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Advances in Water Resources 30 (2007) 148–156

Bedform-induced hyporheic exchange with unsteady flows

Fulvio Boano *, Roberto Revelli, Luca Ridolfi

Department of Hydraulics, Transports, and Civil Infrastructures – Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

IDRAM – Water and Environment Research Center – Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 7 December 2005; received in revised form 10 March 2006; accepted 12 March 2006Available online 27 April 2006

Abstract

The interaction between surface and subsurface water has a crucial influence on the biochemistry of stream environments. Eventhough the river discharge and the flow conditions can seldom be considered to be steady, the influence of this unsteadiness on the hyp-orheic exchange has often been neglected. In this work, a model for the study of hyporheic exchange during unsteady conditions has beendeveloped. The model provides a sound analytical framework for the analysis of the effects of a varying stream discharge on the exchangebetween a stream and the hyporheic zone. The effects of the unsteadiness on the water exchange flux, the residence time of the solutes inthe bed, and the stored mass are quantified. A synthetic example shows the substantial influence of a flood on the hyporheic exchange,and that the application of a steady model can lead to an underestimation of the exchanged mass, even after the flood has ended.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Hyporheic; Unsteady; Floods; Bedforms; Rivers; Exchange

1. Introduction

In fluvial environments, the hyporheic zone is that partof an aquifer that is adjacent to the river and where subsur-face water mixes with stream water. Hyporheic zones havebeen recognized to play a crucial role in fluvial ecologybecause they influence the transport of nutrients and pollu-tants and constitute an important habitat for the benthiccommunity [1]. The interactions between the stream andtopographical features such as bedforms [2,3], slope irregu-larities [4], and channel bends [5], induce a water flow intoand out of the channel bed. This flow allows the nutrientscarried by the river to reach the hyporheic zone, thus con-trolling the growth of microorganisms and the biotic pro-cesses in the subsurface. Moreover, surface water quality

0309-1708/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2006.03.004

* Corresponding author. Address: Department of Hydraulics, Trans-ports, and Civil Infrastructures – Politecnico di Torino, Corso Duca degliAbruzzi 24, 10129 Torino, Italy. Fax: +39 011 564 5698.

E-mail addresses: fulvio.boano@polito.it (F. Boano), roberto.revelli@polito.it (R. Revelli), luca.ridolfi@polito.it (L. Ridolfi).

is influenced by the temporary storage and subsequentrelease of pollutants in the hyporheic zone [6].

During the last decade, researchers have analysed thissurface–subsurface exchange at various scales and adopt-ing different approaches. A number of studies relied ondetailed field measurements of the topographical andhydraulic features of specific sites in order to calibratetwo- or three-dimensional numerical groundwater flowmodels. This approach allows hyporheic paths to be mod-elled, and the interactions between the hyporheic zone andthe surrounding aquifer to be described at scales of theorder of hundreds of meters [4,5,7–11]. The limit of thisapproach is that only site-specific information can beobtained, which is not easily generalizable to different loca-tions. On the other hand, the reciprocal influence of all thekey factors affecting surface–subsurface exchange (e.g.,river slope, meandering pattern, heterogeneity of aquiferpermeability) can be considered, which is crucial to cor-rectly model the hyporheic exchange.

A second kind of approach consists in studying how thetransport of chemicals in streams is influenced by theexchange with the hyporheic zones. In these works [12–

F. Boano et al. / Advances in Water Resources 30 (2007) 148–156 149

14], the in-stream concentration of a tracer injected into ariver was observed at some sampling points, and theseobservations were used to calibrate a one-dimensionalstream transport model. The transient storage model(TSM) [15] has in particular been widely adopted in moststudies. The calibrated parameters could then be relatedto the characteristics of the hyporheic zone. However, ithas been recognized [16] that only the shorter and fasterhyporheic paths in the shallower streambed can be identi-fied with this method, while no information is gained onthe slower transport to the deeper part of the bed. This lim-itation is due to the simplified representation of the hypor-heic zone in the TSM and it cannot be overcome. However,clever strategies to design tracer tests that minimize theseproblems have been developed [17], thus fostering the wide-spread application of the TSM approach.

