The standard assumptions for modeling stream tracer experiments were applied to our experiments, including steady flow prior to the flood, complete tracer mixing in the main channel, along with others thoroughly documented in reviews such as Harvey and Wagner . There were several new aspects of this work. First we injected both solutes and fine particulates to load up storage zones at base flow and then followed with an experimental flood to examine tracer flushing effects. Second we measured tracer dynamics at multiple scales, i.e., at the stream reach scale of hundreds of meters and simultaneously at the geomorphic unit scale in samples collected in situ beneath sandy bedforms. Third we used a transport model with multiple storage zones that could potentially represent dynamics of storage in specific geomorphic units [see, e.g., Briggs et al., 2009; Harvey et al., 2005]. The reach scale measurements characterized the dominant spatial and temporal scales of storage without explicitly separating distinct types of storage. The in situ measurements, on the other hand, quantified storage associated with specific geomorphic features but, due to limited sampling, might not provide statistically valid estimates of whole-stream behavior. Therefore, by comparing transport and storage metrics across spatial scales we assessed the importance of hyporheic exchange beneath bedforms and estimated the cumulative effects on transport and fate of solutes and fine particulates in the stream as a whole.
3.7.2. Reach-Scale Simulations
 The OTIS-2stor model is an extended version [Choi et al., 2000] of the solute and fine particulate transport model of Runkel  including both a faster and more slowly exchanging storage zone. Using two storage zones instead of one typically improves the model by simulating the broader range of storage timescales that have been observed in the field [e.g., Neilson et al., 2010; Briggs et al., 2009; Choi et al., 2000] and that have been demonstrated theoretically [e.g., Cardenas, 2008]. As in all such experiments the longest storage timescales (in this case beyond the timescale of the more slowly exchanging storage zone) will be ignored, but such storage is probably not observable by any type of modeling given the inherent limitations set by injection time, travel time through the experimental reach, and uncertainty in the background tracer concentration [Harvey and Wagner, 2000].
 OTIS-2stor was used to estimate the reach-averaged characteristics of transport including advection and dispersion, groundwater inflow, solute and fine particulate exchange with storage zones, and the rate of removal of fine particulates. The governing equations of the OTIS-2stor model are
main channel concentration [mg l−1]
volumetric flow rate [m3 s−1]
main flow zone cross-sectional area [m2]
longitudinal dispersion coefficient [m2 s−1]
- qLin, qLout
lateral inflow and outflow rate [m3 s−1 m−1]
lateral inflow concentration [mg l−1]
- α1, α2
storage zone 1 and 2 exchange coefficients [s−1]
storage zone 1 and 2 concentration [mg l−1]
- AS1, AS2
storage zone 1 and 2 cross-sectional area [m2]
first-order deposition coefficient for fine particulates in stream [s−1]
- λS1,sed, λS2,sed
first-order deposition coefficient for fine particulates in storage zones [s−1]
 Streamflow discharge estimates from the previously described mass balance analysis were used, and groundwater discharge (qLin) was estimated by differencing streamflow estimates at several locations along the reach. The other transport parameters were estimated by inverse modeling using an extended version of the nonlinear least squares optimization procedure described in the documentation of OTIS [Runkel, 1998] and elaborated in Harvey et al. . The inversely estimated parameters included A, the average area of the main channel, DL, the longitudinal dispersion parameter, which characterizes relatively fast mixing processes that achieve equilibrium in a given transport distance, and rate coefficients and cross-sectional areas (α1, As1, α2, and As2) of the two storage zones.
 Typically the conservative transport parameters are estimated first, by an inverse model run that optimizes those parameters to fit the conservative solute tracer data. The non-conservative transport parameters are then estimated by a second model run that is optimized to fit the non-conservative tracer data. The rate parameters for sediment removal also must be estimated sequentially becauseλsed accounts for total removal from the stream while λS1,sed, λS2,sedeach account for a proportion of the total that occurs in specific zones of the stream. Thus we could not simply estimate all three removal rate parameters in a single step because non-unique results would be expected. Instead our approach was to estimateλsedin the OTIS-2stor simulation (while settingλS1,sed, λS2,sed to zero). This approach is valid for quantifying the sum of all processes contributing to removal. In a second step we independently estimated λS1,sed, λS2,sed by other means using in situ data measured in the hyporheic zone (described in section 3.7.3). The remaining removal processes for fine sediment not accounted for by our in situ sampling in the hyporheic zone (e.g., settling to the streambed) could then be calculated by difference, i.e., λsed − λS1,sed − λS2,sed. The uncertainties of the estimated parameters, when expressed as coefficients of variation, are on the order of 0.1 for A, and between 0.5 and 2 for DL, α1, As1, α2, As2 and λsed, which is typical for environmental modeling and considered good for widely varying and poorly know parameters in natural streams. Uncertainty estimates for λS1,sed and λS2,sed are discussed in section 3.7.3.
 Additional parameters can be calculated directly from the basic parameter values. Those include mean residence time (ts) and physical dimension (ds) of storage zones. The mean residence time (ts) of fluid and solute that enter the storage zones is estimated by the ratio of the storage zone cross-sectional area (As) and the storage-exchange flux (the product ofα and A) (equation (7) in Table 1). The estimate of storage depth determined by reach scale modeling can be directly compared with in situ measurements in the streambed. Estimation of the depth of storage (ds) assumes lateral extension across the full width of the stream. The hyporheic depth, dh, can be computed as As/(w θ) where w is average stream width and θ is average porosity of the streambed, which must be added for calculations that are compared with in situ measurements in the hyporheic zone (equation (6) in Table 1). Methods of directly estimating mean residence times (th) and physical dimensions of the hyporheic zone (dh) at small spatial scales are presented in section 3.7.3.
