## 1. Introduction

[2] The mixing zone between river and groundwater flow systems, often referred to as the hyporheic zone, plays an important role in riverine ecosystems. Water entering the riverbed exchanges energy and solutes with groundwater and returns to the channel with a different chemical signature, depending on the solute mixing, chemical reactions, and residence time. Residence time of solutes is highly variable in space and time and is dependent on physical and hydraulic parameters of the river and underlying groundwater system [*Vaux*, 1968; *Woessener*, 2000; *Haggerty et al*., 2002]. Estimating spatially distributed exchange rates, flow paths, and residence times is important toward understanding the biogeochemical processes in the hyporheic zone [e.g., *Triska et al*., 1993; *Brunke and Gonser*, 1997; *Dahm et al*., 1998; *Zarnetske et al*., 2011]

[3] The spatial distribution of groundwater age is typically estimated using a multidimensional, spatially distributed groundwater flow and transport model, referred to herein as a distributed model. Distributed models are useful for understanding nutrient processing in rivers, such as the effects of river morphology, sediment heterogeneity, and chemical mixing on nutrient fate and transport. Distributed models provide considerable advantages to 1-D modeling approaches as they can account for mixing of flow paths and their influence on mean residence times. Estimating residence times with distributed models has been performed through particle tracking [*Wroblicky et al*., 1998; *Storey et al*., 2003; *Kasahara and Wondzell*, 2003; *Salehin et al*., 2004; *Cardenas et al*., 2004] and through solute-transport methods using the advection-dispersion equation (ADE) [*Goode*, 1996; *Varni and Carrera*, 1996; *Woessener*, 2000; *Lautz and Siegel*, 2006; *Sawyer and Cardenas*, 2009; *Jiang et al*., 2010]. Simulation of solute transport using the ADE has the advantage over advection-only particle tracking methods for estimating residence time because they include mixing of water from different sources through dispersion and diffusion. A novel approach for the direct simulation of the spatial distribution of mean groundwater age is the ADE redefined for groundwater age or “age-mass” [*Goode*, 1996]. Mean age is analogous to the solute concentration calculated by the ADE with a distributed source of unit strength. Thus, the simulated “concentration” at each point in the model domain at steady state is equal to the mean age and represents the combined effects of advection, dispersion, and diffusion [*Goode*, 1996; *Varni and Carrera*, 1998]. Several studies have used direct modeling of groundwater age with the inclusion of dispersion and diffusion in aquifer systems [e.g., *Goode*, 1996; *Varni and Carrera*, 1998; *Engesgaard and Molson*, 1998; *Bethke and Johnson*, 2002], reservoirs [e.g., *Cornation and Perrochet*, 2006], for use in inverse model calibration [*Varni and Carrera*, 1998; *Weissmann et al*., 2002; *Ginn et al*., 2009], and understanding biogeochemical processes [*Ginn*, 1999; *Gomez et al*., 2012]. *Bethke and Johnson* [2002] compared the age-mass concept to the piston flow approach to simulate travel time in a regional aquifer system including aquitards. They found that for an aquifer system that contains a significant amount of aquitard material relative to aquifer material, groundwater age will be significantly older than estimates of water age using piston flow assumptions. This has important implications for understanding residence times in the hyporheic zone where layering is present due to colmation and other fluvial processes that restrict hyporheic flow [*Naranjo et al*., 2012]. River sediment is often heterogeneous and can consist of both fine and coarse-grained deposits that control the distribution of flow paths, mixing of young and old water, and the rate of biogeochemical processes. Inverse model calibration with sediment temperature and pressure observations can be used to characterize the spatial distribution of hydraulic conductivity and minimize uncertainty in residence times. The simulation of residence times can be used to support interpretation of the relationship between age, pore-water chemistry, and biogeochemical reactions [e.g., *Triska et al*., 1993; *Findlay*, 1995; *Zarnetske et al*., 2011].

[4] Previous studies have highlighted the effects of riverbed heterogeneity on hyporheic flow paths and residences times at the 10 m scale [*Salehin et al*., 2004; *Marion et al*., 2008; *Sawyer and Cardenas*, 2009] to the 100 to 2000 m scale [*Fleckenstein et al*., 2006; *Frei et al*., 2009; *Jiang et al*., 2010]. Heterogeneity within the riverbed has been shown to control vertical exchange and residence times by limiting the vertical depth over which exchange occurs and deflecting flows upward. *Vaux* [1968] used simulation modeling to demonstrate that riverbed heterogeneity significantly affected hyporheic flow paths and illustrated that flow paths can be deflected into or out of different configurations of sediment heterogeneity. Representation of riverbed heterogeneity in models is difficult because it requires extensive data collection that is often impractical to collect in large river systems. Heterogeneous representations of streambeds can be developed using interpolation between observations [*Cardenas et al*., 2004], zonal representations parameterized using inverse modeling [*Naranjo et al*., 2012], and through geostatistics [e.g., *Salehin et al*., 2004; *Packman et al*., 2006; *Fleckenstein et al*., 2006]. The formation of sediment structure such as layered deposits has been shown to strongly influence depth of exchange even with the presence of high-permeable sediments near the surface [*Packman et al*., 2006; *Marion et al*., 2008], as well as the direction of seepage between streams and groundwater [*Niswonger and Fogg*, 2008].

