Effect of bed form geometry on the penetration of nonreactive solutes into a streambed

Transcription

1 Effect of bed form geometry on the penetration of nonreactive solutes into a streambed Andrea Marion and Matteo Bellinello Department of Hydraulic, Maritime, Environmental, and Geotechnical Engineering, University of Padua, Padua, Italy Ian Guymer Department of ivil and Structural Engineering, University of Sheffield, Sheffield, UK Aaron Packman WATER RESOURES RESEARH, VOL. 38, NO. 10, 1209, doi: /2001wr000264, 2002 orrection published 10 January 2003 Department of ivil and Environmental Engineering, Northwestern University, Evanston, Illinois, USA Received 19 January 2001; revised 21 November 2001; accepted 28 January 2002; published 23 October [1] The contamination of riverbeds by solutes derived from the surface flow has recently received increasing attention. hannel morphological features such as bed forms are important characteristics of the stream-subsurface interface and represent one control on the rate of solute delivery from the stream to the bed. Generally, larger bed forms are expected to produce greater rates of stream-subsurface exchange. However, the longitudinal dimension (wavelength) of the bed form is also important, and this effect can produce penetration patterns that may be unexpected from a visual observation of the bed surface. Experimental tests in a recirculating flume demonstrate these effects. ommonly used mathematical models do not consider the bed form geometry explicitly and depend on the availability of calibration data to derive exchange parameters for each stream reach. More detailed models that consider the effect of bed form shape are capable of simulating some of the observed experimental results. However, existing physically based models are shown to be insufficient for some bed form geometries that may occur in real streams. INDEX TERM: 1871 Hydrology: Surface water quality; KEYWORDS: solute transport, hyporheic, river contamination itation: Marion, A., M. Bellinello, I. Guymer, and A. Packman, Effect of bed form geometry on the penetration of nonreactive solutes into a streambed, Water Resour. Res., 38(10), 1209, doi: /2001wr000264, Introduction [2] Soluble pollutants entering a river are subject to a number of transport processes. Solutes can both disperse in the surface flow and penetrate permeable channel boundaries. While surface dispersion has been widely studied, following the seminal work of Taylor [1921], exchange between the surface flow and the subsurface water has often been neglected. Only in recent years has stream-subsurface exchange received significant attention, primarily for its role in controlling material fluxes of contaminants and ecologically relevant substances such as nutrients. Stream-subsurface exchange can provide a key control on the net transport of reactive substances because this exchange exposes solutes and suspended matter to subsurface biogeochemical processes. In order to understand the net effect of this exchange on downstream transport, both the physical exchange processes and subsurface reactions must be understood. [3] Penetration of stream-borne substances into permeable streambeds occurs due to several physical processes, primarily advective transport [Bencala and Walters, 1983; Thibodeax and Boyle, 1987; Savant et al., 1987; Harvey and Bencala, 1993; Elliott and Brooks, 1997a]. The bed provides temporary storage of solutes: it receives a net opyright 2002 by the American Geophysical Union /02/2001WR influx during periods of high solute concentration in the flow, and subsequently releases solutes when the concentration in the stream declines [Bencala and Walters, 1983]. The near-stream region where this exchange takes place has been termed the hyporheic zone (Figure 1). It has been observed that the hyporheic zone can be divided into two layers: a superficial layer, where pore water has very similar properties as the water in the main flow, and a lower one where chemical and physical gradients exist [Triska et al., 1989]. [4] Field observations have shown that the hyporheic zone constitutes a distinct ecotone within the larger stream-aquifer system [oleman and Hynes, 1970; Hynes, 1974; Stanford and Gaufin, 1974; Stanford and Ward, 1988]. There has been increasing interest in the transfer of nutrients into and through the hyporheic zone [Triska et al., 1989, 1990, 1993]. Several recent reviews have focused on the importance of hyporheic transport for stream ecology [Allan, 1995; Brunke and Gonser, 1997; Jones and Mulholland, 2000]. A number of field tests have also been performed to quantify the chemical and physical processes involved in stream-subsurface exchange [e.g., Bencala, 1984; erling et al., 1990; astro and Hornberger, 1991; Vallet et al., 1996; Mulholland et al., 1997; Wörman, 1998]. Laboratory experiments have examined the physical exchange processes in detail, and facilitated the development of process-based models for exchange [Thibodeax and

