Notice: Wiley Online Library will be unavailable on Saturday 27th February from 09:00-14:00 GMT / 04:00-09:00 EST / 17:00-22:00 SGT for essential maintenance. Apologies for the inconvenience.
 Solute transport in rivers is controlled by surface hydrodynamics and by mass exchanges between the surface stream and distinct retention zones. This paper presents a residence time model for stream transport of solutes, Solute Transport in Rivers (STIR), that accounts for the effect of the stream-subsurface interactions on river mixing. A stochastic approach is used to derive a relation between the in-stream solute concentration and the residence time distributions (RTDs) in different retention domains. Particular forms of the RTD are suggested for the temporary storage within surface dead zones and for bed form–induced hyporheic exchange. This approach is advantageous for at least two reasons. The first advantage is that exchange parameters can generally be expressed as functions of physical quantities that can be reasonably estimated or directly measured. This gives the model predictive capabilities, and the results can be generalized to conditions different from those directly observed in field experiments. The second reason is that individual exchange processes are represented separately by appropriate residence time distributions, making the model flexible and modular, capable of incorporating the effects of a variety of exchange processes and chemical reactions in a detailed way. The capability of the model is illustrated with an example and with an application to a field case. Analogies and differences with other established models are also discussed.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 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].
where U is the mean flow velocity (m s−1); α is the “mass transfer coefficient” (s−1); A/AS is the ratio of stream to storage cross-sectional areas; CW is the in-stream solute concentration (kg m−3); CS is the concentration of solute in the storage zone (kg m−3); DW is the mainstream longitudinal dispersion coefficient (m2 s−1); and t is time (s). A numerical solution of equation (1) was presented by Runkel and Chapra , which formed the basis of their One-dimensional Transport with Inflow and Storage (OTIS), later extended by Runkel  with a parameter estimation technique (OTIS-P).
 The strong simplifications adopted in the TSM approach have led other researchers to propose different descriptions of the hyporheic exchange. Haggerty et al.  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 . 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. , 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.
 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.
 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 , 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.
2. STIR Model
 The development of a model that mimics the longitudinal dispersion of a solute in a river, coupled with transient storage, requires the schematization of the system and an adequate degree of synthesis of the physics governing the processes. The stream is modeled as a one-dimensional system where x is the longitudinal distance, A is the cross-sectional area, U is the mean stream velocity and DW is the longitudinal dispersion coefficient. It must be stressed that DW accounts only for the effect of the surface flow field, and does not coincide with the “comprehensive” longitudinal dispersion coefficient often used to lump transient storage into the mass balance equation. Since the goal of this modeling approach is to separate the processes, the river is represented as a system composed by distinct physical domains interacting with each other through mass exchanges. The river is divided into the surficial stream in the main channel and different retention domains, such as the surficial dead zones and the hyporheic layer. The downstream transport of solutes is assumed to be controlled by exchanges with N types of storage zones, each one characterized by a given residence time distribution.
2.1. Residence Times in the Surface Stream and in the Storage Zones
 The propagation of a solute along a river is treated as a stochastic process. The time needed for a particle to travel a distance x, indicated with ��, is a random variable with probability density function r(t; x). The time �� is the sum of a time ��W spent on the surface, with pdf rW(t; x), and a time ��S sum of the single residence times within the storage domains, . A particle moving from the main stream into a storage zone follows a certain path and may possibly return back to the mainstream after some time. Particles may be uptaken once, twice or more, resulting in a global behavior that is the sum of individual paths partly in the main surface flow, partly in the retention domains. It is assumed that the longitudinal displacements within the storage zones are negligible compared to the displacement in the surface water. The number of times a particle is trapped in the ith retention domain, ��i, is a discrete random variable (��i = 0,1,2…) with conditional distribution pi(n∣��W = tW). When a particle is trapped in a storage zone, it is released after a time with pdf ϕi(t). Since the trapping events are assumed independent, the time ��Si has conditional density
given ��i = n. Here the symbol (*) denotes time convolution, so that ϕ(t) * ϕ(t) = ϕ(τ)ϕ(t − τ) dτ, where τ is a dummy variable. When n = 0, equation (2) yields rSi∣0(t) = δ(t), where δ(t) is the Dirac delta function (s−1). It follows that the conditional density of ��Si given ��W = tW is
When the transport process is dominated by advection, the uptake probability, pi, can also be thought as a function of the travel distance, x, which is proportional to the mean time spent on the surface. The condition of dominant advection is generally given as [Rutherford, 1994]
which is satisfied in most practical applications in rivers. The spatial dependence could also be more appropriate when there is a low density of retention zones.
