Hyporheic exchange of solutes and colloids with moving bed forms

Transcription

1 WATER RESOURCES RESEARCH, VOL. 37, NO. 10, PAGES , OCTOBER 2001 Hyporheic exchange of solutes and colloids with moving bed forms Aaron I. Packman I and Norman H. Brooks W. M. Keck Laboratory of Hydraulics and Water Resources, Department of Environmental Engineering Science California Institute of Technology, Pasadena, California, USA Abstract. Stream-subsurfacexchange provides the opportunity for stream-borne substances to interact with streambed sediments in the subsurface hyporheic mixing zone. The downstream transport of both solutes and colloids can be substantially affected by this exchange, with significant implications for contaminantransport and stream ecology. Several previous studies have demonstrated that bed form-induced advective flows (pumping) and scour/deposition of bed sediments (turnover) will often be the dominant processes controlling local exchange with the streambed. A new model is presented for combined turnover and pumping exchange due to relatively fast-moving bed forms, i.e., when turnover dominates the exchange in the upper part of the bed where active bed sediment transport occurs. While turnover rapidly mixes the upper layer of the bed, advective pumping produces exchange with the deeper, unscoured region of the subsurface. The net exchange due to these processes was analyzed using fundamental hydraulic principles: the initial exchange was calculated using an existing geometric model for turnover, and then the later exchange was determined by analyzing the advective flow induced under the moving bed form field. The exchange of colloidal particles due to moving bed forms was also modeled by considering the further effects of particle settling and filtration in the subsurface. Experiments were conducted in a recirculating flume to evaluate solute (conservative Li +) and colloid (kaolinite) exchange with a sand bed. The solute and colloid exchange models performed well for fast-moving bed forms, but underpredicted the colloid exchanges observed with lower rates of bed sediment transport. For very slowly moving bed forms it was found that turnover could be completely neglected, and observed colloid exchanges were represented well by a pure pumping model. In the intermediate case where turnover and pumping rates are similar, water carried into the bed by turnover is immediately released by pumping, and vice versa. Thus, while this work further elucidated the basic processes controlling solute and colloid exchange with a bed covered by bed forms and provided a fundamental model for exchange due to fast-moving bed forms, exchange in the intermediate case where turnover and pumping tend to compete can only be bounded by current models. 1. Introduction the stream [e.g., Newbold et at., 1981, 1983; Triska et al., 1993a, 1.1. Relevance 1993b]. Hydrodynamic processes provide the basic mechanism for material transport and generally must be considered when analyzing nutrient cycling in watersheds [e.g., Valett et al., 1996; Stream-subsurface exchange processes control the material Motrice et al., 1997; Mulholland et al., 1997]. A review of the fluxes of ecologically important substances and also signifiimportance of stream-subsurface hydrologic interactions cantly impact in-stream contaminant transport. Much of the within the context of stream ecology was recently presented by food supply for biota in streams is derived from terrestrial Packman and Bencala [1999], which is part of a larger review of sources, and exchange via the subsurface hyporheic mixing the implications of hyporheic processes for stream ecosystems zone is an important pathway for the delivery of these mate- [Jones and Mulholland, 1999]. rials to the stream [Allan, 1995; Brunke and Gonser, 1997]. Downstream contaminant transport is also influenced by Dissolved nutrients, in particular, are primarily transported to stream-subsurface exchange. Exchange processes result in the the stream by subsurface flow [Gregory et al., 1991]. Streamdetention of stream water and thus have a dispersive effect on subsurfacexchange is also important in later cycling of both an introduced tracer. Various idealized models have been denutrients and organic carbon. Both sorption to bed sediments veloped to representhe effect of this exchange using summary and uptake by microbial activity in the bed are important exchange parameters [Bencala and Walters, 1983; Young and processes for the removal of nutrients and organic matter from Wallis, 1986; Castro and Hornberger, 1991]. Moment methods have also been applied to analyze the tracer distribution that Now at Department of Civil Engineering, Northwestern University, results from the passage of a pulse over a porous streambed Evanston, Illinois, USA. [Schmid, 1995; Czernuszenko and Rowinski, 1997; Czernuszenko Copyright 2001 by the American Geophysical Union. Paper number 2001WR /01/2001WR $09.00 et at., 1998; WOrman, 2000]. The effect of reactions with bed sediments on the in-stream transport of nonconservative substances has been demonstrated by geochemical mass balances, 2591

