applicability of the transient storage model to the hyporheic exchange of metals

Journal of Contaminant Hydrology 84 (2006) 21–35

www.elsevier.com/locate/jconhyd

Applicability of the Transient Storage Model to the

hyporheic exchange of metals

Mattia Zaramella a, Andrea Marion a,*, Aaron I. Packman b

a Department of Hydraulic, Maritime, Environmental and Geotechnical Engineering, University of Padua,

via Loredan 20, 35100 Padova, Italyb Department of Civil Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208-3109, USA

Received 3 December 2003; received in revised form 30 November 2005; accepted 2 December 2005

Available online 20 January 2006

Abstract

Stream–subsurface exchange results from a complex ensemble of transport mechanisms that require

different modeling approaches. Field and laboratory experiments show that advective exchange through the

underlying sediments is an important mechanism of solutes transport and storage in riverine systems. Here,

Transient Storage Model parameters are obtained for reactive solute exchange driven by bedform-induced

advection. Consideration of exchange induced by this single mechanism allows specific relationships

between model parameters and system properties like solute reactivity to be identified. This work shows

that when a simplified model like the Transient Storage Model is applied to analyze metal storage in river

sediments, particular attention must be devoted to the choice of modeling parameters.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Hyporheic exchange; Transport model; Metals; Transient storage; Sediment contamination

1. Introduction

Field observations have shown that natural river bed sediments represent a reservoir for

storage of heavy metal contaminants such as cadmium, lead, manganese, zinc and copper (Reece

et al., 1978; Smith et al., 1989. The presence of contaminants in sediments reflects a

combination of aqueous-phase transport of metals, metal sorption to sediments, and sediment

transport. Many researchers have observed and modeled the transport of adsorbing pollutants in

fluvial systems (Zand et al., 1976; Jackman et al., 1984; Bencala et al., 1984; O’Connor, 1988;

0169-7722/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jconhyd.2005.12.002

* Corresponding author. Tel.: +39 049 8275448; fax: +39 049 8275446.

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

M. Zaramella et al. / Journal of Contaminant Hydrology 84 (2006) 21–3522

Runkel et al., 1999; Harvey and Fuller, 1998; Worman et al., 1998). Several models have been

developed for analysis of solute transport in natural streams. The Transient Storage Model

(TSM; Bencala and Walters, 1983; Bencala, 1984) is the most commonly used model for

analysis of stream–subsurface exchange in stream tracer injection studies. Other studies have

used alternative approaches to model this exchange: Jackman et al. (1984) and Worman et al.

(1998) analyzed stream–subsurface exchange with a diffusive model and derived a solution for

transport of reacting solutes.

All of the models cited above require calibration of parameters to the results of solute

injection experiments on a case by case basis, so that results cannot easily be transferred to other

systems. Models like the TSM and the diffusive model are simplified mathematical

representations that do not fully represent the complex natural exchange phenomena. While

such simplified models are necessary to analyze net solute transport based on readily obtainable

data, the inherent limitations of these models require that attention be devoted to identifying the

conditions under which they can be reliably applied to analyze contaminant transport in rivers.

Several studies have focused on the mechanics of advective stream–subsurface exchange.

Advective exchange flow occurs because of flow interactions with a variety of stream features,

such as solid obstacles, meanders, bedforms, and pool-riffle steps along the river (Thibodeax and

Boyle, 1987; Savant et al., 1987; Harvey and Bencala, 1993; Elliott and Brooks, 1997a,b;

Hutchinson and Webster, 1998; Storey et al., 2003). Hart (1995) showed that the Transient

Storage Model represents exchange with the assumption of an exponential distribution of solute

residence times in the sediments. However, Worman et al. (2002) showed that bedform-induced

advective exchange is best represented by a log-normal residence time distribution.

Additional effort has been directed at improving estimation and interpretation of TSM

parameters. Harvey et al. (1996) evaluated the reliability of the TSM for characterization of

stream–subsurface water exchange and Wagner and Harvey (1997) outlined methods for the

design of solute injection experiments to characterize transient storage parameters. Marion et al.

(2003) and Zaramella et al. (2003) analyzed conservative solute exchange induced by bedforms

and showed how TSM parameters can be estimated from the stream properties in this case. All of

these analyses only consider conservative solute transport. Interactions with sediment grain

surfaces also play an important role in determining the transient storage of reactive solutes.

