Model for unsaturated flow above a shallow water

EJ.M.deLaat
Model forunsaturated flowabove ashallowwater-table,
applied toa regional sub-surface flowproblem
Proefschrift
terverkrijgingvandegraadvan
doctorinde landbouwwetenschappen,
opgezagvanderectormagnificus,
dr.H.C.vanderPlas,
hoogleraarindeorganischescheikunde,
inhetopenbaarteverdedigen
opvrijdag22februari1980
desnamiddagstevieruurindeaula
vandeLandbouwhogeschoolteWageningen
Centre for Agricultural
Wageningen - 1980
Publishing
and Documentation
Abstract
Laat,P.J.M,de (1980)Modelforunsaturated flowaboveashallowwater-table,
applied toaregional sub-surfaceflowproblem.Agric.Res.Rep. (Versl.landbouwk.Onderz.)895,ISBN902200725 1,(vii)+ 126p.,42figs,6tables,
182refs,2appendices,Eng.andDutchsummaries.
Also:Doctoralthesis,Wageningen.
Amathematicalmodelisdeveloped tosimulatetransientunsaturated flow
aboveashallowwater-table.Theunsaturated zone,hereextending fromjust
belowthephreatic leveltosoilsurface,isschematized intoarootzoneand
a subsoil.In.therootzonethegradientofthehydraulicpotentialisassumed
equaltozero.Verticalflowinthesubsoilisdescribedbyacombinationof
steady-state situationscorresponding totheupperand lowerboundaryflux,
respectively.Transientflowissolvedbyasequenceofsteady-statesituations,subjecttoboundaryfluxconditionsatthesoilsurfaceandfrombelow
thewatertable.Thesolutionusestimeincrementsoftheorderofdaysand
isefficientintermsofcomputercosts.
Toverify themodelforanactualfield situation,itislinkedatthe
upperboundarytoamodelforévapotranspirationandatthelowerboundary
toamodelfortwo-dimensionalhorizontal saturated flow.Theresulting quasi
three-dimensionalmodelisapplied toafield-sizeflowproblem.Resultsagree
closelywithobservedwater-tableelevations.Thecompositemodel isfurther
usedtopredictconsequencesofgroundwaterextraction.
Freedescriptors:capillaryrise,percolation,saturated-unsaturated flow,,
évapotranspiration,groundwaterextraction,prediction.
Thisthesiswillalsobepublished asAgriculturalResearchReports895.
0 CentreforAgriculturalPublishingandDocumentation,Wageningen,1980.
Nopartofthisbookmaybereproduced orpublished inanyform,byprint,photoprint,
microfilmoranyothermeanswithoutwrittenpermission fromthepublisher.
Modelforunsaturatedflowaboveashallowwater-table,appliedtoaregional
sub-surfaceflowproblem
Curriculumvitae
TheauthorwasborninEindhovenon13May 1944.Afterattending theHogereLandbouwschool inRoermondhestartedhisstudiesattheAgriculturalUniversity inWageningeninNovember 1965.Hegraduated inJune1972withagrohydrologyand catchment
hydrologyasmainsubjectsandmathematicsasasubsidiary.
SinceJuly 1972hehasbeenemployedaslecturerinlandandwaterdevelopmentat
theInternational Institute forHydraulic andEnvironmentalEngineering inDelft.
Thepresentstudyispartofacomprehensive researchprojectcarriedoutbythe
'CommissieWaterhuishouding Gelderland' (CWG).TheCWGinvestigatedqualitativeand
quantitative aspectsofregionalwatermanagementintheProvinceofGelderland.For
theyears 1971-1974atotalamountofonemillionguilderswasmadeavailablebythe
ProvinceofGelderland,StatePublicWorksandregionalwatersupplycompanies.The
authorjoined theactivitiesoftheCWGin1972andparticipatedinthepreparationof
aninterimreportin1975.
Theinvestigationswerecontinuedintheyears 1976-1978.Thefundsforthesecond
researchperiod (1.4millionguilders)wereprovidedbythesameinstitutions.Thework
describedhereinwasthensupervisedbyProfessorW.H.vanderMolen.Afinalreport
covering theentirefieldofinvestigationsbytheCWGis (1979)inpreparation.
Acknowledgements
Myinterestinthepresentsubjectwasraisedwhenwritingareportduringmy final
yearattheAgriculturalUniversityinWageningenin1971.Thestudywas thenguidedby
ProfessorW.H.vanderMolen,whileIreceivedextensivehelpfromDrP.E.Rijtema.
In1972IrH.J.Colenbranderinvitedmetojointheresearchactivitiesofthe
'CommissieWaterhuishoudingGelderland'.Thebasicideasforthepresentworkwereformulatedintheyears1972-1975infullco-operationwith IrC.vandenAkker.Duringthe
difficultearlystagesoftheresearchwereceivedmuchhelpandsuggestionsfromDr
P.E.RijtemaandDrJ.Wesseling.Theresults,firstpublishedatthe1975 (Bratislava)
SymposiumoftheInternationalAssociationofHydrologicalSciences,encouragedme to
pursuetheinvestigations.ProfessorL.J.Mostertman,DirectoroftheInternationalInstituteforHydraulicandEnvironmentalEngineering,supportedthisintentionandprovidedtheopportunitytocarryoutthisstudy.Since 1975theresearchwasguidedby
ProfessorW.H.vanderMolen,whosesuggestionsandcriticalremarkswereveryhelpful
andinstructivewhenwritingthemanuscript.
Theactivitiesofthe 'CommissieWaterhuishoudingGelderland'wereexcellently coordinatedbyIrH.J.Colenbrander,DrTh.J.vandeNesandDrsE.Romijn.Theirinterest
andencouragementwereacontinuoussourceofstimulationthroughouttheentireproject.
Amongmanyotherswhoparticipatedinthediscussions,IwouldliketomentionDrJ.
BoumaandIngH.C.vanHeesenfortheirvaluablecontributionsinthefieldofsoil
science.IndispensableassistancewasrenderedbyIrL.S.T.KranendonkandIrR.H.C.M.
Awaterinwritingandrunningthecomputerprograms.
ThefigureswerewelldrawnattheProvincialeWaterstaatvanGelderlandbyMrY.
Faasen'.AtPudoc,theEnglishtextwasrevisedbyMrsE.M.Brouns-Murray,whilethe
publicationbenefitedfurtherfromtheeditorialremarksofMrR.J.P.Aalpol.
ThetypingwasdonebymywifeMarijke.Herskillandwillingnesstodothisaccuratelyandoftenatthemostimpossiblehoursduringdayandnightisnotheronlycontribution.Withouthercare,cheerfulnessandstimulatinginterestthisthesiswould
neverhave'beencompleted.
Contents
1 Introduction
\
2 Transport of water in soil
3
2.1 Soilwaterpotential
3
2.2 Generalequationofflow
5
2.3 Particularformsofthegeneralequation
6
2.3.1 Saturatedflow
7
2.3.2 Unsaturatedflow
9
2.4 Methodsforsolutionofflowproblems
11
2.4.1 Directsimulationmethods
11
2.4.2 Mathematicalmodels
3 Saturated-unsaturated
14
flow
22
3.1 Thetraditionalapproach
22
3.2 Therigorousapproach
27
3.3 Computationaldifficulties
33
3.4 Alternativesolutions
35
3.5 Scopeofpresentstudy
36
4 Development of a model for unsaturated
flow
38
4.1 Steady-staterelations
38
4.2 Pseudosteady-stateapproach
4.3 Analysisofthepseudosteady-stateapproach
49
58
4.4 Upperboundarysolution
61
4.4.1 Percolation
61
4.4.2 Capillaryrise
63
4.4.3 Rainfallexcessfollowingcapillaryrise
4.4.4 Flowchartfortheupperboundarysolution
4.5 Lowerboundarysolution
4.6 Combinedpseudosteady-statesolution
5 A quasi three-dimensional
approach
64
65
67
71
74
6 Application
and use
79
6.1 Experimentalverification
79
6.1.1 Selectedstudyarea
80
6.1.2 Saturatedflow
82
6.1.3 Unsaturated flow
84
6.1.4 Surfaceflux
85
6.1.5 Simulationresults
87
6.2 Sensitivityanalysis
6.3 Consequencesofgroundwaterextraction
92
97
Summary
104
Samenvatting
1o7
List of symbols
m
Appendix A
115
Appendix B
11g
Literature
1on
1 Introduction
ThereclamationandprotectionoflandfromtheseaandinlandwatersintheNetherlandsduringthepastcenturiesresultedinthedevelopmentoflargepolderareas.The
excellentopportunitiesforwatermanagementintheseareasprovidedoptimum conditions
forcropgrowth,atleastfromaquantitativepointofview.Qualitativeproblemsarose
duetothedeteriorationinqualityofthesupplemented surfacewaterand,particularly
inthelowestpolders,totheintrusionofsalinegroundwater.
Itisonlysincethebeginningofthiscentury thatseriousattentionhasbeen
giventothewatermanagementproblemsintheeasternandsouthernpartofthecountry.
Althoughthelandisabovemeansealevel,floodingwasfrequentinsomeplacesand
largeareassuffered fromtoohighwater-tables.Itiswell-knownthatwetconditions
inthebeginningofthegrowingseasonmayseriouslyaffectagriculturalcropproduction.
Itdelaysthesowingandplantingofcrops,butalsotheseedlingemergenceandgrowth
becauseoflowtemperaturesandhighconcentrationsofcarbondioxideintherootzone
ofthesoil.Toensurefavourableconditionsforcropgrowthatthebeginningofthe
growingseason,thedrainageinmanyofthehigherareaswithmainlyaeoliansoilshas
beendrastically improved.
AlthoughtheaverageannualrainfallexcessintheNetherlandsisbetween200and
300mm,thepotentialévapotranspirationexceedsprecipitationduringthegrowingseason
(ApriltoSeptember)bymorethan100mm.Ifthisamountisavailableforthecropin
therootzone,watersupplyisoptimumforcropproduction.However,mostofthe(sandy)
soilsintheeasternandsouthernpartofthecountryarenotevenabletoretainthe
amountthatisneededinayearforwhich 'averageweatherconditions'apply.Witha
shallowwater-tableaconsiderablepartofthisdeficitmaybesupplementedbythe
transportofsoilmoisture fromthegroundwaterreservoirtotherootzone.Theupward
movementofsoilmoistureintheregionabovethewatertableistermedcapillaryrise.
Thisprocessdependsonthedepthofthewatertable.Itbecomes insignificantforthe
watersupplyofthecropiftheprevailingwater-tabledepthismorethan3-5mbelow
soilsurface.
Therapidexpansionofpopulationandindustryduringthepastdecadesresultedin
aconsiderable increaseinthedemandfordomesticandindustrialwater.Assurface
wateringeneralisofpoorquality,andasthereishardlyanyfreshgroundwaterinthe
west,theamountsextractedfromtheeasternandsouthernpartsofthecountryarerapidlyincreasing.Jn thoseareaswheretheimplementeddrainagesystemis(morethan)
adequate,anadditionalextractionofgroundwaterresultsinanundesirabledrawdownof
thewatertable.Theeffectofadrawdownontheavailibilityofwaterforthecropin
areaswithrelativelyhighwater-tablesistwofold.Itreducestheamountofsoilmoistureinitiallyavailableintherootzoneandithamperscapillaryrise.Asaresultof
thedevelopmentdescribedabove,someoftheareaswhichpreviouslyhadanabundance of
waternowshowashortage.
Thisstudyconcernsgroundwaterflowinshallowwater-tableaquifersandinparticularflowintheunsaturatedregionbetweenthesoilsurfaceandthephreaticlevel.
Althoughthewatermovementinapartlysaturatedsoilmaybedescribedbyone single
equation,theflowregionsaboveandbelowthephreaticsurfacewere traditionally
treatedastwoseparatesystems.Oneofthereasonsfortheseparateapproachisthat
flowintheunsaturated zoneispredominantlyvertical,andinthesaturatedpart ina
horizontaldirection.Moreover,thenumericalsolutionofthegoverning equationrequiresmuchmoreeffortintheunsaturated zonethaninthesaturatedregion.The
availablesolutionsofthree-dimensional,saturated-unsaturatedproblemsusingasingle
equationare,therefore,restrictedtosmall-sizeflowsystems.
A less-rigorousapproachtreatsflowinthesaturatedandunsaturated regionseparatelyandusesaspecialproceduretolinkbothsub-systems.Thepartial differential
equationgoverningnon-steadyunsaturated flowishighlynon-linear.Forstability and
convergencethesolutionrequiresthattimeandspacearediscretized tosmallsteps.
Therestrictionwithrespecttothelengthofthetimeincrementisimposeduponthe
entiresystem.Therefore,fortheless-rigorousapproachtobeattractive intermsof
computercosts,itisnecessarythatthenumericalsolutionoftheequation governing
unsaturatedflowisreplacedbyamoreefficientsimulationmodel.The approximate
solutions,availableatpresent,areunsuitableforacompletetransientanalysis,as
theyconsiderflowinanupwardordownwarddirectiononly.Moreover,mostofthesolutionsassumethatthewatertableisatinfinitedepth.
Forashallowwater-table inasandyaquifer,thecharacteristic timeoftheunsaturatedflowsystemisoftheorderofdays.Withatimeincrementof approximately
thislength,theflowsystemcanbedescribedbyasuccessionofsteady-statesituations.
Thisapproachisusedinthisstudytodevelopamodelforunsaturated flow.
Inordertoverifythismodelforanactualfieldsituation,itislinkedatthe
upperboundarytoamodelforévapotranspiration,andatthelowerboundarytoamodel
fortwo-dimensionalhorizontalsaturated flow.Thecombinedmodelisapplied toanarea
of36km 2 aroundthepumpingsite " t Klooster'intheeastofthecountry.The amount
ofsurfacewaterrunofffromthisareaisrelativelysmall.Itwasselectedforthis
studytoreducetheeffectofthesurfacewatersystemontheverificationofthecombinedsaturated-unsaturated flowmodel.
Finally,themodelisusedtopredictconsequences ofgroundwaterextractiononthe
water-tableelevationandrealévapotranspiration.
2 Transportofwaterinsoil
2.1 SOILWATER POTENTIAL
Inanisothermalsystemthedrivingforcefortransportofsoilwateristhegradientofpotentialenergy.The'InternationalSoilScienceSociety' (Aslyng,1963;Bolt,
1975)usesconceptsbasedonenergyandthermodynamicstodefinetheconditionofwater
insoil (seee.g.Taylor,1968;Hillel,1971).Thetotalpotentialenergyisdescribed
astheamountofworkliberatedbyremovingaunitmassofsoilwaterfromacertain
locationinthesoilintheformofpurefreewateratthesametemperatureandto
transferthisquantityisothermallytoareferencelevelwhereitisdefinedashaving
apotentialofzero.Thecomponentsofthetotalpotentialï arethepressurepotential
ï ,theosmoticorsolutepotential"P andthegravitationalpotential¥.Thus
r =f +f + t
t
The pressure
potential
p
o
(1)
g
*•/
resultsfromapressurethatdiffersfromtheexistingat-2
-3
mosphericpressure.Pressure (N-m )isequivalenttoenergyperunitvolume (J-m ) .
Sincethedensitypismasspervolumeitfollowsthatthepressurepotential (expressed
inenergyperunitmass)
Y = E-
(2)
P P
wherepisthepressurewithrespecttoatmosphericpressure.Intheunsaturatedsoil
thepressurepotentialisnegativeduetotheattractiveforcesofthesoilmatrix.
Buckingham (1907)introducedtheterm 'capillarypotential'toindicatethatthepotentialresults fromcapillaryeffects.Nowadaystheterm 'matricpotential'ispreferred
asthepressurepis,especiallyinclayeysoils,alsoaffectedbyadsorption,byattractionbetweenwatermoleculesandionsintheelectricaldoublelayerofclayparticlesandbysmalldeviationsinthesoilairpressurefromtheexistingatmospheric
pressure.InparticularwithrespecttothelastmentionedeffectseeStroosnijder
(1976).Atthefreewatersurfaceatmosphericpressureexists (bydefinition),sothat
p equalszero.Belowthislevel,inthesaturatedregion,theattractionofthesoil
matrixisnegligible.Pressuremerelyresultsfromthehydrostaticpressure,sothat
valuesfory arepositive.Thepressurepotentialinthesaturatedzonehasbeen
termed 'submergencepotential' (Rose,1966).Althoughthepressureaboveandbelowthe
freewaterlevelresultsfromquitedifferentforces,pisconsideredinthisstudyas
asinglecontinuousquantity,extendingfromthesaturatedtotheunsaturatedregion.
The osmotic
or solute
potential
reducesthetotalpotentialenergyinthepresence
ofamembranewhosepermeabilitytowatermoleculesdiffersfromthattothemolecules
ofthedissolvedsalts.Whendealingwithwatermovementinsoilitisassumedthatthe
solutecanmovefreelywiththesoilwater.Hence
¥ =0
o
(3)
Thisconditionimpliesthatthesoilwaterpotentialisdefinedwithrespecttofree
waterofsimilarchemicalcompositionasthesoilmoisturelocatedatreferencelevel.
The gravitational
potential
istheenergyduetotheearthgravitationalfield.If
gisthegravityconstant,therequiredenergytoliftamassofwatermoveraheightz
abovereferencelevelequalsmgz.Sothegravitationalpotentialperunitmass
¥ =gz
(4)
asgcanbeconsideredaconstantoverthedistancesinvolved.
Thedrivingforcefortransportofwaterinaporousmediumisthengivenby
vy =vY +vy =v(E)+gvz
t
p
g
(5)
6
V
Ifataheightzabovereferencelevelpressurepexists,thetotalpotentialenergyper
unitmassatthisparticularlocationinthesoilmaybewrittenas
P 1
¥ = ƒ
z
j - da+g ƒ d3
0 p
0
(6)
Thepotential¥ trepresentsascalarquantityifitsgradientdescribesavectorfield
withoutarotationalcomponent.Itcanbeshown (DeWiest,1966)that¥ asgivenby
Eqn6generatesanirrotationalvectorfield,providedthatthedensitypisafunction
ofponly.Actually,thedensityofthesoilwateralsovarieswithsoluteconcentration
andtemperature.Inthisstudythesoilwaterisassumedtobehomogeneousandimcompressible,sothatforisothermalsystemsthetotalwaterpotential (energyperunit
mass)isgivenby
* t -jr + § z
CJ-kg"1)
(7)
asherepcanbeconsideredaconstant.MultiplyingEqn7bytheconstantpyieldsthe
pressureequivalentofthewaterpotential (energyperunitvolume)
P =pgz+p
(J-nf3orPa)
(8)
DividingEqn7bytheconstantgresultsinaquantityknownashydraulicheador
hydraulicpotential(energyperunitweight)
*+»z fg
M
C9)
2.2 GENERALEQUATIONOFFLOW
Intheabsenceofotherforces,suchasthermalandelectricalgradients,adifferenceinthetotalpotentialenergybetweentwolocationsinthesoilisthedriving
forcetomovewaterfromthelocationwherethepotentialishightothelocationwhere
alowervalueexists.TheresultingvolumefluxdensityqrelatedtothepotentialgradientisknownasDarcy'slaw,writteninvectorialformas
q =-k(vp+pgvz)
(10)
2 -1 -1
wherethehydraulicconductivityk (m«s «Pa )dependsonthecharacteristicsofthe
soilmatrix,thedynamicviscosityofthefluidandthedegreeofsaturation.Ifthe
valueofkisthesameineachflowdirection,theporousmediumissaidtobehydraulicallyisotropic.Thoughthefluxdensityvectorqhasthedimensionofvelocity
_1
(m-s ) ,thetermvelocityismoreproperlyusedfortheactualvelocityofthewater
intheporespaceofthesoilmatrix.Ingroundwaterhydrologyqispreferably termed
'specificdischarge'.
Intheunsaturatedsoilthepressureofwaterisusuallymeasuredwithatensiometerandbelowthefreewatersurfacewithapiezometer.Bothmethodsmeasurethepressureatacertainlocationinthesoilrelativetoatmosphericpressureasaheightof
awatercolumn,calledpressurehead y. Ifthedensitypintheapparatusequalsthe
densityofthesoilwater,p=pgij>.Itisthereforeconvenienttousethegradientof
thehydraulicheadtowriteEqn10as
q=-K[(^)vp+vz]=-KV*
(11)
wherethehydraulicconductivityK (=pgk)isexpressed inthepracticalunit (m-s ) .
Thecontinuityequationforflowinnon-deformablemedia,statingtheLawofConservationofMass,maybewrittenas
^=-V-pq
(12)
wheretistimeandethevolumefractionofwaterperunitvolumesoilmatrix.Taking
PagainasaconstantandcombiningEqns 11and12resultsinageneralequationofflow,
writteninvectornotationas
ff-=V-(KV<»
(13)
Forflowinanisotropicmediaamoregeneralequationisobtainedbyexpanding
Eqn13asfollows
39_ 3 ,K U i + -L (K ^-) +— (K 1^)
•st""3X"{ \ 3XJ
3yLy 3yJ
(14)
3z <• z 3z'
wherethex,y,zdirectionsarechoseninthethreeprincipaldirectionsofthehydraulic
5
conductivityK ,K andK.Whensolvingmulti-dimensional flowproblems,anisotropy
shouldbetakenintoaccountbecausegenerallythenaturalporousmediumhasastratified
structure.Sincetransformationofisotropicflowproblems intoaproblemforanisotropic
mediaisrelativelysimple,theequationsinSection2.3areconvenientlyderivedfor
isotropicsoils.Forathoroughdiscussiononanisotropyinporousmediathereaderis
referredtoChilds (1969).
2.3 PARTICULARFORMSOFTHEGENERALEQUATION
Whenmodellingcomplicated systemssimplifying assumptionshavetobemade.Someof
thesesimplificationsarenecessaryforamathematicaldescriptionofthesystem.An
exampleistheassumptionthatDarcy'slaw,whichisinaccordancewiththeequation of
Hagen-Poiseuilleforlaminarflowinacirculartube,alsoholdsforflowinporous
media.ThevalidityofDarcy'slawespeciallyinunsaturated soilisstillamatterof
discussion (Swartzendruber,1963and1968;Thames&Evans,1968;Vachaud, 1969).Other
assumptionsarenecessarytoobtainananalyticoradequatenumericalsolutionofthe
problem:forexample,consideringflowinoneortwodirectionsonly,orneglectingthe
variationinhydraulicconductivity.Thesesimplifying assumptions resultinanumberof
differentialequationseachofwhichholdsforacertainclassofflowproblemswhich
arecharacterizedbytheassumptionsmadewhenderivingtheformula.Manyequations have
beengiventhenameoftheauthorwhofirstsuggesteditsuse.Equations frequently
citedwhendiscussingsaturatedandunsaturatedflowwillbedealtwithinthissection.
ItshouldberealizedthatthegeneralequationasformulatedinEqn13isgeneral
insofarasitdescribestheflowinathree-dimensional,non-homogeneous,saturated
orunsaturatedporousmedium,butislessgeneralinsofarasitisrestrictedtoisothermalflowofanincompressiblehomogeneous fluidinarigidsoilwithout other
drivingforcesthanthosedefinedbythehydraulichead.Problemsonmixed saturatedunsaturatedflowinthisstudyandmostoftheproblemsdiscussedinliteratureonthis
subjectsatisfyornearlysatisfytheserestrictions.ThereforeEqn13willbeusedas
thebasicequationforfurtherconsideration.
Thedevelopmentofflowequationsfortransportofwaterinporousmediacame from
twodifferentdisciplines.Saturatedflowproblemshavebeenstudiedbygroundwater
hydrologistsinrelationtocivilengineeringandunsaturated flowhasalwaysbeenthe
domainofthesoilscientistinrelationtoagriculture.Thisseparatedevelopmentmay
beillustratedbythefactthatBuckinghamwhenintroducingthecapillarypotentialin
1907ditnotevenmentionDarcy'slawfrom 1856andittook20yearsbefore Israelson
(1927)notedtheconnection.Thedelayedprogressmadeinthedevelopmentofunsaturated
flowtheorycomparedwiththatofsaturatedflowhasbeenmainlyduetothedifference
inthenatureofthepotentials.Insaturatedmediathepotentials involvingposition
andpressureareeasilyobtainedwhereasitwasnotuntil 1928withtheintroductionof
thetensiometer (Richards,1928)thatunsaturated flowpotentials couldbemeasured.
Moreover,empiricalrelationsbetweenpressureandmoisturecontentandbetweenpressure
andhydraulicconductivityarerequiredforthesolutionofunsaturated flowequations.
Theserelationsaredifficultandtedioustoobtainandarebothsubjecttohysteresis.
Thereforeanalogyofflowthroughporousmediatoheatconductionwasfirstrecognized
forsaturatedflow.
2.3.1
Saturated
flou
Forsaturated flowtheearlierdefinedproportionality factorkintheequationof
DarcyasformulatedinEqn10isafunctionofthepropertiesofthesoilmatrixandthe
fluid.Many investigatorshavetriedtodescribethisparameterintermsofthecharacteristicsofthemediumaswellasthoseoftheliquid.Inthisconnectionusehasbeen
madeoftheexperimentallyderivedequationofPoiseuille.Accordingtothis equation
therateof(laminar)flowthroughatubeofuniformcross-sectionisproportionalto
thehydraulicgradient,whichisessentiallyDarcy'slawforacolumnfilledwithporous
material.Fromconsiderationsontheproportionality constantofbothequations,it
followsthat (Rose, 1966)
k=£j_Ar
2
(15)
whereAisadimensionlessconstant,rthe'effective'radiusoftheporesandnthe
dynamicviscosityoftheliquid.TheconstantAresults fromthefactthattheflow
throughaporousmediumisveryirregularcomparedwithlaminarflowthroughatube.It
containsdimensionless characteristicsonthegeometryofthesoilmatrix.A reliable
expressiontorelatetheconstantAtotheporosity,shapeofthegrains,grain-size
distributionandothergeometricalpropertiesoftheporousmediumhasnotbeenfound.
Muskat (1937)suggestedtolumpAandr intooneparameter thatisafunctionofthe
structureofthemediumaloneandentirely independentofthenatureofthefluid.This
parameterhaslaterbeentermed 'inherent', 'intrinsic'or'specific'permeability.This
conceptofinherentpermeabilityisrathernotusedbysoilscientists,because soils
areingeneralbynomeansinertinthephysicochemicalsense (Childs,1969).Thisis
well-knownfromfarmingpracticewherethestructureofclayeysoilsisimprovedbythe
applicationofcertainfertilizers.However,inthemoreinertsandyporousmediainthe
absenceofair,theconceptofinherentpermeabilityprovedtobeusefulanditisgenerallyappliedbygroundwaterhydrologists.Denotingtheintrinsicpermeabilityby K, the
proportionality constantkisgivenby
k= S.n
(16)
ThehydraulicconductivityK,whichappearsinDarcy'slawexpressedintermsofhydraulicheadmaythenbewrittenas
K=pgk=^ f
Sincephasbeenassumedaconstantandthefluidhomogeneous,thehydraulic conductivity
Kmaystillbeconsideredasacharacteristicofthe(saturated)porousmediumalone.
Laplace'sequation,earlierderivedforthesteadyconductionofelectricityand
(17)
heatwasintroducedforsteadyflowinhomogeneoussaturatedmediabeforetheendofthe
lastcenturybySlichter (1899).With39/3t=0andKisaconstantthisequationfollows
directlyfromEqn13
o8)
\ .ii +i i +ii = o
v
3X
3y
3z
Inahorizontal,completelyconfinedaquiferofuniformthicknessthespecificdischargeinverticaldirectioncanbedisregardedandEqn18reduces to
ii
3x
z
+
09)
ii =o
3y
Forsemi-confinedorleakyaquiferstheverticalfluxisstillsmallenoughtowritethe
continuityequationas
^
•^
3x
=-q.
3y
(20)
H
i
whereDisthethicknessoftheaquiferandq. istheleakagethroughtheupper confining
layer.Substitutingq =-K3<)>/3xandq =-K3<j>/3yintoEqn20andassumingthehydraulic
conductivityKtobeaconstantinverticaldirectionyields
-L(TMi+J_<TMi=n
3x K SxJ
3y*-x3yJ
(21)
q
i
*•
whereT=KDistermedthetransmissivity,afunctionofx andyinnon-homogeneous
media.Thefluxq.maybewrittenintermsofthecharacteristicsoftheconfininglayer
andthehydraulicheadoftheadjoiningaquifer.UsingDarcy'slaw
qi=-K'±^-t =-il^l
(22)
where<f>'isthehydraulicheadintheadjoiningaquifer,K'thehydraulicconductivity
andD'thethicknessoftheconfininglayer.K'andD'areusuallyexpressedas the
resistancec=D'/K'.If<f>'varieswithtime,q iisalsoafunctionoftimeandEqn21
describestransientflowinanon-homogeneous,non-deformable,semi-confinedaquifer.
Animportantclassofproblemsdescribingessentiallyhorizontalflowarebasedon
theDupuit-Forchheimerassumptions.Dupuit (1863)derivedanequationforradialflowin
anunconfinedaquiferassumingthatforsmallinclinationsofthefreewatersurfacethe
streamlinesmaybetakenashorizontal.Furthermoreheassumedthatalongeachvertical
linethehydraulicheadisequaltotheheightofthefreewatersurfaceabovethehorizontalimpermeablebase(thus3qx/3z= 3q/3z= 0 ) .Applyingtheequationofcontinuity
toflowinanycolumnwithafreesurfaceheighthabovetheimpermeablebase,Forchheimer (1886)derivedageneralequationforflowinunconfinedaquiferswithwater
tablesoflowslope.Theequationofcontinuityrequiresthat
•k <h(U + 4y c h V = -y
ft
(23)
whereyisthe 'drainableporosity'or'specificyield',definedasthevolumeofwater
extracted fromthegroundwaterperunitareaandperunitdescentofh.TheDupuit
assumptions allowtheequationsofDarcy tobewrittenasq =-K3h/3xandq =-K3h/3y,
whichcombinedwithEqn23yieldtheequationofBoussinesq (1904)
è^^
+
#<*»#-"$
(24)
AlthoughMuskat (1937)inacomprehensivediscussionstronglytookissuewith the
Dupuit-Forchheimertheoryandpreferredtoawaitthedevelopmentofamore satisfactory
solution,thetheoryhasbecomeverypopularbecauseitiseasytoapply.Theerrors
resulting fromtheDupuit-Forchheimer assumptionsgenerallydependonthecurvatureof
thefreesurfaceandtendtobe largerfortheapproximated shapeofthewater table
thanforthecalculated flowrates.Foroneparticular flowproblemCharny (see
Polubarinova-Kochina, 1962)hasshownthattheDupuit-Forchheimer assumptions leadto
theexactsolutionfortherateofflow.
TherightsideofEqn24represents thechangeintimeofthetotalvolumeofwater
storedinacolumnofunitcross-sectional areaduetoavariationintheheightofthe
watertable.Ithas thedimensionofaflux.Thedimensionlessparameteryisafunction
ofx,yandt.Whenthechangesinharesmallascomparedwiththethicknessofthe
aquifer,Khmaybeconsidered asafunctionofxandyalone.Substituting thetransmissivityT=KhintoEqn24yieldsanon-lineardiffusionequationdevelopedbyJacob
(1950)
A fr3h)+ _L ftÜ1)=u— +rq.
3xll dxJ
3yu 3yJ "3t
(25)
H
x
l
wheretheadditionaltermzq.=q +q_+q 3+.••representssourcesandsinkssuchas
leakagethroughaconfining layer,rainfall,pumpage,etc.ThetransmissivityTisa
functionofxandy,whileq.mayvarywithx,yandt.PositivevaluesofqA represent
asink,negativevaluesasourcefunction.Forsteadyflowconditions thetermy3h/3t
disappearsandEqn25reducestoaformsimilartotheequationforsemi-confined flow
(Eqn21).These typesofequationareknownasPoissonequations.
2.3.2 Unsaturated flow
Considering flowinunsaturatedporousmedia,thehydraulicconductivitybecomesa
functionofthewatercontent,expressibleask=k(e),andthegeneralequationofflow
maybereproducedintheform
|| =v-k(e){vp+Pgvz}
(26)
Equation26canbesolvedonlyifauniquerelationexistsbetweenk andeaswellas
betweeneandp.Haines (1930)wasamongthefirsttoreportexperimentalevidence,using
sandanduniformglassspheres,thate(p)isnotasingle-valuedfunction.Inrigidsoils
uniquerelationsbetweeneandpexistifthechangein9ismonotonie,i.e. themoisture
contentiseithercontinuouslyincreasingordecreasing.Betweenthesetwoextremerelations,knownasthe 'wetting'and 'drying'moisturecharacteristic,afamilyofso-called
'scanning'-curvesdeterminetherelationbetween6andpdependentonthepasthistory.
Hysteresiseffectsintherelationbetweenkand6appeartobelesssizable (Nielsen&
Biggar,1961;Elrick&Bowman,1964;Top&Miller,1966;Poulovassilis,1969),butifk
isexpressedask(p)hysteresisinthemoisturecharacteristicisimposedontherelation
betweenkandp.
ToconvertEqn26intoanequationwithonedependentvariable,theleftsideis
writtenas
If=| S -CCP)H•
C27)
whereC(p)isdefinedasthespecificmoisturecapacity.WritingEqn26asafunctionof
p, Richards (1931)derivedthefollowingequationforunsaturated flowinnon-homogeneous, isotropic,porousmedia
<*»)3 =& CkûOg)•£ CkûO-g)• h MP)g )•pg* g *
C28)
whichisusuallyreferredtoasRichards'equation (Swartzendruber, 1969).Theuseof
Eqn28isrestrictedtotheclassofproblemsinwhichthematricpressurechangesmonotonically,asitfailstotakeintoaccounthysteresiseffectsintherelationsk(p)and
e(p). AmodifiedhystereticversionofRichards'equationhasbeenproposedbyMiller&
Miller (1956),butitsuseislimitedashystereticrelationshipsaredifficulttoobtain
inpractice.
Buckingham (1907)hasexpressedDarcy's lawintermsof6withtheintroductionof
D(e)=k(e) ^
(m 2 ^" 1 )
(29)
whichlaterChilds&Collis-Qeorge (1950)notedasbeingmathematically identical toa
diffusioncoefficient.Applicationofthesoil-waterdiffusivityDrequiresthatdp/de
exists,whichisnotthecaseforsaturatedmediawherepvariesand6remains aconstant.Richards (1931)suggestedthatwritingEqn28intermsoftheotherdependent
variable6isjustamatterofmathematicalexpediencyifp isasingle-valued function
of9.However,VpcanonlybeexpressedintermsofV9when9iscontinuousandthusthe
mediumhomogeneous.Withreferencetotheserestrictions,Eqn28writtenintermsof9
yieldsthetransport-diffusionequation
S "H M*S>+£ CDCe)fi)+ ± cDCo)'gj+pg3!gl •
10
(»J
whichwaspresentedinthisformbyPhilip (1957a).Equation30isanon-linearFokkerPlanckequation.Theclassofflowproblemstowhichitingeneralrefersisabsorption
andinfiltrationintohomogeneousunsaturated soil.Forone-dimensionalhorizontal flow
andotherinstanceswheregravitymaybeneglectedEqn30reducestothenon-lineardiffusionequation
I «& ^ $
(31)
forwhichanalyticalandquasi-analyticalsolutionshavebeenobtained (Philip,1969).
