The flow of river water around large woody debris (LWD) creates pressure gradients along the riverbed that drive a large zone of river-groundwater mixing, or hyporheic exchange. Flume experiments and numerical simulations show that river water downwells into the riverbed upstream of a channel-spanning log and upwells downstream. Exchange rates are greatest near the log and decay exponentially with distance upstream and downstream. We developed equations for bed pressure profiles and hyporheic exchange rates in the vicinity of a channel-spanning log that can be used to evaluate the impact of LWD removal or reintroduction on hyporheic mixing. The magnitude of pressure disturbance along the bed (and thus hyporheic exchange) increases with the fraction of channel depth blocked by the log and channel Froude number. Exchange rates are relatively insensitive to relative depth of the log (gap ratio). At natural densities, LWD in lowland streams drives reach-averaged hyporheic exchange rates similar to a ripple-covered bed. However, the length scales and residence times of hyporheic exchange due to LWD are greater. By removing LWD from streams, humans have altered patterns and rates of hyporheic exchange, which influence habitat distribution and quality for invertebrates and fish.
 Numerous field and laboratory studies have shown that LWD enhances hyporheic exchange. Lautz et al.  showed in a small meadow stream that debris dams and tight meanders promoted hyporheic flow paths with similar length scales and travel times. Lautz and Fanelli  identified zonations in vertical fluid flow and biogeochemical processes in a riverbed upstream and downstream of a log dam restoration structure. Kasahara and Hill  demonstrated that a step constructed with a log induced downwelling upstream and upwelling downstream, and the hyporheic exchange resulted in net nitrate loss. Mutz et al.  conducted a hydraulic tracer study in a low-energy sand bed flume and showed that hyporheic flux doubled after the addition of model wood debris. Wondzell et al.  monitored changes in hyporheic exchange following LWD removal in a small, low-gradient stream and found that hyporheic exchange initially declined but recovered several years later as alternating bars developed.
 Generalized, quantitative descriptions for hyporheic responses to LWD do not currently exist but would be useful for assessing changes in hyporheic exchange due to LWD removal or reintroduction. Where LWD forms steps, sensitivity studies reveal that exchange increases with step height [Endreny et al., 2011a; Hester and Doyle, 2008]. However, LWD forms numerous other structures in rivers, including underflow, deflector, and bar head jams, as well as log rafts, to name only a few [Abbe and Montgomery, 2003; Wallerstein and Thorne, 2004]. Here we quantify hyporheic exchange induced by LWD for the specific case of submerged, channel-spanning logs elevated above the bed. This morphology is representative of the underflow jam of Wallerstein and Thorne , or more generally any individual LWD element that enters the stream perpendicular to flow, spans most of the channel, and is braced above the bed by root wad or branches. This morphology can occur in small, low-to-moderate-gradient streams with unstable beds where channel width and LWD length are similar and peak flows are insufficient to mobilize or rotate LWD [Gurnell et al., 2002; Manga and Kirchner, 2000; Wallerstein and Thorne, 2004]. Though less common in mountain streams [Abbe and Montgomery, 2003; Wohl and Cadol, 2011], this LWD morphology has been widely used in studies of hydraulic resistance and scour [i.e., Gippel et al., 1996; Hygelund and Manga, 2003; Wallerstein et al., 2002], and its simplicity is well-suited for mathematical study. We first present detailed descriptions of hyporheic exchange due to a channel-spanning log based on measurements and simulations for two flume experiments that span subcritical and supercritical flow conditions. Next, we examine how scour morphology that develops near the log influences hyporheic mixing. Finally, we develop empirical relationships for bed pressure profiles and hyporheic exchange near channel-spanning logs based on expanded flume measurements and simulations. These relationships can be used to assess the impact of LWD removal or addition on hydrologic connections between rivers and their adjacent aquifers.
