Coupled 3-D stream flow and hyporheic flow model under varying stream and ambient groundwater flow conditions in a pool-riffle system

Authors

  • Nico Trauth,

    Corresponding author
    1. Department of Hydrogeology, Helmholtz Center for Environmental Research—UFZ, Leipzig, Germany
    2. Water and Earth System Science (WESS) Competence Cluster, Tübingen, Germany
    • Corresponding author: N. Trauth, Department of Hydrogeology, Helmholtz Centre for Environmental Research—UFZ, Permoserstrasse 15, Leipzig DE-04318, Germany. (nico.trauth@ufz.de)

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  • Christian Schmidt,

    1. Department of Hydrogeology, Helmholtz Center for Environmental Research—UFZ, Leipzig, Germany
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  • Uli Maier,

    1. Department of Hydrogeology, Helmholtz Center for Environmental Research—UFZ, Leipzig, Germany
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  • Michael Vieweg,

    1. Department of Hydrogeology, Helmholtz Center for Environmental Research—UFZ, Leipzig, Germany
    2. Water and Earth System Science (WESS) Competence Cluster, Tübingen, Germany
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  • Jan H. Fleckenstein

    1. Department of Hydrogeology, Helmholtz Center for Environmental Research—UFZ, Leipzig, Germany
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Abstract

[1] Exchange of water and solutes across the stream-sediment interface is an important control for biogeochemical transformations in the hyporheic zone (HZ). In this paper, we investigate the interplay between turbulent stream flow and HZ flow in pool-riffle streams under various ambient groundwater flow conditions. Streambed pressures, derived from a computational fluid dynamics (CFD) model, are assigned at the top of the groundwater model, and fluxes at the bottom of the groundwater model domain represent losing and gaining conditions. Simulations for different Reynolds numbers (Re) and pool-riffle morphologies are performed. Results show increasing hyporheic exchange flows (m3/d) for larger Re and a concurrent decrease in residence time (RT). Losing and gaining conditions were found to significantly affect the hyporheic flow field and diminish its spatial extent as well as rates of hyporheic exchange flow. The fraction of stream water circulating through the hyporheic zone is in the range of 1 × 10−5 to 1 × 10−6 per meter stream length, decreasing with increasing discharge. Complex distributions of pressure across the streambed cause significant lateral hyporheic flow components with a mean lateral travel distance of 20% of the longitudinal flow paths length. We found that the relationship between pool-riffle height and hyporheic exchange flow is characterized by a threshold in pool-riffle amplitude, beyond which hyporheic exchange flow becomes independent of riffle height. Hyporheic residence time distributions (RTD) are log-normally distributed with medians ranging between 0.7 and 19 h.

1. Introduction

[2] Interactions between stream water and groundwater are important for the management of water quantity and quality as well as for the functioning of aquatic ecosystems [Stanford and Ward, 1988; Edwards, 1998; Winter et. al., 1999; Fleckenstein et al., 2010]. The main driver for hyporheic exchange is the variation of pressure along the streambed [Thibodeaux and Boyle, 1987; Buffington and Tonina, 2009; Tonina and Buffington, 2009]. These pressure variations are produced by stream flow over dunes, ripples, pool-riffles, or in-stream obstacles. Obstructions can induce variations of the water level causing high pressures at the upstream side and low pressures at the downstream side of the structures. Driven by these pressure variations at the streambed surface, stream water infiltrates into the sediment at the upstream side of the obstruction and exfiltrates back into the stream at the downstream side [Thibodeaux and Boyle, 1987; Harvey and Bencala, 1993; Elliott and Brooks, 1997a, 1997b; Packman et al., 2000; Wörman et al., 2002].

[3] Hyporheic flow paths in natural streams can have large lateral and upstream components as a result of pressure gradients induced by complex three dimensional (3-D) streambed morphology [Tonina and Buffington, 2007, 2011; Angermann et al., 2012]. Large features such as point bars, in-stream gravel bars, and pool-riffle sequences typically show significant lateral variation in morphology and can cause these 3-D patterns of flow [Wörman et al., 2006; Tonina and Buffington, 2007, 2009; Stonedahl et al., 2010].

[4] Similar to the effects of streambed heterogeneity [e.g., Salehin et al., 2004; Sawyer and Cardenas, 2009], this three dimensionality increases the complexity of hyporheic flow paths resulting in distinct zones of up- and downwelling and a wide range of residence times (RT) [Stonedahl et al., 2010]. Patterns of up- and downwelling affect in-stream habitat and areas for fish spawning [Stuart, 1953, 1954; Malcolm et al., 2006; Greig et al., 2007], whereas longer RT foster the transformation of nutrients and other solutes in the streambed [Marzadri et al., 2011, 2012; Zarnetske et al., 2011; Bardini et al., 2012].

[5] Besides streambed morphology also the velocity of stream flow as well as the magnitude of ambient groundwater flow significantly affect the geometry of the hyporheic zone (HZ). Increasing groundwater discharge decreases the depth and volume of the HZ [Kasahara and Wondzell, 2003; Storey et al., 2003; Cardenas and Wilson, 2007b; Boano et al., 2008, 2009], whereas increasing stream discharge generally increases hyporheic exchange flow [Packman et al., 2004; Tonina and Buffington, 2007; Cardenas and Wilson, 2007c].

[6] Given the complex nature of turbulent flow over 3-D bed forms and the induced hyporheic flows, it is challenging to design appropriate flume experiments or field studies of the groundwater-surface water interface. However, numerical experiments have proven to be a viable alternative to improve our mechanistic understanding of flow dynamics and biogeochemistry in these complex environments [e.g., Cardenas and Wilson, 2007a-2007c; Jin et al., 2010; Bardini et al., 2012; Frei et al., 2012].

[7] Stream water flow over complex streambed morphologies and the resulting pressure at the streambed surface can most accurately be represented using computational fluid dynamics (CFD) codes that solve the full Navier-Stokes equations [Cardenas and Wilson, 2007c; Tonina and Buffington, 2009; Endreny et al., 2011; Janssen et al., 2012]. Flow simulations can cover a range of Froude numbers (Fr) and include hydraulic jumps and surface water wave phenomena typical of relatively shallow turbulent flow over larger streambed structures such as pool-riffle sequences. These simulations additionally require a two-phase model [Yue et al., 2005; Polatel, 2006], where the exact water level is derived from the interface of the air-water mixture.

[8] In this study, we use a 3-D, two-phase CFD code, which is sequentially coupled to a groundwater flow model, to systematically investigate the dynamics of hyporheic exchange in a pool-riffle stream. Pool-riffle topography is represented by a simplified approximation of a fully submerged pool-riffle sequence. Hyporheic flow scenarios for five different submerged pool-riffle sequences are systematically evaluated for a range of stream discharges. Additionally, ambient groundwater flow conditions (losing and gaining), which have been disregarded in previous modeling studies of pool-riffle systems [Tonina and Buffington, 2007, 2009, 2011; Marzadri et al., 2010], are considered at different flow rates. This is thought to be a more realistic conceptualization of pool-riffle streams, which are typically embedded in an alluvial aquifer. Hyporheic RT and the 3-D hyporheic flow field are analyzed by means of particle tracking. With this modeling concept, for the first time, we are able to systematically evaluate how variations in streambed morphology, stream discharge, and ambient groundwater flow affect 3-D hyporheic exchange and RT in pool-riffle streams.