A formoftransportofwaterinporousmediathathasnotbeendiscussedisthe
water-vapourmovement.Vapourmovementisaprocessofdiffusionratherthanmass flow
andmayconvenientlybeincludedinthediffusivitytermintheFokker-Planckequation
(Philip,1957a).However,vapourmovementbecomesonlyasignificantfractionofthe
totalunsaturated transportwhenthesoilisverydryandtherateofliquidflowclose
tozero (Rose,1963a,1963b).Hencevapourmovementmaybeneglected (Miller&Klute,
1967).Thisconclusionisonlywarrantedinviewoftheassumptionmadeearlierthat
isothermaltransportofahomogeneous liquidisconsidered.Forconditionsthatareno
longerisothermal,vapourdiffusionbecomesthedominantsysteminthetotalmoisture
transportinverydrysoil (Philip,1957b).Rosema (1974),followinganapproachof
Philip&deVries (1957),showedthatforwetconditionsEqn28cannotbeusedtodescribethediurnalchangeinthetotalmoisturefluxinthetoplayerofabaresoil.
Forananalysisofthesimultaneoustransportofwaterandheatfromthepointofview
.ofirreversiblethermodynamicsthereaderisreferredtoe.g.Cary&Taylor (1962)and
Cary (1963,1966).
2.4 METHODSFORSOLUTIONOFFLOWPROBLEMS
Tosolveproblemsofgroundwaterflowasystem (realorabstract)isderivedto
simulatetheoperationoftheprototypesystemwiththelimitsofaccuracyrequiredby
theproblemunderstudy (Dooge,1973and1977).Suchasimulationsystemistermeda
model.Theprocessofsimulationisthentheoperationofthemodeltopredicttheresponseoftheprototypesystem.Inthissense,differentialequationsgoverninggroundwaterflowaremodels,andsimulationofagroundwaterflowsysteminvolvesthesolution
ofadifferentialequation.Mathematicalmodelsuseanalyticalornumericaltechniques
toobtainthissolution.
Amathematicalmodelrepresentsanabstractsystem.Realsimulationsystems include
physicalandanaloguemodels.Thesedirectsimulationmethodsarefirstreviewedbriefly.Mathematicalmodels,whichareofprimaryinterestforthisstudy,arediscussedin
moredetailafterwards.
2.4.1
Direat simulation
methods
Physicalmodels compriseone-dimensional flowinsoilcolumnsandtwoorthree
11
dimensionalflowinsandtanks.Theporousmedium isusuallyhomogeneous,isotropicand
consistsofartificialornaturalgranularmaterial.Forsaturated flowthemodel isoftenascaled-downversionoftheaquifer,which involves theuseofscalefactors.Since
thesamelawsgoverningflowapplytoboththemodelandtheprototype system,physical
modelsareinparticularusefulforcomparisonwiththeory.Applicationofsand tank
modelstoregionalflowproblemshavenotbeenreported,probablyduetothe restrictions
imposedbythescalefactors (Prickett,1975).
Analoguesolutionsofgroundwaterflowproblemsarebased ontheprinciple that
systemsbelongingtoanentirelydifferentphysical categoryaredescribed by essentially
thesameequationsasthosegoverning flowinporousmedia.SimilarityofDarcy'slawto
theequationforlaminarflowofaviscous fluidthrough acircular tubehasalreadybeen
mentioned.Amodelfortransient,unsaturated,vertical flowbasedonthisanalogywas
builtbyWind (1972).Themodelconsistsofanumberofvesselseachrepresenting one
soillayer.Whenappropriatescale"factorsareused,theshapeofthevessel,its liquid
contentandlevelrepresentthemoisturecharacteristic,moisture contentandmatric
pressure,respectively.Thenon-hysteretic flowprocess issimulatedby theflowofa
viscousfluidthroughanumberoftubesconnecting thevessels.Themodelhasbeen
successfullyusedforflowinheavysoilswithahighwater-table andunderwetconditions.
Aviscousfluidanalogueforsaturated groundwater flowistheparallelplatemodel.
ThismodelisusuallycalledHele-Shawmodel,becauseHele-Shaw (1898)firstnoticed the
analogybetweentheequationfortwo-dimensional laminarflowofaviscous fluid through
anarrowinterspacebetweentwoparallelplatesand theequationofLaplace.Itcanbe
shownthatPoiseuille's lawapplied tothisflowsystemistheanalogue toDarcy's law
forgroundwaterflow (Lamb,1932,p.582).Themodel isused inverticalposition to
simulatetwo-dimensional steadyortransientunconfined flowforavarietyofboundary
conditions (e.g.Awan&O'Donnell, 1972).Non-homogeneity oftheporousmedium isimitatedbyvariationsofthewidthoftheinterspace.Inhorizontalpositionthemodel has
longbeenusedtostudysteadyconfinedandunconfined flowproblems.Santing (1958)
extendeditsusetosimulate thediffusionequationwiththeintroductionofanumber of
vesselsontopofthemodeltoimitatestoragecapacity.Themodel issuitabletosimulatenumerousgroundwaterflowproblems includingsteady,transient,confinedandunconfinedflowinhomogeneous ornon-homogeneousmedia inthepresenceofsources and
sinks,rainfallandevaporation.A disadvantageofthemodel liesinthefactthat the
transmissivities areconstantintimeanddifficult tochangeoncethemodel isconstructedandthewidthoftheinterspacehasbeenfixed.Viscous flowmodelsarerestrictedtosimulatetwo-dimensional flowproblems.Themodelsaredifficult toconstructandthecomplicatedoperationrequires atemperature controlled environment.
TheanalogyofDarcy'slawandOhm'slawgoverningthesteadyflowofanelectricalcurrentthroughaconductivemediumhasledtonumerous electrical analoguemodels
forgroundwaterflow.Themodelmaybeacontinuous ordiscreterepresentation ofthe
porousmedium.Continuoussystemsareusedtostudysteadygroundwater flowproblems.
Theconductivematerialmaybeanelectrolyte inaninsulatedtankorsolidmaterial
fromwhichtheconductiveTeledeltospaperismostcommonlyused.Theshape ofthe
12
conductivemediumisascaled-downversionoftheaquifer.Teledeltospaperisusedto
solvetwo-dimensionalhomogeneous flowproblems.Forthesimulationoftwo-dimensional
flowproblemswith liquidmodelsnon-homogeneityoftheaquifermaybeimitatedby
varyingthebottom levelofthetank.DeJosselindeJong (1962)combinedtwoliquid
tanksbyaresistornetworktostudysteadyflowintwoaquifersseparatedbyaconfininglayer.
Withadiscreteelectrical analoguemodelthepropertiesoftheporousmediumare
simulatedbyanetworkconsistingofelectricalelements.Thenetworkisascaled-down
versionofthehydrologieprototype.At thenodesappropriateelectricalvoltagesand
currentsourcescanbeintroducedtorepresentcorrespondingboundaryconditionsand
sourcesorsinks.Theelectricalelementssimulatingtransmissionandstorageareresistanceandcapacitance.Resistancenetworkanaloguesareusedtosolvesteadyflow
problems.Herbert (1968)showedthatproblemsoftwoandthreedimensionaltransient
flowmaybesolvedbyastepwisesolution,consideringthetime-variantflowprocess
asasuccessionofsteady-states.Thismethodisrathertimeconsumingand introduces
extraerrorsduetodiscretizingthetimeparameter.Transientflowproblemsaremore
convenientlyhandledwithresistance-capacitancenetworks.Resistance-capacitanceanaloguesare themostversatileanaloguemodelsforanalysingsub-surfaceflowsystems,
butthereisalimittothecomplexityoftheflowsystemtheycanhandle (Bouwer,
1967).Thisrefersinparticulartotheinclusionoftransientunsaturated flow (Wind&
Mazee, 1979).
Comparingresultsfromanaloguemodelswithnumericalsolutionsobtainedwith a
digitalcomputer,Prickett&Lonnquist (1969)concludedthatdigitalmethodsareless
timeconsumingformodelconstructionandoperation,andsuperiorfornon-linear
problems.Forthesimulationoflargegroundwaterflowsystemsrequiringmanytimeincrementsandalargecorestorage,analoguemodelsarelesscostlytooperatethan
digitalmodelsbut thedatahandlingismoredifficult.Thisproblemcanbesolvedby
combiningresistancenetworkanddigitalcomputerintoahybridcomputermodel.This
allowsthegroundwaterflowproblemtobeprogrammedasforapuredigitalcomputer
solution,butthenon-linearpartialdifferentialequationissolvedbyaresistance
network.Sincethesolutionwiththeresistancenetworkisalmostinstantaneouslyobtaineditservesasasubroutineinthedigitalcomputerprogramwhichreducesthecomputationaltimedrastically (Vemuri&Dracup,1967).
Apartfromviscousfluidandelectricalanaloguemodelsthereareseveralother
simulationtechniquesbasedonanalogy (Karplus,1958)fromwhichthestretchedmembrane
analoguemodelisworthmentioning.Themodelconsistsofathinrubbersheetstretched
withuniformtension.Theshapeofthemembraneduetoapointloadwhichrepresents a
sourceorsinkisgovernedbyPoisson'sequation.Thetensionofthesheetandtheverticaldeflectionsareanalogoustoaquifertransmissivityandhydraulicheadvariations,
respectively.Themodel is'simpleandinexpensivewhenusedtosimulatesteadyflow
problemsofmultiplewellsinhomogeneousaquifers.DeJosselindeJong (1961)pointed
outthataccuratesolutionscanbeobtainedwithanopticaltechniquefortheobservationofthesimulatedflowpattern.
13
2.4.2
Mathematical models
Mathematicalmodelsdescribetheprototypesystembyasetofalgebraicformulas.
Thenatureoftheformuladependsontheapproachusedtosolvethegroundwaterflow
problem.Thisapproachmayrangefromapureblackboxanalysis,viaconceptualmodels
tothemathematicalphysicsapproach.Strictlyspeaking,itisdifficulttodistinguish
betweenthedifferentapproaches,sincealmosteverymathematicalmodelcontainstoa
certainextentconceptualelements.Themathematicalphysicsapproachresultsindifferentialequations,andtheparticularformsofthegeneralequationderivedinSection2.3
aregenerallyacceptedtobelongtothiscategory.
Mathematicalmodelsuseanalyticalornumericalmethodstosolvethegoverning
equationofflow.Thesolutionrequiresthatthegeometryoftheone,twoorthreedimensionalregioninwhichflowisconsideredisspecifiedaswellastheconditions that
applyattheboundaryoftheflowdomain.Ifattheboundarythevalueofthedependent
variableisgiven,theboundaryconditionisknownastheDirichletcondition.Flux,or
Neumannconditionsrefertosituationsforwhichtheflux (orzeroflux)normaltothe
boundaryisspecified.Iffordifferentpartsoftheboundarydifferenttypesofboundary
conditionsapply,thesystemisknownasamixedboundaryvalueproblem.Theuseof
derivativeboundaryconditionsforthesolutionofasteady-stateflowproblemrequires
thatthenetflowoutoftheflowdomainequalszero.Moreover,toarriveataunique
solutionforatypicalNeumannproblemanadditionalparameterisneeded.Well-defined
boundaryconditionsaresufficienttoobtainaparticularsolutionofasteady-state
flowproblem.Butforthesolutionofatransientflowproblem,theinitialcondition
aswellaschangesinboundaryvalueswithtimehavetobespecified.
Analyticalmethods
Muchefforthasbeenmadetoderiveanalyticalsolutionsofflowproblems.Ingeneralanalyticalsolutionscanonlybeobtainedforhomogeneousmediaandwhensufficient
simplifyingassumptionsaremade.Forsaturatedflowthesehaveledtoagreatnumberof
groundwaterformulas.Well-knownformulasaretheTheisandHantushequationsfortransientradialflowtoawell.Theseequationsareimportantforanapproximationofthe
performanceofwellsandaquiferintheabsenceofsufficientdata.Forthispurposethe
propertiesoftheaquiferanditsboundaryconditionsareidealized.Imaginarywellsare
usedtoreproducethesamedisturbingeffectsastheidealizedgeologicalboundary.A
solutionmaythenbeobtainedbyusingtheprincipleofsuperpositionfortheeffectsof
realandimaginarywellsinaninfiniteaquifer (e.g.Walton&Neill, 1960).
A semi-analyticalsolutionisobtainedwiththeboundaryelementmethod (Brebbia,
1978).Theboundaryofthetwo-dimensionalflowdomainisdividedintoaseriesofelements.VanderVeer (1978)usedacontinuousdistributionofsinks,sourcesandvortices
overeachelementtogenerateaflowpatterninthedomain.Thesolutionfoundbyenforcingtheflowpatterntosatisfytheboundaryconditions,isobtainedbynumerical
techniquesandisexactintheregionenclosedbyanapproximateboundary.
ForthederivationoftheTheisandHantushformulastheBoltzmannsubstitutionhas
14
beenused totransform thepartialdifferential equationintoanordinary differential
equation.Thisreductioninthenumberofindependentvariables isknownas similarity
substitutionandisonlyuseful ifthevariables removedfromtheequationarealso
removed fromthegoverning conditionsbythesamesubstitution.TheBoltzmann similarity
substitutionmay alsobeused tosolvetheFokker-Planck equationforunsaturated flow.
Thisresults inasemi-analytical solutionforwhichanefficientnumericalmethodwas
introducedby Philip (1955).
Pureanalytical solutionswhich arefoundcompletelybymathematical analysis cannotbeobtained fortransientunsaturated flowunless somenon-realistic assumptions are
made.Forinstance,assumingD andk tobeconstants,theone-dimensional Fokker-Planck
equationreduces tothelineardiffusion equation
|f= D 4
(32)
3x
forwhichsolutions foragreatnumberofboundaryconditions arereadily available
(Crank,1956;Carslaw&Jaeger, 1959).
Forsolvingpracticalproblems,analytical andsemi-analyticalmethodsareoften
unsuitable.However,fromsolutions obtainedwithsuchmethodsonecangainabetter
understanding ofthefundamentalstructureoftheflowproblem thanwithan incidental
numericalsolution.
Numericalmethods
Thesolutionofdifferential equationsgoverningflowmaybeapproached numerically
usingafinite elementorfinitedifferencemethod.Withfiniteelementmethods,theflow
problemiseitherreformulatedusingvariationalcalculus (e.g.theRayleigh-Ritzmethod)
orbalancedusingweighted residualprinciples (e.g.themethodofGalerkin).Fortwodimensional flowasolutionisobtainedby firstsub-dividing theflowregionintoelementarysub-areas,theelements.Thesizeoftheelementsmayvary,theshapeisusually triangularorquadrangular.The independentvariable intheinterioroftheelement isexpressed intermsofitsvalueatthecornerpoints.Applicationoffiniteelementmethods
results inasetofsimultaneous equations.Various techniques tosolvesetsof simultaneous
equations arediscussed laterinthissection.
The finite elementmethod isaquiterecentdevelopment inthefieldof sub-surface
hydrology (Zienkiewicz, 1967). Itsrelativemeritscomparedwiththe 'classical'finite
difference techniquehave tobe furtherestablished,asthenumberofcomparisonsbetweenbothmethods isstilllimited.A distinctadvantageofthefiniteelementmethod
istheability togenerateeasilyanyirregulargridtodescribetheflowdomain.Fora
regulargridoftriangular elements,themethodyieldsforthetwo-dimensional equation
ofLaplace thesamesetofsimultaneous equationsasgeneratedbyafinite difference
technique (Remsonet al., 1971).
Fora finitedifference approachagridhastobedefinedwithdimensions depending
15
t
n+1-
n-1-
At
M4
Fig. 1.Thefinitedifferencegrid
forEqn32withdistancexand
timet.
X
0
1 2
i+1
• i-1
m
onthenumberofindependent variables that appear inthepartial differential equation.
Iftheone-dimensional diffusion equation (32)is taken asanexample,thegrid will have
two co-ordinates: distancexandtime t,asshown inFig.1.Every point in this finite
difference grid corresponds toaspecific point inspace ataspecific instant in time.
Itisconvenient tochoosearegular grid with constant AtandAxbutthis isby no means
a requirement (.e.g. Tyson&Weber, 1964). Iftheco-ordinates inthex,tplane are indicatedbyiandn,thesolution atanygiven grid point ornode (i,n)is6 n .Forn=0
initial values for9havetobe given andiftheflow domain isdivided intom equal
intervals,boundary conditions fori=0andi=mhave tobe specified foreach time
level n.
The finite difference approach replaces thederivatives ofthepartial differential
equationby their finite difference analogue.This approachmay lead toanexplicitor
implicit finite difference scheme.An explicit scheme isobtained ifthe time derivative
is replacedbyaforward difference approximation between thenandn+1 time level and
the spacederivatives arereplacedby their finite difference analogues atthen time
level.Applied toEqn32this yields
e
9
i
i
e n-e11,
i+1
AX
At
1
1-1
(33a)
AX
AX
which canbewritten as
„n+1
=en1+D
At
(AX):
C6i-,
i
i+i'
InEqn33btheunknownvalue ofthedependent variable at time level n+1 is explicitly
expressed interms ofknownvalues atthetime leveln.To solve Eqn33bDirichletcon-
16
(33bj)
ditionshavetobespecified.Fluxconditionsinvolveanextraequation.Forinstance,
ifattheboundaryx thefluxq"isspecifiedanimaginarynodeisintroducedasfollows
0n
„
_eii
m+1
m-1
(34a)
\
TÎScJ
tobewrittenas
Ax
n
n11
^n,= ~ s "2v(q~
9"
+)6"
.+e
m+1
D
™
m-1
(34b)
Withtheintroductionofimaginarynodeswhenfluxconditionsarespecifiedatthe
boundary,6canbesolvedattheendofthefirsttimeincrementthrougharepeated
applicationofEqn33b.Oncethesevaluesarecomputed,Eqn33bisusedagaintomove
thesolutionforwardbyanothertimeincrement.Althoughtheexplicitfinitedifference
schemeappearstobeasimplestraightforwardtechnique,ithasfoundlittleapplication
inthefieldofsub-surfacehydrology (Remsonetal., 1971).Thereasonisthatthe
methodisunstableandleadstoameaninglesssolutionduetotheamplificationof
round-offerrors,unlesstheinequality
D At ^ 1
(35)
7772
2
(Ax)
issatisfied (Richtmyer&Morton,1967).Moreover,Eqn35isarequirementforthefinite
differenceapproximationtoconvergetothetruesolutionwheninthelimitAxandAt
approachtozero.Becausestabilityandconvergencecriteriaimposedonanexplicit
finitedifferenceschemeoftenleadtounacceptablerestrictionsonthechoiceofAxand
At,animplicitschemeisusuallypreferred.Suchaschemeisobtainedifthetime
derivativeisreplacedbyabackwarddifferenceapproximationbetweenthen-1andntime
level.IfthisschemeisappliedtoEqn32,theresultingequation
_n „n-1
i -9 i
_
9
At
9
D
-Q n
i-1
9fln
29
i
fln
i+l
e
(36)
(AX) 2 '
containsthree,unknowns.IfforthefirsttimelevelEqn36iswrittenforeachnode,
thisresultsin(m-1)equationswith (m-1)unknowns.Throughasimultaneoussolutionof
thissetofequationsvaluesfor6atthefirsttimelevelareobtained.Theprocedure
isrepeatedtomovethesolutionforwardintime.ThetruncationoftheTaylorseries
whichisusedtoconvertthepartialdifferentialequationintoafinitedifferenceform
resultsinatruncationerror.ThiserrorcanbereducedwiththeCrank-Nicolsonscheme,
whichusesthecentraltimedifferencebyapproximatingthespacederivativeshalfway ,
t ä w leveln-1andn.TheCrank-Nicolsonapproximationofthelineardiffusionequation
(32)is
e^-er
,
1
ifflp,-28ê n ^M 1 Heg_ 1 2e;-' *e£j)
D
-T-= —
i
(37)
~7A^
17
Thecentral (37)andbackward (36)differenceapproximationsleadtosimilarimplicit
schemes,whichareunconditionallystable.ThesecondorderaccuracyoftheCrankNicolsonschemeusuallyresultsinafasterconvergence.Thecoefficientmatrixwhichis
obtainedfromEqns36and37hasatridiagonalform.Itisefficientlysolvedby a
Gaussianeliminationtechniqueknownasthetridiagonalalgorithm.
Iftwospaceparameters (x,y)areinvolvedtheimplicitfinitedifferenceapproximationyieldsequationswithfiveunknowns.Peaceman&Rachford (1955)proposed amethod
whichenablestheapplicationofthetridiagonalalgorithmforthetwo-dimensionalproblem.Themethodisknownasthealternatingdirectionimplicit (ADI)method. Itrequires
twoadvancedtimelevelsforacompleteapplication.Timeleveln isapproachedwithan
equationequivalenttoEqn36wherethefinitedifferenceanalogueof 3e/3y isevaluatedattimeleveln-1.Nextlinesparalleltothex co-ordinatearesolved,oneat a
timeinthedirectionofincreasingy.Forthesecondstepthetreatmentofthespace
parametersisthereverse,i.e.thefinitedifferenceapproximationfor8e/3x is
2
2
2
evaluatedexplicitlyintermsoftheknownvaluesattimeleveln and3e/3y isexpressedimplicitlyfortimeleveln+1.TheADItechniqueisunconditionallystableand
theresultingcoefficientmatrixforeachlinehastheadvantageoustridiagonalform.
AccordingtoRushton (1974),singularities intheflowdomainmayimposerestrictions on
theuseofthemethod.Manysuccessfulapplicationsinthefieldofsaturated (e.g.
Pinder&Bredehoeft,1968)aswellasunsaturated (e.g.Rubin,1968)groundwater flow
havebeenreported.TheADItechniquecanbeextendedtosolvethree-dimensionalproblems (Douglas&Gunn,1964).
Thefinitedifferenceandfiniteelementmethodshaveincommonthattheybothgive
risetoasetoflinear (orlinearized)equations.Forthesolutionofasystemofsimultaneousequationsdirectanditerativemethodsmaybeused.A directmethod isthe
above-mentionedtridiagonalorThomasalgorithm,whichcanbeappliedto coefficient
matricesthatshowatridiagonalform.Thisalgorithmeffectivelyreducestheimplicit
schemetotwoexplicitschemes.Itisobtainedthroughadecompositionofthecoefficientmatrixintoalowertriangularmatrixandanuppertriangularmatrix.Firstthe
lowertriangularmatrixissolvedbyforwardsubstitutionandthentheupper triangular
matrixissolvedbybackwardsubstitution.Sincethismethodgreatlyreducesthenumber
ofcomputationalstepswhencomparedwithotherGaussianeliminationmethods itis
economicalwithrespecttocomputercosts (Isaacson&Keller, 1966).Applicationsofthe
tridiagonalalgorithminthefieldofsub-surfacehydrologyarenumerous,e.g.Hanks&
Bowers (1962),Liakopoulos (1965),Rubin (1969),Jensen&Hanks (1967),Freeze (1969).
Mostofthesub-surfaceflowequationsarenon-linear.Onlyifthecoefficients of
thederivativesinthedifferentialequationareafunctionofthedependentvariable
doestheimplicitfinitedifferenceschemegenerateasetofnon-lineardifference
equations.Thisappliesinparticulartoequationsdescribingunsaturatedflowinwhich
functionsappearsuchask(8),D(e)andC(p).Sincedirectmethodssolvethecoefficientmatrixonlyoncetoadvancethesolutionfromtimeleveln ton+1,thevaluesof
thedependentvariableattheadvancedtimelevelcannotbeusedtoobtaintheaverage
valuesofthecoefficients.Themostobviousandsimpleapproachistheuseofcoefficientsevaluatedfortheknownvalueofthedependentvariableattimeleveln.Since
2
thevaluesofthecoefficientsoftenchangerapidlywithasmallvariationinthevalue
ofthedependentvariable,thisresults inalossofaccuracyunlesssmalltimesteps
areemployed.Thelinearizationtechniquemaybeimprovedifextrapolatedvaluesofthe
dependentvariablefromprevioustimelevelsareusedtoestimatethevaluesofthe
coefficients.This techniqueusedbyRubin&Steinhardt (1963)islesssuitablefor
systemswherethevalueofthedependentvariableisnotmonotonicallyincreasingor
decreasing.Douglas&Jones (1963)proposedapredictor-corrector techniquewhichis
particularlysuited tomildlynon-linear,one-dimensional,parabolicdifferentialequations.Themethod isstablewhenusedincombinationwiththetridiagonalalgorithm.It
involvestwoapplications oftheCrank-Nicolsonscheme.Thefirststep,knownasthe
predictor,solvesthesystemofequationsfortimeleveln+|.Thisfacilitates the
evaluationofthecoefficientsatthistimelevel.Forthesecondstep,knownasthe
corrector,theCrank-Nicolsonschemeisappliedtoadvancethesolutionfromtimelevel
n ton+1,usingthepredictedvaluesofthecoefficientsattimeleveln+J.Withhysteresis thenon-linearitymayrenderthesolutionunstableandlessaccurate.Predictorcorrectortechniqueshavebeenusedbye.g.Molz&Remson (1970),Hornbergeretal.,
(1970),Homberger &Remson (1970).A disadvantageofthemethod isthatitrequires
twiceasmuchcomputertime.Evenmoretime-consuming isamethodusedbyKluteetal.
(1965)wherethesystemofequationsisrepeatedlysolvedtoimprovethevaluesofthe
coefficients inthenon-linearequations.
Withcomplicatedproblemsorwhenthenon-linearitiesaremorepronounced,iterative
methodsarepreferredtothedirectGaussianeliminationtechnique.Moreover,iterative
methodsaretheonlymeanstosolvecoefficientmatriceswhichresultfromdifferencing
ellipticequations.Ifthelineartwo-dimensionalLaplaceequation (19)istakenasan
example,themostsimpleJacobi iterativeschemewhichresultsfromdifferencingthis
ellipticequationiswritten (withAx=Ay)as
•nj-»ï-..j + *ï + ..j + *Lj-. + ^ . v 4
(38)
whereristheiterationindexandi,j indicatesthelocationorthenodeinthex,y
plane.Forthesolutionofanellipticproblemaninitialguessfor^ ^ isrequiredto
starttheiteration.Iftheschemeisexecutedinaspecificorder,earlierimproved
valuesof$canbeusedtospeeduptherateofconvergence.Thistechniqueisknownas
Gauss-SeideliterationandcanbewrittenforEqn19as
•n]-c«! f J + <j-. + ^ . . j + * u . ) / 4
(39)
Therateofconvergenceisgreatlyimprovedwithaschemeknownasthesuccessiveoverrelaxation (SOR)method.Itusesanaccelerationparameter<oandcanbewrittenfor
Eqn19as
wheregenerally 1<u <2.Forcertainproblemsanoptimumvalueforo>maybeobtained
fromtheoreticalconsiderations,forotherproblemsempiricalformulasortrialand
19
errorprocedureshavetobeused.Manyapplicationsofpoint-iterativemethods (GaussSeidelandSOR)totransientandsteadyflowproblemshavebeenreportedinliterature.
Theyincludesaturated (e.g.Remsonetal.,1965;Freeze&Witherspoon,1966;Taylor&
Luthin,1969)aswellasunsaturated (e.g.Watson,1967;Ibrahim&Brutsaert,1968;
Wisleretal.,1968)flowconditions.
Insteadofimprovingthevalueofthedependentvariablefor'eachnodeindependently,ablockorlinesuccessiveover-relaxation (LSOR)methodmaybeused. IfLSORis
appliedtothetwo-dimensionalproblem (19),theiterativeschemeforeachhorizontal
lineofthex,ydifferencegridcanbewrittenas
1,J
L
>J
4
i-l»J
i+l,J
*i,j-M
V
(41)
i,j+H
Thesystemof'equationsgeneratedwithEqn41isefficientlysolvedwiththe tridiagonal
algorithm,since<(£!_,isknownfrompreviouslyobtainedvaluesforthenodesonline
j-1.
Amoreimplicitsolutionisobtainedwiththealternatingdirectionimplicitprocedure (ADIPIT),theiterativevariantoftheADImethod.Eachiterationcycleconsists'
ofsolvingsimultaneoussetsofequationsforrowsandthenforcolumns.Therateof
convergencegreatlydependsonthechoiceoftheaccelerationparameterwhichvariesin
acyclicmanner (Wachspress,1966).ApplicationsofLSORandADIPITmethodshavebeen
reportedbye.g.Bredehoeft&Pinder (1970),Prickett (1975),Vauclinetal. (1975).
Withtheabove-mentionedtechniques,stableandconvergentsolutionscanbe obtained
forrelativelysimple,non-linearflowproblems.ForcomplicatedproblemsStone (1968)
proposedamorepowerfultechniqueknownasthestrongly implicitprocedure (SIP).Howeverdifficultiesarisewhenthefinitedifferenceapproachisusedtosolvemultidimensional,saturated-unsaturatedflowproblemsforheterogeneousmediaorwherethe
geometricboundaryoftheflowdomainisirregular (Vachaudetal., 1975).Thesedifficultiesdonotoccurwiththeapplicationofthefiniteelementtechnique.Thismethod
isflexibleforuseinanirregularflowdomainandallowsattheboundaryachangefrom
DinchlettoNeumannconditionsduringasingletimeincrement.
A recentnumericalapproach,commonlyreferredtoasnumericalsimulationisused
tosolvetransientone-dimensionalunsaturatedflowproblems.Forthispurposethesoil
coumnisdividedintoanumberoflayers.Toeachseparatelayerandforasmalltime
™ 'Da7!l3W «* * » P r i n c i P l e »fcontinuityareapplied.Thisresultsinthe
eaâr"* & T^
71
?
""
-^Pendently ofeachotherthe p ^ c e -
6XPli meth dt0
Whidltheearlier
mentioned i c t i o n s
Zank «Tws?
lltvar
**"*** ""****" ValUe£orthemisture«»tentof
fl0Wm e S a r SC a l C U l a t e d
?
973
°
^
*»»•
^ r ° P ° s e d * numerical simulation technique in which the number of
Z J 97 * r i l y *aCCOrdanœWlththedBn ^m0iStoe P™^- * »it*
2 T onûir V3nK6Ulen ° 9 7 5 ) M ESPeCial C0mpUt6r l a n ^ e developedby
%Z - ~ s - ~ i i , gProg™ W ) ) which greatly reduces
ro_
20
Numericalmethodshaveprovedtobeanimportant toolinthesolutionofcomplicatedflowproblems.Nevertheless,mathematical analysisofflowprocesses isofimportancetogainabetterunderstanding ofthestructureofthesolutionandfor comparison
withresultsobtained throughanumerical approach.Analytical orsemi-analytical methods
areparticularly usefulwhenafirstestimateofquantitative aspectsofaflowsystem
isrequired.
Analoguemethods areused tosolveawidevarietyofflowproblems.The construction
ofaresistance-capacitance networkdoesnotnecessarily requiremoretimethanthesetup
ofanumerical computerprogram.Analoguemodelsare lesscostlytooperate,but computer
methodsaremoreefficient inhandling inputandoutputofdata.Thesizeofthecore
memoryofthecomputer andtherunningcosts arelimitingfactors intheapplication of
numericalmethods tolargeproblems (Freeze&Witherspoon,1968).Howeverthese limits
arerapidlyextendingduetoadvances inthefieldofcomputertechnology.As computer
programs areeasily changed andadapted tootherproblems,theyareinmanycasesconsidered superior todirectsimulationmethods.
21
3 Saturated-unsaturated flow
3.1 THETRADITIONALAPPROACH
Asaresultofthetraditionalapproachtotreatflowinporousmediaofwhichpart
issaturatedandpartunsaturatedseparately,aninterfacebetweenbothflowsystems
mustbedefined.Forthispurposethelevelinthesoilwherethepressureisatmospheric,knownasfreewaterlevel,watertableorphreaticsurfaceismostcommonlyused.
Ithastheadvantagethatitiseasilymeasuredinthefieldandconstitutesaflowline
whenthereissteadyflowwithoutaccretionfromtheoverlyingunsaturated region.The
actualsaturated zoneextendstoalittleabovethefreewaterlevelduetocapillary
rise.Theregionofcompletesaturationabovethewatertablewasoriginally termed
capillaryfringe ('capillairezone')byVersluys (1916).Theheightofthecapillary
fringedependsontheairentryvalue,i.e.thenegativepressureatwhichthesoil
beginstodesaturate.Graduallyalesswell-defineddefinitionhascomeintousetoincludetheheightabovethewatertableatwhichdesaturationbecomes considerableoreven
toincludetheentireregionofunsaturatedflow.Sometextbooksongroundwater flow
(e.g.Verruijt,1970;Bear,1972)misusetheterm 'capillaryrise'fortheheightofthe
capillaryfringe.Capillaryrisereferstoaphenomenon (Breasteretal., 1971)andthe
heightofcapillaryriseisaquantityusedwithrespecttowell-prescribed conditions
ofunsaturatedflow (Wesseling,1957).
Whensolvingsaturatedgroundwaterflowproblems,thephreaticlevelisusually
takenastheupperboundaryoftheflowdomain,disregardingwatermovementintheoverlyingunsaturated zone.Sincetheconductivityintheregionjustabovethewater table
isapproximatelyequaltothesaturatedhydraulicconductivity,someauthors(e.g.
Youngs,1969)includethecapillaryfringeintheflowdomain.Howevertheheightofthe
capillaryfringeisgenerallysmallcomparedwiththesaturatedthicknessoftheaquifer
andforpracticalpurposesthephreatic levelistakenastheupperboundaryofthe
saturatedregion.
Anotherconceptinherenttotheseparateapproachtosaturated-unsaturated flowis
specificyield.Itisoftendefinedasthevolumeofwaterreleasedfromasoil column
ofunitarea,extendingfromthewatertabletothesoilsurfaceifthewater tableis
loweredaunitdistance.Fortheanalysisthatfollowsitisnecessarytodefinemore
preciselythefluxesinthevicinityofamovingwater-table
Considerachangeinthepositionofthephreatic levelAh=h_-h,duringasingle
Urn incrementAt andassumethatflowintheunsaturatedsoilc o l L isinvertical
fc aaT H
?"?"^
aVSrage£1UXd U r l n g
*»timel n C r e m e n t« ™ * ** soil sur-
leVel
todefne
f n*? " ^ ^
'" d e n ° t e d b y q s ( P ° s i t i v e W s ) . Inanattempt
todefinesunilarlythefluxacrossthewatertable,difficultiesariseasitsposition^
22
Fig. 2.Thedifferent levelsatwhichthevertical
fluxesaredefined.Thenumbers (1)and (2)indicatepossible soilmoisturedistributionscorresponding tothewater-tableelevationsh.and h„,
respectively.