 The probability of a particle to be uptaken at a given instant is assumed to be unconditioned by its previous storage history, then ��Si, i = 1,…,N, are mutually independent, and the conditional density of ��S given ��W = tW is
 A particle moving along the stream follows an irregular path because of turbulence. Following well established results from the literature [Taylor, 1954; Elder, 1959; Fischer, 1968], the motion of a particle limited only to surface flow in the main channel can be described as equivalent to a Brownian motion with drift, and the relevant residence time distribution can be inferred from the solution of the advection-dispersion equation (ADE). If it is now assumed that the entrapment within the storage zones does not modify the particle pathways, then the residence time within the surface stream remains unaltered. Thus, when the computational domain is x > 0, with boundary condition at infinity C(x → ∞, t) = 0, the function rW(t; x) is given by
Equation (6) is derived from the solution of the ADE for an input mass pulse, UC − DW∂xC = M/Aδ(t) at x = 0, where M is the injected mass.
 It is now possible to express the overall residence time distribution within a stream reach of length x as
Alternatively, when the total storage time is assumed to be dependent on the travel distance, thus using rS(t; x) instead of rS(t∣tW), the overall RTD is given by
2.2. Uptake Probability for Uniformly Spaced Storage Zones
 Under the assumption of uniform distribution of storage zones along the river, the uptake probability can be expressed as follows. The probability for a particle in the surficial stream to be stored in the ith domain in a time interval δt is assumed to be proportional to the length of the interval. It is expressed as αiδt, where αi (s−1) is the probability per unit time, which is taken to be constant both in time and space (although the temporal constance is not strictly required). The quantities αi represent the rates of transfer or, in other words, the flow rate into the storage zones per unit surficial volume. When hyporheic exchange with the streambed is considered, the relevant rate αB can be expressed as
where qB is the average flow rate into the sediments per unit bed area (m s−1), and d is the flow depth.
 Since the probability for a particle to be caught in the ith storage zone at a given instant is independent from its previous history, the probability for a particle to be caught n times in a time interval tW is given by the Poisson distribution with parameter αitW:
Alternatively, the uptake probability can be thought as a function of the distance from the injection point, x0 = 0, thus
 It is finally noted that, when equation (10) is used for the uptake probability, the Laplace transform (LT) of the overall residence time distribution expressed by (7) can be arranged, after some mathematical manipulations, in the following form [Margolin et al., 2003]:
where the symbol () denotes Laplace transform of the function it is applied to. Equation (12) shows that in Laplace domain the resulting residence time pdf is the same found in the absence of any retention process, but with a frequency shift that depends on the storage time pdf's.
2.3. Example 1: Residence Time Distribution in Surface Dead Zones
 The exchange with surface dead zones is well represented by an exponential RTD. The expression of the single-uptake storage time pdf is then the following:
where TD is a time scale, equal to the mean residence time. In practice the effect of the surface dead zone retention usually acts in a relatively short time scale compared to hyporheic retention, and can often be measured by tracer tests.
2.4. Example 2: Residence Time Distribution of Bed Form-Induced Hyporheic Retention
where the parameter β satisfies the following equation:
Equation (15) is a necessary condition to make ϕB(t) a pdf and is satisfied by β = 10.66. Parameter TB represents a residence time scale in the subsurface. The proposed residence time pdf is a single parameter heavy-tailed distribution that, for t → ∞, decays as a power law, ϕB(t) ∼ πTBt−2. Comparison of the exact and the approximate expression of ϕB(t) is reported in Figure 2.
2.5. Solute Concentration in the Surface Stream
 In this section, a relationship between the in-stream solute concentration and the residence time distribution is derived.
 Consider a stream reach of length x in which a mass M is instantaneously injected at the upstream section, x0 = 0, at time t0 = 0. At any instant t > 0, a part of the total mass is distributed in the main surficial stream, while a part is temporarily retained within the storage domains. The relevant masses are indicated by MW and MS, respectively. The total concentration is then defined as
where δMW(x, t) and δMS(x, t) are the masses contained within the spatial interval [x, x + δx] at time t in the superficial stream and in the storage zones, respectively.