2. Methodology

2.1. Model Geometry and Parameterization

[9] The streambed morphology represented in the numerical model is inspired by a field site with distinct pool-riffle sequences [Schmidt et al., 2012]. The observed wavelength to width ratio for a pool-riffle sequence at the field site is approximately 1, which is relatively low, compared to commonly reported ratios of approximately 3–7 in free formed pool-riffle reaches [e.g., Leopold and Wolman, 1957; Keller,1972] with a tendency to decrease with increasing slope [e.g., Grant et al., 1990]. However, if sediment transport is influenced by local flow obstructions such as bank protection measures, large wood debris, or boulders [Montgomery et al., 1995], significantly smaller ratios may be found. For example, Montgomery et al. [1995] reported ratios of wavelength to stream width of even less than one in the presence of large wood debris. At the field site, bank protection measures (rip-rap) as well as larger wood debris (from fallen riparian phreatophytes) are frequently found likely explaining the observed small pool-riffle spacing. The main features of the gradually varying pool-riffle morphology at the field site could be approximated empirically by the following equation:

display math(1)

where x and y represent the respective longitudinal and transverse planar coordinates in stream flow direction. A is the maximum amplitude of the pool-riffle sequence in the stream center, λ is the wavelength in stream flow direction, and w is the stream width. As can be seen from equation (1), the height of the morphology varies in y direction by twice the wavelength of the x direction. According to this configuration, the minima and maxima in the streambed elevation are located in the center of the stream. In the y direction from the center line, elevations gradually increase or decrease up to the vertical riverbank (gray surface in Figures 1a and 1b). Hence, the streambed morphology is symmetric along the stream center (y = 0 m).

Figure 1.

Wavy water surface of stream flow for (a) high and (b) low discharge scenarios. Maximum Froude numbers are Fr = 1.21 (Figure 1a) and Fr = 1.08 (Figure 1b) at the downstream side of the riffle. (c) Longitudinal cross section of the stream showing water levels at y = 0 m affected by waves induced by undular hydraulic jumps for a high (dark blue line) and a low stream flow (light blue line) scenario. The waves of the low discharge scenario are completely developed within the pool area, while the waves of the high discharge scenario extend into the next riffle structure where they start to interfere with the undular hydraulic jump caused by the next riffle.

[10] We used five different stream water model domains where the bottom geometry differs by the amplitude A of the pool-riffle sequence. The maximum amplitudes A in the stream center are A = [0.1, 0.2, 0.3, 0.4, 0.5] m. The wavelengths are constant with a value of λ = 10 m, as well as the stream width with w = 10 m (Table 1). A constant slope of 2% is assigned to the streambed morphology representing the upper limit of slopes typically occurring in pool-riffle systems [Buffington and Tonina, 2009]. Additionally, our slopes are in the range of values used by Harvey and Bencala [1993] (6.7%), Woessner [2000] (2%), and Marzadri et al. [2010] (0.53–3.3%). Meandering of the stream channel is not considered. When referring to the morphologies in the following sections, the abbreviations A01, A02, A03, A04, and A05 are used, according to the maximum amplitude A. In the CFD simulations, up to 12 different discharge scenarios for each streambed morphology were performed. The lower end of the considered range of discharges was determined by the minimum flow required to fully submerge the pool-riffle structures in the model. The largest discharge roughly represents bank full discharge and is close to the mean of the high flow events observed at the field site over the last 50 years (data obtained from the statistical tables of the agency operating the official gauging stations).

Table 1. Model Parameters Used for the CFD Simulations and the Groundwater Model MIN3P
Stream Configuration
Stream slope2%
Stream width w10 m
wavelength of pool-riffle sequence λ10 m
amplitudes of pool-riffle sequence A0.1, 0.2, 0.3, 0.4, 0.5 m
CFD Code OpenFOAM
Mesh cell size in x direction0.11 m
Mesh cell size in y direction0.11 m
Mesh cell size in z direction0.095 m and smaller
Groundwater Model MIN3P
Hydraulic conductivity K5.0×10−4
Porosity0.3
Depth of aquifer3 m ± amplitude A
Mesh cell size in x direction0.2 m
Mesh cell size in y direction0.2 m
Mesh cell size in z direction0.1 m

[11] The streambed sediment at the field site consists of coarse gravel with finer, mainly sandy sediment fillings in the pore spaces and is covered by an armor layer of cobblestones, which immobilizes the bed for the range of flows considered in our scenarios.

[12] Slug tests were performed to obtain hydraulic conductivity (K) values of the streambed sediments at the field site [Schmidt et al., 2012]. A constant value of K = 5 × 10−4 m/s was set in the groundwater model. The porosity n for the calculation of the pore water velocity was set to 0.3, which is a reasonable value for heterogeneous streambed sediments. The bottom of the groundwater model domain is defined as a flat plane at a depth of 3 m below the baseline of the sinusoidal streambed elevation (Table 1).

[13] The surface water model domain comprises a sequence of five pool-riffle sections to avoid boundary effects. Out of these five pool-riffle sequences only the inner three, where effects on the flow from the inlet or outlet conditions of the CFD simulations are assumed to be negligible, are used in the groundwater model. Finally to avoid boundary effects in the groundwater model, hyporheic flow, extent, and RT are only evaluated in the mid pool-riffle sequence of the porous domain.

2.2. Numerical Modeling and Coupling

2.2.1. Surface Water Model

[14] The OpenSource CFD Toolbox OpenFOAM version 1.7.1 is used to simulate turbulent stream flow over pool-riffle streambeds (http://www.openfoam.org/) in order to derive the hydraulic head distribution at the streambed (H). The software solves the three-dimensional Navier-Stokes equations using the finite-volume approach (FVM) and the PISO algorithm for the pressure-velocity coupling. A two-phase model approach, the Volume of Fluid Method (VoF) [Hirt and Nichols, 1981], is used to simulate both the water and the air fraction of the surface water model domain.

[15] For the turbulence closure model, large eddy simulation (LES) is used, because of its high accuracy when working on flow structures with pronounced eddies [Grigoriadis et al., 2009; Janssen et al., 2012]. For simulating the subgrid scales, the original Smagorinsky subgrid scale model [Smagorinsky, 1963] is used with a filter width, determined by the cubic root from the mesh cell volume.

[16] The mesh consists of hexahedral cells with a width of 0.11 m in x and y directions and less than 0.095 m in z direction. The number of computational cells is about 520 000, varying slightly for the different morphologies. Refinement of mesh cells at the bottom boundary (interface to the groundwater model) revealed no significant changes of the streambed pressure, but vastly increased the calculation time.

[17] By specifying adequate initial conditions, the quasi-steady state can be reached faster. Preliminary simulations with a coarse mesh were used to roughly estimate water level and a reasonable velocity field so that the model converges to a quasi-steady state quickly. Subsequently, the mesh is rotated corresponding to the considered slope of 2% of the stream channel.

[18] The two-phase model requires an inlet boundary condition that defines the fraction of water and air entering the domain. A user-defined application called “GroovyBC” was used to define the inlet condition [Gschaider, 2009, documented at http://openfoamwiki.net/index.php/Contrib_groovyBC]. This boundary condition provides the definition of the entering water and air fraction as well as the magnitude and shape of a velocity profile at the inlet, resulting in a defined stream discharge. The inlet condition is constant over time for all model runs, representing steady state stream discharge. A power law function is assigned to the vertical velocity profile at the inlet. Turbulent flow develops shortly after water and air enter the modeling domain.

[19] The outlet condition is represented by a boundary that allows both water and air to flow out of the domain unhampered, denoted as totalPressure boundary in the OpenFOAM code. The fraction of water and air at the outlet boundary is predicted numerically. The same boundary condition is used at the top of the domain, where air can enter and leave the domain, simulating a natural atmospheric condition. For detailed information on the boundary conditions, the reader is referred to the OpenFOAM documentation (http://www.openfoam.org/).

