Hyporheic flows in stratified beds

Hyporheic flows in stratified beds

Andrea Marion,1 Aaron I. Packman,2 Mattia Zaramella,1 and Andrea Bottacin-Busolin1

Received 1 April 2007; revised 30 May 2008; accepted 14 July 2008; published 25 September 2008.

[1] Surface-subsurface exchange fluxes are receiving increasing interest because of theirimportance in the fate of contaminants, nutrients, and other ecologically relevantsubstances in a variety of aquatic systems. Solutions have previously been developedfor pore water flows induced by geometrical irregularities such as bed forms for the casesof homogeneous sediment beds and idealized heterogeneous beds, but these solutionshave not accounted for the fact that streambed sediments are subject to sorting processesthat often produce well-defined subsurface structures. Sediments at the streambedsurface are often coarser than the underlying material because of size-selective sedimenttransport, producing relatively thin armor layers. Episodic erosional and depositionalprocesses also create thick layers of different composition within the porous medium,forming stratified beds. A series of experiments were conducted to observe conservativesolute transport in armored and stratified beds. An analytical solution was developedfor advective exchange with stratified beds and provides appropriate scaling of thephysical variables that control exchange flows. The results show that armor layers are toothin to significantly alter the advective pumping process but provide significant solutestorage at short time scales. Stratified beds with layers of significant thickness favordevelopment of horizontal flow paths within the bed and change the rate of solute transferacross the stream-subsurface interface compared to homogeneous beds.

Citation: Marion, A., A. I. Packman, M. Zaramella, and A. Bottacin-Busolin (2008), Hyporheic flows in stratified beds, Water

Resour. Res., 44, W09433, doi:10.1029/2007WR006079.

1. Introduction

[2] Exchange between an overlying water column andunderlying sediment bed is known to regulate biogeochem-ical and ecological processes in a wide variety of aquaticsystems, including streams, rivers, wetlands, estuaries, anddeeper marine environments such sandy continental shelves.A variety of hydrodynamic transport processes produceinterfacial exchange greater than the minimum rate resultingfrom molecular diffusion, resulting in enhanced mass trans-fer between the water column and sedimentary pore waters.In systems with relatively permeable sediments and appre-ciable overlying velocities, interfacial transport is normallydominated by pore water advection induced by couplingwith the overlying flow [Packman and Bencala, 2000;Huettel and Webster, 2001].[3] Recently, a variety of solutions have been developed

for the pore water flow field induced by interaction of theoverlying flow with a variety of sedimentary morphologicalforms, including those formed by purely physical processes(sediment transport) and biological activity (bioturbation,etc.) [Elliott and Brooks, 1997; Packman et al., 2000;Huettel et al., 1996; Huettel and Rusch, 2000; Shum andSundby, 1996]. In addition, numerical models have alsobeen used to explore cases with more complex geometries

[Savant et al., 1987; Gooseff et al., 2006; Cardenas andWilson, 2007]. Most of these solutions determine the porewater flow induced by dynamic pressure variations over thebed surface, and have been developed by assuming that thesediments are homogeneous. For a related problem, windpumping through snowpacks, Colbeck [1989] showed thatsubsurface layering substantially alters patterns and rates ofpore fluid flow. Salehin et al. [2004] demonstrated thatlayered sedimentary heterogeneity increases the averageinterfacial flux but tends to limit vertical penetration becauseit favors horizontal transport through high-permeabilitylayers. However, that work used an idealized randomcorrelated permeability field because relatively little isknown about in situ structure in permeable sediment beds.Cardenas and Zlotnik [2003] and Cardenas et al. [2004]used structure-imitating interpolation (kriging) to obtain a3-D reconstruction of modern channel bend deposits on thebasis of constant head injection tests and ground-penetratingradar surveys; they used 3-D numerical modeling to inves-tigate the influence of heterogeneity on the hyporheic fluxes,showing that residence times can decrease or increasecompared to the homogeneous case depending on therelative positions of the heterogeneities and the bed forms.More recently, the influence of aquifer heterogeneity onsolute exchange with the hyporheic zone was investigatedexperimentally by in situ tracer tests and measurements byRyan and Boufadel [2006]. In their study higher concen-trations were found to be in the upper bed sediments whenthe hydraulic conductivity was higher than in the lower bedsediments and vice versa.

1Department of Hydraulic, Maritime, Environmental and GeotechnicalEngineering, University of Padua, Padua, Italy.

2Department of Civil and Environmental Engineering, NorthwesternUniversity, Evanston, Illinois, USA.

