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Parameter Estimation of the Transient Storage Modelfor Stream–Subsurface ExchangeAndrea Marion1; Mattia Zaramella2; and Aaron I. Packman3

Abstract: The transient storage model~TSM! is the most commonly used model for stream–subsurface exchange of solutes. Theprovides a convenient, simplified representation of hyporheic exchange, but its lack of a true physical basis causes its parametdifficult to predict. However, the simple formulation makes the model a useful practical tool for many applications. This work comthe TSM with a physically based pumping model. This comparison is advantageous for two reasons: Advective pumping is knowan important hyporheic exchange process in many streams, and the pumping model can be used to derive dimensionless transieparameters that are properly scaled with important physical stream parameters. Transient storage model parameters are shodependent on both the timescale of observation and the shape of the breakthrough curve, i.e., on the temporal evolution of thconcentration in the surface water. This indicates that the transient storage model can, in practice, lead to incorrect predictions whparameters are obtained without consideration of the stream flow dynamics, the properties of the stream bed, or the process timeswork emphasizes the limitations of simplified models for hyporheic transport, and indicates that such models need to be carefully

DOI: 10.1061/~ASCE!0733-9372~2003!129:5~456!

CE Database subject headings: Models; Subsurface environment; Solutes; Hydraulic transients; Streams.

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Introduction

Stream–subsurface exchange has received significant attentionrecent years, primarily for its role in controlling the fate of con-taminants and ecologically relevant substances. Streamsubsurface exchange can control the net transport of reactive sustances because this exchange exposes solutes and suspematter to subsurface biogeochemical processes. The attentgiven to the exchange of solutes between the surface water oriver and the hyporheic zones has led to the development of vaous types of mathematical models for exchange~Bencala andWalters 1983; Castro and Hornberger 1991; Elliott and Brook1997a; Runkel 1998; Wo¨rman et al. 1998; Wo¨rman et al. 2002!.These models have been applied extensively to solute transpstudies in many streams~Bencala 1984; Castro and Hornberger1991; Rutherford et al. 1993, 1995; Vallet et al. 1996; Mulhollandet al. 1997!. Concerns about the generality of existing models haled to several recent attempts to reconcile existing models~Pack-man and Bencala 2000; Lees et al. 2000!. This work will compare

1Assistant Professor, Dept. of Hydraulic, Maritime, Environmentaland Geotechnical Engineering, Univ. of Padua, via Loredon 20, 3510Padova, Italy~corresponding author!. E-mail: marion@idra.unipd.it

2PhD Candidate, Dept. of Hydraulic, Maritime, Environmental, andGeotechnical Engineering, Univ. of Padua, via Loredan 20, 35100 Pdova, Italy.

3Assistant Professor, Dept. of Civil and Environmental EngineeringNorthwestern Univ., 2145 Sheridan Rd., Evanston, IL, 60208-3109.

Note. Associate Editor: Teresa B. Culver. Discussion open until October 1, 2003. Separate discussions must be submitted for individual ppers. To extend the closing date by one month, a written request mustfiled with the ASCE Managing Editor. The manuscript for this paper wasubmitted for review and possible publication on December 4, 2001approved on June 3, 2002. This paper is part of theJournal of Environ-mental Engineering, Vol. 129, No. 5, May 1, 2003. ©ASCE, ISSN 0733-9372/2003/5-456–463/$18.00.

456 / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / MAY 200

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a simplified model with a fundamentally based model for advetive hyporheic exchange in order to further define the limitatioof the commonly used models.

The compared models are mostly used to analyze hyporhexchange: They are commonly known as the pumping mo~PM! and the transient storage model~TSM!. The TSM providesa way to synthesize the net mass transfer from a stream to its bIts parameters are a storage zone of constant depth identifiethe upper part of the sediments and a coefficient commoknown as ‘‘mass transfer coefficient.’’ The exchange is assumto be proportional to the difference of concentration betwewater and the storage zone through the mass transfer coefficThe TSM is usually formulated as follows@Bencala and Walters~1983!#.

dCb

dt5a

A

As~Cw2Cb! (1)

whereA5cross-sectional area of the stream;As5cross-sectionalarea of the storage zone;a5mass transfer coefficient;Cw5concentration of solute in the stream;Cb5concentration ofsolute in the storage zone; andt5time.