2 27-2 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT Figure 1. Schematic of river ecotons. The hyporheic zone is the most affected by the properties of the flowing water (modified from Triska et al. [1993]). Boyle, 1987; Shimizu et al., 1990; Elliott, 1990; Elliott and Brooks, 1997a, 1997b; Hutchinson and Webster, 1998]. More recently, fundamental models for solute exchange have also been extended to treat the penetration of solutes undergoing sorption onto sediment surfaces [Eylers et al., 1995; Forman, 1998] and the penetration of colloidal particles [Packman et al., 2000a, 2000b]. [5] In most practical situations, the interfacial flux of contaminants is slow compared to the transport by the main flow. In treating boundary exchanges, vertical and transverse mixing processes in the main flow can be normally neglected, as they take place over much smaller timescales, and a one-dimensional advection-diffusion equation can be written for a conservative solute as @x Q AD L ð1þ where t is time, x is the longitudinal coordinate, is the section-averaged solute concentration, Q w is the water discharge, A is the cross-sectional area of flow, D L is the longitudinal dispersion coefficient, m,b is the mass flux from the stream into the wetted perimeter, P is the wetted perimeter. This paper will focus on the evaluation of the exchange term with the permeable streambed, m,b, for the case where there is no lateral inflow or outflow. [6] For field applications, Equation (1) is generally employed with an effective representation of the exchange flux characterized by some lumped exchange coefficient(s). The most commonly used approach is that of the Transient Storage Model, which characterizes stream-subsurface exchange with a mass transfer term a(- s ), where a is an empirical exchange coefficient with units of inverse time and s is the concentration in the subsurface storage zone [Bencala and Walters, 1983]. The model is implemented in a finite difference code for One-dimensional downstream Transport with Inflow and Storage, OTIS [Runkel, 1998]. Other forms of lumped exchange models have also been developed, e.g., the Aggregated Dead Zone model [Young and Wallis, 1986]. However, all of these models lack generality because they depend on curve fitting of experimental solute injection data to derive exchange parameters on a reach-by-reach basis. [7] Physical reasoning and experimental evidence [Savant et al., 1987; Elliott, 1990; Elliott and Brooks, 1997a, 1997b] have clarified that tracer pumping along advective subsurface streamlines dominates local diffusion processes, except for a thin interfacial layer which is still affected by turbulence [Mendoza, 1988; Zhou and Mendoza, 1993, 1995]. The situation is of course different when the sediments are extremely fine and the bed is essentially impermeable [Rutherford et al., 1993, 1995]. In this case, turnover-type exchange will often dominate. The exchange due to turnover consists of the trapping of the stream water within the pores of the particles depositing on the downstream part of a bed form, balanced by the release of an equivalent volume of water from the eroded upstream face of the bed form. This type of interaction becomes important when bed forms move with significant speed [Packman and Brooks, 2001]. It is also worth noting that pumping-type processes are not restricted to bed forms, but may also produce exchange at other spatial scales. Equivalent considerations apply to all cases in which pressure gradients are generated at the boundaries of a porous domain, e.g., at river bends and steps [Harvey and Bencala, 1993; Wroblicky et al., 1998; Wondzell and Swanson, 1996]. The combined effect of different storage zones has been discussed by hoi et al. [2000]. [8] While fundamental models for bed form-driven solute exchange have been validated in small laboratory flumes [Elliott and Brooks, 1997a, 1997b; Packman et al., 1997], a key need for field application is to demonstrate that these models will also work well for the wider variety of bed topography that can be expected in natural streams. This paper describes stream-subsurface exchange experiments that were carried out in a considerably larger flume than any that had been previously used for this type of study. The larger flume allowed experiments to be conducted with natural three-dimensional bed forms and with artificial two-dimensional bed forms that were large relative to the water depth. It will be shown that the existing models do not always adequately represent the exchange due to these large bed features, and thus that additional models are needed to represent the range of morphological features present in natural streams. 2. Theory [9] Two models will be used to evaluate our experimental results. First, a diffusion model will be used to evaluate the exchange with a flat bed. The additional exchange due to the large bed forms will then be assessed with a fundamental pumping model. [10] Richardson and Parr [1988] developed a model for the diffusive release from a sediment due to an overlying water flow. They considered subsurface transport to be due to 1D vertical diffusion only. The resulting exchange is: mðtþ ¼ t ðþd e E2 t t 1=2 D erfc E 1 þ 2E t 1=2 ; ð2þ E D pd where m is the mass exchanged per unit area, t is the time elapsed and E is an exchange coefficient that depends on the

