1. Introduction. In a recent article, Goode [1996] presents an equation for the direct simulation of groundwater mean age. The solution of this equation yields the ...
WATER RESOURCES RESEARCH,
VOL. 34, NO. 12, PAGES 3271-3281, DECEMBER
1998
Simulation of groundwater age distributions Marcelo
Varni
Instituto de Hidrologia de Llanuras,Azul, Argentina
Jes6s Carrera Department of GeotechnicalEngineeringand Geosciences,Schoolof Civil Engineering,TechnicalUniversityof Catalonia(UPC), Barcelona,Spain
Abstract. The objectiveof our work is to examinehow to simulatethe age of groundwaterin sucha way that it can be comparedto actualmeasurements. We start by showingthat computationof kinematicage, the one obtainedby trackingwater along streamlines,is ill posedin heterogeneous aquifers.This, togetherwith its inabilityto accountfor mixingprocesses, makesit inadequatefor comparisonwith age measurements, whichare the resultof someaveragingof the age distributionin the water sample(the type of averagingdependson the measurementprocedure).Thereforewe go on to write the equationsfor the cumulativedistributionfunction of residencetime under transient flow conditions.This allowsus to derivetransientequationsfor the mean age, as well as for the higher-ordermomentsof its distribution,which generalizepreviousresultsby others.These momentscan be usedfor approximatingage measurements, which need not be equal to the mean age of the water sample.Using both a syntheticand a real example, we showthat mean age is an acceptableestimateof radiometricage measurementsin manycases.Includinga second-order correction(varianceof residencetime distribution) alwaysimprovesresults.On the other hand, higher-orderapproximations convergeslowly for old (comparedto half-life) waters,to the point that third-orderapproximations often worsen the results.
rosityare known.Kinematic age considersonly advectionas a transportprocess,which is a limitation. Nevertheless,it is the In a recent article, Goode [1996] presentsan equationfor approachused most often to validate new dating methods,as the direct simulationof groundwatermean age. The solution the ones describedin the above paragraph.This prompted of this equation yields the spatial distributionof the mean Walter [seeNeumanand Neretnieks,1984, p. 830] to state: groundwaterageand includesadvection,diffusion,and dispersion processes. The objectiveof groundwaterage simulations The problemis that in manysystemsthe hydraulicsof the system are not in equilibriumwith the geochemistry. And soit's not really can be to obtain the spatial distributionof groundwaterages right to try and validate the use of an environmentaltracer by and/or information about flow and transportparametersby comparingtravel timescomputedfrom the geochemicaldatawith comparisonbetween simulation and measurements(model thosecomputedfrom the hydraulicmodel. They're two different calibration).Our objectiveis the latter. However,thiscomparthings,and the personusingthe isotopeactuallyhurtshis/hercase ison is not as simple as it might look becausewater samples and in somecasesdiscreditsthe useof his/hertracerwhen actually the tracer representsthe reality and the hydraulicsare just a contain a mixture of waters of different ages and the way in transient artifact of recent history. whichthey are averagedin the measurementprocessneedsnot be identicalto the mean age.This is further complicatedby the It is clear that more sophisticatedmethods are needed for diversityof measurementmethods[Davisand Bentley,1982]: environmental tracer dating of groundwater.This motivated for example,usingdecayof radioisotopeswhich have entered the groundwaterfrom contactwith the atmosphere[Smithet the developmentof differentmodels.Thesewere usedoften in and Zuber, 1982, al., 1976],considerationof certainproductsof radioactivedis- the interpretationof tracer data [Maloszewski 1993;Llamas et al., 1982; Campanaand Simpson,1984; Camintegrationto be an age index [Bathet al., 1979],usingstable isotopicrelationsfor relating their spatial distributionto cli- pana and Mahin, 1985;Zuber, 1986]. However, thesemodels matic changes[Sonntaget al., 1979],and usingdisequilibrium still incorporatesignificantsimplifications.For example,they usuallyneglectspatialvariabilityof aquifer parameters.Morebetweenorigin and productradionuclidesas an age index. The traditional methodfor age simulationis basedon com- over, many of them are solute-specific. The motivation for this work is similar to that of Goode puting the groundwatertravel time from the rechargeto the [1996]:to avoid solute-specific modelingin the hope of taking observationpoint usingthe effectivevelocityfield [Davisand advantage of diverse dating techniques for inversemodeling Bentley,1982].Ages thuscomputedare calledkinematicages [Carrera and Neuman, 1986; Harvey and Gorelick, 1995a,Mein this work. This method allows the simulationof the age dina and Carrera, 1996]. We start by revisiting the kinematic spatial distribution,if mean historicDarcy velocitiesand poageconcept.This leadsus to the need for agedistributions,for Copyright 1998 by the American GeophysicalUnion. whichwe derivethe equationsunder transientflow conditions, thus generalizing previous equations obtained under steady Paper number 98WR02536. 0043-1397/98/98 WR-02536509.00 stateregime.This generalizationis carriedon to the temporal 1.