Copyright 2008 by the American Geophysical Union.0043-1397/08/2007WR006079

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[4] We seek to link these prior studies to the processesthat produce local-scale heterogeneity in active fluvialsystems. Size-selective transport of sediments tends toentrain finer particles from the bed more rapidly thancoarser ones [Sutherland, 1987]. Preferential removal offines from the bed surface by entrainment or winnowingleaves a coarse surface layer characterized by a higherpermeability compared to the parent material. This processis described as armoring, and the resulting layer of coarsesediments is called the armor layer [Sutherland, 1987;Parker and Sutherland, 1990]. The thickness of the armorlayer is normally limited to twice the size of the largestmaterial in the sediment mixture, typically taken as the 90thpercentile of the bed sediment grain size distribution, D90

[Marion and Fraccarollo, 1997a]. Although the armor layeris thin, it can play an important role in the overall interfacialtransport process because interfacial transport rate normallyscales as the square of the grain size [Packman et al., 2004],suggesting that the interfacial flux could be more than anorder of magnitude greater when an armor layer is present.This surficial layer is expected to be particularly importantto transport processes acting at fast time scales and smallspatial scales, such as nutrient delivery to benthic biofilms[e.g., Larned et al., 2004].[5] Until now hyporheic transport through a permeable

layer laying over an impermeable substrate have beenmodeled with simplified mathematical approaches, like theone implemented by the transient storage model [Hays etal., 1966; Day, 1975; Bencala and Walters, 1983] or bythe purely diffusive model [Richardson and Parr, 1988;Marion and Zaramella, 2005]. This kind of approach canadequately represent the exchange process only for somelimited cases of streambed structure, i.e., bed form surfacemorphologies with exchange limited by a shallow imper-meable layer [Marion et al., 2003; Zaramella et al., 2003].On the other hand, analytical solutions for a great variety ofcases provide a suitable physical description of the processitself [Elliott and Brooks, 1997; Packman et al., 2000;Marion et al., 2002; Boano et al., 2006, 2007]. Previously,we have observed that the standard analytical solutions forbed form-induced advective pore water flow fail to repro-duce observed interfacial solute transport at short times[Packman et al., 2004; Zaramella et al., 2006]. Thisdiscrepancy is likely to be related to the fact that the initialmass transfer from the water column to the sediments isaffected by the acceleration of the flow just below the bedsurface by diffusion of momentum and mass that is notincluded in the advective surface-subsurface flow couplingmodels described previously. The interfacial diffusion ofmomentum was originally investigated by Brinkman[1947], and the resulting flow coupling across the sedimentbed surface within the sediment bed due to wall turbulencehas subsequently been studied by a number of authors[Beavers and Joseph, 1967; Ho and Gelhar, 1973; Ruffand Gelhar, 1972; Nagaoka and Ohgaki, 1990; Shimizu etal., 1990; Zhou and Mendoza, 1993]. The subsurface regioninfluenced by diffusion of momentum is now generallyreferred to as the Brinkman layer. A further enhancementof interfacial transport can be induced by the small-scalepressure-driven advective flow around individual sedimentparticles, but little is known about this process because it isvery difficult to observe. Clearly, the presence of an armor

layer will strongly influence all of these momentum andmass transfer processes because the geometry of the sedi-ment grains defines their interaction with the flowing fluid(friction, flow resistance, permeability, etc.).[6] Here we examine the role of layered sediment het-

erogeneity on pore water flow and solute transport. Wepresent new experimental observations of surface-subsur-face exchange with layered sediment beds, including bothflat beds and beds with bed forms. Data collected for thecase of a flat bed with a thin surface layer produced bywinnowing are used to evaluate the extent to which naturalarmoring processes are likely to affect advective pore watertransport over a range of time scales. An analytical solutionis presented for the case of bed form-induced hyporheicexchange when two well-defined layers of different perme-ability are present. The predictions of the model are com-pared to the experimental data.

2. Experiments

[7] Recirculating flumes have been widely used for theirconvenience in isolating and studying single hyporheicexchange mechanisms. These systems allow control of bothstream and subsurface conditions. They are also a closedsystem whose nature is advantageous because exchangeprocesses can then be studied by observing the concentra-tion of a conservative tracer uniformly distributed in therecirculating water. The tracer concentration decreasesbecause of mixing with the initially clean pore water.Exchange can then be measured until the tracer becomeswell mixed throughout the bed. To evaluate solute masstransfer with layered sediment beds, experiments wereconducted in a 7.5-m-long, 20-cm-wide and 40-cm-deeptilting and recirculating flume located in the Environmentaland Biological Transport Processes Laboratory atNorthwestern University.[8] A bed composed of nonuniform sediment was placed

in the channel and a constant volume of water was contin-uously recirculated over the bed. The bed was composed ofa mixture of 50% pea gravel and 50% coarse sand. The peagravel was presorted by the supplier and had a relativelynarrow size distribution (D10 = 2 mm, D50 = 6 mm, andD90 = 9 mm). The mean bulk porosity of the clean gravelwas measured and found to be q = 0.38. The sand hadD10 = 0.2 mm, D50 = 0.8 mm and D90 = 2 mm and alsohad a porosity of approximately 0.38. The mixture had awider size distribution with D10 = 0.4 mm, D50 = 2.3 mm,and D90 = 8 mm. The porosity of the mixture could not beevaluated in situ and was assumed equal to the value forsand. Porosity is shown in section 4 to act only as a scalingfactor in the definition of the dimensionless time t*, whichacts in the same way on model results and experimentaldata. Therefore a correction of the porosity of the mixtureis not necessary for the concepts and results of this study.[9] Sodium chloride (NaCl) was used as a conservative

tracer to evaluate mass transfer from the overlying flow tothe pore water. A concentrated NaCl solution was preparedusing reagent-grade salt and then added to the downstreamend of the flume over one recirculation period in order toestablish a uniform salt concentration in the stream. Thesurface-subsurface exchange flux was determined from therate of change of the in-stream concentration by considering

