Evolution of karst conduit networks in transition from pressurized flow to free-surface flow


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Evolution of karst conduit networks in transition from pressurized flow to free-surface flow

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Evolution of karst conduit networks in transition from pressurized flow to free-surface flow
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Hydrology and Earth Systems Science
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Perne, M
Covington, M.
Gabrovšek, F.
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Models Of Speleogenesis ( local )
Novel Modeling Approach ( local )
Pressurized FLOW ( local )
Pipe Flow ( local )
Free-Surface FLOW ( local )
Open-Channel Flow ( local )
US Environmental Protection Agency Storm Water Management Model ( local )
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serial ( sobekcm )

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Most of the existing models of speleogenesis are limited to situations where flow in all conduits is pressurized. The feedback between the distribution of hydraulic head and growth of new solution conduits determines the geometry of the resulting conduit network. We present a novel modeling approach that allows a transition from pressurized (pipe) flow to a free-surface (open-channel) flow in evolving discrete conduit networks. It calculates flow, solute transport and dissolution enlargement within each time step and steps through time until a stable flow pattern is established. The flow in each time step is calculated by calling the US Environmental Protection Agency Storm Water Management Model (US Environmental Protection Agency, 2014), which efficiently solves the 1-D Saint-Venant equations in a network of conduits. Two basic scenarios are modeled, a low-dip scenario and a high-dip scenario. In the low-dip scenario a slightly inclined plane is populated with a rectangular grid of solution conduits. The recharge is distributed to randomly selected junctions. The results for the pressurized flow regime resemble those of the existing models. When the network becomes vadose, a stable flow pathway develops along a system of conduits that occupy the lowest positions at their inlet junctions. This depends on the initial diameter and inlet position of a conduit, its total incision in a pressurized regime and its alignment relative to the dip of the plane, which plays important role during the vadose entrenchment. In the high-dip scenario a sub-vertical network with recharge on the top and outflow on the side is modeled. It is used to demonstrate the vertical development of karst due to drawdown of the water table, development of invasion vadose caves during vadose flow diversion and to demonstrate the potential importance of deeply penetrating conductive structures.
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Hydrology and Earth Systems Science, Vol. 18 (2014-11-24).

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Hydrol.EarthSyst.Sci.,18,4617–4633,2014 www.hydrol-earth-syst-sci.net/18/4617/2014/ doi:10.5194/hess-18-4617-2014 Authors2014.CCAttribution3.0License. Evolutionofkarstconduitnetworksintransition frompressurizedowtofree-surfaceow M.Perne 1,2,3 ,M.Covington 3 ,andF.Gabrovek 1 1 KarstResearchInstitute,ResearchCentreoftheSlovenianAcademyofSciencesandArts,Postojna,Slovenia 2 JosefStefanInstitute,Ljubljana,Slovenia 3 DepartmentofGeosciences,UniversityofArkansas,Fayetteville,AR,USA Correspondenceto: F.Gabrovekgabrovsek@zrc-sazu.si Received:6May2014–PublishedinHydrol.EarthSyst.Sci.Discuss.:19June2014 Revised:24September2014–Accepted:15October2014–Published:24November2014 Abstract. Mostoftheexistingmodelsofspeleogenesisare limitedtosituationswhereowinallconduitsispressurized. Thefeedbackbetweenthedistributionofhydraulicheadand growthofnewsolutionconduitsdeterminesthegeometryof theresultingconduitnetwork.Wepresentanovelmodeling approachthatallowsatransitionfrompressurizedpipeow toafree-surfaceopen-channelowinevolvingdiscrete conduitnetworks.Itcalculatesow,solutetransportanddissolutionenlargementwithineachtimestepandstepsthrough timeuntilastableowpatternisestablished.Theowin eachtimestepiscalculatedbycallingtheUSEnvironmentalProtectionAgencyStormWaterManagementModelUS EnvironmentalProtectionAgency,2014,whichefciently solvesthe1-DSaint-Venantequationsinanetworkofconduits.Twobasicscenariosaremodeled,alow-dipscenario andahigh-dipscenario.Inthelow-dipscenarioaslightlyinclinedplaneispopulatedwitharectangulargridofsolution conduits.Therechargeisdistributedtorandomlyselected junctions.Theresultsforthepressurizedowregimeresemblethoseoftheexistingmodels.Whenthenetworkbecomes vadose,astableowpathwaydevelopsalongasystemof conduitsthatoccupythelowestpositionsattheirinletjunctions.Thisdependsontheinitialdiameterandinletposition ofaconduit,itstotalincisioninapressurizedregimeandits alignmentrelativetothedipoftheplane,whichplaysimportantroleduringthevadoseentrenchment.Inthehigh-dip scenarioasub-verticalnetworkwithrechargeonthetopand outowonthesideismodeled.Itisusedtodemonstratethe verticaldevelopmentofkarstduetodrawdownofthewater table,developmentofinvasionvadosecavesduringvadose owdiversionandtodemonstratethepotentialimportance ofdeeplypenetratingconductivestructures. 1Introduction 1.1Speleogeneticmodels:ashorthistory,aimsand results Karstaquifersareamongthemostprolicwaterreservoirs. Duetotheirheterogeneityandanisotropy,theirefcientexploitationandprotectionfacemanychallenges.Theroleof solutionconduitsinkarstaquifershasbeenatopicofnumerousstudies.Estimatesshowthatconduitscarryabout99% ofowwithinkarstaquifersandpresentefcienttransport pathwaysforpotentialpollutantsWorthington,1999.However,wehaveonlylimitedinsightintokarstaquifers;the positionofconduitsystemsislargelyunknown,exceptfor thepartsaccessibleforhumanexplorationorencountereddirectlybydrillingorindirectlybygeophysicaltechniques. Speleogenesise.g.,theevolutionofconduitnetworksin karstaquifershasbeenoneofthemaintopicsinkarststudiesofthelastcenturyFordandWilliams,2007.Manyconceptualmodelsofspeleogenesishavebeenproposedbased oneldobservationsAudraetal.,2007;FordandEwers, 1978;AudraandPalmer,2013;Palmer,1991andinferencefrombasicprinciplesofow.However,togaininsight intotheprocessesgoverningspeleogenesis,differentphysicalmodelshavebeenbuiltandfollowedbynumericalmodelsthatarebasedonthephysicalandchemicalprinciplesof ow,dissolutionandtransport. PublishedbyCopernicusPublicationsonbehalfoftheEuropeanGeosciencesUnion.