3 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT 27-3 boundary condition at the interface. The diffusion coefficient, D, is a function of the shear Peclét number, i.e., it is dependent upon the hydrodynamic interaction at the interface and the properties of the porous medium, Pe ¼ u * D m D ¼ 6: D m Pe 2 ; sffiffiffiffiffiffiffiffi K m n ; ð3þ g where u * is the shear velocity, n is the kinematic viscosity of water, g is gravity, K m is the permeability of the porous medium and D m is the molecular diffusion coefficient. Richardson and Parr also obtained an empirical expression for E: p E ¼ 0:103 ffiffiffiffi D ; ð5þ where E is in m/s and D is m 2 /s (converted from original units). [11] Actual exchange processes for a flat bed are likely to be a combination of molecular diffusion plus additional small-scale advective pumping processes [Elliott and Brooks, 1997a]. Nonetheless, the diffusive flux model can readily be used as a framework to evaluate exchanges from flat beds, where an effective diffusion coefficient is derived to represent the combined effects of several transport processes. [12] Elliott and Brooks [1997a] proposed a model for bed form-driven solute exchange that includes a description of the underlying physical exchange processes. The laminar flow of water in a homogeneous and isotropic porous medium is governed by Darcy s law. Once coupled with the continuity equation, Darcy s law leads to Laplace s wellknown equation for the distribution of head in the porous medium. Laplace s equation can readily be solved given either the head or the velocity distribution at the boundaries. Elliott and Brooks employed a simplified bed geometry to obtain an analytical solution of Laplace s equation and the streamlines in the porous medium. These streamlines, but not the resulting interfacial exchange, had previously been calculated by Ho and Gelhar [1973]. Elliott and Brooks assumed that the head at the interface would be sinusoidal with an amplitude dependent on the size of the bed forms present. This is in agreement with the findings of Vittal et al. [1977] and Fehlman [1985]. From the flow pattern in the porous medium, Elliott and Brooks derived the residence time for particles entering the medium at any boundary position using the approach employed by Jury et al. [1986] and White et al. [1986]. They then obtained the total mass stored in the bed with a convolution integral. Elliott [1990] also considered the effects of subsurface longitudinal flow, variable bed permeability, and moving bed forms. This model was tested by observing net stream-subsurface exchange in a rectangular, 15cm-wide, 5m-long experimental flume, with a bed of fine sand (d 50 = 0.47 mm and d 50 = 0.13 mm) [Elliott and Brooks, 1997b]. Additional studies have been conducted in this flume by Eylers et al. [1995] and Forman [1998], and in other similar channels such as the 12m-long flume used by Packman et al. [1997, 2000a, ð4þ 2000b]. In the model, the wavelength of the head perturbation is assumed equal to that of the bed forms, while the amplitude of the perturbation h m is related to the bed form height through an empirical estimate derived by Elliott [1990] from Fehlman s [1985] data: h m ¼ 0:28 U 2 H=h y w y ¼ 3=8 if H=h w 0:34 ; ð6þ 2g 0:34 3=2 if H=h w > 0:34 where U is the mean flow velocity and h w is the flow depth. It is worth noting that equation (6) was obtained from experiments on regular triangular bed forms with a height/ length ratio of 1/7. The equivalent depth of penetration (t), is defined by mt ðþ t ðþ ; and the nondimensional depth of penetration * is defined as ð7þ * ¼ k ; ð8þ q where k = 2p/L f is the wave number (L f is the bed form length). These parameters are convenient representations of overall solute exchange with the bed. The rate of exchange between the surface stream and the pore water in the bed can be characterized by a nondimensional time based on the bed form size and maximum subsurface velocity: t* ¼ k2 K m h m t: ð9þ q [13] Elliott [1990] and Elliott and Brooks [1997a] showed that (t) or *(t*) can be calculated by integrating the residence time function over time. As exchange generally causes the in-stream concentration to change over time, the exchange calculation requires the solution of a convolution integral for (t) or *(t*). Elliott [1990] also derived a simplified solution for asymptotically large timescales. He introduced the well-mixed approximation, i.e., he assumed the concentration in the bed to be uniform and equal to the concentration in the overlying flow. onsidering then the solution at long times and using a Taylor series truncation of the residence function, he obtained a good estimator of the full solution: * ffi lnð0:42t* þ 1Þ; ð10þ valid for t* > 3. This solution is quite intuitive and simple to apply, and it will be used here to evaluate the exchange that was observed in experiments that ran for times t* > 20. The full solution will be used for two other experiments that ran for t* Experimental Setup 3.1. Apparatus and Materials [14] The experiments were carried out in the 2.5m-wide, 25m-long Tilting Flume, at HR Wallingford (UK). The

