One of the mechanisms that greatly affect the pollutant transport in rivers, especially in mountain streams, is the effect of transient storage zones. The main effect of these zones is to retain pollutants temporarily and then release them gradually. Transient storage zones indirectly influence all phenomena related to mass transport in rivers. This paper presents the TOASTS (third-order accuracy simulation of transient storage) model to simulate 1-D pollutant transport in rivers with irregular cross-sections under unsteady flow and transient storage zones. The proposed model was verified versus some analytical solutions and a 2-D hydrodynamic model. In addition, in order to demonstrate the model applicability, two hypothetical examples were designed and four sets of well-established frequently cited tracer study data were used. These cases cover different processes governing transport, cross-section types and flow regimes. The results of the TOASTS model, in comparison with two common contaminant transport models, shows better accuracy and numerical stability.

Comparison of the features of the three models used in this study.

Comparison of numerical methods used in the three models.

*

Error indices of verification by the analytical solution for continuous boundary condition.

First efforts to understand the solute transport subject led to a longitudinal dispersion theory which is often referred to as the classical advection–dispersion equation (ADE; Taylor, 1954). This equation is a parabolic partial differential equation derived from a combination of a continuity equation and Fick's first law. The one-dimensional ADE equation is as follows:

In general, these areas affect pollutant transport in two ways: on the one hand, temporary retention and gradual release of solute cause an asymmetric shape in the observed concentration–time curves, which could not be explained by the classical advection–dispersion theory; on the other hand, it is also affected by the opportunity for reactive pollutants to be frequently contacted with streambed sediments that indirectly affect solute sorption, especially in low-flow conditions (Bencala, 1983, 1984; Bencala et al., 1990; Bencala and Walters, 1983).

Results of the TOASTS model verification by the analytical
solution for continuous boundary condition (

In the literature, several approaches have been proposed to simulate solute transport in the rivers with storage areas, that one of the most commonly used is the transient storage model (TSM). TSM has been developed to consider solute movement from the main channel to stagnant zones and vice versa. The simplest form of the TSM is the one-dimensional advection–dispersion equation with an additional term to consider transient storage (Bencala and Walters, 1983). After the introduction of the TSM, transient storage processes have been studied in a variety of small mountain streams, as well as large rivers, and it was shown that simulation results of tracer study data considering the transient storage impact have good agreement with observed data. Also, it was shown that the interaction between the main channel and storage zones, especially in mountain streams, has a great effect on solute transport behavior (D'Angelo et al., 1993; DeAngelis et al., 1995; Morrice et al., 1997; Czernuszenko et al., 1998; Chapra and Runkel, 1999; Chapra and Wilcock, 2000; Laenen and Bencala, 2001; Fernald et al., 2001; Keefe et al., 2004; Ensign and Doyle, 2005; Van Mazijk and Veling, 2005; Gooseff et al., 2007; Jin et al., 2009).

Error indices of verification by the analytical solution for Heaviside boundary condition.

Results of the TOASTS model verification by the analytical
solution for continuous boundary condition (

Results of the TOASTS model verification by the analytical
solution for Heaviside boundary condition (

Results of the TOASTS model verification by the analytical
solution for Heaviside boundary condition (

Bed elevation contours of the 2-D hypothetical example.

Bed elevation three-dimensional view of the 2-D hypothetical example.

Error indices of verification by the 2-D model.

Results of the TOASTS model verification using the 2-D model.

Comparison of the CTQS, CTCS and BTCS schemes for the pure advection test case.

In this study, a comprehensive model, called TOASTS (third-order accuracy simulation of transient storage), able to obviate shortcomings of current models of contaminant transport, is presented. The TOASTS model uses high-order accuracy numerical schemes and considers transient storage in rivers with irregular cross-sections under non-uniform and unsteady flow regimes. This model presents a comprehensive modeling framework that links three sub-models for calculating geometric properties of irregular cross-sections, solving unsteady flow equations and solving transport equations with transient storage and kinetic sorption.

Properties of the test cases used for the TOASTS model application.

Simulation parameters related to test case 2.