[20] The bottom of the domain is bounded by the fixed streambed, which represents the interface between the surface water and the hyporheic zone. The streambed is treated as an impervious no-slip condition in the simulation, which is commonly used in such simulations [Cardenas and Wilson, 2007a; Tonina and Buffington, 2009; Janssen et al., 2012]. Grain roughness of the streambed is not considered in our model similar to the model approaches of Cardenas and Wilson [2007a-2007c] and Tonina and Buffington [2009]. This assumption is supported by Janssen et al. [2012], who demonstrated that the effects of wall roughness on the near-bed pressure field are negligible. Furthermore, Lane et al. [2004] compared flow over rough gravel beds versus smooth beds and concluded that sensitivity to rough versus smooth conditions was negligible.

[21] The bank (y = 5 m) of the stream channel is treated in a similar manner by a no-slip condition. Due to a symmetric stream channel along the stream center (y = 0 m), only one half of the channel is simulated, which reduces calculation times significantly. Hence, at the stream center a shear-free symmetric boundary condition is used.

[22] OpenFOAM calculates an instantaneous flow field for every time step. For the subsequent statistical analysis of the results, the time-averaged flow field is determined by calculating the averages of the variables over the period of time, after the flow has completely developed and is no longer affected by the initial conditions [Polatel, 2006].

[23] The surface water model is coupled to the subsurface model by assigning H at the streambed surface calculated by the CFD code to the top of a groundwater model as a Dirichlet boundary condition. The mesh cells of the CFD model are approximately half the size of the mesh of the groundwater model, and therefore, an interpolation over data gaps is not required. This one-way sequential coupling approach captures only flow from the surface water domain into the porous domain and does not account for feedbacks from subsurface flow into the surface water domain. However, hyporheic water that enters the stream channel is only a small volume fraction of the total stream discharge and hence has negligible impacts on hydrodynamic flow in the channel [Prinos, 1995; Cheng and Chiew, 1996].

2.2.2. Test of the CFD Model Against Fehlman's Flume Data

[24] To ensure reliable physical behavior of our surface water model, the flume experiments of Fehlman [1985] for flow over triangular bed forms were simulated with the CFD code and the results compared to the experimental data. The bed form geometry of Fehlman's flume, including the flume width, was implemented in an OpenFOAM mesh with a maximum cell width of 0.03 m in all three directions. We adjusted our model parameters to three different runs of Fehlman's experiments, specified by discharge and flume slope (run 4, 8, and 12).

[25] The simulated discharges and water levels above the crest agree well with the experimental data (Figure 2a). Simulated pressure variations across the bed form normalized by the pressure at the crest also compare well to the experimental data (Figure 2b) except at the crest. This is due to the fact that Fehlman did not measure the pressure at the crest and used interpolated values from two adjacent pressure taps located upstream and downstream from the crest. Therefore, the simulated pressure at the crest is significantly lower than the interpolated value from Fehlman, which had already been observed by Cardenas and Wilson [2007c]. However, overall, the simulated bottom pressure agrees well with the measured pressure values (Figure 2b and Table 2). Our test simulations show that the 3-D surface water model can predict flow over 2-D bed forms with reasonable accuracy. However, this corroboration may not fully extend to our 3-D streambed simulations because of the lack of 3-D pressure head measurements as a validation data set.

Figure 2.

Comparison of the simulated data with the measured data of Fehlman's [1985] flume experiment for the runs 4, 8, and 12.

Table 2. Bias and RMSE of the Measured Pressure Data of Fehlman [1985] and the CFD Simulations
 Run 4Run 8Run 12
Bias3.3 × 10−4 m2.0 × 10−3 m2.5 × 10−4 m
RMSE9.8 × 10−4 m2.9 × 10−3 m1.3 × 10−3 m
Table 3. Stream Flow Characteristics of the CFD Scenarios
Morphology-runRe (-)Fr mean (-)Fr max (-)H (m)U (m/s)Q (m3/s)
A 05-14.63E+050.441.250.620.935.38
A 05-27.10E+050.481.080.701.157.40
A 05-31.06E+060.571.110.811.4811.01
A 05-41.31E+060.601.220.881.6712.40
A 05-51.46E+060.631.190.911.7614.78
A 05-61.72E+060.671.210.961.9517.47
A 05-72.16E+060.751.271.052.2722.18
A 05-82.50E+060.771.291.112.4325.54
A 05-92.88E+060.791.281.212.5929.57
A 05-103.27E+060.831.301.292.7834.52
A 03-17.63E+050.651.210.571.467.57
A 03-21.04E+060.741.340.621.7610.33
A 03-31.40E+060.891.470.682.2113.76
A 03-41.51E+060.871.400.722.2214.95
A 03-51.71E+061.021.540.702.5817.22
A 03-61.81E+061.031.570.732.6518.60
A 03-71.99E+061.011.520.772.7019.98
A 03-82.22E+061.011.550.822.8022.05
A 03-92.51E+061.031.550.882.9524.80
A 03-102.85E+061.031.520.933.0927.56
A 03-113.22E+061.121.620.943.3931.80
A 03-123.60E+061.161.650.973.6135.34
A 01-15.40E+050.681.120.421.354.11
A 01-29.01E+050.911.300.471.947.46
A 01-31.41E+061.051.390.572.4711.95
A 01-41.65E+061.151.450.602.7814.93
A 01-52.09E+061.161.430.693.0219.51
A 01-62.47E+061.181.460.773.2424.46
A 01-72.90E+061.191.460.853.4228.53
A 01-83.28E+061.211.440.923.6032.61
A 01-93.59E+061.201.430.973.7035.06

2.2.3. Groundwater Model

[26] The numerical flow and reactive transport model MIN3P was used to simulate steady state water flow in the HZ. MIN3P is a finite volume flow and transport code that solves Richard's equation for variably saturated flow [Mayer et al., 2002].

[27] In a fully saturated domain, as in our case, Richard's equation reduces to the governing groundwater flow equation. The governing equations for incompressible, steady state groundwater flow are:

display math(2)
display math(3)

where q is the Darcy flux, K is the hydraulic conductivity, and H is the gradient of the hydraulic head.

[28] The top of the groundwater model is defined as a Dirichlet boundary with hydraulic head values from the CFD model. At the bottom of the groundwater model domain, at a depth of 3 m below the baseline of the sinusoidal streambed elevation, a Neumann boundary is assigned to control either in- or outflow of water by a Darcy flux denoted as qbot. Gradients, calculated from qbot/K, are within the range of values measured in natural streambeds [e.g., Schmidt et al., 2006; Kennedy et al., 2009; Engelhardt et al., 2011].

[29] For simplicity, we assume that qbot is a uniform flux across the bottom boundary, because of the inherent uncertainty of specifying a realistic spatially varying flux. However, this simplification is supported by results from field studies, which have demonstrated that the variability of vertical flux across the streambed is often small [Shanafield et al., 2010]. Furthermore, gaining and losing exchange fluxes between streams and the underlying aquifer have been shown to be dominated by vertical fluxes across the streambed [Engelhardt et al., 2011], justifying our assumption that lateral fluxes into the stream channel are negligible.

[30] Boundary conditions at the up- and downstream sides as well as at the lateral sides of the model domain are defined by no-flow boundaries.

[31] The mesh of the groundwater model consists of hexahedral cells with maximum vertical extent of less than 0.1 m and 0.2 m in the lateral directions. The elevation of the upper boundary of the groundwater domain follows the streambed morphology.

[32] For each groundwater model scenario, the flux-weighted hyporheic residence time distributions (RTD) and the spatial extent of the HZ were evaluated using forward particle tracking. The proprietary software TECPLOT 360, Version 2011(TecPlot Inc.,) was used to calculate particle tracks based on a second-order Runge-Kutta integration of the steady state pore water velocity fields. The integration stops when the particles leave the model domain (TECPLOT User's Manual, 2011). We released massless particles from each streambed boundary node, in total 2,250 (one per node). In the subsequent statistical analysis, only the particles that first enter and then again exit the streambed domain via the top boundary are considered. This defines the hyporheic flow paths and excludes flow paths that describe the flow of water upwelling from the bottom boundary and exiting at the top boundary as well as of water flowing from the top to the bottom boundary.