Finally, numerous works have dealt with the modellingof the hyporheic exchange at smaller scales. The exchangedue to ripples and dunes has in particular been thoroughlystudied, since such bedforms are frequently present in riv-ers. A physically-based model (advective pumping model,APM) has been proposed [18–20] that identifies two basicprocesses as being responsible for the exchange of soluteswith the hyporheic zone: pumping and turnover. Pumpingindicates the water exchange due to gradients of head onthe bed surface caused by the separation of the flow behindthe bedforms, while turnover denotes the entrapment andrelease of water in the sediments due to the depositionand scour of the bed. Subsequent works examined howthese two processes are influenced by the heterogeneitiesof the streambed hydraulic conductivity [21,22], the physi-cochemical properties of the transported substances[23–26], the geometry of the bedforms [27], and the pene-tration of turbulence in gravel beds [28]. Even though theAPM relies on a simplified representation of the stream-aquifer system, it allows the roles played by the varioushydrodynamic variables (e.g., stream velocity, bedformdimensions, etc.) that govern the phenomenon to be under-stood, thus providing a useful tool to predict the actualhyporheic exchange. Moreover, the APM has been usedto discuss to what extent the hyporheic exchange can bereproduced by existing stream transport models [29–31],and has also constituted a basis for the development of astream transport model that could better represent theexchange with the hyporheic zone [6]. Finally, other modelsexist that consider the exchange of solutes due to turbulentdispersion [32], temperature gradients [33], and wavemotion [34].

The effects of unsteady flow conditions on bedform-driven hyporheic exchange are considered in the presentwork. The river discharge can be treated as constant onlyfor short periods, and natural changes in river dischargeare likely to influence hyporheic exchange. Despite this, onlya small number of studies [7,8,34] deals with unsteady condi-tions. This paper presents an extension of the APM that con-siders the effects of a time-varying discharge. A single fixeddune on a two-dimensional, homogeneous sandy bed was

considered, and a model was developed that allows the hyp-orheic exchange during a typical flood to be evaluated. Anumerical simulation is presented that shows the hyporheicexchange in unsteady conditions compared to a steady case.The model clearly demonstrates the important role playedby the unsteady flow conditions, thus providing a betterinsight into our understanding of hyporheic processes.

2. Model

In order to quantify the hyporheic exchange, the resi-dence time approach adopted by Elliott and Brooks [19]for the case of a steady flow was extended to the case ofa flood hydrograph, Q(t). A stream with a sand bed cov-ered by dunes was considered. The bed was treated astwo-dimensional, isotropic and homogeneous, with con-stant hydraulic conductivity, K, and porosity, h. Sedimentheterogeneity was omitted because it does not alter thequalitative effects of the unsteady flow on the hyporheicexchange. However, the present approach can be extendedto deal with a heterogeneous bed, but it would imply con-siderable additional computational efforts [22]. The bed-forms were characterized by their length, L, and thecrest-to-trough amplitude, A. The exchange of a conserva-tive solute with a known in-stream concentration, C(t), wasthen analysed.

In this work, the bedforms were assumed to be station-ary, that is, their shape and position did not changethroughout the flood. This is a common feature of residualbedforms that have been shaped by a previous major floodevent, and that have then been left in place after the subse-quent decrease in the stage and discharge. The presentmodel can thus deal with time-varying discharges that arenot strong enough to produce significant alterations ofthe bed profile.

2.1. Velocity field in the bed

When the bedforms do not move, the beform-drivenhyporheic exchange is essentially due to pumping pro-cesses. Stream water flowing over bedforms induces a recir-culation flow on the downstream of the lee side of the dune,which results in sinusoidal head fluctuations over the bedsurface [19,35,36] that constitute the driving force for thehyporheic flows. The amplitude of these head fluctuationsis related to the bedform geometry and to the flow proper-ties, while the wavelength coincides with the dune wave-length L.

