Can one identify karst conduit networks geometry and properties from hydraulic and tracer test data?

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Can

one

identify

karst

conduit

networks

geometry

and

properties

from

hydraulic

and

tracer

test

data?

Andrea Borghi

c,∗

_{, Philippe Renard}

a

_{, Fabien Cornaton}

b

a University of Neuchâtel, Center For Hydrogeology and Geothermics (CHYN), 11 Rue Emile Argand, Neuchâtel 20 0 0, Switzerland b DHI-WASY GmbH, Waltersdorfer Strasse 105, Berlin 12526, Germany

c Gocad Research Group, Laboratoire GeoRessources, Université de Lorraine, Site de Brabois, 2 Rue du Doyen Marcel Roubault TSA 70605, Vandoeuvre-Lès-Nancy FR-54518, France

Keywords: Pseudo-genetic Stochastic modeling Karst conduits modeling Inverse problem Finite element simulation

a

b

s

t

r

a

c

t

Karstaquifersare characterizedby extremeheterogeneitydue tothe presence ofkarstconduits em-beddedinafracturedmatrixhavingamuchlowerhydraulicconductivity.Theresultingcontrastinthe physicalproperties ofthe systemimplies thatthe systemreacts veryrapidlytosomechangesin the boundaryconditionsandthatnumericalmodelsareextremelysensitivetosmallmodiﬁcationsin prop-ertiesorpositionsoftheconduits.Furthermore,onemajorissueinallthosemodelsisthatthelocation andsizeoftheconduitsisgenerallyunknown.Forallthosereasons,estimatingkarstnetworkgeometry andtheirpropertiesbysolvinganinverseproblemisaparticularlydiﬃcultproblem.

Inthispaper,twonumericalexperimentsaredescribed.Inthefirstone,18,000flowand transport simulationshavebeencomputedand usedinasystematicmannertoassess statisticallyifonecan re-trievetheparameters ofamodel (geometryand radiusof theconduits, hydraulicconductivityof the conduits)fromhead andtracerdata. Whentwotracertestdata setsare available,thesolution ofthe inverseproblemsindicatewithhighcertaintythatthereareindeedtwoconduitsandnotmore.The ra-diusoftheconduitsareusuallywellidentifiedbutnotthepropertiesofthematrix.Ifmoreconduitsare presentinthesystem,butonlytwotracertestdatasetsareavailable,theinverseproblemisstillable toidentify thetruesolutionas themostprobablebutitalsoindicatesthatthedataareinsufficientto concludewithhighcertainty.

Inthesecondexperiment,amorecomplexmodel(includingnonlinearflowequationsinconduits) isconsidered.Inthisexample,gradient-basedoptimizationtechniquesareprovedtobeefficientfor es-timating the radiusofthe conduitsand thehydraulic conductivity ofthe matrix inapromising and efficientmanner.

Theseresults suggest that, despite the numerical difficulties, inverse methods should be used to constrainnumericalmodelsofkarsticsystemsusingflowand transportdata.Theyalsosuggestthata pragmaticapproachfor thesecomplex systemscouldbe togeneratealarge set ofkarstconduit net-workrealizationsusingapseudo-geneticapproachsuchasSKS,andforeachkarstrealization,flowand transportparameterscouldbeoptimizedusingagradient-basedsearchsuchastheoneimplementedin PEST.

Introduction

Karst aquifers are extremely heterogeneous and difficult to characterize [2]. Their heterogeneity is induced by the presence ofhighlypermeablepreferential flow paths createdby the disso-lution ofthe surrounding rock.Those preferential flow paths are often fracturesand bedding planes that are enlarged by

dissolu-∗ _{Corresponding author. Tel.: +41584690546.}

E-mail address: [email protected] (A. Borghi).

tion,often resultinginto karsticconduits whichare organized in hierarchicalnetworks.

Annable [1] gives an exhaustive overview of the evolution of theconceptualmodelsofspeleogenesisoverthelasttwocenturies. Theconceptualmodelthatisconsideredinthepresentstudy[[63], e.g.]isthefollowing:thekarstaquiferiscomposedby2main hy-drofacies:the matrixwhichrepresentsmorethan 90%ofthe vol-umeoftheaquifer,andhasanimportantstoragerole;andthe con-duitswhichrepresentavery smallvolume, buthaveavery high importance forﬂow, since they are considered to be responsible formoreorless90%ofthetotalﬂow.

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Although karst aquifers have already been studied for more than a century they arestill very difficultto model[1].Overthe last 3decades,several numericalmodeling techniqueshavebeen developed to provide better understanding ofthese aquifers, but alsotomanagewaterquantityandquality.Areviewofthese tech-niques isprovidedbyGhasemizadehetal.[23].Astrikingfeature of this review is that only direct approaches, i.e. modeling flow and transportwhen thegeometryandthe propertiesofthe con-duitsareknown,aredescribed.Followingthesamelineofthought, Salleretal.[52] showthebenefitsofcouplingconduitflow(pipes) withmatrixelements,butpointsouttheuncertaintyregardingthe location oftheconduits(which stronglyinfluencestheflowfield) andtheextremedifficultyofcalibratingsuchmodels.

More generally, among the approaches used to solve inverse problems [57] ingroundwater hydrology,gradient-basedmethods are frequently used.Theyconsist in modifying iteratively the hy-draulic conductivity values either in predefined zones [10] until theerrorbetweenobservedandcalculatedstatevariables reaches a minimumorstabilizesatan asymptoticvalue.Thisisextremely efficient ifzonesofconstant butunknown hydraulic conductivity valuesarepredefined.Ifthespatialdistributionofthepermeability field itselfisunknown,itcanbeinferredaswellusingtechniques such aspilot points,gradualdeformation,orprobability perturba-tionmethods[9,30,48] whicharealsousinggradientoptimization. In the caseof karsticaquifers, the progressivedeformation of the conduit structure and especially topology to reach a maxi-mumlikelihoodorminimalerrorsolutionisparticularlydifficultto achieve. Preliminarytests conductedinthisresearch have shown that the application of methods such as the probability pertur-bation lead to extremely discontinuous behavior of theobjective function,makingtheuseofgradientoptimizationcompletely use-less. Thisisexplained bythe factthat duringtheprogressive de-formationofthegeometry,disconnectionandreconnection ofthe conduits occur leading to abrupt changes in the hydraulic and transportresponse.

Such diﬃculties explain partly why there are so few studies that considered applying inverse methods for karst aquifer dis-tributed parameter models. Notableexceptions areLarocqueet al. [40,41] andPanagopoulos[46] who usedinversemethodsto cali-brate karsthydrogeological models.But,bothgroup ofauthorsdo not includediscrete karst conduits intheir models andrepresent the karstaquiferwitha 2Dequivalentporousmedium.Moreover, inthesepreviousworks,transportdatahavenotbeenconsidered. Inthisperspectiveandbeforeconductingordevelopingany in-versemethodology,itisimportanttounderstandbetterwhich pa-rameters(likethehydraulicconductivityandporosity,ortheshape andthenumberofconduits)mainlycontrolthesimulationresults andthereforethevalueoftheobjectivefunction.

Moreover, the ability of a classical optimization technique to effectively (andpossibly efficiently) calibratethe physical proper-tiesofkarst aquifers hasalsotobe testedwhensimulating com-plexsystems,withDarcylaminarflowinthematrixandturbulent Manning–Stricklerflowindiscretepipes.

In thisperspective,thepresentpaperinvestigates whether in-verse methods could be used to obtain information about the structure ofthekarsticnetworkorthedistributionoftheconduit dimensions.

