Modeling hyporheic exchange with unsteady stream discharge and bedform dynamics

Authors

  • Fulvio Boano,

    Corresponding author
    1. Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy
    • Corresponding author: F. Boano, Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino IT-10129, Italy. (fulvio.boano@polito.it)

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  • Roberto Revelli,

    1. Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy
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  • Luca Ridolfi

    1. Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy
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Abstract

[1] Water exchange between streams and hyporheic zones is highly dynamic, and its temporal variation is related to the hydrologic fluctuations of stream discharge and groundwater levels. Unfortunately, predictions of temporal patterns of exchange are difficult due to the many hydrodynamic and morphodynamic processes that are involved and also to their complex nonlinear interactions. Examples of these processes include the evolution of streambed morphology in response to changing streamflow as well as the feedback on surface flow induced by drag resistance due to evolving bed forms. In this work, we have employed a stochastic method to analyze the temporal dynamics of bed form-driven hyporheic exchange in a stream characterized by subcritical flow and daily discharge variations. The method is an extension of previous studies that includes current-induced alterations of bedform size and celerity and their effect on water exchange. The modeling results show that during high flows, stream water penetrates deeper and for longer times in the sediments. At the same time, the predicted rate of water exchange per unit streambed area decreases because the streambed area occupied by each bed form increases faster than the volumetric rate of stream water exchange induced by the same bed form. This reduction can be compensated by the increase in wetted area with discharge, which may provide additional streambed area for water exchange. One the main finding of the study is that the time-averaged values of exchange flux and depths are quite similar to those modeled for a steady mean discharge, while residence times are somewhat lower. Predicted temporal variations of exchange depths and times around their time-averaged values are moderate compared to steady state values.

1. Introduction

[2] In stream ecosystems, the exchange of water and solutes between the river and the adjacent sediments, the hyporheic zone, enhances the transformation of nutrients by microrganisms attached on sediment grains [Battin et al., 2008; Bottacin-Busolin et al., 2009; Zarnetske et al., 2011] and provides favorable conditions for the sustainment of aerobic macroinvertebrates and fish eggs [Robertson, 2000; Malcolm et al., 2004, 2006]. Groundwater that flows through hyporheic sediments before discharging into the stream also exhibits significant changes in dissolved nitrogen concentrations [Krause et al., 2009]. Maintaining the connectivity between surface water, groundwater, and hyporheic sediments is thus regarded as an effective way to conserve or restore the ecological functioning of the stream corridor [Boulton, 2007; Kasahara and Hill, 2007; Lautz and Fanelli, 2008; Crispell and Endreny, 2009].

[3] Previous modeling [Harvey and Bencala, 1993; Elliott and Brooks, 1997; Cardenas and Wilson, 2007a; Wörman et al., 2007; Revelli et al., 2008; Boano et al., 2010a] and experimental studies [Gooseff et al., 2007; Tonina and Buffington, 2007; Buffington and Tonina, 2009] have identified streambed and landscape topography as the key factors that control water flow across the streambed surface. The results of these studies have clarified that hyporheic exchange occurs as nested flow cells whose sizes span over a broad range of spatial scales [Cardenas, 2007; Bottacin-Busolin and Marion, 2010; Stonedahl et al., 2010]. The size of the smaller flow cells is reduced by discharge (recharge) of groundwater to (from) the stream [Cardenas and Wilson, 2007b; Boano et al., 2009].

[4] The mentioned studies have considerably elucidated the processes that drive hyporheic flow and the resulting spatial patterns of groundwater-surface water exchange. However, the temporal dynamics of hyporheic exchange have received much less attention despite the fact that water flowing in streams and aquifers is seldom in steady state. Temporal variations in the intensity and direction of flow in the hyporheic zone may arise from changes in stream discharge, depth, and velocity, which together influence the distribution of hydraulic head over the streambed that drives water exchange [e.g., Arntzen et al., 2006]. Seasonal changes in regional groundwater flow and in groundwater table levels also control the dynamics of hyporheic exchange [Wondzell and Swanson, 1996; Malcolm et al., 2006]. The problem is further complicated by the fact that streamflow variations also affect hyporheic flow by shaping streambed morphology, resulting in a complex control of stream hydrology on surface-subsurface water exchange.

