WATER
RESOURCES
RESEARCH,
VOL. 35, NO. 12, PAGES 3829-3840, DECEMBER
1999
Reactive solute transport in streams: A surface complexation approach for trace metal sorption Robert
L. Runkel
and Briant
A. Kimball
U.S. GeologicalSurvey,Denver, Colorado
Diane M. McKnight Institute of Arctic and Alpine Research,Universityof Colorado, Boulder
Kenneth
E. Bencala
U.S. GeologicalSurvey,Menlo Park, California
Abstract. A model for trace metalsthat considersin-streamtransport,metal oxide precipitation-dissolution, and pH-dependentsorptionis presented.Linkage between a surfacecomplexationsubmodeland the streamtransportequationsprovidesa framework for modelingsorptiononto staticand/or dynamicsurfaces.A staticsurface(e.g., an ironoxide-coatedstreambed)is definedas a surfacewith a temporallyconstantsolid concentration.
Limited
contact between
solutes in the water column and the static surface
is consideredusinga pseudokineticapproach.A dynamicsurface(e.g., freshlyprecipitated metal oxides)has a temporallyvariablesolidconcentrationand is in equilibriumwith the water column.Transport and depositionof solutemasssorbedto the dynamicsurfaceis representedin the streamtransportequationsthat includeprecipitatesettling.The model is applied to a pH-modification experimentin an acid mine drainagestream.Dissolved copperconcentrations were depressedfor a 3 hour period in responseto the experimentallyelevatedpH. After passageof the pH front, copperwas desorbed,and dissolvedconcentrationsreturnedto ambientlevels.Copper sorptionis modeledby consideringsorptionto agedhydrousferric oxide (HFO) on the streambed(staticsurface) and freshlyprecipitatedHFO in the water column(dynamicsurface).Comparisonof parameterestimateswith reportedvaluessuggests that naturallyformed iron oxidesmay be more effectivein removingtrace metalsthan syntheticoxidesusedin laboratory studies.The model'sability to simulatepH, metal oxideprecipitation-dissolution, and pridependentsorptionprovidesa meansof evaluatingthe complexinteractionsbetweentrace metal chemistryand hydrologictransportat the field scale. 1.
Introduction
Effectiveremediationof streamsand riversaffectedby acid mine drainagerequiresa thoroughunderstandingof the dominant mechanismscontrollingtrace metal concentrations.To gain understanding,mathematicalmodels that quantify the complexinteractionsbetweengeochemical processes andphysical transportare often employed.Runkel et al. [1996a], for example,developedan equilibrium-based modelthat wasused to study metal fate and transport in two mountain streams [Broshears et al., 1996;Runkelet al., 1996b].Theseapplications focusedon the precipitation-dissolution reactionsaffectingthe distributionof iron and aluminumspecies.From a water quality perspectivethesemetalsare oftenproblematicin the upper reachesof mine drainagesystemswhere bufferingby dilute tributaries results in increasedpH and metal precipitation. Hydrousoxidesof iron and aluminumcoat the streambedand adverselyaffect the aquatichabitat of benthic invertebrates and algae[McKnightand Feder,1984;Niyogiet al., 1999].Farther downstream,waters are no longer acidic, and dissolved Copyright1999 by the American GeophysicalUnion. Paper number 1999WR900259. 0043-1397/99/1999 WR900259509.00
concentrationsof iron and aluminum are substantiallyreduced.Despite thesereductions,concentrationsof other metals suchas copperand zinc remain elevated.Removal of these metals is dependenton downstreamdilutions that produce circumneutralconditionsthat, in turn, foster sorption reactions. Owing to the deleteriouseffectsof copperand zinc on aquaticlife, modelsthat considerthe pH dependenceof sorption reactionsare clearlyneededfor many mine drainageapplications. Comprehensivereactive transport simulation of mine drainageremainsinherently complex.Others have consideredsuchparticularcasesas manganeseoxidation[Harvey and Fuller, 1998] and CO2 degassing[Choiet al., 1998]. With respectto sorptionphenomena,two types of models have traditionallybeen used:empirical approachesbasedon distribution coefficients and surface complexation models (SCM) based on electrostatictheory. Empirical models are relativelyeasyto employand are often desirablein site-specific studiesin which pH is spatiallyand temporallyuniform. The disadvantageof the empiricalapproachis that model parametersare highlydependenton the chemicalcompositionof the aqueoussolution [Davisand Kent, 1990]. As such,empirical modelsare of limited utility in mine drainagesystemswhere pH varies spatiallyin responseto groundwaterand tributary inflow.An additionalproblemis that parameterestimatesde-
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velopedfrom observeddata are of little usewhen considering remedial actionsthat causechangesin pH. Surface complexation modelsavoid these pitfalls by explicitlyconsideringthe effectsof pH on sorptionreactions. To date, many investigatorshave applied surfacecomplexation modelsto natural systems[Grossiet al., 1994;All and Dzombak, 1996; Tessletet al., 1996; Wanget al., 1997;Davis et al., 1998;Smithet al., 1998].Theselaboratory-oriented studies provide the fundamental information needed to refine the SCM approachand to develop a viable databaseof sorption parameters.Further, chemicalcharacterization of hydrousiron and aluminum
oxides from acidic streams is consistent with the
SCM approach[McKnightet al., 1992].The next logicalstepis to apply SCMs in field settingswhere stream transport and solutedynamicsare considered.Although the idea of studying sorptionin the contextof transportis not new, field applicationsthat includestreamtransportare limited to the empirical representationsof sorptiondiscussedabove [Bencala,1983; Kuwabaraet al., 1984;Brownand Hosseinipour,1991;Chenet al., 1996]. Groundwatertransportmodelsthat implementthe SCM approachare describedby Cederberg et al. [1985]andYeh and Tripathi[1991]. The primary focus of this paper is the developmentof a reactive transport model for streamsthat includessorption reactionsas defined by a SCM. Model developmentis describedin section2 where the algorithmsunderlyingthe reactive transport model are presented. The model is demonstrated using two applicationsthat describepH-dependent sorptionin an acid mine drainagestream(section3). Model parametersare basedon hydrologicfield data and literature valuesderivedfrom laboratorysorptionstudies.
2.
