The modeling approach and dimensionless parameterization used by Gomez-Velez et al. (2017) can represent most riverbank–aquifer situations and dynamic flood conditions. In our study, we use their conceptual model to set up a baseline case with the same model frame, equations and parameterization metrics. Additional information regarding the implementation of this baseline case can be found in the Supplement. However, where their previous research assumed a vertical riverbank for sinuosity-driven HEF, we consider a sloping riverbank and use the deformed geometry method (DGM) approach to capture the dynamic progression of the sediment–water interface (SWI) along the river course. A constant sloping angle (δ, °) along the alluvial riverbank of a sinusoidal river was implemented in our model (see blue lines of conceptual model in Fig. S1 in the Supplement and the corresponding mathematical model in Fig. S2a), while the SWI was assumed to be always vertical (vertical solid red and green lines in Fig. S2c). As such, the contraction or expansion of the simulated domain, i.e., displacement of the SWI, can be characterized by the sloping angle (there is no movement of the SWI for the vertical riverbank case) and river stage. As the river stage changes, so does the location of the SWI.

When the river stage changes in our model, the sinusoidal boundary will migrate towards or away from the floodplain, meaning that the submerged part of the riverbank is considered contracted, and our model only considers the alluvial aquifer that is not submerged. The changes of the SWI during a flood event can be calculated by considering the river stage and bank slope via 1Y(x,t)=Y0(x)+M(t), where Y(x, t) [L] is the location of the SWI boundary, while Y0(x) [L] is the initial location of the SWI. M(t)=[h(t)-h(0)]/tan(δ) is the displacement of the SWI in the y direction due to river stage fluctuation and the bank slope angle (see the horizontal distance between the vertical solid red and green line in Fig. S2c). In contrast to the vertical riverbank models of Gomez-Velez et al. (2017), M(t) is added in Eq. (1) to simulate sloping riverbank conditions.

To simulate the model domain deformation and mesh displacement, we use the DGM interface in COMSOL Multiphysics (COMSOL) (COMSOL Multiphysics, 1998). In this interface, the deforming feature of a specified domain can be defined as a boundary condition with a given moving velocity or displacement. DGM is based on the arbitrary Lagrangian–Eulerian (ALE) method, which is a hybrid method that allows both the model domain and mesh to move or deform simultaneously in a predefined manner. More details on ALE can be found in Donea et al. (2004). While it has previously been used for simulating general free-surface problems (e.g., Duarte et al., 2004; Maury, 1996; Pohjoranta and Tenno, 2011), to our knowledge, DGM has not yet been implemented to solve moving boundary problems in hyporheic exchange studies. Here we used Eq. (1) as an input to the DGM interface to simulate the displacement of the SWI (water flow) during a dynamic flood event. Infiltration and seepage face before and after the peak time of the flood event, respectively, were neglected (Boano et al., 2006; Cardenas, 2009a, b; Kruegler et al., 2020). Figure 1 illustrates the river stage hydrograph of this study (Fig. 1a, calculated by Eq. (S2) in the Supplement, where t∗=t/td, td is the duration of flood event) and the diagram of the displacement of the SWI (Fig. 1b) during the flood event after coupling DGM into the model. The colored river boundaries in Fig. 1b are corresponding to the times of colored dots in Fig. 1a. Additionally, solute transport and mean RTD were simulated based on the extent of the flow field according to Gomez-Velez et al. (2017), as shown in the Supplement (Figs. S2 and S3, respectively).

Hydraulic conditions used in our numerical modeling study are based on values from Gomez-Velez et al. (2017), who conducted a Monte Carlo analysis. They found that the dynamic variations of HEF and mean RTD are mainly determined by ambient groundwater flow and the ratio of aquifer hydraulic conductivity to the duration of the flood event, referred to as dimensionless constant Γd=Syλ20.5K(1+n0)H0td, (see Table 1 and Fig. S2), where Sy is the specific yield [–], λ is the wavelength of the sinuous river, K is the hydraulic conductivity [LT-1], n0 is the intensity of the flood event [–], H0 is the base river stage [L] and td is the duration of the flood event [T]).

After setting up the baseline model case with a vertical riverbank (δ = 90°), we compared our model results for that case with those obtained by Gomez-Velez et al. (2017) for (a) net HEF represented by Qnet, HZ∗(t); (b) area of HZ, A∗∗(t); and (c) penetration of the HZ, d∗(t) for Γd = 0.1, 1, 10 and 100, and we found that our model simulated those cases with high accuracy (Fig. 2). Parameters A∗∗(t) andd∗(t) are based on modeling the transport of a conservative solute while Qnet, HZ∗(t) is based on modeling water flow. Slight differences between our model and that of Gomez-Velez et al. (2017) might be due to the use of a much more refined mesh in this study as well as different length scales.