[5] Since hyporheic exchange and its temporal variability are influenced by a large number of processes, it is necessary to improve our understanding of how this variability is linked to changes in hydrologic factors such as stream discharge. Thus, the aim of the present modeling work is to investigate the role of hydrologic fluctuations of streamflow on the temporal dynamics of bed form-driven hyporheic exchange. Although our approach is simplified as it does not include factors such as regional groundwater dynamics [Krause et al., 2009] or more complex river corridor settings [Lewandowski et al., 2009; Angermann et al., 2012a, 2012b], these simplifying assumptions allows to better focus on the role of daily streamflow variability on water exchange in streams with subcritical stream flow regimes where dunes are the only bed forms that drive hyporheic exchange. Our work employs a mathematical model that is similar to previous stochastic approaches [Boano et al., 2007, 2010c], which examined the effect of a time-dependent streamflow on hyporheic flow but did not consider the modifications of streambed morphology induced by sediment transport. Here we extend this previous approach by including the effects of changes in bedform size and celerity, and use the resulting physically based model to explore the temporal dynamics of water exchange under a range of hydrologic conditions.

2. Methods

[6] The modeling approach includes three main steps: (1) generation of a time series of stream discharge; (2) evaluation of the corresponding properties of the surface flow and of the bed forms; and (3) calculation of the resulting water exchange and hyporheic flow. For each simulation, these steps are solved sequentially to link surface flow properties and their temporal variability to hyporheic flux, residence times, and depth of the hyporheic zone.

2.1. Stream Discharge

[7] The time series of stream discharge is generated following the stochastic procedure described by Boano et al. [2010c], which is summarized later. The method is based on a schematic description of streamflow dynamics, according to which flood events are represented as rapid increases in stream discharge Q(t) followed by a recession phase:

display math(1)

where ω is a parameter that controls the rate of streamflow recession, ξ is a random white shot-noise term that represents the increases in discharge during floods [Ridolfi et al., 2011], and t is time. The time step adopted in the present analysis is equal to 1 day, but different values can be chosen according to the characteristic timescales of the investigated flood events.

[8] The structure of equation (1) results in an exponential decay of stream discharge between two flood events. The values of the stochastic term ξ are randomly extracted from an exponential distribution with mean value 1/γ, and the time between two consecutive flood events is also derived from an exponential random distribution with mean value 1/λ. In order to avoid vanishing values of stream discharge, a constant value Qb is added to the time series obtained with equation (1) to represent the contribution of base flow. An example of the resulting time series Q(t) is shown in Figure 1.

Figure 1.

Example of temporal dynamics of dimensionless stream discharge Q(t)/Qm (gray line) and hyporheic exchange flux q/qm for simulations 1 (instantaneous equilibrium, continuous black line) and 7 (nonequilibrium, dotted black line). Parameter values of the simulations are reported in Table 1.

[9] The statistical properties of the streamflow time series are summarized by the mean value, Qm, the coefficient of variation, CVQ, and the correlation time, τQ. In particular, the coefficient of variation quantifies the relative variability of stream discharge, and the correlation time is a measure of the typical flood frequency. These statistical quantities can be used to derive the values of ω, γ, and λ, which are required to generate the time series:

display math(2)
display math(3)
display math(4)

[10] It should be noted that streamflow values generated using equation (1) do not account for seasonal discharge variations, which induce temporal changes of Qm, CVQ, and τQ, as the procedure aims to reproduce daily hydrologic fluctuations induced by single flood events, at a scale which is much shorter than the seasonal one. The possible effects of annual variability are discussed later in this work.

2.2. Stream and Bedform Dynamics

[11] We consider a stream with rectangular cross section and constant width, w, and slope, Sb, flowing on erodible sediments with uniform median sediment diameter, D50, porosity, θ, and hydraulic conductivity, K. Additionally, the size of the coarser sediment fraction is described by the 90% quantile of the grain size distribution, D90. This sediment diameter is used to determine the value of roughness height, math formula, which is required to quantify the skin resistance caused by friction between surface stream and individual sediment grains.

[12] Both surface flow properties and bedform geometry are evaluated from the discharge time series. For each day, the stream depth, d(t), is calculated from Q(t) using the form drag predictor approach described by Wright and Parker [2004], which allows consideration of feedback effects on surface flow of the additional drag resistance due to bedform roughness. Then, the average stream velocity is calculated as math formula.

[13] In unsteady flow conditions, dunes adjust their size and celerity to surface flow [e.g., Julien and Klaassen, 1995]. The geometry of bed forms is here summarized by their crest-to-through height, H(t), and wavelength, L(t). Variations of bedform size over time are obtained from the relations proposed by Julien and Klaassen [1995]:

display math(5)

which express an equilibrium condition between dune geometry and surface flow. Equation (5) is only valid as long as dunes are the predominant type of bedform morphology, which is only true in a specific range of flow conditions. In particular, dunes generally exist in subcritical flow with math formula, where math formula is the Froude number, math formula is the dune wave number, and g is the gravity acceleration [Kennedy, 1969]. We have verified that this constraint is always satisfied for the numerical simulations presented in this work, thus ensuring that flow conditions are suitable for the presence of river dunes.