Model Development
Here we develop the equationsand algorithmsneeded to implement the surfacecomplexationapproachwithin a reactive transportmodel. The resultantmodel mustbe flexiblein order to addresspractical problems at the field scale. Acid mine drainage streams,for example, often have iron-oxide-
TRANSPORT
IN
STREAMS
putes the distributionof chemicalspeciesthat existwithin a batch reactor at equilibrium. The mass-balanceand massaction equationsdescribingequilibria form a set of nonlinear algebraicequations(AEs). The coupledsetof PDEs and AEs are solvedusingsequentialiteration [Yehand Tripathi,1989]. The resultantmodel considersa variety of processes including advection,dispersion,transientstorage,the transportand deposition of waterborne solid phases,acid-basereactions,complexation,precipitation-dissolution, and sorption. Governingequationsfor the reactive transportmodel are formulatedin termsof chemicalcomponents,the fundamental building blocks from which all chemicalspeciesare derived [Westallet al., 1976].The total componentconcentration(T) consists of dissolved(C), mobileprecipitate(Pw), immobile precipitate(P•,), mobile sorbed(Sw), and immobilesorbed (S•,) phases.Each phaseconsistsof one or more chemical species.Processesconsideredfor each phaseare depictedin Figure la, where the systemis representedas two compartments. The water column compartment containsthe three mobile phases,C, Pw, and Sw. Pw representssolidsthat form in the water column as the result of mineral precipitation;Sw representsdissolvedmass that has sorbed onto waterborne solids,suchas hydrousmetal oxides.Immobilesubstrate(i.e., the streambedor debris)constitutesthe secondcompartment, containingthe two immobile phasesP•, and S•,. The three mobilephasesare subjectto physicaltransport,as represented by the transportoperatorL. The dissolvedphaseC takespart in precipitation-dissolution and sorption-desorption reactions that occurwithin the water column(interactionswith Pw and Sw). The dissolvedphase is also affected by dissolutionof precipitate from the immobile substrateand by sorptiondesorptionfrom immobilesorbents(interactionswith P•, and S•,). C may increaseor decreasebecauseof externalsources and sinks,as denoted by Scxt.The precipitated and sorbed phasesin the water columnsettle in accordancewith settling
velocity v (L T- 1). Theuseof a singlesettling velocity results in a uniform settlingrate for all waterborne solids•more com-
plexrepresentations of settlingare possible[e.g.,Murray,1970] but are beyondthe scopeof this paper. coated streambed materials that sorb trace metals. Additional Runkelet al. [1996a]developeda generalmassbalanceequasorption reactionsoccur in the water column,where freshly tion for each componentby consideringthe massassociated precipitated metal oxidesprovide sorptivesurfacesthat are with each of the five componentphases.The massbalance subjectto downstreamtransport.The problemis further com- equationfor the total componentconcentrationis givenby plicatedby hydrologicconditionsthat may warrant kinetic or OT/Ot = L(T) - L(Sb + Pb) + Sext, (1) equilibrium-basedtreatment of the sorption reactions.The model describedbelow is thereforeformulatedsuchthat sorpwhere the transport operator is defined in terms of the trantion reactionsmay occur on the streambedand/or within the sientstoragemodel [Bencalaand Walters,1983•Runkel,1998]. water column. Further, the model allowsfor both pseudokiThe immobileprecipitatedand sorbedconcentrations in (1) netic and equilibriumsorption. are governedby 2.1.
Governing Equations
OP•/Ot = • (P- P•)- f• (2) Runkelet al. [1996a]developedan equilibrium-based model for reactivesolutetransportin streams.The reactivetransport model is formed by couplingthe OTIS solutetransportmodel os/ot= (s - &) (3) (one-dimensional transportwith inflow and storage)(http:// webserver'cr'usgs'gøv/øtis) [Runkel, 1998] with a chemical equilibrium submodel.The OTIS solute transport model is wheref•, is the source/sinkterm for dissolutionfrom the imbased on a one-dimensional advection-dispersionequation mobilesubstrate (massL- 3 T-1), g•,is thesource/sink term with additional terms to account for lateral inflow and tranfor sorption-desorption from the immobile substrate(mass sientstorage[Bencalaand Walters,1983].Conservationof mass L -3 T-1), d is the settling depth(L), P is thetotalprecipresultsin a setof partialdifferentialequations(PDEs) describ- itate concentration(equal to Pw plusP•,), and S is the total ing physicaltransport.The chemicalequilibriumsubmodelis sorbedconcentration(equalto Sw plusSo). basedon MINTEQ [Allisonet al., 1991], a model that cornThe governingequationset consistsof three PDEs (for T,
RUNKEL ET AL.: REACTIVE SOLUTE TRANSPORT IN STREAMS
a. Conceptual Systemof Runkelet al. [1996a]
3831
b. Sorptionto a StaticSurface
s•t
water Column •
Water Column
L
L
Immobile Substrate
Immobile Substrate
d. SorptionontoStaticandDynamicSurfaces
c. Sorptionto a DynamicSurface
Se•t
Sext
Water Column •
WaterColumn/-•
L
L
Immobile Substrate
Immobile Substrate
Figure1. Conceptual surface watersystem usedtodevelop thegoverning differential equations. (a) Conceptual system ofRunkel etal.[1996a]. (b)Sorption toa static surface. (c)Sorption toadynamic surface. (d) Sorption ontostatic anddynamic surfaces. Thetotalcomponent concentration consists of dissolved (C), mobile precipitate (Pw),immobile precipitate (P•,),mobile sorbed (Sw),andimmobile sorbed (S•,)phases. Thedissolved phase mayincrease or decrease because of external sources andsinks, asdenoted bySext; dissolved andmobilephases aresubject to transport, asdenotedbyL.