2 2592 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS experimental tracer additions, and manipulations of in-stream stream passage of a tracer pulse. Exchange coefficients are chemistry [e.g., Bencala, 1983, 1984; Harvey and Fuller, 1998; determined by fitting the results of solute transport experi- W6rman, 1998; W6rman et al., 1998; Chapra and Runkel, 1999]. ments conducted in the stream of interest. The Transient Stor- The importance of stream-subsurface exchange has been dem- age Model [Bencala and Walters, 1983] and the Aggregated onstrated most clearly for cases where there is a natural or Dead Zone Model [Young and Wallis, 1986] are two examples anthropogenic input of acidic, metal rich water to streams of this type of approach. Most of the experimental investiga- [McKnight and Bencala, 1990; Kimball et al., 1994, 1999], or tions of stream-subsurfacexchange have relied upon these where large quantities of mine tailings have been released to quasi-empirical models for analysis of the experimental data the stream [Moore and Luoma, 1990; Benner et al., 1995; Nagor- [e.g., DMngelo et al., 1993; Tate et al., 1995; Motrice et al., 1997]. ski and Moore, 1999]. In these cases, coupling of hyporheic While these models are useful for representing tracerexchange flows and surface-chemical reactions in the stre- exchange data, they lump hyporheic exchange with in-stream ambed exert significant control on the watershed response to dead-zone storage and may not include the effect of all subremediation measures [Harvey and Fuller, 1998]. surface flow pathways [Harvey et al., 1996]. Further, exchange Particle transport processes are also important for both coefficients derived from tracer experiment data are not necstream ecology and contaminantransport. Fine particulate essarily representative of the stream because they are depen - organic matter (FPOM) represents an important energy dent on factors such as the timescale of the tracer injection source for the benthic ecosystem, but the in-stream cycling of [Harvey et al., 1996; Wagner and Harvey, 1997]. FPOM is not well-understood [Cushing et al., 1993;Allan, 1995; A more fundamental approach calculatestream-subsurface Brunke and Gonser, 1997; Minshall et al., 2000]. Natural stream exchange fluxes based on the hydraulics of stream and subsursedimentsuch as clays, iron oxides, and humic material have face flows. Ho and Gelhar [1973] derived the stream and subconsiderable potential to carry sorbed contaminants [Stumm surface streamlines for flow over a wavy porous boundary. and Morgan, 1996]. Certain contaminants, notably radionu- Elliott and Brooks [1997a] used this type of fundamental apclides and PCBs, are known to be mobilized by colloid trans- proach to calculate the net exchange resulting from stream port [McCarthy and Zachara, 1989; Ouyang et al., 1996; Dela- flow over a sand bed. Sediment transport produces bed forms ware River Basin Commission (DRBC), 1998; Saiers and (topographical features such as dunes or ripples) on the bed Hornberger, 1999], while other contaminants such as arsenic surface. Flow over bed forms then produces a periodic variaand metals may have a strong particle-bound component [Kim- tion in the dynamic head at the stream-subsurface interface, ball et al., 1995; Environmental Protection Agency (EPA), 1997; which in turn induces a corresponding pattern of advective Schemel et al., 1999]. In some cases the contaminant of interest flow through the porous sand bed. Exchange is produced by may be introduced in a particulate form, for example, mine flow into the bed at regions of high pressure and corresponding tailings with high solid concentrations of arsenic and metals flow out of the bed at regions of low pressure. The net stream- [Moore and Luoma, 1990]. Finally, fine sediments may gener- subsurface exchange due to this processes has been termed ally be regarded as a type of contaminant because they can "pumping." Experiments with a conservative dye both demonpotentially clog the streambed and thereby degrade benthic strated subsurface dye penetration due to pumping and verihabitat [Brunke, 1999; Packman et al., 2000c]. fied that net pumping exchange from the stream could be The goal of our work is to clarify the processes that control predicted using a fundamental hydraulic model [Elliott and the delivery of fine sediments to the streambed and the mech- Brooks, 1997b]. Under typical stream flow conditions, advecanisms for colloid capture, retention, and release by sand-sized tive pumping exchange is 2 orders of magnitude greater than bed sediments. In previous publications we illustrated that diffusive transport [Savant et al., 1987; Elliott and Brooks, stream-subsurface exchange processes can produce trapping of 1997a]. The pumping effect has also been observed for air fine particles in the streambed [Packman and Brooks, 1995], movementhrough snowpacks [Colbeck, 1989, 1997], flow over outlined the methodology needed to conduct controlled col- typical sea bed features such as sediment mounds and shrimp loid-transport experiments in recirculating flumes [Packman et burrows [Huettel et al., 1996, 1998], and in the sand surroundal., 1997], presented a model for fine particle capture due to ing a solid object placed on the bed surface [Hutchinson and stationary bed forms [Packman et al., 2000a], and showed that Webster, 1998]. the model could predict the results of flume experiments on In addition to pumping, bed form motion produces addikaolinite exchange with a stable sand bed in a laboratory flume tional mixing, termed "turnover" exchange. When stream flow [Packman et al., 2000b]. This paper will present a new model is high enough to mobilize bed sediments but the flow is still for fine particle exchange with a sand bed covered by moving subcritical, downstream-propagating dunes and ripples are ofdunes and ripples, i.e., for the case where stream velocities are ten found. In this case, sedimentransport cause scour of the sufficiently high to produce bed-load sediment transport with upstream side of bed forms and deposition on the downstream downstream motion of dune-shaped bed forms. This extends side, producing a net downstream bed form propagation. Pore the range of applicability of the models presented in our pre- water and stream water are exchanged by this alternating scour vious work [Packman et al., 2000a]. We also report the results and deposition of the upper layer of the bed. Elliott and Brooks of flume experiments with moving bed forms, and demonstrate [1997a, 1997b] presented a geometrical model for turnover that the model yields good predictions of exchange when the exchange based on the average depth of scour, but this simple bed form celerity is sufficiently high. exchange prediction underrepresented observed net dye exchange due to moving bed forms Stream-Subsurface Exchange Models Elliott and Brooks [1997a] also presented a model which The typical approach to modeling stream-subsurfacex- applied a Lagrangian coordinate system to analyze pumping change is to represent the exchange rate using one or more under slow-moving bed forms. Turnover was predicted to have idealized exchange coefficients, with the goal of reproducing a negligibl effect in the slow-moving case, because the reprethe effect that stream-subsurfacexchange has on the down- sentative turnover velocity is much less than the pumping ve-

3 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS 2593 locity. This model successfully predicted the initial solute exchange observed in flume experiments but underpredicted the exchange at later times [Elliott and Brooks, 1997b]. One advantage of the fundamental approach is that it allows the explicit consideration of the combined effect of multiple u* = u/kkhm = U/Um, v* = u/kkhm = U/Um, t*/o = k2khmt/o = kumt/o. (6) (7) (8) exchange processes. In this paper, we will establish a new Note that t*/o is time t, normalized by the time required to model for bed form-related exchange that includes both ad- travel a distance 1/k at the characteristic seepage speed Um/O. vective flow driven by dynamic head gradients at the bed sur- The porosity 0 appears because velocities are calculated as face (pumping) and mixing due to scour and deposition of bed Darcy velocities, which requires a factor of 1/0 to yield the sediment (turnover). The new model will be developed for the actual pore water velocities. The normalized velocity compocase of fast-moving bed forms (dimensionless bed form veloc- nents are ity much greater than pumping velocity) and compared with the results of experiments on solute and particle exchange with a sand bed. This model provides new insight into the hydrodynamics of the near-stream subsurface, which explains the u* = -cos (x*) [tanh(d ) sinh (y*) + cosh (y*)], v* = -sin (x*) [tanh(d;) cosh (y*) + sinh (y*)]. (9) (10) limitations of the previous Elliott and Brooks model for ex- The dimensionless velocity profile is very useful because it is change due to moving bed forms. valid for any bed form size or bed sediment composition (pro- The model will be initially developed for solute exchange vided that other assumptionsuch as the validity of Darcy's law with a sand bed, and then extended to predict fine particle are not violated). Small underflow due to the stream slope exchange. Consideration of particle transport will demonstrate (uu = KS) may simply be superimposed on the above velocity a second advantage of the fundamental modeling approach: distribution if necessary [Elliott and Brooks, 1997a, 1997b]. the basic solute transport model can be extended to include Net stream-subsurface exchange due to pumping can be additional nonconservative processes. Particles above 2 m in calculated using the pore water velocity distribution (equations diameter will settle in the subsurface pore water, and all fine (1) and (2) or (9) and (10)). Consider the passage of a tracer particles are subjecto physicochemical filtration when travel- pulse. Some tracer will be advected into the subsurface due to ing through a porous medium. These distinct particle transport the downward pore water velocity component v (or v*). The and immobilization processes will be included in the exchange tracer is carried through the subsurface and eventually returns model and used to predict the results of experiments on ka- to the stream. Net exchange over time can then be determined olinite clay exchange with a sand bed. by appropriate integration of the velocity profile, which is normally accomplished through the use of a numerical particle- 2. Solute Exchange Model tracking model. This type of pumping model has successfully predicted the results of many experiments on solute exchange 2.1. Pumping Model with stationary sand beds [Elliott and Brooks, 1997b; Packman We previously showed [Packman et al., 2000a] that subsur- et al., 2000b]. face velocity field induced under a dune-shaped bed form in a bed of depth d is u = -kkhm cos (kx) [tanh (kdb) sinh (ky) + cosh (ky)], v = -kkhm sin (kx) [tanh (kdb) cosh (ky) + sinh (ky)], where u and v are Darcy velocities in the orthogonal directions x and y, respectively, x is along the bed and y is perpendicular to the average bed surface (positive upwards), k is the bed form wavenumber (k = 2 rr/x), K is the hydraulic conductivity of the bed sediment, and h m is the half-amplitude of the dynamic head variation induced by the bed form. Streamlines for this velocity distribution are given by Packman et al. [2000a]. The dynamic head variation depends on the stream velocity, bed form height, and stream depth. An equation for h m is given by Elliott and Brooks [1997a]. The maximum induced pore water velocity is given by U m = kkhm. This ve- locity occurs at the bed surface, since the induced flow is driven by differential pressures at the stream-subsurface interface. The velocity profile can be nondimensionalized by using scales related to the bed form geometry: (1) (2) x* (3) y* =ky, (4) d* = kd, (5) 2.2. Turnover Model Transport of the bed sediment causes the development of bed forms. In the case of dunes the bed form shape is maintained by bed-load sedimentransport along the upstream side of the dune, and deposition at the downstream face. Consequently, there is a net motion of the bed form downstream and a corresponding exchange of pore water. Elliott and Brooks [1997a] equated turnover exchange to the extent of bed form scour and evaluated the equivalent solute penetration depth geometrically. The motion of a regular, triangular bed form with velocity Ub results in a mean penetration of M= t<u- H )t M= 2 t->u where H, Ut,, and X are the bed form height (trough-to-crest), celerity, and wavelength. Note that turnover can only cause exchange down to the maximum scour depth, Dms, which occurs for regular bed forms of height H at a distance H/2 below the mean bed elevation. When necessary, a distribution of bed form sizes can be also be included in the model. This model provided a reasonable estimate of the average scour depth of sand bed forms in a laboratory flume, but significantly underrepresented the net exchange in turnover-dominated experiments [Elliott and Brooks, 1997b]. We believe that this under-

