Pore water in stream sediments is continuously exchanged with the surface water from the overlying stream. This exchange of water and solutes that occurs across the stream-sediment interface plays an important role for fluvial ecology because of the unique biochemical conditions, rich biodiversity, and high rates of metabolism. While many studies have observed the extent of the hyporheic zone to be modified by changes in the level of the groundwater table, the actual importance of this interaction is still difficult to quantify. Here, we focus on the case of bedform induced hyporheic exchange to show how the the volume of hyporheic sediments that receive water from the stream is significantly reduced by the upwelling of subsurface water. A simple scaling relationship for the assessment of maximum depth of the hyporheic zone is proposed by relating hyporheic flow to the groundwater discharge in an aquifer with given hydraulic properties and head difference between the stream and the aquifer.
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 Streams are connected to the adjacent unconfined aquifers through the river banks and the bed, and the hyporheic zone consists in the part of the aquifer whose biochemical properties are different from both the surface and the subsurface water. The peculiar properties of the hyporheic environment depend on the exchange of water between the stream and the aquifer, and the environmental consequences of this linkage have recently been subject to a growing interest by many researchers [see, e.g., Jones and Mulholland, 2000]. A large number of field studies have confirmed that water and water-borne nutrients and contaminants are frequently exchanged between rivers and aquifers [e.g., Harvey and Bencala, 1993; Wondzell and Swanson, 1996; Wroblicky et al., 1998; Battin et al., 2003]. This exchange exerts a strong influence on the quality of both surface and subsurface waters. Solutes carried by the stream penetrate into the sediments and are retained for times that are typically much longer than the average in-stream advective timescale [Boano et al., 2007a]. As a result, the in-stream concentrations of pollutants are affected by the exchange with the hyporheic zone, as observed in many field studies [e.g., Bencala and Walters, 1983; Harvey and Bencala, 1993; Johansson et al., 2001]. The exchange flux also provides the hyporheic sediments with nutrients and dissolved oxygen from the stream, which determine favorable conditions for the development of a rich microbial community [Brunke and Gonser, 1997; Boulton et al., 1998].
 It is now clear that the exchange between streams and hyporheic zones occurs as local flow systems with water leaving the stream, moving through the subsurface, and finally returning to the stream. Looking to the water flow on a broader scale, it is recognized that the local flow systems are nested inside larger groundwater flow systems, the latter being controlled by the precipitation inputs at the regional scale [Tóth, 1963; Sophocleous, 2002; Hayashi and Rosenberry, 2002; Cardenas, 2007; Wörman et al., 2007]. However, a clear understanding about the influence of the regional groundwater flow on the local hyporheic exchange patterns is still missing. The magnitude of stream-hyporheic exchange in a local flow pattern is mainly controlled by the stream discharge and morphology, but the temporal variability of large-scale groundwater flow systems (i.e., water table elevation) has also been observed to influence the exchange [Harvey and Bencala, 1993; Wroblicky et al., 1998]. High groundwater levels tend to decrease the exchange flux with the hyporheic zone, even though the actual importance of this process differs from stream to stream [Wondzell and Swanson, 1996]. Cardenas and Wilson  have recently demonstrated that the hyporheic exchange induced by bedforms is reduced by the groundwater discharge into the stream, but the study did not provide a way to estimate the groundwater discharge for a particular hydrological condition.
 The present work presents a numerical study of the role of the groundwater discharge in controlling the extent of the hyporheic zone. The coupling between the phreatic aquifer and the stream is considered, and the spatial pattern of groundwater discharge below the streambed is evaluated. The pattern of groundwater discharge is then related to the depth of the hyporheic zone below the streambed dunes. The paper demonstrates that particular groundwater conditions can significantly reduce the volume of sediments influenced by the hyporheic exchange, and sometimes prevent the penetration of the streambed by the stream water. We also show that the upwelling of groundwater induces a variation of the depth of the hyporheic zone across the river section, thus providing spatially-varying habitat conditions for the hyporheic microfauna. Finally, a simple scaling relationship for the maximum depth of the hyporheic zone is proposed.
