Subsurface flow within a single riffle of a low-gradient gravel bed stream was modeled in three dimensions using MODFLOW, a finite difference groundwater flow model. Model simulations showed that exchange flows can only occur in this low-gradient, gaining stream because of a zone of alluvial sediment around the stream that has much higher permeability than the surrounding catchment (K = 10−4 m s−1, compared with K = 10−6 to 10−8 m s−1). The key factors controlling exchange flow within the alluvial zone were identified as the hydraulic conductivity of the alluvium, the hydraulic gradient between upstream and downstream ends of the riffle, and the flux of groundwater entering the alluvium from the sides and beneath. In the study riffle each of these factors changes with season, causing a reversal of flow paths in the alluvium and a reduction in exchange flows from about 0.2–0.5 m3 d−1 per meter stream length in summer to about 0.008–0.04 m3 d−1 per meter stream length during fall to spring. The model also revealed that exchange flows are up to twice as strong, but more variable, at the sides of the stream than near the center, and that vertical flow paths beneath the channel are more persistent under the range of conditions modeled than lateral flow paths into the banks.
 The hyporheic zone is the zone of saturated sediment beneath and lateral to a stream channel that receives input of stream surface water. It can be delimited in many ways [e.g., Triska et al., 1989; cf. Williams, 1989] but essentially is defined by the extent to which surface water invades the subsurface beneath and lateral to a stream and returns to the stream surface farther downstream in a pattern known as exchange flow [Harvey and Wagner, 2000].
 The significance of channel-hyporheic exchange flows to stream ecosystems has been the subject of several recent discussion papers and review articles [e.g., Brunke and Gonser, 1997; Boulton et al., 1998; Jones and Mulholland, 2000]. Complex interactions between surface water and groundwater produce a unique and dynamic set of physicochemical conditions that support a distinct community of invertebrates (the hyporheos [Williams and Hynes, 1974]) and microorganisms [Storey et al., 1999]. As well as a habitat, hyporheic zones are a site for biogeochemical processes that can temporarily store nutrients and significantly alter stream water chemistry [Valett et al., 1997]. Stream energy budgets also are significantly different when the hyporheic compartment is included in addition to the surface community [Grimm and Fisher, 1984].
 Exchange flows occur simultaneously on various scales [Brunke and Gonser, 1997; Boulton et al., 1998], but a basic unit of exchange occurs across riffle-pool sequences. On this scale, surface water enters the subsurface sediments at the upstream end of a riffle (a shallow, fast flowing section of a stream) and returns to the stream channel at the downstream end [Hendricks and White, 1991; Valett et al., 1994]. Riffle-scale exchange flows are of particular significance to the stream ecosystem as they are ubiquitous and likely account for more interaction between surface and subsurface waters than do the longer flow paths [Harvey and Wagner, 2000]. They are especially relevant to the hyporheic biota because the water they carry has a relatively short residence time and thus maintains significant concentrations of dissolved oxygen.
 Laboratory flume studies have provided mechanistic explanations for the formation of riffle-scale exchange flows in terms of hydraulic heads in the surface water [Thibodeaux and Boyle, 1987; Elliott and Brooks, 1997]. Whereas flumes are isolated systems, however, streams typically receive net discharge of water from an underlying aquifer. With sometimes strong hydraulic gradients from the aquifer toward the stream surface, one might expect that exchange flows would be prevented by groundwater discharge flows. Harvey and Bencala  showed, using a modeling approach, that in a steep mountain stream, exchange flows can occur in the presence of a discharging aquifer due to discontinuities in the stream gradient, in that case where the stream gradient steepened from <1% to about 20%. However they did not describe other properties of the streambed or aquifer that are prerequisite for these exchange flows to occur, properties that are likely to be critical in low-gradient lowland streams where such large breaks in stream slope are rarely found. Furthermore, they did not attempt to find relationships between hydrological or geological parameters of the system and the vertical or lateral extent of exchange flows.
 To explain the control of hyporheic zone size using this many factors, however, mitigates against one of the main goals of current hyporheic research, that is, the development of simple frameworks that describe and predict the characteristics of different hyporheic zones [Stanley and Jones, 2000]. Part of the difficulty has been that stream tracer studies are not able to distinguish between surface and subsurface storage zones, and therefore changes in stream properties that are reported simply as affecting “storage zone size” may in fact be affecting primarily the component of exchange that occurs between the main channel and pockets of slowly moving surface water [D'Angelo et al., 1993]. Because stream tracer studies operate from a surface water perspective, they tend to emphasize factors operating in the surface water, such as stream discharge and channel friction. Since flow in the hyporheic zone obeys the laws of saturated flow through a porous medium, for some purposes it is more appropriate to study the controlling factors using a groundwater approach, in which relatively few factors govern flow.