approaches neglectelectrostatic effects,theiruseis P•,, andS•,) for eachcomponent andthe setof AEs repre- coefficient senting chemical equilibria. Thisequation setissolved usinga limited in metal-contaminatedwaters where the primary sorCrank-Nicolson approximation of the differentialequations bentsare hydrousmetal oxidesthat havechargedsurfaces. models(SCMs),in contrast,explicitly andthe sequential iterationapproach. To solve(1), valuesof Surfacecomplexation theeffectsof pH andionicstrength on surfacecharge the statevariablesat the initial time level (n) and advanced consider implications for sorption.Examples of timelevel(n + 1) arerequired. Statevariables at timelevel andthe corresponding n areavailablefromtheprevious timestep,whileestimates of SCMs includethe diffuselayer model [Stummet al., 1970; model[Yates et al., the statevariablesare requiredfor time leveln + 1. Specif- HuangandStumm,1973],the triple-layer two-layermodel ically,estimates ofP, P•,,S, andS•,areneeded, aswellasthe 1974;Daviset al., 1978],andthe generalized source/sink termsf•,, #•,, andSext.Estimatesof P andS are [DzombakandMorel,1990]. 2.2.1. Generalizedtwo-layer model (GTLM). This secprovided directly bytheequilibrium submodel, estimates ofP•, the generalized two-layermodel[Dzombak and andS•,areprovided by solving thecorresponding PDEs,and tiondescribes withinMINTEQ [Allisonet al., the source/sink termsare developedalgorithmically. The re- Morel,1990]asimplemented two-layer model mainder of this sectiondescribeshow a surfacecomplexation 1991].As withotherSCMs,the generalized defines sorption reactions in termsof masslawequations that model is usedto calculate#•, and solve(3). govern theconcentrations of sorbate, sorbent, andsurface sites 2.2. Surface ComplexationModel at equilibrium. The equilibrium constantassociated with a given mass action equation is the product of an intrinsic term Mathematical representations of sorption rangefromsimple the chemical free energyof sitebindinganda distribution-coefficient approaches to morecomplexrepresen- representing termrepresenting thecoulombic freeenergyof binding tationsbasedon electrochemical theory.Becausedistribution- second
3832
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due to the electrostaticallycharged surface. The coulombic whereN s is the site density(molesof sitesper mole sorbent) term actsas a surfaceactivitycoefficientthat accountsfor the and M is the molecularweightof the sorbent(gramssorbent work requiredto moveionsfrom the surfacelayer to the bulk per mole sorbent). solution. 2.2.2. GTLM within the reactive transport model. The Because the coulombic term varies as a function of surface reactivetransportmodel is formulatedsuchthat sorptionmay charge and potential, sorptionmasslaw equationsmust be occuronto staticand dynamicsorptivesurfaces.Staticsorptive rearrangedand expressedin terms of intrinsicsurfacecompl- surfacesare thosefor whichthe concentrationof sorptivesolid exationconstants.For example,considersorptionof a divalent (So) doesnot changein time. Conversely,dynamicsorptive cation: surfacesare thosefor which the concentrationof the sorptive solidis time variable.An exampleof a staticsorptivesurfaceis SOH + M 2+• SOM+ + H +, (4) a streambedarmoredwith hydrousiron oxides[e.g.,Broshears et al., 1996]. In this case the number of sites available for whereM2+ is a divalentcation,H + is a hydrogen ion,SOH is sorptionreactionsis relatively constantthroughoutthe time anuncharged surface hydroxyl group,andSOM+ isa positively period of interest. Other situationsmay arise in which the chargedsurfacespecies.The corresponding massactionequa- number of sites changesin time and sorptionto a dynamic tion is surfaceis applicable.Such is the casewhen hydrousmetal oxides form in the water columnas a result of precipitation {SOM+}{H +} reactions. Within the model theseprecipitatesare definedas K= {SOH}{M2+}, (5) dynamicsorptivesurfaces. To further classifythe sorptioa reactions,it is useful to whereK is the equilibriumconstantand { } denoteschemical subdividethe sorptivesurfacesinto three "pools."Pool 1 conactivity.ExpressingK as the productof the intrinsicand cousistsof the staticsorptivesurface;pool 2 consistsof the dylombic terms and rearrangingyields namic surfacespresentin the water column,that is, associated {SOM+}{H +} with the waterborneprecipitates;and pool 3 consistsof dyg int= (6) namic surfacesthat were initially presentin pool 2 but have settledto the streambedduring the courseof the simulation. The sorbedconcentrations associated with pools1, 2, and3 are whereKintis the intrinsic surface complexation constant, exp denotedby S•, S2, and S3, respectively.Additional assumptionsunderlyingthe useof GTLM withinthe reactivetransport (-•F/RT) is the coulombiccorrectionfactor, ß is surface model are as follows:(1) Sorptionreactionsadhere to the potential(volts),F istheFaradayconstant (96,485C mol-•), R is the molargasconstant(8.314J mol-• K-•), and T is generalizedtwo-layermodel as definedby Dzombakand Morel [1990]. (2) A staticsurfaceand/or a dynamicsurfacemay be absolutetemperature(kelvins). defined.Each surfacemay have high- and low-affinitysites.A Solution of a chemicalequilibriumproblem that includes dynamicsurfaceis distributedbetweenpools2 and 3 asdefined equationssuchas (6) requiresintroductionof a dummychemabove.(3) Specificsurfacearea (S.4) and sorbentmolecular ical componentto accountfor the coulombiccorrectionfactor weight(M) are specifiedfor eachsurface.Site density(Ns) is and a definitionof surfacepotential.Under electricaldouble layer theory, surfacechargeis balancedby a diffuse layer of specifiedfor each site type on each sorptivesurface.Sorbent properties(S.4, M, andNs) are spatiallyand temporallyconcounter chargesin solution;the relationshipbetweensurface stant. charge and surfacepotential is defined by Guoy-Chapman Given theseassumptions, three casesare possible:(1) sorptheory[DzombakandMorel, 1990].Giventhisrelationship,the tion to a static surface, (2) sorption to a dynamicsurface,and total componentconcentrationfor the coulombiccorrection (3) sorption to static and dynamic surfaces. As shownbelow, factor is given by thesecasesdiffer with respectto how (3) is solved,how the S.4Sc equilibrium submodelis used, and how kinetic limitationsare TOTe=o-F ' (7) imposed.
{SOH}{M2+} exp (•) ,
whererr is net surfacechargedensity(C m-2), P = exp (- •F/R T), S.4isspecific surface area(m2 g-1 sorbent), and So is solidconcentration(gramssorbentper liter). Solutionof the chemicalequilibriumproblem alsorequires specificationof componentsthat representthe sorptivesurface. A centralpart of the generalizedtwo-layermodel is the postulationthat eachsorptivesurfacehastwo typesof sitesfor cationbinding.The first type,the high-affinitysite,is generally lessprevalentthan the secondsite type but has a stronger bindingpotential.A secondlow-affinitysiteis in greaterabundancebut hasweakerbindingpotential.Owingto the presence of two site types,two chemicalcomponentsare introducedfor eachsorptivesurface.Total componentconcentration(moles of sitesper liter) for eachsite type is givenby NsSc
TOTsø}•=M '
(8)
2.3.