Typical floods are characterized by relatively small tem-poral and spatial discharge, depth, and stream velocitygradients. In these cases, the amplitude of the head fluctu-ations at each instant t can be considered to be in equilib-rium with the stream velocity and depth at the sameinstant, and can therefore be evaluated with a formulafor steady conditions. In this work, the following empiri-cal expression proposed by Elliott and Brooks [19] wasused

150 F. Boano et al. / Advances in Water Resources 30 (2007) 148–156

hmðtÞ ¼ 0:28U 2ðtÞ

2gA=dðtÞ0:34

� �w

; w ¼3=8 A=dðtÞ < 0:34;

3=2 A=dðtÞ > 0:34;

�

ð1Þwhere hm(t) is the half-amplitude of the sinusoidal headfluctuations, and U(t) and d(t) are the mean velocity anddepth of the stream, respectively, at time t. For the sakeof convenience, the normalized half-amplitude f(t) =hm(t)/h0 is hereafter used, where h0 is the half-amplitudeof the head relative to the base discharge, Q0, and to thecorresponding stream depth and mean velocity. For a givenflood hydrograph, the time evolution of the mean velocityand depth of the stream can be derived, and Eq. (1) can beused to derive the magnitude of the amplitude of the head,that constitutes the driving force of the pumping process.

The flow of an incompressible fluid in a sand bed is gov-erned by the Laplace equation

r2h ¼ 0; ð2Þwhere h(x,y, t) is the hydraulic head in the bed. Eq. (2) issolved on the semi-infinite domain y 6 0, together withthe boundary conditions

h ¼ hmðtÞ sinðkxÞ; y ¼ 0; ð3Þh ¼ 0; y ! �1; ð4Þwhere k = 2p/L is the wave number, and hm(t) is obtainedfrom Eq. (1). A periodic boundary condition is also im-posed, that is, the values of the head are the same atx = 0 and at x = L. It should be noticed that the boundarycondition (3) is applied on y = 0 rather than on the actualbed surface. This simplification was made by Elliott [18],and reflects the assumption that the solute exchange withthe whole bed is little influenced by the unevenness of thebed profile. The solution for the head is

hðx; y; tÞ ¼ h0f ðtÞ sinðkxÞeky : ð5ÞUsing the Darcy equation,~v ¼ �Krh, the velocity field forthe water – as well as for the non-reactive solutes – insidethe bed can be recovered

uðx; y; tÞ ¼ �u0f ðtÞ cosðkxÞeky ; ð6Þvðx; y; tÞ ¼ �u0f ðtÞ sinðkxÞeky ; ð7Þ

where u and v are the horizontal and vertical componentsof the Darcy water velocity, respectively, and u0 = kKh0

is a characteristic scale for Darcy flow in the bed. It shouldbe noticed that the velocity field has the same structure asfor the steady flow case [19], but both velocity componentsare multiplied by the factor f(t). The steady flow case canbe obtained as a particular case for f(t) = 1.

Adopting the normalization originally proposed in [19],that is

x� ¼ kx; y� ¼ ky;t�

h¼ ku0

th¼ k2Kh0

th; ð8Þ

where the porosity, h, is required in order to obtain thepore velocity scale u0/h, the normalized velocity field canbe recovered as

u� ¼ uu0

¼ �ft�

h

� �cosðx�Þey� ; ð9Þ

v� ¼ vu0

¼ �ft�

h

� �sinðx�Þey� : ð10Þ

The effects of the unsteady flow on the seepage velocities istherefore summarized by the function f(t*/h), which in turndepends on the stream flow properties – see Eq. (1).

2.2. Particle paths and front positions

The equations for the particle paths can be derived fromthe normalized velocity field. Using Eqs. (9) and (10), a dif-ferential equation can be stated

dY �

dX �¼ v�

u�¼ tan X �; ð11Þ

where X* and Y* denote the horizontal and vertical posi-tion of a particle, respectively. Eq. (11) can be solvedtogether with the initial condition Y* = 0 at X � ¼ X �0,obtaining

Y � ¼ � lncos X �

cos X �0

� �; ð12Þ

where Y*(X*) is the equation of the path followed by par-ticles that enter the bed at X �0. The most notable featureof Eq. (12) – as well as of (11) – is that the particle pathsdo not depend on time. The particle paths therefore coin-cide with the streamlines, and are the same as in the caseof steady flow [19]. This remarkable result is a directconsequence of (i) the linearity of Eq. (2), and (ii) thehypothesis that the boundary head fluctuations can onlychange in amplitude, but retain their sinusoidal shape.Water – and solute – particles entering the bed at differentinstants follow the same paths, even though they travel atdifferent speeds, as can be deduced from Eqs. (9) and (10).