The exchange parameters are evaluated on the basis of cfitting the results of solute transport experiments conducted instream of interest. The TSM has been applied to a large numbestreams, principally to evaluate metals transport in mining areand the influence of hyporheic exchange on nutrient transp~Triska et al. 1989, 1990, 1993; Winter et al. 1998; Jones aMulholland 2000!. Transient storage parameters can be evaluadirectly from tracer measurements~Hart 1995; Wagner and Har-vey 1997; Packman and Bencala 2000!. Several other modelsexist that are similar to the TSM in that they represent hyporheexchange as mass transfer from the stream to an idealized subface reservoir using an empirical exchange coefficient~e.g.,O’Connor 1988!.

Recent work has probed the underlying processes which drhyporheic exchange. Hyporheic exchange is now known to

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driven primarily by advective processes. Advective exchanflows develop at several spatial scales due to separate menisms such as flow over bedforms, around obstacles, and armeanders~Thibodeaux and Boyle 1987; Savant et al. 1987; Hvey and Bencala 1993; Elliott and Brooks 1997a,b; Hutchinsand Webster 1998; Packman and Brooks 2001!. The bedform-induced PM developed by Elliott and Brooks calculates hyporhexchange within the stream bed based on a fundamental anaof hydrodynamic interactions between the stream and subsurThe case of uniform stream flow over an infinitely deep strebed covered by regular, two-dimensional bedforms, is apprmated by a sinusoidal pressure distribution acting on a flat bConsidering a coordinate system (x,y) parallel and perpendiculato the bed, respectively, the induced pore water flow has Davelocity components

u52kKhm cos~kx!exp~2ky!(2)

v52kKhm sin~kx!exp~2ky!

in the x andy directions, respectively. Exchange is driven by tsinusoidal variation of head at the bed surface with half-amplithm , the bedforms are characterized by their wavelengthl orwave numberk52p/l, and the sediments have hydraulic coductivity K. The half-amplitudehm is a function of the streamvelocity and bed form geometry~Felhman 1985!. The stream–subsurface exchange flux can be found by evaluating Eq.~2! atthe bed surface. Further, solute mixing in the subsurface cacalculated explicitly because Eq.~2! provides the entire subsurface velocity field. The PM has been successfully applied to seral different sets of experimental results~Elliott and Brooks1997b; Packman et al. 2000a,b; Marion et al. 2002!.

The most important feature of the PM is that physical quaties like the stream velocity, permeability of the sediments, achannel geometry are used to evaluate hyporheic exchangecause of this, it has predictive capability and can also be modito be applicable to a slightly different physical system or extento include chemical reactions in a detailed way. The effectslayered subsurface heterogeneity, an impermeable subsulayer, and active bed sediment transport have been includevarious pumping model formulations~Packman et al. 2000aPackman and Brooks 2001!. The PM has also been extendedpredict the transport of reactive solutes and colloidal partic~Eylers et al. 1995; Rutherford et al. 1995; Packman et2000a,b!. However, because it requires detailed knowledgestreambed conditions, the PM has been directly applied tonatural streams~Rutherford et al. 1993, 1995!. Worman et al.~2002! have recently presented a methodology that allows apcation of the PM to stream solute breakthrough data usingment methods, and demonstrated the utility of this approacassessing geomorphic controls on hyporheic exchange in ancultural stream.