Sorption to a Static Surface
A conceptualdiagramdepictingsorptionto a staticsurface is given as Figure lb. When sorptionoccurs,massis transferred from the dissolvedphaseC to the sorbedphaseassociated with pool 1, S •. When only a staticsurfaceis considered, S• is equivalentto the immobile sorbedphase So. During desorption,massis transferredfrom S• to C. The rate of sorption-desorption is governedby the kinetic parameter F. Sorptionto a staticsurfacerequiresseveraladditionalassumptions: (1) The solid concentration(So) doesnot changein time. The sorptivesolidis attachedto the streambedor debris and is thereforeimmobile.(2) The solidconcentrationis allowedto vary spatiallyon a reach-specific basis.(3) Sorptiondesorptionmay be subjectto kinetic limitations. The primarytaskis to solve(3) for the concentrationof the componentsorbedto the staticsurfaceSo. For staticsurfaces the settlingterm in (3) dropsout, yielding
RUNKEL
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SOLUTE
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IN STREAMS
3833
a. EquilibriumSorption
Tn+]
Equilibrium Submodel S n+l
b. Kinetically limited Sorption Pass 1
Tn+1
Equilibrium Submodel sn+l *
Pass 2
T' = T n+ l-Sn+
l ---•
C n+l
Equilibrium Submodel
p•+]
Figure 2. Use of the equilibriumsubmodelfor sorptionto a staticsurface,(a) equilibriumand (b) kinetically limited sorption. 2.3.2. Kinetically-limited sorption. Under the equilibrium approachthe sorbedconcentrationis taken directlyfrom where #/, is estimatedusingoutput from the equilibriumsubthe equilibrium submodel.Here we employ a pseudokinetic model. Two casesof sorptionto static surfacesare now conapproach in which only a fraction of the change in sorbed sidered:equilibriumsorptionand kineticallylimited sorption. concentrationis considered.This kineticlimitationis designed 2.3.1. Equilibrium Sorption. The first step in modeling to reflect the fact that only a portion of the massin the water equilibriumsorptionis to determinethe total componentcon- columncomesin contactwith the staticsurface(i.e., the strecentrations (Tn+ •) usedasinputto thechemical equilibrium ambed).Figure2b depictsuseof the equilibriumsubmodelfor submodel. Forthechemical components, Tn+ • corresponds to kinetically limited sorption. Computationsduring pass 1 are the solutionof (1) from the previoussequentialiteration.Total very similar to the computationsdescribedfor equilibrium
OS•,/Ot= -gb,
(9)
componentconcentrations for the coulombiccorrectionfactor and the high- and low-affinitysiteson the static sorptivesurfaceare givenby (7) and (8). The sorbentconcentration(So) usedin (7) is equal to the temporallyconstant,spatiallyvariablevalueassignedat the beginningof the simulation.Specific valuesof So, S•4, Ns, and M are shownin section3. As shown in Figure 2a, the equilibrium submodeldeterminesthe total componentmassin the dissolved,precipitated,
and sorbedphases(C• +•, P• + •, andS• +•). The sorption source/sinkterm is then calculatedbased on the change in sorbedconcentrationduring the current time step: S n _ sn+•
=
'
sorption; givenTn+ •, thesubmodel determines thetotalcom-
ponent massin the dissolved,precipitated,and sorbedphases
(C• + •*, P• + •*, andS• + •*, wherethe asteriskdenotesthe equilibriumconcentrationin the absenceof a kinetic limitation). The sorptionsource/sink term is now equalto a fraction of the changein sorbedconcentration:
=F
Sn_zXt sn+l* )'
where F is the fraction of the equilibrium quantity that is allowed to sorb-desorbduring the current time step. Solving (9) usinga forwardtime differenceyields
S•+1-- S; q-r(S n+l* - sn).
(13)
whereAt is the integration timestepIT-•]. Giveng/,, (9) is solvedwith a forward time differenceyielding:
During pass1, outputfrom the equilibriumsubmodelreflects solutionchemistryunderthe assumption of chemicalequilibrium. S•+•= Sn+•. (11) The phaseconcentrations from the submodel(C, S, andP) and As shownby (11), the submodelprovidesthe exact quantity the solutionpH thereforedo not includethe effectsof the kine[ic neededfor the solution,S• + •. As such,the model usesthe limitation.To incorporatetheseeffects,the total sorbedconcentration is set equal to the kinetically limited concentration equalitygivenby (11), rather than a formal solutionof (3).
3834
RUNKEL ET AL.: REACTIVE SOLUTE TRANSPORT IN STREAMS
(Sn+l = Sg+l),andpass 2 isinitiated. During pass 2, sorptionC,•+•, p,•+ 1, andS'•+ 1. In thiscase,S'•+ 1 isthetotalconcenreactions arenotconsidered, andthe totalcomponent concen- trationsorbedto the dynamicsurface.The amountof S'•+ 1 on trations are revised to eliminate sorbed mass: thestreambed isdetermined byconsidering thefractionof the dynamicsurface(precipitate)on the streambed: T' = Tn+l- Sn+l.
(14)
GivenT', the equilibriumsubmodel providesthe corrected
S•+1= p-•Sn+l.
