Mixing of surface and groundwater induced by riverbed dunes: Implications for hyporheic zone definitions and pollutant reactions
E. T. Hester,
Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA
Corresponding author: E. T. Hester, Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, 220-D Patton Hall, Blacksburg, VA 24061, USA. (email@example.com)
 The hyporheic zone is often defined as where mixing of surface water and groundwater occurs in shallow sediments beneath and adjacent to rivers. This mixing is credited with creating unique biogeochemical conditions that can attenuate contaminants from either upstream surface water or groundwater under gaining conditions. However, reactions of contaminants upwelling from groundwater may be more dependent on such mixing than contaminants from surface water. We numerically modeled mixing between hyporheic flow paths induced by riverbed dunes and flow paths of adjacent upwelling of deeper groundwater. Results show that only 12.7% or less tracer mass upwelling from deeper groundwater dispersed across into hyporheic flow paths originating in surface water. The spatial extent of a mixing-defined hyporheic zone was smaller than a hyporheic zone defined as hydrologic flow paths leaving and returning to surface water. Mixing-dependent reactions will therefore be localized within a thin mixing zone yet vary considerably with site conditions. For example, mixing in homogeneous sediments was controlled most by variation in hydraulic conductivity and upwelling flow rate which primarily affected mixing zone length. By contrast, introduction of heterogeneity increased mixing primarily by increasing mixing zone thickness, consistent with studies of flow focusing in aquifers. Finally, dispersivity is a critical parameter for which data are needed for shallow sediments. Our results help clarify hyporheic zone definitions and potential for mixing-dependent reactions. In particular, the biogeochemically reactive portion of riverbed sediments from the perspective of upwelling contaminants does not necessarily spatially coincide with more traditional hydrologic conceptions of the hyporheic zone.
 The hyporheic zone is the area where surface water and groundwater meet in sediments immediately beneath and adjacent to streams, rivers, and riverine estuaries [Winter et al., 1998; Jones and Mulholland, 2000; Bianchin et al., 2011]. The hyporheic zone has unique physical, chemical, and biological properties that are different from both overlying surface water and deeper groundwater [Brunke and Gonser, 1997; Boulton et al., 1998; Hester and Gooseff, 2010]. Some parameters, such as flow velocities, residence times, temperatures, and dissolved oxygen (DO) levels have values in the hyporheic zone that are intermediate between those in surface water and deeper groundwater [Hendricks and White, 1991; Hester and Doyle, 2008; Hester et al., 2009]. Others entail the unique superposition of surface water and groundwater conditions, such as the high interstitial surface area of groundwater together with high levels of dissolved, particulate, or sorbed organic carbon (OC) from surface water [Jones et al., 1995a; Schindler and Krabbenhoft, 1998; Sobczak and Findlay, 2002]. Taken together, these unique characteristics can make the hyporheic zone more chemically reactive than overlying surface water or deeper groundwater [Brunke and Gonser, 1997; Boulton et al., 1998]. Such unique hydrologic and biogeochemical conditions in turn support unique microbial communities [Lowell et al., 2009].
 Hyporheic reactions can naturally attenuate pollutants, yet require simultaneous presence of necessary reactants, necessary redox conditions, sufficient time for reaction, and an active microbial community [Schnoor, 1996]. Most studies of hyporheic reactions have focused on contaminants that originate in surface water, where reactants are transported together along the same flow paths with no need for mixing to bring reactants together [e.g., Marion et al., 2002; Packman et al., 2004; Kasahara and Hill, 2007; Boano et al., 2010; Zarnetske et al., 2011]. In such cases the hyporheic zone facilitates attenuation reactions mainly by providing required redox conditions, high interstitial surface area, and sufficient residence times. Contamination sources are also common in deeper groundwater, and frequently surface at streams, rivers, and estuaries [Winter, 1999; Westbrook et al., 2005; Miller, 2007; Landmeyer et al., 2010]. Such contaminants can also naturally attenuate as they cross through the hyporheic zone. For example, high cation exchange capacity combined with high interstitial surface area can enhance sorption, precipitation, or complexation of metals [Benner et al., 1995; Harvey and Fuller, 1998; Gandy et al., 2007; Smith and Lerner, 2008]. In addition, bioavailable organic carbon OC and particulate OC create anoxic zones and sorption sites, which provide conditions that encourage transformation of excess nitrate, retardation and mineralization of organic pollutants, and transformation of metals such as reduction of hexavalent chromium [Moser et al., 2003; Conant et al., 2004; Fischer et al., 2005; Smith et al., 2009; Vervier et al., 2009; Oram et al., 2010]. Some hyporheic reactions can even exacerbate pollutant issues, for example by nitrification or mercury methylation [Jones et al., 1995b; Creswell et al., 2008].
 Contaminants from deeper groundwater often arrive via different flow paths than other critical reactants (e.g., DO from surface water), so mixing of reactants between different flow paths can be critical for reactions to occur. In addition, the rate and extent of microbially mediated attenuation of many contaminant groups (e.g., chlorinated ethenes, nitrate) depends upon redox conditions whose spatial variation within the hyporheic zone is facilitated by mixing of surface water (usually high in DO) and groundwater (often lower in DO). Mixing phenomena have been studied for decades in deeper aquifer settings [De Josselin De Jong, 1958; Pickens and Grisak, 1981; Gelhar et al., 1992; Cirpka and Kitanidis, 2000b], including mixing-dependent reactions which often occur only along the fringes of contaminant plumes [Bijeljic and Blunt, 2007; Bauer et al., 2009; Chiogna et al., 2011; Luo and Cirpka, 2011; Van Breukelen and Rolle, 2012]. In the hyporheic zone, the interaction of upwelling solutes with surface water [Kennedy et al., 2009; Krause et al., 2009] and reactions in upwelling groundwater that are likely mixing dependent have been documented [e.g., Conant et al., 2004; Landmeyer et al., 2010]. Nevertheless, neither the mixing of surface and groundwater upon which such reactions depend, nor the dispersion processes that drive such mixing, have received much attention. Dispersion has been accounted for in a variety of hyporheic models [Woessner, 2000; Lautz and Siegel, 2006; Sawyer and Cardenas, 2009; Bottacin-Busolin and Marion, 2010; Jin et al., 2010; Bardini et al., 2012], but controls on such dispersion was not the focus. Our current understanding is therefore insufficient to accurately anticipate the extent of mixing-dependent contaminant reactions in the hyporheic zone, let alone maximize attenuation via engineering controls. Even more fundamentally, the hyporheic zone itself is often defined as the area of mixing of surface water and groundwater [Triska et al., 1989; Sophocleous, 2002; Lautz and Siegel, 2006; Tonina and Buffington, 2009], or at least an area where significant mixing is expected [Winter et al., 1998; Bencala, 2000]. Yet the spatial extent of such mixing, its controls, and the implications for hyporheic zone definitions have not been rigorously explored.