In this work, the TSM will be compared to the fundamentabased PM in order to examine if their simplified formulations cadequately represent the pumping exchange process. Model slations will be used to examine the behavior of the idealizmodel parameters and to determine how the semiempiricalchange parameters can be estimated from stream properties

Exchange Models: Theory

Model Parameter Normalization

The PM is normally applied in a dimensionless form withparameters scaled appropriately with characteristic physica

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mensions and hydrogeologic properties of the stream system~El-liott and Brooks 1997a!. Exchange is normally represented withthe equivalent depth of penetration,M, which is the volume ofwater exchanged per unit bed surface area. Pumping exchacan then be given in terms of a dimensionless time and dimesionless exchange

t* 5k2Khm

ut

(3)

m* 52pk

uM

where t* 5dimensionless time;m* 5dimensionless penetratedmass; andu5porosity of the bed. The exchange is calculatedthe PM by using the pore water velocity profile given in Eq.~2! todetermine solute penetration into the subsurface. The solutiona semi-infinite bed was found by Elliott and Brooks~1997a,b!,while theory for a finite bed was developed later by Packmaet al. ~2000a,b!. Exchange is found by determining the residenctime function,R(t* ), which is the probability that a particle thatentered in the sediments at timet*50 is still in the bed at ageneral timet* . If the tracer concentrationC(t) in the watercolumn is known, then the dimensionless penetrated mass canevaluated as

m* ~ t* !52pq* E0

t*R~t* !C* ~ t* 2t* !dt* (4)

whereC* 5C/C05concentration normalized with the initial con-centration C0 , and q* 521/(2pKhm)*x50

x5l/2v(x,y50)dx5dimensionless average flux through the bed surface. The chain the in-stream concentrationC* (t* ) can be linked with thehyporheic exchange,m* (t* ), by a mass balance.

A dimensionless formulation of the TSM can be developeFor a wide channel, or one with constant width, Eq.~1! can bewritten as follows:

dCb

dt5

a8ud

~Cw2Cb! (5)

where, ifH5stream depth, the coefficient

a85auH (6)

is the exchange rate with units of velocity andd5depth of thestorage zone. By introducing the PM scaling, the dimensionleexchange rate and storage zone depth are found

a* 5auH

kKhm(7)

d* 5kd (8)

According to these expressions, Eq.~5! becomes

dCb* ~ t* !

dt*5

a*d*

@Cw* ~ t* !2Cb* ~ t* !# (9)

Eqs.~4! and~9! can be applied to any desired boundary and initiconditions. Three different stream concentration curves will bconsidered in the following analysis. These curves represent foing functions that drive hyporheic exchange when the in-streatransport is uncoupled from hyporheic transport. The three funtions have been selected for analysis here because they repreuseful practical cases and have analytical solutions. Analyticsolutions are needed in order to proceed with the numerical mo

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evaluation. It is worth noting that field cases may have a varieof boundary conditions, so the models should be solved for mocomplex boundaries on a case-by-case basis.

Transient Storage Model: Solution for Closed System,Exponential Decay, and Sinusoidal Concentration

The simple formulation of the TSM makes it easily solvable focomplex cases, so that it is particularly suitable for practical aplications. Due to the simplification inherent in the TSM, valuefor the storage zone and the mass transfer coefficient that wfor a particular case may not work for a different one, even for thsame stream and the same bed configuration. This is the reafor evaluating three different boundary conditions representatiof the different kinds of processes that occur in rivers.

Closed SystemsConsider laboratory tests where the volume of surface water~in-cluding the water column and sumps! and the volume of the waterstored in the porous bed are fixed, and water is recirculated. Tsystem is here termed ‘‘closed’’. It is here assumed that the prous bed is thick enough for the exchange process not to befected by the bed thickness. The initial conditionsCw5C0 andCb50 reflect the experimental conditions discussed in this worClosed systems require a relationship linking the transferred mto the concentration in the water. A mass balance providesrequired expression

dm~ t !

dt52d8

dCw* ~ t !

dt(10)

whered85effective stream depth defined as the total volumestream water, including off-channel components~sumps!, dividedby the exchange area, i.e., the area of the interface betweensurface flow and the bed. Eq.~10! is valid for both the TSM andPM.