Here, we analyze the bedform-induced exchange of sorbing pollutants. While the exchange

flow induced by bedforms is only one of the processes that drive solutes into streambed

sediments, this is a useful case to demonstrate how and why idealized Transient Storage Model

parameters can represent reactive solute transport because a physically based analytical solution

exists for this case. Further, good-quality test data are available from experiments conducted in

laboratory flumes. This system provides a particularly useful test case for model evaluation

because experiments can be performed that uncouple longitudinal dispersion and subsurface

exchange. When the experiment design yields negligible longitudinal concentration gradients

along the flume, solute transport models can be greatly simplified, which allows a much easier

evaluation of the exchange parameters. This work will focus on how the TSM performs when

applied to analyze the results of laboratory flume experiments involving the exchange of sorbing

solutes between a stream and a sand bed.

2. Reactive transport in pore water

The reversible, equilibrium adsorption of solutes to sediment surfaces is often represented

in transport models by a retardation coefficient R. This coefficient represents the inverse

M. Zaramella et al. / Journal of Contaminant Hydrology 84 (2006) 21–35 23

fraction of tracer per unit volume remaining in solution in the pore water. R is given by the

expression:

R ¼ 1þ qhkP; ð1Þ

where h is the bed porosity, q is the density of the sediments [M/L3], and kP is the partition

coefficient [L3/M]. The partition coefficient represents the ratio between the solute sorption per

unit sediment mass and the solute concentration in solution. Thus, if CS is the mass

concentration of solute in the pore water, then kPCS is the mass of tracer adsorbed per unit

mass of sediments:

CADS ¼ kPCS: ð2Þ

Equilibrium sorption will be assumed in this work. In the experiments that are used as a test

case here, the adsorption reaction was fast enough to justify use of this assumption (Eylers,

1994).

Under the assumption of reversible equilibrium adsorption, the advective transport in a

relatively thin interfacial zone such as the hyporheic zone, where dispersion along and between

the interstitial flow paths is negligible, is governed by:

RBCS

Btþ us

BCS

Bs¼ 0: ð3Þ

where us is the instantaneous velocity along a pathline denoted by coordinate s. If dispersion

phenomena are negligible and solute–sediment adsorption is both reversible and instantaneous, it

can be deduced from Eq. (3) that reactive and passive interstitial transport can be represented in

the same model, with the difference that time is scaled with the retardation coefficient. However,

the transport of both reactive and passive solutes is identical in the surface water flow above the

sediment bed. As a result, sorbing solutes can be expected to show some peculiar transport

behavior in the coupled stream–subsurface system.

3. Stream–subsurface exchange model for sorbing solutes

The transport of solutes is governed by the one-dimensional advection–dispersion transport

equation with an additional flux term for boundary exchange, UB:

BCW

Btþ B QCW=Að Þ

Bx¼ 1

A

B

BxADL

BCW

Bx

�� P

AUB;

�ð4Þ

where: CW, is the concentration in the main channel [ML�3]; Q, is the volumetric flow rate in

the main channel [L3T�1]; A, is the cross section area of the main channel [L2]; x, is the

longitudinal coordinate [L]; DL, is the longitudinal dispersion coefficient [L2T�1]; P, is wetted

perimeter of the channel [L] ; UB, is the solute flux through the wetted perimeter [ML�2T�1].

The term UBP/A represents the exchange flux with storage zones (bed sediments, dead zones,

side pockets of non-moving water, leakage to or recharge from the aquifer) through the wetted

perimeter of themain section of the channel. This term has beenmodeled in different ways (Hays et

al., 1966; Bencala and Walters, 1983; Jackman et al., 1984; Wagner and Gorelick, 1986; Runkel

and Chapra, 1993; Runkel et al., 1996; Worman et al., 1998; Worman et al., 2002) and has been

found to be significant in modeling solute transport in rivers. Its formulation depends on the


different kinds of solutes or particles transported in the stream water and on the assumptions that

are made regarding their behavior both in the stream and in the underlying porous media.

Two different models are used here to analyze metal and colloid transport into the bed

sediments: an Advective Pumping Model (APM) and the Transient Storage Model (TSM). The

most salient difference between these two models lies in their approach to represent the

exchange process: the APM is based on the physics of a specific form of stream–subsurface flow

coupling, while the TSM is an empirical mass-transfer model.