valuesof C, P, and pH. 2.4. Sorption to a Dynamic Surface
(16)
2.5. Sorptionto Static and DynamicSurfaces
Reactions betweenthe dissolved phaseC andthe dynamic Concurrent simulation of sorptionto staticand dynamic surfacein pools2 and3 are depictedin Figurelc. Dissolved surfacesis depictedin Figure l d, whereinteractionsbetween ionsmaysorbto thewaterborne precipitates in pool2 thereby the dissolved phaseandthevarioussotbentpoolsare shown. increasing thesorbed concentration associated withpool2,S2. As before,sorption-desorption reactionsfor the staticsurface S2is equivalent to themobilesorbedphase(S,,) andisthere- maybe kinetically limited,whereas sorption-desorption reacfore subjectto downstream transportandsettling. After the tionsfor the dynamic surfacearein localequilibriumß reactionoccurs, sorbedmassmaysettle,increasing S3. Alter2.5.1. Equilibrium sorption. Use of the submodelfor natively, desorption mayoccurfrompool2, returning massto equilibrium sorption is shownin Figure3a. Computation of the dissolved phase.Sorption-desorption reactions alsotrans- total componentconcentrations for the coulombiccorrection fer massbetween thedissolved phaseandpool3. Whenonlya factorandthesurface components for thestaticanddynamic dynamic surface isconsidered, S3 isequivalent to theimmobile surfaces is asdescribed above.The submodel provides values sorbedphaseS•,.Assumptions uniqueto the dynamicsurface ofC'•+1 p,•+1 andS•+1 (static surface, pool1) andS•,•1 areasfollows: (1) In pool2 theconcentration of thesorptive (dynamic surface, pools2 and3). The totalsorbedconcentrasolid(So) variesin timeandspaceasa functionof the mobile tion is the sum of the concentrationssorbed to the static and precipitateconcentration of a specifiedhydrousmetaloxide dynamicsurfaces: (pwMe). Thedynamic surface andtheassociated sorbate reside sn+l= S1 nl q-•,2,3 cnl , (17) in thewatercolumnandaresubject to transport andsettling. (2) Pools2 and3 aremodeledusinga singlesurface definedin whereas the immobile sotbed concentrationis the sum of the the submodel. Sorptivesolidsin pools2 and3 thereforehave concentrations in pools1 and 3: identical specific surface areas(S.4),molecular weights (M), andsitedensities (Ns). Massis apportioned between S2 and n • n •,n+l Sb +1 S1 +1-[-• o2, 3ß (18) S3basedon eachpool'scontribution to So. (3) In pool3 the concentration of the sorptivesolid(So) variesin time and spaceasa functionof the immobileprecipitateconcentration 2.5.2. Kineticallylimited sorption. Use of the submodel limitedsorptionis shownin Figure3b. Pass1 of a specified hydrous metaloxide(p•4e).Thedynamic surface for kinetically and the associated sorbate concentration is attached to the
1+1,, provides values of C'•+1,, p,•+1,, S,•
and
S•,•1* . The
limitedconcentration sorbed to pool1 isgivenby streambed or debrisandis thereforeimmobile. (4) Sorption- kinetically desorption reactions in pools2 and3 arein localequilibrium. S•+1._S• -ørlP(S• +1'- S7). (19) Giventheseassumptions, the taskis to solve(3) for the phaseconcentrations frompass1 and concentration of eachcomponent sorbedto the dynamicsur- Aswitha staticsurface, facein pool 3, S•,. As with a staticsurface,a formalsolutionto
solutionpH do not reflect the kinetic limitation;corrected
duringpass2 (3) is not required;all of the sorbedconcentrations maybe valuesof C, P, S2,3,andpH are determined (Figure 3b), where obtained directly fromtheequilibrium submodel. Thesettling term in (3) is considered indirectly,in that the settlingof T' = Tn+l- S7+1 (20) sorbedmassis reflectedin the settlingtermfor the hydrous ß
metal oxide (i.e., sorbedmasssettlesat the samerate as the
In pass2, sorptionto the staticsurfaceis not considered, asthe
precipitate thatmakesupthedynamic surface, asgivenbythe kinetically limitedconcentration forpool1 hasbeencomputed governingequationfor P•,). Use of the equilibriumsubmodel for the dynamicsurfaceis
in (19). Total sorbedand immobilesorbedconcentrations are
givenby (17) and (18).
nearlyidenticalto thatfor thestaticsurface underequilibrium conditions (Figure2a).The onlydifference is computation of the total componentconcentrations for the coulombiccorrec- 3.
Model Application tionfactor(equation (7)) andthe surface components (high- The reactivesolutetransport modelis nowappliedto a andlow-affinity sites,(8));thesolidconcentration (So) isnow pH-modificationexperimentconductedon an acidminedraincalculated basedon the concentration of the precipitate de- agestream. Previous analyses of theexperiment arereported by
fined as the dynamicsurface:
Broshears etal. [1996].Herewerevisittheworkof Broshears et al.
to consider coppersorption. Our (15) andextendtheirsimulations goalin doingsois to demonstrate the sorption algorithms dewhere pMeisthetotalprecipitate concentration forthespec- scribed aboveandto provide a preliminary analysis of copper ified hydrousmetal oxide. As with the static surfaceunder sorption. Detailedanalyses of thepH-modification experiment equilibriumconditions,the submodelprovidesvaluesfor andcopper geochemistry arebeyond thescope of thispaper.
Sc= MPMe,
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REACTIVE
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IN STREAMS
3835
a. EquilibriumSorption C n+l
T
n+l
p n+l
Equilibrium Submodel
Sl n+l
S2,3 n+l b. KineticallylimitedSorption Pass 1
Cn+l,
Tn+1
pn+l,
Equilibrium Submodel
Sln+l*
S2,3 n+ 1'
Cn+l
T' = Tn+I-S• +1
Equilibrium Submodel
p n+l
S2,3 n+l Figure 3. Use of the equilibriumsubmodelfor staticand dynamicsurfaces,(a) equilibriumand (b) kinetically limited sorption.
3.1.
St. Kevin Gulch pH Modification
St. Kevin Gulch is a first-order stream near Leadville, Col-
orado, that is affectedby acid mine drainage.As a result, St. Kevin Gulch is acidic (pH -3.4) and has elevatedlevelsof sulfate,aluminum,iron, and copper.Precipitationand deposition of hydrousiron oxidesis evident,given the orangefloc that coats the streambed.
Characterization
of iron oxides from
St. Kevin Gulch by X-ray diffractionshowan increasein crystallinity and a decreasein reactivityasprecipitatediron oxides age [Hrncirand McKnight,1998].Additional researchrelated to the geochemistryof St. Kevin Gulch includesstudiesof stream-subsurface water exchange[Constantzet al., 1994;Harveyet al., 1996], iron photoreduction[McKnightet al., 1988; Kimball et al., 1992], and nutrient dynamics[Tate et al., 1995]. Broshearset al. [1996] describea pH-modificationexperiment conductedon August25, 1988.Beginningat 0918 hours, a concentratedsolution of sodium carbonatewas injected to increasein-stream pH. The injection rate was increasedat 1154hoursto further elevatepH. The higherinjectionrate was maintaineduntil 1454hours,when the injectionwas stopped. Water sampleswere filtered through0.1 /am filters; aliquots were acidifiedwith HNO 3 for cation and trace metal analysis. Cations and trace metals were analyzedusing an inductively coupledplasmaspectrophotometer; anionswere analyzedusing ion chromatography. The effectsof the pH modificationare shownin Figure 4, where pH, iron, aluminum, and copper are shownfor two samplinglocationslocated24 and 70 m downstreamfrom the injection.The pH profilesreflectthe two-stepadditionof sodium carbonate;pH increasesfrom backgroundto 4.0-4.2
duringthe firststepand increasesto 5.0-5.8 duringthe second step(Figure4a). Dissolvediron concentrations exhibita small decreaseduring the first step that is followed by a larger decreaseduringthe secondstep(Figure4b). Dissolvedaluminum and copper(Figures4c and 4d) concentrations are relatively unaffectedby the first step but decreasesubstantiallyduring the second.