Fig. 3.Typicalrelationsbetweenthe
specificyielduand thedepthofthe
water tablew forthesituationthat
water isreleased fromaninitially
saturated columnatasteadyrateacrossthesoilsurface (Curvea)or
atasteadyrateacross thewater
table (Curveb ) .
isnotstationary.Ifstoragechangesoccurbetweenthelevelshjandh 2 ,thefluxacross
theinitiallevelh.isdefinitelynotequaltothefluxacrossthefinallevelh 2 >To
avoidambiguitiesduetoamovingwater-table,athirdlevelh 3isdefinedjustbelowh 2 ,
sothath 2 -h isverysmall.Thefluxoâcrossthelowerlevelh 3istakenas'the
fluxacrossthewatertable'.Disregardinghorizontalflowcomponentsinthesmallregion
justbelowthewatertable,thefluxacross levelh 2isequaltoq ^
Accordingtothedefinitiongivenabove,thespecificyieldumaybeformulatedas
fe.
q s )At/Ah
(42)
Fig.3showstypicalrelationsbetweenthespecificyield vandw,thedepthofthewater
tablebelowsoilsurface.Curvearepresentsthesituationwhenwaterisreleasedfrom
aninitiallysaturatedcolumnatasteadyrateacrossthesurface (qw=0),andCurveb
forasteadyrateacrossthewatertable (qs=0). Inthefieldofsaturatedgroundwater
f
low,thedefinitionofspecificyieldisgenerallymeanttorefertothelattersitu-
ationwhereinisotropic,homogeneoussoilsuapproachesanapproximateconstantvalue
whenthewatertableissufficientlydeep.Whensolvingunconfinedflowproblemsthe
specificyieldisusuallyconsideredasaconstantproperty,characteristicofthe
W e r . Itsvalueistakenequaltotheaverageaircontentatthesoilsurface (Fig.2,
"hereforthesituations (1)and(2)equilibriumconditionsareassumed).Thefallacyof
23
thisapproachforrapidfluctuationsorshallowwater-tableshasbeenpointedoutby
Childs (1960)andisextensivelydiscussedbydosSantos&Youngs (1969).
Forthesolutionofunconfinedflowproblemsthattakeintoaccountflowfromorto
theunsaturatedzone,yisusuallydefinedas
»=(•%-
%) " / A h
(43)
whereq u (Fig.2)representsthefluxintotheunsaturatedzone (positive)orrecharge
atthephreaticsurfacefromtheoverlyingunsaturatedregion (negative).Thedefinition
of»withEqn43isequivalenttothatwithEqn42ifthelevelh,istakenatthesoil
surface.Inpractice,thelevelforh,ischosensuchthatq uapproachesaconstantvalue
equaltothelongtermaveragefluxacrossthesoilsurface.V i s yieldsanapproximate
constantvalueofv> whichfacilitatesthesolutionofequationsforsaturatedunconfined
flow.
WritingEqn43indifferentialformfortwospacedimensionsgives
v
at + % = V*.y>t,h)
.
(44)
where v andq umaybefunctionsofx,yandt.Ifforconvenience,q isconsideredas
the« l y sourceorsinkfunction,therightsideofEqn25maybereplacedbyEqn44to
'S W>* & * %(T(x,y)f )=qw(x,y,t,h)
(45 )
Equation45describessteadyflowinanon-homogeneousunconfinedaquifer.Transientflow
maybeapproachedbyasuccessionofsteady-statesituations (Muskat,1937).Ifthetime
dependentfunction % isgivenforeachtünestep,theuseofEqn45doesnotrequire
theconceptofspecificyield.
« • O M " S h T d b erCaliZed^ * *6 X a C t f 0 ™ l a t i 0 n °fthesaturatedunconfinedflow
problan1Sfarmorecomplicated.Whenconsideringthree-dimensionalflowinanisotropic
ZST\Z 7\mT"17°*f"PhreatiSCUr£aciepasriori*•—•AffacTz;iiziLz
continuityequatimforasnaneiementa
v
£ -%-%-%§
-qyf ,
^*•*-—
(46)
q h e I q u^ârdT S e n t S ? S P e C i £ i C d i S C h a r g e * ^ «**"**** - o r n a t e direction,
6UnSatUnited Z n e
h 1S the Z C 0 d i
^r
i 1the
Td":
- ^ °f *» fee surrace, s
bince
hydraulic head j f r T , °t i "*
- , i /
follows forz = h that
« * • * * . « " * • vUg and p=0a tthefree surface i t
or
+(x,y,h,t) = h
h =*(x,y,h(x,y,t),t) = *(x,y,z,t)
24
(47)
z=h
(48)
Partial differentiation of h yields
* = 11
3x
ih = i l
3y
*
11 Ui
3z 3x
+
3y
= 11
3t
+
3x
ii ill
3z 3y
+
at
M Hi
3z at
or
°r
or
UI
„r
11 = ih f
H
3x U
3x
3zJ
11 = là h . lij
3y
Ü
at
3y
u
= ^ 1 fl
atu
3zJ
( 4ya )
- -
f49b>
111
azJ
^yD}
CiQrl
( 4ycj
-
SubstitutingDarcy's law (11)forthespecificdischargesintoEqn46givesforthe
elevationh=h(x,y,t)ofthemoving freesurface
u
3h _ K-3£ 3h . -v-34« Hi _ y M. - n
- 3 t _ K 3 l 3 l E + K 3 7 3 y K a i
f501
LbUJ
%
Multiplying Eqn 50 by (1 - | £ ) and substituting Eqn 49 yields the boundary condition at
the free surface
v
| 1 =K(|i) 2 +K(|i) 2 +K(f£)2 - K(|i) - quC1 - |f)
(51)
Equation 51 and the equation for saturated flow, rewritten as
V • (KV*) = 0
(52)
have to be solved simultaneously, subject to appropriate boundary conditions at the fixed
frontiers to determine <Kx,y,z,t) everywhere in the flow domain. Since solutions are only
possible in a very limited number of cases, the Dupuit-Forchheimer assumptions are
generally applied to exclude the vertical flow component. The advantages are that the
number of independent variables i s reduced by one and the solution of the resulting
equation (the equation of Boussinesq (24)) directly yields the position of the free
surface. However the equation i s s t i l l non-linear and two-dimensional analytical solutions have not been obtained. Anumerical approach was presented by Lin (1972) resulting
in a complex finite difference scheme which is efficiently solved using the ADI technique.
There are several methods to linearize either the equation of Boussinesq or the
free surface boundary condition (51). A linearization technique often applied to problems where the change in h i s small compared with the total thickness of the aquifer,
p l a c e s ifc by the average transmissivity T, resulting in the diffusion equation (25)
which is linear in h. The objections to the use of the diffusion equation to saturated"nsaturated flow problems result from the following simplifications:
1
- the assumptions made to f a c i l i t a t e a numerical solution (Dupuit-Forchheimer approximation and linearization),
25
2.theflewisrestrictedtothesaturateddomain,
. 3.thechangeinvolumeofwaterperunitareaperunitchangeinheadisinstantaneous
andconstant,
4.thefluxqîsindependentofthesaturatedflowsystem.
Theobjectionsarelesssevereordisappearforflowinhorizontal,thickaquifersin
whichthewatertableissufficientlydeep(saymorethan5
mbelowsoilsurface) and
wherefluctuationsinthepositionofthefreewaterlevelaresmallandslow.However,
withtransientflowtoagravitywell(pumpingtest)andflowinshallowwater-table
aquifers,theaboveassumptionsareseriouslyviolated.SinceTheis (1935)derivedan
exponentionalintegralfornon-steadyflowtoawellinahomogeneous,isotropicaquifer
whichispumpedatasteadyrate,Theis'formulahasextensivelybeenusedtodetermine
theformationconstants(„andT). Ithaslongbeenrecognizedthat,asaresultofthe
rapiddrawdowninthevicinityofthewelljustafterpumpinghasstarted,deviations
fromTheis'non-equilibriumcurveoccur.AccordingtoWalton(1960)thespecificdischargeattheveryearlystagesofpumpinginanunconfinedaquiferissmallduetoa
delayinyie ld (slowdrainage)andmerelyresultsfromacompressionoftheaquiferand
expansionofthewater.Duringthenextstagethedelayedyieldreachesthewatertable
andtheaquiferbehavesasasemi-confinedaquifertendingtoequilibriumconditions.At
latetimewapproachesaconstantvalueandthetimedrawdowncurvemergeswithTheis'
non-equilibriumcurve.
InthisconnectionBoulton(1955,1963)introducedtheconceptof'delayedyield'.
lesulZl f P a r t°f^ SPeCi£iCyi6ld '*A- instantaneousandthatapart „ ,
uiing romaunitdrawdownattimeTreachesthewatertableaccordingtothXpirV a iS the
Ita
T U ""BeXP{"a C t"T)} 'W h S r e t>Tand
^ y i n d - > - empirical
en
XPreSSedln
h
Z n S n f .!
" * » * » « — • «henwaterisextractedfromawellinan
unconfinedaquifer,thefluxq,resultingfromthedrawdownofthewatertableisgiven
^•'Aïï^/ïï^^fc
v"luatedtooî f j ^ ™ *
^ e
H?r
Z17Z1TI
: l ; ^ d
einT^l \
5
(53)
% =".<«• though Boulton's convolution integral is a
S t6St e V a l U a t i n
° "
haS reCeiVSd i t S S h
- ° f ~ i t i c sm. Boulton
T meth°d 1 S ° n l y C a P a M e ° f e X p r e S S i l * " - * a - t i a n of time
! ' T 1 0 " W l t h d i S t a n C e t 0 * - W e l 1 - S i n c e ** e f f - t s of ver!f S l 0 W d r a i n a g e a r e b ° t h * » * * *»" *e empirical coeffi-
me ng CNeman
sionI^ZJ^Z™^™?,
ceased. Moreover it hasb
also beder vldw Lout ^
Cooleyi cZ Z T
J
r'
,
^
^
^ ^
^
* **** ^
Water
'^••*'—
" t a b l e r e s P ^ e has not yet
^
*™ * * »*» l i n a g emay
^ " s o v a . 1972; Neuman,1972;
s^7:Tre~Zafl°"
'T1118 S h a l l W W a t e r " t a b l e S h a S b - extensively
stricted t f
i l * T T « a g r i C U l t U r a l l a n d s - *** of the studies were rlBdelman, m Ï ^ " £ £ * ™ « * " * * « » * * » fro. rainfall (e.g.
Maasland, 1959). Laborator I T
'K r a i ^ ° « vandeLeur, 1958; Isherôod, 1959;
Laboratoxyexperiments carried out byLuthin »Worstell (1956) and
26
Vachaudetal. (1973)clearly showed thevariablenatureofy.Wherewithpumpingtests
thisvariabilitymerely results fromtherapiddrawdownofthefreewaterlevel,the
specificyield intheshallowwater-table caseratherdependsonthedepthofthewater
tablebelowsurface.ForthisreasonBrutsaert etal. (1961)proposedanapproximate
solutionwhichtreats uasafunctionoftheelevationofthefreewaterlevel.
Thetraditional approach tounsaturated flowconsiderstransportofsoilmoisture
inthevertical direction only.Mostofthepapersdealwithinfiltrationintoahomogeneous,semi-infinitemedium.Theunsaturated zoneextends fromthesoilsurfacetoa
depthatwhich themoisture contentmaybeconsideredastime-invariant,whilethe
phreaticsurface isassumed atinfinitedepth.
Papersonunsaturated flowthatinclude awatertable,whichareparticularly of
interestforthisstudy,aremuch lessnumerous.Exactanalyticalsolutionshavenot
beenpresented.A fewapproximate solutionswereobtainedforthedrainageofaninitiallysaturated soilcolumn.Gardner (1962)assumedthemoisture contenttobealinear
functionofthehydraulichead andYoungs (1960)andYoungs&Aggelides (1976)assumed
aconstantspecificyield.Childs &Poulovassilis (1962)havepresented asolutionto
theshapeofafallingwater-tablemovingwithaconstantvelocity.Capillaryrisefrom
awatertablehasbeensolved forsteady-statesituations (Wind,195S;Wesseling,19S7;
Gardner,1958)orby linearization (Philip, 1966).Thefirstnumerical solutiontoonedimensionalvertical flowwaspresentedbyKlute (1952).Sincethenmanynumericalmodels
forflowintheunsaturated zonehavebeenpublished,butveryfewusethewatertable
asthelowerboundary anduntil 1968noneofthesemodelsincluded interactionwiththe
underlying saturated zone.
3.2 THERIGOROUS APPROACH
AlthoughRichards'equationappliestotransientflowinarigidsystemaboveas
wellasbelowthewatertable,thedifferences inthenatureoftheflowarereasonsto
treatsaturated andunsaturated flowseparately.Intheunsaturated zonethehydraulic
gradientinthehorizontal directionisusuallyanegligiblefractionofthegradientin
thevertical direction sincetheboundaryconditions atthesoilsurface (rainfall,
evaporation)arerelativelyuniformoverlargeareas.Consequentlyflowispredominantly
vertical,oftengovernedby largegradients inthematricpressureincombinationwitha
lowhydraulic conductivity. Belowthewater tablethesoilissaturatedandmatric
pressuregradients donotexist,whilethehydraulicconductivityisalwaysatits
maximum.I nmany saturated flowsystems thehydraulicgradientintheverticaldirection
«•aybeneglected and flow ispredominantlyhorizontal,governedbygravity.Hence,the
Vantage ofa separate treatment isthat,forunsaturatedflow,itisoften s o i e n t
tosolvetheone-dimensionalformofRichards'equation,whereforsaturatedflowthe
relativelysimple two-dimensional formofthe (linearized)Boussinesqequationcanbe
-ed.Adisadvantage isthateffectsofunsaturated flowonunconfined g r o u n d w a r l o w ,
« studiedbyKraijenhoffvandeLeur (1962)inascaledgranularmodel « n n o t t o « n
sidered.Moreover froma fluiddynamicpointofviewthewatertableisan
27
boundaryandthenecessityofaunifiedapproachtosaturated-unsaturatedflowwas
stressedlongago(e.g.Childs,1960;Stallman,1961).Therehavebeenanumberof
multi-dimensionalsteady-statesolutionstosaturated-unsaturatedflowproblems,which
areobtainedbyanaloguemodels(e.g.Bouwer&Little,1959)andnumericalmethods(e.g.
Reisenauer,1963;Luthin&Taylor,1966;Merman,1976).Foratransientanalysis
Richards'equationcanbeusedoracombinationofequationsforsaturatedandunsaturatedflowwhichinsomewayoranotherhavetobelinked.Forthesimulationofnonsteadyflow,viscousanaloguemodelscannotbeusedbecauseofthenon-linearrelationshipsC(p)andk(p).AsolutionofRichards'equationforsaturated-unsaturatedflow
systemswithanelectricalanaloguewouldbeextremelydifficultand,ifeverpossible,
veryexpensive.Intheabsenceofanalyticalsolutionsnumericalmethodsaretheonly
meanstosolvetransientsaturated-unsaturatedflowproblems.Thisapproachwasfirst
appliedbyRubin(1968),whosolvedtransientdrainageofapartlysaturatedslabof
soilintoaditch,aclassicalproblem,knownastheditchdrainagecaseorfalling
water-tablecase.Rubinusedthetwo-dimensionalformofRichards'equation(28)and
expressedtheflowproblemintheverticalplaneintermsofthehydraulicheadas
(54a)
C(P)|f=± (k(p)|±)£
+(k(p)||)
TheflowsystemisschematicallyshowninFig.4.TheheightoftheslabisDandthe
lengtha ,butbecauseofsymmetryonlyhalftheslabisconsidered.Theoriginofthe
Ibl
"LT
L?
I , " " "in^ l0Werle£t—
^ ^ » • * °nanimporX WatCrtable1S ahSightZ
de u i l i b
tede'e^vwh
^
i
*
™ conditioâre
h
turnedeverywherelntheflowregion.Itfollowsfortheinitialconditionatt=0
° KX *L
=z.
0<z <D
(54b)
l
ZLÎZ.TJ?'"""diKheSiSl0Wred*^ • - « * ».- — —tant.
*r t h e v ™ ™ * ^
*™ —
- " — « * 1, also true
«many at x I. Since a seepage face is allowed to develop three
m77T777^77777T7777T7T777777t7mrmm77777777JS7777
28
typesofboundary conditions existatx =0.Betweenthetopoftheseepagefaceandthe
soilsurface thematricpressure isnegativeandoutflowisimpossible,hencethispart
actsasanimpermeableboundary.At theseepage facethematricpressureisatmospheric,
sothehydraulichead *equals theheight zabovetheimperviousbase.Belowthewater
levelintheditchthehydraulicheadequals z .Itfollowsthattheboundary conditions
fort>0maybe formulated as
i -o
S -o
0 <x <L
z=0
0 < z <D
x =L
(S4d)
|i »o
z s <- z < D
x =0
(54e)
z <z<z
w- s
x =0
(54f)
x =0
(54g)
3X
<J> = z
<>
( = z
w
0 <z <z
- w
and
z =D
(54c)
Difficulties insolving theflowproblem (54)donotonlyarisefromitsnon-linearity.
Thegoverningequationisparabolic intheunsaturated zoneandofanelliptical typein
thesaturatedregion,whereC(p)= 0.Thepositionofthefreesurfaceseparatingboth
regionsistimedependent.Moreover,theheightoftheseepagefaceisaprioriunknown
andconstitutespart ofthesolution.A forward finitedifferenceschemethatdetermines
thepositionofthewater tableexplicitly seem?tobeobvious.Taylor&Luthin (1969)
usedanexplicitscheme fortheunsaturatedpartofthesoilwhensolvinganaxisymmetricalflowproblem towardsawell thatcompletelypenetrates theaquiferanddischargesat
aconstantrate.Theboundaryconditions aresimilartothoseusedbyRubin (1968)except
fortheouterradiuswhere aconstantheadisassumed.Anadditionalproblem isthewater
levelinthewellwhich iscontinuously adjustedtoyield theprescribeddischarge.
Applicationoftheexplicit finitedifference schemefornodesforwhichp <0yields
valuesforeand*and isfollowedbyasolutionof•.fornodesforwhichp >0using
SOR.Theexactposition ofthe freesurfacefollowsfromalinearinterpolationbetween
nodesatwhichp changes sign.Thepositionoftheseepagefaceatthewellisobtained
fromextrapolation ofthefreesurface.Thecalculations arerepeatedforanadjusted
waterlevelinthewell if'thecomputedoutflowdiffers toomuchfromtheprescribed
discharge.
'
Explicitnumericalmethodswerenotusedbyotherinvestigators tosolvesaturatedunsaturatedflowproblems,because theyrequire forstabilityreasonsasmallmeshsize
speciallyintheunsaturated zoneinthevicinityofthewell.Itwasfoundthatthe
lengthofthetimestepshouldbesmallenoughtorestrictthechangeinhydraulichead
^ringthesteptovalues lessthan 1mm.Rubin (1968)andlaterVachaudetal. (1975)
s
°lvedtheditchdrainage casewithaniterativealternatingdirectionimplicitpro-
«dure (ADIPIT)inwhich thevaluesofkareevaluatedattheoldtimelevelwhileCis
U * centered.Themethod isrestricted toflowinhomogeneoussystemsandnotsuitable
* *infiltrationproblems inthepresenceofasharpwettingfront.Theunknownposition
29
oftheseepagefacerequiresanadjustmentifafteracompletesetofiterationsthe
computed<f>valuesindicateanupwardorsidewardflowawayfromtheseepagezone.
Vachaud/Vauclinetal.(1975)comparedtheirnumericalsimulationfavourablywithlaboratoryexperiments.Hieresultsindicatedthatthedeclineofthewatertable,thetotal
volumeofoutflowandthedurationofthetransferareseriouslyaffectedbyflowinthe
unsaturatedzone.
Verma&Brutsaert(1970)triedanumberofimplicitmethods (includingADIPIT)to
solvetheditchdrainagecase.Theyfoundthattheunknownpositionofthefreesurface
andtheunknownlengthoftheseepagefaceratherthanthenon-linearityoftheflow
equationweremostcriticalincausingslowconvergence,especiallyformorerealistic
problemsinwhichtheunsaturatedzonewasnotentirelyinornearthecapillaryfringe.
Theimplicitschemetheyfinallyadoptedisprecededbyanexplicitsteptopredict9in
theunsaturatedregion.Têfinitedifferencecorrector,implicitin*,resultsin
aset
ofsimultaneouslinearequationswhicharesolvedbyGaussianelimination.Nextthe
positionofthefreesurfaceandtheevaluesarecomparedwithvaluesobtainedfromthe
explicitstep.Iftheyarefounddifferent,asmallervalueofAtisusedorthelength
oftheseepagefaceisadjusted.Itisobviousthattheuseof
adirectmethodtosolve
1C1
mtrtX
redUC6S
CQm Utation
2 Zf
T,
withiterative
methods.
n0l
tndiCateeffeCt**th
feXpPUCit
Step*t»h
ne« compared
len
At
f
UbeleT
° »*»» ^°
^ °
al n o L T * i g 0 r 0 U ^ P P r 0 a c ht 0the* * *linagecasewasreportedbyHombergeret
al.(1969).Th«implicitfinitedifferenceschemeofRemsonetal.(1967)wasused
aGaUSS-Seidellterati0n N 0
ZTsZnTstnl
- ™ l i S t i C^
c o n ^ n st r e
.unposeaonasmallflowsvstpm
rn 7m~nr-.
lateralhn, m ^ •
mxo.Sm ) ;aconstanthydraulicheadatthevertical
Xr!r;' t i g n o r i n 8 **deveiopment ° £aseepageface-^ *» i™»*
Un tt7T r
-atu a
"
W6re reqUired bSCaUSe n att6mpt
°
^ a c e . Promcomparison with models that donot take into account the
C nClUded t h a t
" f e Ton
o the
t l position
!:•
" the
? water
° table
exrect
of
» i n (,96sT™
™ - d e *>P-dict the posi-
t
rh
Kls
U n S a t U r a t e d fl0W d eS
°
-
h
- * significant
> ™ " n < « Pn*l« are similar tothose usedb ,
- «Uirtr-jrrr^rs^rirric- i s•**""•
- e scheme w h i c h ' i T L ^ "GJa ues sTÎs l i d T ^ " ' ^ ' " ^
"""' "**"
the»11„ u m l y ^ ^
*" "
- * ^ 1 « e r a t ! « procedure. Computationsat
f»nd forpat1 ™ I f r T "
""" ^
" * * U - * > » « » values
concluded that the effects oillT
" ^ * * " ^ t t e sW
face. Theauthors
effects ofhvsteresrs on»ater-table recoveries are negligible.
Thesolvedproblemsdiscussedsofara n ^ 3 l • « . '
s y n d i c a lsaturated-unsaturated Z 7 n M
* - * ™ i * - l plane
oraxiinterestconcernstheeffectofth
water-tableaquifers.Têircommon
effectoftheunsaturatedzoneonunconfinedgroundwaterflowand
thegeneralconclusionisthatthiseffectismorepronouncedforrapiddrawdownorin
thepresenceofsteepgradientsofthefreewaterlevel.
Toinvestigate thedelayedresponseofthewatertableinanaquiferthatispumped
ataconstantrate,Cooley (1971)developedafinitedifferencemodelforaxisymmetrical
flowtoawellthatincludestheunsaturated zoneandtakesintoaccountthecompressibilityofwaterandsoilinthesaturatedregion.Forthispurposethegeneralequation
(13)isrewrittenwithoutneglectingpossiblechangesinpandtheporositynas
V•(pKv*)= -^(Pnsw)
(55)
wheres isthedegreeofwatersaturation.ExpandingthetermontherightofEqn55
yields
£Cpnsw)=Pn^
+pSw
||
+nsw|f
(56)
or
£ (Pnsw) = Pn - ^ * swp2gn(cf • c,) j f
(57)
where the formation compressibility c is defined as
_ _ 1 dn
"f
n op
(58)
and for the compressibility of the water c holds
c
=
w
1 dp
p dp
(59)
Thespecificstorages isgivenby
t
s s=pgn(cf +c w )
\
(60)
andisdefinedasthevolumeofwaterreleasedfromaunitvolumeofporousmediumasa
resultofcompressionofthemediumandexpansionofthewaterwhenthehydraulichead
isloweredoneunit.SubstitutingEqn60intoEqn57yields
3 ,
Ï
3S
^ (Pnsw) = p n
x
(61)
Ü
+
1 F
P V s 3t
WiththespecificmoisturecapacityC(p) =de/dp=(n/Pg)(dsyd*),Eqn55maybewritten
as
n 3*
(62)
V-(pKv*)=p[pgC(p)+s w s s ]-^
^ developmentofEqn62involvestheassumptionthattheformationcompressibilitymay
b
eexpressedinternsoffluidpressureratherthaneffectivestress.Itisfurthermore
turnedthatthecompressibility isconstantwithtimeandthatwithdesaturationc may
b
*neglectedsinceitseffectissmallcomparedwiththatofchangesins„.Cooley (19/ )
31
usedaradial,simplifiedversionofEqn62byneglectingspacialvariationofpand
foundthatforanisotropic,non-homogeneousmediumLSORwas,outofthree different
solutiontechniques,mostefficient.Theboundaryconditionsaresimilartothoseused
byTaylor&Luthin (1969).Theseepagefaceandtheheadinthewellcorresponding toa
constantdischargewerefoundbyiteration.Underrelaxationoftenappearednecessaryto
maintainstability.Itwasfoundthatforunconfinedfloworwhentheaquifer isoverlainbyanaquitard,forwhichanalyticalsolutionsexistatlatetimeandearlytime,
thenumericalsolutionconvergestowardstheanalyticalonewithdecreasing timestep
andmeshsizes.
Neuman(1973)usedthesameequationasCooley (1971)butforanisotropicmedia,so
thatKinEqn62representsatensor.TheequationissolvedbytheGalerkinmethod in
conjunctionwithafiniteelementdiscretizationscheme.Thesolutionofthe coefficient
matrixisobtainedwithaniterativeGaussianeliminationtechnique.After eachiteration
theboundaryconditionsthatinvolvetheseepagefaceareadjusted ifnecessary.As the
; typeofboundaryconditionisallowedtochangeduringthesolutionforasingle time
incrementthemethodissuperiortoafinitedifferencetechnique.Withexamples that
includethetwo-dimensionaltransientseepagethroughanearthdamandalayeredhill
slopecutbyaditch,Neumanshowedaninvertedshapeofthewatertablewhichcouldnot
-beobtainedwiththeclassicalfreesurfaceapproach.Thesamemodelincludingevaporationfromthesoilandtranspirationfromthecropisusedforflowinthevertical
plane (Neumanetal., 1975).
^ A rathercompletetreatmentofthree-dimensional transientflowinsaturated-unsatum e d non-homogeneousporousmediawaspresentedbyFreeze (1971).Theflowequation,
l^Z 1°• *1STUenlntemS°tfhCPreSSUreh6ad* C*=P/PÖ « *
* *esint °
Z7L
7ThyStereSlS**•*elati0nSK(P)"*e ^ '*»f - t edifference
WlthaVertlCally r i e n t e dLS0R
271 1 7 T
°
-^ SChemeist i » centered in*
andvalusforK,•.nandeareextrapolated forthefirstiterationfromprevious time
h u d not
**d a n 8 BlnPreSSUreh6ad^
Tin
aSin le
**»dement
tiœstepdurin*gearly
stagesofp ingis
6
ÏÏL I Z'Z^TuT
for asmalls i
1 a ^ ! Tu
*
At
' ^
-
-
convergence cannot be achieved, the step i s recalculated
^ - ^ - ^ -ample comprises groundwater withdrawal
P r i l l s ^ ; r b e ^ t ^ r <* • * * » -<*. «** *~ - - a of thesize of
finlte
of ^Z^Zl7Z\:uTe'àilKmi0nal
***« ^
to. 64ta reach
of Shards- ^
S
T
^
^
"
** *** * ^
« - Hnearizedform
- * of thetime^ T ^ ^ L C T t S ^
* ? * *" b6gin "
inheadfor ^ s ^ ü Z Z T o Z ^ r
^
^
*» ^
*
*"~
obtainedi s , 0 a», resulting ina^ Z T Z T ' " * " * " *****
"** »
Very
mesh t h a t i s œ e d
m vertical directioncouldnotberef PT
" " ^
SlnCS t h l S
time and storage.
« " ^ « q u i r e too much computer
32
3.3 COMPUTATIONAL DIFFICULTIES
A reviewofavailablenumerical solutionstotransientmulti-dimensional,saturatedunsaturated flowproblemsisgiveninTable1.ExceptforthepaperbyPikuletal.
(1974),allofthepapers listeduseasingleequationtomodel saturated-unsaturated
flow.Itcanbeseenthatthesizeoftheflowproblems solvedissmall.Thereasonis
thenon-linearityoftheflowequationwhichdoesnotallowanefficientnumericaltechniquetobeused.Itiswell-known thatinthepresenceofasharpwetting front,the
changeinmatricpressurecanbeasmuchasafewthousandsofmbaroveradepthofless
than10cm.Butalsoforcapillary riseitisnotunusualtofindpFvalues(pF=lg[-p])
greaterthan4duringadrysummerwithin 100cmabovethewatertable.Thereforethe
meshsizeinthevertical directionmustbeafewcentimetresonly,inordernottolose
asignificantpartofthek(p)ande(p)relations,sincealmostthefullrangeofthese
highlynon-linearrelations appliestoaverticaldistanceoflessthanonemetre. Now,
ifweconsiderashallowwater-table aquiferwithadepthof20mandawatertable
within5mbelowsurface,thenumberofnodesrequiredintheverticaldirectioncould
beestimatedasfollows.Ifthemesh sizeforthefirstmetrebelowsoilsurfaceis
takenas5cm,forthenextmetreas10cm,fortheremainingpartoftheunsaturated
zoneas20cmandinthesaturated zoneas'1m,about60nodeswouldberequired.Freeze
(1971)claimed thatthelargecomputerhewasusingcouldaccommodate 30,000nodal
points,whichrestrictstheflow regioninthehorizontalplanetolessthan23x23
nodes,orafewsquarekilometresifthehorizontalmeshsizeisallowedtobeaslarge
as100m.Becauseoftherestrictioninthemeshsizemostofthepaperslistedin
Table1considertwo-dimensional flowsystemsofafewsquaremetresonly.
Moreseriousisthetimesteprestrictionimposeduponthefinitedifferencesolutiontoobtainconvergence.Thelengthofthetimestepiscloselyrelatedtothemaximumchangeinmatricpressureinanyofthenodes.Insomeofthepaperslistedin
Table1themaximum changeinpressureforwhichconvergencecanbeobtainedisindicatedandappearstobeoftheorderofafewmbarorless.Otherinvestigators report
thatthenumberofiterations requiredtoobtainasolutionisconsiderably increased
forrapidlychanging events.Mostoftheproblemsaresolvedwithavariabletimestep,
whichatthe-startoftheexperimentisoftenlessthanonesecond.Toavoidexcessive
amountsofcomputer timetheconditionsoftheflowproblemsarerelativelywet;the
absolutevalueofthematricpressureneverexceedsafewhundredsofmbar.
Severalauthors (Verma&Brutsaert,1970;GuitJens&Luthin,1971;Freeze, 1971)
suggestedthatfromsimilitude considerationsthesimulationresultsofascaled-down
versionoftheprototypemaybeextrapolatedtotherealsystem.Breasteretal. (1971)
showedthatsimilitude affectsthesoilmoisturecharacteristics.Asmall-scalenumerical
»»del,.basedonrelationsfork(p)and8(p)thatapplytotherealsystem,tendsto
exaggeratetheeffectoftheunsaturated zoneontheflowinthesystem.Moreoverthe
scalefactorforthespace co-ordinatesisalsousedtoreducep.Itfollowsthatthe
s * numericaldifficultiesarefacedwhetherasmall-scaleorreal-sizenumericalmodel
is
usedsincetherangeinmatricpressure thatcorrespondstoasignificantchangein
1Sr e d u
« d proportionally.
33
Table1.Reviewofavailablenumerical solutionstomulti-dimensional,transient,
saturated-unsaturated flowproblems.
Dimension
Typeofproblem Sizeofflow Numerical approximation Solutionmethod
problem
Rubin (1968)
2-D
verticalplane
Ditchdrainage
case
0.3mx0.3m Linearized implicit
finitedifference
scheme
ADIPIT
Taylor &Luthin. (1969)
2-D
Flowtowellin 2.0mx1.2m Explicit finitediffer- Gauss-Seidelin
axisymmetrical shallowwaterenceschemeinunsatu- saturated zone
tableaquifer
ratedzone
Hornberger et al. (1969)
2-D
Ditchdrainage
verticalplane case
Verma&Brutsaert (1970)
2-D
Ditch drainage
v e r t i c a l plane
case
Guitjens &Luthin
2-D
axisymmetrical
0.3mx0.5m Linearized implicit
Gauss-Seidel
finitedifference
scheme
3.0 m x 3.0 m E x p l i c i t p r e d i c t o r
followed by i m p l i c i t
corrector
ADI
(1971)
Flowtowell
3.7mx2.5m Implicitfinitediffer- Gauss-Seidel
(effectof
encescheme
hysteresis)
Cooley (1971)
2-D
_
Flowtowell
20mx396m
axisymmetrical (delayedwatertableresponse)
Freeze (1971)
3-D
General
Neuman (1973/197S)
2-D
Several
verticalplane
Pikul et al. (1974)
quasi2-D
Several
verticalplane
Vaohaud/Vauolin et al.