 Mixing of surface water and groundwater within the hyporheic zone is dependent upon interaction of flow paths originating in surface water with those originating in deeper groundwater. In gaining or neutral streams and rivers, flow paths downwelling from surface water return to the water column within relatively short distances, forming “hyporheic flow cells.” The hydraulic gradients necessary to form such cells are induced by riverbed morphology (e.g., bed forms such as dunes, submerged bars, riffles, and debris dams) through hydrostatic head variations and hydrodynamic forcing [Thibodeaux and Boyle, 1987; Gooseff et al., 2006; Hester and Doyle, 2008; Sawyer et al., 2011; Kaser et al., 2013]. Flow paths of upwelling deeper groundwater are then diverted around, and exit to surface water between, these flows cells [Cardenas and Wilson, 2006, 2007c]. Here we use numerical modeling of hyporheic hydraulics and tracer transport informed by prior laboratory flume experiments to study the extent of, and controls on, mixing between hyporheic flow cells and upwelling groundwater. We focus on dune-type bed forms that are typical of medium and large waterways because of their widespread distribution and because the hyporheic hydraulics they induce are well understood [Thibodeaux and Boyle, 1987; Elliott and Brooks, 1997b; Cardenas and Wilson, 2007d]. The specific objectives of the current study are to (1) determine the overall extent of mixing of upwelling groundwater with hyporheic flow cells and the implications for hyporheic zone definitions, (2) determine the effect of potentially controlling physical parameters on such mixing such as average hydraulic conductivity, heterogeneity of hydraulic conductivity, and upwelling groundwater flux, and (3) compare the effect of such measurable controlling parameters with that induced by variation of nonmeasurable model calibration parameters such as dispersivities.
 We constructed a groundwater flow and transport model using MODFLOW, MODPATH, and MT3DMS within Groundwater Model Software (GMS). MODFLOW is a modular finite-difference flow model that solves the three-dimensional groundwater flow equation [Harbaugh, 2005]:
where Kxx, Kyy, and Kzz are values of hydraulic conductivity (L/T) along the x, y, and z coordinate axes respectively, h is the potentiometric head (L), W is a volumetric flux per unit volume representing sources and/or sinks of water (T−1), Ss is the specific storage of the porous media (L−1), and t is time (T). MODPATH is a particle tracking code that uses the output of MODFLOW [Pollock, 1994]. MT3DMS is a transport model that uses the output of MODFLOW and solves the advection, dispersion, and chemical reaction equation for dissolved constituents in groundwater systems [Zheng and Wang, 1999]:
where θ is porosity of the subsurface medium (dimensionless), Ck is the dissolved concentration of species k (ML−3), t is time (T), xi,j is the distance along the respective Cartesian coordinate axis (L), Dij is the hydrodynamic dispersion coefficient tensor (L2T−1), vi is seepage or linear pore water velocity (LT−1), is the concentration of the source or sink flux for species k (ML−3), and is the chemical reaction term (ML−3T−1).
 The model simulated flow and transport in saturated shallow sediments beneath a riverbed exhibiting a series of dune-like bed forms that are widespread in sand or gravel-bedded rivers [Savant et al., 1987; Garcia, 2008]. We focused on dunes because they are common in larger streams, rivers, and estuaries where pollutant reactions are often of most concern. Interstitial flow and transport in the model was driven by a combination of upwelling groundwater and pressure variation along the dune surfaces due to form drag (hydrodynamic forcing) from the overlying water column [Thibodeaux and Boyle, 1987; Elliott and Brooks, 1997a, 1997b; Cardenas and Wilson, 2006] (Figure 1a). We modeled a two-dimensional vertical slice through a single dune so the z term in equation (1) was set to zero. We adopted the dune geometry of Fehlman  with a model domain length in the downstream direction of 0.914 m and a dune height of 0.138 m. The depth of the model in the vertical direction was 1.138 m (model area = 0.98 m2) including the dune. This model depth was set to eliminate lower boundary effects on the hyporheic flow cell. Our grid size was uniform at 2.5 mm by 2.5 mm, giving 366 model cells in the downstream direction and 456 model cells in the vertical direction. The model domain was 10 mm thick in the third dimension for a total model volume of 0.0098 m3 and total of 156,704 model cells.
2.1. Hydraulic Model Boundary Conditions and Parameters
 We modeled groundwater hydraulics in steady state mode, so the transient term on the right-hand side of equation (1) was set to zero. The surface water-groundwater interface was represented in the flow model as a specified head boundary condition that varied along the length of the interface based on laboratory measurements by Fehlman  for a surface water depth of 0.2540 m above the dune crests and average depth of 0.3228 m (to halfway up the dune face). This depth was chosen as the deepest among those available in Fehlman  and is similar in magnitude to prior studies of this kind [e.g., Cardenas and Wilson, 2007a]. The specified heads along the dune face were set as the average hydrostatic pressure (0.3228 m) plus the dynamic pressure differences across the dune due to form drag measured by Fehlman (Figure 1b). In other words, Fehlman measured the deviation of total dynamic pressure from average hydrostatic pressure and we added the hydrostatic pressure for the chosen depth. The vertical model boundaries on the upstream and downstream sides of the model domain (left and right in Figure 1a) were specified as no-flow, based on purely vertical flow field vectors at these locations when multiple dunes were simulated in preliminary modeling runs. The lower boundary was specified as a constant flow boundary to represent upward groundwater discharge [Cardenas and Wilson, 2006, 2007c]. No additional source or sink terms were specified, so the W term in equation (1) was set to zero. We conducted a sensitivity analysis where controlling factors were varied relative to a homogeneous and heterogeneous base case (Table 1). For all parameters varied, the range of values chosen was either representative of the range of that parameter typically encountered in the field, else limited by modeling constraints; the specifics are discussed for each individual parameter below.