4 2594 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS prediction of exchange is due to the coupling of turnover and pumping, as will be shown in the next section. In general, it is conveniento use this length scale, M, the average depth of solute penetration, to represent net streamsubsurfacexchange. For pumping exchange, M can be derived from solute flux considerations [Elliott and Brooks, 1997a; Packman et al., 2000a] or obtained directly from the particle tracking solution. Following Packman et al. [2000a], mass transfer will be normalized by M* = 2 rkm. (12) For closed systemsuch as a recirculating flume, M can also be related to the in-stream tracer concentration by a mass balance. This approach will be used in section 4 to apply the model to experimental data on net stream-subsurfacexchange Combined Pumping and Turnover Pumping may occur without turnover when the streambed y* 1.0 has residual bed features but the stream velocity is currently below the threshold of bed sedimentransport. This is a typical case in streams, because natural variations in flow rate tend to -1.5 leave residual features which are stable at low flows. In this Ub*: 3 case, the basic pumping model may be used to predict conser vative solute exchange [Elliott and Brooks, 1997b; Packman et Figure 1. Subsurface flows induced under moving bed forms. al., 2000b], and the pumping model may also be coupled with Bed forms are represented by sinusoidal streamwise variation a sorption or other surface-reaction model to predict the exof dynamic head, translating to the right. (a) Pathlines responchange of reactive solutes [Rutherford et al., 1993, 1995; Eylers, sible for flux through the surface of the bed, with and without 1994; Eylers et al., 1995; Forman, 1998], or with a physicochem- turnover. Pathlines for the nonmoving bed case are identical to ical colloid transport model to predict the exchange of fine particles [Packman et al., 2000a, 2000b]. More generally, both pumping and turnover will occur simultaneously, producing complicated interactions between the streamlines. (b) Typical pathlines present deeper in the bed. For truly fast-moving bed forms, U, >> 1, both the surface penetration layer in Figure la and the size of the spirals in Figure lb would be very small. two exchange processes. The rate of pumping from the stream to the bed is highest on the upstream side of the bed form, where the maximum dynamic pressure occurs [Elliott and Brooks, 1999a]. However, this is also a location where the bed pumping' ( 15 ) is being scoured; the local stream velocity and rate of bed sediment transport increases owing to the protrusion of the bed form into the stream. As a result, some of the water (and solutes) pumped into the bed will immediately be released by Scour. Elliott and Brooks addressed this difficulty by scaling turn- The net exchange is then calculated as for the pure pumping model. The effect of moving bed forms is to reduce the solute penetration into the subsurface, as shown in Figure la. This modeling framework (including the pumping model, turnover model, and combined model) yields good predictions for pure pumping cases but greatly underestimates the exchange for over and pumping exchange rates, and then analyzing cases pure turnover cases [Elliott and Brooks, 1997b]. Combined where either turnover or pumping is expected to dominate the exchange is predicted well at short times, but is underpredicted net exchange. The dimensionless turnover velocity U, scales at later times [Elliott and Brooks, 1997b]. the rate of turnover exchange with the maximum pumping exchange rate: We believe that the pure turnover model greatly underpredicts exchange because it does not account for the fact that U = OUb/U m. (13) The 0 appears because U m is calculated as a Darcy velocity. Elliott considered cases with U, << 1 to be pumping dominated, U, >> 1 to be turnover dominated, and U, 1 to be combined. For the dominated cases, Elliott then used only the applicable "pure" model to predict the exchange. For the combined case, Elliott and Brooks used a modified pumping model, in which the bed motion is included by taking a Lagrangian framework that follows the moving bed forms. Effectively, this superimposes the bed and pumping velocities so that U* : U* pumping -- e * b, (14) (a) Y* (b) X ß i '\ ". "'-...' ' ; :,,,, %.. '... \ ', '-....-' /, j pathlines for moving bed Ub = I... pathlines for nonmoving bed X ß m -- bottomof bed pumping to the deeper bed will always be an important exchange mechanism, while turnover only occurs in a thin layer at the top of the bed. Turnover can only be the dominant exchange mechanism at relatively short times, until the entire turnover zone has been mixed. At later times, even a small amount of pumping to the deeper bed will be a significant exchange mechanism. Thus the "pure turnover" case will only occur when pumping is absent or insignificant over the entire subsurface. Examples of pure turnover cases would be bedrock-dominated rivers or isolated sand bars such that practically the entire bed is subjecto scour, or rivers with extremely fine sediment such that the pumping velocities are practically zero (an example is given by Rutherford et al. [1995]). Pumping might also reasonably be eliminated when a fast-moving tracer

5 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS 2595 I v I C1 Turnover region Deep-bed region: pumping, no turnover M2 Figure 2. Zones of turnover and pumping for exchange with moving bed forms, and the relationship between turnover exchange (Mr), pumping exchange (Me), total exchange (M), and stream- and pore-water concentration (C) at two times (denoted by subscripts 1 and 2). The dashed line indicates the average bed surface elevation. pulse passes over the bed. In this case, most of the solute would be rapidly exchanged into and then released from the turnover region, and only an insignificant amount would reach the ignored (for times of the order X/Ub). The initial exchange is then just equal to the equivalent penetration due to turnover alone, M r, given by deeper bed. It is importanto note that U, only identifies which process is primarily responsible for exchange through the streamsubsurface interface. This is a key distinction, because trans- After t = X/Ut, the rapid turnover will ensure that the bed port into and out of deeper layers of the bed is required for stays well-mixed with the stream down to the maximum scour ongoing dilution of a conservative solute in the stream. When depth, y - D ms, which is just -H/2 for regular bed forms. U, is large, turnover controls the local flux through the bed Further net exchange from the stream to the bed must then be surface. However, turnover is limited to the upper layer of the due to pumping exchange, Mp, from the turnover region to the bed, while advective pumping occurs over the entire bed depth deeper bed. Thus the total exchange is given by as shown in Figure lb. As a result, even though U, may indicate that exchange is turnover-dominated, pumping can still be an important process because it produces exchange beyond the maximum scour depth. Essentially, in the model of Elliott and Brooks, transport due to the surface advective flow The exchange due to pumping, Mv(t ), is calculated from the (Figure la) is considered, but further transport due to the flux out of the turnover region into the lower, unmixed portion deeper advective flow (Figure lb) is omitted. of the bed. Our idealized conceptual view of exchange with The spiral pattern in Figure lb results from the superposi- moving bed forms is presented in Figure 2. tion of the periodic flow induced by moving bed forms and the A simple expression can be developed for My(t) in the case constant effect of the channel slope. If some mixing process of fast-moving bed forms. Based on the pathlines shown in (i.e., dispersion) produces exchange between adjacent path- Figure 1 and Elliott and Brooks' observations of dye penetralines, then the net effect will be transport through a series of tion, the net pumping exchange under fast-moving bed forms horizontal layers. When turnover is very high, the pumping can be modeled by assuming that a front of well-mixed fluid is pathlines become very small circles and pumping results in an driven downwards by pumping from the turnover region to incremental exchange to successively deeper layers of the bed. deeper parts of the bed. In this case, the pumping exchange is This is shown quite clearly by the results of Elliott and Brooks, given by the downward flux from the currently well-mixed who observed that dye penetration fronts became roughly hor- portion of the bed: izontal at high bed form velocities. This is an interesting case, because dispersion is not a significant interfacial transport proq (t*) = C*(t*)½*(y*), (18) cess compared to advection (and can thus be neglected in the where C*(t*) is just the in-stream concentration since the calculation of net exchange), but subsurface dispersive mixing is still sufficiento smooth the tracer penetration front. Similar upper part of the bed is assumed to be well-mixed with the stream, and * is the average exchange velocity through a behavior has been observed for advective flows induced in horizontal line at the front depth y. The average velocity can sediments under waves [Shum, 1992], and this process has been termed "rotational dispersion" by Webster and Taylor [1992]. readily be calculated from the velocity profile (equations(9) and (10)). Because the velocity profile is symmetric under each To model this case, we will considerapid turnover exchange bed form, it is only necessary to take the average velocity over down to the maximum scour depth, D ms, and subsequent half of the bed form shape. Also, for the penetration of a pumping to the deeper part of the bed. When U, >> 1, turnover dominates pumping exchange in the region very near horizontal tracer front, exchange flux only occurs due to the velocity out of the well-mixed front. The return velocity into the stream-subsurface interface, and pumping may initially be the front serves to dilute the concentration in the well-mixed m(t) =Mr(t) = - H[ 1- ( 1- )21 t< -. X (16) M(t) = Mr(t) + Mr(t) = -Dms + Mr(t) t > -. (17)