Eq. (12) can be restated in parametric form. Substituting(12) in Eq. (9), it is possible to write

dX �

dðt�=hÞ ¼ �f ðt�=hÞ cos X �0; ð13Þ

where the presence of the porosity h depends on the actualfluid velocity being higher than the Darcy velocity. Thisequation can be solved, with the initial conditionX � ¼ X �0 at t�=h ¼ t�0=h. Integration and substitution in(12) give

X � ¼ X �0 � cos X �0 Ft�

h

� �� F

t�0h

� �� �; ð14Þ

Y � ¼ � lncos X �0 � cos X �0 F t�

h

� �� F

t�0

h

h ih icos X �0

0@

1A; ð15Þ

where F ðt�hÞ ¼R t�=h

0f ðsÞds. By parametrizing with respect to

time t*/h, Eqs. (14) and (15) can be interpreted as the hor-izontal and vertical positions, respectively, of a particlethat has entered the bed at X �0 at time t�0=h. Moreover,Eqs. (14) and (15) can also be parametrized with respect

Fig. 1. Particle paths (continuous lines), and front positions at t*/h =0,1,2,3 (dotted lines). The fronts refer to the steady case with f(t) = 1. Fig. 2. Scheme of the particle paths.

F. Boano et al. / Advances in Water Resources 30 (2007) 148–156 151

to the starting position X �0, and thus describe the front po-sition, at time t*/h, of the particles which entered the bed atthe same instant. The resulting paths and fronts are shownin Fig. 1.

2.3. Hyporheic exchange flux

The exchange flux between the stream and the hyporheiczone can also be evaluated. From Eq. (10), it can be seenthat the water flux per unit bed area q = �vjy=0 =u0f(t)sin(kx) enters the hyporheic zone at 0 < x < L/2(half-wavelength), and is paired by an equal flux from thehyporheic zone to the stream at the other half-wavelength.However, the deeper parts of the bed are subjected only toa fraction of this flux. This fact can have important ecolog-ical consequences (e.g., smaller nutrient fluxes to the deeperpart of the bed), and it would therefore be interesting toquantify the flux that could reach a certain depth.

Let us consider only half a wavelength, �p/2 < x* < p/2,for the sake of simplicity (see Fig. 2). For a given depthd 6 0, a particle entering the bed at 0 < x�0 < p=2 wouldreach the depth y* = d* = kd at x� ¼ x�IN. Substitution inEq. (12) leads to

x�IN ¼ cos�1ðcos x�0 e�d� Þ: ð16Þ

It can be seen from Fig. 2 that only the particles that enterthe bed in the spatial interval ðx�0;p=2Þ eventually reach thedepth d*. The left limit of this range corresponds to the par-ticle that reaches y* = d* at x�IN ¼ 0. Substitution in (16)leads to

x�0 ¼ cos�1 ed� : ð17Þ

Using Eq. (17), the average hyporheic exchange flow perunit bed area can be evaluated

�qdðtÞ ¼R L=4

x0u0f ðtÞ sinðkxÞdxR L=2

0 dx¼ u0f ðtÞ

pekd ð18Þ

and, in normalized form,

�q�d� ðt�=hÞ ¼�qd

u0

¼ f ðt�=hÞp

ed� : ð19Þ

It should be noticed that the magnitude of the hyporheicexchange flux per unit bed surface exponentially decreaseswith depth, as d* 6 0. The effect of unsteady flow condi-tions is still summarized by the term f(t).

2.4. Residence time distribution

Particles that follow different paths have different resi-dence times inside the bed. Since the velocity field isunsteady, the residence time for a single path also changesin time. Moreover, different depths of the bed can be con-sidered, and the residence time distribution (RTD) can bedefined as the fraction of the particles which have enteredthe bed at t0 and that are below the depth d at time t.

