click here full article scaling hyporheic exchange …...biogeochemical reaction rates in streams...

Scaling hyporheic exchange and its influence on

biogeochemical reactions in aquatic ecosystems

Ben L. O’Connor1 and Judson W. Harvey1

Received 14 May 2008; revised 8 September 2008; accepted 25 September 2008; published 17 December 2008.

[1] Hyporheic exchange and biogeochemical reactions are difficult to quantify because ofthe range in fluid-flow and sediment conditions inherent to streams, wetlands, andnearshore marine ecosystems. Field measurements of biogeochemical reactions in aquaticsystems are impeded by the difficulty of measuring hyporheic flow simultaneously withchemical gradients in sediments. Simplified models of hyporheic exchange have beendeveloped using Darcy’s law generated by flow and bed topography at the sediment-waterinterface. However, many modes of transport are potentially involved (moleculardiffusion, bioturbation, advection, shear, bed mobility, and turbulence) with even simplemodels being difficult to apply in complex natural systems characterized by variablesediment sizes and irregular bed geometries. In this study, we synthesize information frompublished hyporheic exchange investigations to develop a scaling relationship forestimating mass transfer in near-surface sediments across a range in fluid-flow andsediment conditions. Net hyporheic exchange was quantified using an effective diffusioncoefficient (De) that integrates all of the various transport processes that occursimultaneously in sediments, and dimensional analysis was used to scale De to shear stressvelocity, roughness height, and permeability that describe fluid-flow and sedimentcharacteristics. We demonstrated the value of the derived scaling relationship by using it toquantify dissolved oxygen (DO) uptake rates on the basis of DO profiles in sediments andcompared them to independent flux measurements. The results support a broad applicationof the De scaling relationship for quantifying coupled hyporheic exchange andbiogeochemical reaction rates in streams and other aquatic ecosystems characterized bycomplex fluid-flow and sediment conditions.

Citation: O’Connor, B. L., and J. W. Harvey (2008), Scaling hyporheic exchange and its influence on biogeochemical reactions in

aquatic ecosystems, Water Resour. Res., 44, W12423, doi:10.1029/2008WR007160.

1. Introduction

[2] Near-surface sediments in aquatic ecosystems arecritical in terms of many biogeochemical reactions such asnitrification and denitrification, as well as gross primaryproductivity and community respiration [e.g., Mulholland etal., 2001; Peterson et al., 2001]. The sediment-waterinterface is an assemblage of mineral particles, organicand biogenic material, biological communities, and voidspaces producing a complex three-dimensional environmentthat serves as a boundary for fluxes of water, solutes, andparticulate matter to occur. Fluid-flow and sediment con-ditions control the transfer of reactive solutes, particles, andbiota across the sediment-water interface to depths ofseveral centimeters, and the gradients of these environmen-tal factors control biogeochemical processes in sediments[Huettel et al., 2003]. Aquatic ecosystems contain a widerange of fluid-flow and sediment characteristics that gener-ate complex transport conditions composed of diffusive,advective, shear, and turbulent processes. These simulta-neous transport mechanisms produce log-scale variability in

hyporheic residence times and spatial variability in theenvironmental factors that control biogeochemical process-es, which is why hyporheic exchange is considered to exertprimary control over the net ecosystem response of biogeo-chemical processes in sediments [Findlay, 1995].[3] Measuring biogeochemical reaction rates is difficult

because of the small-scale over which many microbial-mediated reactions occur. Small-volume pore water sam-plers produce solute concentration gradients on the scale ofcentimeters in stream beds [Harvey and Fuller, 1998]. Forfiner spatial-scale resolution, microsensor technology pro-vides direct measurements of environmental variables innear-surface sediments [e.g., Revsbech et al., 2005]. Con-centration profiles across the sediment-water interface areused to infer biogeochemical reaction rates using modelswith various terms that simulate transport within sediments,which include molecular diffusion, bioturbation, advection,dispersion, and groundwater mixing [Bouldin, 1968;Boudreau, 1984; Berg et al., 1998; Harvey and Fuller,1998]. These models are typically empirical and are noteasily transferable to other aquatic systems, or even the samesystem undergoing changes in fluid-flow and sedimentconditions.[4] Interpreting reaction rates from concentration profiles

requires quantifying the transport mechanism within sedi-

1U.S. Geological Survey, Reston, Virginia, USA.

This paper is not subject to U.S. copyright.Published in 2008 by the American Geophysical Union.

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ments. There has been substantial work in this area forcoastal flat and marine ecosystems where the sediments areoften muddy or sandy, and fairly uniform in particle sizes.For fine particle-sized sediments found in these systems, aswell as many lake sediments, the dominant transport mech-anism is molecular diffusion. The molecular diffusioncoefficient in the sediments (D0

m) has been quantified usingthe molecular diffusion coefficient (Dm) modified to accom-modate the tortuosity between sediment grains, as well asfactors such as bioturbation and dispersion [Boudreau,1984]. The parameterization of the D0

m in this mannerintroduces empirical variables that can be difficult to quan-tify, but its use in interpreting concentration profiles hasprovided insights on the depth distribution of biogeochemicalreactions in sediments [e.g., Revsbech et al., 1981; Nielsenet al., 1990; Thamdrup et al., 1994].[5] Research in the area of hyporheic exchange processes

provides an alternative description of transport in sedimentswith higher permeabilities. Savant et al. [1987], Thibodeauxand Boyle [1987], Harvey and Bencala [1993], and Huettelet al. [1996] demonstrated advective transport throughsediments using solute tracers and hydraulic measurementsthat showed distinct flow patterns entering sediments andreturning to the water column a short distance away. Thisadvective transport mechanism is referred to as ‘‘pumping’’where fluid-flow above the sediments generates a pressuredistribution at the sediment-water interface driving flow intoand out of the permeable sediments. The most frequentlyused model based on advective pumping [Elliott andBrooks, 1997a] assumes a sinusoidal pressure distributionat the sediment-water interface, and has been used toanalyze the results of flume investigations examining hypo-rheic exchange for conditions of fairly uniform sand sedi-ments and regularly shaped bed forms [e.g., Elliott andBrooks, 1997b]. More recent studies have examined thepumping mechanism experimentally under conditions ofvarying bed form geometries [Marion et al., 2002], variablehydraulic conductivity [Salehin et al., 2004], and withingravel sediments [Tonina and Buffington, 2007] whereturbulence is likely to govern transport in the near-surfacesediments.[6] While the pumping model focuses on transport occur-

ring below the sediment-water interface, research on trans-port in porous media has examined the momentum balanceof the overlying flow and inside the sediments. Thisanalysis requires a model that can produce continuity invelocity, shear, and pressure across the sediment-waterinterface. Connecting the momentum equation in the over-lying flow with that in the porous sediment bed has beendone by incorporating a conceptual boundary layer, referredto as the Brinkman layer in the near-surface sediments,which produces additional flow resistance terms and conti-nuity in flow variables at the sediment-water interface[Dade et al., 2001]. This model of hyporheic exchange isreferred to as the ‘‘slip flow’’ model. The boundary condi-tion that connects the overlying flow and the poroussediment bed introduces variables relating to the velocitydistribution within the Brinkman layer that need to berelated to the overlying flow and sediment character-istics empirically. Beavers and Joseph [1967] quantified theslip velocity (us) at the sediment-water interface as beingproportional to the velocity gradient in the overlying flow

with pore-averaged velocities decaying rapidly within theBrinkman layer to the Darcy velocity. The decay of velocitywithin the Brinkman layer is mediated through additionalresistance terms introducing an effective viscosity (ne)within the sediment, which was found not to vary substan-tially with depth in the sediments [Ruff and Gelhar, 1972].More recently, ne was interpreted as an effective diffusioncoefficient (De), which was formulated using the slip flowmodel of Ruff and Gelhar [1972] using velocity profilemeasurements to quantify us [Fries, 2007].[7] Natural aquatic systems tend to have spatially and