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W09433 MARION ET AL.: HYPORHEIC FLOW IN STRATIFIED BEDS W09433

the mass balance between the recirculation stream and thepore water in the sediment bed.[10] Seven tests were performed to observe exchange

with different sediment structures under different flowconditions. The experimental conditions are listed inTable 1. Two series of tests were performed: one at a meanoverlying velocity U = 0.36 m s�1 and the second at a meanoverlying velocity U = 0.21 m s�1. Both of these velocitiesare below the threshold for sediment motion, thus keepingstationary bed topography throughout the solute injectionphase of each test.[11] Four experiments were performed with flat beds

having different subsurface structure. The first experimentwas conducted with a 15 cm deep flat bed composed of thesand-gravel mixture. At the end of experiment 1, thechannel slope was increased from 0.00067 to 0.002 andthe overlying flow rate was increased from 0.009 m3 s�1 to0.012 m3 s�1 in order to selectively remove the finesediment fraction. The dimensionless shear stress (Shields’factor) was below threshold for motion (�0.02) for thegravel and above threshold (>0.1) for the fine sand. Theseconditions were maintained until the grain entrainmentbecame negligible (�24 h). The surface of the bed becamecoarsened by selective removal of the fine fraction, whilethe coarse sediment was not displaced. This armor layer wastherefore produced only by winnowing without imbricationof the coarse particles.[12] After the armor layer was formed, the flume was

readjusted to the original slope and discharge and a secondsolute injection was performed (experiment 2) with thesame overlying mean velocity used in experiment 1 (U =0.36 m s�1). At the completion of experiment 2, the surfacelayer was sampled by the wax method [Proffitt, 1980; Frippand Diplas, 1993]. The composition obtained in terms ofarea by weight was converted into volume by weight usingthe methods developed by Marion and Fraccarollo[1997b]. The composition of the armor layer is shown inFigure 1, along with the grain size distribution of the sand-gravel mixture and the gravel. It is apparent that the armorlayer is almost as coarse as the gravel, D90 � 8 mm.[13] In experiment 3, the bed was composed of two layers:

a 12-cm-thick bottom layer composed of the gravel-sandmixture, and a 3-cm-thick upper layer composed of gravel.This coarse surface layer was approximately four times thethickness of the armor layer developed in experiment 2. Thistest was performed with the same mean overlying velocityused in experiments 1 and 2, U = 0.36 m s�1. Experiment 5was conducted with the flat, layered bed at the lower testvelocity of U = 0.21 m s�1.

[14] The other three tests (experiments 4, 6, and 7) wereperformed with bed forms, and had both layered andhomogeneous sediment structures. The bed forms wereformed manually to approximate a natural dune shape andhad a wavelength of 50 cm and a height of 5 cm. Inexperiments 4 and 6, the sediment structure was similar tothat used in experiment 3: a 3-cm-thick layer of graveloverlying a 12-cm-thick layer of the sand-gravel mixture,but with bed forms formed in the gravel layer instead of aflat bed surface. Experiment 4 had a mean overlyingvelocity U = 0.36 m s�1 while experiment 6 had a meanoverlying velocity U = 0.21 m s�1. Experiment 7 wasconducted with the sand-gravel mixture and with bed formsat a velocity of U = 0.21 m s�1. With no stratification,experiment 7 is used only for comparison with the two testshaving layered bed structure.[15] Because exchange rates normally increase as the

square of the overlying velocity [Packman et al., 2004],experimental results are grouped by the mean overlyingvelocity used in each solute injection. Results for experi-ments 1–4, conducted with U = 0.36 m s�1, are presented inFigure 2. Results for experiments 5–7, U = 0.21 m s�1, arepresented in Figure 3. Data are reported in terms of thechange in the dimensionless in-stream concentration overtime, Cw* (t) = Cw (t)/C0, where C0 is the initial in-streamconcentration, i.e., Cw(0).

3. Hyporheic Exchange With Flat Beds

[16] The hyporheic exchange induced by flat beds can beanalyzed by focusing on experiments 1, 2, and 3 (Figure 2)

Table 1. Summary of the Experimental Conditionsa

Experiment Mean Flow Velocity (m s�1) Water Depth (m) Discharge (m3s�1) Surface Geometry Bed Structure Bed Composition

1 0.36 0.15 0.009 flat homogeneous mixture2 0.36 0.15 0.009 flat armored mixture3 0.36 0.15 0.009 flat layered gravel plus mixture4 0.36 0.15 0.009 bed forms layered gravel plus mixture5 0.21 0.12 0.006 flat layered gravel plus mixture6 0.21 0.12 0.006 bed forms layered gravel plus mixture7 0.21 0.12 0.006 bed forms homogeneous mixture

aTwo series of experiments were performed: one at a mean overlying velocity U = 0.36 m s�1 (experiments 1–4) and the second at a mean overlyingvelocity U = 0.21 m s�1.