An analytical expression for the RTD can now bederived. Again, only half a wavelength, �p/2 < x* < p/2,is considered, because of the symmetry of the problem. Ifthe zone below a given depth d* is considered, a particlethat has entered the bed at x�0 at t�0=h will first reachy* 6 d* at x� ¼ x�IN, and will then leave this deep zone atx� ¼ �x�IN, as shown in Fig. 2. Substitution in (14) gives

Ft�IN

h

� �¼ F

t�0h

� �þ x�0 � x�IN

cos x�0; ð20Þ

Ft�OUT

h

� �¼ F

t�0h

� �þ x�0 þ x�IN

cos x�0; ð21Þ

where t�IN=h and t�OUT=h are the instants at which the parti-cle enters and exits from the zone below d*, respectively,and x�IN is given by (16). For a given depth d* and startingtime t�0=h, these equations give the instant t�IN=h and t�OUT=h,respectively, as a function of the starting position x�0 of thatparticle. The solution can be found by inversion of the inte-gral function F(t).

The first particle that reaches the deep zone, y* 6 d*, isthe one that follows the vertical path from x�0 ¼ p=2. At alater instant t*/h, the deep zone is also reached by the par-ticles that have started in the interval n < x�0 < p=2, where

152 F. Boano et al. / Advances in Water Resources 30 (2007) 148–156

x�0 ¼ n is the solution of Eq. (20) for the considered instant.This interval has the maximum width when n ¼ x�0, that is,when the particle that travels along the tangent path (seeFig. 2) arrives. The correspondent time of arrival t�=h aty* = d* can be found by setting x�IN ¼ 0 in (20) and recalling(17), thus obtaining

Ft�

h

� �¼ F

t�0h

� �þ e�d� cos�1ðed� Þ: ð22Þ

At later times, some particles leave the deep zone and theinterval n < x�0 < p=2 becomes narrower. For t�=h > t�=h,the lower bound n can be found as the solution of (21).It is then possible to write

Rd� ¼1 ðn < x�0 < p=2Þ;0 ð0 < x�0 < nÞ;

�ð23Þ

where n is given by either (20) or (21), depending onwhether t*/h is smaller or larger than t�=h, respectively.

The RTD can be found as a spatial average of Eq. (23),weighted by the local exchange fluxes. Recalling (see Eq.(10)) that water enters the bed only for 0 < x* < p/2, it ispossible to obtain

Rd�t�

h;t�0h

� �¼R p=2

0Rd� ðx�0Þq�0ðx�0Þdx�0R p=2

0q�0ðx�0Þdx�0

¼R p=2

n f ðt�0=hÞ sinðx�0Þdx�0R p=2

0f ðt�0=hÞ sinðx�0Þdx�0

¼ cosn;

where Rd� ðt�=h; t�0=hÞ is the RTD at time t*/h, relative to thestarting time t�0=h and depth d*. It should be clear that bothq�0 and �q�0 are evaluated at d* = 0, as the RTD is the ratiobetween the number of particles at depth d* and the totalnumber of particles that have entered the bed. This expres-sion can be introduced into (20) and (21) with x�0 ¼ n, finallyobtaining

Ft�

h

� �� F

t�0h

� �¼

cos�1 Rd� � cos�1 e�d�Rd�� �

Rd�;

if Ft�

h

� �� F

t�0h

� �<

cos�1 ed�� �

ed�;

Ft�

h

� �� F

t�0h

� �¼

cos�1 Rd� þ cos�1 e�d�Rd�� �

Rd�;

if Ft�

h

� �� F

t�0h

� �>

cos�1 ed�� �

ed�;

ð24Þ

where the second part of each equation depends onwhether t*/h is smaller or larger than t�=h, and derives fromEq. (22).

For a given t�0=h and d*, Eq. (24) can be numericallysolved for Rd� . Because of the unsteadiness of the velocityfield, particles that enter the bed at different t�0=h are char-acterized by different RTDs. Since the argument of the arc-cosine function must be less than unity, it follows thatRd� 6 ed� . This means that Rd� is always less than one whend* < 0 is considered, as some particles never reach thatdepth. Finally, it should be noticed that t*/h is the actual

elapsed time from t*/h = 0, the time spent by a particle inthe bed being t�=h� t�0=h.