To demonstrate the applicability and accuracy of the TOASTS model, results of two hypothetical examples (designed by the authors) and four sets of well-established tracer study data, are compared with the results of two existing frequently used solute transport models: the MIKE 11 model, developed by the Danish Hydraulic Institute (DHI), and the OTIS (one-dimensional transport with inflow and storage) model that today is the only existing model for solute transport with transient storage (Runkel, 1998). The TOASTS model and the two other model features are listed in Table 1. It should be noted that the OTIS model, in simulating solute transport in irregular cross-sections under unsteady flow regimes, has to rely on external stream routing and geometric programs. By contrast, in the TOASTS and MIKE 11 models, geometric properties and unsteady flow data are directly evaluated from river topography, bed roughness, flow initial and boundary conditions data. Another important point is in the numerical scheme which has been used in the TOASTS model solution. The key and basic difference of the TOASTS model is spatial discretization of the transport equation. This model uses the control-volume approach and QUICK (quadratic upstream interpolation for convective kinematics) scheme in spatial discretization of the advection–dispersion equation considering transient storage and kinetic sorption; whereas the two other models employ central spatial differencing. More detailed comparison of numerical schemes used in the structure of three subjected models is given in Table 2.

Error indices of concentration time series in test case 2.

Error indices of concentration longitudinal profiles in test case 2.

As many researchers claim, central spatial differencing is incapable of simulation of pure advection problems and does not introduce good performance in this regard (it leads to non-convergent results with numerical oscillations; Zhang and Aral, 2004; Szymkiewicz, 2010; Versteeg and Malalasekera, 2007). It should be mentioned that, in recent years, the QUICK scheme has been widely used in numerical solutions of partial differential equations due to its high-order accuracy, very small numerical dispersion and higher stability range (Neumann et al., 2011; Lin and Medina, 2003). Hence, usage of the QUICK scheme in numerical discretization of the transport equation leads to significantly better results, especially in advection-dominant problems.

Comparison of the TOASTS, OTIS and MIKE 11 models in test
case 2 for Pe

Comparison of the TOASTS, OTIS and MIKE 11 models in test case 2
for Pe

Comparison of the TOASTS, OTIS and MIKE 11 models in test
case 2 for Pe

There are several equations for solute transport with transient storage, the most well known being the TSM presented by Bencala and Walters (1983). By writing conservation of mass principle for solute in the main channel and storage zone and doing some algebraic manipulation, a coupled set of differential equations is derived:

Numerical solution of Eqs. (4) to (6) in this study are based on the
control-volume method and centered time–QUICK space (CTQS) scheme. The
spatial derivatives are discretized by the QUICK scheme, which is based on
quadratic upstream interpolation of discretization of the advection–dispersion
equation (Leonard, 1979). In this scheme, face values are computed using
quadratic function passing through two upstream nodes and a downstream node.
For an equally spaced grid, the values of a desired quantity,

Finally, the difference equations related to the Eqs. (4) to (6) can be derived as follows:

The Damköhler number is a dimensionless number that reflects the exchange rate between the main channel and storage zones (Jin et al., 2009; Harvey and Wagner, 2000; Wagner and Harvey, 1997; Scott et al., 2003). For a stream or channel this number is defined as:

Uvas Creek (Santa Clara County, California) tracer study site map (Bencala and Walters, 1983).

In this section the TOASTS model is verified using several test cases. These test cases include analytical solutions of constant-coefficient governing equations for two types of upstream boundary condition (continuous and Heaviside) and also by comparing the model results with the 2-D model. Complementary explanations for each case are given below.

In this section, model verification is carried out using analytical
solutions presented by Kazezyılmaz-Alhan (2008).
The designed example is a 200 m length channel with constant cross-sectional
area equal to 1 m

In this case, a solute concentration of 5 mg m

Simulation parameters for the Uvas Creek experiment (test case 3).

Error indices of simulation of the Uvas Creek experiment (test case 3).

Observed and simulated chloride concentrations in the main channel (test case 3).