To test whether our assumption of considering a vertical SWI and using the DGM to characterize the migration of the SWI was appropriate, we compared the vertical 2-D model with a 1-D model coupled with the DGM. Detailed information on this comparison as well as validation results is provided in Sect. S4 in the Supplement. The results show that our approach is reasonable when simulating HEF in a sloping riverbank aquifer.

We then considered a series of riverbank scenarios where the bank slope angle was varied, ranging from δ = 90° (vertical riverbank) to 10° (nearly horizontal case), and Γd values ranged from 0.1 to 100, corresponding to aquifer hydraulic conductivity ranging from 480 to 0.048 md-1, indicating high to low transmissivity. Table 1 presents the parameters used in our numerical modeling study. The finite-element models proposed in this study were set up using COMSOL software. Equations (S1), (S3) and (S6) in the Supplement were implemented using a customized partial differential equation (PDE) interface to include the Boussinesq equation, vertical integrated solute transport equation and equation for calculating residence (travel) time distributions (RTDs), respectively. The model domain was discretized into about 0.5 million variably sized triangular elements, with refinement imposed near the river boundary. Mesh-independent numerical solutions are achieved by limiting grid size (ΔL) to less than 0.2 m. Thus, the transverse and longitudinal Péclet numbers (calculated by Pe=ΔL/αL and Pe=ΔL/αT, respectively) in both advection- and diffusion-dominated zones are less than 1, which is smaller than the upper limit of Pe = 4 to effectively avoid numerical oscillations and instabilities.

Similar to Gomez-Velez et al. (2017), we evaluated the impact of bank slope by comparing the net hyporheic exchange flux (Qnet, HZ∗(t)), area of HZ (A∗∗(t)), penetration distance of the HZ (d∗∗(t)) and mean RTD (μr∗(x,t)) between vertical and sloping riverbank models. A detailed definition of these variables is provided in Sect. S5 in the Supplement.

The progression of spatiotemporal patterns of mean RTD (i.e., travel time of river water in aquifer) is a useful evaluation method for identifying the dynamic variation of aging and rejuvenation of hyporheic water. Here we use the mean RT ratio between a sloping model and a vertical model μr∗(x,t) = log10(μτ-S(x,t)/μτ-V(x,0)) to evaluate the influence of bank slope on the prediction of mean RTD for a given location and time. Figure 8 presents mean RTDs for the initial condition, where μτ0-max⁡ is the maximum RT in the domain. It can be seen that the isolines representing the RT are almost horizontal in the area extending from the river, but RT near the upstream river bend is smaller than downstream because the initial flow direction is towards the negative direction of the x axis. Notably, μ(x,0) grows almost exponentially as y increases, and a positive correlation to Γd at a given location is observed.

Figures 9–12 present five snapshots of μr∗ for different bank slope angles and different aquifer transmissivity values (Γd = 0.1, 1, 10 and 100, respectively). The five snapshots represent the rising limb of the flood event (t∗=0.1), the peak of the flood event (t∗=0.25), the falling limb of the flood event (t∗=0.5) and a time after the flood event (t∗=1, 2.5 and 10). The differences in residence time between sloping and vertical riverbank models are within 12.2 % in the white-colored areas (-0.05 < μr∗ < 0.05) of Figs. 9–12, which indicates a minor effect of bank slope on mean RTD. The colored areas in Figs. 9–12 indicate model results where neglecting the bank slope will lead to an overestimation (μr∗ < -0.05) or underestimation (μr∗ > 0.05) of residence (travel) times.

At t∗=0.1, a smaller bank slope can lead to a shorter travel time of river water in the aquifer (negative values of μr∗) near the SWI compared to the vertical riverbank scenario. The area of shorter travel time caused by bank slope was positively related to aquifer transmissivity. The effect of bank slope is small for Γd = 10 and 100 because the groundwater mound (the raised groundwater stage) piles up around the river boundary, but that small area extended deeper into the alluvial valley for smaller slope angles. Due to the scattered and nested flow paths near the cut bank and point bar, respectively, the area of the negative value of μr∗ at the cut bank of the SWI is larger than that at the point bar. The change of flow direction near the point bar leads to a prolonged flow path for the water in the river channel as well as to forced groundwater mixing with the slightly older water (Fig. 8 shows that the water was potentially older in the y direction compared to the -x direction in the point bar). This effect was amplified with decreasing the bank slope angle, but it is only statistically significant (μr∗ < -0.05 or μr∗ > 0.05) when δ = 10° at t∗=0.1.