 A groundwater modeling approach has been used in several studies [Harvey and Bencala, 1993; Wondzell and Swanson, 1996; Wroblicky et al., 1998; Woessner, 2000] to examine factors controlling hyporheic zone size and flow paths. Harvey and Bencala  were the first to show, through a numerical model, that lateral channel-hyporheic exchange flows could occur in the presence of a discharging aquifer, due to discontinuities in stream slope. Although in their study, exchange flow paths typically persisted throughout the year, they showed that exchange flows were reduced during extremely high precipitation events, because of the high hydraulic gradients toward the stream that were produced in the adjacent aquifer. Wroblicky et al.  showed that lateral exchange flows also may be formed by streambed meanders. Further, they showed that the area bounded by such exchange flows may decrease or increase in size due to changes in groundwater discharge, and may differ between streams due to differences in hydraulic conductivity of the alluvial sediments. Woessner , taking a vertical cross section along a hyporheic zone, demonstrated that small variations in stream surface head and variations in subsurface hydraulic conductivity greatly increase the complexity of exchange flow paths.
 These models have helped develop our understanding of the factors controlling hyporheic exchange flows; however, they have not attempted to use the groundwater modeling approach to evaluate the range of controlling factors suggested by the authors of stream tracer studies. Furthermore, because these models all have represented the stream as a single cell wide, none have been able to compare the flow paths beneath the middle of the stream with those at the sides, and because all were two-dimensional models, none have attempted to compare the behavior of vertical versus lateral flow paths under varying hydrological conditions. Invertebrate and microbial communities in the hyporheic zone exist in three dimensions and respond to fine-scale dynamics of exchange flows, and therefore a high resolution, three-dimensional model is appropriate for increasing our understanding of hyporheic biological communities and the biogeochemical processes they perform.
 Specifically, the aims of the present study are (1) to identify the hydrological and geological conditions that are required for hyporheic exchange flows to occur in the presence of a discharging aquifer; (2) to evaluate the range of potential controlling factors cited by other studies, and condense them to a few key factors that together are sufficient to explain seasonal changes in riffle-scale hyporheic exchange flows and differences between streams; and (3) to describe and explain differences in vertical versus lateral exchange flows and differences in flow paths occurring in different parts of the streambed.
 The Speed River, southern Ontario, is a gravel bed stream that flows across undulating terrain of glacial origin. The low topographic relief produces a low streambed gradient of 2–5 m per km. The primary aquifer is in the dolomite bedrock, which, in the vicinity of the study site, is about 20 m below the ground surface.
 The bedrock is overlain by layers of low-permeability glacial till, kame, and outwash deposits (hydraulic conductivity K = 10−7 to 10−8 m s−1). The stream itself lies in a bed of recent alluvium, 1–1.5 m deep and at least 5–10 m wide on each side of the stream. Salt tracers released up to 60 cm deep in the streambed during the study show that these alluvial deposits have a much higher K of about 2 × 10−4 m s−1 (unpublished data).
 At the sampling site the stream is approximately 6 m wide, varying in depth from about 0.15 to 0.35 m. In summer, base flow discharge is 0.1 m3 s−1, and in winter this increases by a factor of 2–3.
3.1. Field Studies
 Field studies, used to support model results, consisted of measuring hydraulic head distributions in three dimensions in a single 13-m-long riffle site (Figure 1). Hydraulic heads were measured to within ±3 mm, over four seasons from August 1996 to July 1997, with additional measurements during high and low base flow periods until November 1998. Measurements were made using nested minipiezometers (diameter 1.3 cm), each piezometer in the nest having a single 5-mm-diameter opening at 0, 20, 40, 60, 80, or 100 cm below the streambed surface [Fraser et al., 1996]. These were installed about 1 m apart and laid in two transects, one across the stream at the upstream end of the riffle to correspond with the down-welling zone (where surface water recharges the subsurface), and another along the axis of the stream between the upstream and downstream ends of the riffle to capture both the down-welling zone and the up-welling zone (where hyporheic water returns to the channel).
 Salt tracers were used to confirm flow directions indicated by hydraulic head measurements. Fifty-milliliter samples of concentrated NaCl solution were injected through minipiezometers at a depth of 40 cm beneath the streambed. Injection points were located in up-welling and down-welling zones, at the center and near the sides of the stream channel. The tracers were detected in nested minipiezometers with openings at 20, 40, and 60 cm, arranged around each release point in an arc of 25-cm radius. Electrical conductivity was measured in 10-mL water samples withdrawn from these minipiezometers, and the time to the peak conductivity was used to calculate K (K = −(vn)/i, where v is the average linear velocity of tracer, calculated as the flow path length divided by travel time; n is the porosity of sediments; and i is the hydraulic gradient between release and detection points). This method is similar to that used by Harvey and Bencala .
