Common in-stream geomorphic structures such as debris dams and steps can drive hyporheic exchange in streams. Exchange is important for ecological stream function, and restoring function is a goal of many stream restoration projects, yet the connection between in-stream geomorphic form, hydrogeologic setting, and hyporheic exchange remains inadequately characterized. We used the models HEC-RAS, MODFLOW, and MODPATH to simulate coupled surface and subsurface hydraulics in a gaining stream containing a single in-stream geomorphic structure and to systematically evaluate the impact of fundamental characteristics of the structure and its hydrogeologic setting on induced exchange. We also conducted a field study to support model results. Model results indicated that structure size, background groundwater discharge rate, and sediment hydraulic conductivity are the most important factors determining the magnitude of induced hyporheic exchange, followed by geomorphic structure type, depth to bedrock, and channel slope. Model results indicated channel-spanning structures were more effective at driving hyporheic flow than were partially spanning structures, and weirs were more effective than were steps. Across most structure types, downwelling flux rate increased linearly with structure size, yet hyporheic residence time exhibited nonlinear behavior, increasing quickly with size at low structure sizes and declining thereafter. Important trends in model results were observed at the field site and also interpreted using simple hydraulic theory, thereby supporting the modeling approach and clarifying underlying processes.
 The hyporheic zone is the area of mixing of surface and groundwater beneath and adjacent to streams [Triska et al., 1989], particularly that region where hydrologic flow paths leave and return to the stream many times along its length [Harvey and Wagner, 2000]. Exchange of water between a stream and its hyporheic zone (hyporheic exchange) facilitates important exchanges of heat, chemical solutes, and biota between surface stream and subsurface water [Jones and Mulholland, 2000]. These processes affect the distribution and abundance of organisms in streams and the hyporheic zone, ecosystem level processes like nutrient cycling and carbon flux, and water quality [e.g., Boulton et al., 1998; Jones and Mulholland, 2000; Groffman et al., 2005].
 In-stream geomorphic structures (IGSs) such as steps, pools, and log dams are common in natural streams and are known to enhance hyporheic exchange [Kasahara and Wondzell, 2003]. Such structures are also commonly installed in stream restoration projects to recreate habitat for organisms and enhance geomorphic stability [Federal Interagency Stream Restoration Working Group, 1998; Bethel and Neal, 2003], or even to enhance hyporheic exchange [Doll et al., 2003], a stream function that is increasingly recognized as an important goal of restoration [Boulton, 2007]. Nevertheless, the impact of structure form and hydrogeologic setting on induced exchange both in natural and engineered settings is poorly understood.
1.2. Mechanisms of Hyporheic Exchange
 Hyporheic exchange where hyporheic flow paths leave and return to a stream multiple times over a reach [Harvey and Wagner, 2000] requires hydrologically neutral or gaining conditions. Although IGSs induce a three dimensional pattern of hyporheic flow into the bed and banks, similar to recent studies [e.g., Gooseff et al., 2006], we focus here on vertical exchange into the bed for detailed modeling analysis of controlling factors. Mechanisms that drive primarily lateral exchange such as meander bends are not considered (but see Boano et al. ). Further, this discussion focuses strictly on hydrologic exchange mechanisms, ignoring processes such as diffusion that affect only solute flux or heat exchange. In this context, vertical hyporheic exchange can be induced via six basic mechanisms:
 1. Darcy flux due to local channel steepening is nonturbulent flux induced by head gradients created by local steepening of the channel slope relative to average reach channel slope, sometimes called concavity and convexity [Vaux, 1968]. Examples include steps and riffles.
 2. Darcy flux due to backwater is nonturbulent flux induced by head gradients created by water collecting behind obstacles in the channel. Examples include debris dams, large woody debris, boulders, and bars.
 3. Darcy flux due to form drag is nonturbulent flux induced by head gradients created by head loss due to form drag as turbulent stream water flows over bed forms that are either permeable like ripples and dunes [Thibodeaux and Boyle, 1987] or impermeable like partially buried boulders [Hutchinson and Webster, 1998]. This effect has been called hydraulic pumping or pumping exchange [Elliott and Brooks, 1997] because it is induced by pressure differentials along the streambed, but these terms are not used here as they are ambiguous given that localized streambed steepening (mechanism 1 above) and backwater (mechanism 2 above) can also induce pressure differentials.
 4. Darcy flux due to substrate heterogeneity is nonturbulent flux induced by head gradients created by obstructions within the sediments (e.g., areas of lower hydraulic conductivity or shallower bedrock), which induce upwelling upstream and downwelling downstream of the obstruction even in the absence of other exchange mechanisms [Vaux, 1968; Salehin et al., 2004].
 5. Turbulent flux across the bed occurs where turbulent energy of flowing water carries momentum and therefore stream water into the subsurface [Shimizu et al., 1990].
 6. Turnover exchange is the exchange of water as the bed forms move, successively trapping and releasing water [Elliott and Brooks, 1997].
 This study was conducted for baseflow, the most common flow condition. All six hyporheic exchange mechanisms presented above operate in most streams and rivers, but several are relatively insignificant at baseflow. Turnover exchange (mechanism 6) is significant only where bed forms are in motion (i.e., primarily during spates in the absence of large wood or boulders that might otherwise block their advance). Turbulent flux across the streambed (mechanism 5) is insignificant in sands and finer materials and confined to the top 4–6 cm in gravels [Packman and Bencala 2000], although these results potentially overestimate turbulent flux penetration for many gravel or cobble bed streams where sufficient fines are present to fill gaps, or where bed armoring or imbrication has occurred. Further, turbulent flux has been shown to scale with the square of the velocity [Packman and Salehin, 2003], which is generally at a minimum during baseflow conditions [Leopold and Maddock, 1953]. Form drag (mechanism 3) will occur anywhere an obstacle projects into the flow. Debris dams, bars or dunes, partially buried boulders, and large woody debris can all create form drag that is significant under certain conditions. Form drag is maximized for completely submerged conditions and scales with the square of the velocity [Munson et al., 1994] such that it would be least significant at baseflow. Darcy flux induced by local steepening of the streambed (mechanism 1) and backwater behind obstructions (mechanism 2) are therefore generally the primary mechanisms by which IGSs drive water vertically into the hyporheic under baseflow conditions. These mechanisms, along with Darcy flux induced by form drag and substrate heterogeneity (mechanisms 3 and 4), control hyporheic exchange in this setting.
1.3. Existing Approaches for Quantifying Hyporheic Exchange
 Hyporheic exchange due to mechanisms 1–4 at the reach scale has traditionally been quantified in two ways: transient storage and multidimensional approaches. The transient storage approach uses the one-dimensional (1-D) advection-dispersion (A-D) equation with off-channel storage areas to quantify solute movement along a stream and exchange with longer residence time storage areas representing both backwater areas in the channel and hyporheic exchange zones. The most commonly used version of this model is also the simplest with one well-mixed storage zone [Bencala and Walters, 1983]. Other formulations include multiple storage zones [e.g., Gooseff et al., 2004], variable residence time storage zones [e.g., Gooseff et al., 2005], or a diffusive type storage zone [e.g., Packman et al., 2004]. Application of the model generally requires release of a conservative tracer in a stream, measurement of concentration history downstream, and fitting the data to the A-D equation using software such as OTIS [Runkel, 1998] where storage zone parameters are determined via inverse modeling methods. These parameters include (among others, see Runkel ) the cross-sectional area of the storage zone (related to hyporheic zone size), the rate of tracer exchange between free flowing stream and storage zone (related to exchange flux rate), and residence times of tracer in the storage zone.