In order to show the model capability and assess the model accuracy in a
case without transient storage, the model is executed for

In this case a solute concentration of 5 mg m

The main cause of transient storage phenomena is velocity difference between
the main channel and storage zones. 2-D depth-averaged models consider
velocity variations in two dimensions and give more accurate predictions of
solute transport behavior in reality. Hence, they could be used for
verification of the presented 1-D model as a benchmark. For this purpose, a
hypothetical example was designed. To do so, a 1200 m long river,
with irregular cross-sections, is considered. Figures 5 and 6 show bed
topography of the hypothetical river. In order to take into account a
hypothetical storage zone, the distance between 300 and 600 m of the river
has been widened. The flow conditions in the river are considered to be
non-uniform and unsteady. The solute concentration in the main channel and
storage zone, at the beginning of the simulation (initial conditions), is
assumed to be zero. In calculations of both flow and transport models, space
and time steps are considered equal to 100 m and 1 min respectively. The
dispersion coefficient, storage zone area and exchange coefficient are 10 m

Simulation parameters related to test case 4.

In this section, the applications of the TOASTS model using a variety of hypothetical examples and several sets of observed data are presented. Some properties of these test cases are given in Table 6. As shown in this table, the test cases include a wide variety of solute transport simulation applications at different conditions.

In order to show the advantage of the numerical scheme used in the TOASTS
model, for advection-dominant problems, a hypothetical example was designed
and three numerical schemes were applied: CTQS (centered time–quick space), CTCS (centered time–centered space) and
BTCS(backward time–centered space). To do so,
steady flow by velocity of 1 m s

The TOASTS model results for simulation with and without transient storage (test case 3).

Observed and simulated storage zone concentrations computed by the TOASTS model (test case 3).

Observed and simulated strontium concentrations in the main channel (test case 4).

Observed and simulated sorbate strontium concentrations in the Uvas Creek experiment (test case 4).

This example illustrates the application of the TOASTS model in solute
transport simulation by first-order decay. A decaying substance enters the
stream with steady and uniform flow during a 2 h period. The solute
concentration at the upstream boundary is 100 mg m

Error indices of simulation of the Uvas Creek experiment (test case 4).

Error indices of the Athabasca River experiment (test case 5).

In the second case, by increasing the computational space step, all methods show a drop in the peak concentration, that its amount for the MIKE 11 model is more and for the TOASTS model is less than the others (Fig. 10a–c). Figure 10d–f and Table 9 show that the results of the models that use the central differencing scheme in spatial discretization of transport equations show more discrepancy in comparison with the analytical solution.

In the third case, flow velocity increased about four times. As illustrated in Fig. 11c, by increasing the Peclet number, the OTIS model results show more oscillations. This model also shows very intense oscillations in the longitudinal concentration profile in the form of negative concentrations (Fig. 11e), while observed oscillations in the TOASTS model are very small compared to the OTIS model (Fig. 11d). However, the QUICK scheme oscillations in advection-dominant cases are less likely to corrupt the solution. Also the MIKE 11 model results, in comparison with the TOASTS model, have greater difference with the analytical solution.

The main reason for the difference between the obtained results in the three cases is actually related to how advection and dispersion affect the solute transport. The dispersion process affects the distribution of solute in all directions, whereas advection acts only in the flow direction. This fundamental difference manifests itself in the form of limitation in computational grid size.

Simulation parameters related to test case 6.

Huey Creek experiment error indices (test case 6).

Simulation results for the Athabasca River experiment (test case 5).

Huey Creek tracer study site map (Runkel et al., 1998).

Observed and simulated main channel lithium concentrations (test case 6).

Simulated storage zone concentrations (test case 6).

This example shows the TOASTS model application to field data, by using the conservative tracer (chloride) injection experiment results, which was conducted in Uvas Creek, a small mountain stream in California (Fig. 12). Details of the experiments can be found in Avanzino et al. (1984). Table 10 shows simulation parameters for the Uvas Creek experiment (Bencala and Walters, 1983). For assessing efficiency and accuracy of the three discussed models in simulation of the impact of physical processes on solute transport in a mountain stream, they are implemented for this set of observed data. Figure 13a–c illustrates simulated chloride concentration in the main channel. It can be seen from these figures and Table 11 that the TOASTS model simulated the experiment results slightly better than the two other models. Comparison of Fig. 13a and b shows that the TOASTS and OTIS models have good accuracy in modeling the peak concentration and the TOASTS model has a slightly better performance in simulation of a rising tail of concentration–time curve, particularly in the 281 m station. Figure 13c shows MIKE 11 model results. It shows significant discrepancies with the observed data, particularly in peak concentrations. However, at the 38 m station, where transient storage has not still affected solute transport, the results of the three models have little difference with the observed data (Table 11). Figure 14 depicts the TOASTS model results for the Uvas Creek experiment for simulations with and without transient storage at the 281 and 433 m stations. This figure shows that in simulation with transient storage, the results have more fitness with the observed data in general shape of the concentration–time curve, peak concentration and peak arrival time. Figure 15 shows the simulated chloride concentrations in the storage zone. The concentration–time curves in the storage zone have longer tails in comparison with the main channel. That means some portions of the solute mass remain in the storage zones and gradually return to the main channel.