temporally variable fluid-flow and geomorphic conditionsthat generate heterogeneity in the factors controlling hypo-rheic exchange. Parameterizing the variables in the pumpingand slip flow models has only been applied to simplifiedfluid-flow, sediment, and bed geometry conditions, whichmakes application to natural systems difficult. A recentnumerical modeling investigation of hyporheic exchangeused computational fluid dynamic simulations to quantifythe pressure distribution over irregularly shaped bed formscoupled to flow governed by Darcy’s law in the sediments[Cardenas and Wilson, 2007]. This type of analysis hasimproved upon the Elliott and Brooks [1997a] pumpingmodel that assumed simplified conditions at the sediment-water interface (i.e., sinusoidal pressure distribution and aflat bed). While this numerical modeling approach is reveal-ing about the controlling processes of hyporheic exchange, itis not an easily applied tool that can be used to interpretempirical biogeochemical profiles. Another issue with thismodel is that it defines a sharp boundary between the fluid-flow in the water column and advective Darcy flow in thesediments, which does not account for the penetration ofturbulent eddies into the sediment pore space for beds withlarger particle sizes [Nagaoka and Ohgaki, 1990].[8] The reality for complex natural systems is that a

combination of processes including molecular diffusion,advective pumping, shear-driven flow, and turbulence pen-etration control surface and hyporheic water mixing. Thenet hyporheic exchange via these fundamental processescan be represented most simply as an effective diffusionmechanism, where De is quantified by the measurement of aconservative tracer between the water and sediments.Richardson and Parr [1988] and Packman and Salehin[2003] have related tracer-derived De values to variablesdescribing fluid-flow and sediment characteristics in attemptto develop predictive scaling equations for hyporheic ex-change. However, the data sets used in these investigationswere limited to fairly narrow ranges in fluid-flow andsediment conditions. There is also some uncertainty be-tween these studies with regards to which flow and sedi-ment variables best characterize transport. For example, it isnot clear whether fluid-flow conditions should be quantifiedusing the average velocity (U) or the shear stress velocity(u*), or whether sediments should be characterized bypermeability (K), hydraulic conductivity (Kc), or somevariable describing the particle size distribution.[9] The motivation for this study was to quantify hypo-

rheic exchange in complex natural systems where a combi-nation of transport mechanisms control fluxes, and toevaluate conditions for which simple advection-based mod-els are applicable. Additionally, we wanted to develop amodel for hyporheic exchange that could be used to

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interpret reaction rates from biogeochemical profiles innatural systems (i.e., streams where hyporheic exchange ishighly variable, and in advective nearshore marine systems)where the traditionally used molecular diffusion-basedmodels are not always valid. In this investigation, wesynthesized previous laboratory conservative tracer experi-ments and used dimensional analysis to scale De over awide range of fluid-flow and sediment characteristics. Thescaling relationship for De was used to interpret dissolvedoxygen (DO) profiles in sediments from previous flume,open channel, and field investigations. The modeled DOreaction rates were compared to measured flux values toevaluate the reliability of De model. The use of a scalingrelationship to quantify hyporheic exchange sacrifices amechanistic understanding of the individual transportprocesses involved, in order to gain a more simple androbust transport model that can be applied over a widerange in fluid-flow and sediment conditions. The mainbenefit for this more simplified approach is in its applica-tion for examining empirical data addressing coupledbiogeochemical-hyporheic exchange processes in near-surface sediments.

2. Background Information

[10] In this section we describe the basics of threecommonly used hyporheic exchange models: effective dif-fusion, pumping, and slip flow. The effective diffusionmodel lumps together physical transport mechanisms intoa single De coefficient, while the pumping and slip flowmodels primarily focus on advective transport within thesediments. There are alternative methods for quantifyinghyporheic exchange such as a nonlocal exchange models[e.g., Marinelli et al., 1998] and examination of sedimenttracer breakthrough curves [e.g., Reimers et al., 2004],which were not examined in this study.

2.1. Effective Diffusion

[11] Biogeochemical and transport processes across thesediment-water interface are three-dimensional in nature.However, for point measurements of biogeochemical gra-dients, the interpretation of process rates are often viewedfrom a one-dimensional perspective and multiple profiles withinformation on sediment topography can be used to capturespatial variability [e.g., Røy et al., 2002]. Viewing transport asdiffusion provides a direct means to evaluate biogeochemicalgradients using a mass conservation formulation

@C

@t¼ @

@yDe

@C

@y

� �� R ð1Þ

where C is concentration, t is time, and R is the net uptakerate (sum of production and uptake reactions). This type ofmodel has been used to quantify biogeochemical processessuch as photosynthesis and respiration rates by comparingDO profiles in sediments over time between light and darkcycles [e.g., Revsbech et al., 1986]. For many biogeochem-ical processes of interest, a steady state form of equation (1)is suitable for interpretation of reaction rates and anumerical fitting algorithm PROFILE has been developedto evaluate microsensor profiles [Berg et al., 1998].[12] Molecular diffusion has often been assumed to be the

driving transport mechanism when examining biogeochem-

ical gradients in sediments [Reimers et al., 2001]. Themolecular diffusion coefficient (Dm) must take into accountthe diffusion around sediment particles, which has beenexamined theoretically [Berner, 1980], as well as empiri-cally by relating the sediment tortuosity to measured elec-trical resistivity and sediment porosity (q) [Archie, 1942].Several studies have used these methods to derive expres-sions between tortuosity and q [e.g., Boudreau, 1996],which can be used to generalize the expression for themolecular diffusion in sediments as

De ¼ bDm ¼ D0m ð2Þ

where b represents the empirical expression for tortuosity asa function of q, and in this study we used

b ¼ 1

1þ 3 1þ qð Þ ð3Þ

as described by Iversen and Jørgensen [1993]. In additionto molecular diffusion, the effects of bioturbation, turbulentdiffusion, and dispersion have been included into theexpression of effective diffusion

De ¼ b Dm þ Dbð Þ þ Dd ð4Þ

where the biodiffusivity (Db) and dispersion coefficient (Dd)must be obtained empirically or modeled [Berg et al., 1998;Guss, 1998].[13] A measured value for De over a sediment bed can be

obtained directly by use of a conservative tracer. Consider aclosed system (i.e., a recirculating flume) where the tracerconcentration in the sediments (C) is initially zero, and thetracer concentration in the overlying water is Cw = Co (well-mixed system initially). Hyporheic exchange will drive atracer into the sediment pore water resulting in a decrease oftracer concentration in the overlying water over time.Modeling hyporheic exchange as an effective diffusionprocess, the flux of tracer to the sediments is governed byFick’s second law

@C

@t¼ De

@2C

@y2ð5Þ

and with the well-mixed system approximation, theboundary conditions for equation (5) are: the concentrationat the sediment-water interface is equal to Co, and @C/@y = 0deep in the sediments. A solution for equation (5) can beobtained for C assuming a semi-infinite domain and usingLaplace transforms resulting in

C ¼ Co 1� erfy

2ffiffiffiffiffiffiffiDet

p� ��

ð6Þ

as described by Crank [1975]. The flux of solute from theoverlying water to the sediments is governed by Fick’s firstlaw

J ¼ �De

dC

dy

��y¼0

ð7Þ

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and taking the derivative of equation (6) at the sediment-water interface and inserting it into equation (7) gives anexpression of the solute flux

J ¼ Co

ffiffiffiffiffiffiDe

pt

rð8Þ

which is different from the work by Packman and Salehin[2003]; Packman et al. [2004] by a factor of

ffiffiffip

pcaused by

an error in grouping terms in their derivation.[14] The net solute flux can also be obtained by a mass

conservation expression

J ¼ �Vw

As

dCw

dtð9Þ

where Vw is the volume of the overlying water in therecirculating system, As is the surface area of the sedimentbed, and Cw(t) is the solute concentration in the overlyingwater as a function of time. Equating equations (8) and (9),rearranging terms, and integrating results in

Cw ¼ Co 1� 2As

Vw

ffiffiffiffiffiffiffiDet

p

r !ð10Þ

which describes a linear decrease in solute concentrationwith respect to t1/2. Experimental tracer data typicallyfollows this linear behavior for initial time periods, but thedecrease in Cw eventually decays over long time periods[Marion et al., 2002]. The initial slope of a plot of Cw

versus t1/2 can be used to quantify De as

De ¼ffiffiffip

p

2

Vw

As

dC*

d t1=2ð Þ

� �2

ð11Þ

where C* = Cw/Co is the normalized solute concentration inthe overlying water. The error in the expression for J(equation (8)) used by Packman and Salehin [2003] andPackman et al. [2004] propagates through in theirformulation for De, which differs from equation (11) by afactor of p.