Figure 1. Grain size distributions of the armor layer, sand-gravel mixture, and gravel.

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and experiment 5 (Figure 3). Experiments 1, 2, and 3 hadthe same mean flow velocity, but different bed structure(Table 1). Experiment 5 had the same bed structure as inexperiment 3, but a lower mean velocity.

3.1. Thin Surface Layers (Armor Layers)

[17] The effect of a thin bed surface layer of coarsesediment (i.e., an armor layer) can be analyzed by compar-ing the results of experiments 1 and 2, which had identicalbulk sediment composition and overlying flow conditions,but differed in that the originally homogeneous bed inexperiment 1 was subjected to natural winnowing forexperiment 2. In Figure 2, it can be seen that the presenceof the armor layer causes the in-stream solute concentrationto drop much more rapidly at the beginning of the exper-

iment. In the test with the thin coarse layer, the initialconcentration drops almost instantaneously from 1 to 0.987.This reduction matches the decrease in concentration due tofull mixing of the surface water (recirculating volume =0.338 m3) with the armor layer (8 mm thick, equal to D90 ofthe mixture). This indicates that there was very rapid masstransfer into the highly permeable armor layer, resulting inthe entire armor layer becoming well mixed with theoverlying flow. The initial phase of mass transfer can beseen more clearly in Figure 4, which presents solutepenetration, m(t), for the first 100 min of experiments 1and 2. The quantity m(t) is the ratio between the masstransferred across the flow-bed interface per unit area, andthe initial concentration in the flow C0; m(t) is dimension-ally a length and can be interpreted as an equivalentpenetration depth. The initial rate of mass transfer isapproximately four times larger with the armor layer thanwith the homogeneous flat sediment bed. This very fastmixing within the armor layer results from its very smallthickness and by the high rate of turbulent diffusion in theupper layer of the bed.[18] Another important result that can be seen in Figure 2

is that the Cw(t) curves in experiments 1 and 2 becomeparallel in a short time, indicating that the rate of masstransfer to the underlying sediments is similar in armoredand homogeneous beds. The mixing within the armor layeris still very rapid, but this does not have any effect on in-stream solute concentrations after the armor layer becomeswell mixed with the overlying flow. Equivalent behaviorcan be seen in Figure 4. Thus, the armor layer does notappear to significantly reduce the boundary pressure thatdrives deeper advective pore water flow in the bed, and therate of mass transfer into the bed induced by the larger-scaleflow-boundary interaction (form drag over bed forms)matches the homogeneous case. This result shows that the

Figure 3. Dimensionless in-stream concentration Cw(t)/C0

versus time t for experiments 5–7, conducted with U =0.21 m s�1. Experiment 5, flat, layered bed; experiment 6,bed form, layered bed; experiment 7, bed form, homo-geneous bed. Details of the tests are reported in Table 1.

Figure 2. Dimensionless in-stream concentration Cw(t)/C0

versus time t for experiments 1–4, conducted with U =0.36 m s�1. Experiment 1, flat, homogeneous bed;experiment 2, flat, armored bed; experiment 3, flat, layeredbed; experiment 4, bed form, layered bed. Details of thetests are reported in Table 1.

Figure 4. The initial phase of mass transfer; solutepenetration m(t) is presented for the first 100 min ofexperiments 1 and 2. The initial rate of mass transfer isapproximately four times larger with the armor layer(experiment 2) than with the homogeneous flat sedimentbed (experiment 1). The quantity m(t) is the ratio betweenthe mass transferred across the flow-bed interface per unitarea and the initial concentration in the flow C0; m(t) isdimensionally a length and can be interpreted as anequivalent penetration depth.

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processes producing rapid mixing in the armor layer andslower advective transport over the depth of the sedimentbed are essentially independent.[19] The results presented here indicate that surface layers

like armor layers act as surface dead zones of limitedvolume, and have negligible effects on deeper hyporheicflows. This implies that models like the transient storagemodel [Bencala and Walters, 1983] and OTIS [Runkel,1998] are appropriate to treat armor layer storage. Further,this presents a physical mechanism for the multiple storagezone behavior that is sometimes posited when analyzing theresults of in-stream solute injection tests [Choi et al., 2000].However, it is important to note that the armor layerexchange observed here is so rapid that it might not beadequately resolved by in-stream observations in the field,in which case this exchange would be considered as in-stream dispersion.

3.2. Thick Surface Layers

[20] Experiments 3 and 5 show that stratified beds with acoarse upper composition and with a flat stream-subsurfaceinterface act also as surface dead zones at short time scales(Figures 2 and 3). The concentration drops to 0.958 in a fewminutes, which again matches the reduction associated withfull mixing of the surface water with the surface sedimentlayer. This is shown by the rapid decay of the in-streamconcentration, which drops approximately four times morethan with the thin layer. However, the response of thickcoarse surface layers appears quite distinct from the re-sponse of thin layers, as the rate of mass transfer becomesmuch smaller following the initial mixing through thesurface layer. The concentration curves flatten almost im-mediately, showing an inhibition of mass transfer into thefiner underlying sediments. Thus, while thick layers can stillbe modeled as surface dead zones at short times, their effecton longer-term exchange must also be considered. In otherwords, when the coarse surface layer becomes sufficientlythick, then exchange with the surface layer and underlyingsediments become coupled and this coupling must bespecifically accounted for in the stream-subsurface masstransfer model.