Two distinct numerical experiments are carried out. The first considers theinfluenceofchanginggeometryandtopology ofthe karst conduits onthe simulationresults.150differentgeometries with varying karst conduit radius and matrix hydraulic conduc-tivity are considered. All these models are generated using the pseudo-genetic algorithm previously developed by Borghi et al. [8].Inpractice,thetest isperformedbycomparingthe resultsof 18,0002DflowandtransportEPM(EquivalentPorousMedium) fi-niteelementssimulations(Sections3).Thesecondtestinvestigates

theabilityofaninversealgorithmsuchasPEST[17] toretrieve op-timalphysicalparameters whenusingamorecomplexmodel,i.e. a3Dmodelwithkarstconduitsmeshed as1D pipeelementsand nonlinearﬂowdynamicsintheconduits.

1. Literaturereview 1.1. Flowsimulation

Mathematicalblack-boxmodelsconsiderthewholeaquiferasa singlereservoirwhoseglobalbehaviorcanbedescribedwith sim-ple mathematical relations betweenan input signal and an out-put response: e.g. global parameter models [e.g. [42]], orneural networks[e.g.[29]]. Anexhaustive review ofthesekindof mod-els can be found in Ghasemizadeh et al. [23]. Unfortunately, as explained by De Marsily [14], thesemodels maybe suﬃcient to predictspringhydrographs,buttheydonotprovideanyspatial in-formationaboutthekarsticconduitsystem.

Asopposedtoblack-boxmodels,distributedparametermodels arebasedonthediscretizationofthemodeldomainintosub-units. Each sub-unit has homogeneous parameters in the space that it delimits(seeSection2.2).AsGhasemizadehetal.[23] say,the chal-lengeof distributed modelingapproachesto represent karst ground-watersystemsis tocopewiththehighspatial heterogeneityofkarst aquifers.Manyauthorshave alreadymodeledkarst aquifers using parameterdistributed models. Theeasiest wayisto consider the wholekarstaquiferasanequivalentporousmedium(EPM),where matrix,fractures and conduits are brought together inan equiv-alent hydrofacies (e.g.[6,39]). Unfortunately, EPM models havea very low applicability in very kartified fields and may lead to catastrophicsituationslikethecaseofWalkerton(Ontario,Canada) where,inMay2000,7peoplediedfromabacterialcontamination ofthemunicipal watersupplybecausethespringprotectionzone wasbasedonanEPMmodel,whichgavemuchlargertransittime thanwhatwasobserved(later)byfield tracertests(Worthington, [64];Goldscheider,[25];KresicandStevanovic[37]).

Toavoidthiskindofissues,otherauthorshavedeveloped mod-els,whichusetheavailableinformationabouttheconduit geome-trytoaddheterogeneityintheir model.Worthington[62] models the MammothCave aquiferusing MODFLOW, andhedefines the meshelements where karstconduits were explored ascells with higherhydraulicconductivity.Healsohadtoincreasethehydraulic conductivityaccordingtothehierarchyoftheconduitstobe able tosimulaterealisticheaddistributions.Király[33],Királyetal.[35] modeledtheconduitsasdiscrete1Dor2Dfeaturesembeddedina 3D matrixusinga discrete-continuumapproach,flow inconduits beinglaminar. Thediscrete–continuum approachis oftenused to developspeleogenesismodels [1,18,22].Thesespeleogenesis mod-elsareusefultounderstandthecomplexkineticsofkarstaquifers. In addition, as Jeannin[32] pointed out, turbulence is often ob-served inconduit flow. Nowadaysthere are computer codes that allows non linearflow equationsto be usedin discrete conduits, asMODFLOW-CFP[ConduitFlowProcess,[38]] orGROUNDWATER [12].DeRooij[15] developedamodelthatisabletosimulatealso unsaturatedflowinpipes.Recently,Salleretal.[52] showthe ben-efitsoftheapplicationofadiscreteconduitmodelforthe simula-tionoftheMadisonaquiferofSouthernDakota(USA).

1.2. Conduitnetworkmodeling

The models of De Rooij [15] and Saller et al. [52] show en-hanced results with respect to EPM models, because they solve more complex physics, but their authors agree on the diﬃculty thatisposedbytheunknownconduitlocation.Inordertouse re-alisticconduits,onecouldusethenetworksresultingfrom speleo-geneticmodels [1,18,22], butthe computation of thesemodels is 2

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heavyandtheyaredifficulttoconditiontofielddata.Geostatistical basedmodelscanbeusedasanalternativetoobtaindiscrete con-duitmodelslikeFournillonetal.[21].Theyareeasiertocondition to field data, butit is difficult to produce well connected paths. Jaquet etal.[31] proposetouseamodifiedlattice-gasautomaton forthediscretesimulationofkarsticnetworks,whichcanproduce hierarchical structures, butwhich is also difficult to condition to field data.RonayneandGorelick[50] proposetosimulate branch-ing channels(analogous tokarstconduits)usinganonlooping in-vasion percolation model,which produces nicefeatures, but that are based on a pixel discretization to representthe enlargement oftheconduits,andarethereforeverydifficulttousedirectlyfor a flow simulation. Finally, a new approach is to use a pseudo-geneticalgorithm [8,11,28],whichmimics theresultingstructures of karst networks produced by speleogenetical models, without solvingthecompletekineticsofthesystem,i.e.thecombinationof physicaland chemicalprocesses that leadto speleogenesis: equi-libriumstateofcalcite(rateofdissolution/precipitation),transport ofcarbonates,etc.Thesecomputationsareextremely computation-allyexpensive.Ontheotherhand,pseudo-geneticmodelsuse ap-proximatedphysicstogeneratethekarst networks,whichare ex-tremely lighter to compute. The approach of Borghi et al. [8] is basedontheassumptionthatwater(andconsequentlyconduit de-velopment)followa minimumeffortpathassuggestedbyGroves andHoward[26].Theconduitenlargementanddevelopmentdoes not strictly followphysicalandchemical laws duringthe simula-tion,buttheresultingconduitnetworkssatisfyinglymimicthe net-worksobtainedusingfullspeleogeneticmodels.

1.3. Rechargeandepikarstmodel

Karst aquifers present several complex recharge phenomena, which are caused by the presence ofa highlyaltered superficial “skin” (ofafewmeters)onthetopoftheaquifers,calledepikarst [20]. The epikarst has a complex role on aquifer recharge, be-cause it concentrates the rainfall into several “point” inlets (do-lines), whichlead directlytothe karstifiednetwork. Itrepresents roughly the “skin” ofkarst aquifers,that rapidlydrains and con-centratestherainfallintoseveralinfiltration pointsknownas do-lines (concentrated recharge). Moreover, it is quite common that surface waterstreams caninfiltrate directlyintothe conduit net-worksthroughotherpointinletscalledsinkholes.

The conceptualmodel ofMangin[42] considers thata consis-tent partof therainfall isquicklydrained by the epikarsticlayer towardtheconduitnetwork,andtheremaining partofwaterwill beinfiltrateddiffuselyonthelow-permeabilityfracturedvolumes. Király etal.[36] usea nested2Dmodel tosimulatetheepikarst, coupledwitha3Dmodelwherethekarsticnetworkwasmodeled. Theepikarstmodeloutputsareusedasinputfortheconcentrated recharge (sinkholes and dolines). Their recharge function for the flowmodelunderlyingtheepikarstlayeriscomputedasa propor-tion ofdiffuse andconcentrated recharge. Theirfinal conclusions admit that in mostopen karst aquifersmore than 40% ofthe infil-trationshould bedrained rapidlyintothekarst channels.Thiswork showed the benefits of correctly modeling the effect of epikarst onrecharge.AuthorslikeOfterdingeretal.[45] useacomplex hy-drologicalmodelthatcomputestherechargecontributionandthe runoff ofprecipitation onthebaseofseveralparameters,likethe slope,the altitudeandthesoil cover.Theyobtainsatisfactory re-sults.Unfortunately,thiskindofmodelcanbeverydifficultto cor-rectlyparameterize,especiallyifonlyafewmeasurementstations are available formodel calibration. Finally, Weber et al.[61] use black-boxmodelstoestimatetherechargeofakarstaquifer.Their approachshowsverypromisingresultsandtheydemonstratethat agoodestimationoftherechargefunctionisessentialtocorrectly modelarealisticflowbehavior.