2.1. Flume Experiments
 The tilting, recirculating flume has a sediment test section 5.0 m long, 0.30 m wide, and 0.70 m deep (Figure 1). The sediment consists of well-sorted quartz-feldspar granules (d10 = 1.9 mm, d50 = 2.4 mm, d90 = 3.0 mm), with an approximate porosity of 0.40. To represent a channel-spanning log, a 4.0 in. PVC pipe (outer diameter of 11.4 cm) was cut to flume width and mounted 4 cm above the bed. Channel slope was 0.12o (0.002 m m−1), appropriate for a forested lowland stream. Two experiments were conducted: in trial 1, flow was subcritical, mean channel velocity was 14 cm s−1, and flow depth was 20 cm; in trial 2, flow was locally supercritical above the log (but incoming flow was subcritical), mean channel velocity was 12.5 cm s−1, and flow depth was 17 cm. Respective blockage ratios (fraction of channel flow depth blocked by log) were 0.58 and 0.67 (Figure 1). In trial 2, grains beneath the log were near the threshold for mobilization—several grains moved by creep, but no significant scour topography formed.
 We collected three data sets for each trial to characterize hyporheic exchange. Instantaneous pressures were measured and time averaged to remove turbulent fluctuations at eight locations along the sediment-water interface using Validyne DP15 transducers (observed accuracy of ±0.260 mm water). Flow paths in sediment were also mapped from dye injection. Finally, conservative solute tracer tests were performed by injecting calcium chloride (CaCl2) solution into the surface water over approximately one recirculation period and monitoring surface water electrical conductivity as a proxy for salt concentration. Because the flume is a closed system and the tracer is conservative, decline in salt concentration over time indicates exchange between labeled surface water and unlabeled pore water. For both tracer tests, initial concentrations were approximately equal and small (<1 ppt), limiting the effect of density on exchange [Boano et al., 2009].
 To explore the impact of scour topography on hyporheic exchange, we approximately doubled the flow rate after trial 2 to induce scour. The topography approached a steady state within several hours. We mapped the resulting bed profile and included it in a subset of numerical simulations. We did not measure pressure along the scoured bed because sensor installation would have disturbed the topography. No dye injections or tracer experiments were conducted with scour topography.
2.2. Numerical Modeling
 Coupled channel and hyporheic flow was simulated following the computational fluid dynamics (CFD) approach of Cardenas and Wilson : turbulent flow in the channel was linked to porous media flow in the sediment through the pressure distribution at the sediment-water interface [see Cardenas and Wilson, 2007, Figure 2]. Here we include a free surface in turbulent flow simulations, since the water surface deforms near channel-spanning logs. In contrast, Cardenas and Wilson  represented the free surface as a slip (symmetry) boundary.
 We solve for two-dimensional, unsteady, turbulent, multiphase flow of water and overlying air in the channel using the Reynolds-averaged Navier-Stokes (RANS) equations with the closure scheme [Wilcox, 1998] (Figure 2). Liang and Cheng  demonstrated the superior performance of the turbulence closure scheme for flow around a cylinder near a plane boundary. For an incompressible flow,
where and are fluid density and dynamic viscosity, t is time, Ui and are time-averaged and instantaneous velocity components in xi directions, and P is time-averaged pressure. Sij is the mean strain rate tensor:
Reynolds stresses are related to turbulent kinetic energy (k) and specific dissipation rate () by
where is the Kronecker delta and vt is the kinematic eddy viscosity (equivalent to ). The transport equations for k and are
is given by
is defined as
Closure coefficients are the standard values: , , , and . Note that unlike in Cardenas and Wilson , we solve for the unsteady flow field using the unsteady RANS (URANS) approach, which improves numerical stability for two-phase flow. We then time average the pressure at the sediment-water interface.
 The sediment-water interface and log were assigned no-slip boundaries with roughness heights of 3 mm (d90) and 0.010 mm (roughness height for PVC), respectively. The upstream boundary was assigned a fully developed turbulent velocity profile, and the downstream boundary was assigned a hydrostatic pressure profile. Air boundaries were constant pressure outlets. Simulations were run until mean pressure stabilized along the bed.
 Two-dimensional porous flow in sediment was solved using the steady state groundwater flow equation:
where P is pressure, k is sediment permeability, and is fluid viscosity. Darcy velocity or groundwater flux q equals the term in brackets. Pressure at the sediment-water interface was specified from turbulent open-channel flow simulation. The base and sides of the porous flow domain (flume walls) were assigned no-flow boundaries. In all simulations, the sediment was homogeneous and isotropic.