 The quantity r(t; x)dt represents the fraction of mass flowing through the downstream section in the time interval [t, t + dt], and the flux is given by the convolution of r(t; x) with the input flux ϕ0(t). For a mass pulse concentrated in time this is given by ϕ0(t) = M/Aδ(t). The variation per unit time of the total concentration is equal to the opposite of the divergence of the local flux, ϕ0(t) * r(t; x), hence
where the subscript δ is used to denote the solute concentration generated by a mass pulse. The difference between the input and the output flux at the stream storage zone interface is linked to the variation of C − CW according to
If the total concentration is initially equal to the concentration in the surface stream, C(x, t = 0) = CW(x, t = 0), equation (18) can be written in the Laplace domain as
which relates the superficial concentration and the overall residence time distribution. Now, using expression (12), (22) becomes
where CADδ(x, t) is the solution of the advection-dispersion equation (ADE) with the boundary condition given by the same input mass pulse.
 If the residence time in the storage zones is assumed to be dependent on the distance from the injection point, and equation (8) is used instead of (7), an alternative expression can be found for the concentration CW. The balance expressed by (18) now becomes
Combining (24) with (17), with r(t; x) = rW(t; x) * rS(t; x), and integrating over time, we get, for a mass pulse,
Equations (23) and (25) provide a relationship between the system elementary responses in case of pure advection-dispersion and the case with temporary storage. Although these relations have been derived considering a mass pulse concentrated in time, M/Aδ(t), they are also valid for a mass initially concentrated in space and for a concentration pulse. In any case, the elementary response, CWδ, can always be derived from corresponding solutions of the ADE. Once the elementary response is known, the solution to the general case of an initially distributed mass and a given time-dependent boundary condition is readily found by spatial and temporal convolution, respectively.
 It is finally noted that, far from the injection point, when condition (4) holds, the residence time function in the main channel is well approximated by
which provides a direct relation between the overall residence time distribution and the concentration in the surface stream. The validity of (27) was one of the assumptions of the original version of the STIR model [Marion and Zaramella, 2005b].
3. STIR and Other Approaches
 It is now shown that, under certain assumptions, the STIR model converges to other established models. If the transport process in the superficial water is assumed to be Fickian, with additional fluxes due to mass exchanges with the storage zones, and if the downstream transport of the temporarily stored mass is neglected, then the mass balance for the in-stream solute concentration can be written as
Equation (28) is formally similar to the advection-dispersion-mass transfer equation used by Haggerty et al. , extended to account explicitly for different retention processes through the relevant residence time pdf's. When a single type of storage zone is considered, and the single-uptake residence time function ϕ(t) is given by (13) with TD = AS/(αA), equation (28) becomes equivalent to the TSM equations, (1a) and (1b).
where CW0(x) = CW(x, t = 0) is the initial in-stream concentration distribution. It is now observed that, if CAD(x, t) is a solution of the classical advection-dispersion equation (ADE) with initial condition CAD(x, t = 0) = CW0(x), then AD(x, u(s)), with u(s) given by (20), is a solution of (29). Hence, when the uptake process is considered a temporal Poisson process, the stochastic approach of the STIR model leads to the exact solution of equation (28).
 We now consider the continuous time random walk approach as proposed by Boano et al.  for solute transport in rivers. This approach relies on the following generalized master equation (GME):
where M(t) is a memory function. The GME is written in the Laplace domain as
with (s) given by
where ψ(t) is the pdf of the jump durations (or transition rate probability), and = x/U is the average travel time. When ψ(t) is given by an exponential pdf, ψ(t) = exp(−t/)/, the memory function M(t) is a Dirac delta function, and equation (30) reduces to the ADE [Margolin and Berkowitz, 2000]. If solute uptake into the storage zones is assumed to be a Poisson process that only immobilizes the particles without changing the pathways, the Laplace transform of ψ(t) can be expressed as (s) = 0(s + iαi(1 − i(s))) = 0(u(s)), where ψ0(t) is the pdf of the jump durations in the absence of any retention process (an exponential pdf) [Margolin et al., 2003; Cortis et al., 2006; Boano et al., 2007]. For this choice of ψ(t) it is readily seen that, if CAD(x, t) is a solution of the ADE with initial condition CAD(x, t = 0) = C0(x), then
is a solution of equation (31). This relation coincides with (21) for the total concentration. The main difference between the CTRW approach and the STIR model, as for the MRMT formulation, is that the CTRW is more comprehensive, being based on less restrictive assumptions, at least in its general form. The CTRW provides an overall description of solute transport without the need to split the physical domains. This makes it advantageous when a separation between surface transport and storage is not needed. As a counterpart, the CTRW is less explicit when a distinct parameterization of individual processes is required, for example when individual modeling closures are under investigations.