2-D
verticalplane
Implicitfinitediffer- LSOR
encescheme
53mx40m
Implicit finitediffer- Vertically
and6mdeep encescheme
oriented LSOR
Several
ImplicitGalerkin-type Iterativeappli"
finiteelement scheme
cationofGauss
elimination
Several
Predictor-corrector
techniqueappliedto
Richardsaswellas
Boussinesq equation
TridiagonalAlgorithm.Linkage
proceduremay
require iteration
(1975)
Ditchdrainage
case
3.0mx2.0m Linearized implicit
finitedifference
scheme
ADIPIT
Rovey (1975)
3-D
34
Stream-aquifer
system
6000mx6000m Linearized implicit
andvariable finitedifference
depth
scheme
Gausselimination
3.4 ALTERNATIVE SOLUTIONS
Inanattempttofindanalternativeforthesingle-equationmodeltosolvefieldsizeflowproblemsinshallowwater-tableaquifers,itshouldberealizedthatthesolutionspresentedsofarareratheracademic.Theproblemsaresolvedtoshowthesignificanceoftheunsaturated zonetogroundwaterflowandhavebeenchosensuchthatunsaturatedlateralflowisofimportance.However,inthefield,watergradientsarelow,the
Dupuit-Forchheimerassumptionsareapproximatelyvalidinthesaturatedzoneandlateral
unsaturatedflowisinsignificantcomparedwithlateralsaturatedflow.Theseconditions
allowthethree-dimensionalflowsystemtobedescribedintermsofverticalflowinthe
unsaturatedpartandhorizontalflowinthesaturatedregion.Asolutioninthevertical
planetothisquasithree-dimensionalflowsystembasedoncoupledone-dimensional
RichardsandBoussinesq equationswaspresentedbyPikuletal. (1974).Theirmodel
firstsolvedtheequationofBoussinesq,writtenas
K ^ C h g ) = , C x , t ) | | + q u Cx,t)
(")
byusingthepredictor-correctortechniqueofDouglasandJones,wherevaluesforthe
specificyielduandthedischarge (positive)orrecharge (negative)fromtheunsaturated
zoneq oftheprevious timestepareused.Next,ineachofthenodesofthehorizontal
mesh,thesametechniqueisappliedtosolveRichards'equation.Theunsaturatedzoneis
assumedtoextendfromthelowerendoftherootzonewherethefluxisprescribedto
thewatertablewhere zeropressureexists.Theprincipleofcontinuity,appliedtoeach
unsaturatedcolumnyieldsq uwhichisusedinEqn63forthenexttimestep.Thespecific
yieldisderivedfrom
y(x,t)=n-em(x,t)
(64)
wheree is(ratherarbitrarily)definedas 'theminimumsoilmoisturecontentbelowthe
depthfromwhichmoisturemayberemoveddirectlybyévapotranspiration'.Thechangein
theheightofthewatertableappearstobecriticalforthedeterminationofthelength
ofthetimestep.Alargechangeinthewater-tablepositionmakesanadjustmentofthe
lowerboundaxyoftheunsaturatedmodeldesirableaftereachtimestepandmayrequire
aniterativesolutiontobothequationstosatisfytheinternalboundary ^ * - * *
•nodelperformsratherpoorlywhenlateralunsaturatedflowisofimportance Onlywhen
thewater-tablemovementisrelativelysmallandthelengthoftheunsaturatedcolons
canbetakenasaconstant,isthemodelmoreefficientthana
' ^ ^ J ^ ^
Onecoulddrawtheconclusionthatthepresentstateinthedevelopmentofcomputer
-, ^- +„-Fîrl-sizesaturated-unsaturatedflow
technologyprohibitsthenumericalsolutiont f» I d~ e^
sa v a i l a b l e
systems.Ontheotherhandonecouldalsostatethatthenumerica
4
to-dayareinadequatetosolveunsaturatedflowefficiently, ^ ^ J J ^ J
,
sn.llmeshsizethatisrequiredinthe^
^
1
^
™
^
^
^
solutionofafield-sizeproblem,thehorizontalmeshsizesar
foldthemeshsizeintheunsaturatedzoneinverticaldirection.Thisgives
35
discrepancythatontheonehandthehydraulicheadiscalculatedwithanaccuracyofa
fewcentimetreswhereontheotherhanddataareusedthatrepresentan'average'overa
largenon-homogeneousarea.AccordingtoFreeze (1971),complexmodelsareopentothe
chargethattheirsophisticationoutrunstheavailabledata.However,itisrathera
deficiencyofthenumericalmethodthatrequiresthesub-surfaceflowtobecalculated
tosuchahighdegreeofdetailandprecisioninordertoobtainasolution,thatthe
resultsarefaroutofproportiontowhatisusuallywantedinpractice.Thediscrepancy
betweenthenumericalsolutionandtheactualsituationinthefieldisevenmoresevere
closetothesoilsurface.InparticularthevalidityofRichards'equationforflowin
therootzonemaybequestionedinthepresenceofwateruptakebytheroots,non-capillaryporespace,osmoticandtemperaturegradients,non-continuouswettingphase,water
vapourdiffusion,tillage,etc.Moreover,anumberofhydrologicalprocessesthatoccur
abovethesoilsurfaceandgreatlyaffectthesub-surfaceflowareoftenpoorlydescribed,suchasévapotranspiration,interceptionandoverlandflow.Itshouldalsobe
realizedthatthesmalltimeincrementsthathavetobeusedrequireanabundanceof
datawhichareusuallynotavailable.
IfthenumericalapproximationtoRichards'equationforthesimulationofunsaturatedflow1Sabandonedwhatalternativeisavailable?Infactthereisawidevariety
ofpossibilitiesrangingfrompureblackboxanalysestocomplicatedconceptualmodels
basedontheoriginalequationsofDarcyandcontinuity.Thechoicedependsverymuch
onthetypeofproblemtobesolved,theinputthatisavailableandtheoutputthatis
wanted.
3.5 SCOPEOFPRESENTSTUDY
Theobjectiveofthisstudyisthedevelopmentofamathematicalmodeltosolve
fxed-sizesaturated-unsaturatedflowproblemsinrelationtoévapotranspirationfrom
sh owwater-tableaquifers.Itisassumedthatfluctuationsinthepositionofthe
6arC
1 2 e TH'
TCOnParedW l t h^ t0talS a t U r a t 6 dt M c k n e s s ° ftheunconfined
aTy £ r t W 0 d i m e n s i
af t ^ ' i 7 P P e r
°
"
° - 1 horizontalsaturatedflowistakenas
JZ ZT1,
* "l0WSStWater " tablee l e V a t l 0 n0 C C U ™ S - theconsidered
6
S£ ed
rated"! T
inalmdT',
™
a Ï Ï e on;
^ " ^""^ ^ ^ S ^S u**
™
lts
'"aSSUmSdl nV e r t i C a ldireCti0n '™
-
^
^^~
- aquasithree-dimen-
nlTT5)'*"stniCtureofwhichissimilarto^ « *«*—»y
V
" ^
* ^
SCh
edzone At
atodiiTi:zzz—
-
™^™
storage changes in the unconfined
the s o i i
—-—-—-
whi^tflned fZ 1 ST^^ l n C a C h n ° d e ° f * * ^ - d - n s i o n a l horizontal grid,
*-o 1
fel
T i ; ^ SatUrat6d ^
^ the upper and lower boundaryof
ts2l\ZZ s
flow
"eS "
N6Umam C
relatl
°
n t0
° n d i t i 0 n S aPPly- A ^
P-edure is Quired
* » S U b - S > ^ fi- évapotranspiration and saturated
^^z^z~^
P
rï: T^
whcihhasbeen
i-uidge ot neat m the soil i s neglected, estimates
36
ofévapotranspirationaremostaccurateifthemethod isappliedtoperiodsofafew
days.Actualévapotranspirationdependsonmoisture conditionsintheunsaturated zone.
Apartfromaproper linkingproceduretosolvetheNeumannconditions,themodel
forunsaturated flowshouldhavethefollowingproperties.
1.Itshouldcompute thechange inthepositionofthefreewaterlevelforchanging
boundaryconditions.
2.Itshouldsatisfactorily approximate thesoilmoistureconditionsclosetothesoil
surfaceforthecalculationofactualévapotranspiration.
3.Themodelmustallow theuseoftimestepswithalengthoftheorderofdays.
4.Thesolutiontechniquemustbe efficient intermsofcomputertimeandstoragerequirement.
Itshouldbenotedthatexceptinthevicinityoftheupperandlowerboundaryofthe
model,itisnotnecessary forunsaturated flowtobedescribed indetail.
Thougha largenumberofapproximatesolutionstounsaturatedflowproblemsexist,
noneofthesehas the abovementionedproperties.Inordertoarriveatamodel that
solvesfield-sizesaturated-unsaturated flowsystemsinshallowwater-tableaquifers,
asolutiontechnique forunsaturated flowthatmeets theaboverequirements isproposed
andoutlined inthenextchapter.
37
4 Development ofa model for unsaturated flow
4.1 STEADY-STATERELATIONS
InthesamepaperinwhichRichards (1931)derivedthegeneralequation forunsaturatedflow (28),heproposedasolutiontoasteady-statesituationthatincludesawater
table.Forsteadyverticalflowthedifferentialformofthecontinuityequationreduces
to
(65)
whereqistheverticalfluxandztheverticalco-ordinatedirection,both takenpositiveupward.IntegrationofEqn65yields
q=q=constant
-.,,•.
whichappearstobeatrivialresult.Itfollowsforthefluxesacrosstheupperand
lowerboundarythatforsteadyflow
%'*
(67)
whichimpliesastationarypositionofthewatertableasmaybeseenfromEqn42.Tne
rluxqisgivenbyDarcy'slaw
q =-K4i
fo
andthehydraulicheadisdefinedas
(68)
*•P / P g + Z
(69)
-rpractict Z T l 5 t0f0ll0W'*1S C ° nVenient t0 ad0pt> f ™ ^ »"*> ™ * .
Z 1 be^ L 2
It yi r "
S
- \SOm
r
°£ **
qUantiti6S aPPearlng in E
™ ' " ^ **
each f T c 0fd n r r
*
adVantage
^
(Qn) ftenM
'°
itS
^
68
« * 69 The pressure
— - a i value is approxiU
If
* — • *•^
in
in the unt Z L
^
" "*""*
* " • *» * * " * * P « « M • isalso
s ^
t 0 6XPreSS
intheu^t " ^
* " " i n * " » ' «I - d * - most conveniently
Substituting Eqn 69 in Eqn 68 yields
q = -K(p) (-1 4£ + n
Pg dz
38
'J
(70)
Separating the variables in Eqn 70 and solving for z gives
=_ _LL / P _!lEL_ d p
pg
0'
(71)
q + K(p)
wherethereference levelischosenatthestationaryphreaticsurfaceatwhichlevel
z=0andp=0.
Richards (1931)usedalinearrelationbetweenKandptosolveEqn71analytically
forupwardflow.Many otherempiricalK(p)relationshavebeenproposedmorerecently,
someofwhichallowananalytical solutiontoEqn71.AreviewisgivenbyRaats&
Gardner (1971).Therelationbetweenpandzforaparticularsteadyfluxqistermed
pressureprofile z(p,q).Bynumerical integrationofEqn71pressureprofilescanbe
calculatedforanygivenrelationbetweenKandp.
Transportofwaterintheunsaturated zoneinanupwarddirectioniscalledcapillaryrise.Duringthefirsthalfofthiscenturymanyexperimentswerecarriedoutto
determinethemaximumheightofcapillaryriseformanydifferenttypesofsoil.The
definitionofthe'maximumheightofcapillaryrise'wasoftenvagueandcouldreferto
transient,steadyorequilibriumconditions.Well-knownisamethodwheretubesfilled
withair-drysoilaresuppliedatthebasewithwaterofconstantpressure.Themaximum
heightofcapillaryriseisreachedwhentheadvanceofthewettingfrontisnegligible.
Atthisstage,accordingtoMoore (1939),thesumofthemaximumheightofcapillaryrise
andthepressurecorrespondingtothe'moisturecontentofthewettingfront'isequalto
zero.Theexperiment carriedoutbyShaw&Smith (1927)isanexampleofthedeterminationofthemaximumheightofcapillaryriseforsteadyflowconditions.Tubesrangingin
lengthfrom1.2to3.0m,uniformlypackedwithYolosandyloamandYololoamareinitiallywettedandpermanentlysuppliedwithwateratthebase.Capillaryriseismeasured
foraperiodoftenmonths.Fromtheexperimenttheauthorsconcludedthatthemaximum
heightofcapillaryriseequals threemetres,asforthisdepthofthewatertable,evaporationfromthesurfaceduringtheconsideredperiodisnegligible.IntheNetherlands
*»earlycomprehensivedescriptionofwaterintheunsaturated zonewaspresentedby
Versluys (1916).Theunsaturated zoneisdividedfromthewatertableupwardsintoa
capillaryzone (fullysaturated),afunicularzone (unsaturated,continuousliquidphase)
^ dapendularz o n e (unsaturated,discontinuous liquidphase).Versluysdistinguished
betweenheightsofcapillaryriseandfunicularrise.Theratherartificial tripartita
°ftheunsaturated flowregionbecamequitepopularandhasledmanyinvestigatorsto
determineheightsofriseaccordingtotheseconcepts.
AproperdefinitionoftheheightofcapillaryriseasdefinedinEqn71isthe
heightabovethewater tableatwhichagivensteadyupwardfluxcanbemaintainedfora
givenmatricpressureatthisheight.AsystematicapplicationofEqn71tocompute
eightsofcapillaryrisefordifferentvaluesofthefluxqwasfirstcarriedoutDy
Wi
*d (1955).Theanalytical solutiontoEqn71wasobtainedwithanempiricalK(p)rela-
tionwhichmaybeformulatedas
39
K=a(-p)- n
(72)
whereaandnareconstants (n=1.5).Wesseling (1957)usedEqn 72withn=2tocompute
fromEqn71maximumheightsofcapillaryriseforarangeofvaluesofq.Themaximum
heightforaparticularsteadyfluxisfoundbyintegratingEqn 71 fromp=0toavalue
ofpwhichcorrespondswiththeso-calledwiltingpointorbyintegratingtoavalueof
papproachingminusinfinityassuggestedbyGardner (1958). Forpracticalpurposesthe
matricpressureforwiltingcanbetakenequalto-16000mbarorpF4.2,wherepFE lg(-p),
Giventherelationbetweenmoisturecontentandmatricpressure e(p),usually termedthe
soilmoisturecharacteristic orpF-curve,pressureprofiles areeasily transformed into
moistureprofiles z(9,q).
Rijtema (1969)calculatedmoistureprofiles foragreatnumberofsoilsusingdata
availablefromliterature.FromthesedatatheK(p)relationandsoilmoisturecharacteristicofmediumfinesandarepresentedinFigs5aandSb,respectively. Integration
ofEqn71forthisK(p)relationyields thepressureprofilespresentedinFig. 5c.
VanderMolen (1972)showedthatwith"simpleintegrationtechniques acceptable results
maybeobtained.Forq<0(steadypercolation)theprofileshaveadistinctvertical
shape,mergingwiththeequilibriumprofileatthelowerend.Forthedevelopmentofa
modelforunsaturatedflowitisconvenienttoschematize theseprofiles intoavertical
partandintoapartthatcoincideswiththeequilibriumprofileasshowninFig. 5c.
Moistureprofilescomputedfromthepressureprofileswiththeaidofthesoilmoisture
characteristicm Fig.5barepresentedinFig.5d.Thesoilphysicaldataofmediumfine
sandgivenintheFigs5aand5bandthederivedsteady-stateprofiles (Figs5cand 5d)'
willbeusedthroughoutthischaptertoillustrate calculationtechniques.Thesymbols
thatwillbeusedareexplainedinFig.6.Thelowerboundaryoftheunsaturated zoneis
chosenasafixedleveljustbelowthelowestwater-tabledepthoccurringintheperiod
tobeconsidered.Thislevelservesasaninterfacebetweenthesaturated andunsaturatedsub-system.Theverticalco-ordinate çequalszeroatthelowerboundary andis
takenaspositiveinanupwarddirection.Theupperlayeroftheunsaturated zonein
3SrP
erCSen1
tt
Semed r Z0tOnee£rfective t
dhe
ru
.
*
T
°
°
«**
—
I tuar
°
-**— - - e the entire
UnSatoated Z n e1SCall6dSUbSOil
unaturatedzoneistakenashomogeneous althoughwithoutappreciabledifficultiesmost
P P T iff0 f 5^ ^ U S 6 d f ° rS l t U a t i 0 n SW h e r ed i f f e - *«oilphysicaldata
T L r ? ; ! ' T ^ d e P Ü 1° £t h e r ° 0 t — 1SC < ™ «*equLDwhile
b e ThvT T : her ° 0 t Z ° n e^ t h e^
- - * * * * *,rs.Flowisassumed
rn:r e Z T r
n u , a os 2
t i o noniy
^
eZZZZ7Z
rtHo 1 r V
sZ I F
TsZ
level r
40
r
**
istakentobepositive
° 0 t Z ° n e "**"
r '
SUbSOil a t
b0Undai7 b y
and the
^ " * V - P ^ i v e l y . Figure 6 showsan
SOil m i S t U r e f 0 r a
°
^ - *»***—
C=C rs i s denoted by q
* * *W °f the ^ ^
level below sur-
r i b U U O n COrreSPOndS t 0
sH
* * m 0 i S t U r e P ^ " e for q=0as
9 1S 8 i V e n M a f U n C t i 0 n £ t h S h e i g h t 2
°
*™ the wat r table,
OTdl:f"
in
^
T
^
^
**-*• ^
" should be noted thatthe
0 (the phreatic level) changes with time, depending on the value for z r .Y
K(cnvd_1)
m2
m'
m°
m-'
m-2
m-3
m-4
m-5
m-6
-10° -10' -102 -103 -104 -105
p(mbar)
Fig. 5a.RelationbetweenhydraulicconductivityKandmatricpressurepforamedium
finesandysoil.
p(mbar)
!p?"? b 'S o i l moisturecharacteristic orpF-curve fora
nationbetweenmatric pressurepandmoisture contentö.
mediumfinesandysoilgivingthe
41
z(cm)
230
0.010
0.015
0020
-10"
f i g . 5c. Pre
•
p(mbar)
«andnitric'îréssweTfo; dÏÏerenÎâlu^nf ÎV™*" t h e h e i * h t ah°™ ' h e water table
actual profile ( — ) d i f f e r s B l g £ " £ £ e s of the steady fl ux , ( c ^ - I ) . F o r ^ < 0 the
g iyt r o m t h e schematizedprofile.
42
0.010
0.015
0.020
0.05
z and moisture content 6 t o r d u r è r e n t V*J.U<==.
omo"*Qooöoii
O d O ÖÖKJ
-0.005
-0010
Fie
5P
Roi-i.'
f l u x 7J (cm-d 1 ) .
44
,
Zj-g(crn)
ne
subsoil z r s for different values of the steady
0.010
0.015
0.020
-104
prs(mbar)
H g . 5f.Saturationdeficitcurves forthesubsoilgiving the«lationêtween^theâtu
subsoilp r s fordifferent
rationdeficit S s and thepressurent theinterfacerootzone
valuesof thesteady fluxcf(cm-d"
').
S r (cm)
S u (cm)
o o o oOUJ
oi7>"-oqö
r-ö o d o o
II i <
oo
eö
o
5
lp-o
0.015
0.030
-10«
prs(mbar)
Fig. 5g.Saturationdeficitcurvesfortheentireunsaturated zonegiving therelation
betweenthesaturationdeficitS uand thepressureattheinterfacerootzone-subsoil
Prsf o r differentvaluesofthesteadyfluxq"(cm-d -1 ). Saturationdeficitcurve ( )
fortherootzonegivingtherelationbetweenthesaturationdeficitS and thepressure
attheinterfacerootzone-subsoilp r „.
46
Su(cm)
S88ÏS8080 |
ooooootiöo
d
9.
o
Q500
•1.000
150
160
z rs (cm)
Fig.5h.Relationbetween thesaturationdeficitofthe
«
'
thedepthofthewater tablebelowtheinterfacerootzone-«A-oilz__
valuesof thesteady fluxq (cm'd ) .
^
^
subsoil
z r^s fordifferent
l ç rs|p=p rs
z
rs
O*
± O
Eig.6. Schematicpresentation of theunsaturated flowsystem.
thedistancebetweenthelowersideoftherootzoneandthewatertable.At thephreatic
levelp=0andattheheight ç=ç thematriçpressureisdenotedbyp .DeLaat
(1976)showedthatforamovingwater-table 'saturationdeficits'maybeusedtofacilitatecalculationtechniques.Thesaturationdeficitoftheunsaturated zoneS isthe
amountofwaterneededtocompletelysaturatethesoilandequalsthevolumeofair
presentbetweenthelowerboundaryandthesoilsurface.Thesaturationdeficitofthe
subsoil S R maybewrittenas
Ss = ƒ
tn-e(ç)]dç
(73a)
or,sincen-9(c)= 0 for0<ç<? r g -z r s
C
ƒ
tn-e(ç)]dç
rs-Zrs
Substitutingforç=z+ (ç -z )itfollowsthatz=0for ç=ç_„-z_„and
f o r c=
Ç,-c>whence
Ss = ƒ
(73b)
rs
[n-6(z)]dz
rs
z =z.
(73c)
whichisanexpressionforS s inamovingco-ordinatesystem.Thesaturationdeficitof
therootzoneS rmaybewrittenas. . . . .
.
S
Çrs
+Dr
.[n-8(ç)]dç
r - ƒ.
or,applyingthesamesubstitutionforçasabove
48
(74a)
z +D
rs r
[n-6(z)]dz
(74b)
Anditfollows forthesaturationdeficitoftheentireunsaturated zone,S =S +S.
' u
r
4.2 PSEUDO STEADY-STATE APPROACH
Wesseling (1957)usedmoistureprofiles tocalculate themaximumamountofsoil
waterthatisavailableforcropgrowth.Theproceduremaybedescribedasfollows.At
thebeginningofthegrowingseasonequilibriumconditionsareassumed.Thesoilmoisturedistributionequalsthemoistureprofile forq=0andisindicatedbythebroken
lineinFig.7foraninitialdepthofthewatertablew =100cm.Basedondataobtained
byVerhoeven (1953)foralightclaysoilduringtheextremelydrysummerof1947,
Wesseling assumedthatthemoisturecontentatthesurfaceattheendofthegrowing
seasoncorresponds topF4.2 andincreases linearlywithdepthintherootzone.For
smallrootingdepths,asintheexampleinFig.7whereD =30cm,themoisturedistributionisassumeduniformandthematricpressureatadepthD likewiseequals
pF4.2.Themaximumamountofsoilmoisture thatisavailablefromthesubsoilbycapillaryrisetotherootzoneisfoundbyassumingsteadyflowconditionsattheendof
thegrowingseasonbetweenthelowersideoftheroot,zoneandthephreaticlevel.Pressureprofilesareusedtodetermine themagnitudeofthesteadyflowqforagivenfinal
depthofthewater table.Forinstance,ifattheendofthegrowingseasonw =120cm,
thematricpressureataheightz = w - O = 9 0 c m equalspF4.2 andfrominterpolation
inFig.5citisfoundthatq~=0.125 a m d - ' ,Theareabetweentheinitialandfinal
moisturedistributionintheregionbelowtherootzonemaybeintegratednumericallyor
graphically toyield AS =4.6 cm,whichisthemaximumamountavailableduringthe
Fig. 7.Initial (brokenline)andfinalmoisture
distributions fordifferentdepthsof thewater
tablewattheendof thegrowingseason.
s
amount of moisture
available from
the subsoil
(cm)
25-,
Fig. 8.Relationbetween theamountofmoistureavailable from thesubsoilandthedepth
of thewater tablewattheendofthegrowing
season.
T
1
1
1
1
100 120 140 160 180 200
w(cm)
growingseason by capillaryrise trm thesubsoil .If
^
S which ^ r « :: • » " * « * » » - » « « »«,, ™i 1Äle fron ^
subsoil
8 8 tatead f œ
° ° — « — <,9">
« c t «srî'Lch*:^ ' ;
PPlleS t0
depth tat.™««, ,„
!
*" ^ ' « t initial water-table
eB
lnitlal
™ «
^
"
'" " "
•"" £ t o 1 1 " i 5 t m ^ « b u t i o « i„the
s
r
t0t< 1
7172
'
' " K i M a™"t ° f s ° " " « « « . v a i l l e for thecrop
^ e r boundary v
b u t frffl , tte p
r
^
^
^ ^ ^
^
^ ^
^
^
at
led by
9 6 s ) to
tt
ît^rLi
^s":rsoT
?
**°
*"—*•
=*"*«
t
o
«
e
l
a
t
e
« ^
^tc
S
^
!
° " "" "
-'"° "
dePtt f
60,90and120daysass,™*-
inste^ns r* c** «« :IrT ? v *
.feedati« ^
table
rf
availableforcropgrowthduringperiodsof30,
p
r
1 prop sedtes
at tte
° * ^^ «»***.
^ °£ * 2 ™ ^ "»«•• >«»«»-
*ro«hdurinea^ " ' l ^ H T » ~ " ^ " " ^ " * " * " ' & ' °"*
eîlibri» conditions are« ' 1 ? f"™"8 ° f " " *"">»' s » s °» « t i « t • 0
Tl»»trie pressée at Z "
,
^ conditi »» s -V 1»for-üattd asfollows.
- 0toPPP3I ^ ™ " ^
t e i W d s
andthemoisture distribution • ! "
™
- ^
flux
^
'
"
^ ««-
across the lower boundary a = 0
*
ma
n
d« ^ ^ ^ " t T T ^ "** ^ «^
succession of steady-state How
1
^
amountof soil moisture a v a i l a h / V ^ T " " ^
t0
- e r released^
-
"
,
^
^
£
£
£
.
P ? 3 i s assunied
°0mpUte' S t e p ^
- e e n thereactive „oisture profiles after which£
SO
°^ ^
J
*
^
™
^
ste
^
A
P ' *» " ^ i ^
« ~ 2
~
-
^
_ t of
^
"
:
betransported,Atisfoundfromthecontinuityequationappliedtothesubsoil
AS =Atfq
V
(75)
Disregardingthefactthattheboundaryconditionsdifferslightlyfromtheconditions
appliedbyWesseling (1957),Fig.7isusedtoelucidatethecalculationprocedure.For
thefirststepAS g isfoundbyintegratingtheareabetweentheinitialequilibriumcurve
andthemoistureprofileforq=0.125crn-d"1whichyieldsAS s =4.6cm.Since a equals
zero,thetotalamountAS istransportedtotherootzonewitharateofatleast
—l
s
i
0.125cm-d .Ifitisassumedthatq = 0 . 2 cm-d ,itfollowsfromEqn75that
At=23d.Hence,theamountavailablefromthesubsoilafter23dequalsq
xAt=
4.6cm.IntegrationoftheareainFig.7enclosedbythemoistureprofilesforq=0.125
and0.06an«d yieldsAS =3.9 cmforwhichstepanaveragefluxq =0.0925cm-d
applies.NextthelengthofthisstepiscalculatedfromEqn75whichyieldsAt=42d.
Itfollowsthataftert=23+42=65danamountequalto4.6 +42x0.0925=8.5cm
hasbecomeavailableforthecropfromthesubsoil.Continuationofthecalculation
yieldsarelationbetweentheamountavailablefromthesubsoilasafunctionoftime.
InFig.9theresultisshownforamoredetailedcalculation,usingsmallersteps.
Interpolationfort=100dandaddingtheamountavailablefromtherootzonegivesthe
totalamountavailableforthecropduringhundreddays.Althoughaconstantmatric
pressureisassumedatthelowersideoftherootzoneandthefluxacrossthelower
boundaryisnotconsidered,thecalculationproceduremayberegardedasafirstpseudo
steady-statesolutiontocapillaryrise,yieldingthedrawdownofthewatertableand
thechangeinmoisturedistributioninthesubsoilasafunctionoftime.
Feddes (1971)assumedasuddendropinmatricpressureatthelowersideofthe
rootzonefromtheinitialequilibriumvaluetopF4.2andusedthemethoddevelopedby
Wesseling (1957)tocomputeAS (w),similartoFig.8.Theproceduretocalculatethe
amountofmoisture
availablefrom
thesubsoil
(cm)
12-1
T
1 1
P—1
20 40 60 80 100
t(d)
Fig.9.Amountofmoistureavailablefromthe
subsoilasafunctionoftime.
drawdownofthewatertableduetocapillaryriseasafunctionoftimeallows the
phreaticleveltobeloweredbysmallsteps.ForeachstepAwtheamountreleased from
thesubsoilisobtainedfromtherelationAS (w)andthecorresponding average fluxqis
interpolatedinFig.5cforzequaltotheaveragedepthofthewatertablebelowthe
rootzoneandthecorrespondingmatricpressureequaltopF4.2.Theamountreleased
dividedbytheaveragefluxyieldsthetimeforadrawdownofAwcm.
Insteadofafixedmatricpressureatthelowersideoftheroot zone,Rijtema(1971)
usedacalculationprocedureforwhichq
isheldconstantaslongaspossible.Itis
assumedthatflowintherootzoneisgovernedbywateruptakeoftherootsand thatthe
moisturedistributionintherootzoneequalsatalltimestheequilibrium distribution
(d((>/dz= 0).Watermaybeextractedfromthe,rootzoneuntilpF4.2 isreached.Thecalculationprocedureisbasedontheprinciplethatfortheassumedequilibrium conditions
intherootzoneandsteadyflowconditions inthesubsoil,themoisturedistributionis
fullydeterminedforanygivensetofvaluesforqandw.This isshownfortherootzone
extending toadepthD r=30cm,q=0.1 cm-d-1 andw = 120cmasfollows.Themoisture
distributionbetweenthewatertableandtherootzoneequalsthemoistureprofile for
q =0.1 cm-d".Ataheightz=90cmabovethewatertableandq=0.1 cm-d -1 itfollows
fromFig.5cthatthematricpressureequals-500mbar,whichisthepressure atthe
lowersideoftheroot zonep r g .Asaresultoftheassumptionthatintheroot zone
d<)>/dz=0,thematricpressureatthesurfaceequals-530mbarandthemoisturedistributioncorresponds totheequilibriummoistureprofileforp rangingfrom-500to-530
mbar.Figure 10showsthemoisturedistributionforq=0.1 cm-d-1 andw = 120cmtogetherwithanequilibriumdistributionforw =100cm.Withthiscalculationprocedure
thewatertablecanbeloweredstepbystep.Ifattimet=0equilibriumconditions are
assumedforw=100cmandforthefirststepAw=20cmwhileq
=q=0.1 cm-d -1 ,the
situationasdepictedinFig.10occurs.Integrating theincreaseofsaturationdeficits
intherootzoneandthesubsoilyields AS r =3.6 cmandAS =4.1 cm.Iffor convenience
qîsassumedconstantandequalto-0.05cm-d-1 itfollowsfromEqn75thatAt=
A
V ( q r s " %)
=4 1
- /(0.1 +0.05)=27.3dandtheamountavailableforthecropmaybe
e
O 0.10 Q20 O30
- 1' 1 •
\
AS
XJ
r°
r \
-20
*^|sV
-40
-60
\fcjl|
-80
-100
1
-120
w(cm)
52
Fig. 10.Equilibrium soilmoisture distribution
(brokenline)and thesteady-state situation for
q O.l cm-d afteradrawdownof thewater table
of20cm.
amountofmoisture
availablefromthe
entireunsaturatedzone
(cm)
10-
Fig. 11.Total amountofmoistureavailablefrom
theentireunsaturated zoneasafunctionoftime
assumingq r s =q'=0.1 cm'd .
calculatedas AS r +q r s*At= 3.6 •0.1 x27.3=6.3 an.Continuationofthecalculationsforsuccessive stepsduringwhichthewatertableisloweredbyAwcmyieldsthe
totalamount (rootzoneandsubsoil)availableforthecropasafunctionoftimeas
showninFig. 11.Afterthewatertablehasbeenl o w e r e d ^ adepthw«125cm,themaximumheightofcapillaryrise (z=95cm)isreachedforq=0.1 cm-d- (asmaybeseen
fromFig.5c)and thematricpressureatthelowersideoftherootzonehasdroppedto
pF4.2.Fromthisstageonwardsthecalculationsarecontinuedsimilartotheprocedure
describedbyFeddes (1971).Rijtema (1971)usedanempiricalrelationbetweenthedepth
ofthewatertableandthefluxacrossthelowerboundarytodeterminetheaverageflux
q foreachstepAw.
Têabovecalculationprocedureyields thefluxacrossthesurfaceq sasafunction
oftimeforagiveninitialvalue forqandtheassumptionthatq isconstantuntilthe
pressureatthe lowersideoftheroot zoneequalsP F 4.2.A.trialanderrorprocedure
isusedtofindtheinitialvalue forqsuchthatthecomputedvalueforq s equalsa ^
givenextractionratefromtherootzone.As initiallythere isnomoisturedeficitthis
extractionrateequalspotentialévapotranspirationminusrainfall.
DeLaat (1976)usedaconstantvalueforthelengthofthetimeincrementAtand
appliedthecontinuityequationforthesubsoil (75)incombinationwiththecontinuity
equationfortherootzone,writtenas
(76)
AS =Atfa
«W
tosolvethesteady-statesituationofcapillaryriseforgivenvaluesofthefluxacross
theupperandlowerboundaries.AfterRijtema (1971),equilibriumconditionsareassumed
intheroot zoneatalltimes (d*/dz=0sothatdp=-Pgdz).Thisassumptionallowsthe
expressionforS givenbyEqn74btobewrittenintermsofp.Ataheightz=z r 8the
pressurep =p 'sothatthesaturationdeficitS rmaybecalculatedas
P r s -P8 D r
S r =- ^ ƒ
[n-6(p)]dp
p
(77)
rs
Tofacilitatethecalculationprocedure,S iscomputedforanumberofvaluesfor v
toyieldthesaturationdeficitcurvefortherootzone,S(p ) .Thisrelationisshown
inFig.5g(lowercurve)forthesoilmoisturecharacteristic6(p)giveninFig.5band
adepthoftherootzoneD r=30cm.ThesaturationdeficitofthesubsoilS foraparticularsteadyfluxqisfoundfrom
s
zrs
tn-e(z,q)]dz
s= ƒ
(78)
wherethelevelz=0ischosenatthephreaticlevelwhichissituatedatadepthz
belowtherootzone.Themovingzco-ordinatesystemisusedtocalculateS foranumber
ofwater-tabledepthsz r stosetuparelationbetweenS sandz r g .Thisproceduremaybe
carriedoutforanyvalueofthefluxqtoyieldarelationSs(z°s,q)whichisshownin
Fig.Se.Withtheaidofpressureprofiles z(p,q),therelations[(z ,q) istransformed
intosaturationdeficitcurvesforthesubsoilS s (p rs ,q)whicharepresentedinFig. 5f.
Itshowsthatfortheverticalpartofthepercolationprofileswherezisnotdefined
forgivenvaluesofpandq,thesaturationdeficitS sisalsoundefined.Anumerical
approachtothecomputationofsaturationdeficitcurvesfora(heterogeneous)subsoil
isdiscussedinAppendixA.
SincebothS randS gcanbewrittenasafunctionofp thesaturationdeficitof
theentireunsaturatedzoneS uisforanysteadyfluxqcomputedfrom
S
u(Prs'<Ù=S r (p rs )+S s (p rs ,q)
SaturationdeficitcurvesfortheentireunsaturatedzoneS f c . q ) includingadepthof
therootzoneD r=30cmareshowninFig.5g.Finallythesaturationdeficitcurves
S (p q)arecombinedwiththepressureprofileszfe.q)toyieldtherelationS(z,q)
u rs
whichispresentedinFig.Sh.
Thetransientprocessofcapillaryriseduetowaterextractionfromtherootzone
isapproachedbyasequenceofsteady-statesituations.Thecalculationproceduresolves
oreachtimestepAtthesteady-stateprofilesforagivenfluxacrosstheupperand
lowerboundary.SinceSu(prs,q)« *s ( }a r er e l a t e d
f
b 6 t W e e nSu S
r
f
ifully
lfulTT
' '^ 'fqSnr
°c
*.
",.
•«g i V 6 n",* ~ *"u S andï,
tes L d y
state
is
determined.Forexamnlp
U
=
S
Cn
be
fromFig.Sgthatq=0020 Z ^
I'
'
** "* ^ ''' " ^
^ "
P = 1 5 mbar
S -18«* 7-1
" " ° " mterpolationinFig.Shfor
ft
S18.8cmandq=0.02on-d yieldsthedepthofthewatertablebelowtherootzone
" z rs-105cm.
Forthecalculationofthesteady-statesituationforcapillaryriseattime n+l
forgiveninitialvaluesS nanHç11 J U .
^f^dvy
riseattimen-<-2
r
u
C nditionS
+i
thelengthofthe ^
'
" ^ °
< *^ «C thatapplyover
lengthofthetiêincrementAt,thefollowingschemeisused.