At the time of peak flood (t∗=0.25), the river still infiltrates into the aquifer. For Γd = 0.1, results of μr∗ in Figs. 9 show that the bank slope can lead to both overestimated and underestimated RT areas. Both the magnitude of relative RT (μr∗) and the associated area increase with a decreasing slope due to the longer travel distance of river water into the aquifer. As the slope angle decreases, the underestimated travel time area was located closer to the peak of the cutbank. The impact of bank slope on mean RTD for Γd = 1 was rather similar in its pattern compared to Γd = 0.1, but the degree of that impact was reduced. For Γd = 10 and 100, only overestimated travel time area can be seen near the riverbank with a smaller area of impact compared to smaller Γd conditions because the groundwater has not sufficiently propagated into the aquifer due to lower transmissivity.

At t∗=0.5, part of the aquifer that was submerged at t∗=0.25 reemerges due to the decline in river stage. In most cases, smaller bank slopes can lead to wider reemergence of the aquifer, which therefore results in overestimated travel time area near the river boundary; however, this was not the case for Γd = 0.1, where the bank slope can lead to both an overestimated and an underestimated RT area. Furthermore, compared to when t∗=0.25, the impact of bank slope becomes weaker for Γd = 0.1 but more relevant for the larger Γd values.

After the flood event (t∗ > 1), the influence of bank slope on mean travel time is nearly eliminated for Γd = 0.1 and 1 due to the high aquifer transmissivity. However, for aquifers with lower transmissivity (Γd = 10 and 100), bank slope still has a significant effect on RT at t∗=10 and leads to underestimated and overestimated RT areas near the point bar and the cut bank, respectively.

Overall, Figs. 9–12 indicate that the time when bank slope was relevant in predicting RT (travel time of groundwater in aquifer) was determined by the transmissivity of the aquifer. For more transmissive aquifers, the impact of bank slope on the prediction of groundwater travel time cannot be neglected during the flood event (0 < t < td), but that impact will be eliminated after the flood event due to the quick recovery of the aquifer to baseline conditions. For lower-transmissivity aquifers, the bank slope plays an important role in groundwater travel time after t > 0.5∗td and has a more lasting influence on aquifer RT, as more time is required to recover to initial conditions.

Figures 13 shows the progression of the flux-weighted relative RT μout∗(x,t) = n⋅Qout∗(x,t)log10(μτ(x,t)/μτ(x,0)) for different slopes and aquifer transmissivities. Mout∗(x,t) represents the difference in flux-weighted RT of the water discharged into the river compared to the initial condition. At the start of the flood event, there is no μout∗ as river water infiltrates the aquifer. Following the decline in river stage, the aquifer begins to discharge the mixed water with different RT back into the river (see Fig. 4c).

For vertical riverbank conditions (δ = 90°, top row in Fig. 13), upstream (0.5λ < x < λ) and downstream (0 < x < 0.5λ) boundaries of the meander bend discharge older and younger water, respectively. The rejuvenated or aged waters that represent shorter and longer travel times compared to the baseline condition, respectively, were mostly discharged before the flood event (t∗ < 1) due to the greater outflux as shown in Fig. 3a. It can also be seen that water was aged along the upstream bend compared to the more rejuvenated water along the downstream bend. After the flood event, μout∗ gradually disappears along the upstream meander (blank areas) for Γd = 0.1 and 1 because the flow fields were recovering to baseline conditions. Therefore, the upstream meander gradually becomes the inflow boundary.

For cases with lower values of Γd (left columns in Fig. 13), μout∗ reaches equilibrium earlier compared to cases with higher Γd as an increasing bank slope angle causes μout∗ to gradually decrease the travel time of the outflowing water during the flood event. For larger Γd, μout∗ was totally dominated by rejuvenated water during the flood event. Furthermore, smaller bank slope angles can both extend the time that younger water is discharging along the downstream meander and increase the difference in residence times of these younger waters between sloping and vertical conditions.

Tilted riverbanks are common in nature and caused by erosion and bank collapse, as has been observed at multiple scales (Zingg, 1940). Previous studies have shown that bank erosion is stronger where the river planimetry is more sinuous, the river stage varies more frequently, or where the riverbank has larger sloping angles, ultimately leading to a flatter bank (Zingg. 1940; Haggerty and Gorelick, 1995; Mayor et al., 2008; Puttock et al., 2013). Yet, the impact of riverbank geometry and in particular the bank slope on sinuosity-driven lateral hyporheic exchange was ignored in most previous studies. Our results clearly indicate that HZ characteristics (HEF, area and penetration distance of HZ into the alluvial valley) can be underestimated along a meandering river depending on bank slope conditions.

