Solute transport in rivers with multiple storage zones: The STIR model



[1] 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.

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].

Figure 1.

Illustration of the transport processes acting in a river. The downstream transport of solutes is governed by advection and hydrodynamic dispersion in the mainstream and by mass exchanges with different retention zones. These include vertical exchanges with the underlying sediments, where adsorption process may take place; lateral exchanges with surficial dead zones, typically vegetated pockets; and horizontal hyporheic flows induced by planimetric variation of the stream direction.

[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]:

equation image
equation image

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 [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.

2. STIR Model

[8] 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

[9] 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, equation image. 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

equation image

given ��i = n. Here the symbol (*) denotes time convolution, so that ϕ(t) * ϕ(t) = equation image ϕ(τ)ϕ(tτ) , where τ is a dummy variable. When n = 0, equation (2) yields rSi0(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

equation image

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]

equation image

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.

[10] 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

equation image

[11] 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 image

Equation (6) is derived from the solution of the ADE for an input mass pulse, UCDWxC = M/(t) at x = 0, where M is the injected mass.

[12] It is now possible to express the overall residence time distribution within a stream reach of length x as

equation image

Alternatively, when the total storage time is assumed to be dependent on the travel distance, thus using rS(t; x) instead of rS(ttW), the overall RTD is given by

equation image

2.2. Uptake Probability for Uniformly Spaced Storage Zones

[13] 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

equation image

where qB is the average flow rate into the sediments per unit bed area (m s−1), and d is the flow depth.

[14] 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:

equation image

Alternatively, the uptake probability can be thought as a function of the distance from the injection point, x0 = 0, thus

equation image

[15] 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]:

equation image

where the symbol (equation image) 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

[16] 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:

equation image

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

[17] Hyporheic flows are hardly measurable by direct methods, such as tracer tests, unless very long and very expensive techniques are designed [Johansson et al., 2001; Wörman et al., 2002; Gooseff et al., 2003a; Jonsson et al., 2003, 2004]. The Advective Pumping Model (APM) [Elliott and Brooks, 1997a] provides an expression for the cumulative residence time function within the sediments for bed form-induced exchange. Elliott's solution was given in term of an implicit function of time, whereas the application of equations (7), (8) and (12) requires an explicit form of the pdf of the residence time within the sediments ϕB(t). An analytical expression of ϕB(t) that approximates the exact solution for the case of the bed form-induced exchange is [Marion and Zaramella, 2005b]

equation image

where the parameter β satisfies the following equation:

equation image

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.

Figure 2.

Comparison of the exact pdf and the approximate pdf of the residence time within the sediment. The gap between the two curves is visible only at early times and is negligible for practical applications.

2.5. Solute Concentration in the Surface Stream

[18] In this section, a relationship between the in-stream solute concentration and the residence time distribution is derived.

[19] 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

equation image

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.

[20] 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/(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

equation image

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 CCW according to

equation image

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

equation image

and defining the new variable

equation image

we obtain

equation image

By combining (21) with the LT of (17), we get

equation image

which relates the superficial concentration and the overall residence time distribution. Now, using expression (12), (22) becomes

equation image

where CADδ(x, t) is the solution of the advection-dispersion equation (ADE) with the boundary condition given by the same input mass pulse.

[21] 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

equation image

Combining (24) with (17), with r(t; x) = rW(t; x) * rS(t; x), and integrating over time, we get, for a mass pulse,

equation image

[22] 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/(t), they are also valid for a mass initially concentrated in space and for a concentration pulse. In any case, the elementary response, C, 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.

[23] 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

equation image

and therefore, using (25),

equation image

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

[24] 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 image

Equation (28) is formally similar to the advection-dispersion-mass transfer equation used by Haggerty et al. [2000], 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).

[25] Using Laplace transforms, (28) becomes

equation image

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 equation imageAD(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).

[26] We now consider the continuous time random walk approach as proposed by Boano et al. [2007] for solute transport in rivers. This approach relies on the following generalized master equation (GME):

equation image

where M(t) is a memory function. The GME is written in the Laplace domain as

equation image

with equation image(s) given by

equation image

where ψ(t) is the pdf of the jump durations (or transition rate probability), and equation image = x/U is the average travel time. When ψ(t) is given by an exponential pdf, ψ(t) = exp(−t/equation image)/equation image, 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 equation image(s) = equation image0(s + equation imageiαi(1 − equation imagei(s))) = equation image0(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

equation image

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

[27] 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

[28] 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

equation image
equation image

where θ is the sediment porosity and hm is the half amplitude of the sinusoidal dynamic head on the surface given by Fehlman [1985]

equation image

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].

[29] 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.

[30] 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.

Figure 3.

Normalized breakthrough curves of an ideal example 2 km downstream of the injection point. Curve i represents the advection and dispersion processes in the main surficial stream; curve ii was obtained by adding the fast exchanges with the surface dead zones; curve iii represents the combined effects of the surficial transport and the deep exchange with sediments; and curve iv accounts for all the previous processes and for adsorption reactions to the sediment surfaces with a retardation factor R = 2.5.

Figure 4.

Comparison of the concentration distributions predicted by equation (23), which assumes a temporal Poisson process, and by equation (25), which assumes a spatial Poisson process. The two equations give very similar results because the condition of dominant advection (4) is well satisfied in this case.

4.2. Application to a Field Case

[31] 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],

equation image

where U* = equation image 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

equation image

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.

Figure 5.

Stream tracer observations from an RWT tracer experiment and simulation results using (a) the single exponential RTD, (b) the single pumping RTD, and (c) both the exponential and the pumping RTDs.

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
RTD TypeExponentialPumpingExponential Plus Pumping
αD (s−1)7.01 × 10−46.72 × 10−4
TD (s)177179
αB (s−1)1.50 × 10−32.65 × 10−5
TB (s)6721

[32] 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.

5. Conclusions

[33] 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. [2002]), 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. [2000], although their study was limited to the case of multiple storage zones with formally similar RTDs (i.e., exponential distributions).

[34] 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.


[35] 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.