 Tracer experiments were simulated and analyzed following Ren and Packman . To determine the residence time distribution, we solved for transient solute transport in the flume following Cardenas et al.  using the advection-diffusion-dispersion equation:
where C is concentration, t is time, and v is seepage velocity (specific discharge divided by porosity) from pore water flow simulations. Dij is the mechanical dispersion tensor:
where and are transverse and longitudinal dispersivities and is the molecular diffusion coefficient in porous media. In simulations of trials 1 and 2, the porosity was 0.40, and the permeability was 8.5 × 10−9 m2. was 5 × 10−11 m2 s−1, was 1 cm (several grain diameters), and was 1/10 of [Bear, 1972]. At the top boundary (sediment-water interface), downwelling zones were assigned constant normalized concentration (C/C0) of 1.0, and upwelling zones were specified as convective flux boundaries. Flume walls were assigned zero-flux boundaries. Initial solute concentration in the sediment was zero.
 URANS equations were solved using the finite-volume approach implemented in FLUENT [Fluent Inc., 2006] with a fully implicit second-order upwind solution scheme. Multiphase flow was treated with the volume of fluid approach [Hirt and Nichols, 1981]. The domain consisted of more than 120,000 cells, and simulations were checked for grid and boundary dependence. Finer grid spacing was used near the air-water interface, log, and bed. The location of the upstream and downstream boundaries slightly affected the magnitude of head drop over the cylinder (within ±0.5 mm, similar to the error of pressure transducers) but had negligible impact on the downstream head recovery. Groundwater flow and solute transport equations were solved using the finite element approach in COMSOL Multiphysics [COMSOL AB, 2006]. The sediment domain consisted of more than 18,000 triangular elements with node spacing of less than 1 cm at the top and less than 2 cm at the base. Simulations were tested for mesh dependence.
2.3. Sensitivity Study
 We developed an approximate relationship for pressure along the bed due to flow around a channel-spanning log using an expanded set of 27 flume experiments and 49 CFD simulations. From dimensional analysis, controlling variables are the channel Froude number (Fr) with mean channel depth (d) as the length scale, gap ratio (G, the ratio of log height above the bed to channel depth), and blockage ratio (B, the ratio of log diameter D to channel depth) (Figure 1). Flume experiments and CFD simulations together span a range of blockage ratios from 0.33 to 0.67, gap ratios from 0.086 to 0.56, and Froude numbers from 0.026 to 0.23. Flume experiments were performed using three diameters of PVC pipe (8.89, 11.43, and 14.13 cm). Due to the large size of the flume, we did not scale experiments, which precluded larger Froude numbers.
 Equations were fit to pressure along the bed from CFD simulations using a nonlinear least squares method. Measured pressures from flume experiments were only analyzed for magnitudes of pressure drop and recovery near the log (spatial resolution was insufficient to fit all parameters describing the pressure profile), while pressure profiles from CFD simulations were analyzed for all parameters. Fitted parameters were then correlated with Fr, B, and G to produce an empirical relationship for pressure along the bed as a function of channel hydraulics and log configuration.
 To assess the impact of Fr, B, and G on mean hyporheic flow rates, we simulated hyporheic flow over a 10 m long section of stream centered on a single log with sediment depth of 2 m. Pressure along the bed was determined from CFD simulations in 49 cases and our empirical relationship in 67 cases (with no slope adjustment for stream gradient). Mean flux across the bed was computed as the integral of downwelling Darcy flux into the bed divided by total streambed length.
3.1. Two Case Studies of Hyporheic Exchange Near a Channel-Spanning Log
 In trials 1 and 2, surface water accelerates around the log—maximum velocities approach 40 and 50 cm/s, respectively (Figure 2). The free surface deforms toward the log, and a series of standing waves form in trial 2 (Figure 2 and Movies S1, S2, and S3). Numerical simulations show vortex shedding for trial 1, whereas shedding is suppressed for trial 2. Both trials were conducted near the threshold for vortex suppression identified by Bearman and Zdravkovich  in wind tunnel experiments (gap:diameter ratio ∼ 0.3).