[14] The last bedform property that is considered is the celerity of downstream migration, c(t), due to bedload sediment transport. This quantity is estimated with the expression proposed by Kondap and Garde [1973]:

display math(6)

[15] Since equation (6) always provide positive values of dune celerity regardless of stream conditions, the Shields criterion is first applied as an additional check to verify that shear stresses are high enough to mobilize stream sediments. If this criterion is not met we consider solid transport to be negligible, and celerity is coherently set to zero.

[16] The described procedure, hereinafter referred to as instantaneous equilibrium approach, is implicitly based on the assumption that for each day the streambed is in morphodynamic equilibrium with the discharge, i.e., that the characteristic timescale for bedform adaption to stream discharge is much smaller than one day. This assumption could be violated in unsteady conditions, because when stream discharge varies sediment transport and bed geometry require some time to attain equilibrium with the new discharge. This delay between hydrodynamic and morphodynamic processes leads to a nonequilibrium behavior that results in hysteresis in the relationship between bedform size and stream discharge [Julien et al., 2002; Nelson et al., 2011], a fact which implies that streambed geometry retains some degree of memory of previous stream discharge. A complete modeling of these nonequilibrium configurations would require a very detailed description of sediment dynamics, which is outside the scope of the present work. Instead, we have adopted a simple nonequilibrium approach to consider the influence of this nonequilibrium behavior on hyporheic flow. Our approach simply assumes that all bedform properties at the n-th day are in equilibrium with a discharge which is the average of math formula, where Δ is the lag time. In other words, we hypothesize that the streambed retains memory of previous stream discharge over a time window of duration Δ. The implementation of the nonequilibrium approach is identical to the instantaneous equilibrium approach, with the exception that discharge Q(t) must be replaced with a discharge value averaged over the Δ window. The choice of the duration Δ of the averaging window is somewhat arbitrary as there are no clear indications in the literature to guide this choice. For this reason, after some preliminary analyses, we have opted for a window duration of 3 days for the nonequilibrium approach. This value has been chosen because smaller values would not have influenced hyporheic exchange to a relevant extent, while larger values would have been similar to the correlation timescale of streamflow variations and were deemed unrealistic for the stream typologies considered here. Hence, in this work, the nonequilibrium approach has always been applied with Δ = 3 days, while the value Δ = 0 denotes the instantaneous equilibrium approach.

2.3. Hyporheic Flow

[17] Once the time series of the morphological properties of bed forms have been determined, the water flow field in the subsurface can be evaluated. To this aim, we extend the methodology proposed for stationary bed forms in unsteady flow by Boano et al. [2010c] to the case of moving bed forms.

[18] The theory is based on the idea originally proposed by Elliott and Brooks [1997] that the head profile on the streambed can be represented as a sinusoidal wave that migrates downstream with the dune celerity. Adopting a 2D reference system math formula, where x is the streamwise direction and y is the elevation above the mean streambed level, the head distribution on the streambed is expressed as math formula, where

display math(7)

is the amplitude of the head distribution, and α and ψ are two coefficients that depend on bedform geometry and submergence [Boano et al., 2010c].

[19] Hydraulic heads in saturated sediments below the streambed surface rapidly achieve equilibrium with the head distribution over the streambed [Boano et al., 2010c; Pokrajac et al., 2007]. Therefore, the hydraulic head field in the subsurface is found by solving the Laplace's equation, math formula, for an infinitely deep sediment bed. The prescribed boundary condition on the sediment surface is given by the sum of the streambed head distribution plus the contribution of streambed slope, math formula. At the upstream and downstream vertical boundaries, the assigned boundary conditions are math formula and math formula, where ub and db denote the upstream and downstream boundary, respectively.

[20] The typical scales of subsurface velocity, um, and time, Tm, are defined as

display math(8)

where km and hm are the dune wave number and head amplitude, respectively, in equilibrium with mean discharge Qm. Similarly, cm is the celerity of dune migration calculated for math formula, and the subscript m is hereafter used to denote mean flow variables. Using the above definitions, the solution for the groundwater flow field can be written in dimensionless form as

display math(9)
display math(10)

where the dimensionless variables are defined as math formula, math formula, and math formula. Moreover, math formula is the dimensionless dune celerity, math formula is the dimensionless horizontal velocity due to streambed slope only, and math formula, math formula, and math formula are three dimensionless functions that describe the dynamics of k(t), h(t), and c(t), respectively. Each variable is normalized with the corresponding mean flow quantity, so values larger (smaller) than unity represent increases (decreases) of that variable compared to a reference state that corresponds to steady state flow conditions with math formula.