Table 1. Reach and Geomorphic Unit Scale Estimates of Storage and Removal of Solutes and Fine Particulatesa
|Parameter Description||Equation||Equation Number||Variables Not Previously Defined|
|Reach Scale Estimates|
|Depth of storage zones in surface water or streambed sediment|| ||(5)||average width of stream (w), average sediment porosity (θ)|
| || ||(6)|| |
|Residence time of stream water in storage zones|| ||(7)||storage zone and main flow zone cross-sectional areas (As and A), storage- exchange coefficient (α)|
|Uptake velocity of stream water into storage zones|| ||(8)|| |
|Removal velocity of fine particulates by all processes|| ||(9)||first-order deposition coefficient for fine particulates in stream (λsed)|
|Geomorphic Unit-Scale Estimates|
|Residence time of stream water in migrating bedforms|| ||(10)||velocity of bedform migration (ub), bedform wavelength (λbedform)|
|Uptake velocity of stream water into migrating bedforms|| ||(11)||bedform height (H), i.e., vertical height of bedform from peak to trough|
|Uptake velocity of stream water into hyporheic zone|| ||(12)||depth in streambed of measuring point in hyporheic zone (dh), fraction surface water (fsw) in subsurface sample determined from Br tracer, median residence time of solute tracer at sampling depth (th)|
|Removal velocity of fine particulates in hyporheic flow paths||V′h,sed = (Cswfsw − Ch)/Csw)V′h||(13)||plateau concentration of fine particulate tracer in surface water (Csw), plateau concentration in hyporheic zone (Ch), fraction surface water (fsw) in subsurface sample determined from Br tracer, uptake velocity of stream water into hyporheic zone (V′h)|
3.7.3. Comparison of Reach- and Geomorphic Unit-Scale Estimates of Tracer Dynamics
 We used uptake velocity as a simple and general expression of the exchange flux of stream water with storage zones, or alternatively, the removal flux of solute or fine particles (Table 1). Using a common expression allowed us to compare tracer dynamics across spatial and temporal scales. The term “storage” denotes a temporary delay in transport resulting from exchange between the main channel and relatively slowly flowing zones of the stream, such as recirculating flow of surface water or flow through hyporheic paths. Tracer that is still stored at the end of the experiment is referred to as having been “removed” from the flowing stream water. It is recognized that longer timescale storage can occur (e.g., in very long hyporheic flow paths) and that changing flow conditions can quickly remobilize stored tracer. For that reason we focused on quantifying storage and removal rates of solutes and particles at base flow followed by a flood to determine the extent to which tracer “removed” during base flow might suddenly be entrained. To improve our interpretations of the physical processes involved we compared reach-scale uptake velocities estimated from in-stream tracer data with estimates of specific storage processes measured in situ (e.g., migrating bedforms or hyporheic flow). The comparison aids in evaluating the relative importance of specific storage and removal processes.
 Uptake velocities directly estimated at the scale of the stream reach are summarized in equations (8) and (9) in Table 1. Reach-scale uptake velocities were calculated by multiplying the best fit rate parameters from OTIS-2stor simulations (i.e.,α for water or conservative solutes or λ for fine particulates) by the average stream depth (Table 1). For example, Vwis the reach-averaged uptake velocity at which water enters the dominant storage zones, which may include storage in slowly moving surface or subsurface water. Likewise,Vsedis the reach-averaged uptake velocity describing the net rate of removal of fine particulates from streamflow by all processes, including flocculation, settling, and advective transport into and filtration within hyporheic flow paths.
 Some uptake velocities associated with specific geomorphic features could be directly estimated using local in situ observations. Those calculations are summarized in equations (10)–(13) in Table 1. The uptake velocity associated with turnover of migrating bedforms, V′bedform (equation (11) in Table 1) was estimated using measurements of the celerity of migrating bedforms (ub), average bedform wavelength (λbedform), bedform amplitude (hm), i.e., the height difference between bedform peak and mean bed height, and streambed sediment porosity (θ). Average fluid residence time in actively migrating bedforms (t′bedform) was calculated using the same measurements (equation (10) in Table 1). It should be noted that V′bedformapplies both to solute and suspended particulates because mobile bedforms store and release both constituents at the same rate as long as fine particulates are entrained into streamflow when re-exposed by bedform turnover [Packman and Brooks, 2001]. Note that all geomorphic unit-scale uptake velocities are distinguished from their reach-scale counterparts using a prime superscript for symbols.
 Another uptake velocity that could be directly estimated from field data was solute uptake into hyporheic flow paths, V′h. Solute uptake in hyporheic flow was directly estimated using measurements of streambed sediment porosity (θ), the median tracer travel time, t′h, depth to the sampling point in the streambed sediment (dh), and the fraction of surface water in hyporheic flow paths at the measurement depth that was derived from the stream (fsw) (equation (12) in Table 1). The median travel time of the solute tracer was estimated based on the time that subsurface concentration reached 50% of its eventual plateau value at the sampling depth. The fraction of hyporheic flow derived from the stream was computed as (Cs − Ch)/(Cs − Cb) where Cs and Ch are the observed tracer plateau concentrations in the stream and hyporheic zone, respectively, and Cb is the background tracer concentration. Note that results are reported for all depths (1.4, 2.9, 4.9, and 7.9 cm) where hyporheic flow was detected.
 Last, the uptake velocity of fine particulate removal by filtration in hyporheic flow paths, V′h,sed, was directly estimated using equation (13). The approach for estimating uptake of fine particulates follows from Triska et al.  and Harvey and Fuller  and builds on equations for conservatively transported solutes in hyporheic flow paths with an additional parameter for removal (equation (13)). Previous method testing by Harvey and Fuller  indicates that uncertainties are expected on the order of 3% for this method.