Introduction
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VARNI
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SIMULATION
000•
O
O
OF GROUNDWATER
AGE
DISTRIBUTIONS
>E-2
O 0
500m ,
Figure 1. Descriptionof the syntheticexample:(a) flow domain,includingfive high-conductivity lenses;a
recharge of 10mmyr- • takesplaceovertheA portionof theupperboundary; all discharge concentrates on the B portion,whosenodesare markedandwherea mixedboundaryconditionis used;markersindicatethe locationof observation points;(b) finite elementgrid (eachtrianglewasdividedinto four whencomputing
fluxes); (c) flownet;and(d) kinematic agedistribution. Thesizeof theflowdomainis 1500x 500m2. momentsequations,derivedunder steadystateby Harveyand are interestedin exploringthe problemsgeneratedwhen apGorelick[1995b].Momentsof the agedistributionscanbe used plying it to heterogeneousmedia. for approximatingactual age measurements. The validity of suchapproximations is finallytestedby meansof two examples. 2.2. Example Kinematic agesare computedfor a syntheticheterogeneous vertical section which consistsof a medium with hydraulic
2.
Kinematic Age
2.1.
conductivity of 1 m d-1 andfiveembedded lenses withcon-
Definition and Computation
The oldest and most widely used method for computing water age is based on the advectivemodel. Travel times are derivedfrom Darcy'slaw combinedwith an expressionof continuity [DavisandBentley,1982].This requiresestablishing the meanhistorichydraulicgradients,the hydraulicconductivities, and the effectiveporositiesof the aquifer.Then the kinematic age, denotedax, is givenby
a•:(x) =
v fx 4> q xdl dl
o
(1)
o
where 0 is porosity,v andq are seepageand Darcyvelocities, dl is the trajectoryelement,which is taken parallel to q. Distance alongthe trajectoryis definedby x with Xo representing the point at which water entersthe aquifer. Our computation of (1) consists of the followingsteps[Varniet al., 1994]: 1. First, we solvethe flow equationby the finite element
ductivities of 25, 48, 74, 134,and43 m d-1 at zonesnumbered 2-6, respectively(Figure la). Boundaryconditionsare shown
in the Figurela: a 10mmyr-1 recharge wasimposed at zone A, and a mixedboundaryconditionwith a leakagecoefficient
of 10-3 day-1 andexternalzerolevelwasimposedat zoneB. No flowboundaryconditionsare specifiedat the left, right,and bottom boundaries. The effective porosity is 0.1. The flow equationis solvedfor steadystate conditionsusingthe finite elementcodeTRANSIN-II [Medinaet al., 1995]. Figureslb, lc, and ld displaythe finite elementmesh,computed flow net, and age distribution, respectively.Multiple
fluctuations(zoneswith water older than the surroundings) canbe observed,especiallydownstreamof the high-conductivity lenses.This problemis causedby interpolationon a discontinuousfield [Varni et al., 1994]. In essence,the basicproblem with Figure 1 is that the age
field is not continuous.In fact, it is evident that discrete age differencesmay appear between two close stream lines, one method. flowing through a very permeable lens and the other not. 2. We then compute the velocity field using Darcy's law. Therefore smallvariationsin the measurementpoint location Problems caused by discontinuitiesat element boundaries may cause large variationsin age. That is, age computationis were overcomeusing the method of Cordesand Kinzelbach unstable with respect to the measurement point location. [1992]. 3. Finally, we calculategroundwaterage at each node by Moreover,ageis undefinedat stagnationpoints,whereit tends
integrating(1) along the streakline that startsat the node. to infinity(actually,to the ageof the rock formationat most). A problemis saidto be well posedin the senseof Hadamard Notice that this preventsus from having internal sinks and sources,which is indeed one of the limitations of the kinematic [1902]when a solutionexists,and it is uniqueand stable.The age concept.