4. Bed Form-Induced Hyporheic Flows inStratified and Homogeneous Beds

4.1. Experimental Evidence

[21] Comparison of the results of experiments 5, 6, and 7(Figure 3) reveals that heterogeneity can have significanteffects on interfacial solute transport. The initial rate ofmass transfer is much greater in the layered systems than inthe homogenous bed, even when the layered bed is flat(experiment 5) and the homogeneous bed has bed forms(experiment 7). This reflects the fact that the hydraulicconductivity of the gravel is around 2 orders of magnitudegreater than that of the mixture, and the enhancement inexchange because of the pressure variation over bed formsis much less than that. However, the rate of mass transferdecreases much faster with the layered beds than with thehomogeneous bed because of the limiting effect of thehorizontal interface between the two sediment layers. Notethat the composition of the lower portion of the bed is

identical in experiments 5–7, and only the composition ofthe uppermost 3 cm layer differs. The potential to exchangemass across the lower interface is greatly reduced by thedamping of head in the upper sediment layer. This dampingstill affects exchange with a homogeneous bed becauseongoing mass transfer requires mixing into progressivelydeeper portions of the bed, but the rate of decrease is greaterwith a layered bed because of the deflection of the flowpaths at the interface induced by the lower conductivity ofthe underlying layer. This is the source of the effectiveanisotropy that has previously been observed to limitvertical penetration in heterogeneous beds [Salehin et al.,2004]. The overall result is that the rate of mass exchangewith the homogeneous bed becomes greater than withlayered beds at long times, as can be seen from the crossingof the curves in Figure 3.[22] This behavior of layered beds compared to homoge-

neous beds has important implications for practical applica-tions. It shows that using the assumption of a homogeneousbed can lead to underestimation of mass exchange at shorttime scales but overestimation of exchange over long timescales. It is therefore necessary to be cautious whenmaking assumptions regarding the structure of sedimentbeds in the absence of appropriate surveys of sedimentstructure over the depth relevant to the problem beingconsidered.

4.2. Hyporheic Flow Modeling for Two-Layer Beds

[23] The advective pore water flow induced within alayered sediment bed is modeled here starting from theassumption that Darcy’s law applies for the pore water flowwithin the sediments. The upper boundary condition istaken as a sinusoidal variation in dynamic pressure, follow-ing Elliott and Brooks [1997] analysis of pore water flowsunder dune-shaped bed forms. The bed is idealized as beingcomposed of an upper layer of thickness d1 and hydraulicconductivity K1 sitting on top of a lower layer of thicknessd2 and conductivity K2. The bed is considered impermeableat the bottom of the lower layer and periodic in thedownstream direction. The pressure head at the bed surfacecan be written as

h x; y ¼ 0ð Þ ¼ hm sin kxð Þ; ð1Þ

where hm is the half amplitude of the pressure at the surface,k is the wave number and x and y are the horizontal andvertical Cartesian coordinates respectively (y positiveupward).[24] In the subsurface, the hydraulic head follows

Laplace’s equation in two dimensions, r2 h = 0, validfor homogeneous porous media. The solution for theinduced subsurface head distribution can be found bysolving the Laplace equation in both layers, while matchingthe solution at the interface. Symmetry requires that thehydraulic head remains sinusoidal over any horizontalplane with the wavelength of the surface forcing. Thusthe pressure at the interface between the two layers is

h x; y ¼ �d1ð Þ ¼ h12 sin kxð Þ; ð2Þ


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where h12 is the unknown half amplitude of the dynamichead at the interface. For these boundary conditions, thehead distribution can be shown to be

h x; yð Þ ¼ hm sin kxð Þ sinh k d1 þ yð Þ½ � h12=hm sinh kyð Þsinh kd1ð Þ ;

for �d1 < y < 0 ð3Þ

h x; yð Þ ¼ h12 sin kxð Þ cosh k d1 þ d2 þ yð Þ½ cosh kd2ð Þ ;

for � d1 þ d2ð Þ < y < �d1: ð4Þ

[25] This solution agrees with prior solutions for homo-geneous, isotropic beds: the magnitude of the subsurfacehead decays exponentially with depth for a semi-infinitehomogeneous bed [Elliott and Brooks, 1997], and hyper-bolically for beds with finite depth [Packman et al.,2000]. Application of Darcy’s law in the two layers,(u,v) = �K ~rh, leads to the following velocity field:

u x; yð Þ ¼ �kK1hm cos kxð Þ sinh k d1 þ yð Þ½ � h12=hm sinh kyð Þsinh kd1ð Þ

v x; yð Þ ¼ �kK1hm sin kxð Þ cosh k d1 þ yð Þ½ � h12=hm cosh kyð Þsinh kd1ð Þ

8>><>>:for � d1 < y < 0 ð5Þ

u x; yð Þ ¼ �kK2h12 cos kxð Þ cosh k d1 þ d2 þ yð Þ½ cosh kd2ð Þ

v x; yð Þ ¼ �kK2h12 sin kxð Þ sinh k d1 þ d2 þ yð Þ½ cosh kd2ð Þ

8>><>>:for � d1 þ d2ð Þ < y < �d1: ð6Þ

Now the half amplitude of the head at the interface, h12, canbe obtained by matching the vertical velocity at theinterface:

h12 ¼ hmcsch kd1ð Þ

coth kd1ð Þ þ K2=K1 tanh kd2ð Þ : ð7Þ

Defining the nondimensional ratios h12* = h12 /hm and K* =

K1/K2, a dimensionless solution for the subsurface velocityfield can be obtained by dividing equations (5)–(7) by aconstant kK1hm:

u* ¼ u

kK1hm¼ � cos x*ð Þ sinh d1*þ y*ð Þ � h12* sinh y*ð Þ

sinh d1*ð Þ

v* ¼ v

kK1hm¼ � sin x*ð Þ cosh d1*þ y*ð Þ � h12* cosh y*ð Þ

sinh d1*ð Þ

8>><>>:for � d1* < y* < 0

ð8Þ

u* ¼ u

kK1hm¼ � h12*

K*cos x*ð Þ cosh d1*þ d2*þ y*ð Þ

cosh d2*ð Þ

v* ¼ v

kK1hm¼ � h12*

K*sin x*ð Þ sinh d1*þ d2*þ y*ð Þ

cosh d2*ð Þ

8>><>>:for � d1*þ d2*ð Þ < y* < �d1*:

ð9Þ

where x* = kx, y* = ky, d1* = kd1, d2

* = kd2.

[26] The average influx across the stream-subsurfaceinterface is given by the integral of the vertical velocity,�q = 1/l

R l=20

||v(x,y = 0)dx = � 1/lR l=20

v(x,y = 0)dx. Thevelocity is integrated between 0 and l/2 because it is onlyover this part of the bed surface that the flux occurs into thebed. The dimensionless average flux is

q* ¼ �q

kK1hm¼ 1

pcosh d1*ð Þ � h12*

sinh d1*ð Þ

� �: ð10Þ

[27] Figure 5 shows the streamlines and resulting frontpositions for a sample case where the ratio between the twohydraulic conductivities is K* = 10 and the dimensionless-thicknesses of the two layers are d1* = 0.5 and d2* = 0.5.Front positions are drawn for nondimensional time scalest* = 0.5, 1, 2, 4 and 16. Time is normalized using theconductivity K1 and porosity q1 of the upper layer, t* = tk2

K1hm/q1. In the lower layer, the streamlines are less densecompared to the upper layer because of the decrease inpore water velocity and hence in water flux induced by thelower permeability of the second layer. Streamlines in theupper layer of the bed are also constrained by the presenceof the underlying less permeable sediments, and havedifferent patterns than would be found in a homogeneousbed having a depth of either d1 or d1 + d2.

4.3. Residence Time Distributions and Mass Exchange

[28] The effects of surface-subsurface exchange on down-stream solute transport must be evaluated in terms of totalsolute mass transfer and storage within the bed [Bencalaand Walters, 1983; Elliott and Brooks, 1997]. The cumula-tive mass exchange can be determined by solving a convo-lution integral involving the surface water concentrationhistory, the interfacial flux, and the distribution of soluteresidence time within the bed. The complementary cumu-lative distribution function of the residence times, R(t*), isdefined as the fraction of solute entering the bed in at t* = 0

Figure 5. Streamlines and front positions within ahomogeneous layered bed. Results are drawn for a samplecase where the ratio between the two hydraulic conductiv-ities is K* = 10 and the dimensionless thicknesses of the twolayers are d1* = 0.5 and d2* = 0.5. The longitudinaldimensionless distance scales with the bed form wave-length, x* = kx. Front positions are drawn for nondimen-sional time scales, t* = 0.5, 1, 2, 4, and 16.

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still trapped in the bed at time t* [Elliott and Brooks, 1997].The R function was evaluated with a numerical particletracking technique based on the known velocity field(equations (8) and (9)) for five different cases having aninfinite lower layer (d2* ! 1), and d1* = 1, 2, 3, 4 and 5.These five cases were also analyzed with two differentratios between the upper and the lower hydraulic conduc-tivity, K* = 10 and K* = 100. The residence time function Ris plotted in Figures 6a and 6b for K* = 10 and K* = 100,respectively. A distinctive behavior is visible in all cases.The slope of R, which is negative by definition, is very closeto zero at times corresponding to transport across theinterface between the higher- and lower-permeability layers,forming a well-defined deflection point. The time at whichthe deflection occurs increases as the depth of the upperlayer increases, as expected. It is interesting to note that thecurves describing R for any two-layer thicknesses alwayscross (Figure 6). A thicker layer always produces highervalues of R at short time scales and smaller values of R atlong time scales. With a higher conductivity ratio betweenthe upper and lower layers the variation of R becomes moresignificant, as can be seen by comparing the results for K* =10 and K* = 100 in Figures 6a and 6b.[29] The dimensionless form of the cumulative mass