54
(79)
1.CalculateS
forthegivenboundaryfluxconditionsfromthewaterbalanceequation
S n + 1 =S n+At(q n+J -q n+ *)
vn
n
u
u
s
w'
(80)
2.TherelationsS(p ,q)andS(p )maybecombinedtogiveS(q,S) ,sothatfor
S =S thereexistsauniquerelationbetweenS andq
3.Thewaterbalanceequationfortherootzoneiswrittenas
S n + 1 =S n+At(qn+*-q n + i )
vn
n
r
r
s
rs J
(81)
Assumingthatq n+ *=q n + 1 Eqn81providesanotherrelationbetweens£ andq n .Both
rs
n+1
n+1
relationsareusedtosolvegraphicallyorbynumericaliterationS r andq
4. Thewater-tabledepthz isfoundfrominterpolationinS(z ,q)forq=q n and
ss=s n + 1-s n + 1
Intheoriginalscheme (deLaat,1976)thesaturationdeficitS,at_timen+1isused
tocomputefromthesteady-stateprofilestherelationbetweenS^ andq .Therela11
• • as
•b+ + â=çan+ *=S Ün+KlAtfa
*-q* ' ) • Assuming
tionfromthewaterbalanceequationiswritten
4S
2
f
r
IS
thattheaveragefluxduringthetimeincrementacrosstheinterfaceroot« m e - subsoil
equalsthefluxcorrespondingtotheaveragesteady-statesituation (q fs =q ) .te
solutionappliesfromanumericalpointofviewcorrectlyattimen+1.Theuseofaverage
valuesforthesaturationdeficitsS uandS rwillonlyyieldanaveragevalueforqor
q ifthesystemislinear.Theunsaturatedflowprocess,however,isnon-linearandit
isSfoundthatthisapproachmaycausethesolutiontobeinconsistent.Forexamplethe
solutionforS n + *andq n+ *mayresultinavaluefors f ' whichislargerthanthemaximumpossiblesaturationdeficitintherootzone.ThereforetherelationbetweenS and
qisintheaboveschemeevaluatedfromthesteady-stateprofilesforS uattimen+l.
mayeasilybeshownbydecreasingthelengthofthetimeincrementthatthisapproach
doesnotsignificantlyaffectthesimulationresults.Theassumptionintheabovescheme
thatq n+ *=q a + 1 introducesuncertaintyaboutthetimeatwhichthecalculatedsteadystateallies.Thereforethecalculatedsaturationdeficitsaredefinedtoapplyattime
n+1aswellasthecorresponding fluxq.Thecalculatedvalueforq r sappliesattune i
andotherparameters,suchasz r sandp r smaybetakenattimen+1orattimen+1dependingonthetimeatwhichtheinitialvalueisspecified (n-Jor» «spec ivelyj.
Asanumericalexamplethefollowinginitialsituationisassumed:S u- 5.8anand
S»=7.4cm.Otherparameterscorrespondingtotheinitialsteady-statesituationmaybe
obtainedfromFig.5.InterpolationinFig.5gforthegivenvalueofthesaturaion
deficitsyieldsêf-0.06cm-d"'andP « s =-140nfcar.Theinitialdepthofthe w t
tablebelowtherootzoneisinterpolatedfromFig 5hforS u=15.8« an q -0.06
i
„
î n -,n +n =91.5+30=121.5cm.lne
1
n
cm-d" toyieldz =91.5cm.Consequentlyw -z r s+D r »"•»
n+J
boundaryconditio^ thatapplyforthenexttimeincrementAt=10dareq -0.
c m - d - l d <£** - -0-06c d - . Forasolutiontheaboveschemeisappliedas ollow•
1.Tb.saturationdeficitoftheentire
unsaturated zoneiscalculate * o m thewater
• IJ c n + 1 - K S + I O X (0.24 + 0.06) = 18.8 cm.
balanceequation (80)toyieldS - 5.8+ 10 (0
^ ^ ^ . ^ -^
=
2.Figure5gisusedtocomputetherelationSr(q,bujtor:>u
55
constructionoftherelationbetweenS andqanexampleisgiveninFig.5g.Itshows
-1
forS =u18.8cmthatS =7.5cm
forq=0.02
crn-d
.
r
_
^
3.TherelationbetweenS andqresultingfromthewaterbalanceequation (81)maybe
writtenasS =7.4+10x (0.24-q). BothrelationsbetweenS andqareshownin
Fig.12anditappearsgraphicallythatS n + 1 =9.0 cmand(f1*1 =0.08 cm'd -1 .
4.InterpolationinFig.5eforS g =S^ +I -s£ +1 = 18.8-9.0 =9.8 cmandq= 0.08
cm-d yieldsz r g =95.5cm.Itfollowsthatthewatertableduringthistimestep
droppedfrom121.5cmto125.5cmbelowsurface.Thepressureatthelowersideofthe
n+1
rootzoneisfoundfromFig.5gforS =9.0 cmasp
-900mbar.Thesoilmoisture
distributionsatthebeginningandattheendofthetimeincrementaregiveninFig.13.
T
1 1 1 1 1 1 1 1 1 1 r~
O 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.09 0.10 0.11 0.12
q(cm-cH)
Fig. 12.TherelationsbetweenS randqforcapillaryrise.
Curvea:S (q,S)forS = 18.8cm.Curveb:S =7.4 + 10x (0.24-q).
F
ig. 13.Theinitial (brokenline)and finalsoil
moisturedistributionfortheexamplegiveninthe
56
Ifthesameboundary conditionsapplyforsubsequenttimesteps,thesituation
arisesthattheamountofwateravailablefromtherootzoneisexhaustedandtherate
ofcapillaryrisefromthesubsoilisnotsufficienttomaintaintheupperboundaryflux
condition.Consequentlytheflux % mustbereduced.Thecalculationproceduretocompute
thereducedorrealsurfacefluxq « issimilartothatdescribedbyFeddes (1971).
Assumeattimenthefollowing initialsituation:S»=188cmandS r=9.0 cm.Figure5g
showsthatthissituationcorrespondswithq=0.08 cm-d andp r s =-900mbar.^For
At-10dandthesameboundaryconditionsasusedabove (qs -0.24 ando^
1
cm-d" )itfollowsthatS
n+1
-•
= 18.8•10x (0.24+ 0.06)=21.8cm.FromFig.5gitis
seenthatforS =21.8cmandforthemaximumvalueforthematricpressure (pF4.2)
themaxienpossiblerateofcapillaryriseequals0.07cm-d" .TherelationsSr(q,S)
forS -21.8cmandS r -9.0+ 10*(0.24-q)resultingfromthewaterbalanceequation(81)areplottedL Fig.14,whichshowsthatasolutioncannotbefoundfor
q < 0.07 cm-d"1.Themaximumamounttobeextractedacrosstheupperboundaryduring^
timestepn+1 equalstheamountavailablefromtherootzone (Sr
-S -. .
•
0.8cm)andtheamountthatismadeavailablebycapillaryrisefromthesubsoil.The
, , v -1c n-Q n-S n=18.8-9.0 =9.8 cm.Tocompute
initialsaturationdeficitofthesubsoilb g -b u .or
,
t ln.ic
theamountthatismadeavailable fromthesubsoilbycapillaryrisefor10days the
initialvalueforS isincreasedbysmallsteps.Thecalculationscarriedoutforthe
presentnumericalexamplearepresented inTable2.Foreachstepthemaximum r « e o f
capillaryrise< T < isfoundfromFig.5fandthetimerequired oreach tep (Column6)
isfoundfromthewaterbalance equationforthesubsoilasiS s /(q rs - V
S r (cm)
O 0.01
0.020.03 OÓ4 0.050060.Ó7
qCcm-d")
14.RelationsbetweenS_andq
Fig
forq
- 21.8cm.
Curvea:Sr(q,Su)forS
(0.24- q)Curveb:S r=9.0 + 10x
57
Table2.Anexampleforthecalculationofthereducedupperboundary fluxqo (seetext).
(1)
step
number
1
2
3
(2)
Ss
,,
(cm)
9.80
10.12
10.85
11.29
(3)
(4)
(5)
-max
q
(cm-d ')
AS
s
(cm)
0.32
0.73
0.44
0.10
0.09
0.08
-max
q
-%
(6)
(7)
time
total
time
(d)
(cm-d-1)
(d)
0.16
0.15
0.14
2.0
4.9
3.1
0.0
2.0
6.9
10.0
o •amax TU
côijsnôTT*T* m d e available by c a p i l l a ^ r i s e f ™ t h e **»"
P
maximumrato oC
T^ " ^
^ te " * S t e p ( C o l U m n 6> * *»
a :
Mditï n
0.89 an, whichL Z 7 T '
*'
°
° f *» V a l U e S i n C ° l u m n 8 >*elds
With
80
a v
, « x« . „ e n c e T T
*» °' - ^ e * » theroot zone cotises
co.89:0Zo? ;r:^°t a i 7nmfor^ time in ™ nt > <e-
calculatedasS n + 1 *\8 ^ ' s a t u r a t i o ndeficitoftheunsaturated zoneis rethcrootzoneS n + I i. '+10*( ° ' 1 7+°'°6)=2 K 1 ""a n dt h esaturationdeficitof
r « setequaltoitsmaximumvalue.
4.3 ANALYSISOFTHEP S E U D 0STEADy . STATE
^
^
Atruesteady-stntfluxesacrosstheu p t J ^ T " ^ ^ 0btained3 f t e r m in£in itelylongtimewhenboth
magnitudeandinthe
^ ° f^ C O n s i d e r e dsoil columnareofthesame
0
Cons
tionsisana p p a r e n t U ^J"*** "'
< W t l y asuccessionofsteady-statesituaf rm d e l l i n t r a n s
sinceitrequiresthe i h " ^ ° °
g
P ° " intheunsaturatedzone,
»Howachangeins t o m r
^ *"*inCrementt 0b e infinitelylonganditdoesnot
pseudosteady-state *L&1 T, ^ ^ ^ ^ S t a t e di nE q n6 7 ^^ ™^Y
ofthe
conditions,theinitio
0n magnitUdeanddirection f
^
° theboundary flux
Consideranij •, ^ ^ **^
° f« » ^ -rement.
*,.«50on,followedb y^ 1
^
^ * * ** " *° f*»w a t e r t a b l e >
inCm ent
The soilmoistured i s t i l
' " "9 d dUT^ w h i c h %' °andq =0.1 » f
Ume
steady-stateapproachi spresen^ .
"»"»»it resultingfromthepseudo
1
s o l u t i o nis
theactualmoisturecontentd T ' -J** *' ^ ^
basedontheconceptthat
U
spendingtoasteadyf ^ situ ^
° n ^ * " P P " » ^ byamoistureprofilecorresincethefluxrangesf r o m^ ^ for^ ^ ^= %s- The actualsituationisunsteady
boundary.Ratherthana steadT"
^ " * *" f * "t a b l et 0 °"1an ' d_1a tt h e"PP61*
a 0 1 r eP r f i l e f r
ismoreproperlya p p r o ^ ,? '
^
°
° °-=0'1 °n-d"'.theactualsituation
Ca lnatlon
correspondingtoflux es„ ^ * *
°faninfinitenumberofmoistureprofiles
1an d a t
boundary.Howeverthe„ ^ ^ ' T. f ™ °' '
thetoptozerofluxatthelower
approximationofthen ^ ^ ^ J * ™ ? ™ « * t 0* » « W « boundaryfluxis
afair
«lesarelargestnear^
'ƒ" " \ "^ " ^ *"*d i«erencesbetweenmoistureproIndownwardd i c t i o n^ a ^ * * U n d a i ** W « « Ö « approachesthetruesteady-state.
« « actualfluxxncreasinglydeviatesfromtheassumedsteadyflux
(8)
amountavail
ablefrom
subsoil (cm)
0.20
0.44
0.25
Q10 Q20 0.30
J
J
1—r-0
0.10 Q20 030
i ri— L.
(cm)
Fig. 15.Steady-state soilmoisturedistributions,showing thedifference in«turation_
deficit (shaded area)foranequilibrium situation (broken U(broken
n e ) line)and (a)steady capillary
rise (TT-0.1 crn-d"1), (b)steady percolationft--1.0cm-d ).
qbutthedifferencebetweenthesteadymoistureprofileandthenon-steadysoilmoisture
distributiondecreasescontinuouslyandultimatelyvanishescompletelyatthewatertable.
Itisthisphenomenonwhichenablestheuseoftheconceptofasuccessionofsteadystatestoapproachthenon-steadyprocessofcapillaryrise.Thevalidityoftheconcept
improvesif(i)thelengthofthetimeincrementislarge,(ii)q rgchangesslowlyin
timeand(iii)thedifferenceinmagnitudeoftheboundaryfluxconditionsissmall.
(i)Lengthoftimeincrement
,
UnlikethenumericalapproachtoRichards-equationforsolvingone-dimenstonal
transientunsaturated flow(Freeze,1969),thepseudosteady-stateapproachrequires
largevaluesforAt.Thesolutionmayevenbecomeinconsistentifthelengthofthetime
incrementistakensmallerthanthecharacteristictimet.Thecharacteristictimeisti«
appxoximatelagbetweentheinstantaneouschangeintheupperboundaryfluxconditionand
theresponseofthewatertable.Givenachangeintheupperboundaryflaxfromq o
C , thecharacteristic timeequalstheratiooftheamountofw a t e r ybe« « r e dto
reachthesteady-statesoilmoisturedistributioncorrespondingtoq (assunnga
stationarypositionofthewatertable)andAq,,.Fortheexampleused
*™'**J£L
areainFig 15aistheaûntofwatertoberemovedtoreachtheâdy-state « » I t « ,
fromûqrs-0.1cm-d"'whilethepositionofthewatertableremainsunchanged^T h i s ^
a ^ e q u a l s 0.9cm,hence,-0.9/0.1-9days.The * ^ « * * ~ ™
^
f
withthepseudosteady-stateapproachinrelationto* e
^
J
^
^
,
.
(Fig.16a)showsariseofthephreatxclevelforAt<t.This pny
r
forcapillaryriseincombinationwithalowerboundaryfluxcqua1to^ J * f i g furthershowlthattheresponsetoaninstantaneouschange(At*0)islimitedtoarise
° f3 ' 3 m'
• u „ „ . H ™ ofthewater-tabledepthariseswithpercoAsimilarinconsistencyinthesolutionofthewaterw
f
refilled
lationwhenAt<,.TheareashadedinFig.15bshowstheaûntofwter « b o ^
OS --1.2on)iftheinitialequilibriasituationisfollowedbysteadypercolation
s
'
Azrs(cm)
12T
At(d)
At(d)
Inc;eie;tAtPresultinf
f. r t a b l e / e p t hA Z " in" l a t i°ntothelengthofthetime
ationsin L
^ ° m ^ P S e U d °steady-stateapproachforthecorrespondingsituationsinFig.15.NegativevaluesofAz rg indicateariseinthewatertable.
(q.
-1.0cm-d~). Figure16bshowsthattheuseofthepseudosteady-stateapproach
forasituationwithAt<i --1.2/-1.0=1.2dyieldsadrawdownofthewatertable(assumingq =0). TherelationsshowninFig.16dependverymuchontheinitialsituation.
ThismaybeseenfromFig.5e,whichhasbeenusedtoderivetheserelations.
n
(ii)Rateofchangeinq
Applicationofthep'seudosteady-stateapproach
tosituationsforwhich| q
r r
c r e a s e ,-« „„*.<,*.• r.-,
' *•" ^'-"cn-J-uns r o r wmcn q
l6 aS
c l 11rv
T
or : I ' I l l v 1 T
^
CharaCterlStiC
X y i 6 l d S a draWd0Wn
ae-
de-
is
"» * ^ t i v e . Adecrease n'the rate
° f t h e W a t 6 r t a b l e assuming o^ = 0) even
e r duri periodswithevaporation
chL™
i
:
L
t
™
"
— X > «J. rs
- asthemaximumvalueofa iqlimits +„+1, •
rate.Moreover-tK«,^ *
^îtedtothemaximumevaporation
bu£fe s tte
w £
5
4
1Ssmal L
q
^ *TLSLT:« ™ r,: :*? -° <»-— -*the ^
M i t a a t i o T r a t e J™!.
More»er the b u l f ^ "ffe
OÎ a " " ^
° "» "
y be larse œ
^"""k"
1
* ^
% « » « l i l t e d only to
"M"-taMe
depth
* - » a - — t o . » ere, d L " a i ^ « « s T S t e a d " " S t a t e " " ^
»
« zeroorevenb e e « *» e e « i v e (Fif " « « I T , *?°S""'' " ^ '"'" " d " P S
non-steadyandth«m»«™. !
Obviouslytheactualflowsituationishighly
P
typeOlZ „ 2 1
^ ^ ^ ^ " " * * * "* « - « * > •- solvethis
fin
-
(iii)Influenceofboundaryconditions
Hieapproximatenatureofthepseudosteady-stateapproach(asexplainedearlierfor
thesituationthat % =0)improvesifthelowerboundaryflux % ispositive,evenwith
steadypercolation(qrs<0).Forthelattersituationtheupwardfluxacrossthelower
boundaryaffectsthecharacteristictimefavourablywhilethezeroflowconditionsoccurringsomewherebetweentheupperboundaryandthewatertablecoincidewiththelower
(equilibrium)partofthepercolationprofile.Howeverforrelativelylargenegative
valuesofthefluxacrossthelowerboundarythepositionofthewatertableisdominated
bytheshapeofthepercolationprofileprevailinginthelowerpartofthesubsoil
ratherthanthemoistureprofileforcapillaryrise.Assumingzerofluxconditionsatthe
upperboundarythepseudosteady-stateprocedureyieldsanequilibriumsoilmoisture
distributionregardlessofthemagnitudeofthefluxacrossthelowerboundary.Fordeep
water-tablesthesolutionisequivalenttothesituationshowninFig.2.Consequently
thesameobjectionsraisedagainsttheuseofaconstantstoragecoefficienttosolve
saturatedgroundwaterflowproblemsapplytotheuseofthepseudosteady-stateprocedure
whentherearelargenegativevaluesofthefluxacrossthelowerboundary.
inconclusion,thepseudosteady-stateproceduremayonlybeappliedtoperiodsMdth
evaporationexcess(theinconsistencyasdiscussedunder(i)isusuallysmallforcapillaryrise)andincombinationwithalowerboundaryfluxconditionwhichiseitherpositiveorsmallinthedownwarddirection.Inordertoadaptthepseudosteady-stateprocedureforgeneraluse,newconceptshavetobeintroducedtoremovetheexisting;inconsistenciesandtotreatperiodswithrainfallexcessafterasituation « * « £ * *
rise.Forarelativelylargelowerboundaryfluxinthedowârddirection,asolution
ofthepositionofthewatertablecannotbefoundwiththeaidofamoistureprofil^
correspondingtothefluxacrosstheupperboundary.Therefore" - Prosed
J * the^
pseudosteady-stateapproachisappliedtobothboundaryfluxconditionsseparte y ^ e
solutionsfortheupperandlowerboundaryfluxcondition(termedupperorlowerboundary
solution)finallyresultinacombinedpseudosteady-stateprocedure.
4.4 UPPERBOUNDARY SOLUTION
4.4.1
Perao lotion
.
* ^rvmarvrise(page55)reducesthepseudosteady-state
Thecalculationschemeforcapillaryriseipage j
,. reiation
*•,«relationsandtwounknowns(Fig.12).Thefirstrelation
proceduretoaproblemoftworelationsand
^ ^
^
Sr(q,Su)derivedfromthesteady-stateprofiles,1 infactba
cy
^
secondrelationbetweenS randqismerelyanequationofcontinuity b
balancefortherootz o n e W S1).*
>
°
~
^
v
tivevaluesofqtheschememayalsobeusedforpercolation.
Sr(,,Su)foragivenS„value,it f
£
^
somatizationofthepressureprofiles(Fig.5c int
apartthatcoincideswiththeequilibriumprofil
rangeofnegativeqvalues.Forexample,ifS u-10. cm
rô
^
^
^
^
not^ ^
^
^
^
^
^
* ~
^
r ofS isconstantandequalto6.5cm
for-0.03<q<0ncm-d"
thevalueofp andhence
rs
r
(seePQinFig.17). OutsidethisrangeofqvaluesforwhichS risconstanttherelation
betweenS andqis,unlikeforthesituationofcapillaryrise,independentofS.The
entire (independent)relation (theCurveOPRinFig.17)iscomputed fromacombination
ofS r (p rs )andq(p)withp=p r g .Thelatterrelationisequalto-K(p),asforsteady
percolationitfollowsfromEqn71thatq=-Kforlargewater-table depths (z-»•»,and
thusS -*• » ) .
u
'
Foranumericalexampleconsiderthefollowing initialequilibrium situationattime
n:S"=18.5cm,S"=6.8cmandz£ g=110cm.Ifforthenexttimeincrement (At=1d)
thefollowingboundaryconditionsapply:q n+ *=-2.4cm-d-1 andq"+*=0cm-d -1 ,the
totalsaturationdeficitS n + I =18.5-2.4=16.1cm.ForthisS valueS(q.S)is
u
u
r
presentedinFig.17,Curvea (OPQ).TheotherrelationbetweenS andqmaybewritten
asS r=S"+At(q"+i-q)=4.4-qandthesolution,obtainedgraphically fromFig.17
yieldsq""1"1=-0.5cm-d-1andS" +1 =4.9cm.Asexplained earlier,thesolutiontothe
positionofthewatertableisinconsistentifAt<T.Thecharacteristic timemaybe
foundfromFig.5e.Forz r g=110cm,thechangeinsaturationdeficitforqchanging
from0.0to-0.5cm-d"1 isreadasAS s=-1.9cm.ConsequentlyT=AS/Aq =-1.9/-0.5=
tion
of
thewater
tableof
yields
alarge
(for
=S thesolutiontotheposi3.8d.
Since
thelength
thetime
stepdrawdown
usedisone
dayS.
only,
S +1
u
11.2 cm and q = - o . s cm.^"
tionOfthf>irat-oi-»»I1-...i-i-i. - i t isfound from Fig. 5e that z" ,n+l
:1Acm)
1_4Q=
=1211
u
is the assumption of an equilibrium profile in the
S r (cm)
r-10
-9
I' 8
i
'h?
Q
-6
O—
-5
-4
-3
2
M
-0L2-Ö.1
qCcm-d-1)
Fig. 17.RelationbetweenS randqforpercolation.
Curvea:SrCq,Su)forS y» 16.1cm(OPQ)andS
:u-*» (OPR).curveb:S,.=4.4
62
subsoil.Thuswithpercolationthecurveforq=0inFig.5eisalwaysusedtosolvez r s
foragivenS value.ItmaybeseenfromFig.5ethatthisassunçtionisapproximately
correctforshallowwater-tablesorlowpercolationrates.Fordeepwater-tablesthe
resultsareexpectedtobepoor.However,itwasfoundinthisstudythatdeepwatertablesareusuallycomputedbythemodelforthelowerboundarysolution(Section4.5).
Therefore,thewater-tabledepthis,withpercolation,alwayssolvedfromtheequilibrium
Fig.Se.FortheaboveexampleitisfoundforS^ =11.2cmandq=0that
curvein
n+1
zrs =108cm.
Itshouldbenotedthatthevaluefoundforthepercolationrateshouldnotexceed
thesaturatedhydraulicconductivity.Forthepermeable 'mediumfinesandysoil used
heretoillustratecalculationprocedurestheproblemdoesnotarisebutforsoilswith
alowersaturatedhydraulicconductivitysuchasituationmightoccur.Then |q (equals
thesaturatedhydraulicconductivityasaresultofwhichpondingofwaterontheupper
boundaryofthesubsoilmayoccur.Seriouspondingmayresultinasituationw h e r t h e
rootzonebecomeswaterlogged (Sr<0)whilethereisstillasaturationdefiit nthe
subsoil(S >0). Generallypondingoccurswhenthewaterlevelhasreachedthe oil
J
s
.
•-M q =s <0 Thesenegative
surfaceinwhichcasethewaterbalanceequationsyieias u r_
valuesserveastheinitialsaturationdeficitsforthenexttùneincrement^ thea b
senceofsurfacedrainage.Inthepresenceofasurfacedrainagesystem hesaturation
deficitsareincreasedbytheamountthatisdischargedoverlandduringthetime
crement.
•<->, « W r t totheK(p)relation (Appendix A ) ,
Ifthesubsoilisnothomogeneouswithrespectrtotne
MPJ
„ m r a t e dhy_
thepercolationrateattheupperbo.dary
^
^
^
^
^
^
draulicconductivityinoneofthelowerlayers.Thiscauss h
tional
water-tableduetowhichasteady-statesituationmay ^
^
J
Relation
difficultieswithpercolationinaheterogeneoussoil,one(average ) w
mustbeused.
4.4.2
Capillary
L
vise
Asplainedearlier,thepseudosteady-stateprocedure m, «useariseinthe
fluxconditionhutthatthechan8einthephreaticl.veirsgovernedby %jW«ter-tahiedepth2? ,att*eendrfthe«*."
™
™ ^ T o repren d foreouiiihri»cîti«sin* su- . J - - ^ J ^ ^ ^ ^
s# .
sentstherelationbetweenz andtheequiiiDriun
Denotingthiscurvebyz rs (S e )thesolutionofz*sfollowsfrom
dz
z
Z
rs" rs
^Ç
(82)
e
t.at A c = - û f q
wherethechangeinS eresultsfrom % alone,sothatAS e
%
n+!
.Thisprocedure
yieldsastationarypositionofthephreaticlevelforAt<Tandq = 0.For
achanging
lowerboundaryfluxconditionthecharacteristictimewasnotdefined.Itis,however,
assumedthatthesolutionisconsistentifthecomputedvalueforz n+1islargerthan
z*s.Ifthisconditionisnotvalidthefinalwater-tabledepthistakenequaltoz*.
Forexample,foraninitialsituationattimenwithS^=6.2cmandz^g=80cmfollowed
byatimeincrementofonedayduringwhichq£**=0.2on-d"1andq°+J=S0.1cm-d"1,the
saturationdeficitS^ 1 =6.3onandFig.5eyieldsz^ 1=78.1cm,henceariseofthe
phreaticlevel.
AfirstinterpolationinFig.5eforz£s=80cmandq=0yields
S*=6.0cmandasecondinterpolationforS g=S*-A " * q£ +i=6.0-0.1=5.9cmand
q=0givesz*s=79.4cm.Thusz^ 1 =max(z£>,z*8)=max(78.1,79.4)=79.4cm.Without
thecorrectiveproceduretheriseofthewaterlevelwouldhavebeen1.9cm.Thecalculatedriseof0.6cmiscausedbythelowerboundaryfluxconditionwhichispositivein
anupwarddirection.
4.4.3
Rainfall
excess following
capillary
viae
Asituationwithcapillaryrisefollowedbyrainfallexcessrepresentsthemost
extremecaseofachangingupperboundaryfluxcondition;thefluxdoesnotonlychange
inmagnitudebutalsoindirection.Iftherootzoneisdryandtheamountofwaterinfiltratingthroughthesoilsurfaceisrelativelysmall,thesituationattheendofthe
timeincrementmaybehighlynon-steady.Asthepseudosteady-stateapproachisnot
likelytoperformwell,thefollowingprocedureisproposed.Givenaninitialsituation
attimen,thecomputationofthesituationattimen+1consistsoftwosteps.Inthe
tirststep,priortothesolutionofq,thetotalsaturationdeficitisredistributedto
asteady-statesituationcorrespondingtotheinitialwater-tabledepth,takinginto
accounttherainfallexcess.Thusfor^ andS*=S »-q»**„i tt h esteady-statesoil
m sturedistributionissolved.ForthispurposeFig.sh'-aybeusedtoyieldthecorspondingfluxq*.FromFig.5ethesaturationdeficitS*forq*andz» isfound,where
moistd J T ^ i £* <°theCUrVe£ 0 r*"°^ b e- e das-equilibrium
ln
is
defcit- \ T
^ SUbSOil1Sa S S U m e dW h e n^
Percolation.Thesaturation
Z n e1SCOmPUted
=S S
If
cLreslnd UT
°
" '* u- s- * » ««tuai£ ^ e doesnot
1 e ns " , IlnitialSteady - StSte S i t U a t i ° n ^ t 0 *» <*™£* P-edurediscussedinSection4.4.2,S*maybegreaterthanS».ThenS*issetequaltoS*.
s n. ;^
f
StePtHeP S e U d s t o
process tl I> V
°
" * - * " eprocedureL appliedwithS»=S*.
h eglVenl0WSrb0Undaiy C
£oratrai port f
° n d i t i 0 n C J• * » advantage"ofthis
procedure„ thatitdoesnotgiverisetoinconsistencies!Butmoreimportantisthe
T^;:zr;T * ° —*-*•™ J L T T J ^ ^
^C^î:^r—-in the—-—»*—
tribut
o T a c S e T S t a t ey i e l d Sa S i t U a t i ° nW l t h ^ i U ^ ™- m thiswayredistributionaccountsfortheusualdiscrepancybetweentheactualdurationofthlrain
increm^t-^0* Î V ^ ™' **=^ » "* ^ =100an.'Forthenexttime
; C - ; " d ) J . f 0 l l 0 W ^ b ° - ^ editionsapply:q ^ =_0.37».«f« ^
%
0cmd .Tneredistributionofsoilmoistureinthefirststepoftheprocedure
64
iscalculatedasfollows.For z° 8=100cmandS*u=S^•q f*xAt-21.1-0.37x10=
17.4cmitisfoundfromFig.5hthatq*=0.03 cm-d"1.InterpolationinFig.5efor
z£ =100cmandq*=0.03 cm-d"1 yieldsS*=10.0cm,sothatS*=S*-S*=17.4-10.0=
7 " cm.Forthesecondstepthepseudosteady-stateprocedureisappliedwithS u-S*17.4cm,S n=S*=7.4 cm,q n + i=0andthelowerboundaryconditionc£ + s whichhasbeen
givenequaltozero.Itfollowsfromthewaterbalance (Eqn80)thatS f•=U.4_cmfor
whichvaluetherelationS (q.SJ iscomputed.FromthisrelationandS r=S r-qxit
itmaybe foundthatq n+1 =0.022cm-d"1 ands f'=7.2cm.InterpolationinFig.5efor
S n+1 =10.2cmandq n+1 =0.022cm-d"1 yields z£> =102cm.Theexampleshowsthatalmostonethirdoftherainfallexcesshasenteredthesubsoil (ASg=-1.1 cm),while
thereisstillcapillaryriseresultinginadrawdownofthewatertableby2cm.
4.4.4
Flow chart for the upper boundary
solution
TheflowchartinFig.18showsthecalculationschemeofthepseudosteady-state
solutionfortheupperboundaryfluxcondition.Tosimplifythediagramthosesituations
forwhichthephreaticlevelrises intotherootzonearenotconsidered.Beforeapplicationanumberofrelationshavetobecomputed.Thesaturationdeficitintheroot
zoneisintegratedfor 13valuesofp r s mentionedinAppendixA,yieldingSrCp )• The
computationofS,Czr8.«D andS s (p rs ,q)is discussedinAppendixA.CombiningtheserelationswithS(q )givesS u (z rs ,q)andS u (p r s ,q).
Thesteps indicatedintheflowchartareelucidatedasfollows.
1.Givenvalues forS»andS«theinitialsteady-stateisfullydetermined.Asthewatertabledepthmaynotcorrespondtothesteady-statesituation,itsvaluemustbegiven,
thisschemeitisassumedthatz r sapplieshalfwaytheprevioustuneincrement.
2.The lengthofthetimeincrementandboundaryconditionshavetobe
^
^
fluxq mustberegardedasthemaximumpossibleflowrateacrosssoilsurface There
upperboundaryflux< e maybedifferentduetodesiccationorcompletesaturationofthe
rootzone.
. *.-
3.interpolationinFig.5gisrequiredtodeterminewhethertheinitialsituationcorrespondstocapillaryrise(q*>0)orP ^ l ^ f i ^ (qn+i«0)f o l l o wingaperiod
4.Checkwhetherthisisasituationwithrainfallexcess iqg
>
withcapillaryrise(q >0).
., ™,4<.+,,,.<»
5.IfL é Z excessofrainfallfollowingaperiodwithcapillaryrise,soilmoisture
isredistributedasdiscussedinSection4.4.3.
6.ComputeS n + 1 fromthewaterbalanceequation.
u
,. *. c n+I sTiriTT"'forS isdiscussedinbec7.ThecomputationoftherelationbetweenS andq tors>
_ 1 a H o n based
tto,,,te; .»,-*.<^>°—*~,*£• * T , : " ^ J
onthewaterbalancemaybewrittenasS -b r a t w s
H
,.
d
• A *• A *
v *77<71 whereK isthesaturatedhydraulicconductivityand
isdefinedfor-*,.,<q<^ ™ J ^ "
of
r i s e I £ - + > i s outsidethisrange
q* isthemaximumpossiblerateofcapillaryrise,itq
Jmax
.,
qvaluesasolutioncannotbeobtained.
re
8.Checktheupperconstraintofq
m j „ ™ -Fiinrn
9.ApplytheschemeexplainedinTable2tocomputethereducedupperboundaryflux % .
„n-*â
initialdata:S n ,S nandz n*
u' r
rs
\r
boundaryconditionsforAt:q112andq°*
V
interpolateq nforS nandS n
YES
redistribution:
computeS n ,S n
ju r
setqn+*=0
n
s
Û '• s u + ^K* - O l «
n+
setuptworelationsbetweenS r andq l „_J^n+1
r
YESJreductionupperboundary
n+I
ri
.., re
fluxyields:
q ,-n+l
q ,S C
r
11
q1*1
S
-K
sat
YES
+1 S +
r = r AtCq^-q»* 1 )
12 solveS° + 1 andg ^ 1
13 interpolatez"+*fromS(z ,q)for
s =sn+l.sn+l^
O
U
T"
_s^rs_
--^;:f-n+l
q=maxCq^'.O)
15
14
YES
computez*s
n+i
zn+*=max(z
,z*J|
v
rs
rs rsi
16 result:S n + 1 , S n + I ,z"+*andq re
Fig.18.Simplifiedflowchartofthesolutionfor
Forexplanationseethetext.
66
theupperboundaryfluxcondition.
This scheme yields q
pF4.2).
,while S
is setequal to itsmaximumvalue (corresponding to
10.Check the lower constraint ofq11 .
11.Thevalue forq n
•
equals -K
and S n
I
12.Solveq11 and S n
follows from thewater balance.
Sat
V
from the relations setup inStep 7.
13.When interpolating thewater-table depth,q = 0must be used ifq11 < 0 to avoid the
inconsistency as discussed inSection 4.4.1.
14. Check for a situation of capillary rise (q11
>0 ) .
15.Use theprocedure discussed inSection 4.4.2 to correct for apossible inconsistency
inthe computedwater-table depth.
16.The scheme yields initial values for the following time increment and the real upper
boundary fluxq r e . If required othervalues,such as p r andq r g are easily derived.
4.5 LOWER BOUNDARY SOLUTION
The solution for the lowerboundary flux condition isbased on the concept that the
soilmoisture distributionmay be approached by a sequence of steady-state situations
corresponding to the lowerboundary flux a .The solution applies to the lowerpart of
thesubsoil and the situations forwhich the storage coefficient is independent of the
water-table depth.