[5] Simulated hyporheic flow on the basis of distributed models can have great uncertainty because they require parameterization of both the flow and transport equations. Inverse modeling techniques used for heat as a tracer studies are improving for estimating the most uncertain model parameters, specifically hydraulic conductivity [e.g., *Niswonger and Rupp*, 2000; *Niswonger et al*., 2005; *Constantz*, 2008; *Naranjo et al*., 2012], which provide new opportunities for developing distributed models of hyporheic flow in river systems. The use of distributed solute-transport modeling to derive travel times will be uncertain due to errors in hydraulic and transport parameters, boundary conditions, and errors in the conceptual model used to construct the simulation model. Despite the recent controversy (briefly discussed herein), the generalized likelihood uncertainty estimation (GLUE) method has been extensively used in hydrologic models to address predictive uncertainty on the basis of the equifinality concept [*Beven and Freer*, 2001; *Beven*, 2006]. The GLUE method remains an attractive choice for addressing parameter uncertainty in heat as a tracer studies due to the measurement of continuous time series data used in model calibration. Furthermore, by comparing results from standard Monte Carlo and GLUE methods, the reduction in uncertainty provided by observation data (i.e., temperature and pressure) can be quantified. Thus, the value of observation data can be evaluated on the basis of reductions in uncertainty for a particular model result, such as residence time.

[6] Models generally have large uncertainty that stem from the original conceptual model and assignment of hydraulic properties, observational errors, and inadequate representation of the boundary conditions. Equifinality asserts that multiple combinations of model parameters exist, each producing reasonable model performance, which can be addressed through uncertainty analysis. In groundwater model applications, the GLUE method has been used for capture-zone analysis [*Feyen et al*., 2001; *Morse et al*., 2003], stochastic-transport flow models [*Hassan et al*., 2008], and conceptual-model uncertainty [*Rojas et al*., 2010; *Reeves et al*., 2010]. These studies stress the importance of evaluating multiple parameter sets and conceptual models to improve the understanding of hydrological systems.

[7] In this paper, we expand on the work of *Naranjo et al*. [2012] for the purpose of estimating mean residence times and evaluating the effect of parametric uncertainty on solute-transport simulations. The impetus of this work is to develop an approach for estimating model parameters using global methods for simulating mean residence times and their uncertainties. In the present work, residence times are of interest in how they relate to nutrient chemistry sampled from the hyporheic zone. Further, there are few investigations that evaluate 2-D longitudinal hyporheic flow paths and residence times derived from field investigations. The flow model used in this effort was extensively calibrated using measurements of riverbed temperatures and pressure. The value of the observation data was quantified through reductions in the standard deviation of the posterior distributions of residence time using the GLUE approach.

[8] Simulated mean and standard deviation of the spatially distributed residence times with fixed-flow and -transport parameters (base case) was compared to (1) uncertain flow parameters using the GLUE method with fixed-transport parameters and (2) fixed-flow parameters and uncertain-transport parameters using standard Monte Carlo sampling. Uncertainty in the vertical to horizontal anisotropy (*K _{z}*/

*K*) was evaluated with respect to uncertainty in the mean residence time. Anisotropy was evaluated to better understand its effects on longitudinal hyporheic flow, assuming that typical fluvial processes result in layered sediment structure and strongly anisotropic hydraulic conductivity. Three alternative conceptual models were used to evaluate the differences in mean residence times given (1) a two-layer heterogeneous model with anisotropy (CM2), (2) a single-layer homogenous model with anisotropy (CM3), and (3) a single-layer homogenous model with isotropic conditions (CM4). The results of the conceptual models illustrate the need to account for heterogeneity and anisotropy in simulations of hyporheic flow.

_{x}[9] Simulations presented herein involved the estimation of mean residence times and flow paths using a distributed, 2-D longitudinal flow and transport model. The model represents a riffle-pool sequence and the associated underlying hyporheic zone. Here we apply the age-mass approach [*Goode*, 1996] in simulations of the hyporheic zone using a 2-D flow and solute-transport model. Relevance of the work lies in demonstrating the use of solute-transport simulations for estimating mean residence times in a riffle-pool sequence where exchange rates and flow paths were derived from heat as tracer method. Importantly, we show that calibration of hydraulic conductivity on the basis of temperature and pressure significantly reduces the uncertainty in estimates of residence time. However, after calibration, hydraulic conductivity remains the largest contributor of uncertainty in estimates of residence time relative to transport parameters. We demonstrate the use of a rigorously calibrated flow model to estimate the uncertainty in hydraulic conductivity, porosity, and dispersivity in simulating spatially distributed mean residence times.