 Head along the bed generally mimics the free surface elevation but lacks wavelengths much less than the flow depth (Figure 2). The maximum head drop is 7 mm in trial 1 and 10 mm in trial 2. The minimum occurs beneath the log's downstream edge, as in Bearman and Zdravkovich . Head gradually recovers with distance downstream and approaches a value 4 and 6 mm less than upstream head in trials 1 and 2, respectively (Figure 2). Simulated and measured head values are in good agreement.
 Head gradients along the bed cause water to downwell upstream of the log and upwell downstream. Fastest fluxes occur beneath the log and decay rapidly with distance upstream and downstream (Figure 2). The spatial extent of hyporheic exchange is limited only by the flume walls (Figure 3). In simulations for trial 1, the maximum downwelling rate is 3.1 mm s−1 and occurs less than 1 cm upstream of the log's leading edge. The maximum upwelling rate is 5.7 mm s−1 and occurs 5 mm downstream of the log center. The average flow rate across the sediment-water interface is 0.12 mm s−1. In trial 2, simulated fluxes are greater: the maximum downwelling rate is 4.6 mm s−1 and occurs less than 5 mm upstream of the log's leading edge. The maximum upwelling rate is 8.3 mm s−1 and occurs 5 mm downstream of the log center (Figure 3). The average flow rate across the sediment-water interface is 0.18 mm s−1 (50% greater than trial 1). We also compare pore water flow fields using the flushing intensity qf after Zlotnik et al. . The flushing intensity is the horizontally averaged Darcy flux (q) calculated at incremental depths. Flushing intensity is greatest near the sediment-water interface and declines exponentially with depth (Figure 3).
 In general, simulated streamlines agree with observed dye paths (Figure 2; Movies S1 and S2). In simulations for trial 1, upwelling flow paths converge strongly near the log, curving back upstream in some cases. A small, shallow cell of downwelling and upstream-directed exchange overlies deeper upstream-deflected flow paths. If this zone existed in the flume, it probably occurred farther downstream, since upwelling dye streaks near the log did not curve strongly upstream (Figure 2; Movie S1). In models, the existence or location of a shallow, secondary exchange cell is sensitive to pressure gradients beneath the wake zone downstream of the log. Because depth of the cell and exchange rates within it are small, the cell should not significantly impact solute or heat transport within the sediment. In trial 2, simulated flow paths closely match dye streaks (Figure 2). No secondary hyporheic exchange cell occurs, presumably because pressure recovery along the bed is more gradual.
 Simulated travel times based on particle tracking are 80% faster than dye travel times, on average. In trial 1, starting at the upper right injection port and moving clockwise, approximate travel times for each dye path are 15, 510, 240, and 30 s (±15 s). Simulated times are 19, 258, 133, and 16 s. In trial 2, approximate travel times for each dye path were 30, 360, 180, and 15 s (±15 s), compared with simulated times of 14, 159, 98, and 11 s. One source of error may be the model permeability, determined from fitting models with tracer data. Permeability estimated from a constant head test on a column of repacked flume sediment was ∼2.5 times less than the permeability from tracer tests. Actual permeability probably lies between these two estimates. Permeability directly controls dye travel times but has no impact on trajectories. Porosity also controls dye travel times, but the error in our estimate of flume porosity was small (∼5%). Another potential source of error is the flume wall, which may have slowed the flow of pore water and dye. The good agreement between measured and simulated pressure along the bed suggests that the pressure boundary condition is not a significant source of error.
 While dye injections reveal travel times for individual flow paths, a tracer test samples the integrated effect of all hyporheic flow paths. In trial 1, the tracer concentration in surface water approaches a constant value after approximately 30 h, indicating nearly complete mixing between surface water and pore water (Figure 4). Measured concentrations agree well with simulated concentrations, which are based on the residence time distribution from the porous flow model. The longest residence times are on the order of 1 day. In trial 2, surface water concentration declines more rapidly and approaches a constant value after approximately 20 h. Simulated concentrations agree well with measured concentrations at early times but decline more slowly at late times, suggesting that the simulated residence time distribution overemphasizes longer residence times. A small amount of sediment mobilized over the course of the tracer experiment, which may partly explain discrepancies between simulated and measured concentrations.