3.1. Advective Pumping Model

The APM has been developed to study the stream–subsurface induced exchange of solutes

caused by a stream flow over stationary bedforms. Experiments with passive solutes have shown

that an exchange flow is generated by the induced dynamic head variation over bedforms. The

resulting bedform-induced pumping exchange has been determined by analytically solving the

subsurface flow field for a semi-infinite bed (half-plane) (Elliott and Brooks, 1997a,b) or a finite

bed (infinite strip) (Packman et al., 2000a). This model supplies a non-dimensional expression for

the characteristic exchange time and for the penetrated mass of solute into the bed, both obtained

from the basic physical parameters that govern the exchange process:

t* ¼ k2Khm

ht; ð5Þ

m* ¼ 2pkh

m;

where: K, is the hydraulic conductivity of the bed sediments [L/T]; m, is the mass transfer

per unit bed surface area, normalized by the initial concentration in the stream water C0 [L];

k, is the bedform wavenumber, equal to 2p/k, where k is the bedform wavelength [1/L];hm,

is half-amplitude of the sinusoidal distribution of pressure on the bed surface [L].

The non-dimensional accumulation of solute mass in the bed is given by the convolution

integral of the residence time function R(t) and the normalized concentration CW* =CW/C0.

m* t*ð Þ ¼ 2pq*Z t*

0

RR s*ð ÞCW* t*� s*ð Þds*: ð6Þ

In Eq. (9), q*= qh/kKhm, where q is the average exchange flux per bed surface area. The

residence time function R(t) is defined as the probability that solute that entered in the bed at the

time t =0 is still in the bed at a later time, t. The functions R(t) and q* can be found directly from

the analytical solution for the pore water flow field (Elliott and Brooks, 1997a,b). APM theory

has been extended to the case of adsorbing solutes using both the retardation coefficient

approach (Eylers, 1994) and more sophisticated reactive-transport modeling (Ren and Packman,

2004). The simpler residence time approach is adopted here as it sufficiently describes the

experimental data.

3.2. The Transient Storage Model

In the standard TSM formulation for non-reactive solutes, the flux term P/AUB includes

lateral inflow and transient storage, and is modeled as:

P

AUB ¼ � qL

ACL � CWð Þ � a CS � CWð Þ; ð7Þ


dCS

dt¼ � a

A

AS

CS � CWð Þ ð8Þ

where: A, is the cross sectional area of the storage zone [L2]; qL, is the lateral volumetric flow

rate per unit lenght [L2/T]; a, is the stream storage exchange coefficient [1/T]; CS, is the solute

concentration in the storage zone [M/L3].

When applying the TSM to simplified systems where there is no lateral inflow and the

concentration and velocity gradients in Eq. (4) are negligible, there is a simple mass balance

between the water column and the transient storage zone:

dCW

dt¼ � P

AUB: ð9Þ

This equation applies for recirculating laboratory flumes when the experiment design insures

that in-stream concentration gradients are negligible, in which case the flux UB can be directly

evaluated from the rate of change of the in-stream solute concentration. Because the recirculating

flume is a closed system for non-volatile solutes, an additional mass balance equation can be

used to link the accumulated mass transfer to the sediments and the concentration in the

recirculating water:

dCW

dt¼ � 1

dV

dM tð Þdt

; ð10Þ

where dV is the effective stream depth defined as the ratio between the volume of recirculating

water and the bed surface area, V/AB. Defining d*=kdV as the normalized stream depth and

integrating Eq. (10) with the initial condition CW=C0 and CS=0 leads to the following:

CW* tð Þ ¼ 1� h

2pm* tð Þd*

; ð11Þ

where CW* (t)=CW(t)/C0 is the concentration in the water column normalized with the initial

concentration C0. This equation allows the dimensionless accumulated mass exchange, m*(t)

to be found directly from observations of the in-stream solute concentration, CW* (t).

The TSM can be used to represent the bedform-induced exchange of conservative solutes

with the following parameters (Marion et al., 2003):

a* ¼ ahdVkKhm

; ð12Þ

d* ¼ kd;

where dV is the effective water depth in the main channel and d is the depth of the transient

storage zone (i.e., d =AS/b, where b is the width of the stream).

This model can be applied to analyze the exchange of reactive solutes by using the residence

time approach. The quantity RCS is the total solute mass per unit volume of sediment bed, and

the rate of net solute accumulation in the storage zone corresponds to the temporal derivative of

RCS. Thus the TSM exchange term for solutes subject to reversible equilibrium sorption is:

d RCSð Þdt

¼ aA

AS

CW � CSð Þ: ð13Þ

This has exactly the form of the standard TSM exchange term, given in Eq. (8), with the time

scaled by the retardation coefficient. The proposed model formulation represented by Eqs. (12)


and (13) is a simplified expression of the TSM for the bedform-induced advective exchange of

sorbing solutes between a stream and a porous streambed.