For steady flow conditions the storage coefficient isdefined as
(83)
U= ASs/AZrs
where y is a function of z r g and q. The relation v(z r s ,q) derived fromS g (z r s ,q)is
presented in Fig. 19 forq ™ 0.As a result of the schematizationof thepressure profiles (Fig. 5 c ) ,the storagecoefficient for aparticular percolation rate is either
030
q(cnrd~')
0.20-|
0.10-|
180 200 220 240
z r s (cm)
»ig. 19. Storage coefficient » as a function of the„ater-tabledepth z „ for a number
°f steady flow situations q.
67
constantorequaltotheyvaluefortheequilibriumprofile.Forsituationsthatuis
independentofthewater-tabledepththestoragecoefficientisdenotedbyu•Therelationbetweeny andqiseasilyderived,asforsteadypercolationq=-K.Withtheaid
ofthesoilmoisturecharacteristictherelationK(p)istransformedintoK(e).Using
therelationq(9)=-K(6)asitsinverse 8(q),itfollows (Fig.20)thaty =n-e(q)=
—
^
y ( q )whereq =q.Forthemostrelevantvaluesforq (say-1.0<q <-0.01 cm»d )
therelationy( q )mayoftenbeapproximatedby
0.15
-1
0.20
Mq
• " l S Ä t i ^ r n e S v ^ ^ V ^ u cS tt ir va g ie tc y e *f ^f i t h e moisture content 9 used as
lation (broken line) is g ' e n b y " q n £
68
^
°
°
" - t V The approximate re-
yq
(84)
=Ai+Big(-cg
where the constants A and B depend on the soil physical properties. For the medium fine
sandy soil used here A = 0.110 and B= -0.054 yield the broken line in Fig. 20.
The model for the lower boundary solution does not consider flow in the upper part
of the subsoil. The i n i t i a l equilibrium moisture profile serves as the upper boundary of
the model. For example, consider an i n i t i a l situation for z f s = 85 cmand q = 0, followed
bya time increment At = 5 d during which % - -1.0 cm-d'1 . Superposition of the moisture
profile for q = -1.0 cm-d-1 on the i n i t i a l equilibrium curve yields the soil moisture
distribution as shown in Fig. 21a. The moisture profile corresponding to the downward
flux across the lower boundary i s temed -percolation profile'. Since the percolation
profile is at the upper and lower side bounded by the same curve i t may be schematized
to a rectangle (Fig. 21b). The upper boundary of the model is situated at a height
C = c - z , where z i s the i n i t i a l water-table depth (z r s = 85 cm). The shaded
area equals the saturation deficit of the percolation profile S p . The rectangular shape
results from the restriction that the lower boundary solution only applies to situations
for which p = p and therefore i s independent of z r g . I t allows the saturation deficit
q
to be expressed as
S
P-v s -
(85)
ç)
where c is the actual height of the water table, the level for which p = 0. The water
balance of the lower boundary model may be written as
100
120
-140
I .», .u z-1
rs(cm)
qw=-10(cm-d )
Fx
8- 21a.Moistureprofileforq=-1.0cm-d
superimposedontheinitial equilibrium soil
"•oisturedistribution (broken line),where_
J;heshadedareaequalsthesaturationdeficit
5
pofthepercolationprofile (S =5 cm).
qJ.-lOCem-d-1)
Fig. 21b.Schematizationofthepercolationprofileusedforthelower
boundarysolution.
69
n+
n+
S*+1 = S n + Atfa
+At(q i - n *ï
P
?
«O
(86)
whereq istheflux (positiveupwards)acrossthelevel ç.Whensolving çfromEqn85
thesamedifficultiesariseasfortheupperboundarysolution.Thesaturationdeficit
andboundaryfluxconditionwhichdeterminethesteady-statesolutiondonotapplyatthe
sametime.Inviewoftheapproximatenatureoftheanalysislittleofitsgeneralityis
lostwhen zn+ issolvedfromEqn85withS n + *replacedbyS n + 1.Hence,thesolutionfor
Cattimen+ifollowsfrom
cn+l
r n+{
S-
(87)
n»*F
/•„Il+5-v
TheflowchartinFig.22showsthecalculationschemeofthepseudosteady-statesolution
forthelowerboundaryfluxcondition.Whenflowintheupperpartoftheunsaturated
zonecanbeneglected, c^ equalszeroandC p correspondstothephreatic levelatthe
onsetofthecalculations.Thissituationappliesshortlyafterasuddenloweringofthe
levelm openwatercoursesorduringtheearlystagesofapumping test.Ingeneralç
and0^dependonflowintheupperpartoftheunsaturated zone.Asolutionofthese *
variablesisobtainedincombinationwiththemodelfortheupperboundarysolution,as
discussedinthenextsection.
Foranumericalexample,considerthesituationofFig.21toapplyattimen,so
thatS p=5.0cm.Forthenexttimeincrement (At=5d)thefollowingboundaryconditionsareassumed: q° + *
j-1
=0andc£+""J
0.1 cm-d
the
water
balance
(V
fifil
U
ll+l ^
— •.- I
. .t
- follows
j . u j . j . u n ofrom
2.1Ulli
LUC
WctLCI
UÜJ.CI
(Eqn86)thatS p =5.5cm.Calculatingthestoragecoefficient fromEqn84gives
initialdata:S nandç
P
P
boundaryconditionforAt:q n+ *andq n+ *
P
TJ
S*+1 =S n•At(qf
*-< + i )
P
D
p
w
findu q fromu (q^)forq =q"+*
f+l= _s „ + 1
P
P "q
result:S n + 1 andç n + 1
«*a.n»* « £„the W
70
r bomagts „
^
For expUMtion =ee
y =0.110 - 0.054 lg(0.1) = 0.164. Noting ç = 55 cm, i t follows frora Eqn 87 that
n
çn+i = 55 - 5.5/0.164 = 21.5 cm. Since ç ~* = 9.5 cm this corresponds to a rise of the
phreatic level with 12 cm. The r i s e is caused by the decrease in flow rate across the
lower boundary from c£ _ i = -1.0 cm-d-1 to c£+* = -0.1 cm-d-1 . This effect (a rise without
recharge (q = 0) from above) i s similar to the phenomenon of delayed yield (Section 3.1).
4.6 COMBINEDPSEUDO STEADY-STATE SOLUTION
Transient unsaturated flow is approached by a sequence of steady-state situations
corresponding to the upper boundary flux of the subsoil q r g . For capillary rise the
assumption of steady flow i s seriously violated if the flux across the lower boundary
is large in the downward direction so that the actual soil moisture profile has a more
elongated shape than the assumed steady-state profile. Therefore the drawdown of the
water table is recalculated assuming steady flow in the lower part of the subsoil corresponding to the lower boundary flux %. If the lower boundary solution yieldsâ+watertable depth below the level that is found with the steady-state solution for q , a
percolation profile develops. The upper boundary of the percolation profile Cp equals the
phreatic level at the time i t s t a r t s to develop and remains unchanged during the period
the percolation profile e x i s t s . The difference in the calculated phreatic levels xs an
indication to what extent the steady-state profile for q11 is elongated.
Below the upper boundary of the percolation profile the flow is always downwards.
For a solution of the flux q across this level the following conditions can be formuP
lated. The flux q must be
1. downwards in order to satisfy flow conditions in the lower boundary model,
2. equal to q^ 1 for steady percolation in the upper boundary model,
3- approaching zero when the pF in the root zone reaches i t s maximum value,
4- independent of flow conditions in the lower boundary model in order to avoi an
iterative solution.
, , , t u th~
* * • properties are obtained if o f * is taken equal to the steady flux solved « f t t t o
eherne in Fig. 18 for the situation that the water table is at i n f l a t e depth The s
tion uses the relation S (q,S ) for S + - (Curve a (OPR) in Fig. 17) so that q,
andq n + i< ï ï n + l
*W u
q
T? •
Q W that for S ->•0 the waterFor the combined model to be consistent i t xs necessary that
^
Wble depth z r found with the upper boundary solution xs below the leve ç
I * lower boundary solution. Since Ç = Cp for Sp = 0 (Eqn 87) the condxtxon for
Sl
stency may be formulated as
for
z >ç - e
p
rs
rs
p
** v a l i d i t y of this condition is demonstrated with the use of the
^
Produced in Section 4.4.2. If ASe is the increase of the S va u
I l l a t i o n profile started to develop, a positive value of Abe
^ r - t a b l e depth z below the level C . Thus Eqn 88 may be *****
4S
e >S for S . o!SAs a result of t h / c o r r e c t i v e procedure xntroduced
(88)
»
^
^
^ ^ ^
condition
avoid incon.
71
sistency with capillary rise (Section 4.4.2) dS e > - q ^ t . Since q < 0 and dS = (q - qjdt,
i t follows that dS /dt > dS / d t . Hence, if a percolation profile exists (S > 0 and q^< 0)
the condition AS > S is valid during periods with capillary r i s e . For rainfall excess
redistribution causes the saturation deficit in the subsoil S to be equal to Sg at the
beginning of the percolation period. During percolation the equilibrium profile applies
in the subsoil so that S = S and thus dS = (q - q ) d t . Since q > q i t follows that
dSe/dt >-dS /dt.
p Hence the condition AS > S
e i s palways valid,
'
Soil moisture characteristics and K(p) relations are subject to hysteresis. Though
the effects may be considerable, i t was mentioned that they may often be neglectedwhen
both relations are combined (e.g. into a K(9) relation). When computing the saturation
deficit curves for the subsoil, both relations have indeed been used. Therefore hysteresis effects are only considered for the root zone. The use of a hysteretic soil moisture
initial data: S n , S n , S n , zn"^ and ç
u' r '
p'
rs
boundary conditions for At: qn+* and q n + i £
hysteresis: compute S,(p,J •»s ^ z ^ q ) and S
±
upper bounSary
u (P r s ,q)
solution
Fig- 18.yielding
çn+1 „n+1 n+J , r p
u
' r 'z r s andq
10
-Sn = n + 1 I
'
tf
solveqfromSr(q,S)f o rS
%Sr Is;*At(qJf .a U
NO
lowerboundary solution
Fig. 22, yielding
S n + ' and c n + J
F
fg. 23. Simplified flow h
Cha
tion see the t e x t .
"
72
of
«=he combined model for u n s a t u r a t e d flow. For explana-
characteristictocomputeS r (p )causesthisrelationtobecometime-variantandconsequentlyrelationsforS willalsochangeintime.Theeffectofhysteresisonthe
S(p )relationisdiscussedinAppendixB.
TheflowchartofthecombinedmodelforunsaturatedflowisgiveninFig.23.To
obtainasurveyablediagram,situationsforwhichthewaterlevelrisesintotheroot
zonearenotconsidered.Thestepsindicatedintheflowchartareelucidatedasfollows.
1.CalculationsarepreferablystartedforasituationthatS = 0 .Thissituationcanbe(
expectedinshallowwater-table aquifersafteralongwetperiod (withq <q) .Initial
valuesforS andS arefoundfromFig.5foragivenwater-tabledepthandaqvalue
correspondingtotherainfallexcessintheprecedingperiod.ForsituationsthatS
cannotbeneglected,initialvaluesforS andç havetobeestimated.
2.Foragivenlengthofthetimeincrementaconstantfluxattheupperandlowerboundarymustbespecified.
3.ThecomputationofS (p )isdiscussedinAppendixB.Itshouldbenotedthatasa
consequenceofachangingS (p )relation,therelationsforS uarealsotime-variant.
4.Theupperboundary solutionisgiveninFig.18.
5.Checkforthesignofthelowerboundaryfluxcondition.
6.Apercolationprofiledoesnotexist.
7.Thefluxc£ + * equalsthesteady:fluxqforthesituationthatS u*».
8.ThelowerboundarysolutionisgiveninFig.22.
9-Checkwhetherthelowerboundarysolutionyieldsalevel (ç n+i )belowthephreatic
levelthatisfoundwith theupperboundarysolution (ç rs_z r s)•
10.Thetimeindexmaybe increasedifrequired.
73
5 A quasithree-dimensional approach
Forthesolutionofsaturated-unsaturatedsub-surfaceflowinshallowwater-table
aquifers,itisassumedthattheDupuit-Forchheimerassumptionsareapproximatelyvalid.
Thethree-dimensionalflowsystemmaythenbeschematizedintohorizontalflowinthe
saturatedpartandverticalflowintheunsaturatedregion.Ifthefluctuationsofthe
watertablearesmallascomparedwiththetotalsaturatedthicknessDoftheaquifer,
thelattermaybetakenasaconstant.ThevalueofDischosensuchthattheupper
boundaryofthesaturatedzoneisjustbeneaththelowestphreaticleveloccurringinthe
periodconsidered.Sincewaterandsoilareassumedincompressible,storagechangesare
restrictedtotheunsaturatedzone.Takingintoaccountrechargefromtheoverlying
partlysaturatedregion,unconfinedsaturatedflowisdescribedbyEqn45,rewritten
hereforconvenienceas
•kvw i> + i c^y)f) =^./.M)
C89)
wherethetransmissivityT=KD.IfRistheregionforwhichEqn89holdsandS,andS 2
constitutetheboundaryofR,theconditionsvalidattheboundarymaybeformulatedas
onS, :
h=h*(x,y,t)
on S,:
ijl=o
u
r
8n~
wherethephreaticlevelh*onS,issupposedtobegivenandnisthedirectionnormal
totheboundary.
Figure24isthesomatizationofthesaturated-unsaturatedsub-surfaceflowsystem
xnthev e m c a lplane.Itshowsacrosssectionofanunconfinedaquiferboundedbya
streamandagroundwaterdivide(no-flowboundary).Themodelforunsaturatedflowis
presentedatoneparticularlocationonly.Thelowerboundaryofthismodel(ç=0)is
takenatahexghtDabovetheimpermeablebaseoftheaquifer.Atthesoilsurfacethe
upperboundaryf
luxconditionq sissupposedtobegivenasafunctionofx,yandt.
Forthes t a t i o noftransientsub-surfaceflowthetimeisdiscretizedtosmall
steal;T T - ' rinCrementAt'eXtendlngf r ° m **»nt0n+1'fl-i»-sumedtobe
L ?
T T C ° n d i t i 0 n S" * "b 0 U n d a i y° nS.- *t h eN " editionsatthe
PP
7 thS t l m einCrement
Z^tumeTfl
-^ S ° 1 U t i 0 n ° ft h esteady-statesaturatedunsaturatedflowsxtuauonisthenobtainedattime n+i andyieldstheinternalboundary
ste
« T h \ m s o H e d U r e / 7 r i S e S^
P - CD computationofarelationbetweenq„
andh,( 1 1 )soluuonofthesteady-statesaturatedflowsituation,and(iii)solutionôf
74
(90a)
(90b)
v/^/////////;/)/////>//////////////////////AJ///
777?.
Fig. 24.Schematicpresentation intheverticalplaneandboundaryconditionsofthe
quasithree-dimensional approach tosaturated-unsaturated flow.
thesteady-stateunsaturated flowsituation.
(i)relationbetweenq andh
Applicationofthemodelforunsaturated flowforq f*andfordifferentvaluesof
+i
c£ yieldsarelationbetweenthechangeinthepositionofthewatertableAç=
ç*+i-e«"*andq f*.Sinceç=h-D,itfollowsthatAc=Ah.If % iseitherpositive
orsmallinthedownwarddirection,therelationbetweenthelowerboundaryfluxandthe
changeinthephreatic levelisapproximately linear:
,n+i= a . A h+b
(91)
%
whereaandbareconstantstobedeterminedforeachtimestep.THeapproximate linearity
stemsfromtherelationbetweenS uandz r sinFig.5h.Thisrelationgovernsthewatertabledepthintheabsenceofapercolationprofile.Theequilibriumcurveusedwhen
q<0showsthatforasmallchangeinthewater-tabledepthdS u /dz r sisapproximately
constant.ForcapillaryrisethesolvedvalueofqdecreasesslightlyifS uincreasesdue
toq alone,sothat(seeFig.Sh)dS u /dz r s approachesaconstantvalue.SincedS^/dz
isproportionaltodq^/dh,itfollowsforasmallchangeinthephreatic levelduringthe
timeincrementthatEqn91isapproximatelyvalid.
Largechangesinhusuallyinvolvelargenegativeq,,valuesinwhichcasethe
phreaticlevelisgovernedbyEqn87.Introducingd pasthedepthofthewater table
belowtheupperboundaryofthepercolationprofile (Fig.21b)gives
(92)
ç =d +ç
P
P
75
SubstitutingEqn92fortimen-J intoEqn87yields
S
ç n+j= çn-J
ÔP
(93)
ReplacingS n + 1byEqn86,fcq(q£+*)byEqn84andintroducingAh=ç n+ *-çn~*into
Eqn93gives
Q n+J .At-qn+*.At-S n
Ah
T»
-I
P
+
A+Blg(-c£ *)
E +d n ^
P
D
Equation94isvalidforq° + 5<0andS° + 1>0.ItfollowsfromEqn86thatthecondition
n+1
-t.1
1P
S p >0maybewrittenasc£*<q"+*+Sn/At.Hence,Eqn94appliesfor
P
q»+* <min(0,q n+ * +S n /At).
w
P
P
Theimplicitnon-linearexpressionforqînEqn94andtheexplicitoneinEqn91
arecombinedasfollows.Themodelforunsaturatedflowisapplied foranarbitrarynegativevalueofc£ + toyield cÇK InordertosolvetheconstantsaandbinEqn91the
model
forunsaturated flowisusedtwicetocomputeAhfora,smallpositivevalueofc£ + *
andc£ =min(0,qp+ S n /At).Thelattervalueof q^ yieldsawater-table depthbelow
or+equaltoç^whileÊqn94yieldsAh=dj"*,orçn+{=^.F o r deC reasingvaluesof
q w <min(0,c£ +S^/At),Ahdecreaseslinearlyaccording toEqn91 andmorethan
linearlyaccordingtoEqn94.Thepointofintersectionq£isfoundbyaNewtoniterative
procedure.Anexampleofthetime-dependent relationbetweenq .andAhusedtosolve .
Eqn89isgiveninFig. 25.
0.2 q w (cnvd -1 )
«~io
r e l a t i 0 n between
Abased to : o S U 89 ( ShÎliL an T ! fr e tsh e time
" Variant
<w « *
s f
tw
for u n s a t u r a t i n g for I - 0 Z T ^ l
" i f , ™ " ° applications,of theVdel
part (CD) is computed from^qn 9Wi?K I t L , TÎ
^ " ^ b y X)" T h e n ° « " l i n e a r
B- -0.054. Both relations i te e ' t for£ ^-0V^l°' «P =°' A = ° - 1 1 0 a n d
76
(94)
Ax
,i
Ay
i+ 1,j+1
i,j+l
i-1,j +l
D
C
i+1.j
'J
'-1.J
Q
P
i-I.J-l
i.i-1
i+ 1,j-1
Fig. 26.Gridconfigurationfortwodimensionalhorizontalflow.
(ii)solutionofsaturatedflow
ForanumericalsolutionofEqn89theregionRisschematizedtoahorizontalx.y
grid.Ifthenodesinthexdirectionaresubscriptedbyiandthoseintheydirection
byj (Fig.26)thefinitedifferenceequationtoEqn89attimen+Jmaybewrittenas
(T
.+T Oft1?**•-h?i)
2(Ax) 2
2(Ax) 2
CT . +T .)(h?i,-h?i)
(95)
2(Ay)
2(Ay) 2
2
^
1,J
Equation95isappliedforeachofthenodesforwhichhhastobecalculated.The
phreaticlevelisthensolvedwithapointiterativemethod (Gauss-SeidelorSOR).The
rightsideofEqn95isreplacedbyalinear(ized)expressionforo^written (without
thespaceindexi,j)as
\P~\ ) + b
c£ + i=a(h n+i
(96)
wheref
ora >q*thecoefficientsaandbareidenticaltotheconstantsinEqn91.For
q„'<q*t h r a l l ofaandbvaryforeachiterationcyclesothatEqn ^ - p r e s e n
t L tangenttoEqn94.Thetangentforiterationcyclerisobtainedforq„ ,sothat
h
_L - d ( Ad(Ahi
ar " \~K
(97)
xA
%
r-1 „w.= thecoefficienta r (withoutthetimesubscript)
DifferentiatingEqn94forq^ ,gives
as
77
fr-K2
âJ
r-l
1 _n,™,_r-i 1.. — —
At-p^
-O^SBC^" -At-qp-At-STT7TT
^/q^1
C98)
sothat
„rr., .r-l
b r=(£-'r--alr_(Ah)
(99)
NT-1
where (Ah)" isthechangeinthephreaticlevelcalculatedwithEqn94and/"'isthe
q
storagecoefficientaccordingtoEqn84,bothfor a =q*"'.
(iii)solutionofunsaturatedflow
Hiemodelforunsaturatedflowisappliedineachnodeforthegivenupperboundary
fluxcondition % andthelowerboundaryflux ^ calculatedwithEqn96.Thesteadystatesolutionyieldsthesaturationdeficitintherootzoneandthesubsoil,thematric
pressureandthefluxattheinterfacerootzone-subsoil,andtherealupperboundary
s '
Forthesolutionpresentedabovetheinterfacebetweenthemodelsforsaturatedand
unsaturatedflowhasbeentakenatafixedlevel.Equations91and94usedtolinkboth
-dels appeartobeindependentofthislevel,provideditislocatedbelowthephreatic
surface.Theinterfacemaythereforebetakenjustbelowthemovingwater-table,resulting
inavary lng referencelevelforc.Theadvantageofusingsaturationdeficitsinsteadof
saturationsisthatsuchshiftsintheoriginoftheverticalco-ordinate çdonotinvolve
voumetransesacross^ fluctuating. ^ ^ ^ . ^ ^ ^ ^
^
2 Z T l J*"6'™ e r e f 0 r ethem 0 d e l* *c r a t e d flowcanbelinkedtomodelsfor
saturatedflowwhichtakeintoaccountavaryingthicknessofthesaturatedflowregion.
78
6 Application and use
6.1 EXPERIMENTAL VERIFICATION
Inordertoapplythequasithree-dimensional sub-surfaceflowmodeltoanactual
fieldsituation,sinktermsareaddedtotherightsideofEqn89.Theyinvolveaterm
q representinggroundwaterextractionfromwellsandatermq orepresenting groundwater
discharge intotheopenwatersystem.Withdrawalratesfromnodesinwhichgroundwater
isextractedaresupposedtobespecifiedhalfwaythetimeincrement (q" ) .Thedischargeintothesurfacewatersystemiscomputedwiththeaidofalinearizedrelation
betweenthefluxq andh.
ThecomputerprogramusedinthisstudyiswritteninFORTRAN.Thatpartofthe
computermodeldealingwithsaturatedflowwasdevelopedandwrittenbyvandenAkker
(1972).Itusesafiniteelementmethod,basedonthevariationalprinciple (Zienkiewicz,
1967)toapproachthesolutionofEqn89.Ahorizontalx,ygridisusedtodividethe
regionRintoanumberofsub-areas,theelements.Withineachelement [indicatedbya
letterinFig. 26)thetransmissivity isassumedconstant.Usingsquareelements,the
equationfornodei,j attimen+J iswrittenas (vandenAkker,1972)
Vh-j+1
• 2h-))j+1 •hSJfJ) •hoZL +<?,;-> + h Ö-' }+
where i isthemeshwidth (£=A X=Ay).ApplicationofEqn100toeachofthenodesfor
whichhhas tobecalculatedyieldsasetofequationswhichissolvedbySOR.Theoverrelaxationparameteru iscomputedaccording toanempiricalformula
I2
J
whereIandJarethenumberofnodesinxandydirection,respectively.Thetotalnumberofiterationsiscontrolledby themaximumlocaldifferenceinthecalculatedvalue
forhbetweentwosuccessive iterationcycles.Ifthisdifferenceislessthantheerror
criteriume forwhichavalue ischosenatthebeginningofthec a l c u l a t e (xnthis
studye=0.01 an),convergencehasoccurred.Whentestingthecomputerprogramconvergenceproblemswereencounteredresultingfromthediscontinuityintherelation
q„
79
(Fig. 25). The problem arose for the s i t u a t i o n s
r
„,
%<c^<%
r-1
(102a)
and
r
„,
r-i
(102b)
Thefollowingsolutiontotheconvergenceproblemhasbeenadopted.IfoneofthesituationsgivenbyEqn102occurs,Eqn 100isrecalculated fortheconcerningnodewithq*~!
setequaltoq£.Thetworelationsbetween a andhnowapplyingtoq*-1 aretrieduntil
%and % b o t h correspondtoonlyoneofthese.Oncethecomputerprogramproved tobe
internallyconsistent,convergentandnumerically correct,itwas appliedtoanactual
field-sizesaturated-unsaturatedflowproblem.
6.1.1
Selected study area
Thestudyareaselected forsimulationbythesub-surface flowmodelislocatedin
theeastoftheNetherlands aroundthepumpingsite''tKlooster'(Fig.27)nearHengelo
(Gld.)Theareaconsidered forsimulationis6x6km 2 andisdescribedbymeansofa
squaregridwithameshwidthof500m(Fig. 28).Thepumping stationissituated exactly
inthemiddle.Twosmallintermittent streams areschematizedtofollowthenodes.Most
oftheareaisoccupiedbyfarmland (Fig.29),andgrassistheprincipal crop grown
(70*).Theclimateishumidwithmoderatetemperatures.Themeanannualrainfalland
évapotranspirationareabout75cmand45cm,respectively.The regionisgeohydrologicallycharacterizedbyathickcoarsesandyaquifer,overlyingamoreorlessimpermeablelayeroffinesiltysandatadepthofabout35m,andcoveredontopbyafew
metresofaeolianloamysand.Thesurfaceelevationtakenfromadetailed topographical
mapshowsadifferencebetweenthehighestandlowestgridpointintheareaofonly7m.
r
r
twenthe
7,Study area
"\
fvarssel
ƒ
*-v
tinchem » )
winterswijk f
Fig.27.Locationofthestudyarea.
80
Linde»» Laak
ÔostenrijkseVloed
Fig. 28.Grid configuration forthestudy area and the s o m a t i z a t i o n ofboth streams.
x
pumping site,— stream.
grid point
• pumping station
- i — i — i — i — r
—\ stream
| |grass
f53 coniferous forest
Hffl urban area
^ 3 cereals
n *aize
{ ^ potatoes
-I
1 I
I L
Fig. 29.Land use for the growing season
of 1973inthe study area.
Water-table elevations were recorded twiceamonthin28
^
^
f
»
^
^
^
Fig. 30.The deptho£thewater table rangedinthe period considered for s ^ l a t x o n
between zero and4.5mbelow soil surface.
81
1
—k>
+
*
+•
+
+
+
+
+
+
+
+
+
24 •
+
+
+
+
+
+
+
'.5+
+ +
+ +
f
+<J +
7
'
+
+
+
+
26
t-12 +
+
»
2 5
. +
3* •>
+
19
+2 + * +
+
+18 -t.
+
•
20
\
+ + \ '+ +
\
S
+ + + V—H
6.1.2
Saturated
+i
I
e
13»
14 •
+ + +
+
+
+ + +
+
.+
-k + + + +
+
16
•
+ •
+
Fig. 30.Locationofobservationwells inthestudyarea.
flow
IntheeasternpartoftheNetherlandswherethestudyareaislocated (Achterhoek)
rather«tensivegeohydrologicalinvestigationswerecarriedoutinthepast (e.g.Ernst
etal., 1970).Withinaradiusof6kmaroundthepumpingsite 'tKlooster,theresults
of13boringsareavailable.Fromtheboringdescriptionsandgrain-sizedatatransmissivityvalueswereestimated.Thesevaluesaresupportedbyafieldpumping test
carriedoutm 1964bytheInstituteforLandandWaterManagementResearchinWageningen,
theNetherlands.Thetestwasheldonthepumpingsite 'tKloosterbeforethestation
cameintooperation.Basedonresultsfromtheseinvestigationsatransmissivitymapof
? Î o O S r L o o e m C l C O m P i l e d CFig'31)'T r a n S m i S S l v i t yValuesusedinthemodelrangefrom
Duetotheflattopographyofthestudyareathereisnosurfacerunoff,unlessthe
soiliscompletelysaturated.Thesub-surfacedischarge intothedrainagesystemisrelativelysmall.Frominvestigations (Colenbrander,1970)inanearbyexperimentalbasin
Fi
Fig.31.Contours of transmissivity values
(m^-d~')inthestudvarea
82
0,
<1o(cm-cT1)
0.1 0.2 0.3
1 J
L^-h«
Fig. 32.Linearized empirical relationbetweenthewater-table elevationh (•h s -w)
and thegroundwater dischargeq givenby
Eqn 103used inthestudyarea.
(Leerinkbeekarea)alinearrelationwasderivedbetweenthedischarge intothesurface
watersystemq andthewater-tabledepthw (Fig- 32).Sincew =h g -h,thegroundwater
dischargemaybe formulated as
' 0.2
nq
o
= • -0.0013(h g - h) + 0.2
0
for
h -h <0
s
(103a)
for
0 <h -h <150
(103b)
for
(103c)
h -h >150
s
TherelationgivenbyEqn 103applies foreverynode.MorerecentlyErnst (1978)showed
thatfortheeasternpartoftheNetherlands an (approximate)exponentional relation
appliesbetweenq Q andw. Iftherequireddataforthisrelationcanbeobtained it
couldbeused instead ofEqn 103withoutappreciabledifficulties.However,itistobe
expected thatthesimulationresultsarenotnoticeablyaffectedasthetotalopenwater
discharge from themodelareaandthusthegroundwaterdischargeintotheopenwater
coursesarerelativelyunimportant.
For thedischarge ofgroundwater intotheopenwatercourses,schematized innearby
nodes,thedrainageresistance resulting fromasiltlayerattheriverbedandconvergenceofstream linescanbe takenintoaccount (deLaat&Awater, 1978).However,
forthestudy areathisapproachwasnotconsiderednecessary,asthetworiversare
verysmallandonlycarrywaterduringwetperiods (mostlyinwinter time).Forthenodes
inwhichbothstreamswereschematized,thephreatic levelwastakenequaltotheobservedopenwaterlevel.
Dirichlet conditionsapplyforallnodesatthemodelboundary.Theprescribed
phreatic levels foreachsuccessive timestepwerederived fromsixobservationwells
justoutsidethemodelarea.
.
Groundwaterwithdrawal fromwells isrestrictedtothepimpingsitelocatedinthe
centre.Theextractedwaterisalmostentirelyusedfordomesticsupplyoutside the
studyarea.ExtractionratesQ areavailable inn^.d"'.Thefluxq e isobtained foreach
timestep from
83
t+At
I Q(t)
_t
At
6.1.3
Unsaturated
100
,-1
500-500 cm-d
(104)
flow
SoilphysicaldatawerecollectedbytheSoilSurvey Institute,Wageningen,the
Netherlands,in1973(vanHoistetal., 1974).Mostimportantsoilsinthestudyarea
arepodzolsoils (about 501),sandyhydro-earthsoils (about201)andveryoldarable
fields,knownas'Enk'earthsoils (about 201).At,orneareachnodethesoilprofile
wasdescribedfromborings.Thedepthoftheboringswas30cmbelowthelowestwatertablebutnotdeeperthan200cmbelowsoilsurface.Theupper layerofthesoilcontaining 80%oftherootswastakenastheroot zone.ValuesofD r rangedbetween20and
100cmandwererounded (forcomputationalreasons)todecimetres.Basedonthedescribed
texturetheboringswerecomparedwithalargeseriesofsoilprofilesofwhich e(p)and
K(p)relationsareavailable.Thiscomparisonresultedinelevendifferentsoilmoisture
characteristicstobedistinguishedfortherootzone.Forthesubsoiltendifferent
pF-curvesandthreeK(p)relationswereused.Therootzonewastakenashomogeneous.A
typicalsoilmoisturecharacteristic,usedinabout251ofthenodes,isgiveninFig. 33a,
Curvea.Withregardtotheircapillaryproperties,theselectedK(p)relationsmaybe
characterizedaspoor,mediumandgood (Fig.33c).ForeachnodeoneoftheseK(p)relationswasusedtocomputethepressureprofiles.Theseprofileswerecombinedwithtwo
differentsoilmoisturecharacteristicstoobtainthesaturationdeficitcurves.Forthe
upper50cmofthesubsoiloneofthetenselectedPF-curveswasused.Figure33a,Curveb,
showsatypicalsoilmoisturecharacteristicappliedfortheupperpartofthesubsoilin
about45*ofthenodes.Atgreaterdepthone6(p)relationwasusedfortheentirearea.
Thisrelation (Fig.33b)wasactuallymeasuredinthefieldatadepthbetween1.5and2m.
p(mbar)
p(mbar)
-r
J
—r
0.1 0.2 0.3 0.4 0.5
i
e
Fig.33.Soilphysicaldatausedinthestudyarea.
(a)Typicalsoilmoisturecharacteristicsusedfortherootzone (Curvea)andtheupper
partofthesubsoil (Curveb ) .
(b)Soilmoisturecharacteristic usedforthelowerpartofthesubsoil.
84
Themeasurementswerecarriedoutin1977toobtainarelationbetweenKande.Fromthis
relationtheparameterswerederivedforthelowerboundarysolution.TheK(6)relation
wasestablishedbyBouma (1977)usingthecrusttestincombinationwithaninstantaneous
profilemethod (Aryaetal., 1975). Investigationsatdifferentlocationsdonotjustify
avariationintheK(6)relationwithinthestudyarea.Asvaluesforq duringthesummer
periodrangefrom-0.01to-0.05on-d" thecorrespondingrangeoftheK(e)relationis
usedtoderivetheparametersoftheu(q)relation (84)asshowninFig.33d,yielding
A=-0.01andB=-0.06.
6.1.4 Surface flux
Intheabsenceofirrigationinthestudyareaandneglectingsurfacerunoff,the
surfacefluxattime n+\ followsfrom
(105)
E n+i _p n+f
„n+i
re
where E £ * istherealoractualévapotranspirationfluxandP n+ *istheprecipitation
flux,bothtakenasanaverageoverthetimeincrementAt.
Precipitationwasassumedtobeuniformlydistributed.Dailyrainfalldatawere
obtainedfromthreedifferentgaugingstationsoutsideandinthestudyarea (Fxg.27):
Doetinchem (1x),Kervel (1*)andVarssel (2*).Thevaluebetweenbracketsindicatesthe
Ô
—1
q w (cnvcH)0.19 Q21 Q23(X250.270.290.310.33Kfcrrrd )
KCcm-cT1)
1
2
3
4
-10°-10 -10 -10 -10 -10°
p(mbar)
(c)K(p)relationsusedforthesubsoil.
(d)ThemeasuredK(9)relationandthedenvea
i
T
T
0.14 0.120.10O08 0.060.040.02 0
Mq
r e l a t i o n ( b r o ken line)betweenu q andqtf.
85
weightfactorusedforthecalculationofP.Duringtheperiodconsideredprecipitation
wasintheformofrain.Inviewoftheflattopographyofthearea,thelowintensities
oftherainfallandthehighpermeabilityofthesoil,surfacerunoffwasnotconsidered
exceptwhentherootzonewasfullysaturated (S < 0). Thenthewaterremainingonthe
surfacewasassumedtorunoffoverlandduringthesametimeincrement.