[21] When bed forms are not stationary, water is exchanged between stream and sediments due to both pumping and turnover processes. The exchange flux per unit streambed area due to pumping can be found as

display math(11)

[22] The additional turnover contribution to exchange is calculated as [Elliott and Brooks, 1997]

display math(12)

[23] The total exchange flux is simply the sum of the two contributions due to pumping and turnover, math formula. A quantity that is interesting to consider is its temporal average

display math(13)

which describes the average rate of water exchange in unsteady conditions.

[24] The distributions of exchange depths and residence times cannot be found analytically, and a particle tracking procedure that accounts for advective displacement of water particles is thus applied. For each day, 2000 particles are released at the sediment surface along the downwelling area of the dune profile, and their displacements are calculated as the product of instantaneous velocity math formula and integration time step Δt. Subsurface velocities are defined at the same daily timescale as stream discharge, and this timescale may be too large compared to water residence times in the sediments to correctly identify particle paths. Therefore, values of u(t) and v(t) are linearly interpolated between consecutive days to avoid abrupt variations in flow direction of water particles. For each particle path, the maximum depth Y and the residence time T in the subsurface are recorded. Finally, the flux weighted distributions at time t of exchange depths, math formula, and residence times, math formula, are calculated from the 2000 recorded values of Y and T.

[25] The variation over time of the distributions math formula and math formula reflects the different hydrodynamic conditions in the subsurface that derive from the unsteady discharge. In order to compare exchange depths and times in steady and unsteady conditions, it is useful to define the following time-averaged distributions in analogy with equation (13):

display math(14)

and

display math(15)

2.4. Characteristics of the Numerical Simulations

[26] We have performed a total of 15 numerical simulations that explore the influence of different hydrologic conditions on water exchange with the hyporheic zone. The simulations describe the dynamics of two idealized lowland streams, whose low streambed slopes, fine sediments, and relatively high mean discharge favor the development of dunes as the prevailing bedform type. All the statistical parameters that define streamflow hydrology are summarized in Table 1. For each simulation, a discharge time series has been generated with the previously described procedures for a total duration of 5000 days. This long duration ensures that the statistical parameters have stabilized on their correct value.

Table 1. Statistical Properties Adopted in the Numerical Simulations and Model Results About Water Exchange Flux
Sim.Qm (m3/s)Qb (m3/s)τQ (d)CVQΔ (d) math formulaCVq
14020100.2501.050.19
24020200.2501.060.19
34020300.2501.050.18
44020100.5001.190.30
54020200.5001.170.31
64020300.5001.150.28
74020100.2531.050.19
84020200.2531.050.19
94020300.2531.040.18
104020100.5031.180.31
114020200.5031.160.31
124020300.5031.140.29
1342100.5001.130.26
1442200.5001.140.27
1542300.5001.130.26

[27] The first group of simulations (1–12) is chosen to analyze the influence of streamflow variability on bed form-driven hyporheic exchange. The simulations consider a stream with constant width w = 50 m, bed slope math formula, and sediments with characteristic grain sizes math formula and math formula, hydraulic conductivity math formula, and porosity θ = 0.3. These simulations are characterized by the same values of mean discharge Qm = 40 m3/s and base flow math formula, while they consider different values for the other statistical parameters CVQ and τQ. Simulations 1–6 are based on the instantaneous equilibrium approach (Δ = 0), while the nonequilibrium approach has been adopted in simulations 7–12 (Δ = 3 days).

[28] The second group of simulations (13–15) represents the case of a smaller stream, with smaller width w = 10 m, steeper slope math formula, and coarser sediments ( math formula and math formula) with higher hydraulic conductivity math formula. The value of porosity (θ = 0.3) is the same as for the previous simulations. Simulations 13–15 are conceived as a comparison case to show the effect of mean discharge on water exchange. Hence, mean discharge and base flow are assigned lower values (Qm = 4 m3/s, Qb = 2 m3/s) compared to the first group of simulations, while for the other parameters (CVQ and τQ) we have applied the same values of simulations 4–6 (see Table 1) because of their high streamflow variability. The instantaneous equilibrium approach (Δ = 0) has been applied for all the simulations of the second group. It should be noted that the ratio math formula between base flow and mean discharge as well as values of correlation time τQ are the same as in the first group of simulations in order to make the two cases easier to compare.