exchange between the overlying flow and the bed iscommonly expressed as m* = 2pkm/q1, and can be com-

puted in all systems for a constant in-stream concentrationCw with the integral [Elliott and Brooks, 1997]

m* ¼ 2pq*Z t*

0

Cw* t*� t*ð ÞR t*ð Þdt*: ð11Þ

Mass exchanges have been evaluated for the five casesanalyzed in the previous section for the case of masstransfer to an initially clean sediment bed. The initialconditions at t = 0 are: the in-stream concentration Cw(0) =C0 and the subsurface concentration Cs(0) = 0 for y < 0. Theresults presented in Figures 7a and 7b show that the rate ofmass transfer is controlled by the hydraulic conductivity ofthe upper layer at very short times, while at sufficiently longtimes it is controlled by the conductivity of the lower layer;that is, it becomes completely independent of the thicknessof the upper layer. At intermediate times the mass exchangeis highly dependent on both the upper layer thickness andthe conductivity ratio.

4.4. Comparison Between Model and ExperimentalResults

[30] The exchanged mass can be evaluated from the in-stream concentration history using the following relation-ship for the mass balance of solute in the system:

m* ¼ dw* 1� Cw*ð Þ: ð12Þ

Figure 6. Complementary cumulative distribution func-tion of the residence time within the bed, R(t*), plotted for(a) K* = 10 and for (b) K* = 100 in log-log scale. Thecurves are obtained from particle tracking applied to theflow field described by equations (8) and (9).

Figure 7. Dimensionless penetrated mass, m*(t*), withinthe bed plotted for (a) K* = 10 and for (b) K* = 100. Thecurves are calculated from the curves presented in Figure 6using equation (11).


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The parameter dw* is the dimensionless effective waterdepth, equal to 2pkd 0

w /q1, where d0w is the ratio between the

total recirculating volume of water and the bed surface area.[31] The hydraulic conductivity used to model hyporheic

exchange was evaluated with the following empirical rela-tion [McCarthy, 2002]:

K ¼ 0:35D215; ð13Þ

where the hydraulic conductivity, K, has units of cm s�1 andD15 is in cm. The hydraulic conductivity was evaluated tobe K = 3 � 10�4 m s�1 for the sand-gravel mixture and K =0.02 m s�1 for gravel only.[32] The half amplitude hm of the sinusoidal dynamic

head over bed forms was evaluated with Fehlman’s [1985]equation:

hm ¼ 0:28U2

2g

H=d

0:34

3=8

; H=d � 0:34

H=d

0:34

3=2

; H=d > 0:34;

8>>><>>>:

ð14Þ

where H is the bed form height (trough to crest). Inexperiments 6 and 7, the half amplitude of the dynamic headpredicted by equation (13) is hm = 0.85 mm, while inexperiment 4 it is hm = 2.51 mm.[33] The observed and predicted mass transfer for experi-

ments 4 and 6 are compared in Figure 8. A linear time scaleis used in Figure 8a, while a semilog scale is used in Figure8b for direct comparison with Figure 7. The scatter of dataat short times is to be attributed to the difficulty of theadvective model to represent in detail the complex inter-actions affecting early transfer at the flow-bed interface.These tests both had bed forms and layered sedimentstructure, with K* = 67, d1* = 0.38 and d2* = 1.51, butdifferent overlying velocities. The collapse of data from thetwo tests onto a single unique curve shows that the scalingsused to nondimensionalize mass transfer and time arecorrect. Thus, the scaling of the exchange rate with U2 stillholds even with the heterogeneous bed. The first phase ofthe exchange is fast and governed by the 3-cm-deep upperlayer. The inflection point of the curve corresponds to thetime at which the upper permeable layer becomes saturatedwith solute. After this time the exchange process is gov-erned by mass transfer into the lower and less permeablelayer. Therefore it can be seen that the initial phase of masstransfer is regulated by the characteristics of the surficialsediment layer, while tracer exchange rate over longer timescales depends on the characteristics of the underlying lesspermeable material.

5. Conclusions and Implications

[34] The effects of armoring on interfacial transport wereexamined by conducting a solute injection experiment witha flat homogeneous bed and then repeating the soluteinjection following the formation of an armor layer on thebed surface. There was rapid transport through the thinarmor layer, but the subsequent rate of mass transfer wasessentially the same as that observed with the homogenousbed. Thus, turbulent diffusion into the coarse armor layerappeared to be essentially independent from transport in theunderlying porous medium. This was likely because of thethinness of the armor layer (approximately one grain diam-eter) and the very great degree of difference between thepermeability of the armor layer and the bulk sediments (2orders of magnitude). These conditions are normally foundin armor layers when the sediments contain a wide range ofgrain sizes. Thus the net effect of armoring will often be togreatly increase the rate of interfacial exchange and trans-port in the armor layer without significantly affectingexchange with deeper portions of the bed.[35] A mathematical model was developed for the pore

water flow induced in a layered sediment bed by thedynamic pressure variation over dune-shaped bed forms.For the case of a coarser surface layer over a finer and lesspermeable layer, the flow paths in the upper layer arecompressed by the presence of the lower layer and masstransfer into the lower layer is suppressed by the presence ofthe overlying sediments. A comparison of exchange withhomogeneous and layered beds shows that the presence of acoarse near-surface layer increases the interfacial flux andinitial rate of mass transfer, but decreases vertical solutepenetration into the underlying finer sediments and thecorresponding net mass transfer at longer time scales.