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4618M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Themainobjectivesofspeleogeneticmodelingaretotest theconceptualmodels,todetermineandevaluatetherole ofdifferentgeological,hydrologicalandgeochemicalfactors andtondmechanismsthatgoverntheevolutionofconduit networksinkarstaquifers.FewexamplesofdirecteldapplicationhavebeenpublishedEptingetal.,2009. Ewersappliedhardwarephysicalmodelsmade fromplasterofParisorsalt,anddiscoveredseveralkey mechanismsthatwerelaterlargelyconrmedandextended bynumericalmodels.Numericalmodelingofsingleconduit evolutionDreybrodt,1990,1996;Palmer,1991;Dreybrodt andGabrovsek,2000revealedafeed-backmechanismbetweenowanddissolutionratesandstressedtheimportance ofhigher-orderdissolutionkineticsDreybrodt,1990,1996; Palmer,1991;White,1977fortheevolutionofextended conduits.Suchkineticshasbeenprovenexperimentallyfor limestoneandgypsumEisenlohretal.,1999;Jeschkeetal., 2001.Moreelaboratedmodelsof2-DfractureswithstatisticalapertureeldsHannaandRajaram,1998;Szymczakand Ladd,2011showedthatnonlinearkineticsisnotnecessary fortheevolutionofextendedpatternsofsolutionconduits. Theinitialstageofspeleogenesisischaracterizedbyslow enlargementofproto-conduits,whichisacceleratedbypositivefeedbackbetweenowanddissolutionrateunderconstantheadconditions.Dissolutionwideningincreasesthe owratealonganinitialfracture.Thenastheowrateincreases,freshaggressivesolutionpenetratesdeeperintothe fractureandinturnaccelerateswideningandowrates. Thisfeedbackmechanismleadstobreakthrough,whenow andwideningrateincreasebyseveralordersofmagnitude inaveryshorttimeDreybrodt,1990,1996;Palmer,1991; DreybrodtandGabrovsek,2000.Atbreakthroughtheinitiationstageofconduitdevelopmentendsandtheenlargement stagestarts.Thetimeneededtoreachbreakthroughistermed breakthroughtime. 1.2Evolutionofadiscretenetworkunderpressurized owconditions Individualfractureshavebeenassembledintofracturenetworksinordertomodelpatternsofevolvingconduitsystems Lauritzenetal.,1992;GrovesandHoward,1994;Siemers andDreybrodt,1998;KaufmannandBraun,2000;Liedlet al.,2003.Atypicalbenchmarksettingemergedoutofthe Ewers'shardwaremodels.Itincludesaplanepopulatedwith initialproto-channelsfractures/tubeswithinputsandoutputsatdifferenthydraulicheads.Thesemodelsrevealedthe competitionbetweendifferentpathwaysconnectinginputsto outputs,asalreadyobservedbyEwersEwers,1982;Ford andWilliams,2007inthephysicalmodel. Toreviewsomeofthesebasicmechanisms,asimplescenarioisshowninFig.1.Itconsistsofaplanewitharectangulargridoffractures.Theboundaryconditionsareshownin Fig.1a:thesidesofthenetworkaremarkedgeographically N,S,E,Wnorth,south,east,west.No-owconditionsare appliedontheNandSboundaries.WaterentersthenetworkattwoinputsIn,In1andIn2attheWside,initiallyat constanthead H D 5000cm.ThewholeEboundarypresents outputat H D 0m.Initialaperturewidthsoffracturesareset to0.02cm,exceptforthefracturesalongW–ElineconnectingIn1totheoutputboundary,denotedasP1,whichhas slightlylargerinitialaperture.03cmandevolvesfaster thanP2Fig.1a,whichisfeddirectlybyIn2.Figure1shows aperturewidthsaslinewidthsanddissolutionratesasline colors;thebrighterthecolorthehighertherate.EquipotentiallinesarealsoshownonFig.1a–f,whichshowthenetworkatdifferenttimestages,denotedineachpanelinunits ofbreakthroughtime T B . At0.99 T B Fig.1athehighhydraulicheadfromtheinput haspenetratedalongthewidenedfracturesofP1deepinto thenetwork,andsuppressedboththehydraulicgradientand growthofP2.Figure2showstheproleofhydraulichead alongP1dashedandP2fulllineatdifferentstages,as denotedbyarrows.ThegradientbetweenthetipofP1and outputsincreasesintimeuntilthebreakthrough. AfterbreakthroughFig.1b,P1iswidenedwiththemaximumdissolutionratealongitsentirelength.Itbecomesincreasinglyuniformandsodoesthehydraulicgradientalong itseeFigs.1band2.Thegradientbuildsupbetween thehighheadregionalongstillpre-breakthroughplugged P2andpost-breakthroughreleasedP1,whichtriggersthe growthofconduitsconnectingP2toP1.Grayarrowsshow someprincipledirectionsofgrowth. Twopost-breakthroughscenariosareenvisaged: 1.InFig.1candd,theconstantheadiskeptatbothinputs. NewconnectionsbetweenP2andP1evolve,whileP2 alsogrowstowardstheexit.Thenetworkexpandsalong theexistingpathwaysbygrowthofnewbypassessome areshownbygrayarrowsuntilallpossibleowpaths evolvenotshown.Ofcourse,allcatchmentshavelimitsandsuchconditionscannotlastforlong. 2.InFig.1eandf,therechargeatIn1andIn2islimitedto Q max D 500Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 .Inthiscasetheconstantheadconditionsbreak,wheninowattheinputreaches Q max .At 1.5 T B Fig.1e,theheadattheinputofP1isaboutonefthof h max seealsoFig.2andthegradientfromP2 towardsP1ishigh,asIn2isstillundermaximalhead. P2integrateswithP1,butfurtherexpansionofnetwork issuppressedastheheadalongthegrowingexisting pathwaysdecreasesintime.Theinterestedreaderisreferredtoadetailedmodelingstudyontheinuenceof limiteddischargeupontheresultingdistributionofconduitsizesbyHubingerandBirk. Tosummarize:inpressurizedowconditions,theevolution ofthenetworkstartswithcompetitionofpathwaysconnectinginputstooutputsandcontinueswiththeirintegrationand expansionuntilheadgradientsalongun-evolvedpathways arehighenoughforpathwaystobreakthrough.Theevolution Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4619 Figure1. Evolutionof2-Dfracturenetworkunderpressurizedow.Panelsshowaperturewidthsanddissolutionratesatdifferentstageof evolution.Sizeofthedomainis1km 1km,initialaperturewidth a 0 D 0.02cm,exceptforthelineP1,where a 0 D 0.03cm.Linearand forthorderdissolutionkineticsforthelimestoneisusedseeDreybrodtetal.,2005fordetails. iscontrolledbythefeedbackmechanismbetweenthedistributionofhydraulicheadandgrowthofnewconduitpathways.Thisinterplayisaffectedbymanyparameterswhich reectlocalhydrology,geologyandgeochemistry. Manyotherscenariosofearlyspeleogenesishavebeen modeledtostudyfactorssuchastheroleofgeochemical conditionsandmixingcorrosion,exchangeowbetweenthe matrixandconduitnetwork,andtheroleofinsolublerocks intheevolutionofconduitsDreybrodtetal.,2005.Numericalmodelshavealsobeenusedtoassessincreasedleakageatdamsitesorotherhydraulicstructureswhereunnaturallyhighhydraulicgradientscauseashortbreakthrough timeDreybrodt,1996;Romanovetal.,2003;Hilleretal., 2011. Inrealsituationstheavailablerechargecannotsustain pressurizedowwithintheevolvingnetwork,andtheconduitsundergoatransitionfrompressurizedtofree-surface owconditions.Mostaccessiblecavesystemshaveundergonesuchatransition. Thoughmostmodelshaveonlyconsideredpressurized ow,AnnableandSudickyandAnnabledevelopedanelaboratemodeloftheevolutionofasinglepartiallylledconduitembeddedinvariablysaturatedfractured mediaunderlaminarowconditions.Theextensionofsuch amodeltonetworkswithturbulentowremainsafuture challenge. www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4620M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Figure2. ProleofhydraulicheadalongpathwaysP1dashed linesandP2fulllinesfromFig.1.Prolesaretakenatdifferenttimesteps,giveninunitsofbreakthroughtime T B .Graylines showscenariowithconstantinputat1.5 T B Fig.1e. Herewedevelopamodelthatgoesbeyondthedynamics depictedinFig.1byincorporatingthetransitionto,andfurtherevolutionin,afree-surfaceowregime. 