Table 1. Summary Table of Base Case Inputs and Ranges for Parameters Varied in the Sensitivity Analysis
 The homogenous base case model had a homogeneous hydraulic conductivity (K) of 84.41 m/d, which is representative of well-sorted coarse sand and has been used in previous triangular bed form numeric model studies [Cardenas and Wilson, 2007a, 2007c, 2007d]. The porosity is 0.35 [Cardenas and Wilson, 2007a, 2007c, 2007d]. A K of lower than 22 m/d caused the flow cell to completely disappear, resulting in no surface water entering the sediment and therefore no hyporheic flow cell. The choice of upper end for the K range was more arbitrary, but as we increased K, the hyporheic flow cell expanded downward into the model domain. We therefore chose a maximum value (150 m/d) where the shape of the hyporheic flow cell was clearly not being affected by the placement of the lower boundary. This strong effect of K on mixing depth is consistent with our use of a specified flow lower boundary [Cardenas and Wilson, 2006, 2007c], rather than a specified head boundary, condition. This is discussed further in the discussion section.
 The range of values for the lower boundary groundwater (Darcy) flux used in the sensitivity analysis was constrained in two ways. First, a lower boundary flux greater than 8.75 m/d (equivalent to 0.08 m3/d flow rate) would cause the entire hyporheic flow cell to disappear, and so we used a boundary flux somewhat less than this (5.47 m/d) as the maximum. Second, the focus of this study is upwelling groundwater because that is what potentially drives mixing of surface water and groundwater, as well as the reactions of upwelling contaminants. If there is no upwelling groundwater at all, these phenomena do not occur. So we chose a minimum lower boundary groundwater flux (0.55 m/d, equivalent to 0.005 m3/d flow rate) where the placement of the lower boundary was clearly not affecting the shape of the hyporheic flow cell (similar to how we chose our upper K). We kept upper boundary (dune surface) pressures constant between model runs because surface water was not explicitly modeled, and because we felt that the pressure data for the deepest water column in Fehlman  was the most realistic for large rivers. Nevertheless, flow path configurations are driven by the differences in pressures between the upper and lower boundaries, and we have already covered the possible range of such configurations by varying the lower boundary fluxes as described above.
 We used horizontal to vertical anisotropy of 1:1 for the base case similar to Cardenas and Wilson [2007a, 2007c, 2007d]. Salehin et al.  studied hyporheic exchange in heterogeneous streambeds and selected an overall anisotropy of 10:1 and Duwelius  found that the ratio between horizontal and vertical K ranged up to 11:1. We therefore varied anisotropy up to 20:1 in the sensitivity analysis (horizontal K was kept constant and vertical K varied with anisotropy).
 Because riverbed sediment hydraulic conductivity is typically not homogeneous, and because such heterogeneity likely has important impacts on mixing of solutes, a heterogeneous sediment structure was created. K is generally described as a lognormally spatially correlated random field, where the degree of heterogeneity is expressed as the variance of lnK and the structure is described as the spatial covariance of lnK in each direction [Salehin et al., 2004]. We used the turning band method of Tompson et al.  which creates a single realization of second-order stationary, lognormally correlated, multidimensional random K field with user specified correlation scales (Ix and Iy), mean conductivity (lnK), and standard deviation (σlnK). In our sensitivity analysis we varied three parameters, including the variance of the logarithm of K (σ2lnK), the correlation length in the horizontal direction (Ix), and the ratio the horizontal to vertical correlation lengths.
 There are very few data on geostatistical properties of streambeds. As background information, a large sample of glacial outwash sand and gravel aquifers was found to have values of σ2lnK ranging from 0.25 to 4.6 [Gelhar, 1993]. Additionally, these aquifers were found to have an overall anisotropy, with the ratio of horizontal to vertical correlation scale varying from 2:1 to 10:1. For a laboratory flume study of hyporheic flows, Salehin et al.  selected values of σ2lnK corresponding to a moderately high, and high degree of heterogeneity, equal to 1.0 and 2.0, respectively. Additionally, Salehin et al.  selected a correlation scale ratio on the high end of what is typically found in groundwater aquifers (10:1) because sediment transport is dominant in the downstream direction. Based on these data, we used σ2lnK of 1.0 and a correlation scale ratio of 10:1 for the heterogeneous base case (Figure 2). We varied σ2lnK from 0.0 to 2.5, and correlation scale ratio from 2:1 to 10:1.
 Sediment heterogeneity occurs over many nested spatial scales. Cardenas et al.  numerically simulated bed form induced hyporheic exchange in heterogeneous sediment. They did not report their correlation scales, but the size of their areas of constant K vary in dimension from ∼0.5 m up to nearly their full model domain (>20m). Field data from the same site indicated correlation lengths in the 2–3.5 m range [Cardenas and Zlotnik, 2003]. Heterogeneity at these scales would have negligible effect at the scale of our study because the correlation lengths are similar in size to or larger than our model domain. We therefore only investigated heterogeneity at smaller correlations lengths (i.e., considerably less than the length of the dune). Examples of such finer scale heterogeneity from the literature include those of Boano et al. , where heterogeneous bed forms in rivers exhibited correlation lengths much less than the bed form length (few mm). Sawyer and Cardenas  used correlation lengths of 4–20 cm, and Salehin et al.  used 10 cm. We chose a horizontal correlation length within this range (base case 4 cm, varied from 0.9 to 10 cm in the sensitivity analysis). This is similar to these previous studies, but allows for ∼10–100 correlation scales in the sediment domain, thus meeting the ergodicity assumption of the stochastic theory [Salehin et al., 2004].