2.5. Exchanged mass

The existence of a water flux between a stream and ahyporheic zone also implies an exchange of nutrients andcontaminants. Conservative solutes are passively advectedby the velocity field, and are therefore characterized bythe same RTD as the water particles. The total exchangedmass thus depends on both the RTD and the in-stream sol-ute concentration, C(t).

The mass of solute that enters a unit bed area between t0

and t0 + dt0 is �q0ðt0ÞCðt0Þdt0. At the generic time t, the frac-tion Rd is still in the bed at y < d. If the total mass per unitbed area is expressed as m Æ C0, where m is an equivalentdepth of penetration and C0 is a reference concentration,and adopting the following normalization

m� ¼ 2pkmh

;

the total mass per unit bed surface can be evaluated as

m�d�t�

h

� �¼ 2p

Z t�=h

0

�q�0t�0h

� �C�

t�0h

� �Rd�

t�

h;t�0h

� �d

t�0h

� �; ð25Þ

where m�d� ðt�=hÞ is the equivalent penetration depth relativeto the zone below d*, and C*(t) = C(t)/C0. It should be no-ticed that all the factors inside the integral show a depen-dence on time. In particular, �q�0 and Rd� can be evaluatedfrom (19) with d* = 0 and (24), respectively, while the nor-malized in-stream concentration C*(t) must be obtainedindependently. An application of Eq. (25) is proposed inthe next section.

3. Example

A synthetic example is now presented in order todescribe the main features of the unsteady hyporheicexchange. A river with a mild slope, S = 10�4, and with a50-m-wide rectangular section is considered. A Stricklercoefficient ks = 40 m 1/3 s�1, which is a typical value fornatural streams, is adopted. The base discharge isQ0 = 19.5 m3/s, and the correspondent flow depth isd0 = 1 m. A flood characterized by the hydrograph shownin Fig. 3a is considered. The peak discharge is about104 m3/s, which is five times the base discharge, and theduration of the flood is approximately 3 d. After this per-iod the discharge returns to the steady value, and remainsconstant for the rest of the simulation. The in-stream nor-malized time-dependent concentration, C*(t), presented inFig. 3b, has been considered. This concentration historycould derive, for instance, from a contaminant release intothe stream during a flood.

The river bed is assumed to be composed of sand, with arepresentative grain diameter D = 3 mm, a hydraulic con-ductivity K = 10�4 m/s, and a porosity h = 0.3. The bedsurface is covered by dunes of a height A = 0.4 m and

(a)

(b)

Fig. 3. Flood hydrograph (a), and normalized half-amplitude of theboundary head f(t) (continuous line) and in-stream concentration C*(t)(dashed line) (b).

Fig. 4. Exchange flux �qdðtÞ for d = 0 (continuous line), d = �30 cm (dash-dotted line), and d = �1 m (dashed line).

F. Boano et al. / Advances in Water Resources 30 (2007) 148–156 153

length L = 2.8 m (aspect ratio A/L = 1/7). For the exam-ined stream, dunes of such a height would have beenshaped by a discharge of approximately 350 m3/s (see,e.g., [37]), which is remarkably higher than the maximumdischarge for the examined flood. The Shields criterionconfirms that no solid transport occurs during the flood,and thus the height of the residual bedforms is consideredto be constant.

The normalized half-amplitude of the boundary hydrau-lic head, f(t), has been computed with Eq. (1), whichrequires the knowledge of the stream depth, d(t), and themean velocity, U(t). Since the considered flood is character-ized by small spatial and temporal discharge gradients, theChezy formula has been used to estimate d(t) and U(t). Thehalf-amplitude of the head relative to base flow,h0 = 2.8 mm, has also been evaluated with Eq. (1). It canbe seen, from Fig. 3b, that the shape of f(t) is very similarto that of the hydrograph, because it is mainly influencedby the initial increase and subsequent decrease of U2(t) dur-ing the flood. However, the flood is also characterized by areduction – and a subsequent increment – of the relative

dune height A/d(t), therefore the maximum of f(t) is only2, while the ratio between the maximum and the base dis-charge is fivefold. The function f(t) does no longer changefor t > 4 d, after the end of the flood. The presence of twoangular points, which correspond to a minimum valueslightly lower than one, should also be noticed. Theseangular points are due to the sudden changes in the expo-nent w in (1) when the depth d(t) is such that A/d(t) = 0.34.