2.2. Pumping Model

[15] The pumping model presented by Elliott and Brooks[1997a] quantifies hyporheic exchange by examining thesediment domain of the problem. The continuity equationfor steady flow in sediments, assuming isotropic sediments(constant hydraulic conductivity, Kc) follows Laplace’sequation

r2h ¼ 0 ð12Þ

where h is the piezometric head (h = p/rg, where p ispressure and r is the density of water). The pressure fieldwithin the sediments is calculated assuming a sinusoidalhead distribution at the sediment-water interface,

h y ¼ 0ð Þ ¼ hm sin2plx

� �ð13Þ

where l is the bed form wavelength, x is the streamwisecoordinate, and hm is the half amplitude of the piezometrichead variation at the sediment-water interface, which hasbeen empirically derived as

hm ¼ 0:28U2

2g

D=H

0:34

� �g

ð14Þ

where U is the average velocity, D is the bed form height,H is the water depth, and g is an empirical constant (g = 3/8forD/H < 0.34, and g = 3/2 forD/H 0.34) [Elliott, 1990].The resulting solution for the h distribution is

h ¼ hm sin2plx

� �e�

2pl y ð15Þ

where y is the vertical coordinate. Darcy’s law is used tocalculate the seepage velocity field (v, a volume average ofthe interstitial velocity) in the sediments and producesstreamlines entering and exiting the sediments according to

v ¼ �Kg

nrh ð16Þ

where K is the sediment permeability, g is the accelerationdue to gravity, and n is the kinematic viscosity.[16] The net hyporheic exchange to the sediments is

evaluated by assuming that the sediment bed is initiallyfree of solute and the pumping model then tracks theaccumulation of solute mass inside the bed over time. Anaverage residence time function R(x) is introduced, whichrepresents the fraction of solute that has remained in thesediment bed for an elapsed time x (for a more detaileddescription of R(x), see Elliott and Brooks [1997a,Appendix]). This accounts for the fact that solute is enteringand exiting the sediments with a distribution of associatedresidence times. The flux of a solute into the sediment bednormalized by the solute concentration in the overlyingwater, C, is

q xð Þ ¼ v n if v n 0

0 if v n < 0

�ð17Þ

where n is the unit vector normal to the sediment-waterinterface. The average normalized flux, q, is calculated bytaking the average of the absolute value of v n over theentire surface of the sediment bed. The accumulation ofsolute mass per unit area of the bed can be estimated as theconvolution integral of the residence time function

M tð Þ ¼ q

Z 1

x¼0

R xð ÞC t � xð Þdx ð18Þ

which tracks the mass accumulation inside the sediments.To get the net hyporheic exchange, information on C(t) inthe overlying water is measured and equation (18) iscoupled to a mass conservation equation in the overlyingwater.[17] The pumping model can be formulated to represent

mass transfer by effective diffusion in the same mannerused to derive equation (11) where the flux at the

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interface according to Fick’s second law was equated withthe net solute flux derived from a mass balance on therecirculating water in the flume. The analogous expressionto equation (10) describes the solute mass accumulation inthe sediments

M 0

q¼ 2

ffiffiffiffiffiffiffiDet

p

rð19Þ

where M 0 = M/Co and M 0/q is known as the effective depthof solute penetration. A plot of M 0/q versus t1/2 has a linearincrease initially and the slope can be used to quantify De

as:

De ¼ffiffiffip

p

2

d M 0=qð Þd t1=2ð Þ

� �2

ð20Þ

[18] During the initial time period of a tracer experiment,Elliott and Brooks [1997a] showed that the convolutionintegral (equation (18)) can be approximated by

M 0

q¼ 3:5

2p

ffiffiffiffiffiffiffiffiffiffiffiffiKchmt

q

rð21Þ

which is valid up to a point where the bed starts to becomesaturated with tracer. Combining equations (19) and (21)results in

De ¼1

pq3:5

4

� �2

Kchm ð22Þ

which represents an estimation of De using the pumpingmodel with a sinusoidal pressure gradient at the sediment-water interface based only on sediment and fluid-flowvariables.

2.3. Slip Flow Model

[19] The slip flow model depicts the transition in physicsfrom the overlying fluid-flow to the porous media flow inthe sediments. This means connecting the Navier-Stokesequation for the overlying flow and within the sedimentpores at the microscale to Darcy’s law in the poroussediment bed at the macroscale. Mathematically, this is achallenge as the Navier-Stokes equation is a second-order,nonlinear equation and in the overlying flow is expressed as

@u

@tþ u ru ¼ �rp

rþ nr2uþ Fb ð23Þ

where u is the velocity vector field in the overlying flow, pis pressure, and Fb is the net body force (or gravity term infree surface problems). Darcy’s law, which was empiricallydeduced but can also be derived from the Navier-Stokesequations through averaging over large pore volumes, is afirst-order, linear equation

rp

r¼ � n

Kv ð24Þ

which is the same as equation (16) written in terms of pinstead of h. Some researchers have treated this problem

using a single domain (overlying water and sediment bed)[e.g., Basu and Khalili, 1999]. Zhou and Mendoza [1993]started with equation (23) and used ensemble averaging toderive equations describing the macroscale transport withinthe sediment bed.[20] A more basic approach to this two-domain problem

involves only examining the sediment bed portion andincluding additional force terms to equation (24) in thefashion of Beavers and Joseph [1967] and Ruff and Gelhar[1972]. This requires the use of the conceptual Brinkmanboundary layer just below the sediment-water interface. Thecharacteristic length scale for the Brinkman layer is

ffiffiffiffiK

p

[Beavers and Joseph, 1967] over which the additional flowresistance is generated by viscous shear stress and nonlinearform drag. The extended Darcy equation is formed with theaddition of these terms as:

rp

r¼ � n

Kvþ ner2v� CDffiffiffiffi

Kp v2 ð25Þ

The third term in equation (25) represents the Brinkmanterm and introduces an effective viscosity (ne). The fourthterm in equation (25) represents the Forchheimer termwhere CD is a dimensionless form drag coefficient, which isa property of the porous sediment bed [Venkataraman andRama Mohan Rao, 1998]. Ruff and Gelhar [1972] solvedfor the v field using equation (25) with an empiricallyderived value for CD and evaluating ne as both a constantand varying with depth. This solution for v also provided anexpression of the slip velocity, us, tangential to thesediment-water interface (the slip velocity is on the waterside of the interface and is not a volume average of theinterstitial velocity). Their resulting expression for usassuming a constant ne was

us

u*

!3

þ 3

2CDReK

us

u*

!2

¼ 3ReK

2CD

nne

ð26Þ

where ReK = u*ffiffiffiffiK

p/n is the Reynolds number characterizing

flow within the Brinkman layer. Fries [2007] evaluated neas an effective diffusion (ne � De) in equation (26), whichwas rearranged as

De ¼3nRe2K

2CDReKu3sþ þ 3u2sþ

ð27Þ

where the dimensionless slip velocity, us+ = us/u*, wasestimated by fitting measured velocity profiles to theReichardt velocity profile.

2.4. Previous Studies Examining Effective DiffusionModel

[21] Hyporheic exchange is a result of a combination ofprocesses, which is why several transport models have beendeveloped to simulate and analyze empirical data. The threemost commonly used transport models are the effectivediffusion, pumping, and slip flow models presented previ-ously. The pumping and slip flow model are mechanistic inthat they quantify the physics of advective transport bymaking simplifications to the physical conditions of theproblem. While the effective diffusion model uses the


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mathematics of diffusive transport, it is an empirical modelas the transport parameter De represents the net transport byseveral processes and must be obtained from experimentaldata. Tracer tests provide a means to quantify De directly,but the only way to obtain De, a priori, is through thedevelopment of predictive correlations with respect to fluid-flow and sediment characteristics controlling hyporheicexchange.[22] Scaling De to physical conditions governing trans-

port requires thorough, multivariate analysis of the control-ling variables. Previous scaling attempts have focused onrelating De to a characteristic velocity and length scalerelevant to the perceived transport mechanism controllinghyporheic exchange. Richardson and Parr [1988] per-formed tracer experiments in a flume with a flat bed ofglass spheres. The mass transfer was assumed to be dom-inated by shear-induced flow (slip flow model) at theinterface and they found that u* and K best explained thevariability in experimental De values according to De =6.59 10�5Dm(u*

ffiffiffiffiK

p/n)2. Packman and Salehin [2003]

used dimensional analysis and available literature data toscale De to the Reynolds number (Re = UH/n) of theoverlying flow, where U is the average velocity and His the stream depth. This scaling resulted in power lawrelationship of De � Re2 with varying intercepts for studiesusing different types of sediment particle sizes and bedform configurations. The velocity head term (U2/2g) inequation (14) of the pumping model was interpreted asthe cause of the observed Re2 scaling. In a separate studyinvolving a flat bed with gravel sized sediments, De wasscaled to the particle Reynolds number (Rep = Udp/n)[Packman et al., 2004], which also demonstrated Rep

2

scaling but with a better collapse in data from studies usingdifferent particle sizes and bed form configurations.[23] These previous studies that examined the scaling of

De have provided insight on some of the controllingvariables affecting hyporheic exchange. However, thesescaling relationships have not received much attentionbecause they were developed using fairly limited data sets,and they lack validation in their application in predictinghyporheic exchange. More recently, Marion and Zaramella[2005] used a small set of published flume studies anddemonstrated that the effective diffusion model accurately

characterized hyporheic exchange for a broad, intermediaterange of time periods relating to the exchange process. Theyalso concluded that a depth-independent value of De issufficient for the effective diffusion model in predictinghyporheic exchange.