 In contrast, the multidimensional approach uses spatially explicit two-dimensional (2-D) or three-dimensional (3-D) analysis of subsurface flow patterns, usually coupling modeling with field or flume data. Field studies [e.g., Wondzell and Swanson, 1996; Wroblicky et al., 1998] are usually conducted at the reach or segment scale and are often coupled with site-specific groundwater flow and transport models using well-established software (e.g., MODFLOW, MT3D). Model results are used to calculate hyporheic zone size, exchange flux rate, and/or hyporheic residence time for any desired spatial extent. Flume studies are usually conducted at the subreach scale [e.g., Packman et al., 2004] and, where modeling is conducted, often utilize custom-developed numerical codes [e.g., Salehin et al., 2004]. Hyporheic zone size, exchange flux rate, and/or residence time are either measured directly from the flume or output from the model at desired scales. Multidimensional studies have the advantage over transient storage approaches in that hyporheic exchange can be understood as a multidimensional process where individual areas of hyporheic exchange can be isolated and characterized directly for size, flux rate, and residence time. This spatial resolution also allows association of areas of hyporheic exchange with specific geomorphic forms within the reach, allowing correlations to be drawn between form and function [e.g., Kasahara and Wondzell, 2003].
1.4. Motivation and Approach for Study
 The goal of this study was to provide a process-based understanding of how channel form (IGSs) affects hyporheic exchange in streams. This entailed determining the relationships between the magnitude of IGS-induced hyporheic exchange and fundamental characteristics of both the IGSs themselves (size and type) and of their setting (background groundwater discharge rate, sediment hydraulic conductivity, depth to bedrock, in-stream baseflow discharge, and channel slope). The intractability of a primarily laboratory or field approach to creating an array of such relationships required modeling. Multidimensional modeling analysis was chosen over transient storage modeling as the latter is primarily an inverse method where hyporheic response cannot be determined directly for a specified geomorphic form. Unlike previous studies, a sensitivity analysis was conducted to fully quantify the relationships between controlling variables and hyporheic response. We modeled surface and groundwater response to sensitivity analysis perturbations using a simplified hypothetical stream setting to clearly separate the effects of various driving factors. Widely accepted and well tested models were chosen for this purpose for their reliability and to facilitate the use of this general modeling approach by stream restoration or watershed planning practitioners. This study provides the necessary first step toward understanding how individual structures influence hyporheic exchange, which can later be extended to multiple structures. We briefly describe a field study used to support the modeling approach, but primarily focus on the sensitivity of hyporheic exchange to IGS variability. Basic hydraulic theory is then used to interpret field and model results and demonstrate controlling processes.
2.1.1. Conceptual Model of Hypothetical Stream System
 We analyzed coupled surface and groundwater hydraulics in a simplified hypothetical stream reach containing one IGS of varying type. Similar to previous work [e.g., Kasahara and Wondzell, 2003], we analyzed baseflow conditions because such conditions are present the majority of the time and lower discharges tend to dominate many stream ecosystem processes [Doyle et al., 2005]. Steady state conditions were therefore assumed. A net gaining condition (i.e., net discharge from groundwater to stream) at the reach scale was assumed on the basis of our previous definition of the hyporheic zone [Harvey and Wagner, 2000]. We analyzed a small stream (3 m wide) because such first- to second-order streams are the most common, are the focus of considerable restoration effort, and IGSs are most prevalent in this setting. For simplicity, the channel cross section was rectangular with vertical sides and with no floodplain. The channel was straight with a slope of 0.01 m/m for the base case. Sediment and soil were assumed isotropic and homogeneous with respect to the flow of water (similar to Gooseff et al.  and Lautz and Siegel ). Sediment texture for the base case was sand. Channel discharge was 0.2 m3/s for baseflow conditions, giving a normal depth (steady uniform depth away from obstructions) of 0.1 m. The following types of IGSs were analyzed (Figure 1):
 1. Steps were vertical drops of the channel bottom with a consistent slope upstream and downstream.
 2. Weirs were impermeable channel-spanning obstructions rising vertically out of the bottom of an otherwise consistently sloped bed. The structures were perpendicular to flow, with a horizontal top surface. Weirs represented debris dams, log dams, boulder weirs, and log jams that span the channel and create backwater. Flow was assumed to overtop the weir for this study.
 3. Lateral structures were similar to weirs but were of greater height, always exceeding flow depth, and did not completely span the channel. Lateral structures were intended to represent wood or rock structures that do not span the channel and divert water around, but not over, at baseflow.
2.1.2. Numerical Modeling Approach
 The U. S. Army Corps of Engineers (USACE) 1-D river hydraulics model HEC-RAS [U. S. Army Corps of Engineers (USACE), 2002a] was used to simulate stream hydraulics for a 3 m wide, 30 m long reach with a single IGS approximately centered in the reach (Figure 2a). Although this 1-D representation necessarily makes compromises simulating flow across each of the structure types, we considered this acceptable because (1) the purpose of this study was an exploratory sensitivity analysis evaluating basic trends rather than predicting precise magnitudes of system response and (2) the focus was on vertical hyporheic exchange in a straight simplified stream with no floodplain. Manning's n was set to 0.03, reasonable for natural sand or gravel bottoms [USACE, 2002b]. Form drag losses were accounted for in HEC-RAS using the default coefficients for contraction (0.1) and expansion (0.3) except within 1.5 m of the lateral structures, where 0.6 and 0.8 were used [USACE, 2002b], respectively. HEC-RAS assumes an impermeable bed but can accept user-specified lateral inflows at each cross section, which were used to manually couple cross-streambed flows with MODFLOW.