3.2. Influence of Scour Topography
 The flow of river water around LWD preferentially erodes and deposits sediment. This scour morphology also drives hyporheic exchange. Here we compare hyporheic exchange under three scenarios: a channel-spanning log with plane bed, a log with scoured bed, and scoured bed without a log. These three simulations could represent the evolution of hyporheic exchange near a channel-spanning log after initial tree fall, scour development, and log removal or decomposition. Bed topography consists of a scoured zone with maximum depth of 2.6 cm and a bar-like sedimentary deposit with height of 5.6 cm located approximately 60 cm downstream of the log.
 Scour topography locally increases flow depth and decreases blockage ratio. As a result, the total head drop across the log decreases (Figure 5). As in the planar bed case, head recovers approximately 1 m downstream, but a local head maxima occurs on the upper stoss face of the bar, and a local minima occurs at the base of the lee face. Large-scale hyporheic exchange patterns are similar to the plane bed case, but smaller, shallow exchange cells are superimposed near the bar. These small exchange cells slightly increase hyporheic flushing rates within the top few centimeters of sediment, but flushing rates at depth are less for the scoured bed because of the reduction in total head drop, which drives longer flow paths. River water generally downwells in three locations: upstream of the log, near the crest of the bar form, and downstream of the bar form. Maximum upwelling occurs immediately downstream of the log and at the base of the bar's lee face.
 In the absence of the log, scour topography produces a much smaller head drop. Undulations in head along the bed cause water to downwell into the trough and stoss face and upwell immediately downstream of the lee face. Total flux across the bed and hyporheic flushing at all depths are greatly reduced. An important implication is that although scour morphology measurably impacts hyporheic exchange, the log creates the predominant head drop that drives most of the exchange. Additionally, hyporheic exchange is not linearly additive—fluxes due to the log and scoured bed are less than the sum of individual fluxes. The nonlinear behavior is due to the nonlinearity of surface water flow that produces head gradients along the bed.
3.3. A Semiempirical Model for Head Profile Along the Bed
 Head measurements from flume experiments all include a large drop of magnitude h1 along the bed beneath the log (Figure 6). Head partially recovers by magnitude h2 downstream. In Appendix A we heuristically show that the maximum head drop (h1) should scale with blockage ratio and channel Froude number. Height of the log above the bed (or gap ratio) should not impact the head drop. Here we support this argument with pressure measurements and additional CFD simulations. We then develop a simple equation for head along the bed due to flow around a channel-spanning log.
 Measured and simulated head drop normalized by d increases with Fr and B but shows no dependence on G (Figures 7a–7c). Since h1/d is related to both Fr and B via power laws (Figures 7a and 7b), its dependence on both parameters can be combined into one power law. Regression leads to the relationship (Figure 7d)
Head recovery (h2) scales similarly with Fr and B (Figure 7e):
 Lognormal distributions were chosen to represent the pressure variation in the vicinity of the log. These functions match the observed asymmetry in upstream pressure drop and downstream recovery and require only two parameters (expectation and spread, Figure 6). Note, however, that lognormal distributions are used here with no statistical meaning; they are simply convenient. In total, five parameters were fitted for each pressure profile, in addition to the magnitudes of pressure drop and recovery: location of the pressure minimum with respect to log center (x0), expectation and spread of the pressure drop ( and ), and expectation and spread of the pressure recovery ( and ).
 Based on individual correlations, B and G control x0 and the expectation and spread of pressure drop and recovery, normalized by log diameter (Figure 8). Specifically, the pressure minimum shifts further downstream as B decreases and G increases (Figure 8a):
Similarly, the expectation and spread for the pressure drop decrease with B and increase with G (Figures 8b and 8c):
The expectation and spread of pressure recovery are best correlated with G (Figures 8d and 8e):
The correlations are relatively weak, however, suggesting complex behavior in the pressure recovery downstream of the log. In natural settings, scour topography will further complicate the pressure recovery (Figure 5).
 Finally, head h along the bed can be approximated as
Equations (10)–(17) constrain head along the bed for simulating hyporheic exchange and solute or heat transport near channel-spanning logs, eliminating the need for CFD simulations in many cases. These equations do not include the stream gradient, but a slope adjustment could be added to account for other sources of flow resistance.