[29] The influence of streamflow variability on hyporheic exchange can be better understood by comparing the simulated exchange process in unsteady flow conditions with the same process in a reference steady state. The chosen reference state corresponds to a constant discharge equal to the average streamflow value Qm. For simulations 1–12, the average flow conditions correspond to math formula, math formula, and math formula, with bedform size math formula and math formula and celerity math formula. It is important to note that the dimensionless celerity is quite high math formula, which means that dunes move with much faster velocities than pore water. This case hence represents a stream with high solid transport, and it has been chosen to investigate the influence of unsteady sediment transport on hyporheic exchange as it complements our previous analysis that considered a stationary streambed [Boano et al., 2010c]. The hyporheic timescale is math formula, a relatively small value that is an index of short residence times of stream water in the hyporheic zone. For simulations 13–15, the average flow conditions result in math formula, math formula, and math formula, with bedform size math formula and math formula and celerity math formula. Despite the higher celerity of dune migration, the dimensionless dune celerity math formula is lower than for the larger stream. However, since the value is higher than unity this case still represents a case in which dune migration is much faster than water velocity in the subsurface. The combination of lower bedform submergence math formula, shorter length of filtration (Lm), and higher permeability results in a much faster exchange (Tm = 10 min) than in the first group of simulations.

3. Results

3.1. Exchange Flux

[30] The flux has been calculated over the whole time series. The dynamics of the total exchange flux, q(t), is portrayed in Figure 1. The figure shows that the water exchange flux is inversely proportional to stream discharge, with more (less) stream water flushing the sediments during low (high) flow periods. This counterintuitive behavior is the opposite of the one obtained in previous studies with stationary bed forms [Boano et al., 2007, 2010c], and it is determined by the increase of dune length L(t) with discharge predicted by equation (5). When discharge increases, the volumetric rate of water exchange over a single dune increases due to higher head difference (equations (7) and (11)) and dune celerity (equations (6) and (12)). At the same time, the streambed area covered by each bed form increases because of the higher L(t). The net effect is a decrease in the volumetric flow rate per unit streambed area (i.e., the exchange flux) with increasing discharge. If observed separately, both the pumping and turnover flux exhibit the same behavior (data not shown).

[31] Daily values of pumping flux account for approximately 50%–60% (simulations 1–12, larger stream) and 65%–75% (simulations 13–15, smaller stream) of the daily total flux, while the remaining part is due to turnover exchange. The portion of total exchange flux due to pumping is lower for the larger stream than for the smaller one. These different contributions derive from the interplay between the celerity of dune migration and the seepage velocity of infiltrated water, whose ratio is quantified by the dimensionless celerity math formula. The result is a more prominent contribution of turnover to total water exchange in the larger stream math formula than in the smaller one math formula. In both cases, the considerable contribution of turnover to the total water exchange is a consequence of the celerity of dune migration being faster than the pumping-induced seepage velocity, as demonstrated by the high value of dimensionless dune celerity math formula.

[32] The average exchange flux for the case of the instantaneous equilibrium approach (simulations 1–6) is displayed in Figure 2a, which reports the ratio between the time-averaged value of the flux math formula and the corresponding value qm in the reference steady state. The figure shows that for each simulation the ratio math formula is very close to unity, which means that the average value of the exchange flux math formula is almost coincident with the steady state value qm regardless of the intensity (CVQ) and the frequency (τQ) of discharge variability. Similar results are shown in Figure 2b for the nonequilibrium approach (simulations 7–12). Values of math formula almost equal unity even if morphodynamic equilibrium between bed forms and streamflow is not attained. The time-averaged exchange flux is also very close to the steady state value for simulations 13–15 (see Table 1).

Figure 2.

Ratio between hyporheic exchange flux in unsteady math formula and steady (qm) flow conditions as a function of the coefficient of variability of streamflow (CVQ) for simulations (a) 1–6 (instantaneous equilibrium) and (b) 7–12 (nonequilibrium). The simulation numbers are reported over the bars. White, gray, and black bars denote simulations with τQ = 10, 20, and 30 days, respectively. math formula is the time-averaged value of the simulated unsteady flux, and qm represents the constant flux in the same stream with steady streamflow equal to the mean discharge Qm.