Defining

d* 5kd8 (11)

5normalized stream depth, integration of Eq.~10! leads to thefollowing:

Cw* ~ t !512u

2p

m* ~ t !

d*(12)

The effective depth of penetration into the bed can also be writtas a function of the concentration in the storage zone

m~ t !5udCb* ~ t ! (13)

which can be normalized according to Eqs.~8! and ~3!, resultingin

m* ~ t !52pd* Cb* ~ t ! (14)

The dimensionless expression relating the concentrations inbed and stream, found by substituting Eq.~14! into Eq. ~12!, is

Cb* ~ t !5@12Cw* ~ t !#d*

ud*(15)

By combining Eqs.~15! and ~9!, the TSM solution for themass transfer from the stream to the bed is found

Cb* ~ t* !5

12expS 2S ud*d*

11D a*d*

t* Dud*d*

11

(16)

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m* ~ t* !52pd*12expS 2S ud*

d*11D a*

d*t* D

ud*d*

11

(17)

It can be shown from Eqs.~16! and ~17! that the solute con-centration in the pore water increases with an initial rate equal toa* , and the solute concentration everywhere eventually approaches an asymptotic value given by complete mixing throughthe entire transient storage zone

Cf* 51

ud*d*

11

(18)

The case of a sudden change of the solute concentration inriver followed by steady conditions~step change! is now exam-ined. The solution can be derived readily from Eq.~17!, as thelimit for d* →`

m* ~ t* !52pd* S 12expS 2a*d*

t* D D (19)

Sinusoidal VariationAnother important case is when the concentration in the watechanges periodically, which may occur due to diurnal or seasonavariations of the tracer concentration in the water. This case isalso applicable to variations produced by tides or other periodicevents such as industrial discharge cycles. To represent a periodvariation of the concentration in the water, the following expres-sion is introduced

Cw~ t* !5Cm sinS 2p

P*t* D (20)

where P* 5k2Khm /uP5normalized period of oscillation. Eq.~20! should be taken as a periodic deviation from some meanbackground concentration. The normalized depth of penetratiothat occurs due to this periodic forcing is

m* ~ t* !52pd*

S 2pd*P* a* D 2

11H sinS 2p

P*t* D

22pd*P* a* FcosS 2p

P*t* D2expS 2

d*a*

t* D G J (21)

Exponential DecayA fourth interesting case is the exponential decay of the concentration in the water. This is representative of an accidental discharge of a contaminant that degrades exponentially over time.may also represent a substance that is consumed due to a reactthat occurs only in the stream and not in the subsurface, such aphotodegradation. In this case, the concentration in the streamassumed to be

Cw~ t* !5C0 expS 2t*G* D (22)

whereG* 5k2Khm /uG5characteristic decay time. The analyticalsolution of the TSM can easily be found for this boundary con-dition

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The TSM parameters will be evaluated for each of these case

Basis for Model Comparison

Models can be evaluated or compared in several different waThe approach that will be taken here is to fit the TSM to the PMand then to evaluate whether the resulting idealized exchanparameters,a, andd are reasonable. If the idealized model workwell, then its parameters should behave in a manner that is reistic and consistent with model assumptions. The model-fittinapproach is appropriate because the PM is a theoretically badescription of a particular stream–subsurface exchange proceand it has also been validated using the results of laboratory flumexperiments. Thus, we seek to evaluate the ability of the TSto-sensibly reproduce the pumping exchange process. If the idalized model can adequately represent the pumping process, tthis would provide confidence that it has a sound physical basi.e., that its simplifications yield a good approximation of themore-complex natural phenomena that it attempts to representthe TSM cannot adequately model the pumping process, then tmay indicate that it may merely provide a convenient mathemacal description for curve fitting without a firm physical basis.