We show that not accounting for the bank slope and river sinuosity can lead to an underestimation of the infiltration rate of water from the river to the alluvial aquifer (by up to 120 %), as well as the area and penetration distance. This effect is more pronounced for smaller bank slope angles (Fig. 5), which can be more likely found in lowland streams (Laubel et al., 2003), especially in areas with extensive cattle grazing at the streamside (Trimble, 1994).

Doble et al. (2012b), Siergieiev et al. (2015) and Liang et al. (2018) assessed the influence of the bank slope on HEF using a vertical cross-sectional profile. Siergieiev et al. (2015) found that the impact of bank slope on HEF was proportional to the hydraulic conductivity of the aquifer. However, we argue here that the bank slope is more relevant in rivers connected to aquifers with low hydraulic transmissivity (high hydraulic conductivity or low specific yield). Furthermore, we show (Fig. 14 as example) that using only one cross-sectional river profile perpendicular to the river axis does not capture the effect of river sinuosity on HEF as bank storage decreases from point bar to cut bank. This indicates that the accuracy of bank storage estimates can be improved by including river sinuosity, which has often been omitted in the past. In a meandering river with variable bank slope, river geometry thus has a sizable effect on bank storage progression and HEF and should be included in any scenarios.

Under rather stable flow conditions, hyporheic exchange rate and river sinuosity control the biogeochemical zonation (RTD) in the HZ. A higher hyporheic exchange rate (caused, e.g., by a larger hydraulic conductivity in the aquifer or a more sinuous meander) will reduce the mean RTD, promoting biogeochemical reactions (Boano et al., 2010a; Gomez-Velez et al., 2012). However, for a transient flood event, the mean RTD could be both extended and reduced depending on the location with respect to the meander, due to variations in the complex flow paths (Gomez-Velez et al., 2017). Our results indicate that smaller bank slope angles could not only increase HEF and thus lead to increased transport of oxygen and nutrient rich stream water into the aquifer but also alter the location and the residence time of this water within the aquifer system.

Mean RTDs of river water in the alluvial aquifer have been used to evaluate the potential for biogeochemical reactions by comparing the RT with biogeochemical timescales (BTSs) for given solutes (Boano et al., 2010b; Gomez-Velez et al., 2012). Locations where the ratio of RT to BTS (expressed by Damköhler numbers) is small indicate a high reaction potential for that specific chemical species (Gomez-Velez et al., 2015; Pinay et al., 2015). It has been documented that the BTS for dissolved organic matter (DOC) can vary over 10 orders of magnitude (10-1–109 d) (Hunter et al., 1998), while the BTS for oxygen and nitrate has been found to vary over 8 orders of magnitude (10-2–106 d) (Gomez-Velez et al., 2012). Here, we compare mean RTD within the overlapping ranges of these two BTS for vertical and sloping riverbank conditions (δ = 10°) at the peak time of the flood event (t∗ = 0.25) for different aquifer transmissivity conditions and show the zonation of residence times using a BTS range of 10-1–106 d (Fig. 15).

Figure 15 indicates that neglecting the bank slope impacts the prediction of reaction potentials during hyporheic exchange processes, especially for locations with short timescales. The reaction hot spots (areas indicated by the overlapping BTS ranges) for sloping riverbank conditions expanded further into the aquifer compared with the vertical bank conditions, similar to the overestimated areas in Figs. 9 to 12. Note that we did not aim to include specific reaction models in our study but instead used mean RTD as an indicator for various biogeochemical reactions in the aquifer. Furthermore, the wavelength of the river sinusoid in Fig. 15 was λ = 40 m to offer a representative riverbank–aquifer condition. The zonation of BTSs for larger and smaller river sinusoid wavelengths will be reduced towards the river boundary or further expanded into the aquifer, respectively, for both sloping and vertical riverbank conditions. Although the dimensional BTSs for various spatial scales are not shown here, similar patterns between Figs. 9–12 and 15 imply the usability of the mean RTD results (Figs. 9–12) to infer potential biogeochemical reactions.