[33] The temporal variability of stream-sediment exchange is quantified by its coefficient of variation, CVq, which is shown in Figure 3a for the instantaneous equilibrium approach. As expected, the value of CVq increases with increasing CVQ, i.e., streams with highly dynamic flow conditions exhibit stronger temporal variations of the exchange flux. In contrast, the average time between floods, τQ, does not exert any relevant influence on CVq, as the residual variations are too small to be significant and are only due to statistical variations among the different simulations. The same behavior is obtained when the nonequilibrium approach is employed, as displayed in Figure 3b. It should be noted that CVq is always smaller than the corresponding CVQ, which means that the temporal fluctuations of exchange flux are milder than those of stream discharge.

Figure 3.

Coefficient of variation, CVq, of the hyporheic exchange flux in unsteady flow conditions for simulations (a) 1–6 (instantaneous equilibrium) and (b) 7–12 (nonequilibrium). The simulation numbers are reported over the bars. White, gray, and black bars denote simulations with τQ = 10, 20, and 30 days, respectively.

3.2. Comparison Between Steady and Unsteady Model Predictions

[34] Before considering the variability of daily predictions of stream water exchange, we analyze the differences between time-averaged distributions of exchange depths and residence times simulated with the proposed method and the corresponding distributions obtained with a steady state approach with math formula. These distributions are analyzed for the two streams represented by simulations 4–6 and 13–15 (Qm = 40 and 4 m3/s, respectively, see Table 1). The streams also differ by width, streambed permeability, and other properties whose values are controlled by discharge.

[35] The distributions of depths and residence times have been determined with the previously described particle tracking procedure. Since the unsteady particle tracking simulations are computationally expensive, the analysis of depths and residence times of the exchanged water particles has been performed for a subset of 1000 days from the original discharge time series, leading to 1000 distributions of exchange depths, math formula, and residence times, math formula. These distributions have then been averaged according to equations (14), (15) to derive the corresponding time-averaged distributions math formula and math formula for the unsteady flow.

[36] The distributions of exchange depths and residence times are shown in Figures 4 and 5. The comparison between the figures reveals that mean stream discharge has a quite different effect on exchange depths and times. Figures 4a–4c and 5a–5c differ by the value of correlation timescale of streamflow τQ (see Table 1). The similarities between the Figures 4a–4c and 5a–5c indicate that the influence of τQ is not relevant, and therefore the discussion is hereinafter focused on the role of Qm.

Figure 4.

Comparison between distributions of exchange depths, Y, for the larger stream (thin lines) and the smaller stream (thick lines). The time-averaged unsteady distribution (continuous lines) and the corresponding steady-state distribution (dashed lines) are shown for (a) simulations 4 and 13 (τQ = 10 d), (b) simulations 5 and 14 (τQ = 20 d), and (c) simulations 6 and 15 (τQ = 30 d).

Figure 5.

Comparison between distributions of residence times, T, for the larger stream (thin lines) and the smaller stream (thick lines). The time-averaged unsteady distribution (continuous lines) and the corresponding steady state distribution (dashed lines) are shown for (a) simulations 4 and 13 (τQ = 10 d), (b) simulations 5 and 14 (τQ = 20 d), and (c) simulations 6 and 15 (τQ = 30 d).

[37] Figure 4 shows that the maximum depths reached by exchanged water particles in unsteady flow (continuous lines) vary over more than two order of magnitude, with a median depth Y50 around 15 cm. If a steady state model with math formula is used, the predicted exchange depths (dashed lines) are slightly overestimated math formula and their distribution cover the same range of values as the unsteady ones. Even though the overestimation error represents a considerable fraction of the median depth, its magnitude is much smaller than the predicted range of exchange depths.

[38] Figure 4 also shows that the time-averaged distributions of exchange depths math formula (continuous lines) are very similar for the two streams despite the differences in mean discharge and in the related properties. The limited influence of mean discharge on exchange depths results from the interaction between average dune length Lm and dimensionless dune celerity math formula, both of which increase with Qm. In the larger stream, the presence of longer dunes would favor water penetration in the sediments because the size of flow cells is proportionally to Lm. However, this effect is balanced by the higher value of math formula associated with the higher discharge, because it induces a frequent reversal of vertical flow direction in the subsurface that eventually results in a confinement of the hyporheic flow cells and in a limitation of water penetration [Bottacin-Busolin and Marion, 2010]. The combined action of these two opposing elements has an almost negligible influence on exchange depth.

[39] Residence time distributions are displayed in Figure 5 for both unsteady and steady flow. For the unsteady flow, residence times are considerably longer in the larger stream (T50 = 5 h, thin continuous line) than in the smaller one (T50 = 0 17, h ≈ 10 min, thick continuous line). This difference is caused by the higher filtration velocities in the case of lower Qm discussed in section 2.4, resulting in large differences in contact times between exchanged stream water and streambed sediments in spite of the very similar depths of exchange.