Broshearset al. [1996] employedthe model of Runkelet al. [1996a] to simulatethe pH, iron, and aluminum responses describedabove.Model simulationsindicatedthat precipitation-dissolutionreactionswere responsiblefor changesin dissolvediron and aluminum.Changesin pH were attributableto precipitation-dissolution, interactionswith the carbonatesystem, and bufferingby the streambed. In the simulationsthat follow, sorptionreactionsbetween dissolved speciesandhydrousferricoxide(HFO) are governed by the generalizedtwo-layermodelas implementedwithin the equilibriumsubmodel.HFO specificsurfacearea (S.4) and
molecular weight(M) are setat 600m2 g-• and89 g mol-•, respectively[Dzombak and Morel, 1990]. Unless otherwise
noted,surfacecomplexation constants (Kint)are set at the defaultvaluesfrom the HFO database.Additionalprocedures and parameterestimatesare as describedby Broshearset al. [1996]; chemicalcomponentsinclude aluminum,ferrous and ferric iron, copper,carbonate,sulfate,and excesshydrogen;log formation constantsfor ferrihydrite (Fe(OH)3) and gibbsite (AI(OH)3) are set at -3.65 and -9.82, respectively; temperature and ionic strengthare set at 10øCand 0.005, respectively; andphysicaltransportcharacteristics are estimatedfrom conservative
tracer
data.
3836
RUNKEL ET AL.' REACTIVE SOLUTE TRANSPORT IN STREAMS
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Time [hr]
Figure 4. The(a)pH,dissolved (squares) andparticulate (solid diamonds) concentrations of(b)iron,(c)
aluminum,and (d) copperat 24 and 70 m. Particulateconcentrations are set to 0.0 whenthe dissolved
concentration exceeds totalconcentration (aluminum only(opendiamonds)). 3.2. The pH Buffering by the Streambed: Sorption to a Static Surface
streambedand the water column,kinetic limitationswere im-
posed.Their simulation is an exampleof kinetically limited Broshears etal. [1996]used"streambed buffering" to explain sorption to a staticsurface (section 2.3andFigures lb and2b). the downstream attenuation of pH observed duringthe pH Here we presentsimulationresultsfor sulfateto illustratethe modification. In their simulation a streambed with a constant interaction betweensorption andstreamtransport. HFO concentration interacted withthesulfateandexcess hyFollowing Broshears etal. [1996],thestreambed isdefinedas drogencomponents. Owing to limited contactbetweenthe a staticsurface witha solidconcentration (Sc) of 0.2g sorbent
RUNKEL
ET AL.:
REACTIVE
SOLUTE
TRANSPORT
IN STREAMS
3837
occursfollowingthe pH decrease(t > 14.9) as the sulfate B
demand
C
C
1330
I
1325
•
8
I
10
,
I
,
12
decreases
over time.
The simulated sulfate responseis within the range of the sulfate concentrations observed before and after the pHmodificationexperiment(1320-1340/xM). Observedsulfate concentrationsdecreaseto 1230-1280/xM duringthe period of elevatedpH. This decreaseis likely due to the coprecipitation of sulfatewith hydrousferric oxides[Bighamet al., 1996], a processthat wasnot consideredby Broshears et al. [1996].As such, observedsulfate concentrationsare not presented in Figure 5.
1335
A
of the streambed
i
i
,
16
14
i
18
3.3. Copper Sorption: Sorption to Static and Dynamic Surfaces
Time [hr]
Our secondexample extendsthe work of Broshearset al. Figure 5. Simulated sulfate concentrationat 24 m. Arrows [1996] to considersorptionof copper to HFO. As shownin denote three distinct time periods:Preinjectionequilibrium Figure4d, dissolvedcopperconcentrationsdecreaseduringthe (A), initial streamresponse(B), and reestablishment of equi- secondstepof the pH-modificationexperiment.Severalcharlibrium (C). acteristicsof the data point to specificmechanismsresponsible for the decreasein dissolvedcopper.First, formationof copper precipitatesis unlikely giventhe observedpH. Second,particL-•. Sitedensities (Ns) for thehigh-andlow-affinity siteson ulate (total minus dissolved)copper is present in the water the staticsurfaceare set at 0.005 and 0.2 mol of sitesper mole column.The concurrentpresenceof particulateiron (Figure sorbent,respectively[Dzombakand Morel, 1990].Figure 5 de- 4b) suggests the possibilityof sorptionto waterborneHFO. picts the simulateddissolvedsulfateconcentrationsat 24 m. Finally, the decreasein total copper at 24 and 70 m suggests The seeminglyerratic behaviorof sulfateresultsfrom interac- sorptionby the iron-oxide-coatedstreambed. Given these observations,we simulate copper sorptionby tionsbetweenthe pH changeand streamtransport.The sulfate responseto the pH modificationmay be grouped into three consideringtwo surfaces:freshlyprecipitatedHFO in the wadistincttime periods:preinjectionequilibrium,initial in-stream ter column(dynamicsurface)andagedHFO on the streambed response,and reestablishmentof equilibrium. During prein- (staticsurface).As with the sulfateexample,the solidconcenjection equilibrium(t = 8.0-9.4 hours),pH is constant,and tration associatedwith the staticsurfaceis set to 0.2 g sotbent areimposed. The solidconcentradissolvedsulfateis in equilibriumwith the streambed(Figure L -•, andkineticlimitations 5, A arrow). When abrupt changesin pH occur, the initial tion of the dynamicsurfacevaries spatiallyand temporallyas in-streamresponseproducesa rapid changein dissolvedsul- definedby the amount of particulateiron formed by ferrihyfate concentration(Figure 5, B arrows).During the two pH drite precipitation(equation(15)). Sorptionof copper,sulfate, steps(t - - 9.4 and t = - 12.0), sulfatedesorbsfrom the and excesshydrogenis consideredfor both surfaces.Simulastreambed and dissolvedconcentrationsincrease. Similarly, tion of staticand dynamicsurfacesrequiresthe specificationof duringthe pH decrease(t = -14.9) the streambedis sulfate severalparametersto characterizethe sorptivesurfacesand deficient, and dissolved concentrations decrease as sulfate sorptionreactions.Exceptwhere noted below, parametervalsorbsto the streambed.Abrupt changesin pH are followedby ues are set to the best estimatesof Dzombakand Morel [1990] a reestablishmentof equilibriumduringwhich the sulfatede- (Table 1). This exampleillustratessorptionto staticand dymand(or excess)of the streambedis met (Figure5, C arrows). namic surfaces, with kinetic limitations on the static surface (section2.5.2. and Figuresld and 3b). Followinga pH increase(9.4 < t < 12.0 and 12.0 < t < Simulationresultsat 24 and 70 m are shownin Figure 6. At 14.9), dissolvedconcentrationsgraduallydecreaseto preinjection levels as excesssulfate liberated from the bed is ad- both locations, simulated and observed values of dissolved vected downstream.A similar reestablishmentof equilibrium copper are in closeagreementimmediatelyfollowingthe sec-
Table 1. SorptionParameters:Copper Sorptionto HFO Simulation Parameter Estimate
Dzombakand Morel [1990] Parameter