6 2596 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS region above the front. For the case of solute penetration from the stream to the bed, only the downward component of velocity needs to be considered in the calculation of the front propagation because the tracer concentration is zero below the current front position. The same solution will apply for the penetration or release of any horizontal front with a concentration difference A C between the well-mixed front and the rest of the bed. Thus * is given by 1 X*/2 ½*(y*) = - v*(x*, y*) dx*. (19) a0 Distances are nondimensionalized using the wave number, so X* = 2rr. Evaluating for the finite bed velocity profile given in ( 0) (y,) = tanh (d ) cosh (y*) + sinh (y*) Pumping flux results in an accumulated exchange given by a r*/o= X*/OU * y* 0 ' (20) The negative sign is required because downward velocities are negative by definition, while M is positive. Note that the current front position must be used for the depth y* used to evaluate * in (20). The front position is exactly given by the equivalent penetration depth M. Thus the position 1 1 * - - M*(t*/O) = - [-D* + M (t*/o)] (22) Y - 2 ms pumping and turnover would necessitate the use of a finite element or similar numerical model to track solute concentrations at every location in the bed at all times. For high U, we employed the useful simplification that turnover can be as- sumed to control exchange from the stream into the upper layer of the bed. However, pumping may cause some additional exchang even before the bed becomes well mixed in the turnover region. This effect can simply be included in the model by including the pumping exchange term for t < X/Ub; i.e., by changing the limits of integration in (21). The assumption that pumping to the deep bed begins at t = X/Ub results in some underestimation of the early exchange, while the assumption that pumping begins at t = 0 results in some overestimation of the early exchange. In reality, the early pumping exchange will lie between the values given by these two predictions. However, the difference is quite minor for large U, (inducing errors of a few percent) since turnover dominates the early exchange process Correction for Irregular Bed Forms The above analysis is most easily applied if bed forms are completely regular, but there will generally be some distribution of bed form shapes under any given flow conditions. Elliott and Brooks [1997a] developed solutions for the mass exchange for a given distribution of bed form heights (e.g., Gaussian). However, in general, it is not necessary to use this level of modeling detail. Instead, a simple correction can be employed when there is a considerable difference between the height of the average and largest bed form. In other words, it may be worthwhile to account for the bed form size distribution when the distribution is wide. A useful simplification is to just calculate the turnover exchange due to 2-3 size classes of bed is used to calculate the average veloci term *. Equations forms, always including those of the average size and the larg- (20)-(22) can easily be solved by a finite-difference scheme, est size. However, the bed form size distribution will not have using the equivalent penetration depth from the last time step, a significant effect on pumping exchange, and so pumping may M* i- as the front position y* i to evaluate i. * The in-stream be calculated using the average bed forms parameters and the concentration is related to the penetration depth by the geomet of the system; the example of a recirculating flume will be depth of maximum scour (as in equations (20)-(22)). We have employed more sophisticated procedures to estimate the effect considered in section 4. of irregular bed forms on combined pumping/turnover ex- A key aspect of the combined pumping and turnover model presented here was averaging solute transport over a bed form change, but they were found to have an insignificant effect on the overall results [Packman, 1997]. in order to derive an average rate of vertical exchange in the bed. This solution can be applied for both the penetration and release of solutes but is limited to cases where there is no 3. Colloid Exchange Model 3.1. Exchange of Colloids With Stationary Bed Forms significant initial downstream variation in concentration. Further, the pumping veloci field was derived assuming a homogeneous, isotropic porous medium. Streams with heterogeneous sediments and high rates of sediment transport will often develop vertical and longitudinal gradients in the bed sediment size distribution (i.e., bed armoring and sorting), but these effects have been omitted from this analysis. We previously presented a model for the pumping exchange of suspended particles [Packman et al., 2000a], which will serve as the basis for our new combined pumping/turnover model for particles. The colloid pumping model superimposes particle transport behavior on basic advective pumping. The particle transport processes of settling and filtration were parameter Correction for Pumping at Early Times ized appropriately and included in the particle pumping model. Settling of particles in the 2-10/xm size range is not significant The new combined exchange model assumes complete separation be een the turnover and pumping time scales when bed forms are fast moving. As a result, pumping exchange is only considered after the bed has been m ed down to the depth of m imum scour. In reali, however, both pumping and turnover occur simultaneously beginning at t = 0 (i.e., immediately after tracer addition). The actual distribution of the tracer in the turnover region is ve complicated, since tracer brought into the bed by turnover would be subject to in the stream, but is very significant in the bed because pumping velocities are quite low. Colloid filtration is the physicochemical process whereby suspended particles become attached to stationary sediment grains in a porous medium. In flume experiments we observed that particle settling and filtration in the bed caused suspended clay to behave very nonconservatively [Packman et al., 2000hi. In fact, with a typical initial concentration of 200 mg/l, essentially all kaolinite added to the recirculating stream became trapped in a stable pumping, and vice versa. Exact calculation of simultaneous streambed within 24 hours. Thus stream-subsurface exchange