 The U. S. Geological Survey (USGS) 3-D groundwater flow model MODFLOW [Harbaugh and McDonald, 1996] was used to simulate groundwater hydraulics. The model domain was created such that HEC-RAS cross sections were located directly atop MODFLOW nodes along a central column. There were 21 layers in the base case scenario with vertical discretization of 0.25 m, plus one top layer of 0.1 m. Horizontal layers were used rather than dipping layers for simplicity and for improved accuracy of particle tracking in MODPATH [Pollock, 1994]. Horizontal discretization, consistent with the exploratory nature of the study, was fairly coarse with 11 3.0 m square cells in each direction. Hydraulic conductivity (K) was assumed uniform and isotropic at 10−5 m/s to represent fine sand [Freeze and Cherry, 1979] for the base case (Table 1). The water surface profile of the stream (calculated by HEC-RAS) was represented as constant head cells in the top MODFLOW layer using the river package. The presence of different size and type IGSs was expressed in the MODFLOW river package by using a series of different hydraulic head profiles from HEC-RAS and varying the distribution of riverbed thicknesses along the channel. Additional boundary conditions in MODFLOW included constant-head conditions for all model cells at the upstream and downstream ends of the model domain and no-flow conditions for all cells on the sides and bottom [Gooseff et al., 2006]. Constant heads at the ends of the model were set above (0.1 m for base case; Table 1) the normal depth of flow in the stream at the respective HEC-RAS cross sections, creating higher heads in groundwater than in the stream and hence gaining conditions. The no-flow boundary condition on the bottom of the model represents bedrock at depth (5 m deep for base case; Table 1). The USGS program ZONEBUDGET [Harbaugh, 1990] was used to extract hyporheic exchange fluxes from MODFLOW output files for cells at the streambed interface.
Table 1. Parameters Varied in Sensitivity Analysis
used weir-structure-type only; groundwater discharge held constant by raising boundary condition heads as flow depth increased with stream discharge
Background groundwater discharge
Values listed in columns to right are background groundwater discharge rates per streambed area within the patch-scale area in absence of any in-stream geomorphic structures. This flux was varied by adjusting the constant-head boundary conditions in MODFLOW; the difference between this constant-head value and the elevation of the normal depth in the stream (head differential) is also shown in the columns to right.
1.8 × 10−7 m/s (head differential = 0.1 m)
−4.1 × 10−9 m/s (head differential = 0.0 m)
1.8 × 10−6 m/s (head differential = 1.0 m)
used weir-structure-type only
Sediment hydraulic conductivity (K)
homogeneous and isotropic: single K value for entire model domain
10−5 m/s (silty sand)
10−8 m/s (fine silt or clayey silt)
10−2 m/s (gravel)
used weir-structure-type only
background channel slope for reach
used weir-structure-type only
Depth to bedrock
depth from bottom of top model layer (represents bottom of in-stream structure) to no-flow boundary condition at bottom of model domain
used weir-structure-type only; no-flow boundary condition at uniform depth throughout model
 In order to fully simulate the exchange of water across the streambed, the surface and groundwater models were coupled at the surface-groundwater interface. This was accomplished by an iterative process where heads and flows along the streambed are passed back and forth between HEC-RAS and MODFLOW until values converge. An initial set of iterated model runs was conducted to gauge the extent of this feedback on results. Results indicated that the impact of hydraulic feedback is imperceptible on in-stream flows and hyporheic exchange metrics for the base case K (10−5 m/s). The effect of hydraulic feedback was significantly greater for K of 10−2 m/s, but still relatively minor. Iteration was therefore performed only for sensitivity analysis model runs where K was increased to 10−2 m/s.
 The USGS program MODPATH [Pollock, 1994] was used to simulate particle tracking in groundwater and calculate residence times of hyporheic flow paths induced by the modeled IGSs. Particles were released at the surface water–groundwater interface along the centerline of the stream channel. Effective porosity was set to 0.3, a reasonable value for sand [Freeze and Cherry, 1979]. Particle tracks determined by MODPATH were visualized using the USGS program MODPLOT [Pollock, 1994].
 We quantified hyporheic exchange induced by a given IGS as that which downwells within the “patch-scale area” just upstream (within 1 channel width) of the structure. This water then flows downstream beneath the structure and upwells downstream of the structure, forming the “patch-scale hyporheic flow cell,” which is approximately two channel widths long. This is in contrast to the “full hyporheic flow cell” which incorporates all the hyporheic flow induced by the IGS (Figure 2a). While the model domain was chosen to be much larger than the patch-scale hyporheic flow cell, the patch scale was chosen for reporting results because the most intense hyporheic flow occurs in this area, because hyporheic exchange metrics for this area are less sensitive to boundary artifacts of the model than those for the larger full hyporheic flow cell, and because various geomorphic features longitudinally constrain hyporheic flow cells in real streams. In other words, while the full hyporheic flow cell will vary enormously with stream context, patch-scale dynamics are the most important subset that is always present, regardless of geomorphic context (see also section 4). This choice is arbitrary but reasonable as IGSs are often spaced on the order of 2–5 channel widths in natural streams [Montgomery et al., 1995] and presumably at similar distances in restored reaches.
 A variety of fundamental metrics of hyporheic exchange were generated from model output for the patch-scale area (Table 2). Downwelling flux rate and hyporheic volume were defined for all water downwelling within the patch-scale area. Residence time, hyporheic depth, and hyporheic path length were defined for the particle that downwelled at the center of the patch-scale area, representing a median value for the patch scale. While, in reality, there are many flow paths emanating from the patch-scale area that have a distribution of lengths and residence times, for this study it was useful to constrain the results to a single characteristic flow path in order to focus on the trends of system response to driving factors.
Table 2. Hyporheic Exchange Metrics Used to Report Results of Modeling Study
Use of particle originating at center of patch-scale area approximates median value for water downwelling along centerline of channel within patch-scale area.
Downwelling flux rate Qd
downward flux rate of water across streambed within patch-scale area; extracted from MODFLOW results by ZONEBUDGET
Hyporheic residence time tr
travel time of the MODPATH particle that originates at center of patch-scale area between when it enters and exits the groundwater model domaina
Hyporheic zone size: depth dh
maximum depth that the particle in MODPATH, released at center of patch-scale area, reaches below streambed at downstream side of the in-stream geomorphic structures; estimated from MODPLOT visualizations of hyporheic pathlinesa
Hyporheic zone size: path length lh
length of subsurface flow path of MODPATH particle, released at center of patch-scale area; estimated from MODPLOT visualizations of hyporheic pathlinesa
Hyporheic zone size: volume Vh
product of downwelling flux rate Qd [L3/T] and hyporheic residence time tr [T]; representative of the patch hyporheic-scale flow cell
2.1.3. Sensitivity Analysis
 The primary objective of this study was to determine the independent effect of a wide range of potential controls on the vertical hyporheic exchange induced by a single IGS. This was accomplished by a sensitivity analysis where each of the parameters of interest was varied separately. Factors varied included structure type (weir, step, lateral structure) and size (s), represented by weir height (sw), step height (ss), and lateral structure width (sl) (Table 1 and Figure 1). Stream discharge, background groundwater discharge, sediment hydraulic conductivity (K), channel slope, and depth to bedrock (Table 1) were also varied, but because of the extensive number of model runs and volume of output, this was undertaken only for weir-type structures. When each factor was varied, all remaining factors were held constant at base case values. Note that when stream discharge rate was varied, the constant head boundary conditions for groundwater were adjusted to maintain a constant level of background groundwater discharge to the stream. This is appropriate for examining the effects of varying baseflow discharge but not for evaluating response to hydrographs.