The parameterD is defined as

D~a* ,d* ,t* !5mP* ~ t* !2mTSM* ~ t* ! (24)

then the mean-square error of the functionD is given by

S~a* ,d* ,T* !51

T* E0

T*@D~a* ,d* ,t* !#2dt (25)

where T* 5time interval over whichS is evaluated and thus isequivalent to a ‘‘fitting timescale’’. The nondimensional relativemean-square error

S* ~T* !5

1

T* E0

T*D2dt*

1

T* E0

T*@mTSM* ~ t* !#2dt*

(26)

is similarly used to evaluate the performance of the model fodifferent dimensionless timescalesT* .

The model comparison is made by finding the minimum valuof Sas a functiona* andd* . The best-fit values for the exchangeparameters correspond to the points where the gradient ofS is 0,and where the Jacobean is strictly positive

grad~S!

5S ]S

]a*,

]S

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T* E0

T* ]

]a*@D~ t* !#2dt* ,

1

T* E0

T* ]

]d*@D~ t* !#2dt* D

5~0,0! (27)

J~S!5U ]2S

]a* 2

]2S

]a* ]d*

]2S

]d* ]a*]2S

]d* 2

U.0 (28)

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Results will be presented in the next section as the best-values of the exchange parameters for various time scales of coparison.

Results and Discussion

Model Comparison and Parameter Estimation

Fig. 1 presents the best-fit values for TSM parameters for the caof a constant tracer concentration in the stream. ForT* →0, thenormalized depth of the contaminated layer tends to infinitywhile the normalized exchange rate assumes a finite value. Thcan be explained by a direct analytical comparison of the modelThe PM initially has a linear increase of the exchanged mass. Thbehavior can only be modeled with the TSM using an infinitedepth of the storage zone. Near the origin, the residence timfunction has an approximate value of 1~Elliott and Brooks1997a!. By substitutingR(t* )51 in Eq. ~5! and rememberingthat C* (t* )51, then

mP* ~ t* !>2pq* ~ t* ! (29)

as an approximation for the pumping exchange at short timeThis is equivalent to assuming that the normalized TSM exchangrate is equal to the normalized flux through the bed surface,a*5q* . In this case,a* ~0!51/p, as shown in Fig. 1.

As the fitting timescale increases, the pumping model loses ilinear behavior, and a finite best-fit value of the storage zon

Fig. 1. Step change of the stream concentration. Best-fitting valueof ~a! the dimensionless exchange coefficienta* and ~b! the dimen-sionless depth of the exchange layerd* . On the right-hand side ver-tical axis the dimensionless mean square deviationS* is reported.

RNAL OF ENVIRONMENTAL ENGINEERING © ASCE / MAY 2003 / 459

Fig. 2. Step change of the stream concentration. Comparison of thtransient storage model and the pumping model for~a! small dimen-sionless timescales and~b! large dimensionless timescales.

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with constant values ofa* andd* , and can only be reproduced inthe TSM if d* is allowed to increase over time anda* is allowedto decrease.

Optimization over the two different timescales also yields verydifferent values of the TSM parameters even though the streaconditions are the same. The optimal valuesa*50.32 andd*52.05 were found forT* 510, while a*50.08 andd*54.97were found forT* 5500. This result has serious implications forthe use of the TSM in conjunction with tracer experiments tocharacterize hyporheic exchange in streams.

The same optimization procedure has been used to compathe TSM and the PM for the cases where the tracer concentratioin the stream varies sinusoidally or decays exponentially. Resulare shown in Fig. 3 for the sinusoidal case and in Fig. 4 for theexponential case. The different curves in Fig. 3 correspond tdifferent oscillation periods. All of the curves have similar valuesfor short timescales because the PM behaves similarly for diffeentP* in the initial linear exchange. The periodic variation of thetracer concentration in the stream water causes the TSM parameters to approach a steady state at long times. The concentratin the stream and the exchanged mass oscillates with the samfrequency but with different phases. It is important to note thadifferent oscillation periods yield different asymptotic values ofthe TSM parameters even though the magnitude of the concentrtion variation and all physical stream parameters are the samThis again shows that the best-fit TSM parameters obtained frosolute measurements are dependent on the nature of the impossolute breakthrough curve in the stream.