1.3Evolutionofkarstconduitnetworksinthevertical dimension Theverticalevolutionofkarsthasbeenunderdebatefor morethanacentury,startingwithclassicalconceptsof Katzer,Grund,Davis,Swinnerton,RhoadsandSinacoriand othersPalmer,2007.TheFourStateModelofFordand Ewerselegantlycombinestheseconceptsandrelates cavegeometrytothedensityofpermeablessures. GabrovekandDreybrodtandKaufmann modeleda2-Dverticalcrosssectionofakarstsystemto exploretheevolutionofkarstaquifersinthedimensionof lengthanddepthsensulatoFordandEwers,1978.They haveshowntheimportantroleofwatertabledrawdownin speleogenesis.Thesemodelsconsidereddissolutioninthe phreaticpartofanaquiferonlyandpartlymodeledtheformationofdrawdownvadosepassagesFord,1988;Fordand Williams,2007.Conceptualmodelshavebeendeveloped thathypothesizethediversionofvadosewaterandformation ofinvasionvadosesystemsFord,1988;FordandWilliams, 2007;Palmer,2007;AudraandPalmer,2013.However, theseconceptualmodelshavenotbeentestedbynumerical models. Inthefollowingsectionswedescribehowthemodelis builtandpresenttwobasicmodelingscenarios,eachwith severalrepresentativecases.Wefocusonthedescriptionof newmechanismsofowpathwayselectionanddiscussthe resultsinviewoftheexistingconceptualmodels. Figure3. Conceptualframework.Aconduitnetworkwithpoint rechargeatselectedlocationsindicatedbyarrows.Rechargeislimitedbythepositionofthelandsurface h max orbymaximalavailable recharge Q max . 2Themodelsetup 2.1Theconceptualapproach Figure3showsaconceptualframeworkforthemodeling presentedinthiswork.Weassumeaplanepopulatedwith conduitswithwater-solublewalls,similartothatinFig.1. WaterenterstheconduitnetworkatselectedjunctionsindicatedbyarrowsinFig.3.Thedirectrechargeintoajunction islimitedeitherbytheelevationofthelandsurface h max orbythemaximalavailablerecharge Q max ;ifthehydraulic headislowerthan h max ,allavailablerecharge Q max will enteratthejunction,otherwisethehydraulicheadatthejunctionisequalto h max andonlypartoftheavailablerecharge entersthesystem.Asimilarhardwaremodelwasdiscussed byEwerswhousedthetermmultiple-input,multiplerankscenario. Thebasicworkowofthemodelfollowsthesamescheme asinthemodelscitedabovee.g.,Dreybrodtetal.,2005and includesthefollowingsteps: 1.Denethenetworkofconduitsandboundaryconditions waterinletsandoutlets. 2.Calculateowinthenetwork. 3.Coupleow,dissolutionandtransporttocalculatedissolutionratesinallconduits. 4.ChangetheconduitdiameterwithinatimestepaccordingtothedissolutionrateandreturnbacktoStep2or exittheloopwhenastableowpatternisestablishedor nosubstantialchangesinowpatternareexpected. Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4621 Table1. Listofrateconstantsandotherparametersusedinthiswork. Tohavedissolutionratesexpressedasavelocityofwallretreat, concentration[NL )]TJ/F64 7.5716 Tf 5.905 0 Td [(1 ]ismultipliedwithmolarmass[MN )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 ]anddividedbythedensity[ML )]TJ/F64 7.5716 Tf 5.906 0 Td [(3 ]ofthemineralformingtherockandbeing dissolved.Thismakes c eq dimensionless. ParameterNotationValueUnits Diffusioncoefcient D 1.5 10 )]TJ/F64 7.5716 Tf 5.905 0 Td [(9 saltm 2 s )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 1 10 )]TJ/F64 7.5716 Tf 5.906 0 Td [(9 limestone Manningroughnesscoefcient n 0.01or0.0151 Surfacerateconstant 1saltms )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 2 10 )]TJ/F64 7.5716 Tf 5.906 0 Td [(7 limestone Volumeequilibriumconcentration c eq 0.166salt1 1.1 10 )]TJ/F64 7.5716 Tf 5.905 0 Td [(4 limestone Gravitationalacceleration,density g , 9.81ms )]TJ/F64 7.5716 Tf 5.906 0 Td [(2 Densityofwater 10 3 kgm )]TJ/F64 7.5716 Tf 5.905 0 Td [(3 Dynamicviscosityofwater 10 )]TJ/F64 7.5716 Tf 5.905 0 Td [(3 Pas )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 Wealsoassumethat 1.Theowdoesnotdependonthedissolvedload. 2.Timescalesforow,dissolutionandtransportcanbe separatedfromthetimescaleforwidening,i.e.,theevolutiongoesthroughasetofstationarystateswithin whichthewideningisconstant. 2.2Thecalculationofow Weassumethatthenetworkhaspassedtheinitialinceptionstageofspeleogenesisandthatturbulentowhasalreadybeenestablishedinthenetwork.Thereaderisreferred totheworkofDreybrodtetal.forearlyevolutionin thelaminarowregime.One-dimensionalturbulentowis consideredwithinallconduits.Theowcouldbeeitherpressurizedorfreesurface. FlowinpartiallylledconduitsisdescribedbySaintVenantequationsDingman,2002,whicharebasedon depth-averagedconservationofmassandmomentum.SeveralnumericaltechniquesareusedtosolvethemDingman,2002.Ourmodelinvokesanopensourcepackage StormWaterManagementModelabbreviatedSWMMfrom hereon,developedprimarilyforowandtransportsimulationinsewagesystemsbytheUSEnvironmentalProtectionAgencyUSEnvironmentalProtectionAgency,2014. SWMMsolvesthesetofSaint-Venantequationstothedesiredapproximationandaccuracyusingsuccessiveapproximationswithunder-relaxationRossman,2009.Itsusefor thesimulationofowinconduitdominatedkarstsystemshas beendemonstratedbyseveralauthorsPetersonandWicks, 2006;GabrovekandPeric,2006;Halihanetal.,1998.The pressurizedowisaccountedforbyintroductionofactitiousPreissmannslotFig.4atthetopofaconduit'scross sectionCungeandWegner,1964.Inthiswaywetransform Figure4. TheuseofaPreissmannslotenablesuseofthesameset ofequationsforconduitswithfree-surfaceowandconduitswith pressurizedow. apressurizedpipetoanopenchannelwithoutconsiderably changingthehydrauliccharacteristicsandenableuseofthe samesetofequationsforbothowregimes.Frictionlosses inconduitsarecalculatedbytheManningequation V D k n R 2 = 3 S 1 = 2 f ; where S f isthefrictionslope, V theowvelocity, R thehydraulicradiusi.e.,theratiobetweencross-sectionalareaof owandwettedperimeter, n theManningroughnesscoefcient,heretakenintherange0.01
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4622M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow catchments,etc.,whichcouldbeimplementedintofutureupgradesofthemodelspresentedhere. 2.3Dissolutionandtransport Dissolutionratesinkarstenvironmentsaredeterminedbythe reactionkineticsattherock–waterinterfacesurfacecontrolleddissolution,bydiffusiontransportofionicspecies betweentherock–waterboundaryandthebulksolution transportcontrolleddissolution,and,inthecaseofcarbonates,bytherateofCO 2 hydrationKaufmannandDreybrodt, 2007.Eachofthesemechanismscanberatelimitingunder certainconditions. Intheearlyevolutionofconduitnetworks,thewaterin proto-conduitssubmillimeterstofewmillimetersinsize isclosetoequilibriumwiththemineralbeingdissolvedand dissolutionismostlysurfacecontrolled,byhigher-orderkineticsincaseoflimestoneandgypsum.Inturbulentow conditions,forcasesdiscussedinthiswork,dissolutionin limestoneispredominantlysurfacecontrolledbyrstorder kinetics,iftheinputsolutionhasalowsaturationratio.Some issuesrelatedtolimestonedissolutionratesinturbulentow stillremainopen;scallopedwallsoflimestonecavessuggest thattransportcontrolmightplayanimportantroleunderturbulentowconditionsaswellCovington,2014. Forthesereasonswesimplifythedissolutionkineticsby assumingalinearratelawattherock–waterboundary: F s D s )]TJ/F23 9.9626 Tf 4.015 -8.01 Td [(c eq )]TJ/F23 9.9626 Tf 9.431 0 Td [(c s ; where s isthekineticconstant, c eq istheequilibriumconcentrationofionicspeciesoftherockformingmineraland c s theiractualconcentrationatthesurfaceofthemineral. Ionsaretransportedfromthesurfaceintothebulkthrough adiffusionboundarylayerDBLofthickness " Dreybrodt andBuhmann,1991.ThetransportratethroughtheDBLis givenby F t D t . c s )]TJ/F23 9.9626 Tf 9.431 0 Td [(c / ; where t is t D D=": D isadiffusioncoefcient, " thethicknessoftheDBLand c theconcentrationinthebulksolution.EquatingEqs. andgivesanequationfor c s andanexpressionforthe effectiverates: F D )]TJ/F23 9.9626 Tf 4.015 -8.01 Td [(c eq )]TJ/F23 9.9626 Tf 9.432 0 Td [(c I D t s s C t : t dependsonthethickness, " ,oftheDBL,whichisrelated tothethickness, h ,oftheviscoussub-layerbytheSchmidt numberSchlichtingandGersten,2000: " D h Sc )]TJ/F64 7.5716 Tf 5.905 0 Td [(1 = 3 ;Sc D D ; where iskinematicviscosityand Sc theSchmidtnumber, whichrepresentstherelationbetweentheviscousdiffusion rateandmassdiffusionrate.Thethicknessofaviscouslayer overaatwallisgivenbyIncroperaandDeWitt,2002 h D 5 p ! = ; where ! isviscousshearstressatthewalland isthewater density. Viscousshearstressisrelatedtothefrictionslope S f ! D gS f R; where g isearth'sgravitationalacceleration.TakingtheManningrelationEq.1for S f andinsertingEq.intoEq., gives h D 5 R 1 = 6 nV : InsertingEq.intoEq.andfurtherintoEq.,weget anexpressionfor " andforthetransportconstant t : t D n V D 2 = 3 )]TJ/F64 7.5716 Tf 5.905 0 Td [(2 = 3 5 R 1 = 6 : Mostcasesthatwepresentinthisworkassumethat s t , sothat t .Therefore,thedissolutionratesaretransportcontrolled.Usuallyhigherowratesbringwiththem strongermixing,lowerbulkconcentrationsandhigherdissolutionrates.Inmostsituations,theruleofthumbisthefollowing:thehighertheow,thehigherthedissolutionrate. Theionsenteringthewaterincreaseitssaturationstate withrespecttothemineralformingthewalls,anddiminish dissolutionratesalongtheowpathways.Theincreaseof concentrationwithineachconduitisdescribedbyadifferentialequationderivedfromamassbalancewithinaninnitesimalsegmentofconduit: d c d x D F.x/ P.x/ Q ; where F.x/ isdissolutionrateatacoordinate x alongaconduit, Q theowrateand P.x/ ,theconduit'sperimeterat x . IntegrationofEq.alongaconduitgivestheamount ofrockdissolvedwithintheconduit.Thedissolvedloadis addedtothedownstreamjunctionoftheconduitandisthen treatedasaconservativetracerbythepollutantroutingcode ofSWMM. Inmostscenariospresentedinthiswork,transportcontrolleddissolutionprevails.Therefore,dissolutionratesare dependentontheowvelocity.Acase,wherethedissolutionratesarealmostentirelysurfacecontrolled,isalsopresented.Physicalconstantsandparametersusedforcalculationofow,dissolutionandtransportthroughoutthiswork, arelistedinTable1. Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4623 Figure5. Growthofaconduitwithpressurizedowandaconduitwithfree-surfaceow. r isradius, k isthefractionofwetted perimeter, v incisiongrowthrate. 2.4Dissolutionenlargement Dissolutionratesareratesofdissolutionenlargement v in [LT )]TJ/F64 7.5716 Tf 5.905 0 Td [(1 ].Inpressurizedconduits,thecrosssectionchanges uniformlyduringdissolutionFig.5.Inatimestep 1t ,a conduitenlargesby v1t ,whileitscenterremainsattheinitialposition.Foraconduitwithafree-surfaceow,only thewettedpartofthewallisdissolved.Therefore,atransitionfromtubetocanyon-likechannelisexpected.Although SWMMallowsarbitrarychannelgeometries,thetubeshape isusedalsoduringthevadoseconditionsinourmodel.To thisextentanapproximationisused,wherethebottomofa conduitwithafree-surfaceowinciseswiththetruerate v anditsradiusincreaseswithrate k v ,where k isthewetted fractionoftheconduitperimeter.Thecenteroftheconduit lowerswiththerate )]TJ/F23 9.9626 Tf 9.432 0 Td [(k/v . 2.5Themodelstructure Twobasicsettingsarepresented:rstamodelofalow-dip networkispresentedasconceptuallyshowninFig.3.This scenarioisusedtoexaminetheevolutionofconduitnetwork inaplanview.Inasecondscenario,ahighlyinclinedhighdipnetworkismodeledtoexploretheverticalorganizationof owpathways,orevolutionoftheconduitnetworkindimensionoflengthanddepthsensulatoFordandEwers,1978. Figure6introducesamodelstructureforthelow-dipnetwork.Circularconduitswithlength L andinitialdiameter D areassembledinaninclinedrectangulargrid.Theorientation ofthegridplaneismarkedgeographically,N,E,SandW.All conduitsare10mlong,withinitialdiametersontheorderof afewmillimeters.Waterentersthesystemthroughselected junctionsindicatedbyarrowsonFig.6aandowsouton theeasternboundary.Figure6bpresentsjunctiongeometry: eachjunctionisdenedbyaninvertelevation h 0 ,relativeto thebaselevel,aninletoffset h c ,whichistheelevationof theconduitinletrelativetotheinvertelevation,and h max ,the maximaldepthofwaterinthejunction.Ifthehydraulichead atajunctionisabove h max ,thejunctionsurcharges. Figure6cshowsasideviewofthemodel.TheinvertelevationsincreasefromEtoW,1mperjunction.Theslopeof theW–Econduitsistherefore0.1andN–Sorientedconduits arehorizontal.Theinletoffsetdeneshowmuchaconduit canincise.Tokeepconduitsfrombottomingoutastheyincise,theinletoffsets, h c ,aresettoalargevalueof100m. Maximaldepthatjunctions h max is120mforall,exceptfor theinputjunctionswhere h max is111m.Thereisnostorage atthejunctions. EachofthejunctionsontheEboundaryareconnectedto alargeconduit D D 5mthatfreelydrainswatertotheoutfallseeFig.6c.Theseconduitsplaynoroleinthenetwork genesis.TheirroleistoeffectivelydrainallthewaterarrivingtotheEjunctions.Theinvertsofthesejunctionsareat thebaselevelandsoistheinletoftheoutfallconduit.This waythejunctionsontheEboundaryallowafreeoutowof thesystemalongthatface. Inthehigh-dipmodelFig.7,theslopeofthenetwork andthereforetheconduitsis0.99fromtoptobottomand 0.1fromlefttoright.Weusetheexpressionsverticalforthe steepconduitsandhorizontalforthegradualones.Waterentersonthetopsideandexitsattheseepagefaceontheright side.Thebottomandleftboundariesareimpermeable.Inall junctions,gradualhorizontalconduitsarepositioned1m abovethesteepverticalconduits,whichassurespreferentialowalongtheverticalplaneinvadoseconditionssee Fig.7b.Flowalongthehorizontalconduitsisactiveonly whenthejunctionisoodedabovetheirinlets.Theoutow isrealizedasinthelow-dipcase,withlargeconduitsconnectingjunctionstooutfallsontherightboundary. 