7 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS 2597 must be considered when analyzing the mobility or downstream transport of fine particles. The equation for the Darcy velocity of a particle in the bed can be obtained by superimposing advection due to pore-water motion and particle settling. Up = -kkhm cos (kx) {tanh (kdb) sinh (ky) + cash (ky)}, (23) Vp = -kkhm sin (kx) {tanh(kdb) cash (ky) + sinh (ky)} - vso, (24) where Vs is the Stokes settling velocity of the suspended particles. Particles with a diameter <2 m will not settle, but larger clay particles may have a significant settling velocity compared to the pumping velocities. Note that Stumm and Morgan [1996] define colloids as particles with a diameter of <10 m, so both nonsettleable (Brownian) and settleable (non-brownian) colloids may be considered [e.g., Tobiason, 1989]. The relative effects of pumping and settling in the bed may be considered with a normalized settling velocity: V s -- OVs/U m. (25) The 0 appears because u m is calculated as a Darcy velocity. The nondimensional particle paths are given by Up * -- -cos x *(tanh d* b sinhy * +cashy *), (26) Vp *= -sin x* (tanhdocashy * * +sinhy *)- rs. * (27) Filtration was also included by calculating the reduction in suspended particle concentration due to filtration along all subsurface flow paths. This colloid pumping model was quite successful in predicting the results of flume experiments on particle exchange from the stream to the bed [Packman et al., 2000b] Exchange of Colloids With Moving Bed Forms cussed in the review by Ryan and Elimelech [1996].) For these reasons, in the analysis presented here it will be assumed that colloids cannot accumulate in the turnover region. This result might not be true for colloids that become very strongly attached to the bed sediment but will be shown to be valid for kaalinite exchange with a sand bed (section 5). Practically no data are available to make generalizations about the potential for colloid detachment from exposed bed sediment grains in streams. We also previously observed that pumping transport of colloids to the deep bed will often result in nearly all colloids becoming trapped [Packman et al., 2000a, 2000b]. Thus it will also be assumed that exchanged particles are completely removed from suspension when they penetrate below the zone of active bed sediment transport. This assumption will not apply for weakly filterable Brawnian colloids but can be expected to be true for all larger suspended sediments and those that interact even moderately with the bed sediment. Quantitative analysis of the range of applicability of this assumption can be found in the work of Packman et al. [2000a]. Applying the above assumptions, the net pumping exchange of colloids from the stream will be controlled by the flux to the trapping region, i.e., below D ms' qp *(t*)=c*(t*) vp(d* * ms), (28) where C*(t*) is the normalized particle concentration in the stream and v-- p(d *ms) is the average downward particle velocity through the plane y* - D* ms ß Note that the stream and the turnover region are assumed to have the same particle concentration because fast turnover will keep the upper part of the bed well mixed with the stream. In other words, because the turnover exchange rate is much higher than the pumping exchange rate, transporthrough the turnover layer is a fast mass transfer process and net exchange is ultimately controlled by pumping to the deeper bed. Pumping could then be considered the rate-limiting process for particle capture by the bed. The average particle velocity, v, can be calculated from (27)' In extending the colloid pumping model for moving bed forms, the key issue is the calculation of particle trapping both tanh (d;) cash (D*mO + sinh (D*mO v * = - 7r 2' within the zone of active bed sediment transport and in the deeper, unscaured region of the bed. Turnover exchange and colloid filtration are presumably independent, but pumping (29) through moving bed forms makes filtration difficult to quantify because the path length over which filtration occurs is not For particles above 2 / m in diameter, the particle settling known. The problem is quite complicated due to the feedback velocity can result in a dramatically increased flux to the deep between turnover, pumping, and the residence time of colloids bed. The net pumping exchange is then given by in the turnover region. The problem is simplified for the case of rapid bed sediment transport (defined in terms of U,), because in this case pumping transport is small compared to M; = -2rr v;(d*mo 0 (30) turnover and there is correspondingly little opportunity for d r*lo=)t*/ouo* filtration in the turnover region. Further, we found that the colloids used in our experiments were not permanently trapped in the turnover region. The attachment of colloids to This is exactly the same as (21) for conservative tracers, except that the effective depth used to calculate the exchange rate is collector particles is irreversible only as long as the system stays always Dm$ instead of M(t), and settling is included in Vp. unperturbed. In the turnover region, sediment transport results in all the bed sand grains being periodically scoured and transported by the stream flow. As a result, colloids attached to Filtration is not explicitly included because it is assumed that filtration is sufficiently high to result in complete trapping of colloids pumped to the deeper, undisturbed portion of the bed. these mobilized sand grains are subjected to the high-shear If the degree of filtration is lower, some of the particles conditions of the stream flow, which can cause them to become detached and released back to the stream. (Colloids can be detached by both physical and chemical mechanisms, as dispumped to the deep bed will return to the turnover region, and thus the net exchange will be less than that predicted by (30). Equation (30) can then be seen to give the maximum net

8 2598 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS exchange due to trapping in the deep bed. Note that the equivalent penetration depth M is now a virtual penetration that includes the actual penetration downwards in the bed and the filtration of colloids by the bed sediment. This virtual penetration depth is equal to the exchanged mass per unit bed area divided by OC (t). M will always be a virtual penetration when a tracer behaves nonconservatively because the tracer interaction with the bed sediment must be included when determining the net mass exchange from the stream [Packman et al., 2000a]. The combined pumping/turnover model for particles is implemented in a numerical finite difference scheme to account for the changing tracer concentration in the stream, C* (t). Exchange is initially assumed to be completely due to turnover, and (16) is used to calculate the net exchange for t < Exchange for t > X/Ut, is assumed to be attributable to both further turnover and pumping to the deep bed, and the additional exchange is calculated by (21) and (30). Model results will be compared with experimental data for kaolinite exchange with a sand bed in a laboratory flume Comparison With Idealized Exchange Models constant piston velocity or exchange coefficient will work well. However, ongoing solute dilution depends on mixing with ever deeper pore water. Because the advective subsurface flow is driven by pressure gradients at the stream-subsurface interface, pore water velocities and mixing rates decrease with depth in the bed. As a result, the net stream-subsurfacexchange rate is not constant and the simple exchange coefficient models must be modified to account for the variable exchange The combined turnover/pumping model can easily be re- The recirculating flume used for these experiments had a lated to familiar idealized representations of stream- tiltable channel 12 m long, 26.5 cm wide, and 25.4 cm deep. subsurfacexchange. For a box- or layer-type representation of The channel was supported by a steel truss and contained a the streambed, the exchange rate is often represented using a downstream reservoir with a variable speed pump which recirpiston velocity p defined so that the exchange flux = p AC, culated both water and sediments to the upstream end of the where A C is the difference in tracer concentration between the channel via a return pipe. Thus the flume represents a closed stream box and subsurface box. The Transient Storage Model system for water, bed sediments, and any added substances [Bencala and Walters, 1983] uses a similar formulation with a (provided they are nonvolatile). The volume of water and bed mass transfer coefficient for exchange, a, defined such that the material, the channel slope, and the recirculation rate may all rate of change of in-stream concentration is given by -aac. be controlled by the flume operator. In the experiments de- Accounting for the ratio of stream volume to bed area, a = scribed here, the flume flow rate was set to produce the desired p/d, where d is the stream depth. The piston velocity has units bed forms, and the channel slope was then matched to the of length/time, while the Transient Storage "a" just has units of energy grade line in order to establish uniform stream flow inverse time. over a sediment bed of uniform depth. The recirculation rate In general, neither a piston velocity nor a simple exchange was measured with a Venturi/manometer combination in the coefficient will provide a complete representation of streamreturn pipe, and water surface profiles were measured using a subsurfac exchange because the exchanged tracer will not be point gauge mounted on an instrument carriage. well-mixed in the subsurface and the average exchange rate All experiments described here involved natural bed forms will decrease as the tracer penetrates deeper into the bed. However, the idealized formulations apply to the case of colwith ongoing sedimentransport, which caused the bed forms loid exchange with fast moving bed forms because (1) the to advance downstream. The bed sediment was a high-purity moving bed forms tend to produce horizontal exchange fronts, Ottawa silica sand with a geometric mean diameter of 480/ m and (2) complete trapping below D ms causes the particle deand a geometric standard deviation of 1.2. This simple, wellsorted bed sediment was selected in order to examine basic livery to the deep bed to occur at a constant velocity. From (28) it can be seen simply that the piston velocity for stream-subsurfacexchange and colloid filtration without the colloid exchange will be equivalent to the average downward additional complexity of physical or chemical heterogeneity. pumping velocity, (Dms). Equivalently, a = pp/d for the The hydrauli conductivity of this sand is 9.0 cm/min (at 21øC), pumping-driven part of colloid exchange with fast-moving bed and the porosity 0 = [Eylers, 1994]. Experiments were forms (i.e., after the initial period where turnover dominates). conducted in a temperature-controlled laboratory at -20øC. The same considerations apply for solute exchange due to Bed forms were established and allowed to come to equilibmoving bed forms, excep that in this case the penetration front moves downward, which causes b to decrease over time (equarium with the stream flow by recirculating water over the bed for several days. The end result was a set of moving bed forms tions (20)-(22)). Thus, while colloid exchange with fast-moving whose average properties (geometry, velocity) did not change bed forms may be well represented by a constant exchange over time. During the experiments, bed forms that reached the coefficient, solute exchange must be represented by a timedownstream end of the channel fell into the downstream enddependent exchange coefficient. well. This sediment returned to the upstream end of the chan- This is rather interesting because the simplified representa- nel, where it served as the sediment supply for the formation of tions work better for colloid exchange than for conservative new bed forms. While bed forms were continually disappearing solute exchange. Essentially, when colloids are filtered extensively in the bed, the exchange problem can be reduced to one of simply calculating the flux of colloids to the region where they will be filtered. In this case, transport models with a and being produced, the new bed forms always had the same properties as the old ones (statistically speaking) because the system was in equilibrium. Inlet conditions were controlled using an upstream well and a ramp to bring the flow to the level rate. 4. Experimental Methods We used a laboratory flume to conduct experiments on solute and particle exchange with a sand bed. The results of these experiments will be used to evaluate the performance of the new models for combined exchange. The methodology for experiments with stationary bed forms and the material preparations have been described elsewhere [Packman, 1997; Packman et al., 1997, 2000b] and will only be summarized here. The methodology for moving-bed form experiments will be de- scribed in detail.