Fig. 5 shows the TSM behavior for an oscillation periodP*510 and a timescaleT* 550. The TSM provides good resultsin this case because the relatively short oscillation period resul

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Fig. 3. Sinusoidal change of the stream concentration. Best-fittingvalues of~a! the dimensionless exchange coefficienta* and ~b! thedimensionless depth of the exchange layerd* .

depth can be derived through optimization. If the optimization ismade for long timescales, the TSM storage zone depth increasaccording to the net mass exchange, resulting in a logarithmgrowth of d* (T* ). This reflects the fact that pumping exchangeis characterized by mixing to a greater depth over time. Howeveas d* increases, the exchange ratea* decreases over time. TheTSM represents pumping exchange with various degrees of aproximation, as indicated by the value of the normalized meansquare deviationS* .

The simulations in Fig. 2 show the application of the transienstorage model for two values ofT* . For a timescaleT* 510 @Fig.2~a!#, which falls within the region of minimumd* in Fig. 1, theTSM provides a good approximation of the PM. The TSM doenot perform nearly as well for a comparison timescaleT* 5500@Fig. 2~b!#. The TSM still provides a fair representation of the PMcurve, but it underpredicts the initial exchange, asymptotic behaior and intermediate exchange. This behavior results becauseTSM does not provide a good approximation for the pumpingprocess over large timescales. The idealization of a defined trasient storage zone characterized by a given area or depth is nrepresentative of hyporheic exchange in an unconstrained alluvstream. The TSM exponentially approaches a constant tracer cocentration in the storage zone, while advective pumping continueto mix tracer deeper into the bed. Pumping exchange slows cosiderably at long times due to the fact that pore water velocitiedecrease exponentially with depth, but it never actually stopFigs. 2~a and b! indicate that this process cannot be represente

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Fig. 4. Exponential decay of the stream concentration. Best-fittinvalues of~a! the dimensionless exchange coefficienta* and ~b! thedimensionless depth of the exchange layerd* .

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long decay time of the tracer concentration in the water column~corresponding to a persistent contaminant! causes the tracer topenetrate deeply into the bed, and this deep penetration is not wemodeled by the TSM. This is analogous to the behavior of thesteady concentration case at long comparison times.

Comparison with Experimental Results

Experimental data obtained by Marion et al.~2002! will be usedto demonstrate a few key principles. These experiments involvethe observation of stream–subsurface exchange in a relativelarge flume with a coarse sandbed. Four different bed geometriewere used, including both naturally formed and artificially shapedbedforms. The fitting procedure has been applied to analyze thoptimal values of the TSM parametersa* andd* to reproduce theexperimental data. These curves ofa* and d* versus the fittingtimescaleT* are shown in Fig. 7. The results obtained from theexperimental data follow the form of the theoretical curves, especially for the natural bedforms. This example confirms that thebest-fit TSM parameters are dependent on the timescale of thexperiment. This approach shows also that the normalization presented in Eqs.~3!, ~7!, and~8! is capable of simulating bedform-driven hyporheic fluxes. This normalization is a useful way tocompare experimental results and model predictions for differenstream conditions.

Conclusions

The TSM is a very useful tool for the practical analysis ofstream–subsurface exchange. However, it does not fully represethe physics of the pumping process, wherein advective flows arproduced by the pressure gradients at the bed surface associawith the presence of bedforms. The results shown here indicatthat the two calibration parameters employed by TSM are noconstant for a given stream, but instead change depending on thmethod used to obtain them. Comparison of the TSM with thefundamentally based PM shows that the TSM parameters changwith the timescale of the theoretical evaluation and, correspondingly, vary with the duration of experiments used to probe thehyporheic exchange process. The quality of the TSM fit to thepumping model also depends on the timescale of comparisonFurthermore, the model evaluation has been carried out for thredifferent concentration curves, representing common contamination cases. TSM parameters are shown to also be dependent

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Fig. 6. Exponential decay of the stream concentration. Comparisonof the transient storage model and the pumping model.

in only a superficial contamination of the bed. Longer oscillationperiods cause deeper bed layers to become mixed, resultingpoor model performance similar to that shown in Fig. 2 for a highcomparison timescale.