 The extent and velocity of surface water penetration into the down-welling zone were investigated in summer 1997 and spring 1998. This was done by measuring temperature variations over a 24-hour period in the stream channel and across the upper transect at depths of 20, 60, 80 and 100 cm beneath the bed [Silliman et al., 1995]. Measurements were made every 3 hours by inserting the probe of a YSI 400 Tele-thermometer to the base of each minipiezometer on the transect. In any piezometer, a temperature cycle with an amplitude greater than 10% of that in the stream channel was interpreted to mean that surface water had reached that depth in the bed. This figure was based on the observation of Silliman et al.  that in streambed sediments with no vertical water movement, heat conduction could produce temperature fluctuations at 20-cm depth of no greater than 10–15% of the amplitude of surface water fluctuations; therefore fluctuations of greater than 10–15% must be due to advection of surface water. The time delay between the temperature peak in the stream channel and that in each piezometer was used to calculate a first-order estimate of travel time for surface water down-welling, the nonconservative nature of the temperature tracer precluding a more accurate estimate. Daytime temperatures also were measured in surface and hyporheic water during each season from 1996 to 1998.
3.2. Model Description
 The system was modeled using MODFLOW [McDonald and Harbaugh, 1984], a three-dimensional finite difference model, distributed with a graphical user interface by Waterloo Hydrogeologic (Visual MODFLOW).
 The model area was 1000 × 500 m, with the eastern and western boundaries defined by the Speed River catchment boundaries and the northern and southern boundaries following groundwater flow lines (Figure 2a). The ground surface elevation imported into the model was obtained from the 1:50,000 topographic map 40 P/9 (from the Department of Energy, Mines and Resources Canada, 1985) but was refined in the riffle study site with data from a field survey conducted using a surveyor's total station. Grid spacing across the model domain was 8 × 8 m, but was refined to 1 × 1 m at the riffle site.
 Twelve layers were defined, the bottom layer (layer 12) being the dolomite aquifer, 20–40 m thick, with Kx,y = 10−6 m s−1, Kz = 10−7 m s−1. Kx,y was estimated from specific capacity measurements in local water well records (Ontario Ministry of the Environment, Water Well Records Department), and Kz was set an order of magnitude lower to reflect the anisotropy commonly observed in such systems. The overlying layers across most of the model domain represented the glacial till and outwash deposits, with layers 4–11 given K between 10−6 and 10−5 m s−1 and layers 1–3 given K = 3 × 10−8 m s−1 (all isotropic) (Figure 2b). Values for layers 1–3 were obtained from slug tests [Bouwer and Rice, 1976] in piezometers located 20–60 m away from the stream and screened at 1 or 3 m below ground. Values for layers 4–11 were calibrated to produce vertical hydraulic gradients across layers 1–3 that matched those measured in these catchment piezometers and to produce stream discharge in the model that matched stream discharge records [Water Survey of Canada, 1992]. All K values were within ranges given by Singer et al. . In the catchment, layer surfaces followed ground surface contours, and were each 2 m thick; but in the vicinity of the stream, layers 3–9 were made horizontal, with a constant thickness of 0.25 m, in order to allow fine-scale modeling of vertical hydraulic gradients and flow paths. Also in the vicinity of the stream, beneath the channel and up to 6 m on either side, a zone of high hydraulic conductivity (K = 2 × 10−4 m s−1, isotropic), representing the stream alluvium, was assigned to cells in the top layers, up to 1.5 m beneath the streambed. This hydraulic conductivity value was estimated from the movement of salt tracers, released up to 60 cm beneath the streambed surface, which agreed with the value determined from the particle size distribution of bed sediments (using the Kozeny-Carmen equations; Freeze and Cherry ).
 The stream was represented by constant head nodes in the upper layer of streambed cells (Figure 2c), rather than by river boundary nodes, because the streambed conductivity was already included in the K values of the underlying streambed cells. Except in the riffle study area, stream heads were defined as 0.20 m above the ground surface to approximate average stream stage, with the downstream head drop kept at a constant gradient of 0.4%. Within the riffle study area, model heads were defined with greater spatial resolution, using heads measured at the streambed surface along the longitudinal transect, and in this way included the natural discontinuities in water slope at the upstream and downstream ends of the riffle that produce exchange flows (Figure 2d).
 The bottom layer (layer 12), representing the bedrock aquifer, was bounded on all sides by constant heads to permit regional groundwater flow and to define heads for the catchment. Values for the constant head boundaries were taken from a larger-scale model of the same area that had been calibrated using head data from 20 water wells (Ontario Ministry of the Environment, Water Well Records Department). The bottom of layer 12 was a no-flow boundary, and the side boundaries of layers 1–11 (which followed the catchment boundaries and groundwater flow lines) also were no-flow boundaries.
 Aquifer recharge rates in summer and winter were estimated from stream discharge records [Water Survey of Canada, 1992], by dividing summer and winter stream base flow by the total area of the catchment above the gauging station, and assuming that regional groundwater underflow beneath the gauging station was negligible. Recharge was applied as a constant flux to the active area of the top model layer.