The objective of this test case is to demonstrate the capability of the TOASTS model in non-conservative solute transport modeling in natural rivers. For this purpose, the field experiment of the 3 h reactive tracer (strontium) injection into the Uvas Creek was used. The experiment was conducted at low-flow conditions and, due to the high opportunity of solute having frequent contact with relatively immobile streambed materials, solute and streambed interactions and solute sorption into bed sediments were more intense than during the high-flow conditions. Hence, the sorption process must be considered in simulation of this experiment (Bencala, 1983). Some of the simulation parameters are given in Table 12 and the other parameters are the same as those given in Table 10. Figure 16a–c and Table 13 show solute transport simulation results of the three subjected models in comparison with the observed data. According to these figures it could be said that the TOASTS model shows better fitness with the observed data. Figure 16c shows that simulation without taking into account the transient storage and kinetic sorption in the MIKE 11 model leads to very poor results. The zero exchange coefficient at the 38 m station causes reasonable results by this model at this station. Figure 17 illustrates the TOASTS and OTIS model results for sorbate concentrations on the streambed sediments versus the observed data at the 105 and 281 m stations. It is clear from this figure and Table 13 that the TOASTS model is slightly better fitted to the observed data.

This test case shows the TOASTS model application for a river with irregular
cross-sections under non-uniform flow conditions. The real data set for this
test case was collected in a tracer experiment which has been done in the
Athabasca River near Hinton, Alberta, Canada. Details of the experiments can
be found in Putz and Smith (2000). In this study, the simulation reach length
is 8.3 km, between 4.725 to 13.025 km of the river. The main reason for
selecting this reach is that it has common geometric properties of rivers
with storage zones. Total simulation time is 10 h, space and time steps
are considered equal to 25 m and 1 min, respectively. The exchange
coefficient is assumed equal to 6

This test case shows an application of the TOASTS model to simulate solute transport in a stream with irregular cross-sections, under an unsteady flow regime. In most of solute transport models, for simplification, flow is considered to be steady, while in most natural rivers unsteady flow condition is common, and neglecting temporal flow variations may lead to inaccurate results for solute transport simulation.

Tracer study that is used in this section, conducted in January 1992 at Huey
Creek, located in McMurdo valleys, Antarctica (Fig. 19). The flow rate was
variable from 1 to 4 c f s

In this study a comprehensive model was developed that combines numerical schemes with high-order accuracy for solution of the advection–dispersion equation, considering transient storage zones term in rivers. In developing the subjected model (TOASTS), for achieving better accuracy and applicability, irregular-cross sections and an unsteady flow regime were considered. For this purpose the QUICK scheme, due to its high stability and low approximation error, has been used for spatial discretization.

The presented model was verified successfully using several analytical solutions and 2-D hydrodynamics and transport model as benchmarks. Also, its validation and applications were proved using several hypothetical examples and four sets of well-established tracer experiments data under different conditions. The main concluding remarks of this research are as the following:

The numerical scheme used in the TOASTS model (i.e., CTQS scheme), in cases where advection is the dominant transport process (higher Peclet numbers), has less numerical oscillations and higher stability compared to the CTCS and BTCS numerical schemes.

For a specified level of accuracy, TOASTS can provide larger grid size, while other models based on the central scheme face step limitation that leads to more computational cost.

As shown by other researchers, the inclusion of transient storage and sorption in a classical advection–dispersion equation, in many cases, leads to more accurate simulation results.

The TOASTS model is a comprehensive and practical model, that has the ability of solute transport simulation (reactive and non-reactive), with and without storage, under both steady and unsteady flow regimes, in rivers with irregular cross-sections that from this aspect is unique compared to the other existing models. Thus, it could be suggested as a reliable alternative to current popular models in solute transport studies in natural rivers and streams.

In order to access the data, we kindly ask researchers to contact the corresponding author.