9 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS 2599 Table 1. Stream, Bed, Bed Form, and Exchange Parameters of All Flume Experiments a Run Stream Parameters Bed/Bed Form Parameters Exchange Parameters Q, u, d, d ', do, H, )t, U b, h m, Ig m, L/s cm/s cm cm S cm cm cm cm/min cm cm/min U, v * Other Experimental Conditions T, tfinal, U u øc hours arun numbers are consistent with Packman [1997]. Dash indicates variable was not measured. of the bed surface. Inlet effects on the sand bed were limited to the first 1-2 rn of the channel. Bed form geometries and velocities were initially measured by hand, but we used a laser bed profiling system for later experiments. The profiler was composed of a laser displacement sensor (Keyence LB-1101) mounted in a waterproof shell with an optical glass window. The laser sensor has an accuracy on the order of 10/xm over a measurement range of cm from the sensor head. The whole assembly was mounted to an instrument carriage in order to provide measurements of bed surface position versus downstream distance. The profiler was only minimally invasive, as the shell only had to penetrate the water surface sufficiently so that the optical window remained submerged at all times. Exchange experiments were conducted by adding the tracer of interest to the stream uniformly over one water recirculation period. Thus the entire stream was rapidly brought to an initial concentration, Co, while the streambed was initially tracerfree. As the experiment progressed, stream-subsurfacexchange caused stream water to penetrate the streambed, which produced a decrease in the tracer concentration in the stream, C (t), over time. Conservative tracers mix according to dilution of the stream water with pore water, while the concentrations of nonconservative substances are reduced beyond simple mixing due to interactions with the bed sediment. In both cases the rate of stream-subsurface exchange is much less than the rates of recirculation and in-stream mixing. As a result, concentration gradients in the recirculating stream are always small so that the entire volume of recirculating water can always be considered well-mixed relative to the pore water. Lithium ion (Li +, introduced as a LiCI solution) was used as a conservative tracer. Lithium concentrations were measured by ICP-MS, which allowed use of very low concentrations (the initial concentration, Co, for lithium was typically in the / M range). Kaolinite clay was used to investigate colloid exchange, with a Co of either 120 or 200 mg/l in the moving-bed form experiments. This range of initial concentration was selected because it can readily be measured by spectrophotometer, but is not enough to plug the sand bed. Initial runs (1-6) were conducted with tap water. Because the stream chemistry is important, kaolinite suspension parameters were not wellcharacterized in these preliminary experiments [Packman et al., 2000b]. Careful attention was paid in later experiments to the preparation of the kaolinite, sand, and the stream water composition. The improved kaolinite preparation yielded a stable suspension with a mean spherical-equivalent diameter of 7/ m. For experiments 8-16 the stream water had ph between 7 and 8, and a background electrolyte of 5 mm NaC1. Under these conditions, kaolinite can readily become attached to the bed sediment (filtration coefficient of cm- [Packman et al., 2000b]). Both bed sediment and colloids were rapidly recirculated through the pump and return system of the flume. Flow velocities in the return system were kept high by the flume design, which uses a return pipe with considerably lower crosssectional area than the open channel stream flow. As a result, there was no significant deposition of bed sediment and no opportunity for colloid filtration in the return system. A clear section of return pipe was installed in order to confirm that no sediment accumulated in the return system. A total of eight experiments were performed with moving bed forms. The conditions for these experiments are reported in Table 1. Exchange rates (hm, U m) are considerably higher here than in experiments with stationary bed forms [Packman et al., 2000b] because higher stream velocities must be used to obtain moving bed forms. Experimental results will be compared with the predictions of the combined turnover/pumping model in section Results and Model Evaluation The turnover and pumping models may be applied to analyze the flume experiment results by relating the accumulated exchange, M(t), to the observed change in concentration in the stream, C(t). In the flume the lithium or kaolinite added to the recirculating stream is depleted due to exchange with the bed. The total mass of added tracer is given by CoAd ', where Co is the initial well-mixed concentration in the stream, A is the area of the streambed, and d' is the effective depth, the total stream volume (including return pipe, etc.) per unit bed area. If the tracer can be considered well mixed in the bed down to the equivalent penetration depth M, exchange with the bed can then be related to the in-stream concentration by With d' nondimensionalized as d' C* = (31) d' + MO' 2 rkd ' d'* = 0 (32)