[40] A similar behavior is reported for the steady state distributions (dashed lines), which extend over ranges of residence times that are similar to those of the corresponding time-averaged unsteady distributions. The simulated median residence times in the steady cases are 0.24 and 7.7 h for Qm = 4 and 40 m3/s, respectively. Thus, median residence times in unsteady flow are somewhat overestimated by the steady state model. However, the overestimation error is much smaller than the differences in T50 induced by the different values of Qm and also than the overall width of the ranges of residence times shown in Figure 5.

3.3. Exchange Depth Distributions

[41] The simulated results about the depth of stream water penetration in the hyporheic sediments are presented in Figures 6 (instantaneous equilibrium approach) and 7 (nonequilibrium approach). Modeled exchange depths, Y*, are displayed in dimensionless form in order to make the comparison more general. Differences between the curves in Figures 6 and 7 are hardly noticeable, confirming that hysteresis processes in dune growth and evolution do not have any significant influence on the depth of stream water exchange. Median exchange depths are limited ( math formula, which corresponds to about 1%–2% of the dune wavelength Lm) because the fast bedform migration leads to frequent reversals of vertical velocity in the subsurface. As previously discussed, these changes in flow direction reduce the coherence of vertical velocity, thus hampering the penetration of stream water in the sediments [Elliott and Brooks, 1997; Bottacin-Busolin and Marion, 2010].

Figure 6.

Distributions of dimensionless exchange depth, Y*, for simulations 1–6 obtained with the instantaneous equilibrium approach. The time-averaged unsteady distribution (continuous lines) and the corresponding steady state one (dashed lines) are shown together with the 1000 values of the quantiles math formula, math formula, math formula of the individual distributions (gray lines, box: interquartile range, whiskers: extreme values).

Figure 7.

Distributions of dimensionless exchange depth, Y*, for simulations 7–12 obtained with the nonequilibrium approach. The time-averaged unsteady distribution (continuous lines) and the corresponding steady state one (dashed lines) are shown together with the 1000 values of the quantiles math formula, math formula, math formula of the individual distributions (gray lines, box: interquartile range, whiskers: extreme values).

[42] If the time-averaged distribution math formula is considered, Figures 6 and 7 show that this distribution (continuous line) is quite similar to the one relative to steady state flow (dashed line). This similarity is especially evident for lower Q variability (CVQ = 0.25, simulations 1–3 and 7–9), while larger differences can be observed when discharge is more dynamic (CVQ = 0.5, simulations 4–6 and 10–12). Interestingly, the time averaged exchange depths are systematically smaller than steady state ones. Despite these differences, the steady state distributions still provide a good approximation of the average exchange depths in unsteady conditions.

[43] The temporal variability of the exchange depths can be quantified by the 1000 values of the quantiles math formula, math formula, math formula of the individual distributions math formula, shown by the box-and-whisker plots in Figures 6 and 7. The three quantiles have been chosen as representative of shallower, median, and deeper exchange, respectively, and they tend to be higher during high flow periods. In our simulations, the range of variation of each quantile is narrower than the overall range of exchange depths described by the average distribution math formula. Hence, the hyporheic zone expands and contracts in response to flood events, and these variations tend to be smaller than their average value. However, hyporheic depths are obviously more variable in more dynamic streams with higher CVQ, and very large variations can be expected in streams with CVQ > 1. It should be stressed that values of the quantiles are not evenly distributed over their range of variation. Instead, values tend to be grouped around the time-averaged value, while the more extreme values occur with lower frequencies, i.e., only during major flood or drought events.

3.4. Residence Time Distributions

[44] The distributions of dimensionless residence times T* in unsteady conditions are compared to the corresponding steady state ones in Figures 8 and 9. The comparison between the two Figures demonstrates that morphodynamic equilibrium between streambed and discharge has a minor influence on residence times, confirming the behavior already discussed for the exchange depth. Steady state distributions of residence times represent a first-order estimate of the time averaged distributions math formula, an approximation that is less precise than for the exchange depths Y* but still considered acceptable for a first estimate of the contact times between stream water and hyporheic sediments. The magnitude of the deviation increases with discharge variability CVQ.

Figure 8.

Distributions of dimensionless residence times, T*, for simulations 1–6 obtained with the instantaneous equilibrium approach. The time-averaged unsteady distribution (continuous lines) and the corresponding steady state one (dashed lines) are shown together with the 1000 values of the quantiles math formula, math formula, math formula of the individual distributions (gray lines, box: interquartile range, whiskers: extreme values).

Figure 9.