2.2. Transport Model Boundary Conditions and Parameters
 We specified a constant concentration of conservative tracer (10 mg/L) at the lower boundary to simulate upwelling of a groundwater contaminant plume. The upper boundary was a constant concentration of 0 mg/L. The transport model was run until it reached steady state (i.e., the concentration field no longer changed with time). We assumed freshwater in both groundwater and surface water (i.e., no density effects of salt), thereby representing all streams, all rivers, and freshwater portions of estuaries. Reactions and other sources or sinks of solute were not included, so the rightmost two terms on the right-hand side of equation (2) were set to zero.
 Mixing of solutes between adjacent groundwater flow paths in unconsolidated sediments is caused by transverse dispersion [Cirpka et al., 1999; Van Breukelen and Rolle, 2012], which includes both hydrodynamic dispersion due to pore scale tortuosity and larger scale heterogeneity as well as molecular diffusion [Freeze and Cherry, 1979]. Longitudinal and transverse dispersivities (αL and αT (L), respectively) are therefore crucial values with regards to obtaining accurate mixing. Here we seek to model local dispersion rather than macrodispersion. Local dispersivities reported in the literature and used in prior modeling studies exist within fairly narrow ranges despite some range of sediment texture. Sawyer and Cardenas  and Sawyer et al.  use αL = 0.02 m with αT varied simultaneously as 1/10 αL. Cirpka et al.  use αL = 0.01 m with αT varied simultaneously as 1/5 αL. And finally, Gelhar et al. , Cirpka and Kitanidis [2000a, 2000b], and Werth et al.  use αL = 0.01 m with αT varied simultaneously as 1/10 αL. For the base case model, we set αL equal to 0.01 m and αT equal to 0.001 m as within this narrow range of reported values (Table 1). For the sensitivity analysis of the transport model, we varied αL down to 0.0005 m, which is below the lowest value reported from among dozens of studies summarized in [Schulze-Makuch, 2005], and thus brackets the known range of local dispersivities (Table 1). For the upper end of the range we chose 0.05 m. The choice is somewhat arbitrary, as larger values reported in the literature get quickly into the macrodispersion realm. For all variations in αL, αT was varied simultaneously as 1/10 αL. We used 10−9 m2/s for molecular diffusion, which is in the middle of the range for organic compounds in water [Schwarzenbach et al., 1993], and therefore higher than would occur in porous media, yet still more than an order of magnitude less than mechanical dispersion in our model. The 0.0025 m by 0.0025 m grid size minimized numerical dispersion [Cirpka et al., 1999]. This grid size is smaller than those used to model transport in similar settings [Cardenas et al., 2008; Sawyer and Cardenas, 2009] but also makes model runs take >24 h even on a fast PC. This is a key reason why our sensitivity analysis varied parameters one at a time rather than doing Monte-Carlo simulations. Although this allows us to simulation dispersion accurately, we acknowledge that the resulting one-at-a-time variation of parameters reduces the ability to explore parameter interaction.
3.1. Homogeneous Base Case Results
 For the homogeneous base case scenario, the MODFLOW head contours and MODPATH particle tracking (Figure 3a) show water leaving the stream and entering the sediment at high pressure locations along the dune and returning to the stream at low pressure locations along the dune (hyporheic flow cell). Groundwater upwells vertically until it meets surface water downwelling from the stream, and then it diverges either upstream or downstream to discharge into the stream at low pressure zones along the surface water-sediment interface. There is a distinct inflection point where upwelling groundwater meets downwelling surface water. We refer to the depth of this inflection point below the highest dune elevation as the depth of mixing (Figure 3a). Flow paths have longer lengths and longer residence times for particles leaving the stream at the highest pressure locations along the dune. As pressure decreases along the dune, flow path lengths and residence times decrease.
 A mixing zone develops at the boundary between the regions of advected surface water and upwelling groundwater where flow paths from these two source waters occur in parallel (Figure 3a). This mixing zone is created by tracer mass undergoing transverse dispersion from flow paths originating at the lower model boundary (i.e., with tracer) to flow paths originating at the upper model boundary (i.e., initially without tracer). This can be visualized as an area of intermediate tracer concentration such as 10%–90% of the concentration of the lower boundary condition (Figure 4b). This mixing zone is a thin semicircular band, whose area is a function both of the length and thickness of the band. Advective residence times within the mixing zone can be estimated by counting the number of arrows shown along the flow path nearest the mixing zone. For the flow path originating in surface water that comes closest to the mixing zone, there are ∼3 arrows shown, for a residence time of ∼45 min. For the corresponding flow path coming from groundwater, there are ∼4 arrows, or ∼60 min residence time. Note that these are advective residence times, in that they do not account for effects of dispersion.
 A key results metric from our study is the amount of tracer mass that undergoes mixing between flow paths originating in groundwater and flow paths originating in surface water. In other words, we are interested in the amount of tracer mass that undergoes transverse dispersion from flow paths originating at the lower model boundary to flow paths originating at the upper model boundary. Rather than quantify the mixing/dispersion that occurs at individual locations along the mixing zone, we instead quantify the net effect or sum total of mixing/dispersion that occurs along the entire length of the mixing zone. We do this by quantifying the mass fluxes into and out of the model domain (water flow rate times tracer concentration) at various locations across the upper model boundary (Figure 4). We conceptually distinguish groundwater that originated at the bottom boundary (GW) with surface water (SW) that originated at the upper boundary, as well as water that is entering (IN) and exiting (OUT) the model. We use these distinctions to split the upper model boundary into three zones: “GW OUT” is along the upstream and downstream sides of the model domain, where upwelling GW is discharging to the stream, “SW IN” is where SW is entering the sediment bed in the very middle, and “SW OUT” is where SW is discharging to the stream.
 The mass flux for the SW OUT zone was calculated in order to describe mixing. If there was no mixing, all mass of conservative tracer entering the model at the lower boundary would leave the dune surface in the GW OUT zone. In other words, if there was no mixing, no mass would exit the model in the SW OUT zone. The magnitude of mass flux out of the SW OUT zone therefore describes the total cumulative effect of all mixing and therefore transverse dispersive mass flux that occurs along the entire length of the mixing zone. We call this parameter “total mixing flux.” To be clear, total mixing flux is calculated as mass flux out of the model domain, but we are only counting mass flux exiting in the SW OUT region, and mass can only exit at that point if it has previously underdone mixing (i.e., transverse dispersion) within the model domain from flow paths originating at the lower boundary (deeper groundwater) to flow paths originating at the upper boundary (surface water). Note the considerable variability of flux among individual boundary cells, particularly in the GW OUT zones (Figure 4). This is due to the representation of a sloping dune face as a stair-stepped boundary in a rectilinear finite-difference numerical scheme, and is the reason for also including the moving average.