3. Results

3.1. Surface Water Flow

3.1.1. Stream Discharge, Water Level, and Undular Hydraulic Jumps

[33] In contrast to many natural channels that typically show nonlinear relationships between stage (hsurf) and discharge (Qsurf), the rating curve is approximately linear in our model (Figure 3a). However, when Qsurf is approximately 14 m3/s, there is a change in the slope of the rating curve, indicating a change in flow conditions at this point. When Qsurf ≤ 14 m3/s, the water level rises faster with increasing discharge. When Qsurf ≥ 14 m3/s, the water level rises more slowly. This change in the slope of the rating curve coincides with a change of the shape of undular hydraulic jumps that can develop for Fr slightly higher than 1 [Chanson, 2009]. Undular hydraulic jumps are characterized by stationary surface water waves, occurring downstream of the initial water level rise of the hydraulic jump where the flow passes the riffle and enters the pool. The wavelengths and amplitudes of the waves depend on the Fr before the hydraulic jump and the ratio of the average critical depth of the flow over the channel width. Wave amplitudes are damped in the direction of stream flow until they disappear [Chanson and Montes, 1995]. In our simulations, all stream flow scenarios show a wavy water surface with pronounced 3-D structures (Figure 1). Qsurf is positively correlated with the wavelength and the amplitude and negatively correlated with the number of completely developed waves. For a Qsurf < 14 m3/s, the waves of the undular hydraulic jumps in our simulations are restricted to the pool area because the amplitude of the wave is damped to zero before the next downstream riffle starts (light blue line in Figure 1c for Qsurf = 7.4 m3/s). The flow at the upstream side of the next downstream riffle is not affected by the standing wave of the undular hydraulic jump. This behavior changes at Qsurf > 14 m3/s, at which point the wavelength and the amplitude are larger and extend downstream into the zone influenced by the subsequent downstream riffle (dark blue line in Figure 1c). The interference of the waves of the undular hydraulic jump with the flow over the subsequent riffle causes a slower rise in the water level relative to the increasing discharge.

Figure 3.

(a) Stream discharge (Qsurf) as a function of water level (hsurf) and (b) as a function of Reynolds number (Re) for three different streambed morphologies.

[34] Mean Re calculated over all the water columns in the model domain range from 0.5 × 106 to 3.5 × 106. Re numbers increase linearly with Qsurf with a constant slope of approximately 1 × 10−5 for all three morphologies (Figure 3b). The flow characteristics of the stream scenarios are listed in Table 3.

3.1.2. Eddy Geometry and Hydraulic Head Distribution at the Streambed

[35] In the 3-D simulations, two symmetric eddies develop within the pool area, separated by the stream center line (y = 0), as expected from the symmetric streambed configurations (see section 2.2.1). Due to this symmetry, in the following we refer to the eddies and head distributions in one half of the streambed only.

[36] Stream flow over A03 and A05 produces significant eddies, fully developed in three dimensions, whereas the eddies that develop for flow over A01 are very small. The eddies are located in the deepest pool area close to the stream centerline, while close to the stream bank amplitudes in streambed elevation are too small for eddy formation. The pool area covered by eddies is approximately 65% of the stream width for flows over A05 and 50% of the stream width for flows over A03.

[37] Different discharge scenarios (different Re) also influence eddy geometry. Eddies in the low discharge model runs are relatively flat and cover the whole pool area in flow direction (Figure 1a). Under high discharge conditions, the eddies show more circular shapes and are only located deep in the pool, accompanied with lower detachment and reattachment points (Figure 1b). A change between these two general eddy geometries occurs at Re of about 1.4 × 106, where also the slope of the rating curve increases and where the waves of the undular hydraulic jump can affect the flow over the next riffle.

[38] Figure 4 depicts hydraulic head distributions at the streambed for the different scenarios. Generally, maximum heads (Hmax) are located at the upstream side of the riffle and correspond to the reattachment points of the eddy. From the maximum, the head decreases in the direction of stream flow until a minimum (Hmin) is reached at the lee side of the riffle, where the next eddy detaches. The H differences are significantly higher in A05 and A03 than in A01 and increase with Re. This is most notable at the location of the Hmax and Hmin where a shift in their x direction coordinate will depend on changing discharge and morphology: The higher the discharge, the longer the distance between the locations of Hmax and Hmin (e.g., compare Figures 4a and 4c). This is accompanied by smaller eddies in the pool and a lower z elevation of the H maxima. Similarly, the distance between Hmax and Hmin increases with lower pool-riffle amplitude. The same effect is also visible at each individual riffle as a result of the decline in amplitude toward the river bank (y = 5 m). H also varies along the lateral direction of the streambed. The distance between Hmax and Hmin along the x direction increases toward the stream bank (Figure 4). Interestingly, the absolute Hmax and Hmin are not located at the center of the stream (y = 0 m), but rather some distance toward the stream bank. This distance strongly depends on the discharge. For high discharges, H maxima are located closer to the stream center and vice versa (Figure 4). The location of the absolute Hmax at the streambed corresponds to the maximum longitudinal extent of the eddies, which is not located at the stream center, but rather a certain distance toward the stream bank.

Figure 4.

Hydraulic head (H) distribution at the streambed for high, moderate, and low Re scenarios for three pool-riffle amplitudes (columns). Flow is from left to right.

[39] Our results also demonstrate the effects of the undular hydraulic jumps that develop downstream of the riffles on the H distribution at the streambed. The standing waves have distinct wavelengths and amplitudes that increase with discharge. When a wave trough (respectively low hydrostatic pressure) coincides with the upstream side of a riffle, where H is commonly high, the resulting H will be reduced causing local anomalies (clearly visible at x = 19 in Figure 4e). Due to the three-dimensional nature of the waves (see Figure 1), interferences can occur at particular locations at the streambed and do not necessarily extend over the entire channel width.

3.2. Hyporheic Exchange

[40] We define the hyporheic exchange flow (QHZ) as the water volume per time (in m3/d) that enters and subsequently exits the porous domain at the top of the HZ. Infiltrating stream water that exits the domain via the bottom boundary and groundwater that enters the domain via the bottom boundary and exfiltrates into the stream is not included in QHZ. As described in the methods section, QHZ is evaluated for a single pool-riffle sequence (λ = 10 m). In the following sections, the effects of Re, qbot, and the different streambed morphologies on QHZ are described.

3.2.1. Effects of Reynolds Number

[41] The QHZ generally increases with Re for the morphologies A05 and A03 (Figures 5a and 5b), independent of the direction of qbot. This is caused by an overall increase of the hydraulic head gradient at the streambed with increasing Re, respective Qsurf. However, the relation is not monotonically increasing and a meaningful correlation is not derivable. For example, in simulation A03, the QHZ is remarkably reduced for Re of 1.5 × 106 to 2.0 × 106 and reduced even further for very high Re of 3.5 × 106. For A05, variations of QHZ in this Re range are visible, but not as significant as for A03. A possible explanation for this behavior is that the local reduction of H is caused by the undular hydraulic jumps that develop downstream of the riffles. As a result, the head gradient between the upstream and downstream side of the riffle is reduced causing a decrease in QHZ.

Figure 5.

Effect of Re and ambient groundwater flow (qbot) on (a–c) QHZ, (d–f) VHZ*, (g–i) ydist*, and (j–l) MRT. Negative qbot indicate losing conditions and positive qbot gaining conditions. The three columns refer to the streambed morphologies A05, A03, and A01.