3. Methods

3.1. Hyporheic Exchange Data Sets

[24] We compiled data from various flume tracer studiesin order to examine the effects of varying fluid-flow andsediment conditions on transport in permeable sediments.The data sets are summarized in Table 1. The first task wasto get a consistent set of measured De values from the tracerexperiments, as well as fluid-flow and sediment conditions.The procedures used to calculate De vary somewhat be-tween studies, depending upon how the tracer experimentwas conducted. Nagaoka and Ohgaki [1990] used a salttracer in the overlying water and conductivity probes at fourdepths and obtained De using equation (5). This type ofmeasurement was possible because of the large, uniformparticles used to make the bed (Table 1). Richardson andParr [1988] and Lai et al. [1994] used a sediment bedsaturated with a dye tracer of concentration Co,s andmeasured the mass accumulation of tracer in the overlyingwater column as a function of time. For this type ofexperiment, De was calculated using

De ¼ffiffiffip

p

2Co;s

dMw

d t1=2ð Þ

� �2

ð28Þ

where Mw is the cumulative mass of dye transferred fromthe sediment to the water. The rest of the studies used atracer, initially in the water column only, and tracked eitherthe decrease of solute concentration in the water column(equation (10)), or the solute mass accumulation in the bed(equation (19)). For all studies, we used the plots of eithermass or concentration versus the square root of time tocompute the needed slopes in equations (11), (20), and (28)for calculating experimental De values.[25] Reported variables relating to fluid-flow and sedi-

ment conditions included U, bed geometry, particle sizedistribution information, q, and either K or hydraulic con-

Table 1. Summary of Sediment and Fluid-Flow Conditions for Hyporheic Exchange Data Sets

Study dg (mm) K (10�6 cm2) q D (cm) l (cm) u* (cm s�1) U (cm s�1) H (cm)

Richardsona 0.1–3.0 0.17–71 0.36–0.40 0 0 0.3–1.3 3.7–22.9 0.6–1.9Nagaokab 19.0–40.8 500–2300 0.24 0 0 1.1–4.3 8.9–42.8 3.2–7.0Laic 0.5–3.2 2.3–19 0.36–0.38 0 0 0.2–0.6 7.4–15.4 0.5–2.0Elliottd 0.1–0.5 0.08–1.1 0.30–0.33 1.1–2.5 9–30 1.3–2.4 8.6–13.2 3.1–6.5Marione 0.85 5.0 0.38 0–3.5 0–120 1.7–1.8 22.0–28.0 10.9–12.3Packmanf 4.8 150 0.38 0–3.7 0–32 1.1–3.2 9.0–36.1 11.3–20.5Variousg 0.5 0.68–1.8 0.29–0.38 0.8–1.9 15–70 0.5–1.7 12.0–23.7 7.1–12.7Toninah 9.8–10.8 51 0.34 3.6–12.0 515–560 3.8–5.5 28.2–46.0 3.9–10.4

aRichardson and Parr [1988], flat bed with glass sphere sediments.bNagaoka and Ohgaki [1990], flat bed with glass sphere sediments.cLai et al. [1994], flat bed with glass sphere sediments.dElliott and Brooks [1997b], bed forms with sand sediments.eMarion et al. [2002], flat bed and bed forms with sand sediments.fPackman et al. [2004], flat bed and bed forms with gravel sediments.gPackman et al. [2000], Packman and MacKay [2003], Ren and Packman [2004], and Rehg et al. [2005], flat bed and bed forms with sand sediments.hTonina and Buffington [2007], bed forms with gravel sediments.

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ductivity (Kc). Shear stress velocity was either reported orobtained by using the normal flow estimation (u* =

ffiffiffiffiffiffiffiffiffigHS

p,

where S is the slope of the bed). Particle size distributioninformation was used to estimate the geometric meanparticle size (dg) and the particle size such that 90% ofthe particles are finer (d90). The studies used a variety ofsediment bed geometries that included flat beds, as well asthose containing natural and regular bed forms characterizedby the bed form wavelength (l) and amplitude (D). Hypo-rheic exchange is affected by the bed geometry, so it isdifficult to compare values between flat beds and those withbed forms. We used the definition of roughness height (ks)described by van Rijn [1984]

ks ¼ 3d90 þ 1:1D 1� e�25D=l �

ð29Þ

which provided a single variable for examining the effectsof particle size and bed geometry on hyporheic exchangeprocesses. Values of K or Kc were reported in most studies(K = Kcn/g) except for Nagaoka and Ohgaki [1990], whichonly gave information on particle size. For the Nagaokastudy, K was estimated using the Kozeny-Carmen equation

K ¼ 5:6 10�3 q3

1� qð Þ2d2g ð30Þ

which is one of the most widely used equations forpredicting permeability for sediments with fairly uniformsize distributions [Freeze and Cherry, 1979].

3.2. Dimensional Analysis

[26] Buckingham’s P theorem was used to generatedimensionless groupings of the controlling variables thatinvolved three steps: (1) listing the minimum number ofvariables needed to describe hyporheic exchange, (2) gen-erating dimensionless groupings of the controlling varia-bles, and (3) using the compiled hyporheic exchange dataset to determine a power law scaling relationship for De.The variables chosen for dimensional analysis represent therelevant fluid-flow and sediment conditions controllinghyporheic exchange according to both the pumping andslip flow models

De ¼ f D0m;U ; u*; ks;

ffiffiffiffiK

p; n

�ð31Þ

where D0m is defined by equation (2) and includes q; ks is

defined by equation (29) and includes the variables d90, D,and l describing the sediment bed. Equation (31) containsseven variables that are composed of two dimensions(L and T), which resulted in the possibility of 7 � 2 = 5dimensionless groupings. The five dimensionless groupswere simplified to produce four groupings of dimensionlessnumbers known to affect hyporheic exchange

De

D0m

� U

u*

!;

u*ks

n

� �;

u*

ffiffiffiffiK

p

D0m

!ð32Þ

where U/u* = Cz is the Chezy resistance coefficient, u*ks/n =Re* is the shear Reynolds number, and u*

ffiffiffiffiK

p/D0

m = PeKis the permeability based Peclet number. Rewriting

equation (32) with these dimensionless numbers in theform of a power law scaling results in

De

D0m

¼ aCaz Re

b

*PecK ð33Þ

where a is a dimensionless constant. Individual correlationsbetween De/D

0m and the dimensionless numbers (Cz, Re*, and

PeK) were used to determine the scaling exponents a, b, and c.