In homogeneousmedia the kinematicageconceptprovedto
be well posed[Varniet al., 1994;Goode,1996].However,we
aboveexampleshowsthat in heterogeneousmedia, kinematic ages may not exist and they may be unstable. Moreover, uniqueness maybe questionablewhen mixingwatersof differ-
VARNI
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SIMULATION
OF GROUNDWATER
ent ages,as in the caseof recharge.In short, the conceptof kinematicage is ill posed. The need to accountfor dilution processes, suchas dispersionandmatrixdiffusion,whencomparingmeasuredand computed ages,has been pointed out by many authors [e.g., Sudickyand Frind, 1981; Walkerand Cook, 1991;Maloszewskiand Zuber, 1982, 1991]. This, together with the weaknessof the kinematic age concept, makes it clear that such a concept shouldnot be used beyondqualitativeassessments whenever the systemis heterogeneousor dilution processes(e.g., recharge,matrix diffusion,dispersion,and samplingprocedure) take place. In such casesit should not be used for direct comparisonbetween age measurementsand model outputs aimedat estimatingmodelparameters.Rather, realisticassessmentsof age measurementsshouldrecognizethat water samples containa distributionof ages.
3. Age Distribution at a Point 3.1.
Transient
Flow
If a narrow unit-masspulse of a conservativetracer is applied uniformly in the aquifer rechargezones and concentration is monitored at a point in the aquifer, the concentration breakthroughcurvecan be viewedasthe travel time probability distribution[JuryandRoth, 1990].This conceptis analogous to that
of residence
time
distribution
defined
for chemical
AGE
DISTRIBUTIONS
3273
1 if t > r; otherwiseH(t - r) = 0); F• is a Dirichlet boundary,and 1'2is a boundarywith prescribedmassflux; n is the unit vector normal to 1'2;and F v and F 2 are the distributionsof water particlesenteringor leavingthe aquiferthrough internal sink/sourcesand prescribedmassflux boundaries,respectively.F v and F 2 equal H(t - r) when water entersthe aquifer (-q ßn or r > 0) and equal F when water exitsthe aquifer (-q-n or r < 0). Severalremarksshouldbe made regarding(2). First of all, density,porosity,water flux, sink/sourceterms,boundaryconditions, and dispersionparametersmay be time dependent. That is, the flow regime controllingthe distributionof travel timesmay be transient.The time variableis t, somethingthat shouldbe kept in mind becausetravel time is t - r, where r is the time at whichparticlesenteredthe aquifer.That is, (2) doesnot representa singleequationbut a continuousfamilyof equationsfor varying r. Since each equation of this family representsthe evolution of the distributionof particles that entered the aquifer after time r the initial condition for each equationis givenby (2b), where the initial time is preciselyr. One point to keep in mind, however,is that this initial condition only affectsthe solutionof eachF(t, r, x) andbearslittle relationshipwith the actualbeginningof the aquifer. This last issuemaybe importantin geologicallyrecentaquifers.One wouldexpectthat mostshallowaquifers(