1.4.Inverseproblem

The aim of the inverse problem is to identify the geometry, physicalparameters, initial or boundary conditionsof anymodel that describesa systemfromfield observations ofstate variables [51,57].Unfortunately,duetotheirmathematicalstructure,inverse problems are usually ill posed and have either no solution, an infinity ofsolutions, or can be unstable [27]. Manydifferent ap-proacheshavebeendevelopedtoovercomethesedifficultiesover thelast60years.

One of the most generalway tosolve theinverse problemis to frame it into a Bayesian framework. This implies to deﬁne a prior probability distribution for the unknown input parameters anda statisticaldistribution ofthe acceptable erroron the mea-suredstatevariables.Basedonthesetwomainingredientsonecan formulate the expression of a likelihood and deduce a posterior probabilitydistributionfortheunknown parameters.This formal-ismisdescribedindetailin[57].

Then, one can either search only for the parameter set lead-ing to themaximum likelihood usingoptimization techniques or aimtoamorecompletesolutionbysamplingdirectlytheposterior distribution. Sampling the posterior requires using Monte Carlo techniques [53]. Among those, the most simple is the rejection-samplingmethod[59].Itconsistsingeneratingalargenumberof candidate models within the prior distribution and accepting or rejecting them with a probability that dependson the probabil-itydistributionofthe expectedmeasurementerrors [57].The re-sultis a number of models that are proper samplesof the pos-terior probability distribution of the unknown parameters. How-ever, this method is computationally ineﬃcient since it requires runninga verylarge numberofmodels. Itis thereforeusedonly forbenchmarkingmoreeﬃcient methodsorto studyparticularly simplemodelsasitwillbedoneinthefollowingofthispaper.

Inpractice,solvingtheinverseproblemsrequiresingeneralto finda compromise betweencomputational feasibility andproper uncertaintyquantification. It also requires to usesome adequate toolsthat onecan coupletoanyforwardmodel.Oneofthemost flexibletoolsavailable todayforsuchmodelparameterestimation isPEST[17]:itoffersawiderangeofmethodsthatcanbecoupled toanymodel.Itincludesfunctionalitiesfortheminimizationofan objectivefunction using Levenberg–Marquardt method[43,44] as wellasNullSpaceMonteCarlosampling[58] forexample. 2. Karstmodelingmethod

This section describestheforwardmodeling methodsthat are usedinthetwofollowingsectionswhenweinvestigatetheinverse problems.

2.1.Conduitnetworks

In this paper, the Stochastic Karst Simulator (SKS) pseudo-genetic method developed in Borghi et al. [8] is used to model the3Dgeometryofconduitnetwork.Themethodincludesthe fol-lowingsteps:(1) a 3D geological model is built;(2) a stochastic fracturemodelisusedtoaddheterogeneityintothe3Dgeological model[7]; (3)the conduits of thekarstic network are generated using the pseudo-genetic approach, which uses a Fast Marching Algorithm[FMA, [56]]tocompute minimumeffortpathsbetween karsticinlets (dolines andsinkholes) and springs. Thisminimum effortpath computationisbased ontheassumptionthat the wa-ter(and consequentlythe conduits generatedby dissolution)will preferentiallyﬂow insidethemoreconductive discontinuitieslike fracturesand bedding planes,termed asinception horizons [19]. Moreover,theconduitsaregeneratediteratively.Everyconduitthat hasalreadybeensimulatedinﬂuencesthenextones,becauseitis

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Fig. 1. The conduits are meshed as 1D pipes that follow the edges of the 3D voxels (matrix).

considered asa newpreferential ﬂow path.This leads to a hier-archical network, where the conduits that start from several in-let points convergetoward theoutlets of thesystem. SKSallows todistinguishthesaturatedandunsaturatedzones, toaccountfor severalphasesofkarstiﬁcation,andcan generatecyclesand com-plexinterconnectionsbetweentheconduits.

2.2. Meshing

Initscurrentimplementation,themodelismeshedwithbricks in3Dandquadranglesin2D.IfanEquivalentPorousMedium ap-proachisusedtomodeltheflow(asdone inSection 3),thecells corresponding to the conduits are flagged as conduit elements. Otherwise,ifa flowsimulationbasedon laminarorturbulent 1D flowequationfordiscreteconduitsmustbeperformed(Section4), theconduitsaremeshedas1Dlineelements,whicharecomposed by 2 nodes andfollow the edges of the 3D bricks. As shown in Fig. 1, the pipe nodesare consistent with the 3D finite element grid, whichallows usinga discretecontinuum approach to simu-latetheinteractionsbetweenmatrixandpipes.

2.3. Rechargeblackboxmodel

A reservoir black-box model is used to simulate the total recharge function for the ﬁnite element model. Two kinds of recharge patterns are then considered: diffuseand concentrated. Diffuserecharge isdirectlyassignedontopographicsurfacenodes ofthe model,andconcentrated rechargeisassignedtothe nodes connected to dolinesor sinkholes.The proportion

λ

c of thetotal

recharge thatwillbeconsideredasconcentratedisdeﬁnedbythe userandisusedtocomputeboth thetotalconcentratedrecharge RcandthetotaldiffuserechargeRd:

Rc =

λ

cR, Rd= R− R c (1)

where Ris the totalrecharge. Then the concentratedrecharge Ri c

foreachinletiiscalculatedwithrespecttotheirestimated catch-mentsurfaceSioverthetotalNdolinescatchmentsurface:

Ri c= Rc Si N kSk (2) Risafunctionthatvarieswithtimeandthatdependsonthe me-teorologicalconditions.Itcan beestimatedforexampleusingthe

Fig. 2. Model description: in white the matrix elements, in black the conduits; Flow boundary conditions: the blue arrow is the spring (constant head), the inlets have 90% of total recharge, the remaining 10% is distributed on the matrix elements; Transport: injection of 1 kg of tracer in 2 inlets (violet point). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

softwareRS20121 ,whichisahydrology routingsystemdeveloped forwatershedmanagement.

2.4. Flowandtransportboundaryandinitialconditions 2.4.1. Flowboundaryconditions

Springzones, andingeneralthe dischargeareas,aremodeled withprescribed headboundaryconditions:theoutletsofthe sys-temaresupposedtobeatﬁxedaltitudes,andtheoutﬂowsdepend ontheheadvariationsonly.

On the other hand, two kinds ofboundary conditions can be usedforrecharge:

• Prescribedﬂux ofNeumanntype (m/s) atthe boundaryofan element

• Prescribedinﬂow(m3_/_s₎_assigned_to_the_nodes_that_correspond

tosinkholesanddolines

Neumannfluxesareconsideredtobeperpendiculartothe sur-face. GROUNDWATER uses the net inflow projecting the vertical componentofthe inflowingfluxontothe faceofthe correspond-ingelement.

Asource termcan alternatively be usedto model recharge.It corresponds to thedirect injectionofwater intothe elements at thetopographicsurface.

Inthisway, itis possibleto inﬁltratethecorresponding value ofrechargeintothenodes.

2.4.2. Transportboundaryconditions

Intheproposed workﬂow,thetracer testsaresimulated sepa-rately:everyonestartsatinjectiontimeusinganinitial concentra-tionoftracerattheinjectionnode.TheinitialconcentrationCinis

computedfromaknownmasstobeinjectedasfollows:

Cin = M/Vp (3)

whereMistheinjectedmass(kg),andVpistheporousvolumeat

thegivennode(i.e.thenodeoftheinjection).Theporousvolume

1 The reader can ﬁnd the documentation of the software at _{http://e-dric.ch/index.} php/en/software-en/rs2012 .

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Fig. 3. A stochastic fracture network (a) is the basis for the karst conduits model (b) .

ofagivennodeisdeﬁnedas:

Vp = Ni i=1

φ

iVe 8 (4)

where i runs on all the parallelepipedic elements that are con-nected to the given node, Ve is the element volume,

φ

i is the

porosity of each element.If thesimulation uses also1D pipe el-ements, andthatthegivennodeisalsoconnectedwitha 1D ele-ment,Eq.4 becomes:

Vp = Ni i=1

φ

iVe 8 + Nj j=1

π

r2 j 2 (5)

wherej arethe1D elementsconnectedtothenode,andrjisthe

radiusoftheseelements.