3Results 3.1Low-dipnetworks Westartwithasimplescenariowhereallconduitshavethe samelengthm,thesameinitialdiameter.005mand thesameinletoffsets.ThenetworkdipsfromWtowardsthe free-outowboundaryontheEsidewiththeslope0.1.The modelisrunfor50stepsof300s,intotal15000s.Therock usedissalt. Figure8presentssixsnapshotsofthenetwork'sevolution.Fiveinputswith Q max D 1000Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 aremarkedbycirclesanddenotedby1onFig.8a.Theleftcolumnshows owratesandowdirections.Flowratesaredenotedbyline thicknessesandowdirectionsbycolor;redrepresentsow towardsNorWandblacktowardsSorE.Iftheowispressurized,thecolorsaresaturated;palecolorsdenoteconduits withfree-surfaceow.Therightcolumnrepresentschannel diametersbylinethicknessesandgrowthratesbycolors; thebrighterthecolorthehighertherateofconduitdiameterincrease.Theisolinesintheguresrepresentthetotal www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4624M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Figure6. Themodelstructureforthelow-dipnetwork. a Aconduitnetworkwithdiscretewaterinputs,markedbyarrows.Boundariesare denotedgeographically.OutputsarealongtheEboundary. b Geometryandparametersofajunction. c Thesideviewofthemodel,also showingalargeconduitconnectingEjunctionstoanoutfall. Figure7. Themodelstructureforthehigh-dipscenario. a The slopeofthenetworkis0.99infromtoptobottomand0.1fromleft toright.Therightboundaryisaseepagefacewithfreeoutow.Inputsareonthetop. b Junctiongeometry:high-dipverticalconduitsarepositionedbelowthelow-diphorizontalconduits. hydraulicheadswithnumbersgiveninmetersandacontourintervalof1m.Theheadsaredirectlycalculatedatthe junctionsandinterpolatedbykrigingelsewhere.Notethat equipotentiallinesforthejunctionsontheEborderarenot given,astheconduitleadingtotheoutfallisatthebaselevel andlargeenoughtoalwayskeepthewaterinthesejunctions low. Figure8ashowstheinitialsituation.Allinputsareatthe maximalhydraulicheads,andonlyasmallpartofavailable rechargeentersthenetwork.Highgradientdrivesfastgrowth ofW–EconduitsfromIn1andIn2Fig.8bandc.Also, pathwaysheadingNandSfromIn1andIn2evolveinthe pressurizedowregime.TothewestofIn1andIn2,thedevelopmentisstillslow,asthepotentialeldattenstowards W.OnFig.8c,theconduitsdrainingIn1andIn2arepressurizedandexhibitlargeowandwideningrates.Thegradients fromIn3towardstheEboundarybuildupanddrivetheevolutionofpathwaysfromIn3towardstheeast.WhenpathwaysfromIn1andIn2aretoolargetosustainpressurized ow,thehydraulicheadinthemdropstotheirtopographic heightwhichattractsadditionalowfromIn3.Withfurther time,theevolutionprogressesupstream.TheowinpathwaysdrainingIn4andIn5alsoincreases;itpredominantly followsthestraightW–Eline,althoughitisalsoclearlyattractedbyvadosepathwaysleadingfromIn3. Nevertheless,mostoftheowfromupstreaminputsoccursalongadirectlineofW–Eorientedconduits,which evolvemostefcientlyFig.8candd.InFig.8e,theIn3 hasbecomevadoseandinasimilarmannernowattractsow fromIn4andIn5.However,thedirectlineconnectingIn4 totheboundarytakesmostoftheowandgrowsmostefciently.Figure8fshowsthenalstableowconguration. Alltheinputsdraintheavailablerecharge,withthedirect Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4625 Figure8. Sixsnapshotsoftheevolutionoflow-dipnetwork withuniforminitialdiametersandinletoffsets.Left:owrates widthandowdirectionred D owtowardsEortowardsN, black/gray D owtowardsWortowardsS.Right:diameters widthandwideningratescolor.Thecodesbelowshowthicknesses,owratesandwideningrate.Thevaluesatthebarcodes correspondtothethickestlinesintheowrateanddiameterbars andtothewarmestcolorinthebarforthewideningrate.Thescales arelinearwiththethinnestlinesanddarkbluecolorsrepresenting noow,nowideningtheandthesmallestinitialdiameter. pathwaysbetweentheinputsandtheEboundarybeingthe onlyonesthatcontainactiveow. AdetailedlookatFig.8revealsthatatanytime,lookingat theconduitsdrainingaparticularnode,thehighestowrates arealongW–Econduits,whichconsequentlyevolvemore efcientlythanotherconduits.TheinletoffsetsofW–Econduitsincisefasterthanothersandeventuallythewaterlevel atthejunctionfallsbelowtheloweredgesoftheotherconduits,leavingonlytheW–Econduitsactive.ThisisschematicallyshowninFig.9a,wheretwooutletsfromajunctionare compared;outlet1evolvesmoreduringthephreaticstage and,therefore,thebottomoftheconduitreachesalowerelevation.Consequently,outlet1ultimatelycapturesallwater duringthevadoseentrenchment.Severalotherrealizationsof thisscenariowithdifferentrechargeratesattheinputshave endedwiththesamenaldistributionofactiveconduits. Atthispointashortnoteisneededtoexplainwhatis meantbyastableowconguration.Inthecaseofconstantrecharge,thecongurationisconsideredtobestable whenalljunctionsaredrainedbyoneconduitonly,i.e.,there arenodownstreambifurcationsremaining.Thisisthecase inFig.8f.Inmostoftheotherpresentedmodelrunsafew outowbifurcationsremainatthelastpresentedtimestep. Thesebifurcationswouldeventuallydieoutifthemodelwas runlongenough.Wewillusethetermquasi-stabletodescribesuchsituations. Thenextsteptowardslessidealizedscenariosistoassumethattheinitialinletoffsetsofconduitsarerandomly distributedwithintherangeof1m.Figure10showsthenetworkwhenaquasi-stableowpatternhasbeenestablished, whichisnowmorecomplexthaninthepreviouscase.The generalevolutionissimilar,progressingupstream,butsome NandSorientedconduitsmayhaveinitialinletslowenough tokeepthelowestpositionuntilthevadosetransitionoccurs andtheycapturealltheowfromajunction.ThisisschematicallyillustratedinFig.9b.Figure11presentstheevolution ofanetworkwithinitialconduitdiametersdrawnfromauniformdistributionwitharangeof10 )]TJ/F64 7.5716 Tf 5.906 0 Td [(4 to10 )]TJ/F64 7.5716 Tf 5.906 0 Td [(2 m.Initialoffsetsarethesameforallnodes. Generally,theevolutionfollowstheconceptsdescribedin Fig.8.Inthepressurizedphase,theselectionofefcient pathwaysdependsalsoontheconduitdiametersandthe W–Econduitsarenotnecessarilytheoneswiththehighest owrates. Figure12showstheevolutionoftotaldischargefromthe networkovertime.Initially,mostoftheavailablerecharge owsoverthesurface.FirstIn1andIn2integratewithfull recharge,whichaddsupto2m 3 s )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 .Afterthegradientfor In3isincreased,In3integratesandthedischargerisesto 3m 3 s )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 .ThenpathwaysfromIn4andIn5starttocontribute asthesetwopathwaysintegrate. Anotherselectionmechanismbecomesactiveatthetransitiontoafree-surfaceow,whichisshowninFig.13,where afewsnapshotsoftheSWpartofthenetworkshowtheevolutionofseveralcompetingpathwaysevolvingfrominput www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4626M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Figure9. Leftpanel:thegeometryofajunction.Rightpanels a,b :schemeoftwooutowsduringpressurizedowtoppanelsand free-surfaceowbottompanels. a Initialinletoffsetsforbothoutowsareequal. b Initialinletoffsetofoutow2issmallersothat theoutowhasalowerelevation.Bluearrowsindicatetheamountofowdrainedbyeachoutow,andtheblueshadingindicatesthewater table. Figure10. Anetworkwithuniforminitialdiametersandinitialinlet offsetsrandomlydistributedwithinverticalspanof1m. In5.Thejunctionsofinterestaremarkedby1to3and enclosedingraycirclesat4800s.Inthepressurizedow regimes,theN–Sorientedconduits,markedby a , growfasterthantheW–Eorientedconduitsmarkedby b at allthreejunctions,becauseconduits a belongtopathways withsmallerresistancetoow. Whentheowispressurized,theowpartitioningbetweentwocompetingpathways,connectingthesamejunctionsisdividedbasedontheresistancetoow.Notethat conduits b areparalleltothedipofthenetwork,whileconduitsdenotedby a areperpendiculartoit.Theslopeofindividualconduitsandthedistributionofslopesalongthepathwaysplaysnorole.Thisisnotthecaseinafree-surfaceow regime,wheretheslopeoftheconduitthatdrainsthenodeis important.Whenajunctionbecomesvadose,theowoutof thejunctionthroughinitiallylarger,butlesssteepconduits canberedistributedtomorefavorablesteeperconduits.This leadstodownstreamredistributionofowwhichcanmake partofthenetworkinactiveorchangetheowfrompressurizedtofreesurfaceorviceversainsomeoftheconduits.The describedsituationisschematicallyshowninFig.14,where twopathways, a and b connecttwonodes.Pathway a is Figure11. Evolutionofalow-dipnetworkwithrandomlydistributedinitialdiameters. Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4627 Figure12. Thetimeevolutionoftotaldischargefromthenetwork inFig.11. initiallylarger,drainsmoreowandwidensmoreefciently inthepressurizedphase.Whentheconduitturnsvadose,the owratesin a dropduetothelowslopeofthechannelasit leavesthejunction.If,atthetransitiontofree-surfaceow, thewaterlevelintheupstreamnodehasnotdroppedbelow theinletofpathway b ,thesteeperentryintopathway b asit leavesthejunctionwillcause b toincisefasterandprogressivelycapturemoreow. Figure15presentsaquasi-stableowandnetworkpatternforthecaseidenticaltotheonepresentedinFig.11, butwheretheplaneofthenetworkisadditionallytiltedfrom NtoSfor0.3mpernode.Thetiltingmakesowtowards SpreferentialtoowtowardsN,whichisclearlyseeninthe resultingpattern.TheinputIn4nowjoinsIn3.Sinceitisnear theboundary,theinputIn5hasnooptiontodeveloptowards S,exceptthatthepathwayheadingSfromtheinputconduit a atIn5inFig.13nowpersistsmuchlonger. Otherscenarioswithmorecomplexsettings,suchas networkswith50 50nodesandnetworkswithirregular recharge,weremodeledandadditionallyconrmedtheobservationsgivenabove. Finally,weturntoanetworkwheredissolutionrateispredominantlysurfacecontrolled,asissupposedtobethecase forlimestone.Tothisendwehavemodeledanetwork,identicaltotheoneinFig.11,butwith s , c eq and D setsothat dissolutionratesareseveralordersofmagnitudesmallerand almostentirelydependonthesaturationstateofthesolutionratherthanowvelocity.Sincethesystemisinthepostinceptionstagetheratioofdischargetoowlength Q=L inmanyowpathwaysishighenoughthattheyevolvewith themaximalgrowthrates.Allconduitsandchannelsalong thesepathwaysincisewiththesamerate.Figure16showsthe situationat500years,whenaquasi-stableowpatternhas evolvedandthecompletenetworkisvadose.Allactivechannelswithowhavealmostthesameinletoffsetsandthesame incisionrates.Notethatthecolorstelltherateofincreaseof diameter,whichisaproductbetweendissolutionratewhich isveryuniformincaseofsurfacecontrolledratesandthe fractionofconduitbeingooded.Therefore,colorsinthis guremostlytellhowfulltheconduitsare;seealsodiscussioninSect.2.4.Theresultingowpatternis,asidefromthe initialdistributionofdiametersandboundaryconditions,a consequenceoftworules:ateachnode,channelsaligned orientedwiththedip,drainmoreowthanchannelsperpendiculartothedip;ifonlyhorizontalchannelsdrain thenode,owisdistributedevenly.Thepresentedscenario ishighlyidealisticandtheresultsandinterpretationshould betakenwithcare.Innature,thedissolutionrateschange withchanginglithology,theinitialoffsetsarenoteven,sedimentscanplayimportantrole,andwemayquestionifpurely surfacecontrolledratesarereasonable.However,themodel supportstheideasofPalmerPalmer,1991,thatmazecaves developinsituationswhere Q=L islargealongmanyalternativeroutes. 3.2High-dipnetwork Wenowturntothesituationwherethenetworkissteepalmostvertical.Asthisnetworkpresentsaverticalcrosssectionofkarst,weomitthegeographicalnotationandusetop, bottom,leftandrightforthesidesofthenetworks. Similarmodelsforlaminarowhavebeenpresentedby GabrovsekandDreybrodtandbyKaufmann. Thebasicresultofthesepriormodelswasacontinuousdrop ofthewatertableduetoincreasedtransmissivityofthenetworkandtheformationofbaselevelconduits.Ifaxed headboundarywasapplied,competitionbetweenahighconductivityzonealongthewatertableandprominentconduits withinthephreaticpartofthenetworkresultedinacomplexpatternofevolvedconduits.Formanymorescenarios ofthismodelingapproachthereaderisreferredtothebook byDreybrodtetal. 3.2.1Thehomogenouscasewithrechargedistributed overthetopnodes Figure17presentsacasewhereallconduitsare10m longwithinitialdiameterof0.005m.Amaximumpossible rechargeof5Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 isdistributedtoallinputnodesbluearrowsinFig.17aonthetop.Theleftcolumnshowsow ratesaslinethicknessesandcolors,asdenotedinthelegend,atvedifferenttimesteps.Althoughthetermwatertablemightnotbeapplicableforsuchdiscretenetworks,we willuseitforthelinealongthehighestoodednodesdottedbluelinesinFig.17candd.Therightcolumnshowsthe conduitdiametersascodedinthecolorbarforeachgure. Equipotentiallinesintheleftcolumnshowthedistribution ofhydraulichead,giveninmeters. InitiallyFig.17a,asmallpartoftheavailablerecharge entersthenetwork.Atthetop-rightalltherechargeisdrained directlyintotheoutfalljunctionmarkedbyaredcirclein Fig.17a.Theowrateswithintheconduitsaresmalland www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4628M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Figure13. EvolutionofSWedgeofthenetworkinfromFig.11beforeandaftertransitiontofree-surfaceow. Figure14. Distributionofowbetweentwopathwaysdependson theowresistancewhentheowispressurized.Thepathway a has withlowerowresistancegrowsfaster.Afterthetransitiontofreesurfaceow,thepathway b withhigherexitslopefromthejunction cancapturemoreowandincisefaster. Figure15. Quasi-stablestateofnetworkwithsamestructureaspresentedinFig.11,buttheplaneofthenetworkisadditionallytilted fromN–S,for0.3mpernode. dominantalongtheverticalconduitstoptobottom.Flow alonghorizontalconduitsissmallandincreasesfromleftto right. After600sFig.17btheentirenetworkisstillpressurized.Horizontalconduitshaveevolvedsufcientlytodrain moreowbroughtinbyinitiallydevelopedverticalconduits. Accordingly,thepotentialgradientbecomesorientedtothe rightandisthehighestclosetotheboundary.Conduitsatthe top-rightcornerexperiencethefastestgrowthandcapturealmostallrechargefromtheinputs.Theowintheleftpartof thenetworkissmallandthehydraulicpotentialeldisrelativelyatthere.After1200sFig.17cthetop-rightcorner hasbecomevadose.Inthisarea,therechargeiscarriedverticallytothewatertable.Theowratesarethehighestalong thewatertableanddiminishwithdistancefromit. Figure16. Quasi-stablestateforthesamescenarioasinFig.11 withdissolutionkineticsforlimestone. However,wideningisstillsubstantialbelowthewatertablewhichadditionallyincreasesthenetworkpermeability anddownwardsretreatofWT.Theprocesscontinuesuntil theWTdropstothebaselevelandonlyverticalrecharge conduitsandamasterconduitatthebasecontinuetogrow. Theverticalconduitshavebeenwidenedthroughtheentire evolution;theuppermostforthelongesttimeandtheyare thereforelargest.Thediametersdecreasefromtoptobottom.Ontheotherhand,thediameterofhorizontalchannels increasesfromlefttoright,astheyevolveonlybelowthe watertable.Therefore,deeperconduitshavemoretimeto evolve. 