4 27-4 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT Figure 2. View of the tilting flume and cross section of the working reach. flume is equipped with a recirculation system that allowed the use of a constant volume of water throughout each test. At the start of each experiment, a given quantity of solute was added to the flow, and the net exchange of solute was determined from the decrease of the solute concentration in the surface flow. The sand bed was placed in a modified, 65 cm wide working channel, which was contained inside the Tilting Flume by the construction of two 60cm-high parallel walls (Figure 2). The working length of the flume was 18.4 m. At the upstream end, a flare transition was created to lead the flow into the inner channel, avoiding flow separation from the sides. The lateral space (floodplains) was used to access the flume and to aid sampling. [15] After the construction of the inner channel, the bed and the sides were sealed to reduce leakage. A 40cm-deep layer of sand was then prepared in the flume. In the upstream transition, coarse gravel was inserted to develop a boundary layer in the flow upstream of the entrance. A plate was inserted into the bed in order to keep the sand separate from the gravel in the inlet. [16] The recirculating system was composed of a downstream sump of volume 3 to 3.5 m 3 that collected the water exiting the flume. The discharge flowed over a downstream end-gate, which was raised to an appropriate height to maintain uniform flow in the flume. The water was then delivered by a pump (Q max = 100 l/s) through a steel pipe (inner diameter = 174 mm, Volume stored ffi 1.27 m 3 )toan upstream sump having a volume of approximately 1.4 m 3. To calculate the total volume of the system, crucial for the solute mass balance, the following volumes were considered: 1.4 m 3 for the storage in the upstream transition; m 3 in the flowing stream (dependent upon the water depth); and 1.2 m 3 for the downstream storage between the gate and the sump. The total volume of the surface water (i.e., excluding the water trapped in the sediment pores) was estimated to be approximately = 10 m 3, with some variation among tests due to different water depths. [17] The discharge was measured by an orifice plate inserted in the return pipe. The flume was equipped with a manual gantry. The longitudinal position of the gantry was detected by a wire potentiometer. In all tests the flow regime was subcritical and the flow was approximately uniform over the flume length. Three laser bed-profilers (Nippon Automation, resolution 10 mm) were used to provide threedimensional profiles of the bed surface elevation. The profilers were mounted on the gantry in protective glass containers so that the sensor heads could be submerged during operation. The probes were positioned at each quarter of the flume width. The position of the gantry and the readings of the three profilers were logged to a computer. Before each test, the stream water level was measured manually by a point gauge at 2m intervals to check flow uniformity. [18] The upstream end of the flume was equipped with a conveyer belt, which was used to input sediments during the formation of bed forms. All tests were performed with silica sand (SiO 2 : 97.8%; Al 2 O 3 : 1%; organic content: <0.1%). Sand properties were determined by the Sedimentation Laboratory of HR Wallingford Ltd. The sand was uniform with a sieve size d = 0.85 mm, an in situ porosity q = 0.38, and a hydraulic conductivity K m = m/s. The volume of water stored in the pores of the bed in the flume was estimated from the in situ porosity as V p ffi 1.8 m 3, which is approximately 18% of the surface volume in the system. [19] Sodium chloride (Nal) was used as solute. The concentration was measured with a digital temperaturecorrecting conductivity meter (ole-palmer) located at the midpoint of the flume. The instrument was calibrated before every test Bed Form Generation and Geometry [20] The objective of the experimental activity was to measure the effect of natural and artificial bed forms on the exchange of solutes between the recirculating flow and the bed. The simplest bed geometry, a plane bed (test S1), was generated by moving a rectangular timber template along the flume. The bed profile is shown in Figure 3, while the geometrical properties are presented in Table 1. Tests with a plane bed were used as reference data, as discussed in data analysis below. It should be said that the definition of a plane bed here is somewhat arbitrary. In the experiments described here, the term refers to a bed in which the surface fluctuates around its mean value with variations of the order of the grain size, or of a few grain sizes. The scale of the variations from the mean is so small that it can hardly be measured unless grain-scale approaches are used to measure the bed profile [Gomez, 1993].