The results for the exponential decay case are shown in Fig.Asymptotic constant values ofa* and d* are also found in thiscase, and these depend on the magnitude of the decay timescG* . In all cases, the optimal TSM parameters follow the curve fothe steady concentration case until the in-stream degradaticauses them to deviate and approach an asymptotic constvalue. Substances that degrade more quickly have a higher appent exchange ratea* and a lowerd* . Fig. 6 shows the comparisonbetween the models forG*5100 andT* 5600. The relatively

Fig. 5. Sinusoidal change of the stream concentration. Comparisoof the transient storage model and the pumping model.

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Fig. 7. Verification of the parameter estimation for the step changcase using the experimental data by Marion et al.~2002!

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d* 5 dimensionless water depth above sediments, equal tokd8;

H 5 stream depth;hm 5 half-amplitude of the sinusoidal distribution of

pressure on the bed surface;K 5 hydraulic conductivity;k 5 bedform wavenumber, equal to 2p/l;

M 5 net exchanged mass per unit bed surface area;m 5 equivalent penetration depth, equal toM /C0 ;

m* 5 dimensionless equivalent penetration depth, equal to2pkm/u;

R 5 flux weighted, spatially averaged residence time func-tion;

S 5 mean square deviation;T* 5 fitting timescale;

t 5 time;t* 5 dimensionless time, equal tok2Khmt/u;u 5 Darcy horizontal velocity;

u* 5 dimensionless Darcy horizontal velocity, equal tou/(Kkhm);

v 5 Darcy vertical velocity;v* 5 dimensionless Darcy vertical velocity, equal to

v/(Kkhm);x 5 horizontal coordinate;

x* 5 dimensionless longitudinal coordinate, equal tokx;y 5 vertical coordinate;

y* 5 dimensionless vertical coordinate, equal toky;a 5 mass transfer coefficient;

a8 5 exchange rate, equal toauH;a* 5 dimensionless exchange rate, equal toauH/(kKhm);D 5 disagreement between pumping model and transient

storage model;d 5 storage zone depth;

d* 5 dimensionless storage zone depth, equal tokd; andu 5 porosity of bed sediment.

References

Bencala, K. E.~1984!. ‘‘Interactions of solutes and streambed sediment.II. A dynamic analysis of coupled hydrologic and chemical processesthat determine solute transport.’’Water Resour. Res.,20~12!, 1804–1814.

Bencala, K. E., and Walters, R. A.~1983!. ‘‘Simulation of solute transportin a mountain pool-and-riffle stream: A transient storage model.’’Water Resour. Res.,19, 718–724.

Castro, N. M., and Hornberger, G. M.~1991!. ‘‘Surface–subsurface in-teractions in an alluviated mountain stream channel.’’Water Resour.Res.,27~7!, 1613–1621.

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Acknowledgments

The work was partly funded by the University of Padova~Ex 60%funds!, and the Italian National Research Council~CNR!-National Group on Chemical-Industrial and Environmental Hazards. The involvement of one of the writers~A.P.! was supportedby the U.S. National Science Foundation under Grant No. BES0196368 and a supplement from the Office of International Programs.

Notation

The following symbols are used in this paper:A 5 cross-sectional area of the stream;

As 5 cross-sectional area of the storage zone;Cb 5 concentration of tracer in the storage zone;Cw 5 concentration of tracer in the stream;C0 5 initial tracer concentration in the stream, equal to

Cw /C0 ;C* 5 dimensionless tracer concentration;Cf* 5 dimensionless final dilution concentration;d8 5 effective stream depth;

3

d

din

to a

en-e-

h-

e

.-

san

men

r-ter

iu,

-

pro-

rogen

ndas.’’

ten-

eam

sing

nal

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