Figure 8. Observed and predicted dimensionless pene-trated mass, m*(t*), for experiments 4 and 6 plotted in(a) linear scale and (b) semilog scale. In both experiments,the bed was made of two horizontal layers with artificial bedforms at the flow-bed interface. The difference between thetwo tests was the mean flow velocity and the water depth, asreported in Table 1.

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[36] These results emphasize the importance of sedimentstructure on surface-subsurface flow coupling and soluteexchange. The relatively high rates of transport in armorlayers are not likely to be resolved by the tracer methodscommonly used to assess exchange in the field. Where theselayers are present, which will normally be the case insystems with wide size distributions and frequent sedimenttransport, tracer injection methods will tend to substantiallyunderestimate interfacial fluxes, by interpreting fast termretention in the armor layers as a surface process rather thanan hyporheic one. Because the surface of the sediment bedis often characterized by intense biological activity, theinterfacial flux is expected to be important to a variety ofecological and biogeochemical processes, such as microbialutilization of highly labile dissolved nitrogen and carbonderived from the overlying water column. In cases like thiswhere the local-scale interfacial transport is important,particular attention must be paid to resolving the near-subsurface sedimentary environment. Further, the presenceof deeper layers of differing permeability, which are oftenfound in rivers because of episodic sediment transportevents [Leopold et al., 1964], causes rates of mass transferinto the bed to vary over time as the solute penetrates to thedifferent layers of sediments. When such systems areanalyzed using a model that assumes homogeneous subsur-face structure, the instantaneous exchange rate will beoverestimated and underestimated at different stages of theoverall exchange process. The results and theoretical anal-ysis presented here support the recommendation madepreviously by the authors and by others [Harvey andWagner, 2000; Zaramella et al., 2003] that it is importantto combine observations of tracer concentrations in surfacewaters with a characterization of the subsurface structureand direct observations of tracer penetration to particulardepths of interest.

Notation

Cs subsurface concentration.C0 initial tracer concentration in the stream.Cw concentration of tracer in the stream.Cw* dimensionless tracer concentration, equal to Cw/C0.d thickness of the sediment layer

(d1 for the upper layer and d2 for the lower layer).d* dimensionless thickness of the sediment layer

(d1* for the upper layer and d2* for the lower layer),equal to kd.

D10 grain size for which 10% by weight of thesediment is finer.

D15 grain size for which 15% by weight of thesediment is finer.

D50 mean grain size.D90 grain size for which 90% by weight of the

sediment is finer.d 0w ratio between the total recirculating volume of

water and the bed surface area.dw* dimensionless effective water depth,

equal to 2pkd 0w /q1.

H bed form height (trough to crest).h12 half amplitude of the dynamic head at the

interface between the upper and lower layer.

h12* dimensionless half amplitude of the dynamichead at the interface between the upper andlower layer, equal to h12/hm.

hm half amplitude of the sinusoidal distribution ofpressure on the bed surface.

k bed form wave number, equal to 2p/l.K hydraulic conductivity (K1 for the upper layer and

K2 for the lower layer).m equivalent penetration depth (units of depth),

equal to total exchanged mass per unit bed areadivided by the reference concentration in thestream, C0.

m* dimensionless equivalent penetration depth,equal to 2pkm/q1.

�q average influx across the stream-subsurfaceinterface.

q* dimensionless average influx across thestream-subsurface interface, equal to �qkK1hm.

R complementary cumulative distributionfunction of the residence times within the bed.

t time.t* dimensionless time, equal to tk2K1hm/q1U mean flow velocity.u horizontal Darcy pumping velocity.

u* dimensionless horizontal Darcy pumping velocity,equal to u/(kK1hm).

v vertical Darcy pumping velocity.v* dimensionless vertical Darcy pumping velocity,

equal to v/(kK1hm).x horizontal coordinate.

x* dimensionless horizontal coordinate, equal to kx.y vertical coordinate.

y* dimensionless vertical coordinate, equal to ky.l bed form wavelength.q porosity of bed sediment

(q1 for the upper layer and q2 for the lower layer).t* dummy variable for dimensionless time.

[37] Acknowledgments. The participation of the Italian team hasbeen funded by a University of Padua project (title: Measurements andmodeling of hyporheic flows in rivers) granted to the first author. Experi-ments were conducted at Northwestern University with the support of U.S.National Science Foundation grant BES-0196368 to A.I.P. The authorsthank Danilo Giuliani for his assistance in the experimental activity.D. Giuliani’s visit to Northwestern was funded by a fellowship from theA. Gini Foundation.

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��A. Bottacin-Busolin, A. Marion, and M. Zaramella, Department of

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Hyporheic flows in stratified beds