3.2. Magnitude and Variation of Mixing
 Mixing flux was small compared to overall upwelling groundwater flux. Across all homogeneous model runs with base case dispersivities, mixing flux ranges from 1.9% to 10.9% of tracer in upwelling groundwater (Figure 5). Similarly, mixing ranged up to 12.7% for our heterogeneous model runs (Figure 6). These ranges do not include our sensitivity analysis to dispersivity, where mixing flux ranged up to 17.1% for the highest dispersivity (Figure 5d).
 This small percent mixing flux is consistent with mixing zones that were small compared to the model domain. For homogeneous model runs with base case dispersivity, the size of the mixing zone ranged from 1.2% to 3.5% of the model domain, if we define the mixing zone as the area where concentrations are between 10% and 90% of the inflowing boundary condition concentration. Mixing zone size ranged up to 3.6% of the model domain for the heterogeneous case and 8.0% when dispersivity was varied. Even if the definition of mixing zone was broadened to 1%–99% of the lower boundary concentration, mixing zone size range only increased to 2.6%–8.2% of the model domain for base case dispersivity, and up to 20.8% for the highest dispersivity. We normalized the mixing area to the model domain to give a sense of how much shallow sediment (i.e., where hyporheic flow cells typically form) is involved in mixing. We acknowledge that the model domain size is arbitrary in the vertical direction. However, other quantities that we could normalize to (e.g., hyporheic flow cell size or dune size) are less meaningful, and would similarly yield mixing area percentages that are small (<<50%).
3.3. Controls on Mixing in Homogeneous Case
 As hydraulic conductivity (K) increases, the size of the hyporheic flow cell increases, simultaneously increasing the mixing depth and total mixing flux (Figures 7a and 5a). This increase is nonlinear, leveling off somewhat at higher K. The length of the mixing zone simultaneously increases, although its thickness appears to change little. The variation in slope or mild “bumpiness” of the total mixing flux line in this (Figure 5a) and subsequent plots is an artifact of the interaction of a thin mixing zone that only spans a handful of model cells with the stair-stepped discretization of the dune face boundary geometry. This results in mass flux exiting the model from advected surface water (i.e., within the “SW OUT” zone in Figure 4) in only a handful of boundary cells. As parameters varied in the sensitivity analysis such as K are varied, these few cells migrate along the stair-stepped representation of the dune surface boundary condition in the model. This results in some situations where “SW OUT” and “GW OUT” are both occurring in the same model cell at the border of the two flow zones, yet flux from that cell can only be attributed to one of these two flux categories. The resulting bumpiness is evident in all sensitivity analysis plots (Figures 5 and 6). This affects the details of these plots but not the overall trends which are the focus of this study.
 As (Darcy) groundwater flux at the lower boundary increases, the hyporheic flow cell contracts, simultaneously decreasing the mixing depth and total mixing flux (Figures 7b and 5b). This trend is also nonlinear, leveling off somewhat as flux at the lower boundary increases. When a lower boundary flux of 5.47 m/d is reached, the hyporheic flow cell nearly disappears. This trend is opposite of that for K in that increasing K has analogous effects to decreasing lower boundary flux. Similar to varying K, the length of the mixing area changes with variations in lower boundary groundwater flux, while the thickness appears to change little.
 As horizontal to vertical anisotropy increases, the hyporheic flow cell contracts and flattens, simultaneously decreasing mixing depth (Figure 7c). Similar to the cases where K or lower boundary groundwater flux are varied, mixing zone length appears to vary more than mixing zone thickness. Total mixing flux exhibits little overall trend with anisotropy, although the bumpiness discussed earlier in this section is most noticeable here because it is superimposed on a flat trend (Figure 5c). As dispersivity increases, the thickness of the mixing zone increases, increasing the total mixing flux (Figures 7d and 5d). The mixing depth and mixing zone length remain the same.
3.4. Controls on Mixing in Heterogeneous Case
 The results of the heterogeneous model were different from the results of the homogeneous model in several ways. The semicircular mixing zone that was smooth in the homogeneous model (Figure 7) is not smooth in the heterogeneous model (Figure 8). Additionally, the mixing depth in the heterogeneous model is 4.5 cm less (13% different) for σ2lnK = 2.5 than in the homogeneous model for the same K. However, trends of mixing depth and total mixing flux versus K, lower boundary groundwater flux rate, anisotropy, and dispersivity for the heterogeneous model were very similar to the homogeneous model. Similarly, mixing zone sizes as a percent of model domain were similar to those for the corresponding homogeneous models. For these reasons, in this section we present only response to variation of parameters that determine the heterogeneous random K fields.
 As variance of K increases, the thickness of the mixing zone increases slightly, and the mixing depth decreases slightly (Figures 8a and 6a). Total mixing flux increases steadily with increasing variance, indicating that the effect of heterogeneity is to increase mixing. As the horizontal to vertical correlation scale ratio increased from 2:1 to 10:1, mixing depth remained constant at ∼0.3m, and total mixing flux showed no overall trend, fluctuating between ∼8% and ∼10 % of total tracer flux (not shown). As correlation length increases, mixing depth and mixing zone length similarly do not change significantly (Figures 8b and 6b). By contrast, as correlation length increases, total mixing flux increases, along with the thickness of the mixing zone (Figures 8b and 6b).
3.5. Variations in Advective Residence Time Distributions
 Residence times from MODPATH particle tracks can be used to compare the advective regimes of surface water moving through hyporheic flow cells to groundwater upwelling around hyporheic flow cells. To do this, a particle was placed in every cell along the lower boundary of the model as well as the upper boundary/dune surface. Results are shown for a range of hydraulic conductivities in Figure 9. Residence times are much higher for upwelling groundwater (Figure 9b) than within the hyporheic flow cell (Figure 9a). Furthermore, residence times for upwelling groundwater exhibit much less variability than within the hyporheic flow cell, particularly at the low end, where no tailing is observed. This is consistent with all groundwater particles traversing the full model domain height regardless of specific path taken, unlike for the hyporheic flow cell. We note that the model domain height and hence travel times for upwelling groundwater are arbitrary, but this does not affect our conclusion of longer groundwater residence times, as groundwater depths will generally be deeper than our model. While all residence times decrease with increasing K, this effect is much less for upwelling groundwater than hyporheic flow paths because we used a constant discharge lower boundary condition rather than a constant head.