3.2.2. Effects of Ambient Groundwater Flow

[42] Our simulations show strong effects of ambient groundwater flow, represented by the inflow and outflow via the bottom boundary of the groundwater model on hyporheic exchange. For the neutral case (qbot = 0), QHZ and the extent of the hyporheic flow cell are at a maximum. An increase in the magnitude of qbot results in a decrease of QHZ for both the losing and the gaining case. Even low magnitudes of qbot have a significant impact on QHZ. Exchange flows for both gaining and losing conditions are of the same order of magnitude and change similarly with the magnitude of qbot for a given morphology (Figures 5a and 5c). However, under losing conditions, QHZ is consistently slightly lower than under gaining conditions.

[43] Figure 6a shows the relationship between dimensionless qbot* calculated as qbot/K (equals the vertical gradient at the lower boundary condition) and the ratio of QHZ/Qsurf for high and low Re, which can be well approximated by an exponential function. Based on these equations (coefficients for all scenarios are provided in Table 4), the ratio of QHZ/Qsurf can be estimated for a given qbot* and Re. Furthermore, a critical qbot* can be derived, defined as the qbot* value for which all hyporheic exchange is completely suppressed. As the exponential curves are asymptotic toward QHZ/Qsurf = 0, we define a limit for hyporheic exchange at 1% of the maximum QHZ/Qsurf (at qbot* = 0) and denote the corresponding qbot* value as the critical qbot* (shown as a function of Re in Figure 6b). The critical qbot* values increase with Re, because higher Re induces higher H gradients across the riffle, which in turn require higher qbot* values to completely suppress hyporheic flow (except for morphology A01). Under losing conditions, the critical qbot* values do not increase with Re as fast as under gaining conditions. In the high Re scenarios, the critical qbot* is up to twice as high under gaining than under losing conditions.

Figure 6.

(a) Fitted exponential curves for the relationship between qbot* and inline image for high and low Re scenarios of runs A05-3, A05–10, A03-2, A03–11, A01–2, and A01–8 (see Table 3). R2 is >0.97 for all scenarios. (b) Critical qbot* as a function of Re. Critical qbot* is derived from the exponential curves in Figure 6a where inline image< 1% of the maximum inline image (at qbot* = 0).

Table 4. Fitting Parameters for the Relationship Between qbot* and inline image for Exponential Functions of the Type inline image.a
  LosingGaining
 Reabab
  1. a

    For all scenarios R2 > 0.97.

A 054.63E+051.42E-0424.7501.71E-04−36.505
7.10E+051.20E-0422.8281.40E-04−29.016
1.06E+061.04E-0417.0511.17E-04−22.290
1.31E+061.18E-0419.6721.26E-04−18.628
1.46E+068.82E-0517.9379.53E-05−17.537
1.72E+067.46E-0518.6457.84E-05−13.621
2.16E+067.00E-0517.8407.10E-05−10.033
2.50E+065.82E-0519.3985.89E-05−11.066
2.88E+065.69E-0518.0085.69E-05−10.051
3.27E+065.25E-0515.9395.25E-05−8.813
A 037.63E+051.09E-0433.2771.17E-04−35.464
1.04E+061.06E-0423.8331.06E-04−23.833
1.40E+068.28E-0521.9408.70E-05−19.226
1.51E+068.65E-0520.4569.15E-05−21.122
1.71E+065.85E-0523.8795.80E-05−12.167
1.81E+065.60E-0512.0015.60E-05−12.001
1.99E+065.61E-0523.5285.61E-05−12.111
2.22E+066.39E-0522.1346.37E-05−12.034
2.51E+066.96E-0520.0436.85E-05−10.971
2.85E+066.94E-0518.7476.77E-05−10.775
3.22E+065.86E-0518.4425.73E-05−11.520
3.60E+064.89E-0519.0864.74E-05−11.613
A 015.40E+051.65E-0439.7501.85E-04−60.913
9.01E+058.17E-0547.8398.49E-05−59.012
1.41E+064.59E-0549.4464.67E-05−58.069
1.65E+063.32E-0547.9583.41E-05−51.071
2.09E+062.87E-0550.3792.85E-05−42.205
2.47E+062.51E-0549.3932.51E-05−40.536
2.90E+062.11E-0546.6532.21E-05−42.207
3.28E+061.71E-0547.4601.82E-05−45.809
3.59E+061.49E-0549.2971.55E-05−46.100

3.2.3. Effects of Streambed Morphology

[44] As described in section 3.2.1, there is a general positive correlation between Re and QHZ for A03 and A05. In contrast, QHZ for flow of A01 (Figure 5c) does not show the same dependency on increasing discharge, and the QHZ contours depicted in Figure 5c are significantly different for A01 compared to A03 and A05. Also the critical qbot* values (Figure 6b) hardly vary with Re for A01.

[45] At first sight this seems to suggest a general positive correlation between the magnitude of QHZ and streambed amplitude. However, a closer look at the contour plots of A03 and A05 reveals very similar distributions and magnitudes of QHZ for both of these scenarios. This implies that above a certain pool-riffle height, a further increase in height has no significant effect on QHZ, which remains at a quasi-constant level beyond this threshold.

[46] For further evaluation of this possible threshold in pool-riffle height, Figure 7 shows how QHZ is related to the pool-riffle amplitude and qbot under constant Re = 1.4 × 106. The QHZ values on the y axis are normalized to QHZ of morphology A01. The upper graph (Figure 7a) shows that for gaining conditions, QHZ is relatively constant for pool-riffle amplitude heights above A = 0.1 m. For losing conditions, the relationship between amplitude A and QHZ levels of beyond A = 0.2 m. The rate of increase in QHZ with amplitude (the slope of the graphs in Figure 7) before the threshold (beyond which QHZ remains quasi-constant) generally increases with qbot. This effect is more pronounced for gaining conditions than for losing conditions.

Figure 7.

Hyporheic exchange flow as a function of maximum pool-riffle amplitude height (A) under constant stream discharge of Qsurf = 12.0 m3/s. All QHZ values are normalized to QHZ of the scenario with a maximum amplitude height of A = 0.1 m.

3.2.4. 3-D Hyporheic Zone Volume and Lateral Flow

[47] Based on the results of the 3-D particle tracking, the water volume of the hyporheic flow cells (VHZ) enclosed by the outermost flow paths returning to the stream can be derived (Figure 8). The ratio between VHZ and the total water volume stored in the entire porous domain is denoted as the dimensionless hyporheic volume VHZ*. Contour plots of VHZ* versus Re and qbot are shown in Figures 5d–5f. For the neutral case (qbot = 0), VHZ* slightly decreases with increasing Re and shows maximum values for low Re. In contrast, under losing and gaining conditions VHZ* slightly increases with increasing Re or stays relatively constant. The inverse relation of VHZ* to Re under losing/gaining and neutral conditions could be an effect of the no-flow boundary condition at 3 m depth representing the aquifer bottom. Under neutral conditions, this no-flow boundary restricts the full development of the major and minor hyporheic flow cells, which increases with Re. The space between major and minor flow cells also increases with Re and thus, less aquifer volume is affected by hyporheic flow. For gaining and losing conditions, the hyporheic flow cells do not reach the bottom of the domain, so this effect does not occur.

Figure 8.

Hyporheic flow paths for a high discharge scenario of Re = 1.72×106 (H distribution of Figure 4b) of high and low qbot magnitudes for the gaining and losing case. Red colors (negative vertical Darcy velocity) indicate infiltration and blue colors (positive vertical Darcy velocity) exfiltration to the streambed.

[48] When the magnitude of qbot exceeds 2 m/d for both the gaining and losing cases, VHZ* is less than 5% reflecting a small vertical extent of the HZ (Figures 5d and 5f). However, QHZ is still significant even for low magnitudes of qbot (Figures 5a and 5c), indicating significant hyporheic flow even for small HZ volumes. Also, the low VHZ* values for Re between 1.5 × 106 and 2.0 × 106 in A03 indicate a reduction of the HZ volume (Figure 5h), probably caused by the effects of undular hydraulic jumps as described earlier.

