Short-title-ROBUST-SOLUTION-OF-RICHARDS--EQUATION

1RobustSolutionofRichards’EquationforNon-UniformPorousMediaCassT.MillerandGlennA.WilliamsDepartmentofEnvironmentalSciencesandEngineering,UniversityofNorthCarolina,ChapelHill,NCC.T.KelleyandMichaelD.TocciDepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,NCShorttitle:ROBUSTSOLUTIONOFRICHARDS’EQUATION2Abstract.Capillarypressure-saturation-relativepermeabilityrelationsdescribedusingthevanGenuchtenandMualemmodelsfornon-uniformporousmedialeadtonumericalconvergencedicultieswhenusedwithRichards’equationforcertainauxiliaryconditions.Thesedicultiesarisebecauseofdiscontinuitiesinthederivativeofspecicmoisturecapacityandrelativepermeabilityasafunctionofcapillarypressure.Convergencedicultiesareillustratedusingstandardnumericalapproachestosimulatesuchproblems.Constitutiverelations,interblockpermeability,andnonlinearalgebraicsystemapproximationmethods,andtwotimeintegrationapproachesareinvestigated.AnintegralpermeabilityapproachapproximatedbyHermitepolynomialsisrecommendedandshowntoberobustandeconomicalforasetoftestproblems,whichcorrespondtoasand,aloam,andaclayloammedia.31.IntroductionFluidowinunsaturatedporousmediaisoftenmodeledusingRichards’equation(RE)[Richards,1931]andclosedbyconstitutiverelationstodescribetherelationshipamonguidpressures,saturations,andrelativepermeabilities[BrooksandCorey,1966;vanGenuchten,1980].Becauseofthenonlinearitiesinvolved,REisoftensolvedusinglow-ordernumericalapproximationmethods,suchasnitedierenceorniteelementmethods.Thesetypesofsolutionmethodsareusedinmanyoftheexistingunsaturatedowcodes.Theapplicationofthesecodestoawidevarietyofproblemsisconsideredcommonplace[vanderHeidje,1996].Thestandarduseofsuchsimulationmethodsnotwithstanding,problemsexistwithboththerobustnessandeciencyofnumericalsolutionstoRE;advancementsinthesolutionoftheseproblemsisanimportantandactivetopicofresearchinthewaterresourcescommunity.AcommonsetofconstitutiverelationsusedtocloseREisthevanGenuchtenrelationtodescribetheinterdependenceofuidpressuresandsaturationsandtheMualemrelationtodescribetheinterdependencebetweenuidsaturationandrelativepermeability.Theexponent,ornv,inthevanGenuchtenrelationisameasureofpore-sizeuniformity.Formanynaturalporousmedia,typicalvaluesofnvrangebetween1.0to2.0,whendeterminedusingstandardlaboratoryapproachesandttedusingstandardinversetechniques[Kooletal.,1985;vanGenuchtenetal.,1991].UsingthevanGenuchtenandMualem(VGM)constitutiverelationsinexistingREcodes,weexperiencedsignicantproblemsinattainingaconvergentsolutionforcasesinwhichnv2forcertainsetsofauxiliaryconditions.Anexampleofsuchacasewasforinltrationfromapondedsurfaceboundaryconditionintoasystemoriginallydrainedtoequilibrium.Theseexperiencesmotivatedthiswork,whichhadseveralobjectives:(1)todocumentacommonclassofvariably-saturatedowproblemsthatlackrobustnesswhensolvedusingstandardsolutionapproaches;(2)todeterminethereasonwhytraditional4approacheslackrobustnessforthisclassofproblems;(3)toinvestigateavarietyofalternativeapproaches;and(4)tocompareasetofalternativeapproachesforarangeofmediaconditionstotestrobustnessandeciency.2.BackgroundFouraspectsoftheliteratureonunsaturatedowwarrantatleastabriefconsideration:(1)constitutiverelationsusedtodescribepressure-saturation-conductivityrelationsandtypicalparametervaluesfornatural,unconsolidatedmedia;(2)approachestypicallyusedtoapproximateRE;(3)methodsforapproximatingrelativepermeabilitiesinadiscreteapproximationofRE;and(4)strategiesusedtoestimatetherelativelycomplexconstitutiverelationsthatareapartoftheformulationsofconcern.Wediscusseachoftheseaspectsinturn.2.1.Pressure-Saturation-ConductivityRelationsAwell-posedformulationofRErequiresthatconstitutiverelationsbespeciedtodescribetheinterdependenceamonguidpressures,saturations,andrelativepermeabilities,whichwillbereferredtoasp-s-krelations.Severalapproacheshavebeenadvancedtodescribep-s-krelations[BrooksandCorey,1966;Mualem,1976;vanGenuchten,1980],butdeterminingthemostappropriateconstitutiverelationformulationisstillanopenissue.WeusethevanGenuchtenrelationtodescribetherelationshipbetweenuidpressuresandsaturations[vanGenuchten,1980]andtheMualemrelationforthatbetweenuidsaturationsandrelativepermeabilities[Mualem,1976].SeveralcodesdocumentedintheliteratureusetheserelationstocloseRE(e.g.,[Yeh,1987;SimunekandvanGenuchten,1994]).WewillrefertotheserelationscollectivelyasthevanGenuchten/Mualem(VGM)relations.BecauseofthewidespreaduseofthevanGenuchtenrelation,manyexperimentaldatasetshavebeendescribedusingthisapproach,andmanysetsofparametervalues5areavailableintheliterature[vanGenuchtenetal.,1991].Inaddition,aparameterestimationcodeisavailableandhasbeenwidelyusedtodeterminetheseparametervaluesfromexperimentaldata[vanGenuchtenetal.,1991].Theseparametervaluesarerelatedtothemeanporessize(v)andtheuniformityofthepore-sizedistribution(nv).Thestandardrangeofvaluesofnvisofparticularinterest;itcanvaryfromnear1.0[vanGenuchtenetal.,1991]tonearorevengreaterthan10.0[KoolandParker,1987;MayerandMiller,1992],withtheporesizedistributionbeingincreasinglyuni