 The model was run at steady state, and then certain parameters were altered, one at a time, according to the amounts by which they are known to vary in the Speed River. Streambed hydraulic heads along the 13-m riffle were given summer and winter values, according to field measurements made in these two seasons; in summer the head difference between upstream and downstream ends of the riffle was 8 cm, and in winter the difference was 4 cm. Groundwater discharge into the stream was doubled between summer and winter levels by raising the constant heads in layer 12 by 2 m and increasing the recharge to layer 1 from 80 mm yr−1 to 180 mm yr−1. A further 2-m rise in the layer 12 constant heads doubled groundwater discharge again, simulating extreme conditions not observed in field measurements. Discharge to the stream is defined as the volumetric flux of water removed from the model by all constant head cells that represent the stream; this value is given in the “zone budget” function of MODFLOW. Finally, hydraulic conductivity of the alluvial sediments was varied over 2 orders of magnitude, corresponding to the range of variability observed at different places within this single riffle. In order to evaluate the effects of varying each of these three parameters, the model was run through a series of 36 steady state simulations.
 Other parameters, such as the average stream stage and the width and depth of the alluvium, were varied in separate simulations in order to test the effects of these on model behavior.
 While the model was an attempt to reproduce a real system, some simplifications were made and some factors were excluded. Only exchange flows induced by the pool-riffle-pool transition were considered, and thus stream morphological features such as meanders, gravel bars and multiple channels, and obstacles such as boulders and large woody debris, which produce their own exchange flow dynamics, were omitted. Streambed heterogeneities, which can increase the depth of riffle-induced hyporheic flow paths [Woessner, 2000], and evapotranspiration, which can increase surface water/hyporheic exchange by producing hydraulic gradients away from the stream, were also excluded. Further, because the model was run in steady state, it did not consider the effects of transient phenomena such as bank storage [Squillace, 1996].
 Output data that were recorded belonged to three dependent parameters of the model: the volumetric flux of water exchanged between the stream surface and the hyporheic zone (volume per unit time per meter stream length) within the 13-m study riffle, the lateral and vertical extent of stream water penetration into the hyporheic zone, and the travel time of stream water between entry into the hyporheic zone and return to the stream surface. The volumetric flux of water exchanged between the surface water and the constant head cells in the uppermost layer of the riffle site is given by the “zone budget” function of MODFLOW. When the riffle site is defined as a single “zone,” down-welling flux is the total flux from all constant head cells being recharged from surface water, and up-welling flux is the total flux to all constant head cells discharging to the surface water. Exchange flux is the flux of water entering from the surface that also returns to the surface within the defined riffle area; that is, it is the lesser of down-welling and up-welling flux.
 The vertical and lateral extent of stream water penetration into the hyporheic zone and the travel times for exchange flow were determined using the “particle tracking” function of MODFLOW. Particles were released from surface cells in the down-welling zone, and their flow paths were shown graphically by MODFLOW, from which their greatest distance from the stream could be measured. The travel time of each flow path was determined from the number of arrowheads marked on the flow paths, which each represent distance traveled per unit time.
3.4. Modeling Stream Gradient
 A second, simpler model was developed to examine the effects of increasing streambed slope. This model was a two-dimensional longitudinal section, consisting of 50 columns and 20 layers to represent a 50 × 20 m area, and bounded on all sides by constant head boundaries that defined the appropriate horizontal and vertical hydraulic gradients in the stream and groundwater. The head boundary of the top layer represented the stream and included a 13-m-long riffle area that was defined by a head distribution equivalent to that in the riffle area of the main model. An equivalent hydraulic conductivity of 5 × 10−4 m s−1 was assigned to the entire model domain. A series of simulations was run, maintaining the same vertical hydraulic gradient in the groundwater while varying horizontal (stream slope) gradient from 0 to 8%. Across the riffle area, the amount (in cm) that the stream heads deviated from the average stream slope was kept constant. Output parameters measured were the same as those in the main model.
4.1. Field Studies
 Data from the catchment piezometers (not shown) and water well records (Ontario Ministry of the Environment, Water Well Records Department) showed that at all times of year hydraulic heads in the aquifer and in the surficial deposits of the catchment up to 60 m from the stream were 1–2 m higher than in the stream. Thus the stream receives a net discharge of groundwater at all times of year. In the streambed and stream banks, however, the nested piezometers showed that hydraulic gradients reverse with season (Figure 3). From fall to spring, hydraulic gradients in all parts of the study riffle were toward the stream channel from the sides and beneath. However, during summer base flow, gradients at the upper transect were downward and laterally away from the channel, while gradients at the downstream end of the riffle remained up-welling. Thus the down-welling/up-welling pattern indicating riffle-scale exchange flows occurred only during summer base flow. Salt tracers released into the streambed confirmed that hyporheic flow paths were in the directions indicated by the hydraulic gradients.