[49] Major differences in VHZ* are visible between A05, A03, and the shallow morphology A01. VHZ* in A01 shows a sharper decline with increasing Re than in A03 and A05. Furthermore, VHZ* in A01 is more sensitive to qbot, indicated by steeper VHZ*/qbot gradients compared to A03 and A05.

[50] Two distinct flow cells in the HZ can be distinguished: a major and a minor one (Figures S8, S9, and S11 in the supporting information). Flow paths in the major flow cell, located around the crest, are directed downstream with a lateral flow component toward the stream bank. Flow paths that are directed upstream form a minor flow cell with the infiltration zone at a lower location on the stoss side of the riffle and an exfiltration zone close to the upstream pool with a lateral flow direction toward the stream center. This complex 3-D flow field is caused by the spatial variation of H across the streambed (see section above) resulting from the 3-D morphologies. It is additionally influenced by the direction and magnitude of ambient groundwater flow across the bottom boundary. For gaining conditions, the bulk of the stream water is infiltrating at the upstream side of the riffle and subsequently returns to the stream at the downstream side. The general lateral direction of flow in this cell under gaining condition is toward the stream bank. Flow paths that are oriented in the opposite direction (minor flow cell) return to the stream close to the upstream pool center and are laterally deflected toward the stream center (Figures 8c and 8d and 9, right column). Flow cells that develop under losing conditions look quite different. Here large fractions of stream water that infiltrate at the upstream side of a riffle do not return to the stream, but instead leave the model domain at the bottom boundary (Figures 8a and 8b). Areas of infiltration at the streambed with subsequent return to the stream are pushed toward the crest of the riffle (Figure 9, left column). In contrast to the gaining case, the general lateral flow component in this major flow cell is toward the stream center. For high Re (1.72 × 106) or low magnitudes of qbot a smaller hyporheic flow cell can develop toward the upstream pool (Figures 9a–9d), which is also directed toward the stream center. With respect to the impact of qbot, it can generally be stated that: (1) the higher qbot, the smaller the areal extent of the infiltration and exfiltration zones, and (2) for high magnitudes of qbot no distinct minor flow cell develops and only the major flow cell remains but is cropped close to the stream bank (Figures 8b, 9c, 9d, 9g, 9h, 9k, 9l, 9o, and 9p), and (3) the lateral flow directions in the major flow cells show opposite patterns in the gaining (toward stream bank) and in the losing case (toward stream center).

Figure 9.

(a, c, e, g, i, k, m, o) Locations of infiltration and subsequent exfiltration at the streambed in the x–y plane for the two Re scenarios presented in Figures 1, 4b, and 4c for different directions and magnitudes of qbot. The start- and end-points of the flow paths are shown that define hyporheic exchange (infiltration into and subsequent exfiltration out of the streambed). Flow paths that leave or enter the domain at the bottom boundary are not considered. (b, d, f, h, j, l, n, p) Cross sections along the x–z plane at location y = 0.4 m (dashed line in top). The contours represent vertical Darcy velocities and vectors depict the direction of the flow. The bold black lines indicate the maximum vertical extent of the hyporheic flow cells. Stream flow direction is from left to right.

[51] The maximum vertical flow velocities of the particle paths are found at the infiltration and exfiltration points (red and blue colors in Figures 8, 9b, 9d, 9f, 9h, 9j, 9l, 9n, and 9p). Infiltration and exfiltration velocities at the streambed increase with Re, due to higher H gradients and are independent of qbot. Vertical velocities are smallest at the maximum depth of each particle path (white color in Figure 8).

[52] To quantitatively evaluate the lateral flow components, we define the dimensionless distance ydist* as the absolute lateral travel distance ydist normalized by the stream width (10 m). In Figures 5g–5i the mean values of ydist* versus Re and qbot are presented. The magnitude and direction of lateral shifts mainly depend on the hydraulic heads (and thus on Re) at the location of infiltration. Additionally, the direction and magnitude of qbot influences the lateral shift. Overall, ydist is rarely higher than 50% of the longitudinal travel distance. Generally, the lateral travel distance increases slightly with Re for all morphologies. Ambient groundwater flow does not consistently influence ydist*. For A01, ydist*decreases with increasing qbot for both gaining and losing conditions. This relationship cannot be observed for A05 and A03.

3.2.5. Hyporheic Residence Times

[53] Flux-weighted RT in the HZ (only flow paths that enter the HZ from the stream and subsequently return to the stream are considered) were evaluated based on the results of the forward particle tracking. Simulated median RT (MRT) range from 0.7 to 19 h (Figures 5j–5l). Plotted against Re and qbot they show similar spatial patterns as the distributions of QHZ shown in Figures 5a–5c. Figures 5a–5c also show that QHZ MRT is mainly affected by the magnitude of qbot and not by its direction. MRT are generally shorter for increasing magnitudes of qbot. However, MRT are shorter under losing conditions compared to gaining conditions despite relatively similar QHZ. This indicates faster hyporheic flow for the losing case. For all three morphologies, the longest RT occurs in the case of no bottom inflow or outflow.

[54] Increasing Re results in shorter MRT for all three morphologies. The longest MRT occur under low Re, where very low exchange flows appear and vice versa. This indicates a higher hyporheic flow velocity under high Re, accompanied with shorter MRT and higher QHZ.

[55] In contrast to the distribution of QHZ (Figures 5a–5c), the MRT for the three different morphologies are within one order of magnitude, where the longest MRT can be observed for the shallowest morphology A01. Stream flows over steep morphologies produce high QHZ (A03, A05), due to a fast flow through the HZ, which leads to shorter RT but long flow paths. In contrast, for the shallow morphology (A01), flow velocities are slower, and consequently, the RT are larger.

[56] For the scenarios shown in Figure 9, histograms of the RT and fitted log-normal curves are shown in Figure 10. RT for all scenarios are well represented by a log-normal distribution. The RTD do not show significant variations neither between the losing and gaining cases nor between the high and low Re scenarios. Also the RTD for the three different morphologies are very similar (data not shown).

Figure 10.

Histograms of the residence time distributions (RTD) and median residence times (MRT) of the scenarios in Figure 9.

4. Discussion

4.1. Variation of Hydraulic Head at the Streambed and Effects of Undular Hydraulic Jumps

[57] Natural stream flows in channels with pool-riffle structures typically result in complex configurations of the stream water level including (undular) hydraulic jumps [Tonina and Buffington, 2007, 2011]. Such variations in the water level together with pressure effects caused by turbulent flow are the dominant controls of pressure variations at the streambed, which in turn drive hyporheic flow through the streambed [Cardenas and Wilson, 2007a-2007c; Packman et al., 2004; Tonina and Buffington, 2007, 2011]. Hence, an accurate representation of water level and turbulent effects is indispensable when simulating hyporheic exchange flows in river reaches with pronounced pool-riffle structures [Tonina and Buffington, 2007, 2009, 2011; Endreny et al., 2011]. The two-phase CFD approach in this study complies with these requirements. Water levels in our simulations are determined using the VoF-approach and locally deviate by up to 0.45 m from the mean water level. Using a simpler one-phase model would have resulted in significant errors in the pressure distributions at the streambed. By comparing the water level with the hydraulic head, our work provides further evidence that in complex channels the water level is not a good surrogate for the hydraulic head distribution at the streambed. These findings corroborate the results of Tonina and Buffington [2007] from flume experiments. The overall difference between Hmax and Hmin depends on Re, and the location of the Hmax and Hmin on the ratio between dune height and dune length. Furthermore, the distance between the extreme values is larger for lower ratios, as in our results for the different morphologies. However, the Hmax and Hmin at the upstream and downstream side of a riffle vary significantly in the lateral direction and hence also induce lateral hyporheic flow components. At the upstream side, H decreases from the stream center to the bank, whereas H increases toward the stream bank at the downstream side of the riffle. Absolute H maxima are always located closer to the stream center than the H minima. This setting generates a lateral gradient toward the stream bank in addition to the major longitudinal gradient between the upstream side and downstream side of a riffle. A small depression of H located next to the pool close to the stream centre evokes hyporheic flow cells that points upstream and toward the stream center (Figure 4).