4. Application of the Model
 Two examples are now used to illustrate the application of STIR to assess the effects of different transport processes in a natural river. First, an application to an ideal case is presented where surface retention, hyporheic retention and reversible adsorption are sequentially added. Then an application to field tracer tests is reported, where the model parameters are fitted to experimental data.
4.1. Potential Application
 The simulation is performed for a uniform river with depth d = 0.75 m and width b = 20 m. The flow rate is QW = 5 m3s−1 and the longitudinal dispersion coefficient is assumed to be DW = 5 m2s−1. The hyporheic retention is treated using the pumping model (APM) where the exchange parameters are linked to the bed form wavelength L and the sediments permeability K by the relations
where θ is the sediment porosity and hm is the half amplitude of the sinusoidal dynamic head on the surface given by Fehlman 
where H is the bed form height. It is assumed that the sediment permeability is K = 5 × 10−3 m s−1, the porosity is θ = 0.3 and the bed forms have uniform height H = 0.05 m and wavelength L = 10H. The application of the advective pumping theory gives a rate of transfer αB = qB/d = 2.3 × 10−5 s−1 and a time scale for the residence time within the sediments TB = 441 s. When reversible, equilibrium adsorption of solutes to sediment surfaces is present, the net effect on the hyporheic retention can be modeled by simply multiplying this time scale by a retardation factor R > 1 [Zaramella et al., 2006].
 The exchange parameters for the transient storage in the dead zones are here simply defined as follows: the rate of transfer αD is taken to be 2 orders of magnitude larger than αB, while the mean residence time in the dead zones TD is taken to be an order of magnitude shorter than TB. In practical applications these parameters can often be determined by model calibration on the basis of tracer tests.
 An injection of a tracer at a constant rate for 2 h is simulated, and the resulting concentration is evaluated 2 km downstream from the source. Figure 3 shows, in linear space (Figure 3a) and in log-log space (Figure 3b), the normalized concentration distributions obtained by gradually incorporating different transport processes: curve i is the distribution relevant to in-stream advection-dispersion; curve ii represents the distribution obtained by adding the exchange with dead zones; curve iii is obtained by adding the bed form-induced hyporheic exchange; finally, curve iv represents the combined effect of surficial transport, surface and hyporheic retention, and reversible sorption to sediments with a retardation factor R = 2.5. It is clear that the transient storage in the subsurface generates a delay in the downstream propagation of solutes and a longer tail in the breakthrough curves. Figure 4 shows, for case 3, a comparison of the breakthrough curves obtained using equation (23), which assumes a temporal Poisson process, and the concentration obtained with equation (25), assuming a spatial Poisson process. The two equations provide very similar results due to the dominance of advection over surface dispersion in this example.
4.2. Application to a Field Case
 In this section, model parameters are fitted to an experimental breakthrough curve obtained by a slug tracer test. The experimental data were collected during a measurement campaign carried out along the Yarqon river, in Israel, within a technology transfer cooperation project between Italy and Israel. The examined reach is 1084 m long. During the tests the river had an average width b = 3.1 m and depth d = 0.34 m. A slug test was performed with an injection of Rhodamine WT (RWT). During the test the flow rate was measured with a current meter and was found to be steady at 0.21 m3 s−1. Solute concentration values were sampled on a 10 s interval using a portable field fluorometer (SCUFA). The surficial longitudinal dispersion coefficient, DW, can be either calibrated from data, thus adding one more calibration parameter, or estimated by available models. In our example, calibration gave a value very close to the estimate of Fischer's formula [Fischer, 1975],
where U* = is the shear velocity for normal flow depth, linked to the hydraulic radius, RH, and the mean bed slope, S. Models for DW are not always reliable. Care should be given on the estimate of DW when it is not calibrated with data. The exchange parameters, αi and Ti, are estimated by means of a nonlinear least squares algorithm (Levenberg-Marquardt) [Levenberg, 1944; Marquardt, 1963] using a uniform weight for observations and simulated values. The optimization was performed in all cases using log CW values. The data fitting was accomplished using first a single exponential RTD (equation (13)), then only the pumping RTD (equation (14)), and finally both the exponential and pumping RTDs. To provide a measure of the quality of the approximation, the normalized root mean square error (RMSE) was computed for each simulation as
where Csimj are the simulated concentration values, Cobsj are the observed values, and Nobs is the number of observations. The best fit parameters are reported in Table 1 and the relevant curves are shown in Figure 5.