 Measurements of hydraulic head at the streambed surface showed that the head difference between upstream and downstream ends of the riffle (13 m apart) was about 4 cm during fall to spring base flow and 8 cm during summer base flow. The head difference increased in summer because the lower water level caused stream heads to follow the rise and fall of the streambed more closely.
 The temperature cycle data (Figure 4) show that at the upper transect, surface water down-welling in spring was reduced in both lateral and vertical extent compared with that in summer. In spring only the 20-cm piezometers and one 60-cm piezometer in the midstream showed temperature variations greater than 10% of surface variations. Only the three piezometers closest to the middle of the stream showed variations greater than 20% at any depth. In summer, however, all of the piezometers beneath the channel showed variations greater than 20% down to 60 cm, and one showed variations of 10% at 100 cm.
 The spring results seem to conflict with the hydraulic head measurements which indicate that no down-welling occurred at this time. Three explanations are offered: Heat conduction could produce some temperature cycling in shallow bed sediments despite up-welling water flux. However, the results of Silliman et al.  suggest that conduction could only account for a temperature variation of 10–15% at 20 cm depth, even with no vertical water movement. Second, small variations in hydraulic head gradients within a 24-hour period may allow surface water to enter the subsurface occasionally. Previous measurements of hydraulic head in the upper transect repeated at different times of day (unpublished data) suggest that this is a plausible explanation. Finally, down-welling hydraulic gradients may exist on a scale that is too small to be detected by our piezometers, spaced 20 cm apart. Despite the discrepancy, the temperature cycle data indicate a reduction in the size of the hyporheic zone in spring, which agrees with the hydraulic head data.
 The travel time of the temperature peak also was longer in the spring. In spring the temperature peak appeared at 60 cm depth 11–12 hours after it had occurred at the stream surface, whereas in summer the delay was only 4 hours. Both of these travel times are significantly shorter than those shown by the salt tracers, which traveled 25 cm in 19–24 hours. Despite this discrepancy, the temperature data do show that subsurface flow rates were faster in summer.
 Daytime temperatures in the stream channel varied between 0.5° and 4°C in spring and around 20–22°C in summer.
4.2. Modeling: Identifying Key Factors
 Initially the main model was run without assigning a zone of high-permeability alluvial deposits around the stream. In these runs, hydraulic gradients beneath the riffle were always toward the stream (Figure 5a), and no exchange flows were found, regardless of how other properties of the model were changed.
 Once the high-permeability zone around the stream was added, however, it became possible to produce exchange flows within the discharging aquifer/stream system (Figure 5b). Because the exchange flows were never found to extend beyond the boundaries of the high-permeability zone, this zone became considered as the basic unit of study. The question then became, What properties and boundary conditions of the alluvial zone (hyporheic zone) are required for exchange flows to occur?
 During attempts to calibrate the model to measured heads, it became clear that the output parameters (exchange flux, lateral and vertical extent of exchange flow, and hyporheic travel times) are determined by three factors. These factors are the hydraulic conductivity of the alluvial deposits, and two conditions at the boundaries of the hyporheic zone: the head difference between the upstream and downstream ends of the riffle (relative to the average stream gradient), and the flux of groundwater entering the alluvial zone from the sides and beneath.
 Average streambed gradient, examined in the simple two-dimensional model, has more subtle effects on hyporheic flows. Increasing streambed slope (horizontal gradient) from 0 to 8%, while maintaining constant vertical hydraulic gradient in the subsurface, produced no significant changes in exchange flux; that is, a hydraulic head at the upstream end of a riffle that is 4 cm higher than would be determined by the average streambed slope produces the same exchange flux whether the slope is 0% or 8%. However, as the flux of underflow (subsurface water flowing parallel to the stream within the alluvium) increased proportionally with stream slope, hyporheic flow paths became shallower and hyporheic travel times became shorter (Figure 6).
 Other factors that were varied were found to have little or no effect on the output parameters. Widening or deepening the alluvial zone beyond the farthest lateral or vertical extent of stream water infiltration did not affect exchange flux or extent of exchange flows. Raising all stream heads by a constant amount to simulate a higher stream stage, or stream discharge, in wetter seasons also had a negligible effect.
4.3. Quantitative Relationships Between Key Factors and Model Output
 The effects on model output of changing the riffle hydraulic heads and groundwater discharge between summer and winter values, and varying the hydraulic conductivity of the alluvium, are summarized in Figures 7–9.
 Exchange flux, extent of stream water infiltration, and hyporheic travel times are all most sensitive to changes in the head difference between upstream and downstream ends of the riffle, slightly less sensitive to changes in hydraulic conductivity, and least sensitive to changes in groundwater discharge. Over the range of conditions simulated, doubling the head difference increased exchange flux by 150–500%, increased the lateral extent of exchange flows by 2–3 m, and doubled the velocity of hyporheic flows. Doubling the hydraulic conductivity increased exchange flux by 100–400%, increased the lateral extent of exchange flows by 1–2 m, and doubled hyporheic flow velocities. Doubling the groundwater discharge, however, decreased exchange flux by less than 60%, decreased the lateral extent of exchange flows by 1–2 m, and had no effect on hyporheic flow velocities.
