a)thegeometryofthedomaininwhichthephenomenonbeingconsidered takesplace, (b)allphysicalcoefficientsandparametersthataffectthe.phenomenoncon-sidered(e.g.,mediumandfluidparameters), (c)initialcondotionswhichdescribetheinitiaistateofthesystemcon-(d)theinteractionofthesystemunderconsiderationwithsurrounding systems,i.e.,conditionsontheboundariesofthedornaininquestion. 2.5.1Initialcondition LetusnowconsiderthesurfaceintegralinEq.(2.28).Atanyinstantof timet,thereisaninitialdistributionoftpwithintheflowregionbound-arybythe,surfaceS.Thisregionrnaybeapartoralloftheflowregion• ThisdistributionofWformstheinitialconditionforthetransientfluido flowproblem.

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-43-tp(xi,O)=tp.(Xi) (2.29) 2.5.2Boundaryconditions Thefolwregionasawholecommunicatesacrossitsboundarieswithits surroundings.Thenatureofthiscommunication isreflectedintheconditions thatexistontheboundaryoftheflowregion. Thenaturesoftheseconditions .aretermedboundaryconditionsand,ingeneral ,thesearefourtypes.Inaddi-tion,theremanyexist"mixedboundaryconditions "ofspecialtypes,whichwill beomittedforthepresent. a.No-flowboundary Whennofluidentersor!eavestheflow regionacrossitsboundary,the boundaryiscalledano-flowboundary.Influid flow,animpermeablebarrieror aplaneofsymmetryofflowisanimpermeable boundary.)tathematically,thisis identicaltotheconditionthatthereisno gradientinpotentialacrossthis boundaryandhence, "th.(Kij(th)"a..+Ki3)ni=j inwhichfidenotesthenomaltotheboundary. b.Prescribedfluxboundary rtmaysohappenthattheflowregion receivesfluidfromordischarges fluidtoitssurroundingsataknownorprescribed rate.Forexample,asoil maybereceivingrainfallinfiltrationata

constantrateorawellmaybedis-chargingtheflowregionataconstantor variablerate.Suchaconditionis

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knownasaprescribedfluxboundarycondition.Mathematically,thisamounts •tosayingthat,foraunitsurfaceareaoftheboundary,ahlanisknown,and hence, (Kij(tp)-

gall;..+Ki3)ni=-v(xi,t)(2.31)

.SuchaconditioniscalledaNeumannprobleminpotentialtheory.Indeed,the noflowboundarycondition,(a),canbeeonsideredasaspecialcaseof Neumannproblem. c.Prescribedpotentialboundary Thethirdtypeofboundaryconditionariseswhentheflowregioninter- actswithaverylargereservoirofthefluidandreceivesfluidfromordis-chargesfluidintothispracticallyinfinitereservoir.Inthiscase,the potentialontheboundarywillbedeterminedbythefluidpotentialinthein-finitereservoir,whichmayfluctuateintimeinaknown,hence,

ip(Xi,t)=tpb(Xi,t).(2.32)

ThistypeofboundaryconditioniscalledaDirichletprobleminpotential theory. rnmostproblemsofpracticalin'terestdifferentpartsoftheboundary ofaflowregionmayexperiencedifferenttypeaofboundaryconditionsandthus theseproblemsaremixedinitial-boundary-valueproblems. d.Seepagefaces Aseepagefaceisanexternalboundaryofthesaturatedzonewherewater leavesthesystemandtpisuniformlyzeroasshowninFig.2.5.Undertransient conditions,thelengthoftheseepagefacevarieswithtimeinamannerthat cannotbepredictedapriori.Ifonetreatstheseepagefaceasaprescribed pressureheadboundarywithtp=o,thelengthofthisfacerernainsfixed,and

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