3.3. Dissolved Oxygen Data Sets

[27] We examined previous studies measuring DO pro-files in sediments and compiled available information onfluid-flow and sediment conditions (see auxiliary material,Tables S1–S4).1 Many of the DO profiles were measured ina stagnant or mixed chamber systems where estimating u*was not possible. Also, it was uncommon to list detailedparticle size distribution and in some cases, neither K or Kc

was reported.[28] Data from four investigations was used to assess the

scaling relationship for quantifying De in interpreting uptakeand production rates from DO concentration profiles. Theexperimental conditions for these four investigations in-clude recirculating flumes [Guss, 1998; O’Connor andHondzo, 2008b], a large-scale flow-through channel[O’Connor and Hondzo, 2008a], and field measurementstaken within the south fork of the Iowa River located incentral Iowa (B. L. O’Connor et al., unpublished data,2007). The relevant sediment and fluid-flow conditionsfor these studies are listed in Table 2. The DO concentrationprofiles from the Iowa River were performed in a similarfashion as those described by O’Connor and Hondzo[2008a] using a Unisense OX-N needle sensor (tip diameterof 1.1 mm) with a PA2000 picoammeter (Unisense, Aarhus,Denmark) sampling at a frequency of 1 Hz. A Rickly pointgage (Rickly Hydrological, Columbus, OH) with 1-mmprecision was attached to a tripod and used to traverse theneedle sensor. Information regarding the Iowa River sam-pling site is included in the auxiliary material.[29] The depth zonation of DO production and uptake

rates were estimated from the concentration profiles withinthe sediments using the PROFILE model [Berg et al., 1998].This model fits concentration data to equation (1) describingone-dimensional, steady state diffusion and includes a sta-tistical algorithm to minimize the number of zones needed tomatch the measured concentration profile. The DO concen-tration at the sediment-water interface and a zero flux at adepth below the DO penetration depth (d, where C = 0) wereused as boundary conditions for the model. The PROFILEanalysis was performed twice for each profile, first usingmolecular diffusion (D0

m) described by equation (2), andthen using effective diffusion (De) described by the scalingrelationship developed in section 3.2. The net DO flux (JDO)to the sediments was calculated using a sediment-sideestimate from the production/uptake zones according to

JDO ¼Z d

i¼0

Rdy ð34Þ

1Auxiliary materials are available in the HTML. doi:10.1029/2008WR007160.


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where R is the net uptake rate, so a negative value indicatesDO production. This estimate of JDO was compared tomeasured DO fluxes (Jm) that were obtained using water-side flux estimation methods as described in each study, andthe method described by O’Connor and Hondzo [2008a]was used for the Iowa River data.

4. Results

4.1. Scaling Effective Diffusion Using HyporheicExchange Data Sets

[30] The experimental De values derived from the hypo-rheic tracer data were compared to individual variables

describing fluid-flow and sediment conditions. The compar-isons exhibited positive correlations (Figure 1), and allcorrelations were statistically significant (p < 0.05) exceptfor the correlation with stream depth, H (Figure 1c). Nosingle variable serves as a good predictor for De asexpected. These individual correlations were used as guide-lines in selecting variables for dimensional analysis with thenotion that a variable with greater explanatory powerindividually, will facilitate a better collapse of the experi-mental De data in a multivariate dimensionless expression.[31] The variables listed in equation (31) for scaling De to

fluid-flow and sediment characteristics is the final result ofthe dimensional analysis. Several iterations of the three

Table 2. Summary of Sediment and Fluid-Flow Conditions for Dissolved Oxygen Profile Data Sets

Study Experiment dg (mm) K (10�6 cm2) q u* (cm s�1) U (cm s�1) H (cm)

Gussa (flume) E-2 0.4 7.0b 0.50 0.21 2.8 3.0E-1 0.4 7.0b 0.50 0.41 10.5 3.0D-2 0.4 7.0b 0.50 0.61 23.3 3.0

O’Connorc (channel) OC-1 6.8 2.2 0.35 0.15 1.0 8.0OC-2 6.8 1.6 0.35 0.41 5.0 16.0

O’Connord (flume) 2 0.4 2.0 0.60 0.23 4.0 7.55 0.4 2.0 0.60 0.44 8.2 7.56 0.4 2.0 0.60 0.56 11.2 7.510 0.4 2.0 0.60 0.46 9.0 7.511 0.4 2.0 0.60 0.53 11.0 7.5

Iowa River (field) MP2 25.0e 0.7 0.32 4.80 58.3 52.0MP4 40.0e 0.04 0.32 2.70 27.2 34.5

aGuss [1998].bEstimated using equation (30).cO’Connor and Hondzo [2008a].dO’Connor and Hondzo [2008b].ed90 values are reported.

Figure 1. Correlations between effective diffusion values, De, and variables describing fluid-flowand sediment conditions: (a) average velocity, U; (b) shear stress velocity, u*; (c) flow depth, H;(d) permeability, K; (e) roughness height, ks; and (f) geometric mean particle size, dg.

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steps described in section 3.2 were performed, and thevariables listed in equation (31) provided the best collapsein the experimental De data. Correlations between thedimensionless groups of equation (32) that were used todetermine the scaling exponents a, b, and c in equation (33)are shown in Figure 2. Positive correlations were observedfor De/D

0m with Re* and PeK with similar slopes (�2) and

R2 = 0.89 and 0.87, respectively. The correlation betweenDe/D

0m and Cz depicted a negative correlation with a high

degree of scatter in the data (R2 = 0.32) and a slope ofapproximately �5. This weak correlation with Cz is partiallycaused by the fact that Cz values only range between 2 and40 whereas the values for De/D

0m range over six orders of

magnitude.[32] The Cz term was dropped from equation (33) because

its weak correlation with De/D0m did not provide explana-

tory power. A plot of De/D0m versus Re*

7/4PeK9/4 had a slope

of 0.55, which was combined with the scaling exponentsb = 7/4 and c = 9/4 to give the final De scaling equation as

De

D0m

¼5 10�4Re

*Pe

6=5K for Re

*Pe

6=5K 2000

1 for Re*Pe

6=5K < 2000:

8><>: ð35Þ

where the inverse of the scaling constant (a = 5 10�4)provided a threshold value in transport conditions (Re*PeK

6/5 =2000) below which transport was governed by moleculardiffusion resulting in De/D

0m = 1 (Figure 3). All but one of

the experimental De data points were above the thresholdcondition for molecular diffusion. The scaling relationshipdescribed by equation (35) had a slope of one and explained95% of the variance with 95% confidence intervals of theslope between 0.93 to 1.02, and a between 2.4 10�4 and1 10�3.[33] The experimental tracer De data was compared with

the estimation expressions using the pumping model, slipflow model, and the scaling relationship described byequations (22), (27), and (35), respectively (Figure 4). Thepumping model estimation is a function of hm, whichimplies the presence of bed forms. For flat bed experiments,D was assumed to be dg in equation (14). The estimation

expression of the slip flow model requires a normalized slipvelocity, us

+. None of the hyporheic exchange investigationsprovided depth profiles of the streamwise velocity, so themethod used by Fries [2007] could not be used. Instead,we used the expression of Engelund [1970], us

+ = 1.9 +2.5 ln(H/k0s) where k0s = 2.5dg is the equivalent sandroughness height, which resulted in us values that rangedbetween 3–15 u*. The pumping model estimation matchedexperimental De values for the experiments of Elliottand Brooks [1997b], Marion et al. [2002], and Packmanet al. [2004], but overestimated De for the other studies(Figure 4a). The slip flow model underestimated experi-mental De values for all the studies examined.

4.2. Effective Diffusion Model for InterpretingDissolved Oxygen Uptake

[34] The DO concentration profiles listed in Table 2 wereanalyzed using the effective diffusion model (De, equation (35))and the traditional molecular diffusion model (D0

m,equation (2)). The PROFILE model was used to determinethe depth distribution of the net DO uptake rate (R inequation (1)). Experiment D-2 from Guss [1998] depicteda different pattern in the distribution of R between themolecular and effective diffusion models (De/D

0m = 3.5),

yet both models matched the DO concentration profile. Formolecular diffusion, R = 118 g m�2 d�1 from the sediment-water interface down to 1.3 mm, whereas the effectivediffusion model had R = 0 down to 0.6 mm and R =950 g m�2 d�1 between 0.6 and 1.3 mm (Figure 5a).Experiment 6 from O’Connor and Hondzo [2008b] dem-onstrated similar patterns in the distribution of R betweenmolecular and effective diffusion models but with varyingrates. From the sediment-water interface down to 0.8 mm,R increased from 555 to 1040 g m�2 d�1 (Figure 5b).[35] The DO profiles from the experiments by Guss

[1998] and O’Connor and Hondzo [2008b] were measuredin flumes that prevented light penetration, which inhibitedphotosynthesis. For the experiments of [O’Connor andHondzo, 2008a] and the Iowa River, the DO concentrationprofiles were performed in sediments with algal presence,so DO production was a factor. The molecular and effectivediffusion models produced similar patterns in the distribu-

Figure 2. Correlations between the dimensionless effective diffusion, De/D0m, and the dimensionless

groupings from equation (33): (a) Chezy coefficient, Cz = U/u*; (b) the shear Reynolds number, Re* =u*ks/n; and (c) the permeability Peclet number, PeK = u*

ffiffiffiffiK

p/n. The scaling coefficients a, b, and c are the

slopes of the individual plots.