 Hyporheic travel times are related to both flow velocity and distance between down-welling and up-welling points. Decreasing head difference and hydraulic conductivity and increasing groundwater discharge caused water to reemerge closer to its source (Figure 9); thus in some simulations with slower water velocities, travel times were actually shorter due to the shorter flow paths. In simulations with high down-welling fluxes, some water particles did not reemerge at the upwelling zone, but continued tens of meters beneath the stream before emerging.
 Reducing K to 10−5 m s−1 or lower effectively eliminates all exchange flow under the range of head and groundwater discharge values considered here.
 In this low-gradient stream, model output shows exchange flux to be a small proportion of stream discharge. If summer stream discharge is 0.1 m3 s−1 (i.e., 8640 m3 d−1) and exchange flux with summer heads and groundwater discharge is about 0.2–0.5 m3 d−1 per meter stream length, then only 0.03–0.07% of stream discharge is exchanged with the hyporheic zone within this 13-m riffle. Assuming that winter stream discharge is 3 times summer discharge and exchange flux with winter heads and groundwater discharge is about 0.008–0.04 m3 d−1 per meter stream length, then in winter only 0.0004–0.002% of stream discharge is exchanged within this 13-m riffle.
4.4. Vertical Versus Lateral Exchange Flows
 Under the various simulations performed, vertical exchange flows within the channel occurred more consistently than lateral flows into the stream banks. Down-welling extended to the bottom layer of the alluvial deposits (1.25 m below the particle release layer) in 21 of the 36 simulations, and vertical exchange occurred to some extent in 30 of the 36 simulations (Figure 8). Considering only those simulations that represented observed seasonal changes in the Speed River, average vertical hydraulic gradients within the channel varied between −0.010 in summer and 0.001 in fall to spring. In contrast, lateral exchange occurred in only 21 of the 36 simulations and extended farther than 1 m in only 14. Lateral hydraulic gradients across the stream banks, from the channel to 3 m into the banks, varied from −0.015 in summer to 0.025 in fall to spring.
 Field measurements of hydraulic head agreed well with this pattern. Lateral hydraulic gradients across the stream banks, from the channel to 2.5 m into the banks, varied between an average of −0.016 in summer and 0.019 in fall to spring. In contrast, vertical gradients within the channel, from the bed surface to a depth of 1 m, varied between an average of −0.01 in summer and 0.014 in fall to spring. While these data do not indicate persistence of vertical flow paths, they do show that vertical gradients vary less than lateral gradients in the study riffle.
4.5. Differences Across the Width of the Stream
 In the model, groundwater discharge to the stream is always concentrated at the sides of the stream. In the riffle study area the stream is six cells wide, and although each cell of a row was given the same hydraulic head, the two outermost cells receive 4.5–16 times the net groundwater discharge of the two inner cells. This means that in some simulations with limited exchange flow, in the predominantly down-welling area at the head of the riffle, up-welling occurred at the sides while down-welling persisted in the center of the stream (Figures 10 and 11). In fact it was only in simulations with the very lowest K that no down-welling at all was seen. In simulations with high exchange flow, down-welling flux was up to twice as great in the outermost cells than in the center cells (Figure 10). Thus hyporheic exchange flows are more variable at the sides of the stream than at the center.
 Field data also reflected this pattern. Temperature fluctuations indicating down-welling (Figure 4a) were much higher in the shallow sediments near the center of the stream than near the sides at a time when up-welling predominated; at 20 cm depth, temperature fluctuations in piezometer 4 were 90% as great as fluctuations in the surface water, whereas those in piezometers 3, 7, and 8 were less than 20% as great. Figure 12 shows that vertical hydraulic gradients in the center of the stream (piezometer 5) were significantly less variable over all seasons than were vertical gradients at the sides (piezometers 3, 7, and 8); Levy multiple comparison procedure for nine piezometers (k = 8) resulted in q = 6.89, n = 17, 16, p < 0.001; q = 9.65, n = 17, 13, p < 0.001; q = 6.02, n = 17, 16, p < 0.001 for piezometers 5 and 3, 5 and 7, and 5 and 8, respectively [Zar, 1996]. Figure 12 also shows that near the center of the stream (piezometer 4) the vertical gradient remained, on average, down-welling, while at all other points the average gradient over all seasons was up-welling.