Distributions of dimensionless residence times, T*, for simulations 7–12 obtained with the nonequilibrium approach. The time-averaged unsteady distribution (continuous lines) and the corresponding steady state one (dashed lines) are shown together with the 1000 values of the quantiles math formula, math formula, math formula of the individual distributions (gray lines, box: interquartile range, whiskers: extreme values).

[45] Another difference between the distributions of residence times and those of exchange depths is that the former exhibit larger temporal variations compared to the latter. The ranges of the quantiles math formula, math formula, and math formula (box-and-whisker plots) overlap significantly with each other, which implies that sediment regions where water is characterized by short residence times are occasionally flushed by older stream water with much longer residence times. Since residence times may be associated to concentrations of dissolved oxygen and other chemicals [Bardini et al., 2012; Boano et al., 2010b; Marzadri et al., 2011], this result points out that hyporheic exchange may control temporal variations in subsurface water chemistry. However, it should be kept in mind that quantiles of residence times are frequently closer to their mean value than to the extremes of the ranges shown in Figures 8 and 9, and hence these temporal variations of subsurface water chemistry may only be occasional. For all quantiles, our simulations predicted higher values of residence times with increasing streamflow.

4. Discussion and Conclusions

[46] The proposed stochastic method has been applied to analyze the temporal dynamics of hyporheic exchange in a stream characterized by subcritical flow with unsteady discharge. The performed numerical simulations have been specifically chosen to enhance the effect of streambed modification induced by unsteady sediment transport. Comparison with previous approaches [Boano et al., 2007, 2010c] has provided insights on the role of morphodynamic adaptation of bedform size and celerity to changing streamflow as well as on the influence of unsteady turnover. Our modeling results indicate that water exchange is minimally affected by the rate at which the streambed attains morphodynamic equilibrium with discharge.

[47] In general, our findings suggest that the mean stream discharge is a good indicator of the average values of unsteady exchange properties. Steady state simulations with mean discharge have provided accurate predictions of the exchange flux and of the hyporheic zone depth, while the times spent by stream water in the hyporheic sediments are more sensitive to fluctuations in flow conditions and have been quantified only as order-of-magnitude estimates. Given that the implementation of unsteady approaches requires considerable computation efforts, the possibility to obtain an approximate but reasonable description of exchange processes with steady state methods certainly represents a great advantage for future modeling studies. Even though our method does not consider variations at the seasonal scale of streamflow as well as of other process (e.g., groundwater dynamics, bioclogging), the results suggest that hyporheic exchange processes should adapt themselves to the seasonal mean discharge as a consequence of these low-frequency discharge fluctuations.

[48] The major difference with the case of stationary bed forms is that the hyporheic exchange flux decreases with increasing discharge because the increase in the volumetric rate of exchanged water is compensated by the larger streambed area for a single dune. Yet our simulations have neglected increases in stream width at higher flow, and it is possible that in real streams the increase in wetted area with discharge may provide additional streambed area for water exchange, thus partially compensating or overcompensating the reduction of exchange flux.

[49] The analysis of the temporal dynamics of the exchange flux has shown that changes in bedform size result in stronger fluctuations of exchange compared to the case of stationary bed forms. Our analysis has also revealed that the intensity of this temporal variability is strongly influenced by the relative magnitude of discharge variations (CVQ), while the correlation time (τQ) plays a negligible role on water exchange.

[50] The depth of stream water penetration and its residence time in the sediments are usually higher during high streamflow periods. In particular, exchange depths exhibit moderate variations during floods, although occasional flushing of deeper sediment zones may occur as a result of extreme flood events. Residence times are more variable, but their fluctuations are comparable to the already wide range of exchange timescales in steady state conditions. The range of residence times in steady state would be even wider in streams with lower celerity of dune migration due to sediment transport. We hence expect that temporal variations of residence times would be relatively less important in these streams. It is finally worth noting that the predicted framework of deeper exchange and longer residence time during high flow is coherent with observed patterns of surface-subsurface exchange [Malcolm et al., 2006; Kaser et al., 2009], even though fluctuations of the groundwater table (not considered in the present study) may also contribute to the short- and long-term dynamics of hyporheic exchange by enhancing bank storage and by hampering the penetration of stream water in the sediments. Future modeling studies aimed to describe the influence of large-scale subsurface flow due to groundwater table dynamics on exchange flow induced by unsteady stream discharge will undoubtedly advance our understanding of hyporheic zone processes.

Acknowledgments

[51] This research has been partially funded by the Italian Ministry of University and Research (MIUR—PRIN 2008A7EBA3).