The average water fluxes between the stream and thehyporheic zone are presented in Fig. 4. The time evolutionof the flux has the same shape as f(t) for each analyseddepth. This is a direct consequence of Eq. (19), from whichit follows that the maximum water flux is twice the baseflow value. The exchanged flux quickly decreases withdepth, reducing to a half of the surface value at 30 cm indepth, and to a tenth at one meter in depth.

Fig. 5a shows two examples of residence time distribu-tions, R0ðt � t0; t0Þ, for t0 = 0 and t0 = 6 d, respectively.The functions are plotted against t � t0, that is, they repre-sent the cumulative probability functions of the actual timespent by a particle inside the whole bed (d = 0). It canbe observed that the particles that enter the bed duringthe flood (t0 = 0) experience shorter residence time insidethe sediments than the particles that have entered after theend of flood (t0 = 6 d). This is due to the higher filtrationvelocities during the flood, as can be seen from Eqs. (9)and (10), which cause particles to leave the bed earlier thanin the case of hyporheic exchange due to base flow. Theunsteadiness of the flow thus results in shorter contacttimes between the solutes and sediments, which can havean important ecological relevance for nutrients and otherreactive solutes.

The permanence of the solutes in the stream bed alsodepends on the considered depth. Fig. 5b compares threeRTDs Rdðt; t0Þ for d = 0, �0.3 and �1 m, respectively. Allthe three curves refer to t0 = 0. The lag between the startof the particle motion at t0 and the arrival at the different

(a)

(b)

Fig. 5. Residence time distributions R0ðt � t0; t0Þ for t0 = 0 (continuousline) and t0 = 6 d (dashed line) (a). Residence time distributions Rdðt; 0Þ ford = 0 (continuous line), d = �30 cm (dash-dotted line), and d = �1 m(dashed line) (b).

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (d)

m0/

θ (m

)

Fig. 6. Equivalent depth of penetration of the solute in the bed in the caseof a steady C*(t). Comparison between the unsteady case (continuous line)and the steady cases with Q = 19.5 m3/s (dash-dotted line) andQ = 104 m3/s (dashed line).

154 F. Boano et al. / Advances in Water Resources 30 (2007) 148–156

depths is evident. It takes approximately one and twentydays, respectively, for the first particles to reach a depthof 30 cm and 1 m. After these arrival times, the RTDsgradually increase to a maximum value, then they tendasymptotically to R0. The gradual increase reflects the dif-ferent arrival times for particles that travel along differentpaths, while the maximum values, 0.5 and 0.1 ford = �0.3 and �1 m, respectively, correspond to the valuesof the exchanged flows at the same depths.

The solute exchange between the stream and the hypor-heic zone has been evaluated for the normalized in-streamconcentration C*(t) = H(t), where H(t) is the Heavisidestep function. Even though this case is not representativeof the solute transport in real streams, it has repeatedlybeen used to describe the hyporheic exchange in laboratoryflumes (e.g., [20,23]). Moreover, the adoption of a steadyC*(t) allows for a better understanding of the influence ofthe unsteady discharge Q(t) on the hyporheic exchange.The mass exchanged between the stream and the sedimentsis reported in Fig. 6, expressed as an equivalent depth of

penetration, m0/h. It is compared to the exchanged massin two steady discharge cases, that is, the case of base flow,Q = 19.5 m3/s, and the case of a constant discharge equalto the peak value, Q = 104 m3/s. During the first phaseof the flood the discharge increases up to the peak value,and the penetration depth m0/h is similar to the one thatwould result for a steady flow with 104 m3/s. As the dis-charge gradually returns to the base value the net soluteflux into the bed decreases, and the exchanged mass asymp-totically tends to the values relative to the steady case with19.5 m3/s. The long time after the end of the flood (t � 3 d)that is required for the exchanged mass to return to thesteady state curve can be observed. It should also benoticed that the unsteady penetration depth always liesbetween the two steady cases.