Evapotranspirationrateswerecomputedforeachnodeindividually.Neglectingthe
storageofheatinthesoil,theformulaofPenman(1948)forthecalculationofevaporationofawetsurfaceE tmaybewrittenas
wet
E
wet=
'
sR
- 2 + YE
— —
(106)
S +y
wheresistheslopeofthetemperature-saturatedvapourpressurecurve,R isthenet
radiation,Listhelatentheatofvaporization,yisthepsychrometricconstantandE
istheaerodynamicevaporation.Usingturbulenttransporttheories,theoriginalempiricalexpressionforE a ,proposedbyPenman,waslaterimproved,toincludethegeometry
oftheevaporatingsurface (seeFeddes,1971)
_ ep a (es-e a )
a "p
r —
*a
a
whereeistheratioofmolecularweightofwatervapouranddryair,p isthedensity
oftheair,P g istheatmosphericpressure,e g isthesaturatedvapourpressureforthe
airtemperatureat2mheight,e aistheactualvapourpressureat2mheightandr ais
thediffusionresistancetowatervapourintheair.Valuesforr ainrelationtocrop
heightandwindvelocityweretabulatedbyFeddes (1971).Standardmeteorologicaldata
wereusedtocalculateE w e t fromEqn106.Theycomprisewindvelocity,relativehumidity,
temperatureandrelativesunshineduration.Thedailyvalues (24hoursmeans)wereprovidedbytheRoyalDutchMeteorologicalInstituteandobtainedfromthefollowingstastions:Almen,Diepenveen,TwentheandWinterswijk (seeFig.27).
F
(107)
Takingintoaccountthediffusionresistancergofbothcropandsoilandneglecting
evaporationofinterceptedwater,therealévapotranspirationE ofacroppedsurface
withlimitedwatersupplymaybewrittenas (Monteith,1965;Rijtema,1965)
E
re= s+Y(1V r /r)
(108)
sa
AfterRijtema (1965),thediffusionresistancer gisexpressedas
r Br +S
s c c<ri*V
wherer cisthediffusionresistancedependingonthefractionofsoilcovered,r is'the
resistancedependingonlightintensity,r istheresistancedependingonsoilmoisture
conditionsandflowintheplantwhileS cisthefractionofthesoilcoveredbythecrop.
86
O09)
Foracropwithamplewatersupplyr =0anditfollowsforthepotentialévapotranspiration
S +v
pot= s+Y(1 + Cr +SrJ/r ) E wet
c
ct
f 110 )
a.'
TheexpressionproposedbyRijtema (1965)fortheresistancer ofthesoilplantsystem
andvaluesforr candr asfunctionsofS andmeanshort-waveradiation,respectively,
canbefoundfromvanBakel (1979).
Thelinkingofthemodelsforévapotranspirationandunsaturatedflowrequiresthat
—2 -1
—1
évapotranspirationrates,hereexpressedinkg-m -s areconvertedtoan»d .Thereal
évapotranspirationE dependsonsoilmoistureconditionsthroughtheresistancer
whileunsaturatedflowdependsonE throughtheupperboundaryfluxq s .Thereforefew
iterationsofthecalculationofbothmodelsarenecessarytosolveq n+ iandE n + '.
Therehasbeenlittlechangeinthecroppingpattern(Fig.29)duringtheyears
consideredforsimulation.Thesmallurbanareaintheregionistreatedbythemodelas
ifitweregrass.
6.1.5 Simulation
results
Theabilityofthemodeltocorrectlysimulatewater-tableelevationsforanactual
field-sizesub-surfaceflowproblemwastestedinthestudyareaoveratimeperiodof
almost6years.ThesimulationstartedatthebeginningofApril1971andendedinDecember1976,usingatimeincrementof10days.Ascomparedwithaverageweatherconditions,
thesummerof1972wasextremelywetandthegrowingseason (theperiodfromAprilto
September)oftheyears 1971,1973and1975wasdry.Extremelydrywastheyear1976,
whilethewinterof1974/1975wasverywet.
Theinitialsteady-statesituationwascalculatedseveraltimesfordifferentpercolationratesintheunsaturatedregion.Forq^ (=q)=-0.1cm-d-1 calculatedphreatic
levelscomparedfavourablywithobservedwater-tableelevationsattheonsetofthesimulationperiod.Theyears1971,1972and1973weresimulatedseveraltimesduringthe
developmentofthemodel.Resultsofearliermodelversionsarepublishedelsewhere
(deLaatetal.,1975;deLaat&vandenAkker, 1976).Forthecalibrationofthepresent
modelthegrowingseasonof1971wasused.Calibrationwasnecessarytoestimatethe
hysteresisfactorusedfortherootzoneandtotesttheempiricalrelation(103)between
q Qandw.Thetestrunsdidnotgivereasonstoaltertheq Q (w)relationadoptedoriginally.Furthermoreitappearedfromtestingdifferenthysteresisfactors (0,0.5and1.0)
thatavalueof0.5wasmostsuitable.
Computedwater-tableelevationswereinterpolatedintimeandspacetobecompared
withobservedvaluesinthe28wellsshowninFig.30.Fromthedifferencebetweenthe
measuredandsimulatedwater-tableelevationAh(cm), theaverageÄEandtheaverage
absolutedifference Jtàï\ arecalculatedforthetotalnumberofobservations.Valuesfor
un",]ÄhTandthestandarddeviation aofAharepresentedinTable3a.Inhydrologyan
efficiencyfactorRîsoftenusedforthecomparisonofrainfall-runoffmodels.The
efficiencyfactormaybedefinedas(Nash&Sutcliffe,1970)
87
Table3. (a)Comparisonofobserved andsimulatedwater-table elevations, (b)Idem,with
TO settozero.
b
a
WellNo.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Ah
|âh|
1
2
15
-4
-5
-2
-9
2
12
19
10
9
-8
-4
28
25
0
0
7
-10
6
14
16
-48
10
12
16
7
6
7
16
7
9
9
13
10
13
20
13
10
13
11
28
25
11
8
12
11
9
16
19
50
11
13
17
11
0
h
8
10
12
8
8
10
12
14
10
11
11
8
12
12
12
12
15
10
17
6
10
12
15
17
9
11
12
11
0.97
0.97
0.87
0.97
0.96
0.96
0.92
0.94
0.92
0.79
0.90
0.95
0.92
0.94
0.71
0.76
0.87
0.96
0.78
0.95
0.94
0.82
0.83
0.28
0.94
0.91
0.86
0.91
WellNo.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
|Ah|
6
8
9
6
7
8
9
10
7
8
8
6
10
10
10
10
11
8
11
5
7
9
8
10
7
7
8
8
h
0.97
0.97
0.95
0.98
0.98
0.97
0.95
0.94
0.97
0.95
0.95
0.97
0.95
0.95
0.95
0.95
0.87
0.96
0.81
0.99
0.96
0.92
0.92
0.92
0.97
0.96
0.95
0.93
Fig.34.Simulated water-table
elevationcontoursand observed
valuesat theend ofAugust 1973
inthestudy area.Thequantities
areexpressed incm.
88
00 0) •
•rt J3 O
2
o—o
o
Ä
—
O O <0 V CN
O O Ö O O
O 00 <0 ^- CM
i -o
o
o.
UJ
E
o
£ co
o
m
f
90
o
o
V
o
10
o
a>
3
a
r-l
>
•a
X>
o
H
a
c*
oo
•f-i
Pu
91
R -1
SO7-F M 2
whereFrepresentsthemeasuredwater-tableelevations,F'thesimulatedvaluesandFthe
meanoftheobserveddata..IfsimulatedandobserveddatafullyagreeR^=1,whileIL=0
ifthesimulatedvaluesequalthemeanoftheobservedvalues.Theefficiencyfactorfor
eachoftheobservationwellsispresentedinTable3a.ThevalueforAhislargelygovernedbythedifferencebetweenactualandmodelsurfaceelevationattheobservation
well.InordertoeliminatetheeffectofÂÏÏonthecomparisonofthesimulatedandobservedfluctuationofthephreaticlevel,valuesfor |^h|andR werecomputedforobserveddatawhichwere'corrected'forAh(Table3b).
Simulatedwater-tableelevationcontoursandobservedvaluesattheendofAugust
1973areshowninFig. 34.Toillustratethegoodnessoffit, observedandsimulated
water-tableelevationsforobservationwellNo.12areplottedinFig.35.Alsogivenin
thesamefigurearethegroundwaterextractionrates,precipitationdata,calculated
potentialandrealévapotranspirationratesandtheresultingactualfluxacrossthesoil
surface.Simulatedwater-tableelevationsinfourobservationwells (NosS,14,21and
28)arecomparedinFig. 36withobservedvalues,whichare 'corrected'forÂE.
o b t a i ^ "I thSnatUTe° f*"m0del f ° r«»«toted flow,leastaccurateresultsare
S
WlthC a P i U a r yrlSe1Sf0ll0Wed
occlrstT?
*y « * * * 1 excess.Thissituation
b y ! re!
'**gr0Wlng S e a S ° n ° £1972-* - attempttoimprovetheresult
At
affeclZ
,'"aPPeared that redUCing* » l e n S t hofthetimeincrementdoesnot
Wat6r tableeleVati0n
ratedfitT
"
significantly.Theeffectofnon-steadysatureSUltlng £ r ma
Variati0nin
inL3T TTZT
° ^
»heextractionrateisshown
Figs35and36bythecalculatedphreaticlevelinthesummerof1976.
6.2 SENSITIVITYANALYSIS
Parti T a n X T ^ ^ 1 ^ ^ ^ ^
"* ** " ^ *
^
W
n d e nly
C0VerstheentirCSUU
lationperiod(vanBakel lQ7<n 4
"^ ° ° **
"
a
thequasithree-dimensional
PP roxima terelationbetween q^ andh,derivedfrom
oftheunsaturatedf l ^ Z d ^ T ^ T e S U l t S ' ^ ^ **thel0Werb0Undai7 C ° n d i t i ° n
1S
effectoncalculatedrealan/
"'t h e r e £ o r e > limitedprimarilytothe
Anotherpartoft Lse
^ ^ ^ V ™ ™ »tes.
butonlyfortheyears1 9 7 3 ^ 0 7 !T ^ * COnsidersflowintheentirestudyarea,
usedthesameperiodtostudy H ^ T
I "^ ^
**"" &" *"**
^
caldataonthesimulationresultsT
G è r e n t W S ofcollectingsoilphys
simulatingregional saturad
investigationsareimportant,asthecostsof
detailtowhichsoiiD h v J Ü Ï' unsatura tedflowproblemslargelydependonthedegreeof
Inthisstudythesenî ata" ^*^ measuredvaluesofsevenparameterT^ 1 * 7 ^-^ r6SUltS° fsimulationt0avariationinthe
reasons.Thehysteresisf a c ^lnVeStlgated-TheParameterswereselectedfordifferent
intherelationsK(p)a n d ^ M ^ C h ° S e n**U SValUehadt0beestimated.Variations
V Jwereanalyzedasnoactualmeasurementsoftheserela-
(1979)
r S i-
of
tions were conducted in the study area. The transmissivity, the prescribed phreatic
levels for the nodes at the boundary and the relation y q (<g were chosen because comprehensive data were not available. Finally the depth of the 'effective root zone' was
included in the analysis as i t s value is not well defined. The parameter Dr results from
the somatization of the unsaturated region into a root zone and a subsoil. In the root
zoneupward flow is governed by the water uptake of the roots and moisture is available
for the crop until P F 4.2 applies over the entire depth. The root zone may, therefore, be
considered as a reservoir, the size of which depends on V Although rooting depths were
extensively measured in the study area, the effective rooting depth D r , which is assumed
to comprise 801 of the roots, had to be estimated.
,
The sensitivity analysis for the seven parameters is restricted to results o m n d
for a period of one year (37 time increments of 10 days each) starting at the b ginning
of April 1973. First a run of the model was made with the parameters set equa to the
values used for the six-year simulation period. This run was then repeated witf> n o ^ n g
changed except the value of the parameter under consideration. The effec of parameter
variation was investigated for the simulated water-table elevations and ^ e j a ^ l a t e
•
-™ „f thP sensitivity may be obtained by
real évapotranspirations. A global impression of the sensitiv y j
houndarv
p a r i n g average values. To reduce the effect of the prescribed levels « * £ £ £ >
s e a t e d w a t J t a b l e elevations and real évapotranspiration values were ave^ge d o v r
the interior of the model area. The interior comprises 49 nodes located in the
a distance of more than 1000 mfrom the model boundary.
original and
Average water-table elevations resulting from ^ ^ ^ ^ ^ ^ f f e c t of a
changed parameter value were plotted. Figure 37 shows, as an ******' ^ ^
^ ^ ^
variation in the hysteresis factor on the average water-table
A'MVJ'A'S'O'N'D
1973
J'F M
1974
« «i«=u- A l l A T *
8 . 37.Comparison of calculated w a t e r " " D ^ r .table
" i r f a1
cC
tT
o r fromits
"eaforsensitivity toachange inthe hysteresis
Fl
0,5
H tozero( ).
93
Table4.Summaryofsensitivityanalysisresults.
Parameter
Change
Effectofchanging parametervalue on
water-table elevation
realévapotranspiration
Hysteresis
factor
Settozero
At thebeginning of the
secondhalfyear 12cm
lower,thereafter 2cm
higher.
Overestimated by0.3cm,
but locallymorethan
1cm.
Hydraulic
conductivity
relationK(p)
'good'->•'medium'
'medium'-*'poor'
(Fig. 33c)
Inthefirsthalfyear
2cmhigher.Inthe
secondhalfyear,at
firstmore than 10cm
higher,laterdecreasing
tonochange.
Underestimated by 1.8cm,
but locallymorethan
5 cm.
Groundwater
discharge
relationq(h)
SeeFig.38
Varying from20cm lower
forthehighest levels
to2cmlowerfor the
lowestlevels.
Underestimated by0.6cm,
but locallymorethan
4 cm.
Transmissivity
T
Increasedby25%
High levels2cmlower.
Localeffect (except
forthewell site)
rangesfrom+3to-8cm.
Underestimated by0.1cm.
Prescribed
phreaticlevels
attheboundary
Raisedby5cm
Highlevels2cmand low
levels4cmhigher.
Overestimated by0.1cm,
but locally almost 1cm.
Storage
coefficient
relationu (q)
SeeFig.39
Up to 10cmhigher in
summerand 10cmlower
inwinter.
Underestimated by0.4cm
(at somelocationsby
Icmbutalsooverestimated by 1 cm).
Depthofroot
zoneD
Decreasedby 10cm Inthesecondhalfyear
(whenthewater tableis
rising) 10to 15cm
higher.
Underestimated byalmost
2.8 cm. (Thelocaleffect
ranges from2to5cm.)
ofthestudyarea.Theresult*fm- „ n
SCVen a r a
table ai<=n „,-, *u r
P ^ t e r s are summarized in Table 4. This
umie also gives the pffpn-n-e
calculated fnr «,
Parameter variation on the total real évapotranspiration
*r ^ J I ^ : ^ : p r d ofoneyear-êffect-iiesto*• mevaiue
below.
v e r l o r . The results of the sensitivity analysis are discussed
Hysteresis factor Neelecti™Wo*
torainfallexcessatt L " d ofT ^ " ^ ******** ^ ^ ° £** **"
SUmBT
rainfallexcessduringthe< ™ !
**"**beS6en" Fig'37"SinCe" ^ °
S6aS0n
transpirationratesareM g n e T ^
"^ *"**rO0t Z ° n e '**calculated *"**
Hydraulic conductivity Thecanin»«
pr perties f
describedbythreedifferent Ir\
°
° thesubsoilinthestudyareaare
» 'Poor'appliestoonlyeLht J ^ * " 0 ^ ^ ^ K(P) " ^ ^ à""*****
snxnodes.Forthesensitivityanalysisthesenodeswere
94
table
ofthe
left unchanged. Nodes i n i t i a l l y characterized as 'good' became 'medium' and the capillary
properties of nodes i n i t i a l l y 'medium' were changed to 'poor'. The calculated real évapotranspiration proves sensitive to a variation in the K(p) relation. At places where the
water-table depth may be considered as ' c r i t i c a l ' , the calculated values are largely
reduced. As a result of the poor capillary properties, less water becomes available for
the crop due to a decrease in capillary rise and an increase of the pF value in the root
zone. Consequently, the saturation deficit at the end of the summer is smaller, resulting
in higher levels when r a i n f a l l excess causes the water table to rise.
Groundwater discharge The relation between groundwater discharge and water-table depth
was drastically changed (Fig. 38). The change represents an 'improvement' of the drainage
system affecting primarily the most shallow water-tables. The effect on water tables
deeper than 200 cm i s almost negligible. Calculated real évapotranspiration values are
lower as the drawdown of the water table hampers the process of capillary rise and reduces (at some places considerably) the amount of moisture available in the root zone at
the beginning of the growing season.
TvanemUsivity A large change in the transmissivity values has negligible effect on the
simüated water-table elevations and calculated real évapotranspiration rates.
Prescribed phreatia levels at the boundary The prescribed phreatic
^
^
^
serie
from the same data in two different ways, independent of each other. Bo
for 1973were compared for two arbitrarily selected nodes. The 951 confxdence « x t t t f
^
the average value appeared to be 4 and 2 cm, «
^
^
^
^
^
s i t U ation.
Prescribed levels were raised by 5 cm. The change also applies
Asthe water table in the study area is relatively deep (the average dePtf> to 197^ ^
the interior is 190 cm) the calculated real évapotranspiration is not very
a
change in the prescribed levels at the boundary.
Ol
0
qo(crrvd-1)
0.2 0.3
w(cm)
Fi
38.The
Toriginal
he
S8-- 38.
(-)andchanged(-).
tiv- betweenq0andhusedforthesensixt
y analysis.
Theoriginal
andand
changed
( )\
• 'rial(-)
(—)
cnangeu
J
between
*• ••"*
^ Ite.,
n T ,«*
*— <v " ~**
"~. forthesenitivityanalysis.
sitivityanalysis.
/
*£J^
9S
Storage
coefficient
Thechangeintherelationbetweenu and q^ (Fig.39)effectively •
increasesthestoragecoefficientby0.03.Asaresultofthelargeru valuethefluctuationofthewatertablein1973isreducedby20cm.Thechangehastwoeffectswhich
actinoppositedirectionsonthecalculatedrealévapotranspiration.Ontheonehand
capillaryrisebenefitsfromthehigherphreaticlevelsinthegrowingseason,whileon
theotherhandlessmoistureisavailableinthesubsoilduetoalargerdownwardflux
acrossthelowerboundaryofthemodelforunsaturated flow.
Depth of the root zone AdecreaseofD resultsinanunderestimationofE f e forthree
reasons: (i)lesswaterisavailableintherootzoneatthebeginningofthegrowing
season,(ii)lesswateriskeptintherootzoneduringperiodswithrainfallexcess,
and (iii)thesupplybycapillaryriseishamperedduetolargerz values.Thecalculatedlowerrealévapotranspirationvaluesresultinsmallersaturationdeficitsatthe
endofthesummeryieldinghigherphreaticlevelsduringthetimethewatertableis
rising.
Thesensitivitywas,apartfromtheseveninputparameters,alsoinvestigatedfor
achangeinthecalculationprocedure.TheupperboundaryconditiongivenbyEqn105
requiresaniterativesolutionofthemodelsforévapotranspirationandunsaturatedflow
tocalculateE n .InsteadofsolvingE n + * by iteration,E n + * isused tocomputethe
surfacefluxas
n+i= E n+J _p n+J
n
s
(112)
pot
Theactualsurfacefluxq r e computedbythemodelforunsaturatedflowisthenusedto
calculatetherealévapotranspirationrate
E n+J _(q rejn+i+ p n+J
(113)
Asaresultofthechangeinthemodel,thecalculatedrealévapotranspirationrateequals
itspotentialvalueuntilthepressureintherootzonereacheswiltingpoint,because
q " =q g forpFvalues lessthan4.2.TheuseofEqn112insteadofEqn105andthe
calculationofE r ewithEqn113ratherthanbyiterationdidnothaveanyeffectonthe
simulatedwater-tableelevations.Thecalculatedrealévapotranspirationfortheinterior
ofthemodelareawasoverestimatedbyonly0.1 cm,butafter130dayswhenpF4.2 is
reachedinmostpartsoftheregionby0.5 cm.Localeffects largelydependonthetype
oflanduse.Potatoesappearedtobeverysensitivewhilethecalculatedrealévapotranspirationofgrasswashardlyaffected.
Theresultsofthesensitivityanalysismaybesummarizedasfollows.Thefluctuationofthesimulatedwater-tableelevationdependslargelyontherelationbetweenu
andq wwhichisderivedfromtheK(e)relationapplyingtothelowerpartofthesubsoil.
Theaveragewater-tableheightinsummerispredominantlygovernedbytheprescribed
phreaticlevelsattheboundary,whileinwintertheempiricalrelation (103)between
96
q andhappearstoprevail.
Thecalculatedrealévapotranspirationissensitivetothewater-tableelevationat
thebeginningofthegrowingseason,thehydraulicconductivityrelationK(p)andthe
depthoftheeffectiverootzoneD .ThesensitivitytotheparameterD r ismostpronouncedasitdirectlyaffectstheamountofwateravailableforthecrop.A similar
sensitivitywasnoticedbyFeddesetal. (1978)usingasinktermfunctiontodescribe
wateruptakebyroots.Theyreportedthatarelativelysmallchangeinthesinkterm
functionaffectsthesystem.
A finalrunofthemodelshowedthatthedifferencebetweentherealandpotential
évapotranspirationofgrassisnotgovernedbythediffusionresistancer p (seeSection
6.1.4)butresultsfromadeficiencyofavailablewaterintherootzone.
Whenevaluatingthesensitivityanalysis,itshouldberealizedthattheresults
applyforoneparticularsituation.Inanotherperiodorregionforwhichconditions
differsignificantly fromthoseinthestudyareain1973,theforegoingconclusionsmay
notbeapplicable.
6.3 CONSEQUENCESOFGROUNDWATEREXTRACTION
Themodelhasbeenusedinthestudyareatopredictconsequencesoftheimplemented
groundwaterextraction.Sincethemodelwasverifiedonlywithrespecttowater-table
elevations,intheabsenceofotherpossibilities,aninvestigationoftheseconsequences
shouldberestrictedtothepredictionofthedrawdownofthephreaticlevel.Neverthelesstentativeconclusionswillbedrawnwithrespecttootherhydrologicalconsequences
forthefollowingreason.Assumingthatthegeohydrologicaldata,thegroundwaterextractionratesandtheprescribedphreaticlevelsattheboundaryarecorrect,thesimulatedwater-tableelevationisgovernedby % andq Q .Becausethedischargeq oisvery
small,inparticularduringthelast M yearofthesimulatedtimeperiod,thefluxq„is
correctlysimulatedconsideringtheexcellentagreementofcomputedandobservedwatertableelevations.Therechargeofthesaturatedzonefromtheoverlyingunsaturatedregion
depends (inparticularattheendofthegrowingseason)verymuchonthesaturation
deficitandthusthesurfacefluxv
Assumingthatrainfallrateswereaccuratelymeasured
therealévapotranspirationduringthegrowingseasonmusthavebeenapproachedproperly.
Potentialévapotranspirationratesareindependentofsoilmoistureconditionsandverifiedforlysimeterexperiments (Rijtema,1965)andfieldexperiments (Feddes,1971).
Thesimulationmodelwasappliedinthestudyareainexactlythesamewayasfor
verificationbutwithoutgroundwaterextractionfromthewellsinthecentre.A condition
forthisapplicationisthattheeffectofthechangeintheactualsituationonthe
boundaryconditionsiseithernegligibleorpredictable.Withregardtothestudyarea,
theboundaryofthemodelwaschosenatsuchadistancefromthepumpingsitethatthe
prescribedphreaticlevelsarenotappreciablyaffectedbytheimplementedextraction,
whiletheeffectonthewaterlevelsinbothstreamsisassumedinsignificant
Thewaterbalancesresultingfromsimulationoftheactualsituationandthe s t ationwithoutextractionarepresentedinTable5.Thequantitiesare« * » » ^
»
andrefertoaperiodofapproximatelyoneyear (exceptfor1976)startingatthebegin
97
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99
Fig. 40.Contours for thesimulated situationin
thestudyareaattheendofAugust 1973.
(a)Differenceinwater-table elevation (cm)for
thesituationswithandwithout the implemented
extraction.
(b)Relativeévapotranspiration (%) contours.
(c)Difference inrelative évapotranspirationbetweenthesituationswithandwithout extraction.
ningofApril.Thewaterbalances forthesummerhalfyear (170days)aregivenin
Table6.Thistableshowsthatthetotalamountofgroundwater,leavingtheregionduring
thegrowingseasonassurfacewater ('Surfacewaterdischarge')isrelativelysmall.Most
oftherainfallexcessinthestudyareaisdischargedacrossthemodelboundaryas
groundwater ('Groundwaterdischarge').
Thedifferenceinthecalculatedwater-tableelevationforthesituationswithand
withoutgroundwaterextractionattheendofAugust 1973isshowninFig.40a.Asa
resultofthedrawdownlesswaterbecomesavailableforthecropbycapillaryrise,which
mayresultinareductionoftheévapotranspiration.Mostofthereductions occurduring
thesummerhalfyear.Therelativecontributiontothesupplyoftheimplemented groundwaterextractionofeachofthetermsofthewaterbalanceduringthesummerhalfyear
ispresentedinFig.41.Theresultsshowthatasimplerelationbetweenreductionof
évapotranspirationduetogroundwaterextractionandtheprevailingclimatologicalconditionsduringthegrowingseasondoesnotexist.Otherimportantfactorsmustbeconsidered,suchaswater-tabledepthandsoilmoistureconditionsatthebeginningofthe
growingseason (whichareveryfavourablein1975)andthedistributionofprecipitation
overtheseason.
Ifcropproductionisnotrestrictedbywatersupply,thetotalactualévapotranspirationattheendofthegrowingseasonSE equalsthetotalpotentialévapotranspirationIE . Theproductioncapacityofthecropisoftenexpressedintermsof
100
75
55
25
231
1 18
H
M
sw
1972 (wet)
69
1971 (dry)
14
1974 (normal)
1973 (dry)
m Evapotranspiration
Wfà Surface water
I
73
IGroundwater
K 3 Storage change
1976 (verydry)
ixxyi
! i 8 ; Al. Relative c o n t r i b u t i o n (%) of the different sources to groun
d
urini8 the growing season (170 days).
dwater extraction
101
3
O
J3
•O
a
ta
J,
tt)
3
u
01
J5
O
o
4J
•
rt c
01 o
M -H
. 0 -U
O. O
<0
T) H
Cl W
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rH
U
2
to M
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<o« 1333
O
O H
to 00
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o ai
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ai
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r-i
a.
-* • eH
öS CI
•ri .c
t» u
102
relativeévapotranspiration (Feddes&vanWijk, 1976),definedas (ZE /iE
)-100%.
Relativeévapotranspirationcalculatedfromsimulationresultsforthesituationwithout
extractionispresented inFig.40b.Reductions inrelativeévapotranspirationandconsequentlyincropproductionarelikelytooccurinareasshowinghighévapotranspiration
ratesandsituatednottoofarfromthepumpingsiteasmaybeseenfromFigs40band
40c.
Theeffectofgroundwaterextractiononthecalculatedphreatic levelinwellNo.12
isshowninFig.42.
Applications ofthequasithree-dimensional approachtosaturated-unsaturated flow
asdescribed inthisstudywerereported forthefollowingregions.
2
-Leerinkbeek area (141 km ) . DeLaat&vandenAkker (1976)studiedconsequences of
groundwaterextractiononthewater-table elevationandcropproductionforathree-year
period.
-Dinxperlo area (57.5km ) . Awater (1976)investigatedpossibilities toreducethedrawdownresulting frompumpagebymeansofsurfacewaterinfiltration.Thesimulationperiod
covered 4Jyears.
-Glindhorst area (156km 2 ). The 'WerkgroepWateronttrekking GelderseVallei' (1979)
studiedhydrological consequences fordifferentgroundwater extractionratesforathreeyearperiod.
-Achterhoek area (701 km 2 ).Awater&deLaat (1979)investigated forathree-yearperiod
theeffectofsprinkling anddifferentextractionpatternsonthewater-table elevation
andrealévapotranspiration.
Foreachoftheabovementioned applications thelengthofthetimestepusedwas
tendaysandthemeshwidthofthetwo-dimensionalhorizontalgrid 1000m,exceptforthe
Dinxperlo areawhere thedistancebetweenthenodeswas 500m.
103
Summary
Themostimportantdrivingforcesfortransportofwaterinsoilaredifferencesin
elevationandpressure.Theseforcesareusuallycombinedintothehydraulicpotential
gradient.Darcy'slawrelatesthisgradienttothefluxdensityorspecificdischarge.
TheproportionalityfactorofbothquantitiesisthehydraulicconductivityK.CombinationofDarcy'slawandtheprincipleofcontinuityleadstoageneralequation (13)with
twodependentvariables (6and p ) . Inthisstudythegeneralityisrestricted toisothermalflowofanincompressiblehomogeneous liquidinanisotropicrigid soil.
Theparticularformsofthegeneralequationapplyingtosimplified flowproblems
areessentiallydifferentforasituationofcompletesaturationandforapartlysaturatedflowsystem.Forcompletesaturationthenumberofdependentvariablesreducesto
one,andthehydraulicconductivityisafunctionoftheindependentvariablesalone.The
solutionofdifferentialequationsgoverningflowinunsaturatedporousmediarequires
thesoilmoisturecharacteristic,therelationbetween 6andp,tobe specified.Moreover,
thehydraulicconductivityisafunctionofeorp.Sincebothempiricalrelations {e(p)
andK(p)orK(e)}aredifficulttomeasureandsubjecttohysteresis,solutionsofpartialdifferentialequationsgoverningsaturatedflowareoftenmoreeasilyobtainedthan
ofthosegoverningunsaturated flow.
Thealgebraicformulationoftheflowproblemresultsinanabstract simulation
system,ormathematicalmodel.Realsimulationsystemscomprisephysicalandanalogue
models.Aviscousfluidanaloguemodelforsimulatingverticalunsaturated flowwas
developedbyWind (1972)andaspecialpurposeelectricalanaloguebyWind &Mazee (1979).
Themostversatilemodelsforsaturatedflowaretheresistance-capacitanceanalogues.
Althoughthesedirectsimulationmethodsarecapableofsolvingcomplexflowproblems,
mathematicalmodelsare,duetorecentadvancesinthefieldofcomputer technology,
consideredsuperiorinmanyways.
Thenumericalsolutionofthegoverningpartialdifferentialequationsmaybeobtainedbyfiniteelementorfinitedifferencemethods.Theuseoffiniteelementmethods
isadvantageousiftheflowdomainistobedescribedbyanirregulargridorwhencomplicatedsaturated-unsaturated flowproblemsaretobesolved.Finiteelement techniques
arearecentdevelopmentinthefieldofsub-surfacehydrology.Mostoftheavailable
solutionsofgroundwaterflowproblems indeeduseafinitedifferencemethod.SomecurrentfinitedifferencetechniquesarediscussedtowardstheendofChapter2.
Forhistoricalreasonsandinviewoftheabovementioneddifferences inthenature
ofthepartialdifferentialequationsgoverningflowincompletelysaturatedandpartly
saturatedporousmedia,flowaboveandbelowthewatertablewastraditionally treated
separately.Asitisoftensufficienttoconsiderflowintheunsaturated regionverticallyand,inthesaturatedpartinhorizontaldirectiononly,theseparate approach
largelyreducesthecomplexityoftheflowproblem.However,seriousobjectionsare
raisedifwatertablesareshalloworrapidlyfluctuate,astheeffectofunsaturated
flowonthesaturatedsystemmaybeconsiderable.
Aunifiedapproachtosaturated-unsaturated flowwasfirstreportedbyRubin(1968).
InChapter3areviewisgivenofanumberofpapersusingasingleequationtomodel
flowinpartlysaturatedflowsystems.Theproblemssolvedbythisrigorousapproachdeal
withpumpingtestsandflowinshallowwater-tableaquifers.Fortheseflowproblemsthe
effectoftheunsaturatedsystemonunconfinedgroundwaterflowismostpronounced.
Theuseofasingleequationtosolvesaturated-unsaturated flowproblemsintroduces
numericaldifficulties.Thegoverningequationisparabolicintheunsaturatedzoneand
ofanellipticaltypeinthesaturatedpart,whilethepositionofthephreaticsurface
separatingbothregionsisaprioriunknown.Thenumericalsolutionrequiresasmallmesh
sizeintheregionabovethewatertableandinthevicinityofthewell,becausethe
valueofthedependentvariablemaychangedrasticallyoverashortdistance.Moreover,
thenon-linearityofthecoefficientsintheunsaturatedpartoftheflowdomainrequires
forreasonsofstabilityandconvergencethattimeisdiscretizedtosmallsteps.Dueto
thelimitedcapacityofthecorememoryofthecomputerandtheextremelyhighrunnxng
costs,applicationstoregionalproblemshavenotbeenreported.
AnalternativesolutionproposedbyPikuletal.(1974)linksRichards'equation
forverticalunsaturatedflowtotheequationofBoussinesqforhorizontal»Juniled
flow.Theefficiencyoftheresultingquasithree-dimensionalapproachforsoving»turated-unsaturatedflowproblemsdoesnotimprovesignificantly,mainlybecauseofth
timesteprestrictionforthesolutionoftheequationforunsaturatedflow,which
imposedupontheentiresystem.
rmnnt.r
Am i l forverticalunsaturatedflowbeingmoreefficientin ten* * " * £ *
costsisdevelopedinChapter4.Themodelsimulatestransientflowby s u c c ^ n of
steady-statesituations.Steadyupwardflowin
"
^
^
£
£
£
?
metreabovethewatertablewasfirstcomputedbyRichards IWIJ. V
tionofsteady-staterelationswascarriedoutbyWind
^
]
^
^
J
^
^
puteforvariouswater-tabledepthsthemaximaamountofsoil «
^
^
^
^
crop.Feitsma(1969)usedasuccessionofsteady-statesituationstosimulatethetran
sientprocessofcapillaryriseandthedrawdownofthewatertable
inthisstudythepseudosteady-stateapproachtocapilaryri i - X« *•
approximatevaluedependsonthelengthofthe* - « £ « ! £ ^ t T
2 waterrelationtothecharacteristic* - oftheunsatur«rff £ « * - ^
^
tableinasandyaquifer,thecharacterise« ,
he°
Y
^
ofthepseudosteady-stateapproachbecomeinconsisentif^ ^ »
£ o ra
ment.edissmallerthanthe*
«
^
^
£
^
^
decreasingfluxacrosstheupperboundaryandforthes
followedbypercolation.F u r « thepositiono, ^
^
£
latedproperlyifthefluxacrossthelowerb o u n t y- aJ
Inthisstudytheunsaturatedzoneextendsfromjustbelowt V
thesoilsurface.Theregioniss c h e m e d intoaroo z subo
therootzoneislargelygovernedbythewateruptakeoftheroots,
*
L
»
L
levelt 0
^
^ ^
105
hydraulicpotentialintherootzoneisassumedequaltozero.Itisshownthatthe
steady-statesituationisfullydeterminedbyonlytwoparameters (e.g.the saturation
deficitoftheroot zoneS r andthesteadyfluxinthesubsoil q ) . Theuseofsaturation
deficitsreducesthesolutionofthesteady-statesituationtoaproblemoftworelations
withtwounknowns (Sr andq ) . Thesteady-statesolutioncorresponding totheupperboundaryfluxofthesubsoilistermedupperboundarysolution.Proceduresaredevelopedto
accountfortheabovementionedinconsistencies andtotreatperiodswithrainfallexcess
followingcapillaryrise.Iftheroot zonedesiccates towiltingpoint the calculation
procedureyieldsfurthermoretheactualfluxacrossthesoilsurface.Whenthereisa
largedownwardfluxacrossthelowerboundary,theupperboundarysolutionisunsuitable
forcomputing^thewater-tabledepth.Foradownward lowerboundary fluxconditionthe
positionofthephreaticlevelisthereforesimulatedbyapseudosteady-state approach
topercolationapplyingtothelowerpartoftheunsaturated zone.Thesteady-statesolutioncorrespondingtothelowerboundaryfluxofthesubsoil istermed lowerboundary
solution.Theupperandlowerboundarysolutionsarecombinedintoonesimulationmodel,
takingintoaccounthysteresisandheterogeneity.However,withpercolation capillary
propertiesareassumedhomogeneous,sothatthemodeldoesnotallowforthe formation
ofperchedwater-tables.