2.2. Field Experiment
 A simple field experiment was conducted to discern if trends and patterns observed in the model results were also observed in the field despite more heterogeneous conditions. We focused on groundwater (and hence MODFLOW) behavior due to higher uncertainty relative to surface hydraulics. We constructed a single, fully spanning, variable height weir in the upper portions of Craig Creek in the Jefferson National Forest near Blacksburg, Virginia. This stream reach (width 1–2 m, baseflow discharge 0.5–5.0 L/s, gravel and cobble substrate) was somewhat smaller than the hypothetical stream used in the model to allow easier weir construction. As weir height was varied, vertical head gradient (proportional to downwelling flux rate) was measured within the downwelling zone just upstream of the weir using a 1.25” polyvinyl chloride (PVC) piezometer with a single screen 20–25 cm below the streambed. Simultaneously, the residence time of hyporheic flow beneath the structure was estimated by injecting a concentrated salt slug into separate piezometer just upstream of the weir (0.5” PVC with a single screen 10–15 cm below the streambed) and determining the inflection time (time to reach half the peak electrical conductivity) of the resultant salt breakthrough curve in a piezometer just downstream of the weir (1.25” PVC with a single screen 10–25 cm below the streambed) using an electrical conductivity probe. All three piezometer screens were located (horizontally and vertically) within the area equivalent to the patch-scale hyporheic flow cell from the modeling (Figure 2a), allowing the most rigorous comparison with patch-scale modeling results. Field data were evaluated as a function of weir height, and resulting trends compared to model output. The parameters measured in the field were chosen because of their primary importance among the results of the modeling and also for ease and accuracy of measurement in the field.
3.1. Modeling Sensitivity Analysis
 Modeling indicated that an IGS in a gaining alluvial stream will induce downwelling into the subsurface upstream of the structure if the structure is of sufficient size. Much of this downwelling water moves downstream beneath the structure within the alluvium and then re-emerges (Figure 2a). This IGS-induced flux manifests as a characteristic pattern of downwelling and upwelling along the channel bed (Figure 2b). Model results indicate this pattern varies in magnitude with IGS size and type, as well as with hydrologic and geologic setting, but overall shape remains similar.
3.1.1. Effect of Structure Size and Type
 For all structure types and patch-scale hyporheic exchange metrics, there were minimum structure sizes required to overcome the background gaining conditions. In the plots of hyporheic exchange metrics versus structure size (Figure 3), this effect manifests as a section at the left (small structure) end of each curve that does not rise above zero exchange. All results discussed below refer to exchange induced by structures larger than this minimum size.
 As weir and step height increased (sw and ss), downwelling flux rate (Qd, Table 2) increased linearly, but the flux rate for steps was always about half as high as that for weirs (Figure 3a). For lateral structures, structure width (sl) had little effect on Qd until sl exceeded approximately 50% of total channel width. For all IGS types, hyporheic residence time (tr, Table 2) varied nonlinearly with structure size, initially increasing steeply and then decreasing asymptotically (Figure 3b). The trends for steps and weirs were similar, with peak tr at approximately the same structure size, but tr for weirs was approximately double that for steps.
 The trends of the three hyporheic size zone metrics, depth (dh), path length (lh), and volume (Vh) (see Table 2 for calculation methods), were similar and mostly nonlinear (Figures 3c–3e). For both weirs and steps, all three metrics increased steeply at small s, and then became independent of s at larger s. Hyporheic zone size metrics for weirs exceeded those of steps for all s, with dh and lh for weirs about twice that for steps. The dh asymptotes were considerably shallower than depth to bedrock because these pathlines originated within the patch-scale area (i.e., close to the structure), and therefore did not penetrate as deeply as particles originating further from the structure (Figure 2a). For lateral structures, hyporheic zone size metrics also increased with s, but either increased approximately linearly with sl (dh and lh,Figures 3c and 3d), or increased roughly exponentially with sl (Vh, Figure 3e). Interestingly, the ratios of lh to dh for each of the IGS types approached approximately 3 at larger structure sizes.
3.1.2. Effect of Hydrologic and Geologic Setting
 The effect of hydrologic and geologic variables on patch-scale hyporheic exchange is presented in two ways. First, the plots of exchange metrics (Qd, tr, and dh) versus sw are expanded to each include three curves: the original base case curve along with two additional values of the hydrologic or geologic variable being evaluated. This type of plot is shown to quantify response over the full range of structure sizes. Second, additional plots are created of Qd, tr, and dh versus the hydrologic or geologic setting variable itself. These are included to directly quantify the impact of the hydrologic or geologic variable. dh is the only metric of hyporheic zone size included because our focus here is the shapes rather than the magnitudes of trends, and trends for lh and Vh share the same basic shape as dh.
 The shape of the relationships between sw and Qd, tr, and dh changed little with baseflow discharge (Figure 4, left). As baseflow discharge increased (with background groundwater discharge held constant) at a representative sw of 0.2 m, Qd increased approximately 51%, tr decreased approximately 15%, and dh increased approximately 10% (Figure 4, right). All three relationships between hyporheic exchange and baseflow discharge were nonlinear, becoming independent of discharge at higher discharges.
 Hyporheic exchange metrics responded significantly to background groundwater discharge rate (i.e., degree of gaining). The shape of the relationships between sw and the various hyporheic exchange metrics were all similar (Figure 5, left), and Qd, tr, and dh all decreased to zero as background groundwater discharge increased (Figure 5, right).
 Some metrics of hyporheic exchange responded substantially to changes in hydraulic conductivity (K) while others did not (Figure 6). Both Qd and tr responded dramatically over many orders of magnitude as K increased from 10−8 m/s to 10−2 m/s at a representative sw of 0.1 m (Figure 6, right). Specifically, Qd increased linearly with K (Figure 6 shows trend as linear on a log-log plot, but trend would also be linear on a linear-linear plot), and tr decreased logarithmically with K (Figure 6 shows trend as negatively linear on a log-log plot, which would look approximately negatively exponential on a linear-linear plot). On the other hand, dh changed little with increasing K, decreasing approximately 9%, with all of the decrease occurring at a K of 10−2 m/s (Figure 6, right). In all cases, the shape of the relationships between exchange metrics and sw varied little if at all with K (Figure 6, left).
 Hyporheic exchange metrics responded significantly to channel slope (Figure 7). The shape of the relationships between sw and Qd, tr, and dh changed relatively little with slope, except for tr which peaked at increasing sw with increasing slope (Figure 7, left). The relationships between Qd, tr, and dh and slope are mostly nonlinear, increasing at low slopes and decreasing at higher slopes (Figure 7, right). Peak exchange generally occurred at slopes of 0.005–0.01 m/m. The one exception was tr for sw = 1.0 m, which increased slightly with slope.
 Finally, hyporheic exchange responded significantly to depth to bedrock (Figure 8). The shape of the relationships between sw and Qd, tr, and dh varied somewhat with depth to bedrock but were generally consistent (Figure 8, left). The largest change was for tr, which peaked at increasing sw as depth to bedrock increased. For most weir heights, Qd, tr, and dh increased with increasing depth to bedrock, although all three trends were also reversed at small sw (< ∼0.1–0.2 m). Two different curves are therefore shown for Qd, tr, and dh versus depth to bedrock (Figure 8, right). All such relationships were nonlinear, exhibiting a rise (moderate to large sw) or drop (small sw) at shallower bedrock depths and a leveling off at greater bedrock depths.