4. Transient storage parameter analysis

The TSM is based on the hypothesis that the exchange between the main channel and the

storage zones is governed by first-order mass transfer. On the other hand the APM is based

on the assumption that stream–subsurface exchange is an advective process driven by head

variations over the bed surface, and it has been successfully applied to analyze the results of

laboratory experiments conducted with different kind of tracers (Eylers, 1994; Elliott and

Brooks, 1997b; Packman et al., 2000b; Marion et al, 2002) and to field experiments

(Worman et al., 2002). Because analytical solutions exist for both models, Marion et al.

(2003) showed that advective mass transfer of conservative solutes could be represented in

the TSM framework. An equivalent dimensionless solution for advective mass transfer of

sorbing solutes can be found by integration of Eq. (13), yielding:

m* t*ð Þ ¼ 2pd*R1� exp � hRd*

d*þ 1

�a*Rd*

t*

� ��

hRd*d*

þ 1

: ð14Þ

Eq. (14) is an analytical form of the TSM applicable to laboratory flume data on the bedform-

induced exchange of sorbing solutes.

TSM parameters for the exchange of dissolved metals with a sand bed will be evaluated both

by comparison with the APM and by calibration to laboratory flume data. Because the APM

model is based on the mechanics of flow coupling across the stream–subsurface interface and

has been verified by extensive experimentation, it is taken as a reference model for evaluating

the TSM’s ability to represent the bedform-induced exchange. The experimental evaluation will

use the data of Eylers, who observed the exchange of several different metals with a sand bed in

a laboratory flume (Eylers, 1994; Eylers et al., 1995). Eylers (1994) also showed that the APM

could be used to predict the observed metal exchange.

4.1. TSM–APM comparison

The TSMwas fit to the APM, in the form of Eqs. (14) and (6), respectively, in order to analyze

the ability of the first-order mass transfer model to reliably predict bedform-induced advective

exchange. The models were calibrated using a root-mean-square minimization technique,

following the methodology we used previously for passive solutes (Marion et al., 2003; Zara-

mella et al., 2003). Exchange was evaluated over a range of timescales, T. For each dimen-

sionless fitting timescale T*=2pkT/h, best-fit values of a* and d* were obtained numerically by

solving for the minimum mean square deviation between the two models. This fitting timescale

simply represents the elapsed time over which the model agreement is evaluated.

Fig. 1 (continuous line) presents the best-fit values of the non-dimensional exchange rate a*and the normalized bed depth d* over a range of normalized fitting timescale T*/R for the case

of a steady concentration in the water column CW* , that can be directly obtained by assuming an

infinite effective water depth d*Yl in Eq. (10) and consequently in Eq. (14). The pair of

values a*(T*/R), d*(T*/R) represents the non-dimensional set of transient storage parameters

that make the TSM best fit the Pumping Model at T*/R. The dimensionless time T*/R used here


includes the retardation coefficient, so that just one fitting curve is found even though the

sorption coefficient, kP, varies. There is evidence of strong variability in a* and d* over the time

range for which the fitting computation is made. These parameters depend on the fitting

timescale in the same way found for passive solutes by Marion et al. (2003), but this is true only

for the simple boundary condition of steady concentration in the stream water, CW* =1 for all T*.

In most practical cases, CW* depends on the accumulated mass transfer to the bed (Eq. (11)),

and results obtained for passive solutes cannot be generalized to sorbing solutes. This behavior is

illustrated by simulations of the exchange of sorbing solutes with three different retardation

coefficients, R =1 (passive solutes), R =5, and R =10, with a representative dimensionless stream

depth d*=3. Results of the fitting of the TSM to the APM for these cases are presented in Fig. 1

and compared to the case of steady concentration in the water column. The effective water depth

Fig. 1. Variation of the best-fit non-dimensional exchange rate a* and normalized bed depth d* over a range of fitting

timescales T*/R for the theoretical case of steady solute concentration in the water column and for the case of closed

system with an effective stream depth d*=3 and three different retardation coefficients. Note that R =1 indicates a non

reactive (conservative) solute and that the case of steady concentration in the water column is equivalent to the case of a

closed system with effective stream depth d*Yl.

-


affects the choice of the exchange rate, especially for highly reactive solutes. Higher temporal

variability in the exchange parameters is expected for lower values of d* (results not shown).