Theseequationsshowthattheinitialconcentrationstrongly de-pendsonthediscretization,ontheelementsthatareconnectedto thegivennode,andalsoontheirproperties(i.e.porosity,radius). Moreover,ifrechargeisappliedtotheinjectionnode,itmay addi-tionallydilute thesolute.Toaccount forthisdilution, the inﬂow-ingrechargeQtatatimetwithdurationofdtissummedoverthe

porousvolumeVpasfollows:

Vt p= Vp +

Qt

(

t

)

dt (6)

whereVt

p representsthe“porous” volumeattime t,i.e.thewater

volumethatdilutesthetracerattimet. 3. Retrievingconduitgeometry

In the present section, we analyze if geometric or physical properties ofa karsticsystem can be identiﬁed using an inverse approach. To answer this question, a numerical experiment is performedinwhich an ensembleofdifferentgeometries and pa-rameter setsare generated. Then we search systematically which othergeometriesorparametersetsproducesimilarresponses. 3.1. Modeldescription

The case study considered in this numerical experiment is a synthetic2D finiteelementmodelwith200x200msizeand2m resolution (dx=dy). The flow and transport equations are solved usingthefiniteelementcodeGROUNDWATER[12].

The model considers 2 hydrofacies: “karst” and “matrix” (NB: karst elements are those that contains a karst conduit). Flow in

karst andmatrix elements followDarcy’s law (laminar flow); an EPM2 is therefore used within the “karst” elements. The model assumes steady-state flow and transient transport. The synthetic casemimics a small karst aquifer, for which only 2 inlets (sink-holes)and1springare known.Forboth inlets,themeanhead is supposedtobe knownandatracer testhasbeenperformed.The tracertestis apunctualinjection3 ofa unitarymassattime 0of simulation.The flow boundary conditions are: no flow boundary onevery side ofthe model,a fixed headatthe springnode and a constant recharge over the model. 90% ofthe total recharge is directlyinfiltrated intheinlets ofthemodelandonly10%onthe “matrix” elements(i.e.

λ

c=0.9,inEq.1).Leaving10% ofrecharge

ontheotherelements allowsthereproductionofa gradientfrom everyelementofthemodeltowardthespring.Fig.2 summarizes themodelsetup.

3.2.Geometricalkarstrealizations

150geometricalrealizationsaregeneratedusingthepseudo ge-neticmethodologydescribedinBorghietal.[8] (Fig.4).All realiza-tionscanbegroupedinto3familiesbasedonthenumberofinlets (50realizationsforeveryfamily).Theﬁrstfamilyhasonly2inlets (i.e.the 2 known inlets),the second 7 (i.e. 2 of knownlocation, andtheotherswithrandomlocations) andthethird 14inlets(of which2aredeterministic).NotethatSKSgeneratesconduitsfrom every inlettoward the spring.It means that a simulationwith2 inletswill correspondto geometrieswith2main conduits. These conduitscanmergeandcreateahierarchicalstructurethatcanbe seenasa treewiththreebranches, butforthesake ofsimplicity, wewilldescribethesekarstnetworksasmadeoftwoconduitsand similarlyforthosewith7or14inlets.Thepseudogenetic method-ologyrequirestheexistenceofan initialfracturenetwork to con-trolthevariabilityoftherealizationsofkarstnetworks.Here,a dif-ferentequiprobable realizationofafracture modelisusedfor ev-erykarstrealization.Thesamefracturestatisticsareusedforevery realizationso that theﬁnal karst realizationsstill remain statisti-cally comparable. Fig.3 showsone realizationofthe fracturation model(a) andthecorresponding stochastic karst conduits model (b).

2 _{Appendix B}_{details the EPM computation.}

3 _{Appendix D}_{explains the necessary conditions to ensure the numerical stability} of the tracer test simulation.

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Fig. 4. Stochastic karst realizations used for this study, 50 realizations with (a) 2 conduits, (b) 7 conduits, and (c) 14 conduits.

3.3. Flowparametersforeachkarstrealization

For every karst realization, 120 different combinations of the hydraulic conductivityof thematrix Kand conduitradius rwere tested. The hydraulic conductivity of the matrix K varies from 5· 10−7 _m/s _to ₁₀−3 _m/s₍₈ _values _with _regular _steps _in _a _log

10

space) andtheradiusroftheconduitsinkarst elements,varying from 0.1to 0.8m (15 valueswithan incrementof 0.05m). It is important to note that changingthe radius ofthe conduits leads to differences inboth porosity n (Eq.B.1) and equivalent perme-ability

κ

eq (Eq.B.4). Fig.5 showsthevariationofboth porosity n

andequivalenthydraulicconductivityKeqwithanincreasingradius

rfrom0.1mto0.8in2Dcellsof2mlength.

3.4. Referencesimulationselection

After having solved the direct problem for both tracer tests andfor every combinationof parameters for every karst realiza-tion(which means150x120x2=36,000 simulations)onereference simulationischosen. Itis selectedfromthe18,000ﬂow parame-ter ﬁelds(150x120). This referencesimulation isthen considered asourreality.The resultsofthissimulationare 2observationsof headHi _at_both_inlets, _and₂_tracer_breakthrough_curves_Ci_at_the

spring.

Theseobservationsarethen comparedtotheresultsofall the othersimulations.Thecomparisonisbasedonthefollowing objec-tivefunction:

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Fig. 5. Variation of the equivalent porosity [-] and hydraulic conductivity [m/s] as a function of the radius r of the conduits .

e =

3.5. Inverseproblem

Havingagivenreferencedataset,solvingthe inverseproblem consistshereinﬁnding theensembleofconﬁgurations(geometry + parameters) that match the observations.In practice, theerror eis computedusingEq.7 forall theother models.Onlythe one forwhicheislowerthanagiventhresholdareselected.Theresult is an ensemble of realizations that match the data and describe statisticallytheremaininguncertainty.

Fig.6 illustratesthisprocedure.In Fig.6a,theensemble ofall the18,000recoverycurvesaredisplayedwiththeonethatis con-sidered asthe reference. Hugedifferencesbetween thereference andmost ofthe other breakthrough curvesare observed. Fig.6b showstheselectedacceptablesimulations,i.e.thosedisplayingan errore(Section3.4)lowerthan1%.

Fig.7ashowsthehistogramofec andFig.7bthehistogramof

eH inthiscase.Theresponsefortransport(Fig.7a)ismuchmore

discriminantthantheone forflow(Fig.7b).Thehistogramofthe errorsontheflow responsesshowsthatitisquiteeasy toobtain agoodfit,asmanysimulationshaveaverysmallvalueofeH.The

reason forthis behavior is that withonly few observations, it is easy toobtainseveralmodelsthat matchtheobservations within agiventhresholdofconﬁdence.Onthecontrary,transportresults showawiderdistributionoftheerrorsec,relatedtoahigher

sen-sitivityofthesolutionto thegeometryoftheﬂowpaths.Indeed, the erroron thebreakthrough curvesishighlyinﬂuenced by the path followedby the streamlines.Fig.7c showsthehistogramof the objectivefunction witha value of

λ

equal to1/2 (Eq.7), i.e. givingthesameweighttobothecandeH.

3.6.Statisticalanalysis

Theprevioussectionhasshownthatitispossibletoﬁnd mod-els matching a certain reference. In the present section, the ex-periment is repeated using systematically all models as a refer-ence.Theinverseproblemissolvedforeachone (asexplainedin Section3.5).Thisallowsanalyzinginastatisticalmannerifthe pa-rametersarewellidentiﬁable.