 The Cartesian coordinate system pictured in Figures 1a and 1b is adopted, where x, y, and z denote the streamwise, transversal, and vertical directions, respectively. In the present analysis, the vertical extent of the hyporheic zone is defined as the maximum depth reached by the advective flowpaths, and it is determined in two sequential steps. First, the coupled stream-aquifer system sketched in Figure 1a is considered – where only one half of the system is examined because of the symmetry of the problem – and the groundwater discharge through the streambed is evaluated. Then, the groundwater discharge is superimposed on the hyporheic flow field induced by the bedforms, and the hyporheic exchange represented by the bold arrows in Figure 1b is obtained.
 The geometry of the gaining stream in Figure 1a is summarized by the river half-width, L, and the depth, d. Groundwater discharge to the stream is induced by the head difference, Δ, between the groundwater table and the stream surface. This flow is exemplified by the bold arrows in Figure 1a, that show the upwelling of subsurface water through the streambed. For the sake of simplicity, the distance between the river bank and the point of observation of the groundwater level is chosen to be equal to the river half-width, L. For a given Δ, the influence on the results of the shape of the groundwater table has been verified to be negligible. Thus, a parabolic shape is assumed for the groundwater table, and a homogeneous, isotropic value of the hydraulic conductivity, K, is assigned to the sediments.
 Water flow in the aquifer is governed by the Laplace equation, ∇2h = 0, where h is the hydraulic head in the subsurface [e.g., Bear and Verruijt, 1992]. The Laplace equation is applied to the semi-infinite 2D domain ABCDEF shown in Figure 1. In order to solve the equation in a domain of infinite vertical extension, a no-flow boundary, AB, is introduced that is deep enough to avoid any influence on the results. No-flow boundary conditions are also imposed on the right boundary, BC, because of symmetry, and on the water table, EF. The head on the river bed and bank, CD and DE, is assumed to be equal to the free surface level in the stream, and the left boundary, AF, is assigned a head equal to the water table level. The Laplace equation is solved using a finite-element method, and the velocity field in the sediments is obtained from the Darcy's law. In particular, the velocities at the stream-sediment interface, z = 0, represent the profile of the upwelling velocity through the streambed, vup(y). This velocity profile is used to assess the impact of the groundwater discharge on the hyporheic exchange.
 The bedform-induced flow pattern in a plane x-z normal to the stream cross-section is modeled extending the approach proposed by Elliott and Brooks [1997a]. The upwelling velocity, vup, is added to include the effect of groundwater discharge
where x is the streamwise coordinate, u and v are the Darcy velocities in the longitudinal and vertical directions, respectively, u0 = kKh0 is a typical velocity scale for the hyporheic flow, k = 2π/λ is the bedform wavenumber, λ is the bedform wavelength, and h0 is the head perturbation determined by the presence of the bedform [see Elliott and Brooks, 1997a]. According to our simulations, the vertical variations of the upwelling velocity are small compared to the transversal ones. Thus, the upwelling velocity vup in (2) is treated as a constant along z.
 Extensive numerical simulations have been carried out using the approach described in the previous section. The profiles of the upwelling velocity, vup(y), are displayed in Figure 2a for two typical cases. Figure 2a shows that the most of the groundwater is discharged near the river banks, whereas the central part of the streambed receives a lesser amount of groundwater. This result has been obtained for the case of a rectangular river cross-section, but the qualitative behavior is supposed to hold even for different geometries of the stream section.
 The penetration of the surface water in the sediments is hampered by the groundwater upwelling, as suggested by the comparison of the two terms on the right-hand side of equation (2). The downward velocity induced by the head perturbation on the streambed decays exponentially with depth, and it is eventually equaled by the upwelling velocity, vup. Since vup(y) is not constant across the stream, the depth of the hyporheic zone, zH(y), varies over the stream width as well.
 For a given lateral position, y, the flow pattern in the x-z plane is described by equations (1)–(2). The lowest point that can be reached by the hyporheic flowpaths in this plane occurs at x = λ/4, where equation (2) predicts the highest downward velocities. The vertical extent in the x-z plane of the hyporheic zone, zH, can be evaluated putting v∣x=λ/4 = 0 in equation (2)
where the aforementioned dependence of zH(y) from the distance from the river bank, y, is explicitly shown. Equation (3) shows that the increase of the groundwater discharge results in a shallower hyporheic zone and thus reduces the volume of sediments that exchange water and solutes with the stream. It is also interesting to notice that according to equation (3) the depth of the hyporheic zone scales with the dune wavelength.