The exchange rate, a*, is higher than in the case of steady concentration, and its value

increases with R. This behavior occurs because of the extensive removal of solute mass from the

water column by the process of stream–subsurface exchange and sorption to the streambed

sediments. Greater affinity of the solute for the bed sediment (larger kP and R) yields a greater

rate of removal and thus a greater overall exchange rate. Thus the Transient Storage Model

parameters are not completely scalable with the retardation coefficient. While a single curves for

a*(T*/R) and d*(T*/R) were found for the case of CW* =1 (Fig. 1, continuous line), the curves

for a*(T*/R) and d*(T*/R) can be seen to progressively deviate for increasing T*/R. This

complexity makes it difficult to estimate a and AS for reactive solutes based on the results of

tracer injections performed with conservative tracers. In practice, it is convenient to conduct a

field injection only with a conservative solute, such as sodium bromide, and to analyze sorption

behavior by conducting batch experiments with the bed sediments and reactive solute(s) of

interest. In this case, the effects of sorption cannot properly be represented with a retardation

coefficient (as in Eq. (13)), and the error introduced by this approximation increases with time.

4.2. Analysis of experimental data

The experimental results of Eylers (1994) provide good-quality, well-defined test cases for

the exchange of metals with a sand bed. All of Eylers’ experiments were conducted by injecting

one or more dissolved metals in a fixed volume of stream water recirculating over an initially

uncontaminated bed and then measuring the decrease in the metals’ concentration in the stream

water over time. These experiments meet the assumptions used in deriving the theory presented

in Section 3.

Experiments were conducted with two kinds of sand, Ottawa 30 (O30) with a geometric

mean diameter of 500 Am, and Nevada 70 (N70) with a geometric mean diameter of 195

Am. Five different bivalent metals were used: zinc, copper, calcium, and magnesium. These

metals each had a different affinity for the bed sediments, and increasing sorption was

Table 1

Conditions in Eylers (1994) experiments on the exchange of metals with stationary sand beds

Run

number

Metal Retardation

coefficient, R

pH Sand Wavelength,

k(cm)

Permeability,

K(cm/min)

Effective water

depth, d*

1 Zn2+ 15 7.1 O30 19.7 9.0 4.14

2 Zn2+ 12 7.0 O30 19.7 9.0 4.14

5 Ca2+ 5 7.0 O30 21.9 9.0 4.62

6 Zn2+ 12 7.3 O30 20.6 9.0 5.34

8 Zn2+ 12 7.3 O30 24.0 9.0 5.05

10 Zn2+ 3 6.1 O30 18.6 9.0 4.34

Ca2+ 2

11 Zn2+ 20 7.3 N70 27.3 2.4 3.65

12 Zn2+ 12 7.3 O30 21.0 6.6 3.85

Mg2+ 3

Ca2+ 2

14 Zn2+ 7 7.3 O30 19.0 6.6 4.25

Mg2+ 3

Cu2+ 15


observed in the order Ca2+bMg2+bZn2+bCu2+. Experiments were conducted in buffered

(bicarbonate) background solutions at different pH, yielding a wide range of retardation

coefficients between 2 and 20. An additional favorable feature of Eylers’ work was that his

experiments were conducted with a regular series of bedforms, and injections of a

conservative solute (lithium chloride) showed that the APM adequately predicted the basic

hydrodynamic exchange in all experiments. Experimental conditions for these experiments

are presented in Table 1.

Fig. 2 shows the exchange parameters obtained by fitting the model to the experimental data

on the exchange of dissolved metals. These results are presented in dimensional form, a(T),d(T), and in the dimensionless space used previously for the model comparisons, a*(T*/R),

Fig. 2. Values of (A) the exchange rate, a, and (B) the bed depth, d, obtained by fitting the TSM to the experimental data

of Eylers (1994). Plotting normalized values of (C) the exchange rate, a*, and (D) the bed depth, d*, against thedimensionless timescale, T*/R, collapses the experimental curves.


d*(T*/R). Each point in Fig. 2 corresponds to an individual measurement of metal concentration,

CW(t). The measurement time is used as the fitting timescale, T. Different fitting timescales yield

different values for the TSM exchange parameters. Normalization collapses the experimental

data, but there is still considerable variability in the dimensionless parameters between

experimental runs. This behavior follows results obtained in previous work on the exchange of

conservative solutes (Marion et al., 2003).

Results for metals with different sorption behavior are shown in Figs. 3 and 4. Individual

data based on measured metals concentrations are compared with theoretical curves obtained

from fitting the TSM to the APM. Fig. 3 presents results for weak adsorption behavior, R

between 2 and 7, while Fig. 4 presents results for stronger adsorption, R between 12 and 20.

The experimental observations generally follow the behavior predicted by the APM. The early

exchange behavior, T*/RV10, shows greater a* and smaller d* than expected from the model.