The impact of the bank slope on RT is basically controlled by aquifer transmissivity. For higher aquifer transmissivity conditions, the impact of the bank slope appears to be more pronounced when the river stage rises during a flood event. For lower aquifer transmissivity conditions, the bank slope seems more relevant for mean RTD after the flood event, and its impact is more long-lasting. Smaller bank slope angles could extend (near the point bar) or reduce (near the cut bank) pore water travel times throughout the flood event, compared to the non-sloping (vertical) riverbank condition. This indicates that compared with the vertical riverbank condition, point bars with bank slopes are more favorable for removing dissolved organic carbon and for nitrification, while cut banks with bank slope may have adverse effects on the groundwater quality near rivers. The vertical profile modeling study of Derx et al. (2014) suggested that for riverbank restoration projects, increasing HEF by reducing the slope angle may have a negative effect on restoration. The mean RTD results of this study also suggest that the impact of bank slope on groundwater quality is determined by the location with respect to the meander (near the point bar or cut bank). As such, our analysis of mean RTDs can provide valuable information on whether and where riverbank slope can induce biogeochemical hotspots and hot moments and help guide choices to be made in biogeochemical field surveys regarding location and sampling time under dynamic river stage conditions, especially when the connected aquifers have low hydraulic transmissivity.

In this study, we propose a parsimonious reduced-order, idealized horizontal 2-D model that simplifies the variation of the river–aquifer interface using the moving boundary method to depict the displacement of the SWI along a sloping riverbank. An advantage of this approach is reduced model complexity as compared to a three-dimensional model, which greatly reduces time and data requirements during model building and computational demand during the simulation of HEF and especially residence time distributions. Thus, our reduced-order model acts as a first step to gaining insight into the patterns of hyporheic exchange, riverbank storage and mean RTD in settings with more complex riverbank morphology and dynamic forcing. Future efforts should be focused on optimizing the computational method applied here and on including more detailed morphology and hydrodynamic characteristics.

In our simulations we assume a constant bank slope angle along the entire meandering river, while natural riverbanks often change their slope angle from reach to reach as well as with time. This variability could lead to more complex SWI travel distances and residence time distributions and new conceptualizations that account for the contribution of bank slope on time-varying RTD and HZ extent are needed. In our simulations we tested the model using a range of aquifer hydraulic conductivities. Although hydraulic conductivity (or transmissivity) is a critical parameter in the quantification of exchange fluxes and RTD between the two systems under varying slope conditions, other parameters such as valley water head fluctuation; water abstraction, e.g., for agriculture or drinking water supply; peak flood event characteristics; or larger-scale groundwater head fluctuation, e.g., due to changing groundwater recharge in the context of changing rainfall patterns, have not been considered here but might also impact HZ extent, RTD and river–aquifer exchange flux. For example, valley water head fluctuation and water abstraction in the aquifer will lead to a lower groundwater table, increasing the hydraulic gradient between river and aquifer. This will lead to the formation of a larger HZ area as well as longer travel distances and times of river water in the aquifer. Thus, reducing the slope of the riverbank could reduce the infiltration of polluted river water into the riparian aquifer.

The current study assumes a perennial stream and unconfined (phreatic) conditions in the connected aquifer as well as changing hydraulic gradients, leading to gaining and losing conditions in the river. Where there is no hydraulic gradient between the river and aquifer, no large-scale infiltration of river water into the riverbanks will occur, while local turbulent flow (e.g., due to obstacles in the river channel) might lead to localized infiltration over short distances and short timescales (Sawyer et. al., 2011; Stonedahl et al., 2013; Käser et al., 2013). Where the unconfined layer is small (e.g., in mountainous headwater streams with a rather small sediment layer overlying a hard-rock aquifer with relatively low hydraulic conductivity), the HZ is limited in its maximum extent, and travel times and distances are considerably shorter. However, in mountainous settings, slope angles are often much steeper due to erosion (here rivers incising into the bedrock), and further simulations are required to better understand the feedback between the bank slope angle, hydraulic gradient and maximum extent of the unconfined layer, allowing for hyporheic exchange processes. These simulations will also help us better understand the impact of the bank slope on quantitative and qualitative water supply to abstraction wells, e.g., used for the production of drinking water.

While using the Boussinesq equation neglects the influence of the vadose zone, this approach as well as the assumption of a vertically integrated distribution of hydraulic head has been widely used in the literature and proven adequate when simulating sinuosity-driven HEF patterns (Boano et al., 2006, 2010a; Cardenas, 2008, 2009a, b; Gomez-Velez et al., 2012, 2017; Kruegler et al., 2020). While we found differences in HEF patterns when comparing simple models using the Boussinesq with those using Richards' equation (Eq. S4 in the Supplement), these differences exist independent of using the DGM. However, we recommend in future studies to more systematically consider these two different approaches with respect to their advantages and limitations, e.g., in terms of computability or efficiency in predicting HEF under various conditions. While in an ideal scenario a 3-D modeling approach includes the vadose zone and the riverbank slope angle (both variable in time and space), at the moment, the implementation of such detailed models in practice suffers from limited computing capabilities.