3.2. Field Experiment
 Field data confirmed that principal trends observed in the modeling results (for weirs) were also observed in a more heterogeneous field setting. Specifically, vertical head gradient (proportional to downwelling flux rate, Qd, via Darcy's Law) in the patch-scale area increased linearly with weir height for both model and field results (Figure 9a). Similarly, hyporheic residence time for water downwelling within the patch-scale area (tr) decreased roughly exponentially with weir height across most weir heights (Figure 9b). Field and model tr results diverged at the smallest weir heights, likely because the salt injection port had to be set deep enough to prevent salt solution escaping to the stream which prevented measurement of the shortest flow paths created during the lowest weir heights.
4.1. Model Confidence and Spatial Scale
 Issues of scale and model confidence are discussed together because the issues are related and many lines of evidence available to address one issue also address the other. There are four lines of evidence supporting the validity of the model results as presented in this study. First, model software (MODFLOW and HECRAS) is well-developed and has been extensively tested and refined. Second, the models were used for comparative purposes (i.e., determining trends of system response in a sensitivity analysis) in a simplified hypothetical setting rather than for predictive purposes at a heterogeneous real site. Third, the field experiment indicated the model was accurately quantifying trends of hyporheic response to varying weir height. These field results do not explicitly confirm model performance with other structure types, and additional field or flume studies would be useful. However, the field experiment for weirs is at least somewhat relevant to steps and lateral structures because these other structure types, like weirs, drive hyporheic flow by creating hydraulic head perturbations in the stream that propagate into the subsurface. Fourth, trends from the modeling and field experiment appear reasonable when simple hydraulic theory is used to interpret the hydraulic processes behind them.
 These lines of evidence also inform the application of these model results to other scales. First, the field experiment and modeling results show similar trends of system response to weir height, despite differences in the size of the streams. This supports the idea that modeled trends are not specific to a particular size stream. Second, modeled trends of system response are congruent with application of hydraulic theory where no scale was specified (see sections 4.3 and 4.4). This further supports the idea that modeled trends are not specific to a particular size stream, but also indicates that modeled trends are not specific to a particular scale of observation (e.g., patch-scale versus full hyporheic flow cell). We acknowledge that the very shortest flow paths induced by the structure were outside the scope of the field experiment, and that the modeling did not address turbulent flux and scour that might affect these short flow paths under certain circumstances. However, the overall Darcy flux hyporheic response characterized in these results appears relevant to a range of scales.
 These scale issues then bear on the rationale and justification for choosing the “patch-scale area” and the corresponding “patch-scale hyporheic flow cell” for presenting the results. Note the patch scale was merely the scale for reporting the results, whereas the scale of the model domain was considerably larger (see section 2 for details). To begin, the patch scale was not chosen as an expedient compromise, but rather was intentionally implemented as both a necessary and highly useful way to broaden the applicability of the results. Specifically, we originally evaluated results for the full hyporheic flow cell induced by the IGS, but this proved untenable, because as structure size increased, the flow cell would eventually encompass the entire MODFLOW model domain in the longitudinal direction, which artificially constrained hyporheic metrics at greater structure sizes. To remedy this problem, the length of the model domain for weir scenarios was increased several times up to approximately 200 m, but the larger weirs still induced a hyporheic flow cell that pressed up against at least one end of the model domain. While this effect might be somewhat less for other structure types or steeper channel slopes, for much of the sensitivity analysis presented in this study, the full hyporheic flow cell was an ill-defined concept. Furthermore, such long model domains devoid of constraints from other hyporheic flow cells induced by other geomorphic forms are meaningless when applied to real streams. We further discovered that patch-scale hyporheic exchange metrics did not change significantly as the length of the model domain changed, and were therefore independent of the arbitrary model boundary conditions. This fact, combined with the typical presence of other structures up and downstream in real streams, rendered patch-scale results more generally applicable than results for the full hyporheic flow cell. In other words, while the full hyporheic flow cell will vary enormously with stream context, patch-scale dynamics are the most important portion that is always present, regardless of geomorphic context.
 We acknowledge that the patch-scale hyporheic flow cell takes up a variable percentage of the full hyporheic flow cell as the full hyporheic flow cell varies in size as controlling factors are varied in the sensitivity analysis. This effect is probably most significant for hyporheic zone size metrics, but even then, this issue affects the details, not the overall shape, of the relationships between driving factors and hyporheic response. For example, hyporheic depth levels off with increasing structure height for both patch-scale and full hyporheic flow cells, just at different values. An alternative approach to a fixed size patch-scale area would be to vary the patch scale as some percentage of the full hyporheic flow cell. However, as discussed above, the full hyporheic flow cell is not defined for larger structures in our modeling context, so this approach would place significant limits on the structure sizes included.
4.2. Magnitude of Structure-Induced Hyporheic Exchange
 The degree of hyporheic exchange induced by an IGS can be better understood when set within its stream context. For example, the magnitude of downwelling flux (Qd, Table 2) induced by an IGS can be viewed as a portion of baseflow stream discharge. This percentage varies with hydraulic conductivity (K) and degree of gaining, but will often be much less than 100% for natural sediments. We calculated this percentage from our modeling results (presented previously in Figure 3a) by dividing the downwelling flux rate by in-stream discharge rate. Qd values from the base case (K = 10−5 m/s, appropriate where the finest sediment size fraction is medium sand) were all less than about 0.015% of surface discharge (Figure 10, top). For K = 10−2 m/s (appropriate where the finest sediment size fraction is gravel), induced Qd can approach 25–50% of the stream discharge. The latter values, while substantial, reflect the largest size of the most effective type of IGS (weirs), and most structures would therefore induce significantly less flux, even in coarse substrate. This is consistent with most hyporheic studies on streams and rivers with either hydrologically gaining or approximately net-neutral conditions which report a small fraction of surface flow moving through the hyporheic owing to relatively fine sediments [e.g., Kasahara and Wondzell, 2003; Storey et al., 2003; Lautz and Siegel, 2006]. Only a few studies report K values high enough that relatively large fractions of surface flow may move through the hyporheic zone [e.g., Poole et al., 2004].
 Hyporheic residence times (tr, Table 2; Figure 3b) for the base case scenario (sandy sediment; see Table 1) were mostly in the range of days to weeks and much greater than typical surface water residence times, generally in the range of seconds to hours. Like Qd, tr is highly dependent on K. These results are similar to those found by others for sandy bed streams [e.g., Lautz and Siegel, 2006; Cardenas et al., 2004], and greater than those for streams of generally coarser sediments [e.g., Kasahara and Wondzell, 2003].