4.1. Implications of a Thin Mixing Zone for Hyporheic Zone Definitions
 An important conclusion of our work is that mixing between dune-induced hyporheic flow paths and upwelling groundwater in gaining or hydrologically neutral rivers occurred in a thin zone (up to 8.2% of model domain) and involved a small minority upwelling solute (up to 12.7%). Although this quantitative result has not to our knowledge been reported previously, it is qualitatively consistent with a few earlier studies which have shown thin mixing zones beneath dunes [Sawyer and Cardenas, 2009; Jin et al., 2010]. This has substantial implications for conceptions of the hyporheic zone, which is often defined as the area of mixing of surface water and groundwater [Triska et al., 1989; Sophocleous, 2002; Tonina and Buffington, 2009], or at least an area where significant mixing is expected [Winter et al., 1998; Bencala, 2000]. Our results indicate that the spatial extent of a mixing-defined hyporheic zone, at least beneath dunes, is generally much smaller than a hyporheic zone defined as where flow paths leave and return to the water column within relatively short distances [Harvey and Wagner, 2000] (Figure 10). The latter definition defines the hyporheic zone as the presence of surface water in the subsurface, while the former defines it as mixing of two different water sources. Some definitions are hybrids between these two extremes, and consider both such regions as the hyporheic zone [e.g., Packman et al., 2004; Lautz and Siegel, 2006].
 Although we did not model analogous mixing that would occur beneath weir-type structures (e.g., debris dams or boulder weirs) or step-pool sequences of similar size to our dune-induced hyporheic flow cell (i.e., ∼1 m), mixing zones may be thin in such systems as well. The size of mixing zones in larger hyporheic flow systems such as meander bends, pool-riffle sequences, and large bars [Storey et al., 2003; Boano et al., 2006; Burkholder et al., 2008] is less certain given the longer flow paths involved. Results of some field studies such as [Triska et al., 1989] show substantial ranges in concentration of channel-injected tracers in riparian wells which seem to indicate significant mixing of surface and groundwaters, but may instead be due to well screens intercepting multiple flow paths of differing concentrations. This effect would be heightened by widespread and increasingly recognized preferential flow paths [Fox et al., 2011; Gormally et al., 2011; Menichino et al., 2013].
 It is important to distinguish true mixing, where solutes from different source waters are present simultaneously in an area of overlap [Kitanidis, 1994] from macrodispersion where solutes from one source water interfinger among areas of the other source water due to flow path and sediment texture heterogeneity. True mixing occurs as a result of small-scale concentration gradients, and fingering can increase true mixing indirectly by increasing the length or surface area of porous media with high concentration gradients, but fingering itself is not mixing. Instead, fingering increases the overall size of a solute plume (“spreading”), and is often represented in coarsely resolved numerical models via large macrodispersion coefficients that scale with the size of the model domain [Gelhar et al., 1992; Xu and Eckstein, 1995]. In contrast, true mixing occurs only along the margins of such fingers, as described by a much smaller local dispersion coefficient, as used in our modeling. Mixing-dependent reactions in turn are predicted by this much smaller true mixing along the fingers [Cirpka et al., 1999]. For this reason, hyporheic zones defined by hydrochemical mixing simulated in numerical models that use macrodispersion coefficients such as those in Woessner  or Lautz and Siegel  which are 10x what we used, quantify spreading rather than the extent of true mixing. We conclude that true mixing does occur in the hyporheic zone, but it is comparatively minimal for the scenarios we modeled. Mixing-based definitions of the hyporheic zone should therefore be more clearly articulated to indicate that there is colocation of different source waters within the subsurface, but not necessarily much true mixing.
 Our modeling and this discussion consider only steady state hydraulic conditions and relatively fine sediments where Darcy flow dominates (i.e., finest sediment fraction is sandy with K ∼10−3 m/s or finer). The steady state conditions are reasonable for many hyporheic settings, particularly at base flow. Nevertheless, significant stage fluctuations do occur both during storms and cyclically due to hydropower operations, snowmelt, and evapotranspiration. Under such conditions, hyporheic flow paths are expected to shift over time [Wondzell and Swanson, 1996; Peterson and Connelly, 2001; Loheide and Lundquist, 2009], and mixing may be greater. Darcy flow is also a reasonable assumption for many hyporheic settings. Nevertheless, sediments that are coarse enough or shallow enough for turbulent flux to occur [Nagaoka and Ohgaki, 1990] would also need to account for turbulent diffusion.
4.2. Implications of Mixing Zone Size for Reactions of Upwelling Groundwater Pollutants
 A clear implication of the thinness of hyporheic mixing zones (Figure 10) and the associated relatively small mixing flux is the expectation of restricted opportunity for mixing-dependent reactions [Cirpka et al., 1999; Werth et al., 2006]. Ultimately, reactions of many upwelling groundwater contaminants in the hyporheic zone will be controlled by mixing (total mixing flux) and amount of time spent (residence time) within the hyporheic mixing zone. This is in contrast to reactions of many surface water pollutants in the hyporheic zone, which will be controlled by the flow rate of surface water through the hyporheic zone (i.e., percent of river discharge that cycles through hyporheic zone), as well as its residence time. While interaction of upwelling contaminants with hyporheic zones has been studied in the field [Kennedy et al., 2009; Krause et al., 2009] we are not aware of previous field measurements that have rigorously identified the thin mixing zones described here. Yet several field studies have shown heightened reaction rates in hyporheic zones relative to upgradient aquifer plumes, suggesting that mixing-dependent reactions may be significant under certain circumstances. For example, Conant et al.  showed significant reductive dehalogenation of perchloroethene (PCE) occurring in shallow riverbed sediments unlike in the upgradient plume. Because such dehalogenation is an anaerobic process, the greater rates occurring in the hyporheic zone could be due to advection of dissolved organic carbon (DOC) from surface water, subsequent mixing of that DOC with upwelling groundwater flow paths, and ultimately microbially mediated DO consumption. Whether the DOC from surface water has a significant effect on PCE degradation would depend on whether there is sufficient DOC and reaction time to consume the DO from surface water to maintain the required anaerobic conditions.