 The profiles of the depth of the hyporheic zone, zH(y), for the two cases have been evaluated using (3), and are shown in Figure 2b. Figure 2b demonstrates that the discharge of groundwater constitutes a factor that contrasts the penetration of water in the sediments, reducing the volume of pore water that comes into contact with surface water and with the water-carried solutes.
 The comparison between the cases of lower and higher groundwater discharge – solid and dashed line in Figures 2a and 2b, respectively – reveals that a strong upwelling of groundwater can prevent the exchange of stream water through the parts of the streambed closer to the banks. Equation (3) eventually predicts the complete disappearance of the bedform-induced exchange when the groundwater discharge per unit streambed area, vup, is everywhere higher than u0. Thus, the supply of oxygen and nutrients from the stream to the sediments is expected to vary according to the different surface and subsurface water levels, determining periodical disturbances of the environmental conditions of the hyporheic ecosystem.
 As shown in Figure 2, the maximum depth of the hyporheic zone, zH* = zH(y = L), occurs at the center of the stream where the upwelling velocity is lower. The value of the upwelling velocity at the center of the streambed, vup* = vup(y = L), is influenced by the hydraulic conductivity of the sediments, K, and the parameters L, Δ, and d, defined in Figure 1. It follows from dimensional considerations that the dimensionless upwelling velocity vup*/K must depend only on the average slope of the groundwater table, Δ/L, and on the stream aspect ratio, L/d.
 A number of simulations have been performed in order to represent different streams with aspect ratio L/d ranging between 5 and 30, and for average slopes of the groundwater table, Δ/L, up to 10−1. The upwelling velocities have been evaluated and the results are plotted against the groundwater table slope, Δ/L, in Figure 3. Figure 3 clearly shows that the stream aspect ratio, L/d, has a negligible influence on the amount of groundwater discharge. Furthermore, the values of the upwelling velocity in Figure 3 exhibit an almost perfect linear correlation with Δ/L. This relationship can be be expressed as
which provides a very good fit to the data in Figure 3, with errors that are less than 2%. This simple relationship links the upwelling velocity to the hydraulic characteristics of the stream-aquifer system depicted in Figure 1.
Equation (4) can be used to assess the depth of the sediments that is affected by the hyporheic exchange. The combination of (3) and (4) yields
 This expression provides a simple and rapid tool to estimate the maximum depth of sediments that is influenced by the exchange with the surface water.
 The present analysis has demonstrated how the stream-aquifer interaction can significantly alter the exchange of water and solutes between the stream and the hyporheic zone. The head difference between the aquifer and the stream induces the discharge of groundwater through the streambed. This discharge contrasts the penetration of stream water into the sediments and limits the maximum depth that can be reached by the hyporheic flowpaths. Since the upwelling of groundwater is stronger near the banks than at the middle of the stream, the depth of hyporheic zone is not constant across the stream width. When the groundwater discharge increases, the hyporheic exchange is limited to the central part of the stream cross-section and to the shallower sediments. Very high discharge of groundwater (vup > u0) completely prevent the flow from the stream to the hyporheic zone.
 The limitation of the surface-subsurface interactions has many important implications for the fluvial environment. First, it influences the supply of oxygen to the sediments, which constitutes the main limiting factor for aerobic microorganisms in the hyporheic zone. Second, since rivers commonly receive wastewater from sewage systems and industrial facilities, information about the extent of the hyporheic zone would also be valuable for the study of contamination of river sediments.
 The role of groundwater discharge is particularly relevant because of its variability in both space and time, that implies spatial and temporal variability of the depth and flow conditions of the hyporheic zone as well. The presence of spatial gradients of pore velocities and nutrient concentrations determines the diversification of the physical and chemical conditions of the subsurface environment that can enhance the hyporheic biodiversity. Variations in time that follow the dynamic behavior of the coupled stream-aquifer system are also likely to exert a major influence on the hyporheic ecosystem. These considerations underline the need for a deeper integration between the work of hydrologists, biologists, and ecologists in order to improve our understanding of the fluvial environments.
 The financial support for this research has been provided by Regione Piemonte and Fondazione CRT.