5.1. Prerequisites for Hyporheic Exchange Flows
 According to our model, a significant hyporheic zone will develop around a pool-riffle sequence only where hydraulic gradients toward the stream from the sides and beneath are less than, or close to, the longitudinal hydraulic gradient between the upstream and downstream ends of the riffle. For the Speed River, which drains a moderate- to low-permeability catchment (K = 10−6 − 10−8), hydraulic gradients toward the stream will be this low only if the alluvial sediments surrounding the stream, in which the hyporheic zone is to develop, are highly permeable (K > 10−5). Other studies of surface water-groundwater exchange also have been in streams surrounded by high-permeability alluvial zones [e.g., Castro and Hornberger, 1991; Harvey and Bencala, 1993; Morrice et al., 1997], and it is likely that in most stream systems the hydraulic conductivity of the catchment material is too low to allow hyporheic exchange flows without the existence of such a zone.
5.2. Controlling Factors
 The size of the hyporheic zone and the flux of hyporheic exchange depend on the strength of the surface water hydraulic gradient between the two ends of the riffle, relative to the strength of the groundwater hydraulic gradients perpendicular to the stream. The three factors investigated in this model work together to control the relative strength of the surface water and groundwater hydraulic gradients. Groundwater hydraulic gradients adjacent to the stream are controlled by the hydraulic conductivity of the alluvial sediments and the flux of groundwater to the stream, in accordance with Darcy's law [Freeze and Cherry, 1979]. The surface water hydraulic gradient is independent of both of these factors, being controlled by the amplitude of undulations in the streambed.
 The essential characteristic of the riffle hydraulic gradient is its deviation from the average stream slope (Harvey and Bencala  refer to this as “stream slope variability”), the head at the upstream end of the riffle being higher than the average slope and the head at the downstream end being lower than average. This was shown by our two-dimensional model, in which the average stream slope was varied between 0 and 8%. Although, in absolute terms, the hydraulic gradient between upstream and downstream ends of the riffle varied according to the stream slope, the deviation of the heads in the riffle from the average stream slope was held constant, and the result was that exchange flux remained constant.
 The main model shows that typical seasonal changes in the riffle head gradient, alluvium hydraulic conductivity, and groundwater discharge are sufficient to explain the seasonal changes in flow direction that were found by field measurements in this Speed River study area. In the field, all three factors change between summer base flow, when exchange flows occur, and the rest of the year, when groundwater discharge almost eliminates exchange flows. First, between summer and fall-spring, the hydraulic gradient between upstream and downstream ends of the riffle decreases by half, since the heads in the stream do not follow the rise and fall of the streambed as closely when the stream stage is high as when it is low. Second, higher aquifer recharge rates following autumn rains and spring runoff increase groundwater discharge to the hyporheic zone. Finally, the drop in water temperature from about 20°C in summer to close to 0°C in winter causes a calculated 40% decrease in hydraulic conductivity, due to the higher viscosity of the water at lower temperature [Freeze and Cherry, 1979]. The model shows that these three changes, that were measured directly or derived from field measurements, together almost eliminate exchange flow in this riffle.
 Our results agree with the findings of Wroblicky et al. , who also identified hydraulic conductivity, discontinuities in streambed slope, and changes in groundwater discharge as primary factors controlling hyporheic exchange flows. Other authors also have recognized the strong influence of hydraulic conductivity [e.g., Wondzell and Swanson, 1996; Valett et al., 1996] on hyporheic flows, though few except Constantz et al.  have included the role of water temperature in changing hydraulic conductivity.
5.3. Other Factors
 The average gradient of the streambed was shown, by the two-dimensional model, to have no effect on the amount of exchange flux. However, it does affect the velocity and shape of the exchange flow path. This is because underflow, i.e., groundwater flowing beneath the stream that does not exchange with the surface, increases in flux and velocity with increasing streambed gradient. Exchange flow is entrained into underflow, and therefore its path is narrower and more streamlined where the streambed is steeper.
 Several other factors that have been associated with differences in storage zone characteristics in stream tracer studies were not found, in our models, to be important factors controlling hyporheic flows. Stream stage and discharge have been correlated with reduced storage zone area and decreased residence time [D'Angelo et al., 1993; Morrice et al., 1997]. Stream stage, in groundwater terms, is equivalent to hydraulic head, but increasing all stream heads uniformly in this steady state model had a negligible effect on exchange flows, since groundwater heads simply reached equilibrium with the higher stream heads.
D'Angelo et al.  also found a negative correlation between stream velocity and storage zone size, and they and Hart et al.  found positive correlations between stream velocity and storage zone exchange rate. Stream velocity is another property of the surface water and is not a definable parameter in groundwater models. Since the stream tracer approach used in these two studies incorporates both surface and subsurface storage zones, it is likely that the correlation they found reflects mainly increased interaction with surface storage zones, or with near-surface zones, which may be influenced by turbulence. Alternatively, stream velocity may have been correlated with other factors such as groundwater discharge, which have been shown to influence subsurface flows.