5 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT 27-5 Table 2. Bed Form Generation Description Formation Time Bed Slope Water Discharge, l/s Sediment Input, g/s Ripples 5 min Dunes and ripples 1.5 hours Figure 3. Bed profiles for the tests. [21] The generation of natural bed forms that would remain immobile under experimental flow conditions was not a trivial problem. It was not possible to develop bed forms at high flow and then reduce the flow below the critical condition for sediment motion, because this would have limited the experimental condition to extremely small discharges. In particular, the water discharge and the sediment input by the conveyer belt could not be raised above some equipment-dependent limits. It was also necessary to maintain a sufficient water depth at low flow, i.e., at tests conditions. It was therefore decided to develop the bed forms at high slopes, and then to reduce the slope to obtain both sediment stability and reasonable water depths. The expected response of the system for given discharge, slope and sediment input was calculated using Van Rijn s [1984a, 1984b, 1984c] method, and compared with other models for sediment transport. However, the formulae used gave somewhat different results and it was decided to use them only for general guidance and then to proceed by trial and error. [22] At the start of the generation process, the bed slope was set at the design value (Table 2). Then the conveyer belt was set at the appropriate speed to provide the desired sediment input. After some time, the flow velocity was reduced by raising the downstream gate, and the conveyer belt was stopped. The bed profile was then measured with the laser probes. Typical ripples were generated for test S2, as shown in Figures 3 and 4. Test S3 was run over a bed made of dunes covered by ripples (Figures 3 and 5). Bed form height, defined in terms of two times the standard deviation of the longitudinal bed profile to account for the irregular pattern, and bed form wavelength are reported in Table 1. [23] Tests were also conducted with artificial sinusoidal and triangular bed forms. Sinusoidal bed forms were formed by hand using a timber template. The sinusoidal bed forms were two-dimensional (extending completely across the flume) and had a regular length. Triangular bed forms were generated by producing spatially periodic disturbances of the bed (two-dimensional sediment heaps), which were then worked by the flow until they achieved a stable shape (Figure 6). The sinusoidal bed forms are therefore completely artificial in the sense that they were never shaped by flow processes, while the triangular bed forms represent a natural shape produced by the streamflow from an imposed initial bed geometry. 4. Experimental Procedures 4.1. Test onditions and Duration [24] All tests were performed with constant discharge and uniform flow conditions. In order to conduct experiments Table 1. Bed Form Properties Description Bed form height H, m Wavelength L f,m L f /H Plane Natural ripples Natural dunes and ripples Artificial sinusoidal dunes Artificial triangular dunes Figure 4. Natural ripples as used in test S2.

6 27-6 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT measurements that the solute mixed fully in the system in approximately 30 min, i.e., over about 3 times the average surface water recirculation time of the system. The conductivity meter was placed on the gantry approximately midway along the flume length. The concentration was measured every 10 min, using triplicate readings for each measurement. Tests lasted up to 10 hours. [27] At the completion of each test, the surface water was easily removed by draining the downstream sump. The bed sediments were then mixed to release all trapped water, and the flume was refilled with clean water for the next experiment. oncentration measurements confirmed that no significant amount of solute was retained from one experiment to the next. Figure 5. Natural dunes/ripples as used in test S3. without sediment motion, both the slope and discharge were greatly reduced from the conditions used for bed form generation (Table 3). It could be argued that the shape and properties of the bed form might never at that discharge under natural conditions. However, as the objective of the present work was to evaluate the performance of existing exchange models, the use of these bed forms is justified. [25] The bed profile was surveyed before the start of each test. Then the system was filled with water. Particular care was required in filling the system, as the front generated by inflow can alter the bed form shape. Therefore, the flume was filled slowly (Q = 2 l/s) from the downstream end, with the tailgate fully raised to store water and reduce the effects of front propagation. Once the flume was full, the fill pump was switched off and the recirculating system was started and slowly brought to the design discharge. Then the tailgate was slowly lowered until uniform flow conditions were reached, i.e., until the water surface slope equaled the average bed slope. The operating position of the tailgate was marked, and then the tailgate was raised in order to produce a good flow condition for the solute injection. [26] During the solute injection, the flow velocity in the flume was greatly reduced by a strong backwater effect in order to reduce the mass penetration into the bed during the transition to a well-mixed concentration in the surface water. A lower velocity implies much lower pressure gradients, as the amplitude of the pressure perturbation scales with the velocity squared (see Equation 6). This minimized any exchange during the initial surface mixing. The salt was first dissolved at high concentration (20 to 22 kg of salt in 150 l of water, Table 3) in a separate water tank, and then the concentrated solution was poured into the downstream sump. It was estimated from concentration 4.2. Evaluation of System Losses [28] The plane bed test (S1) showed that significant solute penetration occurred in the absence of bed forms. The plane bed test S1 was used as a reference test or baseline for the other cases reported here. This allowed the evaluation of solute detention in the gravel which filled the transition reach at the entrance of the flume or in any stagnation (or dead) zones that might exist in the sump or other parts of the recirculation system. [29] The plane bed results were first compared with Richardson and Parr s [1988] model. Equations (3) and (4) were used to calculate the Peclét number and the effective diffusion coefficient, using the following physical parameters [Weast, 1988]: D m = m 2 /s, n = m 2 /s at T = 10. For large times, equation (2) can be simplified to: " mt ðþ¼ t ðþd 1 þ 2E 1 # 1=2 ; ð11þ E pd which, for the conditions of test S1, applies for times larger than 55 min. Figure 6. Artificial triangular dunes, as used in test S4.