 In another example, a plume of petroleum hydrocarbons and fuel oxygenates were characterized as they upwelled to a river (Carmans River example in Landmeyer et al. ). Similar to Conant et al. , degradation was minimal in the upgradient aquifer and yet concentrations decreased substantially upon reaching the riverbed. Dilution and mixing with surface water was reported as the dominant contributor to reduced concentrations in the shallowest sediments, although mixing processes themselves were not rigorously characterized. Degradation was also confirmed to occur, and because these contaminants degrade primarily aerobically, it seems highly likely that such degradation represents a mixing-dependent reaction that requires oxygen supplied from surface water. Nevertheless, degradation was reported as only a few percent of the upwelling contaminant mass flux. This high degree of reported mixing in combination with a low amount of degradation could be explained simply by insufficient residence times to accomplish substantial degradation. However, another explanation could be that the low observed concentrations in the shallowest sediments, which was assumed to be mixing, was instead movement of surface water through sediment (i.e., hyporheic flow cells). This would imply that mixing was actually quite small, similar to our study, and such small amounts of mixing would also explain the minimal degradation.
 Like all reactions, mixing-dependent reactions also require appropriate residence times. We can evaluate potential for reactions by comparing mixing zone residence times to reaction times. For example, residence times we observed within the mixing zone for the base case (∼1 h) may be insufficient for some reactions such as denitrification (e.g., >10 h in Zarnetske et al. ). Because the focus of our study was not residence times, we did not estimate mixing zone residence times for other model scenarios, but they clearly would be higher for lower K and possibly in the range of denitrification. Longer hyporheic flow paths due to larger geomorphic forms such as pool-riffle sequence or larger dunes may also have longer residence times. Finally, residence times might also increase with time-varying hydraulic boundary conditions such as due to hydropower operations which induce oscillations in flow path trajectories [Peterson and Connelly, 2001]. Future modeling studies that include reactions of important contaminants, as well as laboratory and field studies of hyporheic zone mixing and reactions, should clarify which reactions are mixing dependent, and how important they are for hyporheic natural attenuation.
4.3. Mixing in Homogeneous Case: Controls and Effects
 We found that increases in K and decreases in lower boundary groundwater flux rate both increased total mixing flux by similar amounts over the ranges of K and groundwater flux that we examined (Figures 5a and 5b). These effects on mixing must act either through changes to the dispersion rate along the interface between advected surface water and upwelling groundwater as quantified by the transverse dispersion coefficient DT (L2/T), or through the amount of time over which the dispersion process is able to occur t (T). For the case of varying K, it is intuitive that increases in K lead to increases in pore water velocity v (L/T) which lead to increases in DT
which could explain increases in total mixing flux. Dm is the molecular diffusion coefficient (L2/T), αT is transverse dispersivity (L), and n is effective porosity (L3/L3). Dm is multiplied by n to empirically account for reductions in diffusion that occur due to tortuosity of flow paths around sediment grains [Grathwohl, 1998]. However, this explanation does not work for the case of varying lower boundary flow rate, where increases in lower boundary flow rate lead to increases in v along the interface, yet lead to decreases in total mixing flux. So effects of these parameters on t must also be important.
 In order to understand the effects of K and lower boundary groundwater flux rate on total mixing flux, we examine their effects on mixing zone size (both length and thickness of the mixing zone). Mixing zone size exhibits trends with K and lower boundary groundwater flux rate that are very similar to trends of total mixing flux (Figures 11a and 11b), and must therefore be capturing the net effect of both D and t. In other words, mixing flux is linearly correlated with mixing zone size (Figure 11c). This makes sense because t is a function not only of v, but also the length of the interface, which is part of mixing size. Because mixing zone thickness does not vary substantially with either K or lower boundary groundwater flux rate (Figures 7a and 7b), we propose that mixing zone length is the primary aspect of mixing zone size that affects total mixing flux. The length of the mixing zone is in turn determined by the depth of the hyporheic flow cell (mixing depth), which is in turn determined by K and lower boundary groundwater flux rate that control the relative balance of flows coming from surface water and groundwater. This purely hydraulic impact on mixing is consistent with a variety of prior modeling studies, including those showing an increase in depth of hyporheic flow paths (equivalent to mixing depth in our study) with increasing K when the lower boundary condition is a specified flow rather than a specified head [Cardenas and Wilson, 2007b] as well as those showing a decrease in depth of hyporheic flow paths with increasing lower boundary groundwater flux rate [Cardenas and Wilson, 2006, 2007c]. By contrast, modeling studies that vary K in the context of a constant head groundwater boundary condition show little effect of K on mixing depth [e.g., Boano et al., 2008; Hester and Doyle, 2008]. As such, the strong effect of K on mixing we observed is largely due to our choice of lower boundary condition, and may be less important in field conditions relative to variations in lower boundary groundwater flux rate.
 To understand why mixing zone thickness varies little with K or lower boundary groundwater flux rate, it helps to examine in more detail the case of variations in K. This lack of effect may initially seem nonintuitive because increases in K lead to increases in v and therefore to increases in D. And increases in D clearly lead to increases in dispersion rates and therefore increases in mixing zone thickness, all else equal (Figures 7d and 5d). Here we use simple hydraulic and transport equations to explain this phenomenon. We start by estimating the mixing zone thickness δ (L) as
where t is time (T) [Freeze and Cherry, 1979; Werth et al., 2006]. We then note that K enters into equation (4) only through the linear pore water velocity v (L/T) from Darcy's Law,
where i the hydraulic gradient (L/L). In particular, D in equation (4) is the product of αT and v per equation (3), assuming Dm is small compared to αTv. The latter is true for our modeling with more than an order-of-magnitude difference even at the lowest velocities observed in our lowest K scenario, and more than 2 orders of magnitude when factoring in n in equation (3). Time (t) in equation (4) is also a function of velocity, being equal to total length of mixing front Δs (L) divided by v. When αTv is multiplied by Δs/v, the velocities cancel, leaving
 Equation (6) shows that mixing zone thickness is independent of K and v and dependent only on the dispersivity α and the length of the mixing zone (Δs). The implication is that if increases in K do not increase the length of the mixing zone, there is no effect on mixing zone thickness and therefore on total mixing flux.