 Hyporheic path lengths (lh, Table 2) and depths (dh, Table 2) (Figures 3c and 3d) were in the range of a few meters, with lh generally in the range of three times dh. Corresponding hyporheic volumes (Vh, Table 2) ranged over a few to a few tens of cubic meters. To allow an order of magnitude comparison with other studies, these hyporheic zone size metrics can be expressed in transient storage terms, where storage zone cross-sectional area is divided by stream cross-sectional area (As/A). We calculated As as Vh/lh, which estimates the average cross-sectional area (normal to flow) of Vh. We calculated A as the channel width times the normal depth used in the HEC-RAS simulations. The resulting patch-scale As/A estimates for our model simulations range with structure size up to 5–25, depending on structure type (Figure 10, bottom). As values used in Figure 10 are for the patch-scale area, and those for the full hyporheic flow cell would be larger. However, given tr is in the range of days to weeks for our hypothetical stream (Figure 3b) and the hours to days duration of most transient storage experiments, transient storage experiments run on our hypothetical stream would probably detect a patch-scale or smaller hyporheic contribution to transient storage. Nevertheless, transient storage analyses on real streams would generally also detect a surface water storage component [Ensign and Doyle, 2005]. We did not attempt to estimate a surface storage component for our hypothetical stream, but it would presumably be significant for the weir and lateral storage cases, and insignificant for the step case. Our As estimates therefore would be greater than or equal to the reported values if we accounted for surface storage. Overall then, given that our method of calculating As may both tend to overestimate As by including longer residence time hyporheic path lengths than are typical of tracer studies and tend to underestimate As by not including surface storage, we would expect literature values from field studies to agree with our modeling results, at least at an order of magnitude level. This is indeed the case, as literature values tend to range from 0 to nearly 10 [Harvey and Wagner, 2000] and our results range from 0 up to a maximum anywhere between 5 and 25, depending on structure type (Figure 10, bottom).
4.3. Effect of Structure Size and Type
 Structure size was a critical factor in determining the relationship between IGS morphology and hyporheic exchange. Relationships between IGS size (s; weir height sw, step height ss, and lateral structure width sl) and the hyporheic exchange metrics (Figure 3) were of similar shape for weirs and steps, but different for lateral structures. Qd increased approximately linearly with sw and ss, but increased nonlinearly (approximately exponentially) with sl (Figure 3a). This is consistent with our field experiment for weirs (Figure 9), and also with Kasahara and Wondzell , who reported that larger steps induce greater flux than smaller steps. Similarly, Storey et al.  found an increase in downwelling flux with head drop across a riffle. This increase in head drop is analogous to an increase in s in our modeling study, because increasing s increases in-stream head drop and associated hyporheic Darcy flux processes owing to channel steepening, backwater, and/or form drag mechanisms (mechanisms 1, 2, and 3 in section 1.2). In fact, this connection between structure size, head drop, and induced hyporheic flux can be understood by applying Darcy's Law to the hyporheic flow originating within the patch-scale area. Because s is varied in this study under conditions of constant K, Darcy's Law can be simplified,
where Ac is the cross-sectional area of the subsurface (hyporheic) flow path and Δh is the in-stream head drop across the structure. Because the relationships between Δh and s (Figure 11a) have the same shapes as the relationships between Qd and s (Figure 3a), it is clear that Δh in equation (1) is more important than Ac or lh in determining Qd and that an IGS controls Qd primarily by controlling Δh.
 In contrast to Qd, tr varied nonlinearly with s for all three IGS types, with large initial increases and then asymptotic decreases (Figure 3b). This result is less intuitive than the linear increase of Qd with s. However, while only the falling portion of this curve was reproduced at our field site (Figure 9), the processes giving rise to the entire relationship can be understood for weirs and steps by applying equation (1) to the flow path originating at the center of the patch-scale area. To start, average linear groundwater velocity (v) along the flow path can be derived from equation (1) by dividing Qd by Ac and the effective porosity (ne):
 Further, velocity is defined as
where Δt is the travel time along a flow path, that is, the residence time, tr. Equations (2) and (3) can be combined and solved for tr, and recognizing that ne and K are constant for this analysis,
 Examining the numerator and denominator of equation (4) in turn, we focus on how each varies with s, and how their behavior differs between small and large values of s. For the numerator, our model results indicate that lh increases with s for small values of s but levels off to a constant value at large values of s (Figure 3d). Because such an asymptotic curve retains its basic shape when all the ordinate values are squared, the relationship of lh2 with s will also have rising and constant sections that correspond to the same ranges of s. For the denominator of equation (4), Δh increases linearly with s for weirs and steps (Figure 11a), and thus Δh−1 relates inversely to s (Figure 11b). Comparing these trends for the numerator (Figure 3d) and the denominator (Figure 11b) which combine per equation (4) to produce the tr versus s relationship for weirs and steps (Figure 3b), lh appears to dominate the relationship at small s and Δh appears to dominate the relationship at large s (Figure 12). This seems to clarify the processes behind the relationship observed in the modeling, indicating that under gaining conditions, weirs and steps control tr at small s primarily by controlling lh and control tr at large s primarily by controlling Δh. Even at small s, however, lh is ultimately controlled by Δh (Figure 3d), tying the observed pattern back to the hyporheic Darcy flux processes (mechanisms 1, 2, and 3 in section 1.2). Further investigation would be useful to confirm this hypothesis in real streams, particularly for the rising portion of the curve where dynamics at small s are influenced by scour patterns downstream from the weir and turbulent flux in coarser sediment.
 All hyporheic zone size metrics (dh, lh, and Vh) relate nonlinearly to sw and ss, increasing rapidly at small s, and then becoming independent of s at greater s (Figures 3c–3e). The processes behind this pattern can be understood by examining how the effect of geologic and geomorphic constraints increases with structure size. First, under gaining conditions, no hyporheic exchange will be induced in the absence of a structure, such that dh, lh, and Vh → 0 as s → 0. Second, as s increases, dh, lh, and Vh will eventually be constrained both vertically and longitudinally by nearby geologic or geomorphic features. Specifically, as s increases, the vertical extent of the hyporheic zone induced by the structure will eventually be constrained by geologic units such as bedrock or clay layers. At the same time, the longitudinal extent of the hyporheic zone will eventually be constrained by other IGS-induced hyporheic zones up and downstream, other geomorphic features, or bedrock outcrops. The net effect is a relationship between Vh (and therefore dh and lh) and s that increases from zero and then asymptotes. By contrast, hyporheic zone size metrics (dh, lh, and Vh) do not appear to level off with sl, (Figures 3c–3e), but this is expected because lateral structures in this study are infinitely tall (never overtopping), so Δh increases rapidly at large sl to an undefined condition at sl = 1.0. Literature data for comparison are relatively rare. Gooseff et al.  showed that mean hyporheic depth (analogous to dh) both increased and decreased with reach-averaged step height in various modeling studies informed by field data [Gooseff et al., 2006, Figure 6a], although they did not explain the contradiction. Lautz and Siegel [2006, Table 2] showed that for small dam structures (analogous to weirs), hyporheic volume increased with head drop. This increasing trend is similar to ours, but because they used a “hydrochemical” (>10% of water from stream) delineation of the hyporheic zone and other parameters varied between structures, the results may not be directly comparable. Because little existing work is directly comparable to our hyporheic zone size results, additional field or flume studies in this area, particularly those that more realistically address turbulent flux and scour in shallow sediments downstream of the structure, would be beneficial.