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tion of DO uptake and production, but with large differ-ences between rates (Figure 6). Experiment OC-2 from[O’Connor and Hondzo, 2008a] demonstrated DO produc-tion from the sediment-water interface down to 0.85 mmwith R = �184 and �1.2 104 g m�2 d�1 for the molecularand effective diffusion models, respectively. Below the DO

production zone, DO uptake rates were slightly greaterbetween R = 204 and 1.3 104 g m�2 d�1 down to1.7 mm in the sediments. The DO uptake rates thendecreased to R = 5 and 318 g m�2 d�1 down to the DOpenetration depth (d = 4.25 mm, Figures 6a and 6b). Asimilar pattern of a sharp transition between DO production

Figure 3. Resulting scaling relationship of the effective diffusion coefficient, De, from dimensionalanalysis. The change from molecular based diffusion described by equation (2) to effective diffusiondescribed by equation (35) occurs at the threshold value of Re*PeK

6/5 = 2000.

Figure 4. Comparison between experimentally derived effective diffusion values, De, and estimatesbased on the (a) pumping model, (b) slip flow model, and (c) scaling relationship described by equations(22), (27), and (35), respectively.

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and uptake was observed for experiment MP2 for the IowaRiver study. However, the maximum DO production waslocated 6 mm below the sediment-water interface with R =�4.1 and �4.8 104 g m�2 d�1 for the molecular andeffective diffusion models. The maximum DO uptake wasslightly lower than the maximum DO production with R =3.8 and 4.4 104 g m�2 d�1 from 12 to 18 mm within thesediments, followed by R = 0.8 and 1.6 104 g m�2 d�1

down to d = 24 mm (Figures 6c and 6d).[36] The differences in DO uptake and production between

the effective and molecular diffusion models (Figures 5and 6) demonstrates a range in De/D

0m values between 1 and

3750 for all the studies examined. The net DO flux for themolecular and effective diffusion models, estimated by theintegration of the DO uptake profiles (JDO, equation (34)),was compared to an independent measurement of the flux(Jm) for each experiment (Table 3), with the ratio of the fluxes

shown in Figure 7. For the experiments of Guss [1998], boththe effective and molecular diffusion models underestimatedJm, except for the effective diffusion model in experimentD-2, which closely matched Jm. For experiments 5, 6, 10, and11 ofO’Connor and Hondzo [2008b] the measured Jm valueswere between the molecular and effective diffusion modelfluxes, yet the effective diffusion model provided betterflux estimates based on percent difference. In general, forDe/D

0m < 3, the molecular and effective diffusion models

bounded the measured flux while for De/D0m 3, the

effective diffusion model closely match the measured fluxand the molecular model underestimated the flux by as muchas a factor of 105 for De/D

0m = 3750.

5. Discussion

[37] Studying biogeochemical reactions in aquatic sedi-ments is challenging because of the complexity of theindividual physical, chemical, and biological components.The complexity is one reason why numerical modelingapproaches are becoming a popular tool for addressingcoupled biogeochemical-hyporheic processes [e.g., Meys-man et al., 2007]. However, measurements of hyporheicexchange and biogeochemical gradients need to be contin-ued because simple kinetic expressions used for modelingstudies are not directly transferable to systems with complexsediments where multiple controlling factors such as redoxpotential, organic carbon, microbiological gradients, andcoupled biogeochemical processes all interact with transportprocesses to control reaction rates. Quantifying biogeo-chemical reaction rates on the basis of measured concen-tration profiles provides a method for examining the depthdistribution of the process in relation to gradients of con-trolling environmental factors in the sediments. This em-pirical approach requires lots of data collection to capturethe inherent spatial and temporal variability in controllingfactors. The data collected could be used for quantifyingbiogeochemical reaction rates in a similar manner that wasapplied to quantifying hyporheic exchange in this study,through the development of scaling relationships betweenthe reaction rates and controlling environmental factors,for which some progress has been made [e.g., O’Connor etal., 2006]. However, further work is needed in identifyingall relevant variables affecting the individual biogeochem-ical reactions and determining the appropriate methods fortheir quantification.[38] Measurements of biogeochemical gradients of inter-

est are relatively simple to obtain (e.g., microsensors andpore water samplers), and the De scaling relationshipdeveloped in this study can be used to infer reaction ratesacross a wide range in sediment and fluid-flow conditions.This tool will allow researchers to address several of the keyquestions regarding the complex interactions among con-trolling factors, such as the debate over organic matterquality versus transport control on carbon cycling in sedi-ments [e.g., Rothman and Forney, 2007; Boudreau et al.,2008], as well as extend research on coupled biogeochem-ical processes, like DO uptake and denitrification [e.g.,Arnon et al., 2007; O’Connor and Hondzo, 2008b] insystems with more complex transport conditions. Theexamples of DO uptake rates based on scaled De transportvalues presented in this study were all under steady stateconditions; however, this type of analysis could be extended

Figure 5. Depth distribution of measured dissolvedoxygen (DO) concentrations, as well as modeled concentra-tions and uptake rates, R, for (a) experiment D-2 from Guss[1998] and (b) experiment 6 from O’Connor and Hondzo[2008b]. The DO concentration and uptake profiles weremodeled using both molecular diffusion, D0

m, and effectivediffusion, De.


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to address nonsteady processes as well. For example,photosynthesis rates are controlled by solar radiation thatvaries over the course of the day. Typically, DO concentra-tion profiles can be obtained on the order of minutes, somultiple profiles can be measured to address the nonsteadynature of photosynthesis. The value of De used to interpret

these profiles will be steady over time scales where theindividual sediment and fluid-flow variables defining De aresteady.[39] Accurately quantifying the transport in aquatic sedi-

ments is essential to this approach for obtaining biogeo-chemical reaction rates. As was shown in this study, the DO

Figure 6. Depth distribution of measured dissolved oxygen (DO) concentrations, as well as modeledconcentrations and uptake rates, R, using both the molecular (D0

m) and effective (De) diffusion models for(a and b) experiment OC-2 from O’Connor and Hondzo [2008a] and (c and d) site MP2 from the IowaRiver study. Note the change in the x axis scale between the molecular and effective models.

Table 3. Comparison of Molecular (D0m) and Effective (De) Diffusion Models With the Measured Dissolved

Oxygen Flux (Jm)

Study Experiment De/D0m JDO (D0

m) (g m�2 d�1) JDO (De) (g m�2 d�1) Jm (g m�2 d�1)

Guss (flume) E-2 1 0.114 0.114 0.658a

E-1 1.5 0.156 0.281 0.666a

D-2 3.5 0.185 0.794 0.733a

O’Connor (channel) OC-1 3 0.028 0.230 0.290b

OC-2 25 0.030 1.830 1.858b

O’Connor (flume) 2 1 0.340 0.340 0.353c

5 1.2 0.338 0.541 0.441c

6 2.1 0.473 0.864 0.577c

10 1.2 0.558 0.892 0.839c

11 1.6 0.744 1.172 0.996c

Iowa River (field) MP2 3750 3.96 10�5 2.904 2.905b

MP4 304 2.21 10�3 2.098 2.982b

aCalculated from dispersion model described by Guss [1998].bCalculated from similarity model described by O’Connor and Hondzo [2008a].cCalculated using a mass balance described by O’Connor and Hondzo [2008b].

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uptake rates followed the same depth distribution patternbetween molecular and effective diffusion models, but withsubstantial differences in the DO uptake rates (Figures 5 and6). The transport was verified by comparing DO fluxestimates using equation (34) with measured flux values.An additional check on the accuracy of the scaled De

transport term could be obtained by examining multicom-ponent profiles (e.g., combined DO and nitrate concentra-tion profiles) where the De value should be similar (only asmall difference in D0

m values affecting the value of De) andthe interpreted uptake rates can be compared to measuredsurface fluxes for each component. Additionally, the De

scaling relationship developed in this study was based onflume tracer experiments, which can be improved upon byfurther advancements in measuring transport in aquaticsediments, such as the use of ultrasonic velocity profiler(UVP) technology [Manes et al., 2006].