D'Angelo et al.  and Wondzell and Swanson  cited alluvial thickness and floodplain width as important factors determining the location and extent of the hyporheic zone. In our model, increasing the width and depth of the alluvial zone beyond the extent of hyporheic flows has no effect on hyporheic flux or flow paths. It is probable therefore that the effects found by these two authors were on larger-scale flow paths, which extend to the limits of the alluvial zone at the sites they studied.
5.4. Flows at the Sides Versus the Center of the Stream
 The finding in our model and field data, that both groundwater discharge and surface water down-welling are concentrated at the sides of the stream, agrees with a general pattern found in lakes. McBride and Pfannkuch  showed through both numerical models and field data that groundwater discharge to lakes is concentrated near the shoreline, with the seepage rate declining exponentially with distance from shore. Pfannkuch and Winter  showed that the “crowding” of groundwater flow lines becomes more pronounced as the width of the lake increases relative to the depth of the underlying aquifer, and that some crowding will occur provided the lake is as wide as the underlying aquifer is deep. Streams in general, and our study riffle in particular, are typically narrow compared with the depth of the underlying aquifer; however, it is likely that the depth of the highly permeable alluvial zone, in which the exchange flows occur, is the relevant parameter, rather than the entire depth of the aquifer. This would explain why our model and field results showed crowding of exchange flows near the sides of the stream, despite a deep aquifer beneath. Pfannkuch and Winter  also showed that anisotropy in the bed sediments produces a more even seepage flux across a lake bed, and therefore among different streams, exchange flows may be more or less uniform across the streambed depending on the history of processes that have formed the bed.
 The tendency for groundwater discharge to concentrate at the sides of the stream also explains why vertical exchange occurred more consistently than lateral exchange under the range of conditions simulated in the model. Groundwater discharge is the force that opposes surface water down-welling beneath a riffle. If this force is concentrated at the sides of the stream, where lateral exchange occurs, vertical exchange can continue beneath the middle of the stream under conditions that prevent lateral exchange.
5.5. Biological Implications of Small-Scale Patterns
 The significance of hydrological exchange between streams and hyporheic zones is most often expressed in terms of its implications for subsurface organisms and biogeochemical processes [e.g., Harvey and Wagner, 2000]. In our model, such implications arise from the small-scale patterns that were identified. If lateral hyporheic exchange is more variable seasonally than vertical exchange, the biological community lateral to the stream channel will experience a more variable physicochemical environment than the community beneath. Microorganisms with more flexible metabolism therefore may be favored in more variable parts of the hyporheic zone. Similarly, if the sides of the channel experience stronger fluxes, both down-welling and up-welling, than the center, this may create spatial and temporal patterns in biogeochemical processes and faunal distributions in response to the renewal rate of resources such as oxygen. Future ecological studies therefore might look for patterns in the microbial and meiofaunal communities that reflect responses to variability in flow conditions at different parts of the hyporheic zone.
In a low-gradient gaining stream system, riffle-scale exchange flows are possible only when high-permeability materials (K > 10−5 m s−1) surround the stream channel. In a moderate- to low-permeability catchment (e.g., K = 10−6 to 10−8 m s−1) this may be achieved by a zone of alluvial deposits around the stream.
The amount of exchange flux, the lateral and vertical extent of surface water penetration, and the travel times through the hyporheic zone are determined by three measurable parameters: the hydraulic conductivity of the alluvium, the groundwater flux to the alluvium, and the hydraulic gradient between upstream and downstream ends of the riffle. These together provide a simple framework to describe differences in the size of the hyporheic zone between stream systems and between seasons. Average stream gradient, stream stage, and the thickness of the alluvium do not affect hyporheic exchange flux, although stream gradient does make subsurface flow paths shallower and subsurface residence times shorter.
In temperate streams, the hydraulic conductivity of the streambed can vary by up to 40% with season due to changes in water temperature; the hydraulic gradient between upstream and downstream ends of the riffle can decrease by half in winter because the higher stream level conforms less closely to the contours of the streambed; and the ground water discharge may increase by a factor of 2 or more due to higher aquifer recharge. The combined effect can be a tenfold to thirtyfold reduction in hyporheic exchange flux with season in low-gradient, gaining, temperate-zone streams.
Exchange flows tend to be stronger but more variable at the sides than at the center of the stream channel. Under conditions that limit exchange flow, groundwater discharge is 5–20 times greater at the sides of the stream, and under certain conditions groundwater discharge prevents vertical and lateral exchange flows at the sides while vertical exchange still occurs beneath the center of the stream. Under conditions that allow more exchange flow, down-welling and out-welling flux at the sides of the stream is up to twice as great as that at the center.
 We thank the Mackenzie family for permission to use their stream and water well, Marilyn Smith for help with field work, and Rick Gerber, Brewster Conant Jr., Steve Holysh, and E. J. Wexler for advice on modeling and the hydrogeology of the area. We especially thank the three anonymous reviewers for their valuable and constructive comments on an earlier version of the paper. This research was supported by a National Sciences and Engineering Research Council of Canada grant to D. D. Williams.