A quasithree-dimensionalapproachforsimulatingtransientsub-surface flowin
shallowwater-tableaquifersisoutlinedinChapter 5.Thesolutionusesatwo-dimensionalhorizontalgridtodescribesaturatedflow.A specialprocedure isdevelopedto
linkineachnodeofthegridtheunsaturated flowmodelwiththesaturated system.
Anareaof36km intheeastoftheNetherlandswaschosenforexperimentalverificationofthequasithree-dimensionalmodel.Theareawasdescribedbya rectangular
gridwithameshsizeof500m.Thelengthofthetimeincrementusedwas tendaysand
thesimulationperiodcoveredalmostsixyears.Simulatedwater-tableelevations compared
favourablywithobservedvalues.Lessaccurateresultswereobtained inperiodswithan
alternatingévapotranspirationandrainfallexcess.Thesensitivityofthe simulated
water-tableelevationsandcalculatedrealévapotranspirationratestoavariationinthe
valueofseveralparameterswasinvestigated.Theresultsdiscussed inChapter6show
thatthecalculatedrealévapotranspirationismostsensitivetotheconceptual approach
forwateruptakebytheroots.Fortheapproachtoévapotranspirationasused inthis
study,itappearedfurthermorethatthedifferencebetweenthecalculated actualand
potentialévapotranspirationofgrassdependsontheamountofsoilmoisture available
intherootzoneratherthanupontheempiricaldiffusionresistance forsoilandcrop
(
V
Anapplicationofthemodelforsaturated-unsaturated flowisgiven.Consequences
oftheimplementedgroundwaterextractioninthestudyareaonthecalculatedphreatic
levelsandwaterbalancesarepredictedforthesameperiodusedfortheverification
ofthemodel.
106
Samenvatting
Debelangrijkste drijvendekrachtenvoordebewegingvanwaterindegrond zijnverschilleninhoogteendruk.Hetisgebruikelijkdezekrachtentecombinerenindegradientvandehydraulischepotentiaal.Hetverband tussendezegradiëntendefluxdichtheid
ofhetspecifieke debietstaatbekendalsdewetvanDarcy.Hierinisdehydraulische
doorlatendheidKdeevenredigheidsconstantevanbeidegrootheden.Combinatievandewet
vanDarcyenhetcontinuïteitsbeginsel leidttoteenalgemenestromingsvergelijking(13)
mettweeafhankelijkevariabelen (9enp ) . Indezestudieisdealgemeenheidbeperkttot
isotherme stromingvaneenonsamendrukbarehomogenevloeistofineenisotrope rigide
grond.
Vandealgemeneformulering afgeleidevergelijkingenvoorvereenvoudigdestromingsproblemenvertonenwezenlijkeverschillenvoorzoverzijbetrekkinghebbenopeengeheel
ofeengedeeltelijkverzadigd systeem.Bijvolledigeverzadigingisernogslechts sprake
vanéénafhankelijkevariabeledienietvaninvloedisopdehydraulische doorlatendheid
K.Voorhetoplossenvandifferentiaalvergelijkingen voorstromingineengedeeltelijk
verzadigd systeemmoethetverbandtussen6enpwordengespecificeerd.Bovendienisde
hydraulische doorlatendheideenfunctievan9ofp.Beideempirischerelaties (e(p)en
K(p)ofK(9)}zijnmoeilijktebepalenenonderhevigaanhysteresis.Vandaardatoplossingenvanpartiële differentiaalvergelijkingen inhetalgemeeneenvoudigerwordenverkregenvoorstromingineenvolledigverzadigd systeemdanvoorstromingineengedeeltelijkverzadigd medium.
Dealgebraïsche formuleringvanhetstromingsprobleem resulteertineenabstract
simulatiesysteemofmathematischmodel.Daarnaastbestaanerookfysischeenanalogemodellen.Wind (1972)ontwikkeldeeenhydraulisch anàlogonenWind&Mazee (1979)een
elektrisch analogonvoordesimulatievanverticalestromingindeonverzadigde zone.
Demeestveelzijdigeelectrische analogonsvoordesimulatievanverzadigdegrondwaterstromingbestaanuiteennetwerkvanweerstandenencondensatoren.Ofschoonmetdeze
directesimulatietechnieken gecompliceerde stromingsproblemen zijnoptelossen,worden
mathematischemodelleninvelerleiopzichtalssuperieurbeschouwd.Hieraanheeftvooral
derecenteontwikkelingophetgebiedvandedigitalecomputertechniek bijgedragen.
Voorhetnumeriekoplossenvanstromingsvergelijkingenwordeneindigeelementen-en
eindigedifferentiemethoden gebruikt.Deeindigeelementenmethodebiedtvoordelenbxjhet
oplossenvangecompliceerdeverzadigde-onverzadigde stromingsproblemenenxn « « » " «
waarbijhetgebruikvaneenonregelmatignetwerkwenselijk is.DemethcKle
^ J ? ™ £
korttoegepastvoorhetoplossenvanstromingsproblemeninporeuzemedxa Van& bestun
denumerieke oplossingenisdanookhetgrootstedeelverkregenmetbehulp™
differenties.L
" ^
aantalgangbareeindigedifferentietechniekenwordtbesprokenaanhet
eindevanhoofdstuk2.
107
Omhistorische redenen, maar ook vanwege de genoemde verschillen tussen stromingsvergelijkingen voor volledig verzadigde en gedeeltelijk verzadigde systemen, werd de
waterbeweging boven en beneden het freatisch vlak vanouds gescheiden behandeld. Deze benadering vereenvoudigt de oplossing van het stromingsprobleem aanmerkelijk, omdat in de
onverzadigde zone veelal volstaan kan worden met het in beschouwing nemen van stroming
in verticale richting en in de verzadigde zone met stroming in het horizontale vlak. Maar
in het geval van ondiepe of snel fluctuerende grondwaterstanden bestaan er ernstige bezwaren tegen deze aanpak vanwege het effect van de onverzadigde stroming op het verzadigde systeem.
Een integrale benadering van verzadigde-onverzadigde stroming werd voor het eerst
gerapporteerd door Rubin (1968). In hoofdstuk 3 wordt een overzicht gegeven van modellen
die gebruik maken van slechts één vergelijking voor het oplossen van stroming in een gedeeltelijk verzadigd medium. De toepassingen van deze rigoureuze benadering hebben betrekking op de simulatie van pompproeven en stroming in watervoerende pakketten met een
ondiepe grondwaterstand. Voor deze stromingssituaties i s het effect van het onverzadigde
systeem op de stroming in het verzadigde freatische pakket het meest geprononceerd.
Het gebruik van slechts één vergelijking voor het simuleren van verzadigde-onverzadigde stroming introduceert numerieke problemen. De stromingsvergelijking i s namelijk
parabolisch in de onverzadigde zone en van een elliptisch type in het verzadigde deel,
terwijl de ligging van het freatisch vlak tussen beide gebieden a p r i o r i onbekend i s .
Omdat de afhankelijke variabele in de onverzadigde zone en in de buurt van de put aanzienlijk kan variëren over een geringe afstand, moet gebruik gemaakt worden van een netwerk met een kleine maaswijdte. Bovendien vereist de n i e t - l i n e a r i t e i t van de coëfficiënten die betrekking hebben op het onverzadigde deel van het stromingsgebied dat on.reJ Ü T IT " t a b l l l t e i t e n u r g e n t i e de tijdstappen tot een kleine grootte worden teruggebracht. Voor stromingsgebieden van enige omvang leidt d i t tot exorbitante rekentijden
een tekort aan beschikbare geheugencapaciteit van de computer. Vandaar dat tot op
heden geen toepassingen op regionale schaal bekend zijn.
g e l i i k i n e a v t e m a t l e V e ° P l 0 S S i n g CPikUl G t a 1 - ' 1 9 7 4 ) i s d e k ° P P e l i n g v a n R i c h a r d S ' Ver"
horizontale Z ^ ^ J ™ ^
^ ^
»* d e v e ^ ^ ^ van Boussinesq voor
ng DS r e s u l t e r e
niet t o t T
t
'
n d e quasi drie-dimensionale aanpak blijkt
onverzadiX "f"*"™
^ ^ e r e doelmatigheid te leiden bij het oplossen van verzadigded i ^ d e f t e e S t r 0 m i n g S p r 0 b l e - n - * belangrijkste oorzaak hiervan i s dat de beperkingen
opgelegd aan ^ Z J T y s t e T ^
1
v e r z a d L S r ! ^ * * " " ^ ** °
strommj w o ^ Z
r
Z ™ T
Opwaartse stationair
^
ntwikkelin
^
^
* ****** " " ^
^
s ™
8
'
""**
g beschreven van een model voor verticale on^
^
^
* » ' ^stationaire
^szMileerd met een opeenvolging van stationaire toestanden,
freatisch vlak w e r d ^ T ^ " ^ g r ° n d k o l o f f l m e t e e n h o °gte van een meter boven het
rekening van s t a t i o n l ^
^ ^ b e r e k e n d d o o r Schawls (1931). Een systematische beTelatleS
Uitg6VOerd d o o r Wind
toegepast bij de bepaHn
^
(19SS) en later door anderen
in relatie tot de die te^ ^ ^ V °° r ^ *****m a x i m a a l beschikbare hoeveelheid vocht
opeenvolging van s t a « & ^ ^ g r ° n d w a t e r s t a n d - Feitsma (1969) gebruikte hierbij een
naire toestanden omhet niet-stationaire proces van capillaire
108
opstijging en grondwaterstandsdaling te simuleren.
In deze studie i s de pseudo stationaire benadering van capillaire opstijging geanalyseerd. De resultaten die met deze aanpak worden verkregen, blijken afhankelijk te
zijn van de lengte van de gebruikte tijdstap in relatie tot de karakteristieke tijd van
het onverzadigde systeem. In het geval van een ondiepe grondwaterstand in een zandige
grond ligt de waarde van de karakteristieke tijd in de orde van grootte van dagen. Indien
de lengte van de gebruikte tijdstap kleiner is dan de karakteristieke tijd, worden met de
pseudo stationaire benadering resultaten verkregen die fysisch gezien onjuist zijn. Hetzelfde geldt in geval van een afnemende flux door de bovenrand van het model en voor de
situatie waarbij capillaire opstijging wordt gevolgd door percolatie. Bovendien is gebleken dat voor een grote neerwaartse flux door de onderrand de positie van het freatisch
vlak niet goed gesimuleerd kan worden.
In deze studie strekt de onverzadigde zone zich uit van juist beneden het freatisch
vlak tot aan maaiveld. Het gebied i s schematisch verdeeld in een wortelzone en een ondergrond. Omdat stroming in de wortelzone in hoge mate wordt bepaald door de
^
^
Z
dewortels, is de gradiënt van de hydraulische potentiaal in de wortelzone f ^ J ^ 1
aannul. Er i s aangetoond dat de stationaire stromingstoestand volledig is bepaald door
slechts twee parameters (b.v. het verzadigingstekort van de wortelzone S en de stat
naire flox in de ondergrond q). Door gebruik te maken van verzadigingstekor* w
oplossing van een stationaire situatie ~
^
£
£
^
J
Z
£
Z
• » t»ee onbekenden (S en 5 ) . De oplossing van de stationaire si
procedures
~
* f t a door de bovenrand van de ondergrond «ordt ' » - ™ * ^ J ^ 1 ^
zijn ontwikkeld om de hierboven genoemde onjuistheden te corrigeren e^ ^ ^ s i r a u l e r e n .
neerslagoverschot volgend op een situatie met capillaire opstijg?.ng^ ^
^
^ wer.
Indien de wortelzone uitdroogt tot verwelkingspunt, berekent
^^ ^
^
olijke flux door de bovenrand. In het geval van een F0^™™
^ grondwaterstand,
onderrand is de bovenrandoplossing ongeschikt voor de ere ^ ^ ^ ^ ^ w o r d t gesimuVandaar dat voor een neerwaartse flux door de onderrand e ^ ^ ^ ^ ^ ^
van
leerd met behulp van een pseudo stationaire benadering van^ e^s ^ ^ ^ & s t a t i o .
ie ondergrond dat j u i s t boven het freatisch vlak gelegen ï s . . ^ ^ o n v e r z a d i g d e zone wordt
•«ire situatie overeenkomend met de flux door de onderran van
^ . ^ ^ i n é én
onderrandoplossing genoemd. De boven- en onderrandoplossingen ^ ^ ^ . ^ I n g e v a i
simulatiemodel waarbij rekening i s gehouden met hysteresis en
v e r o n d e r s t e l d > zo dat
e c h t
vanpercolatie worden de capillaire eigenschappen
<™j; ^ ^
•et het model geen schijngrondwaterspiegels gesimuleer
^ ^ _ drie -dimensionale beIn hoofdstuk 5 wordt een uiteenzetting gegeven van ee ^ ^ ^ ^ stroming in waternadering voor de simulatie van niet-stationaire verzadigê^ w o r d t g e b r u i k geurende pakketten met een ondiepe grondwaterstand. Bij
b e s c h r i j v i n g v an de verzak t van een twee-dmensionaal horizontaal netwerk v o o r
^ m ^ i e d e r knooppunt van
di
gde grondwaterstroming. Een speciale procedure is ontwi
^ ^ verzadig.
d
it netwerk het model voor stroming in de onverzadigde zon
de
systeem.
.
x t e v e r ifiëren werd in het
Teneinde het quasi drie-dimensionale model experimen^ ^ ^
^ b e s c h rijving van
°°sten van N ^ - H ^ »«, <**ied gekozen ter grootte van
109
hetgebiedisgebruikgemaaktvaneenrechthoekignetwerkmeteenmaaswijdtevan500m.
Meteentijdstapvantiendagenwerdeenperiodevanbijnazesjaargesimuleerd.Deovereenkomsttussendegesimuleerdegrondwaterstandenendewaargenomenwaardenisbevredigend.Mindernauwkeurigeresultatenzijnverkregeninperiodenmeteenafwisselendverdampings-enneerslagoverschot.Degevoeligheidvandegesimuleerdegrondwaterstandenen
deberekendeevapotranspiratie tenaanzienvaneenvariatieindewaardevaneenaantal
parameterswerdonderzocht.Uiteenbesprekingvanderesultateninhoofdstuk6blijkt
datdeberekendewerkelijkeverdamping inhogematewordtbepaalddoordeconceptuele
benaderingvandewateropnamedoordewortels.Voordeindezestudiegevolgdebenaderingswijzevandegewasverdamping isverdergeblekendathetverschil tussendeberekende
werkelijkeenpotentiëleevapotranspiratievangrasvrijweluitsluitend afhankelijkis
vandebeschikbarehoeveelheidvochtindewortelzoneeninveelminderematevande
empirischediffusieweerstandvoorbodemengewas (r) .
Meteentoepassingvanhetmodelvoorverzadigde-onverzadigde stromingwerdende
gevolgenvangrondwateronttrekkingvoordeberekendegrondwaterstandenenwaterbalansen
voorspeld.Detoepassingheeftbetrekkingopdeonttrekkinginhetmodelgebied ende
periodedieookvoordeverificatievanhetmodelisgebruikt.
110
List ofsymbols
a
a
CoefficientinEqn72
CoefficientinEqn91
A
CoefficientinEqn15
A
CoefficientinEqn84
b
CoefficientinEqn91
cm«d-1«mbar"
d~'
cm-d"1
B
CoefficientinEqn84
c
Hydraulicresistanceofconfininglayer
s
f
Compressibilityofthesoilmatrix
Pa
w
Compressibilityofwater
Pa
C
Specificmoisturecapacity
Pa ,mbar
8
Partialdifferentialoperator
d
Depthofwatertablebelowupperboundaryofpercola- cm
c
c
tionprofile
D
Thicknessof (saturatedpartof)aquifer
m,cm
D
DiffusivityasdefinedbyEqn29
m -s"
9 —1
D'
Thicknessofconfininglayer
Dj.
Thicknessof 'effective'rootzone
ea
Actualvapourpressureat2mheight
e
Saturatedvapourpressurefortheairat2mheight
E
Aerodynamicevaporation
E
Potentialévapotranspiration
E
Realévapotranspiration
Ewet
Evaporationofawetsurface
m
cm
bar
bar
-2-1
F
Observedwater-tableelevation
F
Meanofobservedwater-tableelevations
F'
Simulatedwater-tableelevation
kg'm *s
—2 —i
kg'in *s ,an*d
-2 -I
kg*m -s ,cm»d
-2 -1
—1
-1
kg-m -s
cm
cm
cm
-2
g
Accelerationduetogravity
h
Water-tableelevation
h*
Water-tableelevationatboundarySj
hg
Soilsurfaceelevation
i
kI
Spaceindexinx direction
Proportionality
factor
inDarcy'slaw(Eqn10)
Totalnumberofnodes
inxdirection
Kj
Hydraulic
conductivity
Spaceindex
inydirection
J
Totalnumberofnodesinydirection
m-s
m,cm
cm
m,cm
m -s -Pa
2 -1 -I
m-s ,cm-d
111
K
Hydraulicconductivityofanaquifertakenasacon- m-s ,cm-d
stantinverticaldirection
Hydraulicconductivityofconfininglayer
m-s
Principalcomponentsofthehydraulicconductivity
m-s
tensor
Saturatedhydraulicconductivityintheunsaturated
cm'd
K'
K ,K,K
K
p
p
P
P
q
q
qe
q.^
q„
zone
Meshwidth
Latentheatofvaporization
Horizontaldistanceusedinproblem (54)
Indexforboundarynodeinxdirection
Timeindex
Porosity
CoefficientinEqn72
Directionnormaltotheboundary
Hydraulicormatricpressure,relativetoatmospheric
pressure
Atmosphericpressure (p =1.013)
Matricpressureatinterfacerootzone-subsoil
Pressureequivalentoftotalsoilwaterpotential
Precipitation
Fluxdensityorspecificdischarge
Fluxincaseofsteadyunsaturatedverticalflow
Sinktermduetogroundwaterextraction
Sourceorsinktermfunction
Groundwaterdischargeintosurfacewatersystem
a
q
Upperboundaryfluxofpercolationprofile
Fluxacrossinterfacerootzone-subsoil
S.
L
L
m
n
n
n
n
p
cm
J-kg
m
Pa,mbar
bar
mbar
Pa
cm-d"
m-s ,cm-d"
cm'd cm'd cm-d"
cm-d"
-
-1
cm-d'
cm-d
rs
-1
qs
qsr e
Maximumpossiblefluxacrosssoilsurface
Realfluxacrosssoilsurface
m-s ,cm-d
cm-d
q
upwardfluxfromthesaturatedregionintotheunm-s
saturatedzone
Verticalfluxacrossaleveljustbelowthewater
m-s ,cm'd
tableorlowerboundaryoftheunsaturatedflowmodel
Fluxforwhichbothrelations (91)and (94)apply
cm-d"
-1
a
q*
—1
q ,q,q
y
Q
r
r
ra
rc
i"4
112
Fluxintherespectiveco-ordinatedirections
m-s
z
3-1
Groundwaterextraction
Iterationindex
'Effective'poreradius
Diffusionresistancetowatervapourintheair
Diffusionresistancedependingonthefractionof
soilcovered
Diffusionresistancedependingonlightintensity
m «d
m
s-m"
S'rn"
s-m"
-1
Diffusionresistancedependingonsoilmoisture
s«m
conditionsandflowintheplant
Totaldiffusionresistanceofcropandsoil
«E
s»m~
Efficiencyfactor
Netradiation
J*s" •m~
Slopeofthetemperature-saturationvapourpres-
bar-K-1
surecurve
_-l
Specificstorage {s =Pgn(c,+c)}
S
S
.> S 2
I
w
Degreeofwatersaturation
Partofboundaryforhorizontalsaturatedflowfor
whichh=•h*,andthefluxnormaltotheboundary
equalszero,respectively
Fractionofthesoilcoveredbythecrop
Saturationdeficitinthesubsoilforq=0
Saturationdeficitofpercolationprofile
Saturationdeficitofrootzone
Saturationdeficitofsubsoil
t
T
T
A
w
x,y,z
Saturationdeficitofentireunsaturatedzone
(S =S +S)
u
r s'
Time
cm
cm
cm
cm
cm
Depthofwatertablebelowsoilsurface
s,d
m 2 s , cm'd
2
2j-1
cm'd
m,cm
Cartesianco-ordinatedirectionsordistancealong
m,cm
Transmissivity
AveragetransmissivityofelementA
therespectiveco-ordinatedirections
z.,z,z
i w's
z
rs
Variousheightsusedinproblem (54)
m
Distancebetweenphreatidlevelandinterfaceroot
cm
z*
rs
Depthofwatertableresultingfromthelowerbound-
zone-subsoilapplyingtotheupperboundarysolution
cm
aryfluxalone
a,e
a
Integrationdummies
Reciprocalofdelayindex
Y
A
Psychrometricconstant
e
Ratiomolecularweightofwatervapouranddryair
.-1
bar^K-1
Increment
(e=0.622)
Heightofthewatertableabovethelowerboundary
cm
oftheunsaturatedflowmodel
Elevationofupperboundaryofpercolationprofile
cm
inthemodelforunsaturatedflow
rs
Distancebetweentheinterfacerootzone-subsoil
cm
andthelowerboundaryoftheunsaturatedflowmodel
Fluiddynamicviscosity
kg«m~ 'S
113
e
e
Fractionalvolumetricmoisturecontent
MoisturecontentusedinEqn64
m
K
\i
V.,VT,
2
Intrinsicpermeability
Specificyieldorstoragecoefficient
Short-termandlong-termspecificyield,respective-
m
AB
y
p
p
a
T
T
$
$'
¥t,v >VQ>\
u
V
114
ly,usedinEqn53
Storagecoefficientwhichisindependentofwatertabledepth
Densityofsoilwater
kg>m
Densityofair (p =1.2047)
kg«m
Standarddeviationofthedifferencesbetweensimucm
latedandobservedwater-tableelevations
Time (T<t)usedinEqn53
s
Characteristictime
d
Hydraulicheadortotalsoilwaterpotentialexm,cm
pressedasenergyperunitweight
Hydraulicheadinadjoiningaquifer
m
Total,pressure,osmoticandgravitationalsoilwater J'kgpotential,respectively,expressedasenergyperunit
mass
Over-relaxationparameter
Operatorforgradientordivergence
_3
_3
Appendix A
Computation of Sß and z
as a function
of p
and q for a heterogeneous soil
profile
IfdifferentK(p)relationsapplytodifferentlayersinthesubsoilpressureprofilesdonotexist.Insteadofz(p,q),arelationcanbecomputedbetweenthepressure
attheinterfacerootzone-subsoilp andthedepthofthewatertablebelowthis
interfacez rs
foranumberofpositivevaluesforq,yieldingz (p ,q).Anumerical
rs vs
approachtothecomputationoftherelationsS(z ,q)andS(p ,q)foraheterogeneous
s rs
srs
subsoilisgivenbelow.
Thesubsoilisdividedintolayerswithadepthof1cm.ForeachlayerasoilmoisturecharacteristicandK(p)relationmustbespecified.Givenawater-tabledepthz
(integerincm)thelayerindex i runsfrom1toz ,where i =1forthelayerofwhich
thelowersideisatadepthz (Fig.A1). Foragivensteadyfluxqandwater-table
depthz r thecomputationofS andp proceedsasfollows.Thematricpressuredistributionisnumericallyapproachedforthesuccessivelayersstartingatthephreaticlevel
inupwarddirection.Thevariablesareinitializedasfollows:p=0;Ap=-1mbar;S g=0
ands,=1,whereApisafirstestimateforthechangeinpoverlayer i.
Step1: Theaveragepressurepinlayer l isestimatedasp=p+jAp.
Step 2:InterpolatethehydraulicconductivityKforp=pfromtheK(p)relationthat
appliesforlayer i,.(Itmayoftenbenecessarytocarryoutthisinterpolationan
adoublelogarithmicscale,duetothenon-linearityofthisrelation.)
Step3:ComputetheincreaseinheightAzfromDarcy'slaw,writtenas
l=zrs
-rs
I
4
3
2
1=1
Fig.Al.Theuseofthelayerindex I foraparticularwater-tabledepthz r g .
115
A z =__!
p
^ _ ip
2 q+K
Step4:Improvetheestimateforphalfway layer %to
+
P =P2§
Step5:InterpolateKforp=p fromtheK(p)relationthatappliesforlayer l.
Step6:Computethechangeinpoverlayer i (forwhichAz= 1cm)fromDarcy'slaw,now
writtenas
A =_
P Pg— j p * Az
(ComparisonwithanalyticalsolutionsshowsnoneedtorepeatSteps 1to6to
improvethesolutionforAp.)
Step7:Interpolate forp=p+Âpthemoisturecontent efromthesoilmoisturecharacteristicforlayer l andincreaseS withthesaturationdeficitofthis layer
S s=S +sn-9
wherenistheporosityoflayer I.
Step8:Computethematricpressureattheuppersideoflayer i
p=p +Ap
Step9: Increasethelayerindex
s.= i +1
Step 10:If i <z r ggotoStep 1.Ifnot,thecomputations arecompleted andp =p.
Theaboveschemeisexecutedforvaluesofz r g increasing fromzerowithstepsof 1cm
untiltheabsolutevaluecalculated forp r s isgreaterthanorequalto 16000mbar.If
thesoilishomogeneous thecomputedrelationbetweenpandz (andbetween 9andz)does
notchangewithz r g .Thenvalues for.p r g andSfiareeasilycomputed forz iftheabove
schemeisappliedwiththefollowing initialdata: a =z andvalues for"p,S and
_
*"s
P -p r s ascomputedforthepreviouswater-table depth (z rg -1).Fromthe calculated
relationsbetweenz r s ,p r g andS svaluesforz r g andS g areinterpolated fordifferent
valuesofp r s ,yieldingS g (p rs )andz rs (p ) .
Theaboveprocedure iscarriedoutforanumberofvalues forqresulting inthe
relationsS s (p rs ,q)andz r s (P r s ,q).InthisstudyS g andz r g arecomputed forthefollowing13valuesforp ^ : 0,-10,-20,-31,-50,-100,-250,-500,-1000,-2500,-5000,
-10000and-16000mbarandthefollowing 18valuesforq: 0,0.001,0.005,0.010,0.015,
116
S
0.020,0.030,0.040,0.060,0.080,0.100,0.125,0.150,0.200,0.300,0.400,0.500and
1.000 cm-cf1.
Asforahomogeneousprofilethevaluescomputedforzasafunctionofpareindependentofthewater-tabledepth,therelationz (p ,q)maybewrittenas z(p,q),
whicharethepressureprofilesinFig.5c.ThesaturationdeficitcurvesS (p ,q)are
presentedinFig.5f,whilethederivedrelationS (z ,q)whichresultsfromacombinationofS (p ,q)andz (p ,q)isshowninFig.5e.
s irs
rs rs
117
Appendix B
Hysteresis in the Sp(Pps)
relation
Considerthehystereticrelationbetween6andpgiveninFig.B1.Thesolidlines
representtherelationfordrying,thebrokenlinesforwetting.Themostextremecurve
fordryingcorrespondstothesoilmoisturecharacteristicgiveninFig.5b.Disregarding
thescanningcurves,therelationsfordryingandwettingareusedtocomputeS (p ) .
TheresultisgiveninFig.B2foradepthoftherootzoneD =30cm.Thebrokenline
representsthesituationforwhichp continuouslydecreasesfrompF4.2tozeroassuming
equilibriumconditionsintherootzone.Thisprocessmaybeapproximatediftherootzone
isslowlywettedbycapillaryrisefromthesubsoilwhileq = 0 .Generallywettingis
causedbyrainfallexcess,resultinginahighlynon-linearflowprocesswhichiscomplicatedbyhysteresis.Asthepseudosteady-stateproceduredoesnotconsiderflowin
therootzone,thetotaleffectmustbelumpedintotheS(p )relation.Itshouldbe
notedthatthenatureofthepseudosteady-stateprocedurehampersthepressureatthe
lowersideoftherootzoneobtaininglowpFvalues.Evenafteralongwetperiodthe
p r s valuemaynotdropbelowpF1.5. Thereforeitisassumedthathysteresiseffectsin
therootzonehaveceasedifthematricpressurep hasreachedavalueofe.g.pF1.5.
TheresultingnumericalrepresentationofthehystereticS(p )relationforthe13
valuesofp r g mentionedinAppendixAisgiveninFig.B3.
Dataonhysteresisinthesoilmoisturecharacteristicareusuallynotavailable.
Thereforea 'hysteresisfactor'isintroduced,definedasthenumberoflogarithmcycles
p(mbar)
-10'
118
Fig.Bl. Soilmoisture characteristic
showinghysteresiswith6(p)relations
fordrying (—)andforwetting ( )•
T—m—i—i—i—i—i—i—n
1013151.72J0 2.42.7ao 3.43.74.04.2
pF-lg(-prs)
'g(-Prs)
Fig. B2.Saturationdeficitcurveforthe
rootzone (D =30cm)showinghysteresis
withS r (p rs )relations fordrying (-) and
forwetting ( ) .
Fig. B3.Numericalrepresentationofthe
S r (p r „)relationshowinghysteresis (hysteresis factorequals0.5)withS r (p rs )
relations fordrying (-) and forwetting
(
).
overwhichtheS(p )curvefordryingisshiftedalongthep r g axistoobtainthe
wettingcurve.ThehysteresisfactorapplyingtoFig.B3equals0.5.Intheabsenceof
datathehysteresisfactormustbecalibrated.
AnumericalprocedureisdevelopedtocomputetheS r (p rs )relationatthebeginning
ofeachtimeincrement.Fortimestepn+1thescanningcurveconnectingthecurvesfor
dryingandwettingiscomputedsuchthatitjoinsthedryingcurveforvaluesofS r>S r
andthewettingcurveforvaluesofS r<S r .
119
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Stellingen
1.Eennumerieke oplossingvandevergelijkingvanRichards
isthans nog ongeschiktvoorhet simulerenvanregionale
stromingsproblemen.
Ditproefschrift.
2.Bijhetgebruik vanvochtgehalte-envochtspanningsprofielen
wordtdevoorwaardevanhomogeniteit tenonrechtevaakniet
vermeld.
R.A. Feddes, 1971.Water,heat and cropgrowth.Med.Landbouwh.
Wageningen 71-12.
W.H.van der Molen, 1972.Stroming indeonverzadigdezone.
LH, afd. Cultuurtechniek.
G.P.Wind, 1979.Analog modeling oftransientmoisture flow
inunsaturated soil. Versl. landbouwk.Onderz.894.Pudoc,
Wageningen.
Ditproefschrift.
3.Voorhetnumeriek oplossenvan stromingsvergelijkingenwaarbijhetgebied wordtbeschrevenmeteenregelmatignetwerk,
ishetgebruik vaneindigedifferentiemethoden inhetalgemeen
teprefererenbovendatvandeminderdoorzichtigeeindige
elemententechnieken.
4.Geziendedoor hetgebruikvansimulatiemodellengestimuleerde
vraagnaarbodemfysische parameters,dientmeeraandachtte
wordenbesteed aanverbanden tussendezevaakmoeilijKmeerbaregegevens eneenvoudigvasttestellen bodemeigenschappen
zoals textuur,structuur,organischstofgehaltee.d.
A.W.Warrick,G.J. Muilen &D.R. Nielsen,1977.Soilwater
flux based upon field-measured soil-waterproperties.
Soil Sei. Soc.Am.Proc. 41:14-19.
5.Hetgebruik vanpersputtenbijdeinfiltratievanoppervlaktewater inhetNederlandse kustgebiedwordtopdenduurernstig
bedreigd doorhetontstaanvanbrakwateralsgevolgvan
menging endispersie.
A.J.Roebert, 1979.Werkgroep hydrologie van persputsystemen
geïnstalleerd. H 2 0 (12),nr. 15:341-343.
6.Hetontbrekenvaneengeïntegreerd beheervanoPPf^vlaktewaterengrondwater heefterinbelangrijkematetoe013
gedragendatverschillende gronden,zowelu ^ Jetoogpunt
vanlandbouwproduktie alsnatuurbeheer,tediepziDnontwatera.
7-Bijhetonderzoek naarderelatie tussenvegetatietypeen
waterhuishouding is-voordehogergelegenjonden een
meeruitgesprokenverband teverwachtenmethetjochtleveren
vermogenvandegrond danmethet grondwaterstandsverloop.
8.Verlagingvandegrondwaterstand inveengronden veroorzaakt
een zakkingvanhetmaaiveld dooreenversnelde oxydatie
vanhetveenenmoetderhalve beschouwd worden alseen
maatregel totontgronding.
9.Hetprofijtbeginsel inhetGelderswaterschapsreglement is
instrijdmetdeintentievanhet indatreglement eveneens
toegepasteomslagstelsel.
10.Projectenmethetdoelvisueelgehandicapten deel telaten
nemenaanhetregulierebasisonderwijs dienen gestimuleerd
teworden.
11.Aandiploma'svoormachineschrijven en stenografie kan
weinigwaardeworden toegekend zolang geen garantiesbestaan
tenaanzienvandekwaliteitvandeopleiding endeexamens
ineigenofonderlingbeheerwordenafgenomen.
12.Degedachtedathetnachtelijk zuurstofverbruik vanbloemen
enplantendesamenstellingvandeatmosfeer inziekenhuisofslaapkamermerkbaarbeïnvloedt,berustopeenmisvatting.
ProefschriftvanP.J.M,de Laat
Model forunsaturated flow above a shallow water-table,applied
to aregional sub-surface flow problem
Wageningen 22februari 1980

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Model for unsaturated flow above a shallow water

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