Specific surface area(Sc), m2 g-• Molecular weight of HFO, grams HFO per mol HFO Site density(Ns), molessitesper moles HFO
Log intrinsicsurfacecomplexation
constant for copper, g int*
Site Type Both Both
Low affinity High affinity Low affinity High affinity
Range 200-840
0.1-0.3 0.001-0.01
2.49-3.40
Best Estimate 600 89
0.2 0.005 0.6 2.89
Static Surface 600 89
0.2 0.005 0.6 1.85
Dynamic Surface 600 89
0.2 0.01 0.6 3.40
*Intrinsic surfacecomplexationconstantsfor sulfateand excesshydrogenare set equal to the default databasevaluesfrom the equilibrium submodel.
3838
RUNKEL
24 rn
4.4
3.6
2.8
2.0
4.4-
3.6
2.8
ET AL.: REACTIVE
Dissolved
Total
Observed
[]
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....
SOLUTE
TRANSPORT
IN STREAMS
the copper lossduring the pH modificationis attributableto errorsin modelingpH. As noted by Broshearset al. [1996], a number
of factors
make
concurrent
simulation
of iron
and
aluminumprecipitationand pH a difficult task. Here we have made no attempt to improve the simulationsof Broshearset al.; simulationsof pH at 24 and 70 m underestimateobserved pH by 0.1-0.4 units. This underestimationgenerallyincreases during the secondstep of the pH modification.The correspondingcoppersimulationtherefore overestimatesdissolved copper concentrations,with greater errors occurring during the later stagesof the secondpH step (Figure 6). Another
feature
of the simulations
is the overestimation
of
the copperspike.The spikeoccursfollowingterminationof the pH modificationas copper desorbsfrom the HFO. Owing to the immobilenature of the streambed,desorptionof copper from the staticsurfacerepresentsthe greatestcontributionto the copperspike (much of the coppersorbedto the dynamic surfaceduring the pH modificationis advecteddownstream and is therefore unavailablewhen the spike is formed). To minimize overestimationof the copper spike, simulationparameterswere adjustedto favor the dynamicsurfaceover the static surface:the high-affinitysite density(Ns) for the dynamic surfaceis set at the upper value reported by Dzombak
andMorel[1990];surfacecomplexation constants (Kint)describingsorptionof copper to high-affinitysitesare set below the lower reportedvalue for the staticsurfaceand at the upper reportedvalue for the dynamicsurface(Table 1). These adk ' 1'0 1'2 ' 1'4 ' 1'6 ' 1'8 justments are warranted given that the dynamic surface is Time [hr] composedof freshlyprecipitated HFO that has greater sorpFigure 6. Total (diamonds),dissolved(squares),simulated tive potential than aged oxideson the streambed[Hrncir and total (dashed curve), and simulated dissolved(solid curve) McKnight, 1998]. Hydrologic factors also favor the dynamic copperconcentrationsat 24 and 70 m. surface as fresh precipitatesin the water column have close contactwith dissolvedcopper. An alternate method of favoring the dynamicsurfaceis to ond pH step.In accordancewith the sulfateexamplethe sim- increase the kinetic limitation for the static surface. This alterulation exhibitsa sharpdecreasein dissolvedcopperduringthe native was not pursuedbecauseof the effect of the kinetic period of rapidly changingpH. Simulatedvaluesgenerallyex- limitation on the pH modeling conductedby Broshearset al. ceed observedvaluesduring the latter part of the pH modifi- [1996]. Changesto the kinetic limitationwould thus affect not cation.As the pH modificationis discontinued,desorptionof only coppersorptionbut alsopH and the precipitationof iron copperproducesa spikethat is in excessof preinjectioncopper and aluminum. levels.This spikeis overestimatedby the simulationat both 24 The importanceof favoringthe dynamicsurfaceis shownby and 70 m. consideringan additionalsimulationin whichboth surfacesare modeled using the best estimates of Dzombak and Morel [1990]. As shownin Figure 7, the additionalsimulationpro4. Discussion ducesa dissolvedcopper decreasethat is comparableto that 4.1. Copper Simulations and ProcessIdentification producedwhen the dynamicsurfaceis favored.The difference Prior to obtainingthe simulationresultsshownin Figure 6, betweenthe two simulationsbecomesevidentwhen the pH a number of simulationswere conductedto verify the obser- modification is terminated; use of the best estimates increases vation that sorptionoccurredboth in the water columnand at the degreeto whichthe copperspikeis overestimated.Use of the stream-streambedinterface. An initial simulation using parametersthat favor sorptionto the dynamicsurfacesubstanfreshlyprecipitatedHFO as a dynamicsurfacegrosslyunder- tially reduce the overestimation. estimatedthe lossof dissolvedcoppertherebyindicatingthat Useof Ns andKintvaluesfor the dynamic surfacethatare sorptionto fresh precipitatesalone could not accountfor the equal to the upper values reported by Dzombak and Morel copper loss during the pH modification.Additional simula- [1990] (Table 1) is consistentwith recent studiesof metal tionsthat treated the streambedas a staticsurfaceand ignored sorption using naturally formed iron oxides. Websteret al. sorption to fresh precipitates improved the correspondence [1998], for example, concludethat iron-rich precipitatesin betweenobservedand simulatedcopperduringthe pH modi- mine drainage systemsare more effective in sorbingcopper ficationbut greatlyoverestimatedthe magnitudeof the copper and zinc than syntheticiron oxidesbecauseof the presenceof sulfate in natural HFOs. Oxides formed in the sulfate-rich spike. Although similarproblemsof over and under estimationare waters of St. Kevin Gulch should have similar propertiesas evidentin the simulationshownin Figure 6, considerationof suggested by Smithet al. [1998].Usingwaterscollectedin July both static and dynamicsurfacesprovidesthe best reproduc- 1989, Smith et al. [1998] found good agreementbetween obtion of observeddata. The simulation'sfailure to reproduce servedcoppersorptionand that predictedby GTLM usingthe 2.0
RUNKEL
ET AL.:
REACTIVE
SOLUTE
4.4
• 3.6
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2.0
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TRANSPORT
IN STREAMS
3839
dynamicsurfaces.Broshears et al. [1996]consideredsorptionto the streambedusingonly the staticsurface,whereasother data setsmay require exclusiveuse of a dynamicsurface.Furthermore, by includinga separatedatabasefor each surface,the model can be easilycustomizedto reflect specificgeochemical conditions.For coppersorptionthe HFO databaseswere modified to favor sorptionto fresh precipitates.Other situations may arisein which sorptionreactionsfor specificcomponents are omitted from a given database,for example, modeling metal sorptionin the water columnand pH-bufferingreactions at the stream-streambed interface.A final feature is the ability to considerhydrologicconditionsby placingkinetic limitations on the static surface.
