1.1. Relevance and Previous Work
 A “hyporheic zone” (or HZ for brevity) is an area where water infiltrates from streams then flows through streambed sediments and stream banks and returns to the surface after relatively short pathways. These zones are important for two major reasons. They provide hyporheic and riparian organisms critical solutes, including nutrients, and dissolved gases [Triska et al., 1989, 1993; Findlay, 1995; Harvey and Fuller, 1998; Doyle et al., 2003]; they also control the distribution of solutes and colloids from bed form to watershed scales [Elliot and Brooks, 1997a; Woessner, 2000; Packman and Brooks, 2001; Sophocleous, 2002; Kasahara and Wondzell, 2003]. Understanding of HZ exchange improves through integrated modeling and field observations, supported by laboratory experiments.
 Several methods have been proposed for modeling hyporheic exchange and are reviewed by Packman and Bencala . Among the simplest of models are those that describe the exchange of solutes between rivers and adjacent transient storage zones as linear first-order mass transfer processes with lumped exchange coefficients, such as those by Bencala and Walters  or Young and Wallis . Exchange models based on one-dimensional diffusive processes, transverse to the channel, are slightly more sophisticated [Worman, 1998; Jonsson et al., 2003]. Other models consider the effects of early non-Fickian transport of solutes from the substratum to the stream [Richardson and Parr, 1988]. Parameters in these models are determined by empirically matching model output to actual solute breakthrough curves in laboratory experiments [Marion et al., 2002] and detailed field experiments [Harvey et al., 1996; Choi et al., 2000; Jonsson et al., 2003]. In certain settings, empirical determination of these parameters is not straightforward, and sometimes not possible, suggesting that these models are conceptually inconsistent with some environments [Harvey and Fuller, 1998; Harvey and Wagner, 2000]. A major source of discrepancy is that these simple models do not completely and realistically represent the hydrodynamics involved in hyporheic exchange.
 In some cases, exchange processes between streams and aquifers are dominated by advection rather than diffusion. Two important mechanisms for this type of hyporheic exchange have been proposed. The first is driven by advective flow induced by head gradients which are in turn generated by streambed topography due to bed forms or other irregularities such as logs and boulders and/or water surface topography. The second is due to the dynamic behavior of bed forms which temporarily trap and release water as they migrate. These two mechanisms are referred to as “pumping” and “turnover” [Elliot and Brooks, 1997a; Packman and Brooks, 2001].
 Several studies have investigated the mechanics of pumping from a theoretical perspective, often supported by experiments. Ho and Gelhar  present results of analytical and experimental studies on turbulent flow with wavy permeable boundaries. Thibodeaux and Boyle  propose a simple physically based model supported by laboratory observations. Shum  examines the effects of the passage of progressive gravity waves on advective transport in a porous bed. Savant et al. , applying the boundary element numerical method, replicate flume observations of flow along a vertical plane induced by head fluctuations. More sophisticated analytical models, supported by flume experiments and numerical modeling, consider the transfer of solutes and colloids through mobile bed forms [Elliot and Brooks, 1997a, 1997b; Packman and Brooks, 2001]. Worman et al.  presents a model that couples longitudinal solute transport in streams with solute advection along a continuous distribution of hyporheic flow paths. All of these theoretical, experimental and numerical studies are confined to two-dimensional (2-D) vertical domains, taken longitudinally along the channel, either due to their experimental setup or to enable simpler theoretical or numerical analyses.
 There has also been considerable work on 2-D essentially horizontal flow models. Examples of reach-scale 2-D numerical modeling of hyporheic exchange are given by Harvey and Bencala , Wondzell and Swanson , and Wroblicky et al. . The first example conceptually studies the impact of stepped-channels on surface-subsurface exchange. The last two examples are based on extensive data sets that allowed calibration of the flow models. All three cases demonstrate the viability of using numerical models to simulate horizontal flow into, through and out of channel banks while neglecting vertical exchange.