This behavior can be attributed to the presence of topographical features distinctly smaller than

the mean dune size, which produce greater interfacial flux over short time scales (for an

illustration of these effects, see Fig. 2b of Elliott and Brooks, 1997b). At longer observation

times, the primary bedforms dominate the exchange process and the experimental data

Fig. 3. TSM parameter sets, a*(T*/R), d*(T*/R), obtained by fitting data from experiments with relatively low

retardation coefficients, compared with the theoretical curves obtained by fitting the TSM to the APM based on the model

input parameters given in Table 1.

Fig. 3 (continued).


collapse to the APM curves. The enhanced near-surface exchange induced by small

topographical features has important implications for the distribution of strongly sorbing

solutes in the bed. When metals sorb strongly to the bed sediments, nearly all of the metal

mass in the bed will reside on sediment grain surfaces. In this case, the higher exchange flux

produced by the secondary bedforms will result in a nearly complete retention of the

exchanged metals in a thin layer at the bed surface. This corresponds to a greater exchange

rate, a, and thinner storage zone, d, in the TSM.

5. Conclusions

The Transient Storage Model has been widely applied to analyze the exchange of reactive

solutes with streambeds, with transient storage parameters obtained by numerical fitting

procedures. Exchange is known to occur by a number of different mechanisms, many involving

advection induced by stream-boundary interactions over a range of spatial scales. Here, we

evaluated TSM parameters for bedform-induced advective exchange of metals using both

theoretical analysis and comparison to experimental data sets. The results both provide an

improved framework for relating reactive transient storage parameters to basic system

characteristics and also show some important limitations of the TSM.

Fig. 4. TSM parameter sets, a*(T*/R), d*(T*/R), obtained by fitting data from experiments with relatively high

retardation coefficients, compared with the theoretical curves obtained by fitting the TSM to the APM based on the model

input parameters given in Table 1.


The analysis presented here shows that TSM parameters obtained with from fitting solute

breakthrough curves over a particular period of observation cannot reproduce advective-

reactive solute exchange occurring over different timescales. Instead, best-fit transient storage


parameters depend on both the exchange timescale of interest and the degree of solute sorption

to the bed sediments. For example, different TSM parameters will be found for the rapid

passage of a solute pulse over an uncontaminated bed than for a prolonged event (like a

plateau of concentration) involving the same solute in the same system. This is true for both

conservative and reactive solutes, but solute sorption to streambed sediments produces

additional deviations from the TSM assumptions. Overall mass-balance between the stream

and subsurface causes the sorption model parameters (kP, R) to influence the net exchange

parameters (a, AS). Equilibrium sorption in pore water is known to produce a spatial

fractionation of solutes according to their reactivity, with less reactive solutes propagating

farther than more reactive ones. Because the TSM parameterizes solute storage in terms of an

exchange rate (a) linked to a well-mixed zone of specified size (AS), differences in the relative

degree of penetration of various reactive solutes necessarily requires that the transport of

solutes of different reactivity be represented with different TSM parameters. Further, because

advective exchange generally occurs over a range of spatial scales, the exchange of solutes of

different reactivity is also expected to be controlled by different streambed topographical

features. Ultimately, solutes with a very high degree of affinity for the bed sediments are

expected to only exchange with a thin layer of the bed surface. These solutes are expected to

be exchanged with a higher overall rate because flow interactions with small-scale streambed

features produces an enhanced local exchange.

The net implication of all of these complexities is that it is very difficult to use TSM

parameters obtained with a conservative solute to model the behavior of a reactive solute. The

TSM is a simple model designed to quantitatively represent observed non-advective-dispersive

solute transport behavior, i.e., tailing of solute breakthrough curves. While the TSM has been

shown to be a robust model for this purpose, it does not explicitly represent the subsurface and

cannot be used to assess reactive transport in pore waters. As a result, it is not generally suitable

for representing reactive solute exchange with streambeds over a wide range of time scales.

Wherever possible, reactive transport modeling should be based on experiments conducted with

the solute of interest and with data collected over the time scales of interest. In addition, in situ

observations are normally required to assess local transport within the hyporheic zone (Harvey

and Wagner, 2000). The scaling framework presented here can also assist with generalizing best-

fit TSM parameters, but caution must be used because only one exchange process is formally

considered here and natural conditions can be expected to vary much more widely.