 Taken together, the impact of IGS size on the various hyporheic exchange metrics, at least for weirs and steps under gaining conditions where turbulent hyporheic flux is minimal, appears to be determined by its effect on both Δh induced in the stream, and on the size of the induced hyporheic flow cell (dh, lh, and Vh) relative to the available subsurface flow domain that is subject to various geologic, geomorphic, and hydraulic constraints. In particular, for Qd, changes in Δh are always the dominant controlling factor that translates changes in structure size to changes in hyporheic Darcy flux processes due to structure-induced channel steepening, backwater and/or form drag (mechanisms 1, 2, and 3 in section 1.2). On the other hand, for hyporheic zone size, Δh is the dominant factor for small values of s but geomorphic constraints are dominant for large values of s, yielding an overall nonlinear trend. Finally, for tr, Δh is dominant for large values of s, but only because hyporheic zone size has been constrained in this range, which allows Δh to operate along a relatively fixed length flow path.
 The type of IGS was also a critical factor in inducing hyporheic exchange. Channel-spanning structures (weirs, steps) were generally more effective than partially spanning lateral structures. Lateral structures had relatively little influence on hyporheic exchange until they spanned a substantial portion of the stream width (∼50%). In addition, hyporheic exchange metrics for weirs exceeded those for steps at all structures sizes (Figure 3). These results are somewhat nonintuitive, as greater Qd for weirs than steps of the same height should be accompanied by greater v and lesser tr, everything else being equal. Instead, greater tr is observed for weirs, owing in part to greater lh, but also owing to greater divergence of the flow paths originating in the patch-scale area for weirs relative to steps, leading to greater reduction in velocities along the weir flow paths, and therefore greater values for tr. The larger lh for weirs is itself due also to this divergence of flow that does not occur in the step case, which overwhelms the extra path length increment for steps that results from the shallower water column and hence deeper sediment upstream of the structure. Some existing studies are relevant to portions of these results. For example, Lautz and Siegel  found that debris dams (weirs) were more effective at driving downwelling flux than meander bends in a real stream. Further, Kasahara and Wondzell  found that steps generally induced more hyporheic exchange flow (analogous to Qd) of shorter residence time than geomorphic features such as secondary channels and sinuosity [Kasahara and Wondzell, 2003, Figures 4, 6, and 8]. Although these results do not directly address weirs versus steps, they seem to indicate that IGSs may induce more downwelling flux than planform features, but such a conclusion is probably premature given the full range of feature configurations and sizes were not evaluated.
4.4. Effect of Hydrologic and Geologic Setting
 The hydrogeologic context of an IGS was generally important in determining the degree of hyporheic exchange induced, but this importance varied among specific contextual parameters. In-stream baseflow discharge had a relatively minor effect on hyporheic exchange (Figure 4). As discussed earlier, variation in Δh controls the process of hyporheic Darcy flux due to structure-induced channel steepening, backwater and/or form drag (mechanisms 1, 2, and 3 in section 1.2), reflected in this case in the similarity in the shape of the relationships between these two parameters and baseflow discharge (Figure 13a). This minor effect implies that hyporheic exchange might not vary that much while groundwater heads and hence baseflow discharge vary throughout their annual cycles. This variation of baseflow discharge should be distinguished from discharge variation in spates, where stream stage can be temporarily elevated without commensurate increases in groundwater levels. Although discussion of the flood pulse cycle is beyond the scope of this study, others have found a more significant hyporheic effect in that situation [e.g., Saenger et al., 2005].
 Background groundwater discharge had a substantial effect on all hyporheic exchange metrics that was consistent across weir heights (Figure 5). As groundwater discharge increased, downwelling flux rate (Qd), hyporheic residence time (tr), and hyporheic depth (dh) all decreased markedly. This result is reasonable because higher groundwater discharge entails a greater rate of background upwelling within the reach which would tend to oppose structure-induced hyporheic flux. This is consistent with Storey et al. , who found that increased groundwater discharge led to decreased downwelling flux, and with Lautz and Siegel , who found that increased areal recharge to groundwater (i.e., increased precipitation, which would increase background groundwater discharge rate) decreased hyporheic volume. Similarly, Cardenas and Wilson  found that increasing ambient groundwater discharge decreased both exchange flux and residence time of bedform-induced subsurface flow. Taken together, these results imply that hyporheic exchange would be less prevalent in areas with strong gaining conditions (e.g., wetter climates and areas of topographic convergence) and more prevalent elsewhere.
 Increases in hydraulic conductivity (K) significantly affected exchange, inducing increases in Qd, decreases in tr, but having little impact on dh (Figure 6). The impact of K on Qd is consistent with Darcy's Law (equation (1)), which indicates that Qd is proportional to K where Δh,lh, and Ac are constant, yielding a linear relationship. The inverse relationship between K and tr is also expected because as Qd increases, average linear groundwater velocity (v) increases, and tr decreases, all else being equal. Because the sediment was homogeneous, what little variation occurred in dh occurred at higher values of K owing to stream-groundwater feedback (see section 2.1.2), and K had no affect on any hyporheic zone size metric for K of 10−5 m/s or less. These results (Figure 6) generally agree with those of Storey et al.  for downwelling flux and hyporheic depth across much of the range of K, but were more complicated for residence time, possibly because they varied other parameters between runs in addition to K (e.g., groundwater discharge). Contrary to our findings, Lautz and Siegel  reported that hyporheic volume increased with K, but they defined the hyporheic zone “hydrochemically” which is dependent on transport parameters (some of which can vary with K), and their site was heterogeneous with respect to K. Overall, our results imply that hyporheic zones in areas with coarser substrate (e.g., steep mountainous regions, glacial outwash) may have greater downwelling flux rates whereas areas with finer substrates (e.g., lower gradient plains or coastal regions) may have greater hyporheic residence times, all else being equal. Hyporheic zone size will depend on the delineation method, and the degree of substrate heterogeneity.