3.2.2Inhomogeneouscase InthecaseshowninFig.18weassignamorecomplexdistributionofinitialconduitdiameters.Theinitialdiameter d o ofeachconduitisconstructedasasumofagroupcontribution d g whichisgiventoallconduitsalignedalongthe sameline,andanindividualcontribution d i .Theseareboth random,sampledfromauniformdistribution,where d g 2 [0, 0.005m]and d i 2 [0,0.01m].Theprobabilitythatconduits alongacertainlinegettheindividualcontributionsis0.5. Usingthisgroupcontribution,weenhancethepotentialimportanceofconductivestructurallines. Theinitialdiameterofthetophorizontallineofconduits is0.1m. Arechargeof100Ls )]TJ/F64 7.5716 Tf 5.905 0 Td [(1 isintroducedtothetop-leftjunctionseethebluearrowinFig.18a.Thetwogivenlegendsforowratesanddiametersarevalidforallgures.At 3000sFig.18a,aboutonefourthoftheavailablerecharge Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4629 Figure17. Evolutionofhomogenoussub-verticalnetwork.Bluearrowson a denoteinputs.Isolinesandvaluespresentthehydraulic potential[m]. www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4630M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow iscapturedanddraineddirectlytotheoutfallbythetopline ofhorizontalconduits. Pathwaysalongtheconduitswithinitiallylargerdiameters evolveefcientlyandcaptureanincreasingamountofow. At9000sFig.18babout70%oftheowiscapturedby thejunctionmarkedbyabluetriangleanddenotedby1in Fig.18b.Itfeedsalineofverticalconduitthatdischargesinto outowsthroughhorizontalconduits.Numbersontheconduitsinthetop-rightregiondenoteowalongtheconduitsin Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 .Thedischargetotheoutowdiminishesdownwards. However,theseconduitswideneffectivelyandcannotsustainapressurizedregime,sothatthepositionofthehighest outowmigratesdownwards. By24000s,theoutowpositionhasretreatedtothebottomFig.18c.Whentheverticalpathwaydownwardsfrom point1becomesvadose,itprovidesafree-outowboundaryandtriggersthedevelopmentofpathwaysdrainingsink points2and3Fig.18bandc,whichsooncaptureall theow.InFig.18c,theowalongthetoplinehasretreatedtopoint3,andthroughouttheremainderofthesimulationcontinuestoretreattowardsthelefttopoints4and5 Fig.18d.Ultimately,theowiscapturedbythenodeat point5Fig.18e.Similarly,theowmigratesfromtopto bottom,towardsthedeeperconnectingpathways.Figure18e showsthestableowsituationat75000s,whereallthe owfollowsonesinglepathway.Downwardandleftward progressisslowbecausesomeoftheconduitstotheleft areinitiallysmallandthepermeabilityislow.Incomparison withauniformnetworkwithdistributedrecharge,thedevelopmentfollowsinitiallyprominentpathways,withprogressiveupstreamowcapturing.Soonafterapathwaybecomes vadose,theowisovertakenbytheevolvingpathwaystoits left. 3.2.3Theroleofprominentstructures Theprogressionmechanism,describedabove,isdemonstratedclearlybyanalidealized,buttelling,example.We assumethreeverticalconduitswellswithaninitialdiameterof0.2m,extendingcompletelythroughthedomaininthe verticaldirection. Theseareconnectedwithveevenlyspacedhorizontalconduitswithinitialdiameter0.005mextendingacross thedomain.Allotherconduitsareeffectivelyimpermeable, withadiameterof10 )]TJ/F64 7.5716 Tf 5.905 0 Td [(5 m.Amaximumpossiblerecharge of100Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 isavailabletotheprominentverticalconduits wellsasmarkedbythearrowsatthetopofFig.19a. InitiallyFig.19a,allconduitsarepressurized.Thereis almostnogradientleftofW3,whereevolutionisslowor none.HighgradientsexistbetweenW3andtheoutfalls,the highestbeingalongthedeepesthorizontalconduit,which hasthehighestowandevolvesmostefciently.AsW3 becomesvadose,itpresentsafree-outowboundaryforthe owfromitsleftandthegradientalongthehorizontalconduitsconnectingW2toW3buildsup.Theseconduitsnow Figure18. High-dipnetworkwithrandominitialdistributionof conduitdiameters.Flowentersatthetop-leftedgeofthenetwork aspointedbyabluearrow.Valueson b showowratesalongthe selectedindividualconduits. experiencefastevolutionwithratesincreasingfromthetop tothebottomFig.19b.Themechanismprogressesleftwards:whenW2becomesvadose,W1connectstoitas showninFig.19c.InFig.19d,astableowconditionis shown,wherealltheowfollowsthewellswhichfeedthe baselevelchannel. Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4631 Figure19. High-dipnetworkwiththreeprominentconduitswells, markedbyW1toW3.Arechargeof100Ls )]TJ/F64 7.5716 Tf 5.906 0 Td [(1 isavailabletothe prominentconduits. 4Discussion 4.1Low-dipscenario SensulatoPalmerthispaperconsidersthehydrologicalcontrolofcavepatterns,particularlythoseleadingto branchworkcavesystems.Inthepressurizedphasethemodel givessimilarresultsastheotherexistingmodels.Thismodel introducestheselectionofowpathwaysonalocalscale, i.e.,ataparticularjunction,whichoccurswhenajunction becomesvadose.Inalong-termperspective,onlyoneoutlet conduitdrainsthenode.Innature,down-owbifurcations arenotcommoninopenchannels. Inthepressurizedphase,theowoutfromajunction isdistributedtotheoutletconduits,accordingtotheirresistancetoowandthedistributionofhydraulicheads. Thisalsodenestherateoftheirinletincision.Whena junctionbecomesvadose,theconduitwiththelowestinlet entryelevationhasanadvantageandisacandidatetotake alltheow.However,undervadoseconditionstheconduit's alignmentwithrespecttothedipofthenetworkbecomes important,ashigherslopegenerallyinvokeshigherenergy grade,higherowvelocityandfasterincision.Aconduitthat gainsadvantageinpressurizedconditionscanbesurpassed byaconduitwithahigherslope,whichhasanadvantagein free-surfaceconditions.Oncethestableowpatternisestablished,theowfollowsasystemofconduitsthatalloccupy thelowestpositionintheirupstreamjunctions. 4.2High-dipscenario Inahomogenousscenario,theevolutionisfocusedtothe transitionalareabetweenpressurizedandfree-surfaceow, thewatertable.Theowfromthesurfaceisgravitational alongthevadosechannelsdowntothewatertable.There, itislargelyfocusedtotheconduitsclosetothewatertable. Thescenariodemonstratesarelativelysmoothdrawdownof thewatertableduetoincreasingpermeabilityinthephreatic zone.Theendresultisarelativelyuniformnetworkwitha growingbaselevelconduit.Similarresultswereobtainedby GabrovekandDreybrodtandbyKaufmann, whereonlydissolutioninthephreaticzonewasconsidered. Theinhomogeneouscasedemonstratestheevolutionofinvasionvadosecavesbasedonowdiversion.Thedrawdown ofthephreaticzoneisirregular,followingfastevolutionof prominentpathwaysandprogressiveupstreamowcapturing.Suchascenariocanproduceanextendednetworkof steepvadosepassages. Deeplypenetratingconductivestructurescanplayanimportantroleastheytransfersurfacewaterdeepintothemassifandredistributehydraulicgradients.Thiswayfastevolutionalongdeephorizonscanbetriggered. 5Conclusions Thepresentedmodelclosessomeoftheopenquestions, whichhavenotbeenaddressedbytheolderexistingmodels.Thenalowpatternresultsfromallstagesofnetwork development,startingwiththeinitialstage,continuingwith thegrowth,integrationandexpansionunderpressurizedow aswellas,whatisdemonstratedbythismodel,withthenal selectionofstableowpathwaysonalocalscaleduringand aftertransitiontofree-surfaceowregime. Ontheotherhand,themodelopensnewchallengesrelated toevolutionofkarstaquifersinvadosesettings.Furtherwork isneededtoimproveestimationofdissolutionratesandthe relatedroleofsedimenttransportandmechanicalerosion. Furtherstepstowardsmorerealisticmodelingdomainand boundaryconditionsarealsoneeded.Infact,asinglelowdipplaneisascenariowhichisnotcommoninthenature.A carefulsteptowards3-Dmodelsthatsimulatespeleogenesis, inbothphreaticandvadoseconditions,isthereforeneeded. www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014