Notation

A Cross sectional area of the stream

AS Cross sectional area of the storage zone

CADS Adsorbed mass per unit mass of sediments

C0 Initial tracer concentration in the stream

CS Concentration of tracer in the storage zone

CW Concentration of tracer in the stream

CW* Dimensionless tracer concentration in the stream, equal to CW/C0

d Stream depth

d* Dimensionless water depth above sediments, equal to kdVdV Effective water depth above sediments

db Pumping model bed depth

db* Dimensionless pumping model bed depth, equal to kdbDS Groundwater dispersion coefficient


hm Half-amplitude of the sinusoidal distribution of pressure on the bed surface

k Bed form wavenumber, equal to 2p/kK Hydraulic conductivity

kP Partition coefficient

M Net exchanged mass per unit bed surface area

m* Dimensionless equivalent penetration depth, equal to 2pkm/hq* Non-dimensional averaged flux toward the sediments

R Retardation coefficient, equal to 1+q/hkPR Flux weighted, spatially averaged residence time function

S Mean square deviation

t Time

t* Dimensionless time, equal to k2Khmt/hT* Dimensionless fitting timescale

uS Mean groundwater flow velocity

a Mass transfer coefficient

a* Dimensionless exchange rate, equal to ahH/(kKhm)

d Storage zone depth

d* Dimensionless storage zone depth, equal to kdq Bulk density of the sediments

h Porosity of bed sediment

Acknowledgements

The participation of Mattia Zaramella and Andrea Marion was funded by the Italian Ministry

of the Environment, project on bAn Integrated Approach to the Remediation of Polluted River

SedimentsQ, and by the Italian National Research Council (CNR), National Group on Chemical-

Industrial and Environmental Hazards. Part of the material is based upon work supported by the

National Science Foundation under grant BES-0196368.

References

Bencala, K.E., 1984. Interactions of solutes and streambed sediment: 2. A dynamic analysis of coupled hydrologic and

chemical processes that determine solute transport. Water Resources Research 20 (12), 1804–1814.

Bencala, K.E., Walters, R.A., 1983. Simulation of solute transport in a mountain pool-and-riffle stream: a Transient

Storage Model. Water Resources Research 19, 718–724.

Bencala, K.E., Kennedy, V.C., Zellweger, G.W., Jackman, A.P., Avanzino, R.J., 1984. Interactions of solutes and

streambed sediment: 1. An experimental analysis of cation and anion transport in a mountain stream. Water Resources

Research 20 (12), 1797–1803.

Elliott, A.H., Brooks, N.H., 1997a. Transfer of nonsorbing solutes to a streambed with bed forms: theory. Water

Resources Research 33 (1), 123–136.

Elliott, A.H., Brooks, N.H., 1997b. Transfer of nonsorbing solutes to a streambed with bed forms: laboratory

experiments. Water Resources Research 33 (1), 137–151.

Eylers, H., 1994. Transport of adsorbing metal ions between stream water and sediment bed in a laboratory flume. Report

No. KH-R-56. California Institute of Technology, Pasadena, California.

Eylers, H., Brooks, N.H., Morgan, J.J., 1995. Transport of adsorbing metals from stream water to a stationary sand-bed in

a laboratory flume. Marine and Freshwater Research 46, 209–214.

Hart, D.R., 1995. Parameter estimation and stochastic interpretation of the Transient Storage Model for solute transport in

streams. Water Resources Research 31 (2), 323–328.

Harvey, J.W., Bencala, K.E., 1993. The effect of streambed topography on surface-subsurface water exchange in

mountain catchments. Water Resources Research 29 (1), 89–98.


Harvey, J.W., Fuller, C.C., 1998. Effect of enhanced manganese oxidation in the hyporheic zone on basin-scale

geochemical mass balance. Water Resources Research 34 (4), 623–636.

Harvey, J.W., Wagner, B.J., 2000. Quantifying hydrologic interactions between streams and their subsurface

hyporheic zones. In: Jones, J.B., Mulholland, P.J. (Eds.), Streams and Groundwaters. Academic Press, San Diego,

pp. 3–44.

Harvey, J.W., Wagner, B.J., Bencala, K.E., 1996. Evaluating the reliability of the stream tracer approach to characterize

stream–subsurface water exchange. Water Resources Research 32 (8), 2441–2451.

Hays, J.R., Krenkel, P.A., Schnelle, K.B., 1966. Mass transport mechanism in open channel flow. Sanitary and Water

Resour. Eng. Tech. Rep., vol. 8. Vanderbilt University, Nashville, Tennessee.

Hutchinson, P.A., Webster, I.T., 1998. Solute uptake in aquatic sediments due to current-obstacle interactions. Journal of

Environmental Engineering 124 (5), 419–426.