 There are a few fully three-dimensional (3-D) simulations of hyporheic exchange. For example, there are channel-scale (hundreds of meters) studies by Storey et al. , who investigate key factors controlling hyporheic exchange, and by Kasahara and Wondzell , who examine the impacts of morphologic features. Storey et al.  demonstrate that the homogeneous hydraulic conductivity (K) of the alluvial deposits controls the rate and extent of hyporheic exchange; no hyporheic exchange will occur if the K of the streambed is below a certain threshold.
 Some of the models mentioned above consider heterogeneity at larger spatial scales. For instance, Kasahara and Wondzell  interpolated slug test data by assigning K values to regions around wells using the Thiessen Polygon method. Storey et al.  employed spatially variable aquifer and streambed hydraulic properties that varied at a scale on the order of tens of meters. However, owing to their scales and resolution, all of these models ignore the finer-scale heterogeneity typical of streambeds [Bridge, 2003]. This limitation is widely recognized by investigators of hyporheic processes, and is best summarized by Packman and Bencala [2000, p. 51]: “Some additional complexities typically found in the natural environment, such as heterogeneity in the bed sediment, have also been omitted from the current models. Thus, even though these models are useful because they include process-level understanding, their application has been limited.” Even earlier Harvey and Bencala [1993, p. 96] stated “…the influence of heterogeneous hydraulic properties of the alluvium on surface-subsurface water exchange is a high priority to be considered in future research.” Is there field evidence to confirm this speculation on the importance of heterogeneity? White's  observed temperature distributions at a site in the Maple River, northern Michigan, from which he inferred HZ geometry, appear to confirm this importance. Stronger confirmation comes from field tracer tests by Wagner and Bretschko  which suggest that bed-scale variability of K results in a complex 3-D network of flow paths, and from which they deduce that heterogeneity is responsible for the patchy distribution of benthic invertebrates at their study site in Austria. What modeling has been done to test the importance of heterogeneity? A recent compilation of research on modeling of HZ processes listed no efforts addressing issues relating to streambed heterogeneity [Runkel et al., 2003]. However, there are a few ongoing investigations that tackle these issues [Matos et al., 2003; Salehin et al., 2003].
 Conceptual understanding of hyporheic processes can only be further broadened if multidimensional analyses including heterogeneity are pursued [Sophocleous, 2002]. Numerical modeling of hyporheic flow is a viable solution to this impasse since it allows flexibility in the parameters and processes that can be investigated [Packman and Bencala, 2000]. Previous modeling efforts by Woessner  elucidated this. He introduced high K rectangles set in a matrix of lower K. A linear head gradient was then imposed on the top boundary of the two-dimensional vertical section. A no-flow boundary was set at the downstream end of the domain in order to generate return flow to the river. This resulted in flow lines that are similar to field observations [see Woessner, 2000, Figures 5 and 6] although the model conditions, i.e., no flow at the downstream end and a binary K field, are only a crude approximation of natural conditions.
1.2. Purpose of This Study
 Previous studies have not exploited the capability of groundwater flow models to explicitly consider bed-scale 3-D spatial variability in hydraulic properties of the sub-channel HZ. Partly, this owes to the extensive fieldwork necessary for the data intensive sedimentological models necessary to represent realistic spatial heterogeneity of streambed hydraulic conductivity. Thus several fundamental questions remain unanswered. Under what conditions does heterogeneity induce substantial hyporheic exchange? Is the influence of heterogeneity on hyporheic flow comparable to the control exerted by bed or water surface topography, including the effects of bed forms and channel curvature? How are HZ geometry, streambed flux, and the HZ residence time of surface water controlled by each of these influences? In particular, when can we neglect and when should we consider heterogeneity, and channel curvature, in models of hyporheic processes? How do these answers change during the dynamic events of a flood with its evolving boundary condition at the streambed? The purpose of this paper is to provide some tentative answers to these questions based on modeling efforts using previously published field observations of heterogeneous streambed conductivity.