Torepresenttheresults,the probabilitydistribution ofthe es-timatedparameter valuesiscomputed andcomparedto the true parameter values in the form of log ratios: log10(rest/rref) and

log10(Kest/Kref).Theseprobabilitiesarerepresentedconditionaltoa

certainvalueofthereferenceparameters:

P

(

log ₁₀

(

rest/rre f

)

,log 10

(

Kest/Kre f

)

|

Nre f= x

)

(10)

wherex is the possible numberof conduits. When log10(rest/rref)

orlog10(Kest/Kref) are equal to 0,it means that the estimated

ra-diusrestandtheestimatedmatrixconductivityKestareequaltothe

reference.WealsocomputetheprobabilityofhavingNest number

Kest/Kre f

)

|

Nest =y,Nre f =x

)

(12)

wherexandyarethepossiblenumberofconduits.

Figs.8and9 showtheresults.Theysuggestthattheconduit ra-diusislikelytobe betteridentiﬁedbytheinverseprocedurethan thehydraulicconductivityofthematrix.Theprobabilityofhaving anestimatedradiuslargerorsmallerthantwicetherealradiusis verysmall;aratiointherange[1/2,2]betweentherealand esti-matedradiuscorrespondstoalog10valueintherange[−0.3,0.3].

Weseethatoutsideofthisrange,theprobabilitiesareverylow. Ontheopposite,theprobabilitytohaveanestimatedhydraulic conductivityforthematrixaround3ordersofmagnitudelargeror smallerthanthetruthislarge.Thisisconsistentwithour concep-tualmodel,forwhich 90%of theﬂow andpossiblyall thetracer aresupposedtopassthroughtheconduits.

One interesting thing that can be seen on Fig. 8 is that when the referencehas only 2 conduits (Nre f=2), the

probabil-ityP

(

Nest=2

|

Nre f =2

)

isextremelyhighascomparedtoP

(

Nest=

7,14

|

N_{re f}=2

)

.Inthiscasetheinverseproblemisquiterobustand ableto identify precisely that the unknown reality has indeed2 conduits.Thisresultisofcoursevalidonlyunderthetested con-ﬁguration(2inlets,2conduits,2tracertests).

Ontheopposite,whenthereferencehas7conduits (Nre f =7),

theprobabilitytoestimate thatthemodelshouldhave7inletsis only slightly higher than the probability of having 2 or14 con-duits.This iseven morediﬃcultwhen thereferencehas14 con-duits(Nre f =14).Thisisrelatedtothefactthatonly2tracertests

areavailableinthissyntheticexample.Butitshowsthatwhen suf-ﬁcientdataareavailable,onecanexpectthattheinverseproblem willallowtoidentifyproperlysomemajorfeaturesofakarstic net-work(suchasthenumberofmainconduitsortheirradius).

Looking more indetail atFig.9,we noticethat the estimated radius rest and matrix hydraulic conductivity Kest are very well

identiﬁedforthesimulationsforwhichNest=Nre f (Fig.9a,e,and

i),i.e.both log10-ratios arecloseto0.When Nest <Nref,the

esti-matedvaluesofrest andKest are overestimated.Thismakes sense

because the model needs to drain more water relatively to the numberofconduits.Forthesamereason,whenNest >Nrefthe

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Fig. 6. Tracer test results (cumulated mass at the spring). (a) In red we see the tracer test result for tracer one on the reference simulation, in blue the results of all the other simulations. (b) selection of the “best ﬁt” simulations, i.e. the ones under 1% of error. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

Fig. 7. pdf of normalized error, (a) on tracer test results ( e c ). (b) on ﬂow results ( e H ). (c) using both ﬂow and transport ( e ), with λ= 0 . 5 .

4. Retrievingconduitpropertiesusinggradient-basedmethods

Intheprevioussection,theacceptableparametersareobtained by rejectionsampling andsystematicsearch implyingto run the modelformanypossiblegeometriesorparametervalues.Thiswas possiblebecausethemodelwassmallandsimple.Foralargescale

model(regionalscale,millionsofnodes)thisisgenerallynot pos-sibleandafastermethod isrequired.Thisiswhywe testinthis section the feasibilityto use a gradientbased approach to accel-erate the identiﬁcation of the physical parameters when the ge-ometryis ﬁxed.Forthatpurpose,we run asystematicparameter search to knowexhaustively theobjective function andthen test 8

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Fig. 8. Joint probability distribution functions obtained using rejection sampling: (a) case with a reference having 2 conduits ( N re f = 2 ); (b) case with N re f = 7 ; (c) case with

Nre f = 14 .

theeﬃciencyofthe gradientsearch usingthesoftwarePEST [17] onasyntheticcase.

4.1. Modeldescription

Karst networksoften show some degree of hierarchy [34]. As Annable [1] demonstratesby speleogeneticalmodeling,karst con-duitscanshowanorganizationofthemeanconduitdiameterthat increases accordingto the hierarchyof the network, i.e. that the biggest conduits are located downstream asthey collect the wa-teroftheiraffluentconduitsinasimilarwayasrivers.Differently fromstreams,karstconduitscanalsodiverge,andinthiscasethe flowisdividedbetweenthedifferentdiffluentconduits[1].To con-sider these particular characteristics of karst networks, in a pre-vious work [60] we have proposed a simplemodel inwhich the conduitradiusrisestimatedusingapowerlawsimilartoHorton’s law:

r

(

u

)

=

α

euβ (13)

where

α

and

β

aretwoparametersthathavetobecalibrated,and u is the order of the conduit. However, unlike Collon-Drouaillet

etal.[11] andVuilleumier etal.[60] who used a Horton’s order basedontheriverStrahlerclassiﬁcation,heretheorderuofeach conduitiscomputedastheratioofthecatchmentsurfaceSthatis drainedby the givenconduitover the totalcatchment surface as suggestedinBorghietal.[8]:

ui= j∈AiSj N k=1Sk (14)

whereNisthetotalnumberofinletsofthesystem, Skrepresents

thetotalsurfaceofthecatchmentthatisdrainedbykarstconduits, andAi representsthe set ofindices that draininto theconduit i

(Figs.10 and11).

SKSuseswalkerstocomputethepathsoftheconduits.To con-siderthedivergingconduits,itstorestheinformationoftheorigin ofthewalkersforeachcomputedpath.Inthisway,itis possible tokeepthedivergenceofconduitsconsistentwiththeir radius.In theproposedapproach,theideaistocreatetheminimalhydraulic necessarydimensionstoallowthesystemtodrainrecharge water outofthesystem.

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Fig. 9. Joint probability distribution functions of the estimated conduit radius r est and matrix hydraulic conductivity K est as a function of the estimated N est and reference

Nref number of conduits.

Fig. 10. The order of each conduit u is the ratio between the total catchment sur- face S that it drains and the total karstic catchment ( Eq. 14 ).

Inthissection,we consideronlyonesingle3Drealizationofa synthetickarsticaquifergeneratedusingSKS.Itcovers anarea of 3km2.The3Dmeshisrathercoarsetoreducecomputingtime;it iscomposed of40 elements inx direction, 30iny directionand 3 in z direction. The cells are cubic, with a width of 50 m. The conduits are meshed with1D pipe elements4 , located along the edges of matrix elements that are meshed as 3D cubes (Fig. 1). The referenceparameters are

α

=0.2,

β

=1.5 (Eq.13) andKM=

10−5.5_[_m/s_]_(the_hydraulic_conductivity_of_the_matrix)._The

geome-tryanddistribution oftheconduitradius forthereferenceis dis-playedinFig.11.

Theﬂowisthensimulatedintransientstateusing GROUNDWA-TER[12].Theparameterschosenforthereferenceallowthe repre-sentationofarealistickarstbehaviorwithrapidresponsetopulse eventintherechargefunction.Thesyntheticdatausedforthe in-verseproblemincludetheheadvariationsinﬁveobservationwells placedatthebaseofthemodelatcoordinatesgiveninTable1 and thespringdischargeatlocation(1000;500)wheretheheadis pre-scribedatavalueofonemeter(headisequaltoaltitude).

