## 1. Introduction

[2] Environmental quality and health safety assessment often requires prediction of solute transport in rivers. The downstream propagation of the transported substances in a natural stream is influenced by exchanges between the surface water and the surrounding retention zones, typically vegetated pockets, dead zones and permeable subsurface, as illustrated in Figure 1. Filtration through the porous boundary of the river bed leads the dissolved substances within the porous media where sorption onto the sediment surface, deposition of the finer suspended particulate matter and other biogeochemical reactions may significantly affect their fate. The near-stream region of the porous boundary affected by the concentration of solutes in the stream, known as the hyporheic zone, is a natural habitat for the fluvial micro fauna, and hence it is extremely important for the evolution of a riverine ecosystem. Exchange between the stream and the underlying hyporheic zone is known to be primarily driven by advective processes which develop at several spatial scales because of separate mechanisms such as flow over bed forms, around obstacles and through bars and meanders, as shown by several recent studies [*Thibodeaux and Boyle*, 1987; *Savant et al.*, 1987; *Harvey and Bencala*, 1993; *Elliott and Brooks*, 1997a, 1997b; *Hutchinson and Webster*, 1998; *Packman and Brooks*, 2001; *Marion et al.*, 2002; *Boano et al.*, 2006a, 2006b; *Cardenas and Wilson*, 2007; *Tonina and Buffington*, 2007].

[3] The attention given to the exchange of solutes between the surface water of a river and the hyporheic zone has led to the development of various types of mathematical formulations. One of the most commonly used models is the transient storage model (TSM), presented by *Bencala and Walters* [1983]. This model has been widely applied for field experiments conducted both in small streams and large rivers [*Bencala*, 1984; *Castro and Hornberger*, 1991; *Vallet et al.*, 1996; *Mulholland et al.*, 1997; *Harvey and Fuller*, 1998; *Runkel et al.*, 1998; *Choi et al.*, 1999; *Fernald et al.*, 2001]. In the TSM the net mass transfer from the stream to the retention domains is assumed to be proportional to the difference of concentration between the surface water and a storage zone of constant cross-sectional area. The mathematical formulation of the TSM for nonreactive solutes is usually given as follows [*Nordin and Troutman*, 1980; *Bencala and Walters*, 1983; *Czernuszenko and Rowinski*, 1997; *Lees et al.*, 2000; *De Smedt and Wierenga*, 2005; *De Smedt*, 2006]:

where *U* is the mean flow velocity (m s^{−1}); *α* is the “mass transfer coefficient” (s^{−1}); *A*/*A*_{S} is the ratio of stream to storage cross-sectional areas; *C*_{W} is the in-stream solute concentration (kg m^{−3}); *C*_{S} is the concentration of solute in the storage zone (kg m^{−3}); *D*_{W} is the mainstream longitudinal dispersion coefficient (m^{2} s^{−1}); and *t* is time (s). A numerical solution of equation (1) was presented by *Runkel and Chapra* [1993], which formed the basis of their One-dimensional Transport with Inflow and Storage (OTIS), later extended by *Runkel* [1998] with a parameter estimation technique (OTIS-P).

[4] The simplification of the physical processes involved in hyporheic exchange which is inherent in the TSM is a cause of uncertainty in the parameter estimation. Recent studies and field observations have demonstrated that when advective pumping is a significant exchange process the best fit TSM parameters are dependent on the time scale of the process and the upstream boundary condition (incoming concentration) [*Harvey et al.*, 1996; *Harvey and Wagner*, 2000; *Wörman et al.*, 2002; *Marion et al.*, 2003; *Zaramella et al.*, 2003; *Marion and Zaramella*, 2005a]. This uncertainty of the TSM parameters often interferes with the observation of important results, such as the relationship between transient storage and the fluxes of reactive substances of interest (e.g., nutrients, contaminants) [*Hall et al.*, 2002; *Zaramella et al.*, 2006].

[5] The strong simplifications adopted in the TSM approach have led other researchers to propose different descriptions of the hyporheic exchange. *Haggerty et al.* [2000] used an advection-dispersion mass transfer equation (ADMTE), which is the core equation of the Solute Transport and Multirate Mass Transfer-Linear Coordinates (STAMMT-L) model of *Haggerty and Reeves* [2002]. The ADMTE is characterized by an additional source/sink term that represents the mass exchanges with the storage zones through a convolution integral of the in-stream solute concentration and a residence time distribution. This model has recently been applied to describe the late time behavior of breakthrough curves in natural streams [*Haggerty et al.*, 2000, 2002; *Gooseff et al.*, 2003b, 2005, 2007], and has been found to yield good agreement to experimental observations when a power law residence time distribution (RTD) is used. A similar mathematical formulation was suggested by *Wörman et al.* [2002], who used the advective pumping theory [*Elliott and Brooks*, 1997a, 1997b] to express the hyporheic residence time distribution, and proved that this approach yields a better description of the solute concentration within the sediments.

[6] Recently, a few authors [*Deng et al.*, 2004, 2006; *Kim and Kavvas*, 2006] have suggested the use of a fractional advection-dispersion equation (FADE) to provide solutions that resembles the highly skewed and heavy-tailed breakthrough curves observed in rivers. The FADE is based on a generalized Fick's law that takes the flux to be proportional to the fractional derivative of the solute concentration [*Chaves*, 1998; *Metzler and Klafter*, 2000; *Schumer et al.*, 2001]. However, this approach does not fully describe the underlying physics of the stream-subsurface exchange and, as a consequence, it is difficult to give a physical interpretation to the model parameters. More recently, the continuous time random walk (CTRW) theory [*Montroll and Weiss*, 1965; *Scher and Lax*, 1973] has been applied to stream transport of solutes [*Boano et al.*, 2007]. In the CTRW theory solute particles move by discrete jumps that are described by a joint probability density function (pdf) of the jump length and duration. This conceptualization of particle motion leads to a master equation that generalizes the classical advection-dispersion equation for the case of non-Fickian transport. In this modeling framework the transient storage of solutes is represented by a waiting time pdf.

[7] This paper presents an alternative conceptual model for Solute Transport in Rivers (STIR) that provides a physically based description of the stream-subsurface interactions on river mixing. The first version of the STIR model was presented by *Marion and Zaramella* [2005b] as a multiple process extension of the single process stochastic model proposed by *Hart* [1995], and has later been applied to a few field cases of heavily polluted natural streams in Israel and Serbia. In this paper, the STIR model is presented in a comprehensive mathematical framework that extends the original formulation. It is shown that, under specific assumptions, STIR converges to other models, such as the TSM, the multirate mass transfer (MRMT) and the CTRW approach. For practical applications STIR can be seen as an extension of TSM in which general forms of the storage time statistics can be implemented. The capability of the model is illustrated with a theoretical example, and an example of tracer test data is used to demonstrate the applicability to field cases.