Table 1. Parameter Estimates of the STIR Model Using the Single Exponential RTD, the Single RTD Derived From the Advective Pumping Model, and Both the Exponential and the Pumping RTDs
Exponential Plus Pumping
7.01 × 10−4
6.72 × 10−4
1.50 × 10−3
2.65 × 10−5
 The lowest value of the RMSE was obtained in the simulation with both the exponential and pumping RTDs, as expected. When a single distribution is used, results are very different. With the sole exponential RTD only the initial part of the breakthrough curve is reproduced in an acceptable way (Figure 5a). The model is completely inadequate to fit the tail of the curve. With the sole pumping RTD (Figure 5b) a better, although still not very good, fit is achieved compared to the exponential RTD, but the parameters of the distribution are unreasonable. The best fit time scale for hyporheic retention is TB ≈ 6 s which is clearly unacceptable. The two-domain model, instead, produces an almost perfect fit of the breakthrough curve (Figure 5c) with calibration parameters that account for a larger flux and a shorter retention in the surface dead zones compared to the hyporheic zone: the time scale of the hyporheic storage is TB ≈ 12 min, which is about 4 times greater than the surficial transient storage, and the ratio between the transfer rates is αD/αB ≈ 25. Thus, when hyporheic and surface retention processes are separated using distinct RTDs, the relevant parameters assume more plausible values from a physical point of view. It should also be noted that RWT is known to sorb slightly on sediments [Bencala et al., 1983; Gooseff et al., 2005]. The estimate of the long-term residence time may therefore account for the combined effect of predominant hyporheic flow and sorption. However, these two processes act in the same physical domains (the sediment deposits) and can be coupled consistently with the assumptions of STIR. A separation of the two effect is only possible on the basis of direct sorption data that are unfortunately not available in our field case.
 Solute transport in rivers is influenced by complex interactions between the overlying stream and the sediment bed. Direct measurements by tracer tests usually allow the evaluation of the short-term exchange processes only. Exchanges with surface dead zones and with the hyporheic zone generate an overall effect characterized by the superposition of processes acting at different time scales. Here a model is presented that simulates the effect of temporary retention on longitudinal solute transport in rivers. A relation between the in-stream solute concentration and the residence time distributions in different storage domains has been derived by representing the mainstream storage zone exchange as a stochastic process. This formulation allows for each process to be represented separately by a physically based RTD and uptake probability. When the hyporheic exchange is primarily driven by pressure variations on the bed surface induced by irregularities such as bed forms, the advective pumping theory can be used to model the temporary detainment of solutes into the bed. An approximate explicit form for the pumping residence time pdf has been proposed. Adsorption processes onto the sediment surface are easily included by applying a retardation factor to the time scale of hyporheic retention. On the other side, the fast exchanges with the dead zones can be represented by an exponential RTD. In both cases solute uptake into the storage zones can be taken to be a Poisson process. Other retention phenomena, such as the horizontal hyporheic flows induced by meanders, can similarly be modeled if an appropriate RTD is provided. STIR has also the potential to include the effects of other parameters such as heterogeneity [e.g., Cardenas et al., 2004; Marion et al., 2008]. Nevertheless, it must be stressed that the model complexity, and the number of parameters, may always be adapted to the practical problem on the basis of the available information and on the objectives of the analysis. A single exponential distribution, or a power law RTD (as suggested by Haggerty et al. ), can be used when there is no need to distinguish between the different storage processes. Conversely, problems requiring an estimate of the hyporheic contamination could be more adequately solved by using a distinct parameterization for shallow and deep retention. Because the storage within the surficial dead zones and the storage within the sediments are generally characterized by very different time scales, parameter estimation is expected to yield values that are more representative of the physics of the processes. This is partly confirmed by the work of Choi et al. , although their study was limited to the case of multiple storage zones with formally similar RTDs (i.e., exponential distributions).
 Finally it is shown that, under specific assumptions, STIR, MRMT and CTRW yield the same solutions. This leads to a deeper understanding of their respective theoretical formulations and applicability.
 This work has been carried out through Institutional Project of the University of Padova (Progetto di Ateneo) titled “Measurements and modeling of hyporheic flows in rivers” and by funds from a Italian-Israeli cooperation project on environmental technology transfer (Project 5, “An Integrated Approach to the Remediation of Polluted River Sediments”) funded by the Italian Ministry of the Environment through CUEIM. The authors thank Steve Wallis for suggesting the model acronym. STIR was also the name of an informal contact group of European scientists who contributed valuable discussions in the past few years.