7 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT 27-7 Table 3. Test onditions Test Duration, hours Description Mean Bed Slope S Discharge Q, l/s Water Depth h w,cm Mass Added Nal, kg S1 10 plane S2 10 natural ripples S3 10 natural dunes and ripples S4 9 artificial sinusoidal dunes S5 8 artificial triangular dunes [30] In a recirculating system, if the change in solute concentration in the stream only occurred due to exchange with the bed, the following mass balance equation would apply: t ðþ ; ð12þ t ðþþmt ðþa b 0 where is the total volume of surface water in the system (exclusive of the pore water in the bed), A b is the area of the plane projection of the bed (for a rectangular flume, A b = LB), and again m is the mass of solute exchanged per unit bed area. Introducing the equivalent depth of penetration (t) defined in (7), Equation (12) becomes: þ ðþ t LB : Inserting (11) in (12), the following equation is obtained: 0 1 n 1 þ D E 1 þ 2E t pd o 1=2 : LB ð13þ ð14þ which reproduces the experimental curve for test S1 (Figure 7). Additional confidence in this correction is provided by the fact that the asymptotic system loss was 11% of the system volume, which closely corresponds to the ratio between the permeable volume of the inlet region and the total system volume. Therefore, the additional solute detention can be fully attributed to the effect of the porous volume occupying the upstream end of the flume. 5. Results and Model Evaluation [31] The mass of solute entering the bed was derived from the measurement of the concentration in the surface volume of water. The results for all experiments are shown in Figure 8, plotted as a nondimensional concentration (normalized with initial concentration 0 ) versus time. As described previously, time was measured from the moment the solute reached a uniform concentration in the flume. As expected, all tests show a decay of the solute concentration in the surface flow. The rate of decay was high at the omparison between the model (14) and the experimental data is very poor, due to the fact that there were losses of solute within the recirculation system. These solute losses were relatively large, particularly to the gravel transition at the entrance of the channel. The plane bed case was therefore used to quantify the system losses according to the following procedure. onsidering the volume of water that would generate an equivalent system loss, V p (t), Equation (12) can be modified as follows: þ t ðþ LB þ Vp ðþ t ; ð15þ and V p (t)/ is derived as: V p 0 ðþ 1 t LB ðþ: t ð16þ The function V p (t)/ was obtained interpolating the data with an exponential curve (R 2 = 0.997): V p ðþ¼0:11 t 1 e 0:1505t ; ð17þ where t is in hours. Therefore, the following equation is obtained: 0 1 o 1=2 n 1 þ D E 1 þ 2E t pd LB þ Vp ; ð18þ ðþ t Figure 7. Evaluation of system losses by comparison of the concentration loss predicted by Richardson and Parr s [1988] equation and the experimental results with a flat bed.