Jackman, A.P., Walters, R.A., Kennedy, V.C., 1984. Transport and concentration controls for chloride, strontium,

potassium and lead in Uvas Creek, a small cobble-bed stream in Santa Clara county, California. Journal of Hydrology

75, 111–141.

Marion, A., Bellinello, M., Guymer, I., Packman, A.I., 2002. Effect of bedform geometry on the penetration of passive

solutes into a stream bed. Water Resources Research 38 (10), 1209.

Marion, A., Zaramella, M., Packman, A.I., 2003. Parameter estimation of the Transient Storage Model for stream–

subsurface exchange. Journal of Environmental Engineering 129 (5).

O’Connor, D.J., 1988. Models of sorptive toxic substances in freshwater systems: III. streams and rivers. Journal of

Environmental Engineering 114 (3), 552–574.

Packman, A.I., Brooks, N.H., Morgan, J.J., 2000a. A physicochemical model for colloid exchange between a stream and

a sand streambed with bed forms. Water Resources Research 36 (8), 2351–2361.

Packman, A.I., Brooks, N.H., Morgan, J.J., 2000b. Kaolinite exchange between a stream and streambed: laboratory

experiments and validation of a colloid transport model. Water Resources Research 36 (8), 2363–2372.

Reece, D.E., Felkey, J.R., Wai, C.M., 1978. Heavy metal pollution in the sediments of the Cour d’Alene river, Idaho.

Environmental Geology 2 (5), 289–293.

Ren, J., Packman, A.I., 2004. Multi-phase contaminant exchange between streams and streambeds: theory and numerical

simulations. Environmental Science and Technology 38 (1), 2901–2911.

Runkel, R.L., Chapra, S.C., 1993. An efficient numerical solution of the transient storage equations for solute transport in

small streams. Water Resources Research 29 (1), 211–215.

Runkel, R.L., Bencala, K.E., Broshears, R.E., Chapra, S.C., 1996. Reactive solute transport in streams: 1. Development

of an equilibrium based model. Water Resources Research 32 (2), 409–418.

Runkel, R.L., Kimball, B.A., McKnight, D.M., Bencala, K.E., 1999. Reactive solute transport in streams: a surface

complexation approach for trace metal sorption. Water Resources Research 35 (12), 3829.

Savant, S.A., Reible, D.D., Thibodeax, L.J., 1987. Convective transport within stable river sediments. Water Resources

Research 23 (9), 1763–1768.

Smith, K.S., Ranvilee, J.F. and Macalady, D.L., 1989, Predictive modeling of copper, cadmium, and zinc partitioning

between stream water and bed sediment from a stream receiving acid mine drainage, St. Kevin Gulch, Colorado,

USGS Toxic Substances Hydrology Program, Water Resources Invest. Report 91-4034, 380–386.

Storey, R.G., Howard, K.W.F., Williams, D.D., 2003. Factors controlling riffle-scale hyporheic exchange flows and their

seasonal changes in a gaining stream: a three-dimensional groundwater flow model. Water Resources Research 39

(20), 1034.

Thibodeax, L.J., Boyle, J.D., 1987. Bedform-generated convective transport in bottom sediment. Nature 325, 341–343.

Wagner, B.J., Gorelick, S.M., 1986. A statistical methodology for estimating transport parameters: theory and application

to one-dimensional advective–dispersive systems. Water Resources Research 22 (8), 1303–1315.

Wagner, B.J., Harvey, J.W., 1997. Experimental design for estimating parameters of rate-limited mass-transfer: analysis

of stream tracer studies. Water Resources Research 33 (7), 1731–1741.

Worman, A., Forsman, J., Johansson, H., 1998. Modelling retention of sorbing solutes in streams based on a tracer

experiment using 51Cr. Journal of Environmental Engineering 124 (2), 122–130.

Worman, A., Packman, A.I., Johansson, H., Jonsson, K., 2002. Effect of flow-induced exchange in hyporheic zones on

longitudinal transport of solutes in streams and rivers. Water Resour. Res. 38 (1), 1001.

Zand, S.M., Kennedy, V.C., Zellweger, G.W., Avanzino, R.J., 1976. Solute transport and modeling of water quality in a

small stream. United States Geological Survey Journal Research 4 (2), 233–240.

Zaramella, M., Packman, A.I., Marion, A., 2003. Application of the Transient Storage Model to analyze advective

hyporheic exchange with deep and shallow sediment beds. Water Resources Research 39 (7), 1198.

applicability of the transient storage model to the hyporheic exchange of metals

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