[58] For the first time in pool-riffle systems, we show that surface water waves produced by undular hydraulic jumps can cause local anomalies in streambed pressure. We found that when the trough of the standing wave coincides with the region of highest streambed pressures in the reattachment zone of the eddy, H at the streambed is significantly reduced. This effect alters the H distribution across the streambed and in turn affects the 3-D hyporheic flow field. The severity of these effects depends on the magnitude of stream discharge.

[59] Also, the complex head distribution simulated by the CFD model results from the specific, field-inspired, 3-D streambed morphology. Streambed morphology at the site is affected by bank protection measures (e.g., rip-rap) and large woody debris, which constrains sediment dynamics resulting in a small ratio of pool-riffle wavelength to stream width (∼1) [Montgomery et al., 1995]. To our knowledge, hydraulic head distributions and hyporheic flows in such pool-riffle systems have not been studied systematically. Probably, higher ratios of pool-riffle wavelength to stream width would diminish the effect of standing wave interference. Furthermore, stream flow could adapt more smoothly to the morphology, likely resulting in less lateral variations of hydraulic head and, hence, a decrease in the lateral hyporheic flow component.

[60] However, further investigations regarding the impact of morphology on hyporheic flow are needed to evaluate the dependence of QHZ on the ratios between wavelength to width as well as ratios between wavelength and amplitude.

4.2. 3-D Hyporheic Flow Paths

[61] The pronounced variations of H at the streambed induce complex 3-D hyporheic flow paths with flow components in both longitudinal and lateral directions (Figures 8 and 9). Lateral flow distances (ydist) were on average 20% of the longitudinal flow distance. The ydist increased only slightly with Re; however, the effect of Re is evident in longitudinal direction where increasing Re results in deeper and longer flow hyporheic flow paths. We suspect that pronounced lateral flow components may support an enhanced mixing of stream water and groundwater in the hyporheic zone.

[62] The 3-D hyporheic flow paths were strongly affected by ambient groundwater flow. Most notably regarding the shapes of the hyporheic flow cells and the spatial patterns of infiltration and exfiltration zones, which were found to be significantly different between the gaining and losing case (Figures 8 and 9). Under gaining conditions, one single infiltration area exists, and flow paths are separated into two distinct hyporheic flow cells, one in downstream and one in upstream direction (Figures 9e, 9f, 9g, 9h, 9m, 9n, 9o and 9p). For the losing case, two noncontiguous infiltration areas with completely separated flow cells develop, particularly pronounced for high Re (Figures 9a and 9b). This separation is forced by the fast downward flow located close to the maximum in H at the upstream side of the riffle (Figures 8a, 9a, and 9b). This general separation of the infiltrating stream water into flow cells pointing in upstream and downstream directions has also been shown in several 2-D studies investigating stream flow over structured streambeds and the effects of ambient groundwater on flow in the HZ [Boano et al., 2008; Cardenas and Wilson, 2007b]. In addition to this general flow separation shown in our 3-D simulations, the lateral in- and exfiltration patterns are also affected by ambient groundwater flow. The variations in the hyporheic flow field result from competing pressure gradients either caused by the H gradients along the streambed or vertical gradients imposed by ambient groundwater flow. For increasing magnitudes of qbot, the hyporheic flow cell first disappears at the stream bank. A plausible explanation for this is that the lower bed form induces a gradient along the smaller bed form amplitude in the vicinity of the stream bank. For strong gaining conditions, the vertical gradient evokes groundwater upwelling in the pool area but also close to the stream bank and thus hampers the formation of a hyporheic flow cell that would span over the entire stream width (Figure 8d). Additionally, this high vertical upward directed gradient close to the stream bank induces flow paths in the hyporheic flow cell that have strong lateral components toward the stream bank (Figures 8d, 9g, 9h, 9o, and 9p at y = 3–4 m).

[63] For the losing case, strong downward vertical gradients (high negative qbot) cause stream water to predominantly downwell at the upstream Hmax location (Figure 8b). Additional downwelling occurs close to the stream bank along the crest area, where the bed form-induced gradient does not exceed the vertical gradient. Here similarly to the gaining case, hyporheic flow cells are truncated close to the stream bank and shifted in downstream direction from the crest (Figures 9c, 9d, 9k, and 9l). In contrast to the gaining case, the flow paths in the hyporheic flow cells close to the stream bank have a strong lateral flow component toward the stream center, induced by the downward vertical gradient.

[64] For both the losing and the gaining cases higher magnitudes of qbot induce higher vertical gradients and, hence, a stronger lateral shift of the hyporheic flow paths close to the stream bank. Although hyporheic flow paths are strongly altered by qbot, the average lateral flow component expressed as ydist remained relatively unaffected (Figures 5 and 6g–6i). Hyporheic flow paths that deviated from the general downstream direction but rather showed complex patterns of in- and exfiltration in directions lateral and opposed to the direction of stream flow have also been observed in a field study by Angermann et al. [2012]. These patterns could potentially be explained by the interplay between 3-D streambed morphology and ambient groundwater flow.

4.3. Effects of Ambient Groundwater Flow and Varying Discharge on the Integral Characteristics of Hyporheic Exchange

[65] Our results show that the integral characteristics of hyporheic exchange QHZ, VHZ*, and MRT are similar between the gaining and losing cases for the same magnitudes of qbot and Re. This relative insensitivity of the magnitudes of QHZ, VHZ*, and MRT to the direction of groundwater flow in our model is in line with the results of Cardenas and Wilson [2007b], who found QHZ, hyporheic depth, and MRT to be almost equal between the gaining and the losing cases. In contrast to Cardenas and Wilson [2007b], however, QHZ, VHZ*, and MRT in our simulations are not absolutely identical in both cases. Under losing conditions, QHZ is slightly lower, VHZ* is lower and the MRT are shorter compared to the gaining case. Also the critical qbot* values, where all hyporheic exchange is suppressed, are different for similar magnitudes of qbot*. Critical qbot* is up to 50% smaller under losing conditions, suggesting that smaller vertical gradients between the streambed and the bottom of the groundwater domain are needed to completely inhibit hyporheic flow compared to the gaining case. This can be caused by the different spatial extents of the infiltration areas at the streambed and the separation of hyporheic flow into an up- and a downstream flow cell, when comparing the gaining to the losing case, described in the section above.

[66] Therefore, we conclude that under losing conditions smaller and thus shallower flow cells develop, accompanied with faster hyporheic flows, and respective shorter MRT. Furthermore, in the losing case hyporheic exchange is more sensitive to the magnitude of qbot.

[67] The ratios between hyporheic exchange flow (QHZ) and stream discharge (Qsurf) decrease for increasing Re. Hence, for high stream flows the fraction of total stream flow that becomes hyporheic flow and is in turn exposed to the higher reaction rates in the HZ becomes smaller. This in turn diminishes potential effects on overall stream water concentrations of constituents such as nitrates. The ratios are generally higher for low magnitudes of qbot, indicating a higher fraction of the infiltrated water of Qsurf under low ambient groundwater flow. Losing conditions result in slightly lower QHZ/Qsurf ratios compared to gaining conditions due to the overall lower values of QHZ. The ratio QHZ/Qsurf is generally low (1 × 10−5 to 1 × 10−6 per meter stream length) and agrees with values of hyporheic exchange found in other studies [Battin et al., 2003, Boulton et al., 1998; Saenger et al., 2005].