8 27-8 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT distribution over a porous bed with a plane interface between the flow and the bed. Equation (13) can be rewritten as þ q A b k * ðþ t ; ð19þ and, using the dimensionless time t* introduced in Equation (9), Equation (15) can be written as follows: ðt* Þ ¼ þ * ðt* Þ q k LB þ Vp ðt* Þ ; ð20þ where V p / (t*) is the equivalent to (17) written in terms of t* and represents the system losses evaluated from the plane bed case. The durations of the experiments are reported in Table 4 in terms of t*. For tests S2 and S3, with durations of t* = 90.4 and 29.3, respectively, Elliott and Brooks [1997a] model has been used in the simplified form (10). In this case, equation (20) becomes: 1 ðt* Þ ¼ 0 1 þ lnð0:42t* þ 1Þ q k LB þ Vp ðt* Þ : ð21þ Figure 8. flow. Decay of solute concentration in the surface beginning of the test and gradually decreased as the experiments progressed. [32] Inspection of the results shows that the solute penetration due to natural bed forms was larger than the penetration obtained with a plane bed and smaller than the penetration due to large artificial bed forms. The results also confirm that bed form amplitude is not the only significant control on exchange, as tests S2 and S3 gave very similar penetrations even though they had bed forms of very different sizes (Figure 8). Also, triangular bed forms produced a larger initial exchange rate than was observed with sinusoidal bed forms, but the rate decreased more rapidly with the triangular case. This caused the in-stream concentration curves to cross after 6 7 hours. This behavior was also observed in other penetration tests using fluorometry, which are not reported here [Bellinello, 1999]. [33] Bed form tests S2 S5 were compared with Elliott s [1990] model, which was derived for a sinusoidal head For tests S4 and S5, with durations of t* = 7.9 and 8.6, respectively, the convolution integral was used for * and Equation (20) was evaluated numerically. The model predictions are compared with experimental data in Figures It can be seen that the model represents well the exchange due to natural bed forms and, for large times, that of the artificial bed with sinusoidal bed forms. However, it does not simulate well the exchange due to artificial triangular bed forms that occupy a significant fraction of the flow depth. 6. Discussion [34] Elliott and Brooks idealization of the streambed consists of a plane bed with a sinusoidal dynamic head distribution [Elliott and Brooks, 1997a]. The model applies the head perturbation that results from flow over bed forms as a boundary condition at the surface of the plane bed. Elliott and Brooks [1997b] demonstrated that this model successfully predicts the exchange due to flow over dunes when the bed forms have only a small deviation from the mean bed elevation. In their experiments, the bed form height was on the order of 1.5 cm while the stream depth was on the order of 10 cm. In this case, the assumption of a flat bed with the correct head distribution was clearly reasonable, as the model predicted experimental results quite well. This is expected to often be applicable in nature because dunes are known to scale with the Table 4. Surface System Volume and Hydraulic Quantities for Tests Test System Duration, t* Description Mean velocity Volume,m 3 U, m/s Amplitude of Head Variation a h m,m S natural ripples S natural dunes/ripples S artificial sinusoidal dunes S artificial triangular dunes a See equation (21).

9 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT 27-9 flow depth, and have a height on the order of 1/10 of the flow depth. Indeed, the naturally formed bed forms in cases S2 and S3 have bed form height : flow depth ratios of 1/22.4 and 1/10.7, respectively, and the pumping model predicted the resulting exchange well. However, the cases S4 and S5 reported here protrude much farther into the streamflow, with bed form height : flow depth ratios of 1/ 3.5 and 1/4.5, respectively. Protruding bed forms of this type can be generated in nature by high streamflow stages, and then left on the streambed as relict features at lower flow stages. This extensive protrusion clearly alters the pumping rate and causes the idealized pumping model to misrepresent the observed exchanges, particularly at small times. [35] Two effects can explain the deviation of the observed exchange from the idealized pumping model prediction. In the triangular case (S3), a distinct flow separation takes place at the top of the bed form, which presumably produces higher differential pressures across the bed form. Further, the flow paths through the exposed tip of the bed form are shorter than that occur through a plane bed with a sinusoidal dynamic head distribution. As a result, the exchange through protruding bed forms is expected to be greater than that predicted by the Elliott and Brooks [1997a] model. However, it is only possible to observe this effect early in recirculating flume experiments, as the protruding portions of the bed forms will rapidly become well mixed with the streamflow. This effect can account for the higher initial exchange rate in experiments S4 and S5. At later times, the shorter flow paths through the triangular bed forms also Figure 9. omparison of experimental results versus the modified Elliott and Brooks [1997a] model in the test with natural ripples. Figure 10. omparison of experimental results versus the modified Elliott and Brooks [1997a] model in the test with natural dunes and ripples. apparently hinder the extension of a head gradient deep into the bed. This limits the exchange with triangular bed forms to a shallower layer than would be predicted by Elliott and Brooks [1997a] pumping model, as reflected by the crossing of (t) curves in Figure 8. [36] The implications of this work are that additional models will be necessary to predict stream-subsurface exchange in the field. Several types of bed forms are known to exist, and considerable variation in the bed topography is expected within