Theobjectivefunction

φ

isdeﬁnedasthesumofthesquared weighted residuals in the same manneras implemented inPEST [17]:

φ

= no i=1

(

wi· r i

)

2 (15)

4 _{Appendix C}_{details the equations used for 1D pipes.}

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Fig. 11. Distribution of the radius of the conduits in a synthetic case. The radius increases with the order of the conduits. For a visual purpose, the apparent conduit radius has been exaggerated.

Table 1

Observation wells coordinates.

ID X Y H1 10 0 0 750 H2 650 500 H3 650 10 0 0 H4 1350 500 H5 1350 10 0 0 Table 2

Ranges of parameter values for the systematic search. min/max : bounds for the parameter; nb : number of inter- vals. αand K M increments have been computed in a log10

scale.

Parameter Reference Min Max nb

α[ −] 0.2 0.05 1.5 16

β[ −] 1.5 0.1 5 14

KM [ m/s ] 10−5 .5 10−7 10−4 16

whereno isthenumberofobservations,riisthe residualsofthe

ithobservationandwiitsweight.Theobservationsaretakenfrom

6 differenttime series.The ﬁrst one is thespring discharge, and the other 5 are the heads in the observation wells. To give the same weight to the discharge observation and to the head ob-servation, a weight of 1/

σ

is given to the dischargeobservation (where

σ

is the variance of each time series) and a weight of 1/

σ

× 1/5 is givento thehead observationsbecause thereare 5 observationpoints. Theobservationdataarepresentedin Fig.12: thespringdischargeandthehydraulicheadobservedinboreholes reactrapidlywithrechargeimpulsesasexpectedinarealistickarst system.

4.2. Systematicsearch

Inaﬁrststep,theobjectivefunction

φ

issystematically evalu-atedforallpossibleparametervalues(

α

,

β

,Km),inordertoknow

precisely whether the problem has a unique or an ensemble of possible solutions and to get some information about the shape oftheobjectivefunction.Thetestedparameterrangesaregivenin Table2.NotethatKM issampledinalogarithmicscale,becauseit

canvaryoverseveralordersofmagnitude.Forthisstep,atotalof 3584ﬂowsimulationshavebeenmade.

Fig. 12. Reference simulation: (a) simulated spring discharge (b) simulated hy- draulic head in observation wells.

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Fig. 13. Systematic search result: isosurfaces of the objective function φas a function of the 3 parameters α,β, and K m . The axes of the parameter αand K m are in

log scale for improved clarity.

φ

foreachparameter(requires sev-eraladditionalrunsoftheﬂowmodel)

5. update the parameters using the Levenberg–Marquardt method[43,44]

Theresultspresentedherearepartofwideresearch ﬁeld aim-ingatbetterunderstandingkarstaquifers,andproviding forecast-ingandmodelingtoolsforthesecomplexhydrogeologicalsystems. Inthis work,many assumptions were madeto easethe compre-hensionoftheinverseproblemappliedtokarstaquifers.Inthe fol-lowing,we summarizeanddiscusstheimplicationsofourresults (Section5.1)andcoveralsosomemoregeneralquestionsaboutthe modelingofkarsticsystems(Section5.2).

5.1. Resultsofthisstudy

The tests that have been performed in the scope of this re-searchshowtwoencouragingresults.Thefirsttestshowsthatitis possibletoinfersomegeneralinformationsuch asthenumberof mainconduits,ortheconduitradius,usingrejectionsamplingand systematicsearch methods.The problemassociatedwiththis ap-proachisthatitisextremelydemandingintermsofcomputer re-sources.UsingadesktopPC(2GbRam,2.8GHzCPU),ittakesabout 30minto runone flow andtransportsimulation,while the gen-erationofthe karstnetwork takesabout5 s.Runningthe 36,000 simulationsrequiredforthefirstexperimentwaspossibleonly be-cause we used the high performance linux computing cluster of theUniversityofNeuchtel.

However, the flow model in this first test still remains very simple: it uses an equivalent porous media approach within the conduit elements insteadof a moreaccurate but non-linearflow equations.High Reynoldsnumber values(reaching 3000 insome elementsandinsomesimulations)indicatethatDarcy’slawisnot applicableforthoseelementsandforsomesimulations.Further re-search should considerthe inertialeffects inthose elements, but 12

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Fig. 14. (a) Eight parameter estimation runs using PEST [17] , displayed in the parameter space. 7 runs have given reasonable results and found a parameter set close to the reference. (b) combination of Fig. 13 (with transparency) and (a). We notice that some optimal parameter sets are trapped in local minima of the objective function.

this was not possible inthe framework of thisproject. Nonethe-less, thestatisticalanalysisoftheerrorfunctionshowsthatusing tracer testdatadeeplyenhancestheabilitytoinferthegeometry andequivalenthydraulic radiusofthe karst system.Solute trans-portdependsonparticletrajectories;headdistributionandspring hydrographsarelesssensitivetoparticularﬂowpaths,resultingin ahigherdegreeofuncertaintyintheposteriordistribution.

This suggests that any information obtainedfrom tracer tests (alsonaturaltracers)shouldbeusedtosolvetheinverseproblem in karsticaquifers. The objective function should be a composite functionwithastrongweightontransport.Weobservedthatwith our modelsetup therewas a much largeruncertainty associated withtheestimationofthehydraulicconductivityofthelimestone matrixincomparisontotheconduitradius.Wenoticedthatkarst systemswithsmallernumberofdrains(2conduits)wereeasierto identifythan others(e.g.7 or14conduits)when onlytwo tracer testsareavailable.

Thesecond test showedthat itispossibletousestandard pa-rameterestimationtechniquestooptimizethephysicalproperties associated with a geometrical SKS realization. This enhances the computational efficiency of the search by one or two orders of magnitude(10xto100xlessmodelrunsareneeded).Thissecond test didnotincludetracer test simulation,butincludedmore so-phisticated physics forthe flow problem. Basedon the resultsof thefirst test,itcan bereasonablyguessed thattheresultswould benefitoftheinclusion oftracertest resultstocompute theerror function.Unfortunately,thedirectproblemruntimewouldalso in-creasesignificantly5 ,thisiswhyitwasdecidednottocompute so-lutetransportinthistest.

Based on the previous results, we foresee that one possible pragmaticapproachtosolvethe inverseprobleminakarstic sys-tem could be based on the combination of rejection sampling andgradientbasedoptimization.Theprocedurecouldfollowthree steps:(1)severalSKSrealizationsaregenerated,(2)theparameters ofeachoneofthemareoptimizedusingagradientbasedmethod, and(3)onlytheonesthatprovideanacceptableerrorarekeptfor uncertaintyanalysis.

Suchamethodwouldhavetheadvantageofreducingthe com-putingtimeverysigniﬁcantlyascomparedtoapurerejection sam-pling,itwouldhoweverprovideonlysomemodelsthatreproduce

5 _{Appendix A}_{details a model-run workﬂow to minimize the necessary cpu time} of direct problems.

theobservation,butnotapropersetofmodelswithin the poste-rioruncertaintydistribution since combiningthesamplingofthe geometry and the optimization step may introduce a statistical bias.Thereforemoreresearch isstill neededtoclarifythose prob-lems.

5.2.Openquestionsandfurtheroutlooks 5.2.1. Questionsrelatedtothedirectproblem

Inthisstudy,theconduitsare simulatedasstraightpipes, and their radius follows a power law that is dependent ofthe topo-logicalorderoftheconduits(Eq.13).Itisclearthatthisapproach corresponds to a strong simplification of naturalconduits. In re-ality,the shape of the conduits vary significantly. As a result of anisotropic preferential dissolution, some parts of the same con-duit usually presentlocal variation oftheir radius.Assuming the conduitsascylindricalpipesleadstoastrongsimplification.These simplifications are necessary because currently no rule exists to characterize theselocal variations. The use of saturated conduits onlyresults inanother significant simplification.Some numerical schemes which allow for the use of variably saturated conduits [16]exist and could be used.