3.4. Hyporheic Exchange From LWD Geometry and Flow Parameters
 Hyporheic exchange rates increase with Froude number and blockage ratio:
where is the mean flux across the sediment-water interface normalized by sediment hydraulic conductivity (Figure 9a). An important point is that hydraulic conductivity can vary by orders of magnitude in riverbed sediment, while B and Fr have limited ranges. Sediment permeability therefore exerts the greatest control on hyporheic exchange due to logs. Gap ratio has a limited impact on mean hyporheic exchange rate. Including gap ratio in the predictive relationship for hyporheic exchange does not significantly improve the regression, and the exponent associated with gap ratio is significantly smaller than exponents associated with Froude number and blockage ratio (Figure 9b). Gap ratio does influence spatial distributions of flux across the bed, although maximum downward and upward fluxes always occur near the log (i.e., Figures 2 and 3).
 The flow of surface water past a channel-spanning log induces a pressure drop along the bed that drives downwelling upstream of the log and upwelling downstream. Rates of hyporheic exchange are fastest near the log and decay exponentially with depth and distance upstream and downstream. These patterns of hyporheic exchange are fundamentally similar to exchange induced by weirs and steps, which also create an abrupt pressure drop along the bed [Endreny et al., 2011a, 2011b; Hester et al., 2009]. Our numerical simulations suggest that a small, shallow zone of downwelling and upstream-directed recirculation can form downstream of the log within the broader zone of upwelling. Flume observations did not confirm the existence of this recirculation cell, which may depend on subtle pressure recovery behavior beneath the log's wake. Similarly, Endreny et al. [2011b] showed that shallow recirculation cells can form in broader zones of upwelling downstream of steps beneath the plunging nappe and in the zone of pressure recovery at the end of the hydraulic jump. These shallow recirculation cells are difficult if not impossible to measure in field settings.
 Hyporheic exchange due to channel-spanning logs increases with blockage ratio, Froude number, and sediment permeability. The dependence on blockage ratio resembles the behavior for weirs, where exchange increases with weir height [Hester and Doyle, 2008]. The dependence on Froude number and permeability resembles the behavior for planar and rippled beds, where exchange increases with the shear Reynolds number and permeability Péclet number [O'Connor and Harvey, 2008]. In our study, shear velocity was not apparent or intuitive for open-channel flow around a single log, and Froude number was a logical choice for scaling (see Appendix A). Because we did not explore Reynolds number dependence, we caution against applying our results to cases with significantly different Reynolds numbers. Gap ratio does not significantly impact hyporheic exchange for the large blockage ratios considered here (>0.25) because the log interacts with the water surface, which controls pressure gradients along the bed. For smaller blockage ratios, pressure gradients along the bed may vary more with gap ratio because local inertia and drag contributions near the log may dominate free surface contributions. Gippel et al.  and Hygelund and Manga  similarly showed that drag coefficients on logs do not vary strongly with gap ratio (or relative log depth).
 Depending on blockage ratio and Froude number, LWD in natural densities can drive reach-averaged hyporheic exchange rates similar to rippled bed topography. As an example, hyporheic exchange rates averaged over the 5 m flume in trial 2 are slightly greater than rates due to bedforms under the same flow conditions (Figure 3). Mean flux is 1.32 × 10−4 m s−1 for bedforms and 1.87 × 10−4 m s−1 for the single log, which can be considered a reasonable approximation for the reach-averaged flux in a lowland stream with log spacing of 5 m. This spacing is well within the reported range for natural lowland streams in central Europe (defined using log diameter >0.10 m) [Hering et al., 2000]. Notably, exchange beneath logs is much deeper than beneath ripples. Two hours after introduction of conservative tracer to surface water, the sampled hyporheic zone area is 1.9 m2 for the log and only 0.074 m2 for bedforms (defined by the area where C/C0 > 0.1) (Figure 3). The implication is that LWD and ripples can drive similar reach-averaged hyporheic exchange rates, but the length scales and residence times of LWD-induced hyporheic exchange are greater. Where LWD exists, it is a key driver of hyporheic mixing at the scale of meters.
 Our analysis is restricted to simple LWD configurations. Specifically, we did not consider partially buried logs but only suspended logs that span the entire channel at orientations normal to the flow direction. We anticipate that hyporheic exchange should increase with Froude number, blockage ratio, and permeability for more complex LWD morphologies such as channel-spanning logs at oblique angles or logs with scoured beds. The scoured bed in our numerical simulation did not significantly alter exchange rates or large-scale flow patterns but only created additional shallow, nested exchange cells (Figure 5).