a stream system. The fundamental pumping model developed by Elliott and Brooks [1997a, 1997b] and extended by Eylers et al. [1995], Packman et al. [2000a, 2000b], and Packman and Brooks [2001] considers only typical two-dimensional dunes and ripples. This model is powerful in that it can predict exchange based on fundamental hydrodynamic interactions between the stream and bed. The properly scaled dimensionless variables in the pumping model indicate the relationships between the key variables that control exchange. As the experimental results presented here demonstrate, this model can explain observations that would otherwise be confusing, e.g., that short bed forms and tall bed forms can produce the same net exchange because bed form height is not the only controlling variable and, instead, the parameter grouping k 2 K m h m /q controls the rate of exchange. However, currently available models are only applicable for a limited range of bed form shapes. [37] Similarly, existing models do not realistically consider subsurface heterogeneity in a realistic fashion, and cannot predict the interfacial momentum or mass transfer due to turbulence. Thus, much work remains to be done to

10 27-10 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT bring modeling capability up to a sufficient level of complexity to consider field cases directly. In particular, care must be exercised in the application of existing models to low-order mountain streams, where extreme flow events and fresh sediment supply produce a fluvial environment with large bed features compared to base flows, and highly permeable and heterogeneous bed sediments. Nonetheless, existing fundamental models provide considerable insight into the functioning of the stream/hyporheos/aquifer corridor, indicating that field experimental programs should routinely consider the characteristics of the stream-subsurface interface when analyzing stream-aquifer connections. 7. onclusions [38] Solute exchange from a flowing stream to a permeable sediment bed is dominated by advection of water produced by head gradients at the porous boundary. The presence of bed forms in rivers induces an advective pumping flow that is largely responsible for mixing in the hyporheic zone. The subsurface flow is affected not only by the amplitude of bed topographical features, but also by their wavelength. This leads to cases where very different bed configurations can produce the same amount of solute penetration. The model proposed by Elliott and Brooks [1997a] provides a direct explanation of this behavior. [39] The experimental results also show that existing fundamental models become insufficient when extreme bed form sizes are present, even when these features are regular and two-dimensional. We attribute the discrepancy between the model prediction and the observed results to the Figure 11. omparison of experimental results versus the modified Elliot and Brooks [1997a] model in the test with artificial sinusoidal dunes. Figure 12. omparison of experimental results versus the modified Elliott and Brooks [1997a] model in the test with natural artificial triangular dunes. fact that the Elliott and Brooks [1997b] model does not account for storage in the part of the bed form that protrudes above the mean bed elevation. [40] The results presented here demonstrate that it may not be straightforward to translate visual observations of stream patterns into estimates of hyporheic exchange. However, an understanding of the fundamental hydraulic processes that drive exchange can indicate the parameters that control net transport. In the field, a reasonable survey of the distribution and size of bed features can be used to estimate the head fluctuations at the stream-subsurface interface. Further analysis of bed sediment properties, most notably hydraulic conductivity, is then required to estimate exchange fluxes. Notation A cross-sectional area of flow. A b area of the plane projection of the bed. B flume width. section-averaged solute concentration. 0 initial solute concentration. * nondimensional solute concentration. s concentration in the storage zone. D diffusion coefficient. D L longitudinal dispersion coefficient. D m molecular diffusion coefficient. E exchange coefficient. g acceleration due to gravity. H bed form height. head perturbation height. h m

11 MARION ET AL.: EFFET OF BED GEOMETRY ON HYPORHEI TRANSPORT h w flow depth. k bed form wave number. K m hydraulic conductivity. L working length of flume. L f bed form length. m mass exchanged per unit area. P wetted perimeter. Pe Peclét number. Q w water discharge. T temperature. t time. t* nondimensional time. U mean flow velocity. u * shear velocity. V p volume of water stored in the pores. volume of surface water in the system. x longitudinal coordinate. a exchange coefficient. mass flux from the stream into the wetted perimeter. y exponent. q porosity. equivalent depth of penetration. * nondimensional equivalent depth of penetration. n kinematic viscosity of water. m, b [41] Acknowledgments. Funding for the experimental part of this work was provided by the E ommission, Directorate General for Science, Research and Development, as a part of the HM Large Installation Programme, Sediment Transport at High River Stage, under ontract ERBHGET Data analysis has been partially funded by the University of Padua (funds e x 60% ) and by the Italian National Research ouncil (NR), National Group on hemical-industrial and Environmental Hazards (Project Leader: S. Gabriele). Aaron Packman s involvement in this project was supported by the National Science Foundation through AREER award BES and travel grant INT Special thanks are due to R. Bettess of HR Wallingford Ltd (UK) for supervising the use of the Tilting Flume. Technical support from Robert Potter, Mattia Zaramella, and by B. Willetts s team is gratefully acknowledged. The comments of Jungyill hoi and two anonymous reviewers helped us to improve the manuscript. References Allan, J. 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