## 1. Introduction

Hyporheic exchange—the exchange of stream and shallow subsurface water—is controlled by pressure gradients along the streambed surface and subsurface groundwater gradients. Over multiple scales, the bedform induced hyporheic exchange was identified as a crucial process for the biogeochemistry and ecology of rivers [

1,

2,

3,

4,

5,

6,

7,

8,

9,

10]. On large and intermediate scales, stream stage differences, meander loops or bars can generate hyporheic exchange. Accordingly, it is possible to control surface water-groundwater exchange by river stage manipulation e.g., to manage the inflow of saline groundwater into a river [

11]. A decrease of the groundwater level, in turn, impacts surface water infiltration up to a maximum where groundwater and surface water are disconnected. This condition is achieved when the clogging layer does not cross the top of the capillary zone above the water table [

12]. On small scales, river sediments usually form topographic features such as dunes or ripples. The flowing fluid encounters an uneven surface on the permeable streambed, which results in an irregular pattern in the pressure along that surface and induces hyporheic exchange [

11,

12,

13].

Within theoretical, experimental, and computational studies the general mechanics of the bedform induced hyporheic exchange were examined over the past decades. By manipulating streambed morphology, stream discharge, and groundwater flow, experiments have been used to study driving forces for the hyporheic exchange intensively [

14,

15,

16,

17]. At submerged structures such as pool-riffle sequences or ripples, turbulences, eddies or hydraulic jumps may occur. Packman et al. [

15], Tonina and Buffington [

18], Voermans et al. [

19] and other studies showed, that turbulence influences hyporheic exchange and should not be ignored. Facing these complex three-dimensional flow dynamics at the sediment-water interface, it can be challenging to establish suitable flume experiments or field studies. Computational fluid dynamics has proven to be a viable alternative. The majority of these studies have focused on surface-subsurface coupled models. Reasons for the application of different models for the surface and the subsurface are for example the strong temporal variability in streams including relatively high velocities, whereas the velocities and temporal variabilities in the groundwater are usually several orders of magnitude smaller, leading to different applied equations for the stream and the subsurface. Often, the two computational domains are linked by pressure. Pressure distributions from a surface water model are consequently used for a coupled groundwater model [

20,

21,

22,

23,

24,

25,

26]. However, also fully coupled models such as the Integrated Hydrology Model [

27] or HydroGeoSphere have already been successfully applied [

28,

29,

30]. Within these models, open channel flow is described by the two-dimensional diffusion-wave approximation of the St. Venant equations, whereas the three-dimensional Richards equation is used for the subsurface. Water and solute exchange flux terms enable to simultaneously solve one system of equations for both flow regimes.

For many coupled surface-subsurface models, the Darcy law is applied within the sediment. However, especially for coarse bed rivers, this law may cause errors in the presence of non-Darcy hyporheic flow [

15]. Following Bear [

31], the linear assumption of the Darcy law is only valid if the Reynolds number does not exceed a value between 1 and 10. Applying Darcy’s law in non-Darcy-flow areas leads to an overestimating of groundwater flow rates [

32]. Packman et al. [

15] investigated hyporheic exchange through gravel beds with dune-like morphologies and applied the modified Elliot and Brooks model [

33]. They realized that the model did not perform well—among other reasons—due to non-Darcy flow in the near-surface sediment which was not considered in the model. One possible solution to model groundwater in non-Darcy-flow areas is e.g., to use the Darcy-Brinkmann equation instead of the Darcy law. However, there is an additional parameter—the effective viscosity—which has to be determined.