 Another limitation of this study is the artificial flow boundary created by flume walls and included in numerical models. In natural streams, competing interactions between LWD and other geomorphic features will determine the spatial extent of the hyporheic zone. In simulations, the hyporheic zone fills the domain, regardless of boundary position or conditions, because the single channel-spanning log is the only source of exchange. Domain size also impacts mean hyporheic flux estimates (Figure 9; equation (18)). We selected a domain size for our sensitivity study (10 m long and 2 m deep) to minimize the influence of boundary conditions on hyporheic exchange patterns while maintaining a reasonable representative elementary volume for examining the effects of a single channel-spanning log. At the upstream and downstream boundary locations, hyporheic flow rates are orders of magnitude lower than in the vicinity of the log, so these boundaries do not significantly affect exchange near the log. However, because domain-averaged exchange rates are dependent on domain size, equation (18) cannot be used to predict average hyporheic exchange rates in reaches with LWD. Instead, equations (10)–(17) can be used to predict head along the bed and to simulate hyporheic flow. In this case, wake interaction between logs should be negligible for separation distances greater than 10 diameters [Ranga Raju et al., 1983].
 Channel-spanning logs drive significant hyporheic mixing at the meter scale characterized by downwelling upstream and upwelling downstream. Local scouring of the bed near a channel-spanning log induces additional shallow, nested exchange cells, but the log contributes the majority of exchange. Rates of exchange are fastest near the log and decay exponentially with depth and distance upstream or downstream. Exchange increases with blockage ratio and Froude number and is relatively insensitive to vertical location of the log in the channel.
 To our knowledge, this study is the first to analyze the mechanics of hyporheic exchange due to in-streamflow obstacles suspended within the channel and to show that exchange generally increases with obstacle size and channel Froude number. The observed dependence of hyporheic exchange on LWD blockage ratio and Froude number likely applies to other LWD morphologies besides channel-spanning logs, but more studies are needed. The channel-spanning LWD morphology examined here is typically found in small, lowland streams with mobile beds and relatively constant discharge.
 Under natural LWD densities for lowland streams (in our example, one log every 5 m), reach-averaged hyporheic exchange rates due to LWD and ripples are similar. However, hyporheic exchange due to LWD has longer path lengths and residence times. Our findings suggest that in LWD-depleted streams, hyporheic exchange may be severely reduced (especially in straightened channels with relatively simple morphologies). LWD reintroduction is a promising tool for restoring hydrologic connectivity. Our model for bed pressure profiles and hyporheic exchange rates in the vicinity of channel-spanning logs can be used to inform the design of LWD reintroduction and restoration projects.
 Here we show from first principles how the head drop due to flow around a log should scale with blockage ratio and Froude number. Consider energy at two locations: the log center where the flow constriction is maximized and far upstream. Energy conservation for an inviscid, incompressible, irrotational fluid subject to gravity at these two locations leads to the Bernoulli equation:
where h is hydraulic head, which is assumed to vary only in the downstream direction, and v is mean flow velocity. Conservation of mass requires
where d is flow thickness. Noting that B = dlog/dupstrm and combining (A1) with (A2) yields
Dividing by dupstrm and noting that ,
where h1 is and d is dupstrm. For an inviscid, irrotational flow subject to no drag, h1/d should increase with the square of the Froude number and approach infinity as blockage ratio approaches unity. Equation (A4) resembles empirically developed scaling in equation (10).
 This research was supported by the National Science Foundation (EAR-0836540), an AGU Horton Research grant to AHS, and the Geology Foundation at the University of TX at Austin. Audrey Sawyer was supported by a Graduate Research Fellowship from the National Science Foundation. Wade Chaney, Katy Gerecht, Jesus Gomez Velez, Frank Hucks, Michael Markowski, Derek Sawyer, and Yao You provided assistance in flume experiments. We thank David Mohrig and Ben Hodges for fruitful discussions on turbulent channel hydraulics. The manuscript benefitted from comments by the Associate Editor, Eric Hester, and two anonymous reviewers.