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4632M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow Whatwemeanbycarefulisthegradualaddingofcomplexity,sothatateachnewstepallmechanismsfromprevious stepsarewellunderstood.Thepresentedmodelallowssuch extensions. Atthesametime,wehavetokeepinmindthatthemodelingresultsarenotstandalone,i.e.,theyshouldprogress handinhandwithconceptualmodelsbasedontheeld observations. Acknowledgements. Thisworkwouldnothavebeenpossible withoutthesupportofSlovenianResearchAgency,whonanced thedoctoralworkofM.Perne.TheworkofF.Gabrovekwas fundedbytheprojectJ2-4093.M.CovingtonandM.Pernewere alsosupportedbytheNationalScienceFoundationundergrant no.1226903.WethanktoDerekFordandSteffenBirkforthe valuablecomments,whichhelpedtoimprovethemanuscript. Editedby:S.Attinger References Annable,W.K.:Numericalanalysisofconduitevolutioninkarstic aquifers,Ph.D.,UniversityofWaterloo,Waterloo,139pp.,2003. Annable,W.K.andSudicky,E.A.:Simulationofkarstgenesis:hydrodynamicandgeochemicalrock-waterinteractionsin partially-lledconduits,Bulletind'Hydrogeologie,16,211, 1998. Audra,P.,Bini,A.,Gabrovsek,F.,Hauselmann,P.,Hoblea,F.,Jeannin,P.,Kunaver,J.,Monbaron,M.,Sustersic,F.,Tognini,P., Trimmel,H.,andWildberger,A.:Caveandkarstevolutionin theAlpsandtheirrelationtopaleoclimateandpaleotopography, ActaCarsolog.,36,53,2007. Audra,P.,andPalmer,A.N.:6.17TheVerticalDimensionof Karst:ControlsofVerticalCavePattern,in:TreatiseonGeomorphology,editedby:Shroder,J.F.,AcademicPress,SanDiego, 186,2013. Covington,M.D.:Calcitedissolutionunderturbulentowconditions:aremainingconundrum,ActaCarsologica,43,159, 2014. Cunge,J.A.andWegner,M.:NumericalintegrationofBarrde Saint-Venant'sowequationsbymeansofanimplicitschemeof nitedifferences,LaHouilleBlanche,33,1964. Dingman,S.L.:Physicalhydrology,PrenticeHall,UpperSaddle River,N.J.,646pp.,2002. Dreybrodt,W.:Theroleofdissolutionkineticsinthedevelopment ofkarstaquifersinlimestone–Amodelsimulationofkarstevolution,J.Geol.,98,639,1990. Dreybrodt,W.:Principlesofearlydevelopmentofkarstconduits undernaturalandman-madeconditionsrevealedbymathematicalanalysisofnumericalmodels,WaterResour.Res.,32, 2923,1996. Dreybrodt,W.andBuhmann,D.:AMass-TransferModelForDissolutionAndPrecipitationOfCalciteFromSolutionsInTurbulentMotion,Chem.Geol.,90,107,1991. Dreybrodt,W.andGabrovsek,F.:Dynamicsoftheevolutionof asinglekarstconduit,in:Speleogenesis:Evolutionofkarst aquifers,editedby:Klimchouk,A.,Ford,D.C.,Palmer,A.,and Dreybrodt,W.,NationalSpeleologicalSociety,Huntsville,Alabama,184,2000. Dreybrodt,W.,Gabrovsek,F.,andRomanov,D.:Processesof speleogenesis:Amodelingapproach,Carsologica,editedby: Gabrovsek,F.,ZalobaZRC,Ljubljana,375pp.,2005. Eisenlohr,L.,Meteva,K.,Gabrovsek,F.,andDreybrodt,W.:TheinhibitingactionofintrinsicimpuritiesinnaturalcalciumcarbonatemineralstotheirdissolutionkineticsinaqueousH 2 O-CO 2 solutions,Geochim.Cosmochim.Acta,63,989,1999. Epting,J.,Romanov,D.,Huggenberger,P.,andKaufmann,G.:Integratingeldandnumericalmodelingmethodsforappliedurbankarsthydrogeology,Hydrol.EarthSyst.Sci.,13,1163, doi:10.5194/hess-13-1163-2009,2009. Ewers,R.:Caverndevelopmentinthedimensionoflengthand breadth,McMasterUniversity,Hamilton,Ontario,1982. Ford,D.C.:Characteristicsofdissolutionalcavesystemsincarbonaterocks,in:Paleokarst,editedby:James,N.P.andChoquette, P.W.,Springer,NewYork,25,1988. Ford,D.C.andEwers,R.:Thedevelopmentoflimestonecavesin thedimensionsoflengthanddepth,CanadianJournalofEarth Sciences,15,1783,1978. Ford,D.C.andWilliams,P.:KarstHydrogeologyandGeomorphology,JohnWiley&Sons,Chichester,562pp.,2007. Gabrovek,F.andDreybrodt,W.:Amodeloftheearlyevolutionof karstaquifersinlimestoneinthedimensionsoflengthanddepth, J.Hydrol.,240,206,2001. Gabrovek,F.andPeric,B.:Monitoringtheoodpulsesinthe epiphreaticzoneofkarstaquifers:ThecaseofRekariversystem, Karstplateau,SWSlovenia,ActaCarsolog.,35,35,2006. Groves,C.G.andHoward,A.D.:Earlydevelopmentofkarstsystems,1.Preferentialowpathenlargementunderlaminar-ow, WaterResour.Res.,30,2837,1994. Halihan,T.,Wicks,C.M.,andEngeln,J.F.:Physicalresponseofa karstdrainagebasintooodpulses:ExampleoftheDevil'sIceboxcavesystemMissouri,USA,J.Hydrol.,204,24,1998. Hanna,R.B.andRajaram,H.:Inuenceofaperturevariabilityon dissolutionalgrowthofssuresinkarstformations,WaterResour.Res.,34,2843,1998. Hiller,T.,Kaufmann,G.,andRomanov,D.:Karsticationbeneath dam-sites:Fromconceptualmodelstorealisticscenarios,J.Hydrol.,398,202,doi:10.1016/j.jhydrol.2010.12.014,2011. Hubinger,B.andBirk,S.:Inuenceofinitialheterogeneitiesand rechargelimitationsontheevolutionofaperturedistributions incarbonateaquifers,Hydrol.EarthSyst.Sci.,15,3715, doi:10.5194/hess-15-3715-2011,2011. Incropera,F.P.andDeWitt,D.P.:Fundamentalsofheatandmass transfer,J.Wiley,NewYork,981pp.,2002. Jeschke,A.A.,Vosbeck,K.,andDreybrodt,W.:Surfacecontrolled dissolutionratesofgypsuminaqueoussolutionsexhibitnonlineardissolutionkinetics,Geochim.Cosmochim.Acta,65,27, 2001. Kaufmann,G.:Modellingunsaturatedowinanevolvingkarst aquifer,J.Hydrol.,276,53,2003. Kaufmann,G.andBraun,J.:Karstaquiferevolutioninfractured, porousrocks,WaterResour.Res.,36,1381,2000. Hydrol.EarthSyst.Sci.,18,4617–4633,2014www.hydrol-earth-syst-sci.net/18/4617/2014/

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M.Perneetal.:Evolutionofkarstconduitnetworksintransitionfrompressurizedowtofree-surfaceow4633 Kaufmann,G.andDreybrodt,W.:Calcitedissolutionkineticsinthe systemCaCO 3 -H 2 O-CaCO 3 athighundersaturation,Geochim. Cosmochim.Acta,71,1398,2007. Lauritzen,S.-E.,Odling,N.,andPetersen,J.:Modellingtheevolutionofchannelnetworkincarbonaterocks,ISRMSymposium, Eurock'92,Chester,UK,57,1992. Liedl,R.,Sauter,M.,Huckinghaus,D.,Clemens,T.,andTeutsch, G.:Simulationofthedevelopmentofkarstaquifersusingacoupledcontinuumpipeowmodel,WaterResour.Res.,39,1057, doi:10.1029/2001WR001206,2003. Palmer,A.N.:Originandmorphologyoflimestonecaves,Geol. Soc.Am.Bull.,103,1-21,1991. Palmer,A.N.:Cavegeology,CaveBooks,Dayton,Ohio,vi, 454pp.,2007. Peterson,E.andWicks,C.:Assessingtheimportanceofconduitgeometryandphysicalparametersinkarstsystemsusingthestorm watermanagementmodelSWMM,J.Hydrol.,329,294, 2006. Romanov,D.,Gabrovsek,F.,andDreybrodt,W.:Damsitesinsolublerocks:amodelofincreasingleakagebydissolutionalwideningoffracturesbeneathadam,Eng.Geol.,70,17,2003. Rossman,L.A.:StormWaterManagementModel,Version5.0, USEnvironmentalProtectionAgency,Cincinnati,266pp.,2009. Schlichting,H.,andGersten,K.:Boundary-LayerTheory,8thEdn., Springer-Verlag,NewYork,802pp.,2000. Siemers,J.andDreybrodt,W.:Earlydevelopmentofkarstaquifers onpercolationnetworksoffracturesinlimestone,WaterResour. Res.,34,409,1998. Szymczak,P.andLadd,A.J.C.:Theinitialstagesofcaveformation:Beyondtheone-dimensionalparadigm,EarthPlanet.Sc. Lett.,301,424,2011. USEnvironmentalProtectionAgency:StormwatermanagementmodelSWMM:http://www2.epa.gov/water-research/ storm-water-management-model-swmm?,lastaccess: 20November2014. White,W.B.:Theroleofsolutionkineticsinthedevelopmentof karstaquifers,in:KarstHydrogeology,Memoir12,editedby: Dole,F.L.andTolson,J.S.,InternationalAssociationofHydrogeologists,Huntsville,Alabama,503,1977. Worthington,S.:Acomprehensivestrategyforunderstandingow incarbonateaquifers,in:Karstmodeling:Specialpublicattion5, editedby:Palmer,A.,Palmer,M.,andSasovsky,I.,TheKarst WatersInstitute,Charlestown,WestVirginia,30,1999. www.hydrol-earth-syst-sci.net/18/4617/2014/Hydrol.EarthSyst.Sci.,18,4617–4633,2014


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