10 2600 PACIGMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS o o? t [] "o' u [] to Ex. 8 Ub* =0.35 oo.o.?g */ { ttt i t ofi [] t O o [] / [] Ex t Ex. ll Ub* -' :...,... :...:.. '/0 Figure 3. Comparison of model predictions with the observations of conservative solute (Li +) and colloid (kaolinite) exchange in a laboratory flume, presented in order of increasing dimensionless bed form velocity, U,. Solid lines are the prediction of the combined turnover/pumping model for colloid exchange. Dotted lines are model predictions for solutexchange. Dashed lines (uppermost curves in the first three graphs) are predictions of the pure pumping model for colloid exchange [Packman et al., 2000a]. Model predictions used only measured parameters as inputs and are not fit to the data in any way. The relevant experimental conditions are given in Table 1. Note varying timescales in horizontal axes. and M* as given in (12), this becomes preparations, etc., as described by Packman et al. [1997]). If d t desired, the particl exchange model could be fit directly to the C* = d'* + M* ' (33) experimental data by using the particle settling velocity as a tuning variable, which would account for poor estimation of To model experimental results, a prediction for M*(t*) is particle characteristics (an example is given by Packman et al. calculated from the model by numerical integration, as out- [2000b]). lined in sections 2 and 3, and then C*(t*) is obtained using The fundamental solut exchange model generally predicted (33). The parameters given in Table 1 were used as model lithium exchange very well. Experiment 9 is a notabl excepinputs, so that a model prediction of exchange was obtained for tion, and we conclude that there was an error in the conceneach experiment based solely on the measured experimental tration measurement in this experiment (some possible reaconditions. Exchange data for each moving-bed form experi- sons for such a poor measurement may be found in the work of ment are presented in Figures 3 and 4, along with the predic- Packman et al. [2000b]). The general high quality of the model tions of the combined turnover/pumping models for solutes performance indicates that we have appropriately synthesized and colloids. In addition, similar solute-transport data for fast- the pumping and turnover processes in our model. Because moving bed forms from Elliott and Brooks [1997b] are analyzed turnover and pumping exchange compete in the upper layer of with the new model in Figure 5. the bed, dimensionless exchange is less for the combined case The model curves in Figures 3-5 are based solely on mea- than for pumping alone. However, the net exchange is much sured variables and are not fit to the exchange data in any way. greater than that simply predicted from scour alone. Thus modest disagreements between model predictions and The combined model for colloid exchange begins to perform exchange data can generally be attributable to experimental reasonably well for fast turnover (U, > 1.5) but only does a error or misestimation of input variables. We previously esti- good job of predicting the exchange in Experiment 11, with mated the typical error in experimental variables to be of the U, = 8.4. This indicates that settling and filtration causes order of 10% [Packman et al., 2000b]. The expected error is colloids to be trapped in the bed, and confirms our assumpconsiderably worse for kaolinit exchange in early experiments tions that (1) turnover causes colloids to be detached due to (runs 1, 2, 4, and 5 shown in Figure 4), because these early high shear, and (2) filtration is high enough to remove all experiments utilized tap water and only limited cleaning and particles from suspension once they travel below the region of preparation procedures [Packman and Brooks, 1995]. The un- active bed scour. Our approach of adding nonconservative reliable performance of these early experiments led us to de- processes to a fundamental hydraulic transport model clearly velop methods to carefully control the conditions that influ- allows detailed evaluation of the importance of various transence particle-particle interactions (water chemistry, particle port and capture mechanisms. Essentially, the turnoveregion

11 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS o 0.4,? / Ex._I ,. Ex U *=0.76 t' [] u [] Kaolinite o Li + 0.0, ' O 25 3O '/o Ex d */o Ub* = 2.0 [] 0 [] [] [] */0 */0 Figure 4. Results of early flume experiments on kaolinite and Li + exchange with a sand bed, conducted in a laboratory flume with tap water. Experiments 1 and 2 involved only kaolinite exchange. Because materials preparations and water chemistry were not well-controlled, particle-particle interactions were not well defined in this system and the colloid transport model is not expected to reliably predict kaolinite exchange. Nonetheless, the colloid exchange observed in experiments 1, 2, and 5 is consistent with the prediction of the exchange models. Li + exchange is not dependent on the background water chemistry, and was modeled well. represents a barrier to the irreversible deposition of suspended estimate the extent of filtration that occurs within the slowsediments in the bed. Colloids that make it through the region moving bed forms. of active turnover can then be trapped in the deeper, stable part of the bed. A corollary to this is that colloids would be mobilized from the deep bed if this region were scoured, e.g., 6. Discussion and Conclusions due to a large increase in stream flow. We observed this behavior when we allowed particles to be deposited in a stationary bed and then increased the stream flow to induce bed sedimentransport (results not shown). The release would be difficult to model because the spatial distribution of attached colloids (i.e., the initial condition) is unknown. Models were developed to predict the exchange of conservative solutes and colloidal particles between a stream and a streambed for the case of active bed sediment transport. Our previous work [Elliott and Brooks, 1997a, 1997b; Packman et al., 2000a, 2000b] indicated that two transport processes are important in this case: trapping and release due to the scour The model underpredicts the observed colloid exchanges and deposition of the bed sediment (turnover), and advective with slow (U, << 1) and intermediate (U, 1) turnover. exchange due to induced dynamic head variations at the This indicates the limited applicability of our new combined model to cases with slow-moving bed forms (recall that the stream-subsurface interface (pumping). Our new models are based on an assumed spatial separation of these processes: model was derived by assuming U, >> 1). Where applicable, turnover controls the exchange through the region of active predictions of the pure pumping model for colloids [Packman et al., 2000b] are included in the graphs for comparison. The pure pumping model provides reasonably good predictions for the exchange due to slow-moving bed forms (U, < 0.4). These results indicate that turnover can be neglected when the rate of bed sedimentransport is low. In all cases the results are observed to be between the predictions of the pure pumping and combined turnover/pumping models. The combined model begins to give reasonable predictions of colloid exchange when U, is only somewhat greater than unity, so that current models only fail to work well in the region U, 1 (roughly, ). An analytical solution for this case will be sedimentransport, and pumping then produces exchange with the deeper, unscoured part of the subsurface. In this case, turnover exchange can be calculated using the Elliott and Brooks model, and the additional advective pumping exchange can be calculated based on the average downward pumping velocity. This model view is supported by experimental observations that dye penetration fronts become approximately horizontal at high rates of bed sediment transport, indicating that horizontal variations in the advective pumping flow field are averaged out due to the rapid downstream progression of the induced dynamic head pattern at the bed surface. Experiments were conducted in a 12-m-long recirculating difficult to obtain because it is characterized by competition flume to test the model. The flume was run with sufficient between pumping input and turnover release, as described in section 2. Further, because the advective flow paths in the turnover region are quite complicated, it is very difficult to stream flow over a sand bed to produce active bed sediment transport. A conservative solute (LiC1) and colloidal particles (kaolinite) were added to the recirculating stream, and the