 In general, all hyporheic exchange metrics exhibit a maximum at a channel slope of 0.005–0.01 m/m (Figure 7). This is true across the full range of weir heights evaluated, with the sole exception of tr at large weir heights. As discussed earlier, variation in Δh controls the hyporheic Darcy flux processes induced by the structure (Qd), reflected in this case in the similarity in the shape of the relationships between these two parameters and channel slope (Figure 13b). Minor deviations in shape between these two curves likely result from variations in the other parameters besides Δh on the right-hand side of equation (1), but the overall correspondence confirms Δh is the primary driver. It then becomes important to understand why Δh peaks at an intermediate slope of 0.01: Δh decreases as slope decreases from 0.01 because the decreased slope decreases the Froude number further below 1 and therefore normal flow depth increases in the stream (discharge is held constant). As this occurs, the weir, which remains at constant height, becomes more drowned, decreasing Δh. On the other hand, as slope increases above 0.01, while normal depth does continue to decrease, this effect is overwhelmed by the increasing channel slope and corresponding down-valley groundwater head gradient, which progressively reduces the effective head drop by reducing the perturbation that a constant height weir can have on the subsurface flow field. Although not evaluated in this study, it is expected that the slope at which hyporheic exchange peaks for a given structure type and height may vary with certain conditions such as baseflow discharge and background groundwater discharge. The results for slopes less than 0.01 m/m have not, to our knowledge, been previously reported. However, the results for slopes greater than 0.01 are consistent with those of Storey et al. , who found that increasing channel slope surrounding a riffle from 0.01 to 0.08 m/m decreased the residence time through the riffle and decreased hyporheic depth. In addition, Gooseff et al.  reported that the length of the hyporheic flow cell induced by steps and riffles decreased as channel slope increased from 0.04 to 0.10 m/m in simulations of both synthetic and field profiles of streams [Gooseff et al., 2006, Figure 3a]. These results would be consistent with our results if hyporheic size decreased at the same time. On the other hand, Storey et al.  found little effect on downwelling flux at the head of a riffle as slope increased from 0.01 to 0.08 m/m. Overall, these results indicate that, all else being equal, hyporheic exchange is probably less prevalent in steep streams (e.g., slopes of 0.08–0.10 m/m) than in shallower ones, but more research is necessary to confirm this finding, corroborate our results at slopes less than 0.01 m/m, and determine how universally applicable they are.
 Varying depth to bedrock had arguably the most complex effect on weir-induced hyporheic exchange (Figure 8), although all metrics increased with depth to bedrock for most structure sizes. The increase in hyporheic exchange with depth to bedrock makes sense because as depth to bedrock increases, hyporheic flow is progressively less vertically constrained, as discussed earlier. This implies that hyporheic exchange should increase in prevalence when moving from locations of shallow bedrock to areas of deeper bedrock, all else being equal (e.g., moving from steep mountainous locations to lowland areas). To our knowledge the impact of this parameter has not been isolated in previous work, and makes our results difficult to generalize. For example, depth to bedrock was varied with stream order in the work of Gooseff et al. , but not independently.
 In discussing the effect of each of the setting descriptors in turn, we note that many geographic variations observed in these parameters are at least somewhat related to watershed position within the stream network hierarchy. For instance, in some cases, steep headwater areas tend to have steeper channel slopes, greater groundwater discharge, and coarser substrate relative to areas further downstream. In such cases, headwater areas should, according to our results, generally exhibit smaller hyporheic zones of shorter residence time relative to areas further downstream. In the end, however, each of these controlling factors can vary in complex ways throughout the stream network hierarchy, and broad generalizations concerning the effect of position within that hierarchy are probably premature.
 We did not explicitly quantify the relative impacts of each driving factor on overall hyporheic exchange. However, on the basis of our sensitivity analysis we can conclude qualitatively that IGS size, background groundwater discharge rate, and hydraulic conductivity are most important when considering all hyporheic exchange metrics together. Somewhat less but still important are IGS type and depth to bedrock. Channel slope appears to be of low to moderate importance, and baseflow discharge was found to be relatively unimportant. These results generally agree with those of other studies. Both Lautz and Siegel  and Kasahara and Wondzell  indicate that structure type is important in controlling exchange. Storey et al.  conclude that hyporheic exchange through a riffle is controlled primarily by head drop across the riffle (related to riffle size), sediment hydraulic conductivity, and background groundwater discharge rate (in that order), and secondarily by channel slope. Saenger et al.  report that K is more important than stream discharge for controlling hyporheic exchange flux. Finally, Woessner , Cardenas et al. , and Salehin et al.  show heterogeneity to be significant, a factor that we did not address. We acknowledge that natural processes like colmation and erosion in real streams will cause structure types and induced exchange to change over time. These important processes are beyond the scope of this study, but deserve serious consideration in future work.
5. Summary and Conclusions
 Hyporheic exchange is increasingly recognized as important in streams and rivers for the ecologically relevant functions it provides. The in-stream geomorphic structures (IGSs) analyzed in this study (weirs, steps, lateral structures) drive significant hyporheic exchange in streams under baseflow conditions mainly by inducing Darcy flux through both local steepening of the streambed and creating backwater behind obstructions. A multidimensional modeling approach was necessary to rigorously evaluate hyporheic response to a suite of controlling factors associated with IGSs. Sensitivity analysis results yielded many insights relevant to understanding IGSs under baseflow conditions. Structure size, background groundwater discharge rate, and hydraulic conductivity appear to be the most important factors controlling hyporheic exchange, followed by structure type, depth to bedrock, and channel slope. Downwelling flux rate and hyporheic zone size generally increase with structure size, while hyporheic residence time peaks at a small or intermediate size. Nonlinear elements of these trends appear to be related to how the size of the induced hyporheic flow cell increases with structure size until it is constrained by geologic, geomorphic, or hydrologic constraints. Hydrogeologic setting appears to be important, with reduced background groundwater discharge, increased depth to bedrock, and low to intermediate slopes tending to maximize hyporheic exchange, while the impact of substrate hydraulic conductivity varies depending on the exchange metric of interest. Structure types vary in their ability to induce hyporheic flow with channel-spanning structures (weirs, steps) generally more effective than partially spanning structures (lateral structures), and weirs more effective than steps. A field experiment determined that our modeling approach anticipates key trends of hyporheic response to driving factors observed in a more heterogeneous field setting, at least for one structure type. Trends observed in both field and model results appear reasonable when interpreted with simple hydraulic theory, which indicates that structures modulate hyporheic exchange mainly through their effect on head drop in the stream. While further testing of these results with field or flume studies is recommended, many lines of evidence support their basic form.
stream cross-sectional area, m2.
flow path cross-sectional area, m2.
storage zone cross-sectional area, m2.
hyporheic depth, m.
in-stream head drop across structure, m.
substrate hydraulic conductivity, m/s.
hyporheic path length, m.
effective porosity, dimensionless.
downwelling flux rate within patch-scale area, m3/s.
structure size, m or cm.
lateral structure width, m.
step height, m or cm.
weir height, m or cm.
hyporheic residence time for water downwelling within the patch-scale area, s or d.
travel time along flow path, s or d.
average linear groundwater velocity, m/s.
hyporheic volume, m3.
flow path length, m.
 This work was supported by an EPA Science to Achieve Results (STAR) graduate fellowship to E. T. Hester and by an NSF grant to M. W. Doyle (CAREER-BCS-0441504). Geoffrey C. Poole and Jeffrey J. Clark provided valuable advice on this study, and Myles Killar, Meredith Harvill, and Frank M. Smith provided valuable field assistance. Tamao Kasahara, John Stofleth, Jay P. Zarnetske, Michael N. Gooseff, and three anonymous reviewers provided detailed and helpful comments on this manuscript.