In the present study, an extended version of the three-dimensional Navier–Stokes equations after Oxtoby et al. [

34] is used for the whole system comprising the stream as well as the subsurface. For the application in the groundwater, sediment porosity, as well as an additional drag term, are included into the Navier–Stokes equations. The model is consequently also applicable for high Reynolds numbers within the subsurface where the Darcy law cannot be applied. To our knowledge, this solver was never used for the hyporheic zone before. We apply the new integral solver to evaluate the effect of ripple geometries and surface hydraulics on hyporheic exchange processes, based on the study by Broecker et al. [

35] who investigated free surface flow and tracer retention over streambeds and ripples without considering the subsurface. In Broecker et al. [

35] the three-dimensional Navier–Stokes equations were solved in combination with an implemented transport equation. In that study, ripple sizes, spacing as well as flow velocities affected pressure gradients and tracer retention considerably. Seven simulation cases were examined varying ripple height, length, distance, and flow rate. The investigated ripple geometries and flow rates are mainly transferred to the present study. Only case 6 is not used for the present study, as the irregular distance between the ripples gave no significant new findings compared to equal distances [

35]. In contrast to Broecker et al. [

35], the present study examines both free surface flow and subsurface flow. The aim of the present study is to evaluate the impact of ripple dimensions, lengths, spacing and surface velocity on flow dynamics within the hyporheic zone using a new integral model.

## 4. Conclusions

CFD simulations were designed to simultaneously examine both surface and subsurface flow processes with an extended version of the three-dimensional Navier–Stokes equations. Based on two simulations for seepages through dams, it was shown that the applied model can describe the interaction of groundwater and surface water. The validated CFD model was applied to investigate the impact of bed form structures, grain sizes and surface flow discharges on hyporheic exchange processes. The examined ripple structures changed the streambed pressure and created in- and outflowing fluxes at the interface which were calculated for each case study for a representative ripple in the middle and play a significant role in biogeochemical processes within the hyporheic zone. It was shown that not only the surface water influences the flow within the sediment, but also the sediment properties lead to a change of the flow field within the surface water. Consequently, we claim, that a simple coupling of surface water with a closed boundary at the streambed, which is commonly used, is not appropriate at least for coarse sediments. Moreover, non-Darcy-flow areas were observed for all cases within the sediment. For the sandy sediment, the non-Darcy-flow areas are restricted to the upper layer and the crest of the investigated ripples. For the gravel non-Darcy-flow was observed almost down to the bottom of the model. The application of the Darcy law in these areas would lead to an overestimation of flow rates. For equations that can be applied in non-Darcy-flow areas in the subsurface such as the Darcy-Brinkmann-equation, additional parameters such as the effective viscosity have to be determined.

Comparing the extended Navier–Stokes equations with the commonly used coupling of surface water with a Darcy-law-model, the integral model is definitely more time consuming than the coupled models. The model shows direct feedbacks from surface to subsurface and vice versa, is applicable also in non-Darcy-flow areas and provides high resolution results. The applied LES turbulence model gives additional insights about the turbulence at the interface which has a high impact on hyporheic exchange.

The general flow paths were the same for almost all simulations. Upstream of the crest, in high pressure areas, surface water flows into the ripple. Subsurface water flows out of the ripple towards the crest as well as upstream of the inflow area. Only for the ripple with the highest dimension multiple inflow areas were recognized upstream of the crest. Higher turbulences were generally observed for sandy ripples compared to gravel ripples. Gravel ripples always showed higher hyporheic exchange fluxes compared to sand ripples. Three ripple dimensions with the same height to length ratios were examined, but no clear relationship to exchange fluxes was found. For longer ripples, the exchange was slightly smaller due to less turbulence, while distances between the ripples increased the hyporheic exchange fluxes drastically. Also, in the flat streambed sections exchange was observed. Decreasing flow rates lead to decreasing exchange fluxes.

Numerous of the observations of our simulations were already seen in laboratory experiments. Our simulations allowed to get a deeper understanding of the present velocity and pressure distribution at the interface and to determine in- and outflowing fluxes, which can be important for the understanding and prediction of hydrological, chemical, and biological processes. In contrast to other coupled models, it is applicable in non-Darcy-flow areas and allows to simultaneously simulate the surface and subsurface with one system of equation for surface and groundwater. We can develop upscaling approaches where we quantify the exchange rates depending on the ripple geometry and other variables with the high resolution three-dimensional integral model to serve as sink/source terms in one- or two-dimensional shallow water flow models. The shallow water equations are based on vertical averaged velocities (not discretizing the vertical dimension) and are generally applied on coarser scales. In a next step, also transport equations will be included in the presented integral model.