The data presentedin this and other studiesclearlysupport the role of pH as a mastervariablethat influencesthe sorption of copperand other trace metals.Despite the uncertaintiesof Figure 7. Dissolvedcopperconcentrations at 24 m: observed modelingsorptionin naturalsystems the prospectsfor studying (squares);simulated,usingparametersthat favorthe dynamic pH-dependentsorptionin a transportsettingare promising. surface(dashedcurve);and simulated,usingbestestimatesof With minor modifications to the estimates of Dzombak and Dzombakand Morel [1990] (solid curve). Morel [1990] (Table 1) the reactivemodel reproducesthe general featuresof the experimentaldata. Especiallynoteworthyis the timing of the observedand simulatedcopperresponse:the databaseof DzombakandMorel [1990].Relativelypoor agree- effect of the first pH stepis minimal,whereasthe secondstep ment betweenobservedsorptionand model predictionswas causesa sharp drop in dissolvedcopper. The mechanistic, found usingwatersfrom October 1989,when the sulfatecon- pH-dependentmodelthusprovidesa valid first-approximation centrationsof unfiltered stream water were 25-60% higher of copper fate and transport.As noted here and elsewhere than those observedin July. Smith [1996] also found poor [e.g., Websteret al., 1998], additionalrefinementsto the HFO agreementbetweenmodel predictionsand copperconcentra- databaseare neededto further quantifytracemetal sorptionin tionsobservedat 24 m duringthe 1988 experiment. natural systems. As demonstratedabove,overestimationof the copperspike may be reduced by attributing more of the copper loss to processes that occurwithin the water column.Here we have 5. Conclusion taken the approachof increasingthe parametervaluesfor the We present a modeling framework for trace metals that dynamicsurfacesuch that more copper is sorbed to HFO. considersboth in-stream transport and pH-dependent sorpAlthough not includedherein, considerationof other water- tion. When coupledwith the ability to model precipitationof column processeswould produce further reductionsin the metal hydroxidesand pH [Broshears et al., 1996;Runkelet al., simulatedcopperspike.Potentialprocesses includesorption 1996b], the model representsa useful tool for studyinggeoonto hydrousaluminumoxides(Figure 4c) [Karthikeyan et al., chemicalprocesses in streamsaffectedby acid mine drainage 1997],coprecipitation of copperwith HFO [Karthikeyan et al., and acid rain. Analysisof a highly dynamicfield experiment 1997], and enhancedsorptiondue to organicacids[Ali and demonstratesthe need for suchmodelsthat considerthe pH Dzombak, 1996]. dependenceof sorption.To further developprocesssimulaDespite our successin reproducingthe general response tionsrelated to acidmine drainageremediation,more detailed characteristics of copper, additionalwork is needed to fully field measurements will be neededto definethe rolesof freshly characterizetracemetal sorptionin St. Kevin Gulch and other precipitatedmetal oxidesand agedstreambedcoatingsin trace acid mine drainagesystems.Simulationresultsfor dissolved metal sorption. copperat 251 and 498 m, two additionalsamplinglocations, greatlyoverestimate the dissolved copperlossobservedduring Acknowledgments. Fundingfor thiswork wasprovidedby the U.S. the pH modification.This overestimationis due to the high GeologicalSurvey'sToxic SubstancesHydrologyProgram. The ausolidconcentrations (S c for the staticsurface)usedby Bros- thorsappreciatethe helpful commentsprovidedby Robert Broshears, hearset al. [1996]to buffer the pH changeat thesedownstream Kathy Smith, Durelle Scott, JamesA. Davis, Jim Szecsody,and an locations.Another area of concernis the potentialcompetition anonymousreviewer. betweencopperandothermetalsfor sorptionsites.Additional simulationsthat considersorptionof zinc, cadmium,and calReferences cium result in a small reductionin coppersorptionat 24 and 70 m. These simulationsare not includedin the final analysis, Ali, M. A., and D. A. Dzombak, Effects of simple organic acids on sorption of Cu2+ andCa2+ ongoethite, Geochim. Cosmochim. Acta, however, as observed concentrationsof these metals were un1•2
Time [hr]
affectedby the pH modification. 4.2.
Model
Formulation
Theseapplicationsillustrateseveralkey aspectsof the reactive transportmodel. The ability to model static and/or dynamic surfacesprovidesa flexiblemodelingframework.Here we have modeled copper sorptionby consideringstatic and
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McKnight, D. M., K. E. Bencala,G. W. Zellweger,G. R. Aiken, G. L. Feder, and K. A. Thorn, Sorptionof dissolvedorganiccarbonby hydrousaluminumand iron oxidesoccurringat the confluenceof
(ReceivedMarch 19, 1999;revisedAugust16, 1999; acceptedAugust19, 1999.)