 Within the range of each parameter varied in our modeling study, varying dispersivity had among the largest effects on total mixing flux of all parameters varied (Figures 5 and 6). Because our range of input parameter values was representative of the full range of field conditions for many of these parameters, this means that variation in model dispersivity can potentially overwhelm control of mixing by other parameters that represent field variation. We therefore emphasize the importance of accurately specifying dispersivity in numerical models of hyporheic zone transport, particularly where mixing-dependent reactions are of interest. Our modeling relied on dispersivity data that are attributable to aquifer settings much deeper than the hyporheic zone. Determining dispersivity in hyporheic zones is therefore critical to further progress in understanding solute migration and natural attenuation within the hyporheic zone.
4.4. Mixing in Heterogeneous Case: Controls and Effects
 The effect of heterogeneity of K was to increase total mixing flux relative to the homogeneous case (Figure 6a). Horizontal correlation length and variance in the heterogeneous case (Figures 6a and 6b) had effects that were similar in magnitude to those of K and lower boundary groundwater flux rate in the homogeneous case (Figures 5a and 5b). However, because the effect of K on mixing that we observed was in large part an effect of our using a constant flow lower boundary condition, it appears that overall, heterogeneity and degree of upwelling controlled mixing of surface water and upwelling groundwater in hyporheic sediments. Nevertheless, we only evaluated heterogeneity of the type that can be described by the geostatistical model used by Tompson et al. . The effects of other types of heterogeneity, such as cross bedding, layering, or preferential flow paths [Sawyer and Cardenas, 2009; Song et al., 2010; Menichino et al., 2013], are also observed in near-stream environments, and are worth evaluating. Furthermore, we only evaluated the effects of small scale heterogeneity (horizontal correlation lengths <1m), so the total effect of heterogeneity, including heterogeneity at larger scales [Cardenas and Zlotnik, 2003; Cardenas et al., 2004], would likely be greater.
 The effect of heterogeneity of K on mixing in saturated porous media has not received much attention in the hyporheic literature, yet has been the topic of a growing number of studies in the general hydrogeology literature. The increase in total mixing flux that we observed with degree of K heterogeneity (i.e., variance of K, Figure 6a) is consistent with effects of flow focusing observed in such general groundwater studies, which has been shown to increase transverse mixing [Werth et al., 2006; Rolle et al., 2009; Chiogna et al., 2011; Van Breukelen and Rolle, 2012]. This phenomenon has the effect of increasing the thickness of the mixing zone (Figure 8a), similar to changing the dispersivity in the model (Figure 7d). This leads to greater rates of mixing-dependent reactions and reduction of steady state plume lengths in aquifers [Cirpka et al., 1999; Chiogna et al., 2011], and may have similar effects in hyporheic zone.
 We conducted numerical modeling of conservative tracer transport within shallow riverbed sediments. In particular, we simulated mixing of tracer upwelling from deeper groundwater with surface water downwelling through hyporheic flow cells induced by riverbed dunes. This represents upwelling of contaminants or other solutes toward rivers under hydrologically gaining conditions. We defined mixing as (1) the total mixing flux across the interface between hyporheic flow cells and upwelling groundwater and (2) the size of the mixing zone between these two flow regions. We found that mixing flux is a small proportion of overall upwelling groundwater flux, ranging up to at most 12.7% of the incoming tracer. Similarly, the mixing zone is thin, ranging up to at most 8.2% of our model domain. This has implications for conceptions of the hyporheic zone, which is often defined as the area of mixing of surface water and groundwater. Among our results, the spatial extent of a mixing-defined hyporheic zone was generally much smaller than the extent of a hyporheic zone defined as where hydrologic flow paths leave and return to the water column within relatively short distances. We therefore expect restricted opportunity for mixing-dependent hyporheic reactions. Nevertheless, several prior studies show heightened reaction rates in hyporheic zones relative to in upgradient aquifers, even in cases where reactions are likely mixing dependent, suggesting that mixing-dependent reactions may be significant under certain circumstances. This may be explained by the greater residence times of upwelling groundwater in shallow sediments than for surface water moving through hyporheic flow cells. Future modeling studies that include reactions of important contaminants, as well as laboratory and field studies of hyporheic zone mixing and reactions, will clarify which reactions are mixing dependent, and how important they are for hyporheic reactions.
 We found that variations in both hydraulic conductivity (K) and lower boundary groundwater flux rate had substantial effects on mixing in the homogeneous case, but that anisotropy of K had little effect. The effects of K and lower boundary groundwater flux rate were mediated by their effects on the length of the mixing zone (i.e., the depth of the hyporheic flow cell), rather than its thickness. The effect of introducing heterogeneity was to increase mixing relative to the homogeneous case. This has not been reported previously for the hyporheic zone but is consistent with effects of flow focusing, which has been shown to increase transverse mixing in general groundwater studies. Variance of K and horizontal correlation length had greater effects on mixing than varying correlation scale ratio. Overall, degree of heterogeneity as well as parameters that control the length of mixing zones within sediments (K and lower boundary groundwater flux rate) had the greatest effect on mixing. We note that varying dispersivity had among the largest effects on model-simulated mixing of any parameter varied, yet variations in dispersivity are artificial modeling constructs rather than representations of variations in real field settings. As a result, accurately specifying dispersivity in numerical models of hyporheic zone transport is critical if mixing is to be simulated, particularly where mixing-dependent reactions are of interest.
 We thank Andy Tompson for providing the turning bands code as well as assistance with its use. We thank the Charles E. Via, Jr. Endowment at Virginia Tech for providing support to Katie Young. We thank Fulvio Boano, Adam Wlostowski, and two anonymous reviewers for helpful comments on this manuscript.