Anadditionalquestionisrelatedtotheimportanceof simulat-ingthelimestonematrix.Itwouldbeinteresting toinvestigatein whichcasesitispossibletoexcludethematrixfromtheproblem withoutlosing significant information.This wouldlead to a sim-plificationofthewholeproblem,butcoulddrasticallyenhancethe computingefficiency.Using this approach,it could be interesting totest theapplicationofhydrologicalsoftwareforurbanconduits networktosolvethisproblem[47].Inthecontextofinverse prob-lem,itmaybepossibletouseonlytheconduitsforflowduringthe firstestimationoftheparameters, andthenperformthefull cou-pledsimulationonlyonafewrepresentativemodels.Thiscouldbe usedasaproxy(approximatedphysicssolver)forflowsimulation. Saller et al. [52] use MODFLOW-CFP and transfer functions to simulate the interactions between the matrix and conduits. They identify the difficulty in correctly assessing the values of thesetransfer functions, but show also that thisgives a realistic behavior to theseinteractions. In ourstudy, we use a direct hy-draulic connection between matrix and conduits. Further stud-ies should be achieved to include transfer functions in the in-verseproblemframework.Thiswouldprobablyresultsinaneven more complex inverse problem because more parameters would needto beidentified. But,onthe other hand,theuseoftransfer

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functionscouldenhancethenumericaleﬃciencybecausethe con-trasts ofhydraulicconductivitywouldbesmoother,andthedirect ﬂowproblemwouldprobablyreachconvergencemorerapidly, re-ducingtheCPUtimeneeded.

5.2.2. Questionsrelatedtotheinverseproblem

Severalaspectsoftheinverseproblemhavenotbeentakeninto accountinthepresentwork.Forexample,therechargemodelused inthisstudyisbasedonareservoirmodel.Theidentiﬁcationofits parameters was not included in the inverseproblem framework. As Weber etal.[61] show,the recharge modelhasa verystrong impact on springhydrographs forreal applications.Similarly, the matrixandtheradiusoftheconduitshavebeendescribedwitha verysmallnumberofparameters,whileweexpectthatthematrix will be heterogeneous withparameters varying in spaceand we expect thedistributionoftheconduitdiameterstobemuchmore complexthanthesimplepower-lawproposedhere.Allthese addi-tional complexitywill needtobe facedwhen dealingwitha real case.

In terms ofmethodology, a methodthat could be used to in-creasethenumericaleﬃciencyofthewholeinverseframeworkis to employ approximate physics solvers (so called “proxies”) and classiﬁcation techniquesasa preliminarystep toselectcandidate models (parameter sets)andto minimize the number ofruns of thehighlydemandingandaccuratedirectproblem[55].Afurther technique isproposedbyGinsbourgeretal.[24] whousedkriging to interpolatethe errorfunction usingproxies.Such an approach could beusedforkarstaquifersaswell,butadditionalresearchis requiredtoidentifygoodproxiesforkarstaquifers.

Conclusion

In this paper, several aspects of the inverse problem in karst modeling have been studied.Our numerical experimentsdid not aimatsolvingtheinverseproblem,butratheratprovidingabetter understandingoftheunderlyingchallenges.Indeed,theestimation ofthe3Dgeometryofakarsticnetworkandofthecorresponding physicalparameterscanbeimpossibletoachievebyamanualtrial and errorapproach. Therefore, thesemodels should beneﬁt from availableautomaticparameterestimationtechniques.Furthermore, theinverseproblemneedstobeappliedwiththeaimtoevaluate theuncertaintyrelatedtothesemodelsandtheirparameters.

The numerical experiments describedin thispaper show two veryencouragingresults.

Thefirsttesthasshownthatbothphysicalandgeometrical pa-rameter can be identified pretty accurately in an inverse frame-work ifadequatedata isavailable.Byrepeating theexperimenta verylargenumberoftimeswithdifferentreferenceshaving differ-entnumberofconduits,differentmatrixpermeabilityanddifferent conduit radius, we could statisticallyevaluate how efficientlythe inversemethodcanretrievethefeaturesofthereferencefromthe headortracerdata.Thistestindicatedthatdatafromtracertests hada verystrong impactonthe identificationofthegeometrical propertiesofthekarstnetwork.

The second test has shown the possibility of improving the numerical efficiency of the inversealgorithm by accelerating the search foroptimal parameters when the geometryis fixed using gradient-based methods. The model that has been used for this test considers morecomplicatedphysicsthan theone ofthefirst test. Asystematicanalysisoftheobjectivefunction

φ

showsthat severalcombinationsoftheparametersthatinﬂuencetheradiusof theconduits(

α

and

β

)cangivemoreorlessequallysatisfactorily results. Thehydraulic conductivity ofthematrix isvery sensitive tothelowconductivityvalues,butassoonasitissuﬃcienttolet therechargeinﬁltrateandletthewaterreachtheconduits,

φ

isfar

lesssensitivetoits variations.Thistest alsoshowsthat PEST[17] canbeusedeﬃcientlytosolvethisproblem.

Theseresultsindicatethatonecouldsolvetheinverseproblem in karsticaquifers in a pragmatic manner using a two steps ap-proach.First,asetofmanydifferentkarstgeometricalrealizations aregeneratedusingthepseudo-geneticmethodology[8],and sec-ondlyforeachoneofthemthebestphysicalparametersarefound usingaparameterestimationtechnique.Theresultingsetof simu-lationscanbeusedforfurtheranalysis.

Inboth tests, we useda combinationofalreadyexisting tools to simulateflow and transport in karst aquifers with distributed parameter models. The karstic conduits are meshed as1D pipes that follow the edges of the 3D elements used for the matrix. Turbulent flow equations are used to simulate flow in the con-duits.Theradiusoftheconduitsdependsontheirhierarchical or-der. Major conduits have largest radiuses, while secondary con-duitsaresmaller. Theconduits arehierarchicallyclassifiedby the ratiooftheir catchment surfaceover thetotal catchmentsurface. Therecharge functionappliedontheinlets alsodependsontheir catchment surface. The greatest part of recharge is applied di-rectlyto theinletsandonlya smallpercentageisappliedon the restof themodel.This allows foran approximation ofthe effect of epikarst in convoying water to the inlets. We argue that this methodologyisprettygeneralandcouldbeusedformanykarstic systems.

Finally,ifweconsider againthequestionstatedinthetitle of thepresentstudy“isitpossibletoidentifykarstconduitnetworks geometryandpropertiesfromhydraulicandtracertestdata?”,the answerthat issuggestedis “Yes,itseems possibleto bracketthe valuesoftheradiusoftheconduitswithinareasonablerange, pro-videdthatenoughcomputingpowerandsuﬃcientdataare avail-able”.In practice,thisstudy showsvery promisingresults,but it is based on a limited set of numerical experiments and restric-tive assumptions. Further research is needed to enhance the in-verseframework,andespeciallythesolvingofthedirectproblem. Asstatedinthediscussion,maybetheuseofapproximatephysics tosimulatethedirectproblemwithin theinverseproblem frame-workcouldallowtohighlyspeedupthesearchofvalidkarst real-izations,andthereforeallowsuchamethodologytobeappliedon realsizeaquiferproblems.

Acknowledgments

ThefundingofthisresearchwasprovidedbytheSwissNational Foundation forResearch,onfundno.P2NEP2-151935.Thanks also to E-Dric.ch company for providing their watershed management softwareRS2012 forthisresearch. Partialfundingofthisresearch wasprovidedalsobySchlumbergerWaterServices.