12 2602 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS o Dye E&BRun 19 Turnover + Ub* = 16 Pumping -... Tumove r Only øø I o Dye E&B Run 20 -I - Turnover + Ub* = 8 -I Pumping _.- _l... Turnover Only o,,,.-. _ ,--.,......, Figure 5. Comparison of model predictions with the results of experiments conducted by Elliott and Brooks [1997b]. Open diamonds represent data for conservative dye penetration into a deep sand bed. The solid line is the exchange predicted by the combined turnover and pumping model. The dashed curve represents the model prediction for average scour depth, which is equivalento the exchange due to turnover (scour) alone. Note that the exchange scale is exaggerated relative to Figures 3 and 4 in order to clearly show conservative dye exchange. exchanges of these substances with the bed were observed over time. The model generally did a good job of predicting the results of flume experiments on solute exchange. No curve fitting was required to reproduce the observed exchanges for moderate- to fast-moving bed forms. This indicates that we have appropriately synthesized the turnover and pumping pro- cesses in our combined model. Colloid exchange was modeled by considering particle settling and filtration. For the fast moving case we did not explicitly include filtration but instead employed assumptions that there would be no permanent filtration in the turnover region, but complete particle capture below this region. These assumptions were found to be applicable for the kaolinite-sand system, but may not necessarily apply to all colloid-collector pairs. Settling is not a significant transport mechanism for particles of less than 2-/xm diameter (sometimes called "true" colloids), but can be significant even for fine silts (2-10/xm diameter) because subsurface advective velocities are so low. Observed colloid exchanges were predicted well for fast-moving bed forms. Essentially, ongoing bed sedimentransport in a defined surface layer hinders the deposition of colloids in the bed. Further, scour can release colloids that were previously trapped in a stable bed. The combined turnover/pumping model did not perform well for cases with slowly-moving bed forms. However, colloid exchanges with slow-moving bed forms were bounded by the predictions of the pure pumping model and the new combined model. While the dimensionless turnover velocity (U, = OUt,/ U m) does not completely represent the relative effects of pumping and turnover because it does not account for exchange below the region of active bed sedimentransport, this parameter can still be used to predict whether the pure pumping model or combined model should be used. Pure pumping occurs only for stationary bed forms, U, = 0. Cases with U, << 1 can be considered slow turnover, cases with U, >> 1 are fast turnover, and cases with U, 1 are intermediate. The deviation from the pure pumping model is predicted by U* b, with decreasing net dimensionless exchange for increasing U,. Experimental results show that the pumping model still works well for cases with very low turnover U, < 0.4, while the combined turnover/pumping model begins to work reasonably well for U, > 1.5, and very well for U, > 8. The intermediate case 0.5 < U, < 1.5 is difficulto analyze because it involves competition between pumping and turnover transport, and the exchange for this case can only be bounded with current models. Also, the pure turnover model only applies when pumping is eliminated, for example, when the bed is impermeable. While the model presented here was shown to be sufficient to predict solute and colloid exchanges in a laboratory flume, a considerable number of additional complexities will have to be included before the model can be applied to the transport of solutes and suspended sediments in streams and rivers. The laboratory experiments were restricted to a simple, well-sorted sand bed in order to examine the basic processes that control hyporheic exchange of solutes and colloids. In nature, a wide variety of bed sediment types can be found. New models will have to be developed for much coarser and finer sediments. Non-Darcy flow is expected in the upper layers of gravel beds, and colloid filtration is not well understood for this case. Fine, silty bed sediments admit only very low pore water velocities and may also produce straining of suspended sediments (instead of the deep-bed filtration modeled here). In addition, fluvial environments are often very heterogeneous, and characteristic gradients in bed properties develop owing to sediment armoring and sorting. It may be quite difficult to determine "average" filtration behavior of suspended sediments in any given stream. Further, the current model includes only the most basic description of colloid filtration in the streambed. Weakly filterable colloids can return from deeper regions of the bed to the stream flow and may also show slow detachment and a corresponding long-term release of deposited colloids from the bed. Biological processesuch as biofilm growth can alter the properties of the streambed and may control colloid attachment. Finally, the long-term dynamics of suspended sediment transport involves multiple feedbacks between sediment supply, deposition in the bed, biological growth, and alteration of aggregate bed sediment properties. Many aspects of these feedbacks are not well understood at the present time. In the context of the preceding discussion, the work presented here can be seen as a framework for the analysis of hyporheic exchange and interactions between suspended and bed sediments. The process-based, physicochemical model was shown to adequately represent solute and colloid exchange with a simple sand bed in a laboratory flume. As we develop improved methods for the analysis of stream system heterogeneity, complex biogeochemical processes, and nonideal colloid

13 PACKMAN AND BROOKS: HYPORHEIC EXCHANGE OF SOLUTES AND COLLOIDS 2603 filtration, the model presented here can be extended to be more directly applicable to natural systems. Notation,4 area of the streambed C tracer concentration in the stream, measured by analysis of a grab sample. Co initial tracer concentration in the stream. C* dimensionless tracer concentration, equal to C/Co. A C difference in concentration between stream water and pore water (when both are assumed to be wellmixed). d stream depth (in flume experiments, average of measurements every half meter). dt depth of sand bed (average of measurements every half meter). d* b dimensionless bed depth, d* b = kdt. d' effective water column depth, the total volume of stream water divided by the bed area. In the flume, d' is the stream depth plus an additional effective depth corresponding to the volume of water in the return pipe and end sections of the flume. d' * dimensionless effective water column depth, equal to 2,rkd' / O. Drn s depth of maximum scour, which could easily be observed through the glass window in the side of the flume. D* ms dimensionless depth of maximum scour, equal to kd rns. h half amplitude of the sinusoidal distribution of dynamic pressure head at the bed surface, calculated using an equation developed by Elliott and Brooks [1997a]. H average bed form height (trough to crest), a large number of bed forms (20-50) were measured in all experiments. k bed form wave number, equivalent to 2,r/X. K hydraulic conductivity of bed sediment, equal to 9.0 cm/min for all experiments (at T 21øC). M equivalent penetration depth, a measure of tracer exchange with the bed. M* dimensionless equivalent penetration depth into the bed, equal to 2,rkM. M penetration depth due to pumping. M* P dimensionless penetration depth due to pumping, equal to 2,rkM. Mr penetration depth due to turnover. qp exchange flux of solutes or colloids due to pumping. qp ß dimensionless pumping flux, equal to qp/um ß p piston velocity for exchange with the bed, equivalento exchange flux/ac. Q recirculation flow rate in the flume as measured by a calibrated Venturi meter/manometer combination. S slope of the energy grade line. t time. t* dimensionless time, equal to k2khm t. t an time of the final clay concentration measurement in the experiment, provided to indicate the overall duration of each experiment and to facilitate conversion of Figures 3 and 4 into dimensional units. T temperature. u horizontal Darcy velocity (of water, tracer, or particle). u* dimensionless Darcy horizontal velocity, equal to U/Urnø u m,maximum Darcy velocity in the bed calculated from the pumping model, equal to kkhm. Up horizontal Darcy particle velocity. Up dimensionless horizontal Darcy particle velocity, equal to Up/Urn. u, underflow velocity, the horizontal velocity induced by the slope of the stream. u, dimensionless underflow Darcy velocity, equal to U u/u rn. U average stream velocity over the cross section of the channel. Ut bed form velocity, taken as the average propagation rate (Ax/At) of many bed forms; bed form travel over a Ax of m was observed through a glass window in the side of the flume. U dimensionless bed form velocity, equal to Ut /(urn/ 0). v vertical Darcy velocity (of water, tracer, or particle). v* dimensionless vertical Darcy velocity, equal to v/ (y) * Urn, average downward velocity through a horizontal plane at the depth y. dimensionless average downward velocity, equal to v,p vertical Darcy particle velocity. vp dimensionless vertical particle velocity, Vp = vp/ Urn. v (y) average downward particle velocity through a horizontal plane at the depth y. Z dimensionless average downward particle velocity. Vs particle settling velocity, usually taken as the Stokes settling velocity. V*s dimensionless particle settling velocity, equal to Ors/Urn. x longitudinal coordinate along the streambed. x* dimensionless longitudinal coordinate, equal to kx. y vertical coordinate. y* dimensionless vertical coordinate, equal to ky. a exchange coefficient from the Transient Storage Model (units of inverse time). X dune wavelength. X* dimensionless dune wavelength, equal to kx or 2,r. 0 porosity of bed sediment (equal to for all experiments). % '* dummy time variables for integration. Acknowledgments. The authors gratefully acknowledge financial support for this research through NSF grants BCS and BES , and an NDSEG/ONR fellowship which supported the graduate study of the first author. We also greatly appreciate the suggestions of three very knowledgeable reviewers, Markus Huettel, Jud Harvey, and Kit Rutherford, which helped us to improve the manuscript. References Allan, J. D., Stream Ecology: Structure and Function of Running Waters, Chapman and Hall, New York, Bencala, K. E., Simulation of solute transport in a mountain pool-andriffle stream with a kinetic mass transfer model for sorption, Water Resour. Res., 19(3), , 1983.

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