[68] Our results demonstrate that the connection of the stream to the ambient groundwater system cannot be disregarded when examining hyporheic exchange. Even low vertical gradients (qbot* in Figure 6a) between stream and groundwater can significantly alter the hyporheic flow field and diminish hyporheic exchange flow and extent under both losing and gaining conditions.

4.4. Threshold of Streambed Amplitude

[69] The influence of streambed amplitude on hyporheic exchange is characterized by a threshold behavior in our simulation. Above a certain threshold in amplitude, QHZ and MRT become insensitive to further increases in amplitude (see Figure 7). Our results generally show that a distinct streambed morphology is necessary to produce sufficient pressure differences at the streambed for advective pumping to occur. If this condition is met, additional effects of the height of the streambed morphology on hyporheic exchange are minor. Tonina and Buffington [2011] showed data from flume experiments with partially submerged pool-riffle sequences of four different amplitudes. In their Figure 5, q* (comparable to our QHZ) is smallest for the lowest amplitude and practically equal for the three higher amplitudes. Although they did not explicitly describe this relationship, their results are in line with the threshold of bed form amplitude observed in our study. Furthermore, Packman et al. [2004] showed in a flume experiment that exchange flow in structured beds is generally higher than in a flat bed and that higher bed forms only slightly increased hyporheic exchange. Although, the study of Packman et al. [2004] was conducted in 2-D, the lack of a significant increase of hyporheic exchange with increasing bed form height is similar to our results based on 3-D modeling.

[70] Hence, the most important controls of hyporheic exchange in the studied pool-riffle systems are stream discharge, and ambient groundwater flow, while the height of bed form amplitude has minor effects beyond a defined threshold.

4.5. Residence Times

[71] Many biogeochemical reactions in the HZ are time dependent and thus directly related to the RT of infiltrated stream water in the HZ [Zarnetske et al., 2011]. For instance, long RT lead to higher consumption of redox-sensitive compounds (oxygen, nitrate), provided that a sufficient carbon source is available [Brunke and Gonser, 1997; Zarnetske et al., 2011; Marzadri et al., 2011-2013]. Therefore, hyporheic RT is a crucial parameter to evaluate the potential for biogeochemical reactions in the HZ. However, Bardini et al. [2012] showed that the transformation of several redox-sensitive compounds increased with stream velocity because the higher amount of solutes advected into the HZ was more important than the overall decrease in residence time.

[72] Our RT calculations show that with increasing Re, MRT becomes shorter due to faster flow velocities in the HZ, accompanied with higher QHZ and vice versa, also in agreement with the findings of Tonina and Buffington [2011]. Hence, low hyporheic flows (small QHZ) cause longer exposure times to the hyporheic sediments and hence a higher potential for certain biogeochemical reactions to occur.

[73] Furthermore, the flow over less-pronounced streambed morphologies (e.g., A01) produces lower H gradients that result in low QHZ but high MRT. The effect of longer RT for shallow morphologies is in agreement with the findings of Marzadri et al. [2010], where flows over shallow bed form amplitudes resulted in longer mean RT. However, when comparing our RT data and streambed configuration to those of Figure 8a in Marzadri et al. [2010], our data do not fit to the proposed relationship, because normalized mean RT are too low, compared to the normalized streambed amplitude.

[74] The RT for all our scenarios are well represented by a log-normal distribution. This agrees with several previous studies that have examined hyporheic RT induced by flow over dune-shaped bed forms [Cardenas et al., 2004], submerged pool-riffle structures [Marzadri et al., 2010], and partially submerged pool-riffle sequences [Tonina and Buffington, 2011]. If we were to include the full complexity of a field site, where morphological features at nested scales and sediment heterogeneity occur, our RTD would likely show more tailing. However, the MRT would likely not change very much. Interestingly, the RT in our model cover the range of RT over which Zarnetske et al. [2011] observed a biogeochemical regime shift in a streambed of similar texture from net nitrifying to net denitrifying. Hence, our results demonstrate that pool-riffle sequences can cause RT that might facilitate biogeochemical regime shifts in the hyporheic sediments.

5. Summary and Conclusions

[75] In this 3-D modeling study the effects of stream discharge, streambed morphology, and ambient groundwater flow on hyporheic flow in pool-riffle streams were systematically investigated. We combined two-phase CFD simulations of turbulent stream flow over 3-D pool-riffle streambeds with flow modeling in the underlying porous streambed (HZ) for different stream discharges and streambed morphologies. Ambient groundwater flow was represented by either gaining (upward flow of groundwater into the streambed) or losing (downward flow of stream water into the aquifer) conditions. 3-D flow paths in the HZ and hyporheic RTD were derived from particle tracking.

[76] Our CFD simulations showed the formation of 3-D eddies and complex water surfaces including undular hydraulic jumps at the downstream side of riffles (for Fr between 1 and 1.2) resulting in distinct pressure variations across the streambed surface. If a trough of the standing wave in the undular hydraulic jump coincided with the high pressure in the reattachment zone of the eddy, streambed pressure was locally reduced affecting the developing hyporheic flow cell. Due to the three-dimensional nature of the bed forms, the Hmin at the downstream side of the riffles was shifted toward the stream bank relative to the location of Hmax at the upstream side. This separation effect induced significant lateral flow components in the HZ. Additionally, a constricted zone of lower H in the upstream pool region caused secondary hyporheic flow cells in the upstream direction with a slight shift toward the stream center. Lateral flow components in the major hyporheic flow cell were also affected by the direction of ambient groundwater flow and showed a shift toward the stream bank for the gaining case and a shift toward the stream center for the losing case.

[77] Our results highlight the importance of ambient groundwater flow for HZ extent and the characteristics of the 3-D hyporheic flow field in pool-riffle streams, both under gaining and losing conditions. The extent of the hyporheic water volume within the aquifer (VHZ*) strongly decreased with increasing magnitude of groundwater flow accompanied with a lower hyporheic exchange flow rate and shorter RT. Neglecting the vertical hydraulic gradients induced by typical rates of ambient groundwater flow would lead to a significant overestimation of the hyporheic exchange flow.

[78] Total hyporheic exchange flow (QHZ) increased with stream discharge while MRT decreased, due to faster hyporheic flow, despite longer flow paths, as also shown by Tonina and Buffington [2011]. The ratio between hyporheic exchange flow and stream discharge (QHZ/Qsurf) was on the order of 1 × 10−5 to 1 × 10−6 per meter stream length and decreased with increasing stream discharge. However, for the relationship between QHZ and pool-riffle amplitude, we found a threshold in the pool-riffle height, beyond which QHZ remained constant for even higher pool-riffle amplitudes. This threshold is also observed in the results of the flume study of Tonina and Buffington [2011] and comparable to the findings of the 2-D study of Packman et al. [2004].

[79] Hyporheic RT showed unimodal log-normal distributions and MRT ranged from 0.7 to 19 h over the series of investigated scenarios. Over a similar range of MRT, shifts from a net nitrifying to a net denitrifying biogeochemical regime have been observed in comparable field settings [Zarnetske et al., 2011]. This suggests that the streambeds investigated in this study may have the potential to attenuate redox-sensitive solutes.

[80] Our results elucidated distinct 3-D patterns and dynamics of hyporheic exchange in pool-riffle streams, and for the first time, highlighted how ambient groundwater flow affects these hyporheic flow characteristics. These dynamics will likely affect the biogeochemical turnover of solutes in pool-riffle streams and their impact should be addressed in future work.

Acknowledgments

[81] The research was funded by the Ministry of Science, Research,and Arts of Baden-Wuärttemberg (AZ Zu 33-721.3-2) and the Helmholtz Centre for Environmental Research (UFZ) in Leipzig. We thank the associate editor, Bayani Cardenas, Ted Endreny, and two anonymous reviewers for their useful comments that greatly improved the quality of the paper.

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