Finally, the solute exchange has also been evaluated forthe unsteady in-stream concentration C*(t) shown inFig. 3b. The exchanged mass for the unsteady case andthe two steady cases, Q = 19.5 m3/s and Q = 104 m3/s,respectively, are reported in Fig. 7a. The most notable fea-ture is that the penetration depth, m0/h, for long times ishigher for the unsteady case than for both the low- andhigh-flow steady cases. During the first phase of the flood,namely, t < 2 d, the coupling of the high exchange flux �qðtÞand in-stream concentration C(t) determines an accumula-tion of solute in the streambed. The maximum penetrationdepth is about 8.5 cm, that is, nearly twice the depth thatwould result from a steady base flow and slightly less thanthe value due to the steady high discharge. At later times,as the flow conditions gradually return to the base values,the exchange flux decreases, and the accumulated massbegins to be washed away at a slower rate than the accumu-lation rate. If a pollutant is considered, this would result inhigh levels of contamination that would still be relevant fora long time after the end of the flood. This example clearlyshows that it is not possible to model the amount of

(a)

(b)

Fig. 7. Equivalent depth of penetration of the solute in the bed.(a) Comparison between the unsteady case (continuous line) and thesteady cases with Q = 19.5 m3/s (dash-dotted line) and Q = 104 m3/s(dashed line); (b) depth of penetration at the surface (continuous line),30 cm depth (dash-dotted line), and 1 m depth (dashed line).

F. Boano et al. / Advances in Water Resources 30 (2007) 148–156 155

exchanged mass in unsteady conditions using an equivalentdischarge, as neither the base flow value nor the peak valuecompletely describes the evolution of the accumulatedmass.

The equivalent depth of penetration, md(t)/h, is pre-sented in Fig. 7b for different depths of the bed, namely,d = 0, �0.3, and �1 m. The accumulated mass decreaseswith increasing depth, while the peak value shifts towardslater times. It should be noticed that m0/h is the depth ofpenetration that would occur if the bed was well-mixed,while the solute flux is not uniform in space – see the frontsin Fig. 1. Even though the maximum penetration depth m0/his about 8.5 cm, some solute eventually reaches deeper zonesof the bed. At t = 20 d, while the overall mass in the bed isdecreasing, the mass at one meter below the surface is begin-ning to increase. This spatial unevenness can have importantimplications on the stream ecology, e.g., on the availabilityof nutrients in the deeper sediments. Although the estimatedvalues of the solute mass in the shallower part of the bed are

somehow inaccurate as the actual bed profile is neglected, itshould be noticed that the depth interested by the hyporheicexchange is much greater than the dune height. The modelpredictions are thus reliable for long enough times, whenmost of the mass is stored in the deeper bed.

4. Conclusions

A model for the bedform-induced hyporheic exchange inunsteady flow conditions has been described in this work.The model has demonstrated the influence of the unstead-iness of the flow on the hyporheic exchange. While the flowpattern is the same as in the steady case, both the waterexchange flux and the residence times in the bed are influ-enced by the changes in the stream discharge and depth.The exchange flux increases during a flood, but the ratiobetween the maximum and the minimum flux is less thanthe ratio between the maximum and the minimum dis-charge. This happens because the higher stream velocitiesare coupled with smaller relative dune heights, and viceversa. At the same time, the greater filtration velocity actu-ally reduces the residence time of the solutes in the bed. Theincreased exchange flux is thus likely to be important forfast-reactive solutes, which would not be very influencedby the reduced contact time with the sediments.

Given the in-stream concentration history, the modelalso evaluates the overall mass stored in the bed. If the sol-ute concentration is in phase with the flood hydrograph, ithas been shown that the unsteady flow results in higher lev-els of accumulated mass in the streambed, with respect tothe exchange due to both the minimum and maximum dis-charge. This storage is still remarkable for a long time afterthe end of the flood, as the release of solutes by the bed ismuch slower than the inward flux during the flood. Themodel thus shows that the adoption of a constant valuefor the discharge can lead to a substantial underestimationof the hyporheic exchange.

Acknowledgements

The authors would like to thank the CRC Foundation(Fondazione Cassa di Risparmio di Cuneo) and RegionePiemonte for the financial support of this research.

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