5.1. Flow and Sediment Variables Important toHyporheic Exchange

[40] The first step in developing a scaling relationship forDe was to determine the characteristic velocity and lengthscales that best predict hyporheic exchange over a widerange in conditions. To the best of our knowledge, thecombined hyporheic exchange data set used in this study isthe most comprehensive data set assembled to date in termsof the ranges in individual sediment and fluid-flow variables(Table 1). Figure 1 shows the correlation between measuredDe values and individual variables describing fluid-flow andsediment conditions. Previous scaling attempts using rela-tively limited data sets by Packman and Salehin [2003] andPackman et al. [2004] used the variables U, dg, and H toscale De, whereas Richardson and Parr [1988] used the

variables u* and K. The compiled data set used in this studyshowed that u* was slightly better than U at predicting De,and that ks, K, and dg had stronger correlations with De thanH (Figure 1).[41] The final De scaling relationship (equation (35))

includes the transport term Re*PeK6/5 depicted by the x axis

of Figure 3. The 6/5 exponent on PeK is a result of fittingexperimental De data using dimensional analysis, but is veryclose to a value of one. Using this approximation De �Re*PeK implies that

De � u2*; ks d90;D;lð Þ; n�1;ffiffiffiffiK

p;D0

m Dm; qð Þ ð36Þ

which has similar features to the previous scaling attemptsby Richardson and Parr [1988], Packman and Salehin[2003], and Packman et al. [2004]. However, the scalingrelationship developed in this study correctly compareseffective to molecular diffusion in the sediments byincluding the effects of tortuosity, as well as including ameans to collapse data between flat bed and bed formgeometries. The use of ks and D0

m introduces empiricisminto the dimensional analysis through equations (2) and(29). However, these terms also allow for the relativelysimple scaling relationship (equation (35)) to include theeffects of q, D, l, and d90 that are variables known to affecthyporheic exchange. In the end, the resulting De scalingrelationship explained 95% of the variance in the compre-hensive data set, over a wide range in fluid-flow andsediment conditions. This high degree of predictive powerindicates that the key transport mechanisms were repre-sented by the variables chosen and the formulation of the De

scaling relationship.

5.2. Transport Mechanisms Involved With HyporheicExchange

[42] Quantifying hyporheic exchange in natural systemsis complicated by the many transport mechanism involved.The pumping and slip flow model assume advective trans-port governed by Darcy’s law in the sediment. The differ-ence between the two models is the driving force at thesediment-water interface, where the pumping model focuseson pressure (head distributions) and the slip flow model onshear (velocity distributions). However, both models do notaccount for molecular diffusion and turbulent transport thatlimits their applicability across a broad range of fluid-flowand sediment conditions.[43] Figure 4 compares the pumping, slip flow, and

effective diffusion models in their prediction of measuredDe values. The pumping model estimate of De assumes asinusoidal head distribution at the sediment-water interface,which is why it worked well for experiments with regularlyspaced bed forms. The experiments of Richardson and Parr[1988] and Lai et al. [1994] were performed using flat bedswith particle sizes <3 mm. The pumping model overesti-mated De for these experiments because it is unlikely thatthe smooth beds generated substantial pressure gradients atthe interface, so the flux was dominated by momentum fluxinduced by shear. For the experiments of Nagaoka andOhgaki [1990] and Tonina and Buffington [2007], theoverestimation of De was caused by the large particle sizesand irregular bed forms (large l relative to D values) thatcreated a nonsinusoidal head distribution at the interface. Its

Figure 7. Comparison of the PROFILE model estimate ofDO flux, JDO, using equation (34) to the measured flux, Jm,using molecular diffusion (D0

m, open symbols) and effectivediffusion (De, closed symbols).


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not clear why the pumping model overestimated De for thestudies of the ‘‘Various’’ investigations as they had similarexperimental characteristics as those given by Elliott andBrooks [1997b], with exception of a lower range in u*values (Table 1).[44] The slip flow model underestimated measured De

values because the Engelund [1970] expression for us+

generated slip velocities at the sediment-water interface thatwere 3–15 u*, which are very unlikely. This analysis isstill valuable in assessing the role of the slip flow model onhyporheic exchange. The flat bed experiments of Richard-son and Parr [1988] and Lai et al. [1994] closely matchedmeasured De values if us

+ was reduced by a factor of 3, butall other studies were still underestimated. The slip flowmodel assumes that transport is dominated by shear forcesacross the interface, which occurs over a smooth boundary.

A simple calculation of the dimensionless hydraulic rough-ness height (ks

+ = dgu*/n) resulted in hydraulically smoothbed conditions (ks

+ < 5 [Nezu and Nakagawa, 1993]) for theexperiments of Richardson and Parr [1988] and Lai et al.[1994], as well as a few of the experimental runs with bedforms in the studies of Elliott and Brooks [1997b] and of the‘‘Various’’ investigations. This suggests that shear-inducedhyporheic exchange is relevant for hydraulically smooth flatbeds, but for conditions with larger particle sizes and bedforms there are pressures forces generated at the sediment-water interface that dominate over shear-induced momen-tum flux.[45] Experimental conditions that produced good agree-

ment between modeled and measured De values in Figure 4can be used to assess ranges in fluid-flow and sedimentconditions that are relevant to molecular diffusion, shear-induced flow, advective pumping, and turbulence penetra-tion. Figure 8a depicts the qualitative ranges in sedimentand fluid-flow conditions (represented as Re*PeK

6/5) for themechanisms involved with hyporheic exchange. Moleculardiffusion dominates transport between 100 < Re*PeK

6/5 <1.4 104, followed by overlapping regions consisting ofshear-induced flow between 103 < Re*PeK

6/5 < 2 106,advective pumping between 3 105 < Re*PeK

6/5 < 1.6 109, and turbulence penetration for Re*PeK

6/5 > 1.3 107.This analysis shows that for a given set of transportconditions, there are multiple exchange processes involved,which agrees with the conclusion of Packman and Salehin[2003] for why using an effective diffusion model isappropriate.

5.3. Hyporheic Scaling Relationship for FieldInvestigations

[46] There are two main factors that need to be consideredfor applying the developed scaling relationship for assessinghyporheic exchange in the field: groundwater dischargeand scale. For the flume experiments used to develop thescaling relationship of De, solute flux was controlled byphysical forcing from the overlying flow. However, innatural streams, physical forcing on hyporheic exchangecan also occur on the sediment-side of the domain fromdischarging groundwater [Woessner, 2000]. The effect ofdischarging groundwater on hyporheic exchange can beviewed as the ‘‘underflow’’ component in the advectivepumping model, which was shown to reduce the net massexchange [Elliott and Brooks, 1997a]. In this study, weexamined the effect of groundwater discharge in relation tothe scaling relationship for De using nested piezometer datafrom tracer studies conducted at Pinal Creek (located incentral Arizona), which was characterized by unusuallyhigh groundwater discharge rates for a small stream [Harveyand Fuller, 1998; Fuller and Harvey, 2000]. The tracer-based values of De measured in Pinal Creek fall slightlybelow the line of the scaling relationship, as shown inFigure 8b. This is the result of groundwater dischargegenerating a physical limitation of hyporheic exchange,which is not represented in the variables incorporated intothe scaling relationship.[47] The flume investigations used to develop the pump-

ing, slip flow, and scaling relationship models of hyporheicexchange used a wide range in fluid-flow and sedimentcharacteristics, but for each experiment the transport varia-bles were fairly uniform. Thus the application of these

Figure 8. (a) Ranges in transport conditions (fluid-flowand sediment characteristics represented by Re*PeK

6/5) forthe various mechanisms involved with hyporheic exchange.(b) Comparison between effective diffusion scaling relation-ship based on the combined flume data sets and field tracerdata from Pinal Creek, AZ [Harvey and Fuller, 1998; Fullerand Harvey, 2000]. Pinal Creek had large groundwaterinputs that limited hyporheic exchange.