AppendixA. Modelrunworkﬂow

To be used in an inverseproblem framework, the model run workflowisdefinedinordertominimizethetotalnecessaryCPU time. Theworkflow must take intoaccount transientflow condi-tions, andtransienttransport.When studyingkarst aquifers, sev-eral tracer testsmay be performed. Therefore, the modelshould matchall of them.Unfortunately, the transport simulation, espe-cially with strong heterogeneity in the flow medium, may lead toveryhighcomputational times.The modelrunworkflow is in-tended to reduce the necessary CPU time as much as possible, whensimulating flowandtransport attransientstate. Itis sepa-ratedinthreemainsteps:

1. steadystateﬂowsimulation

2. longtransientstate(years)ﬂowsimulation

3. short transient state (months) ﬂow and transport simula-tionsforeachtracer

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Fig. A.15. Flowchart of the model run.

The first stepof themodel run isa steadystate flow simula-tion usingthe average flowconditions (i.e.the average discharge ofthesprings) asinput forthe model.Thesteadystate hydraulic headresultisthenusedasinitialconditionsforthetransientflow simulation.Thesteadystatesimulationhastoberun inthesame hydrological conditions asthe beginning ofthe transient simula-tion,e.g.ifthetransientsimulationstartsinalow-flowperiod,the steadystatesimulationmustberununderlowflowconditiontoo, andvice-versa.

The transient simulation is done for the whole period of in-terest, which must include the periods of the tracer tests, and whichmaylast afew years.Duringthissimulation,the hydraulic head results at every tracer test injection time is stored to ini-tialize the flow field of the short flow and transport simula-tions. The transport simulations are performed onlyover a short period of time (a few days or weeks, depending on the dura-tion of the experiments), because they are heavier to compute. The advantage of this workflow is that the tracer simulations are consistent with the whole transient flow simulation, with-out needing to solve the whole transient simulation (years) in-cluding transport. Fig. A.15 shows the flowchart of the model run.

AppendixB. Simulatingﬂowandtransportbyequivalent medium“karst” elements

Thefirsttestswiththehierarchicalconduitnetworkshavebeen doneusingan equivalentporousmedium(EPM)approach,evenif thisdoesnotallowthesimulationofthefullcomplexkarsticflow (e.g.turbulentflow inthe conduits).Still,some authors have ob-tainedsatisfactoryresultsinsomecasessuchasScanlonetal.[54] whouseequivalentpropertiesforentireregionsoftheirmodel.In thissection,the EPMapproach isused byassigning specific flow andtransport properties to the karst conduit elements, similarly toWorthington[62].Thismethodhastheadvantageofbeing sim-pleandfasttosolve ascomparedto theuseofdiscreteelements withnonlinearflowequations(SectionAppendixC).

Thepropertiesoftheconduitelementsarederivedfromthe ra-diusofthe conduits andthemesh sizein thefollowingmanner: theequivalentporosity

φ

eq(-)iscomputedbyestimatingthe

vol-umeofvoidinthecelldividedbythevolumeofthecell:

φ

eq = Vv oid

Vcell

= Vconduit+

(

Vcell− V conduit

)

·

φ

matrix

Vcell

(16)

whereVconduitisthevolumeoftheconduit

(

m3

)

,i.e.

π

r2d,with

d thelength in thecell (m),

φ

_matrix isthe porosity ofthe matrix (−).

Theequivalentpermeabilityiscomputedsuchthatthetotal dis-charge Qtot

(

m3/s

)

ﬂowingthroughone cellinagivendirectionis

identicalintheequivalentporousmediumorinthepipeplus ma-trixmodel.Thetotaldischargeisgivenby:

Qtot = Qm + Qc

|

Qc = −

π

r4 8

ρ

g

μ ∇

H, Qm = −κm

ρ

g

μ

·

(

A−

π

r 2

₎

A

∇

H (B.2)

whereA[m2_]_is_the_area_of_the_cell_side_crossed_by_the_ﬂux,

_π

_r2_the

area oftheconduit,Qc[m3/s]isthedischargethroughtheconduit

thatcrossestheelement(usingthePoiseuillelaw),Qm[m3/s]isthe

dischargeﬂowingthroughthematrixsurroundingtheconduit,

κ

m

isthepermeabilityofthematrix(m2_),_g_is_the_acceleration_of_the

gravity (m· s−2_),

_ρ

_is _the _water _density ₍_kg_{· m}−3₎ _and

_μ

_is _the

water dynamic viscosity (kg· m−1· s−1) and

∇

by:

κ

eq =

κ

m+

π

r2 A r2 8 −

κ

m (B.4)

The useofan equivalentporous mediumapproach isastrong approximation, especially for transport problems, as we need to putunrealisticallylowporositytothekarstelements(theelements containing akarstconduit)sothattheporevelocitybecomes sat-isfactoryintermsoftransport[37].

AppendixC. Simulatingﬂowandtransportin1Dpipes embeddedina3Dporousmatrix

The wholemodelismeshed usingaregulargrid (Section 2.2). Pipeselements arelocatedontheedgesofthe3Delements (hex-ahedrons)ofthekarsticmatrix.Thereisadirecthydraulic connec-tionbetweenthepipesandthematrix,asthenodesofbothtypes ofelementsarethesame.

Tomodeltheflowintheconduitstwodifferentapproachescan beused:laminarandturbulentflow.Thelinearflowequationused inpipesistheDarcy–Poiseuilleformula,thehydraulicconductivity forthepipeselementsinthiscaseis:

κ

=

ρ

g

μ

r

2

8 (C.1)

The flow equation used in pipes for turbulent flow is the Manning–Strickler (Turbulentflow conditions).Thehydraulic con-ductivityforthepipesistherefore:

κ

=

φ

r2

||∇

H

||

(C.2) where

φ

is thefriction coeﬃcient (m1/3_s−1_)._Both_cases _are_used

withthesamelawforﬂow(

v

=−K

∇

H)whereonlythedeﬁnition ofKvariesfromonecasetotheother(K=

κ

ρg

μ).

AppendixD. Transportsimulation

Theinjectionofthetracerismodeledasaninitialconcentration atthebeginningofthetransportsimulation.Theinitial ﬂowﬁeld

ofthesimulationisextractedfromthetransientlong periodflow simulation.Inthisway,allthetransportsimulationsareconsistent with the long period flow simulation. Ideally, the transientflow fieldofthetransientflowsimulationshouldbe usedasflowfield forthetransportsimulation,toreducethecomputationaltime,but thisisnotpossible,becausethetime-stepsevolutiondependsalso fromthetransportsimulation.

Accordingtomanyauthors [3–5],it isnecessarytopay atten-tion to two parameters: the Peclet number Pe and the Courant numberCr[13].ThePecletnumberisdeﬁnedas:

Pe =

v

dx

D (D.1)

where v is the ﬂow velocity, i.e. the pore velocity vp multiplied

δ

tmax

dx (D.2)

where

δ

tmaxismaximaltimestepsize(s)forthetransient

simula-tion.Thenumericalsolutioncanbeconsideredstableifthesetwo conditionsaresatisﬁed:

Pe< 2 , Cr< 1 (D.3)

Thesetwoinequalitiescanbeexpressedinordertoisolatethe dis-persionDandthemaximumtimestep

δ

tmax:

v

dx

2 <D,

δ

tmax<

dx

v

(D.4)

The dispersion is a parameter that depends on the scale of the problem, as it can be used to assume the dispersion ofa pollu-tantinducedbypore-scaleheterogeneitiesinanequivalentporous medium.AsexplainedinRauschetal.[49] theeffectofotherkinds ofheterogeneities(likekarstfeaturesinthiscase)maystrongly in-fluencethe behaviorof thesolute inthe simulation.Inthiscase, thedispersionhastobeinterpretedasthedispersionoccurringat cellscale,anddependsthereforestronglyonthemeshdimensions. Moreover, following the conditions ofEq. D.4, it appears that particularattentionhastobepayedtomeshsizeandonthevalues to the dispersion coefficient. The time discretization is alsovery important.However, when simulating karst aquifers, witha very strongheterogeneity,itcanbedifficulttocomputethevalueofthe maximalflowvelocity inthemediumapriori,whichcanbe very fastinsome cases.Toguesstheseparametersbefore runningthe full transienttransport simulation,a steadystate flow simulation canbeperformedinhighflowconditions.

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