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hyporheic exchange models should be considered relevantat the patch scale. We use the term patch scale to indicatethat the hyporheic exchange models are relevant for scalesover which variables describing transport such as u*, ks, andK can be considered characteristic. Natural streams consistof geomorphic features such as pools, riffles, and meandersthat generate heterogeneity in variables describing bedgeometry, fluid-flow, and sediment characteristics. In orderto apply hyporheic exchange models to field investigations,the issue of scale in the controlling fluid-flow and sedimentvariables must be considered.[48] The dimensional analysis used in this study gener-

ated the dimensionless terms Cz, Re*, and PeK, which arequalitatively known to affect hyporheic exchange as theydescribe flow resistance, as well as advective and diffusivetransport processes. The weak correlation between De andCz (R

2 = 0.32) was surprising, but also indicates the scaledependency for hyporheic exchange models. For reach-scale tracer measurements of transient storage, the Darcy-Weisbach friction factor (f = 8gHS/U2) has been shown tobe a good predictor for reach-averaged hyporheic exchange[e.g., Harvey et al., 2003; Zarnetske et al., 2007]. Mathe-matically f and Cz are similar (Cz =

ffiffiffiffiffiffiffiffi8=f

p) if uniform flow

is assumed in the channel. So, while flow resistancequantified as a friction factor appears to explain hyporheicexchange at the reach scale, it does not factor into thetransport mechanisms incorporated into the patch-scalehyporheic models.

5.4. Interpretation of Dissolved Oxygen Uptake Rates

[49] Our main purpose for quantifying hyporheic ex-change as an effective diffusion process was to interpretbiogeochemical reaction rates from concentration profilesin systems with complex fluid-flow and sediment condi-tions. DO uptake and production occurs in near-surfacesediments and is controlled by oxidation, respiration,decomposition, and photosynthesis, for which the complexinteraction of these processes can be represented as zero-order kinetics like the R term in equation (1) [DiToro,2001]. Examining the differences between the molecularand effective diffusion models for interpreting DO con-centration profiles demonstrated large differences in netuptake rates (Figures 5 and 6). With exception to the DOprofile of experiment D-2 of Guss [1998], the depthzonation of DO uptake and production did not changebetween the molecular and effective diffusion models, onlythe magnitude of the volumetric uptake and productionrates changed.[50] The experiments of Guss [1998] and O’Connor and

Hondzo [2008b] were done in shaded flumes, so photosyn-thesis was inhibited. The values of R were largest directlybelow the sediment-water interface except for the effectivediffusion model of experiment D-2 of Guss [1998], whichdepicts transport dominating over DO uptake down to 0.6mm (Figure 5). This interpretation implies that the DO fluxto the sediments exceeded the intrinsic DO uptake capabil-ity of the sediment, which is attributable to the transitionbetween water- to sediment-side control of the net DO flux.The experiments of O’Connor and Hondzo [2008a] andthose performed in the Iowa River occurred in sedimentswith algae, so photosynthesis was a factor on the DOdynamics. DO production occurred directly below thesediment-water interface and there was a sharp transition

between the maximum DO production and DO uptakezones (Figure 6). This suggests that the DO production inthe surface layer sediments stimulates DO uptake in thesediments below, which is consistent with the findings ofalgal-induced respiration hot spots described by Jones et al.[1995].[51] The accuracy of the molecular and effective diffusion

models interpretation of the sediment DO concentrationprofiles was checked by calculating the net DO flux usingequation (34) and comparing it to a measured flux (Jm). Themethods used to calculate Jm varied between experiments,but all of them used a water-side method of determiningflux, which have been traditionally favored over sediment-side approaches like equation (34) because of the difficultyof quantifying the transport in porous sediments [Gundersenand Jørgensen, 1990]. For De/D

0m 3, the effective diffu-

sion model closely matched the measured flux while themolecular diffusion model greatly underestimated themeasured flux (Figure 7). Matching water-side flux valuesgives confidence in the ability of the effective diffusionmodel for interpreting biogeochemical profiles. Whileseveral studies have examined the importance of thezonation of uptake and production rates on biogeochemicalreactions under molecular diffusion transport [e.g., Revsbechet al., 2005], the results of this study demonstrated that thelocation of the reaction zones did not change betweenmolecular and effective diffusion models, but the reactionrates were increased by several orders of magnitude.Therefore, using the proposed De scaling relationshipfor interpreting the zonation of biogeochemical reactionscan provide an accurate depiction of the net biogeochem-ical reaction rates in ecosystems with complex transportmechanisms.

6. Summary

[52] Hyporheic exchange is a result of many transportmechanisms over a range in fluid-flow and sedimentconditions. The scaling relationship for De developed inthis study is an empirical model that retains features of thedeveloped theories regarding hyporheic exchange. Thiseffective diffusion representation of hyporheic exchange isdirectly compatible with models used to interpret biogeo-chemical reaction rates on the basis of chemical profilesacross the sediment-water interface. Comparisons betweenmodeled and measured DO fluxes from a variety of systemspresented in this study show good agreement over a widerange in fluid-flow and sediment conditions. Applyinghyporheic exchange models to complex natural systemsneeds to address factors relating to variability in the con-trolling fluid-flow and sediment characteristics, as well asthe potential for resistance to hyporheic exchange within thesediments (e.g., groundwater discharge). The scaling ap-proach used in this study can be easily adapted to includegroundwater discharge and other factors affecting hyporheicexchange once experimental evidence becomes availablethat can be incorporated into dimensional analysis. Thisempirical approach of examining hyporheic exchange andbiogeochemical reactions serves as a methodology that canbe used to study the coupling of these processes in complexnatural systems. Future work addressing theoretical advance-ments, on either the hyporheic exchange or the biogeochem-


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istry, can benefit from using a scaling approach as a means toexamine the key variables and processes involved.

Notation

a, b, c scaling exponents.As surface area of sediment bed (L2).C solute concentration (ML�3).

CD Forchheimer form drag coefficient.Cz Chezy resistance coefficient (U/u*).Co initial solute concentration (ML�3).Co,s initial saturated bed concentration (ML�3).Cw water solute concentration (ML�3).C* normalized concentration (C/Co).Db biodiffusivity (L2 T�1).Dd dispersion (L2 T�1).De effective diffusion (L2 T�1).Dm molecular diffusion (L2 T�1).D0m sediment molecular diffusion (bDm) (L

2 T�1).dg geometric mean particle size (L).d90 particle size for which 90% of sediment is finer

(L).Fb net body force (MLT�2).f Darcy-Weisbach friction factor.H stream depth (L).h piezometric head (L).

hm piezometric head at sediment surface (L).J solute flux (ML�2 T�1).

JDO DO flux, equation (34) (ML�2 T�1).Jm measured DO flux (ML�2 T�1).K permeability (L2).Kc hydraulic conductivity (LT�1).ks roughness height (L).k0s equivalent sand roughness (L).ks+ hydraulic roughness height (dgu*/n).M mass accumulation in bed (ML�2).Mw mass accumulation in water (ML�2).M 0 effective penetration depth (M/Co) (L).n unit vector normal to bed.p pressure (ML�1 T�2).

PeK permeability Peclet number (u*ffiffiffiffiK

p/D0

m).Pe* shear Peclet number (u*ks/D

0m).

q normalized solute flux (LT�1).R net uptake rate (ML�3 T�1).R residence time function.Re Reynolds number (UH/n).ReK permeability Reynolds number (u*

ffiffiffiffiK

p/n).

Re* shear Reynolds number (u*ks/n).S bed slope.t time (T).U average velocity (LT�1).us slip velocity (LT�1).us+ normalized slip velocity (us/u*).u* shear stress velocity (LT�1).u velocity vector (LT�1).

Vw water volume, flume experiments (L3).v seepage velocity vector (LT�1).x streamwise coordinate.y vertical coordinate.a dimensionless scaling constant.b sediment diffusion correction term.g empirical constant for equation (14).

D bed form height (L).d DO penetration depth (L).q porosity.l bed form wavelength (L).n kinematic viscosity (L2 T�1).ne effective viscosity (L2 T�1).x time lag (T).r water density (ML�3).

[53] Acknowledgments. This research was performed while the firstauthor held a National Research Council Research Associateship at the U.S.Geological Survey in Reston, Virginia. Funding was provided through theNational Water Quality Assessment (NAWQA) and National ResearchPrograms of the USGS. Additional support for J. W. H. was provided bythe National Science Foundation through award EAR0408744. Fieldassistance was provided by Dan Nowacki, Lauren McPhillips, and LeannaWestfall for the data collected from the Iowa River. Earlier versions of themanuscript were improved thanks to comments from colleague reviewersKen Bencala and Heinz Stefan, as well as journal reviewers BernardBoudreau, Aaron Packman, and two anonymous reviewers. Any use oftrade, firm, or product names is for descriptive purposes only and does notimply endorsement by the U.S. Government.

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��J. W. Harvey and B. L. O’Connor, U.S. Geological Survey, Mail Stop

430 National Center, Reston, VA 20192, USA. ([email protected];[email protected])


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