A comprehensive one-dimensional numerical model for solute transport in rivers

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.


Introduction
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: where A is the flow area, C the solute concentration, Q the volumetric flow rate, D the dispersion coefficient, λ the firstorder decay coefficient, S the source term, t the time and x the distance.When this equation is used to simulate the transport in prismatic channels and rivers with relatively uniform cross-sections, accurate results can be expected; but field studies, particularly in mountain pool-and-riffle streams, indicate that observed concentration-time curves have a lower peak concentration and longer tails than the ADE equation predictions (Godfrey and Frederick, 1970;Nordin and Sabol, 1974;Nordin and Troutman, 1980;Day, 1975).Thus a group of researchers, based on field studies, stated that to accomplish more accurate simulations of solute transport in natural rivers and streams, the ADE equation should be modified.They added some extra terms to it for consideration of the impact of stagnant areas that were so-called storage zones (Bencala et al., 1990;Bencala and Walters, 1983;Jackman et al., 1984;Runkel, 1998;Czernuszenko and Rowinski, 1997;Singh, 2003).Transient storage zones mainly include eddies, stream poolside areas, stream gravel bed, streambed sediments, porous media of river bed and banks and stagnant areas behind flow obstructions such as big boulders, stream side vegetation, woody debris and so on (Jackson et al., 2013).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 condi- tions (Bencala, 1983(Bencala, , 1984;;Bencala et al., 1990;Bencala and Walters, 1983).
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).
In this study, a comprehensive model, called   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.
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.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.

Governing equations
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: where ĈS is the background storage zone solute concentration, C sed is the sorbate concentration on the streambed sediment, K d is the distribution coefficient, λ and λ S are the main channel and storage zone first-order decay coefficients respectively, λ and λS are the main channel and storage zone sorption rate coefficients respectively, ρ is the mass of acces- sible sediment/volume water and L and S are the right-hand side differential operator of Eqs. ( 2) and ( 3) respectively.

Numerical solution scheme
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, ϕ, on the cell faces are given by the following equations: where P denotes an unknown node with neighbor nodes W (at left) and E (at right).It should be noted that the corresponding cell faces are denoted by the lowercase letters, w and e. Gradient at cell faces can be estimated using the following relationships: Finally, the difference equations related to the Eqs.( 4) to (6) can be derived as follows: Writing Eqs. ( 12) to ( 14) for all control-volumes in the solution domain and applying the boundary conditions, a system of linear algebraic equations will be introduced: where a W W , a W , a P , a E and R P are the corresponding coefficients and the right-hand side term.Solving this system, main channel concentrations in n + 1 time level will be computed.Having main channel concentration values, the storage zone and streambed sediment concentrations could be calculated.

Damköhler Index
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: where L is the main channel length, u the average flow velocity and DaI the Damköhler number.When DaI is much greater than unity (e.g., 100), the exchange rate between the main channel and storage zone is too fast and it could be assumed that these two segments are in balance.Accordingly, when DaI is much lower than unity (e.g., 0.01) the exchange rate between main channel and storage zone is very low and negligible.In other words, in such a stream where DaI is very low, there is practically no significant exchange between the main channel and storage zone, and transient storage zones do not affect downstream solute transport.Therefore, for reasonable estimation of transient storage model parameters, the DaI value must be within 0.1 to 10 range (Fernald et al., 2001;Wagner and Harvey, 1997;Ramaswami et al., 2005).

Model verification
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.

Verification by analytical solutions
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 2 .The flow discharge, dispersion coefficient, storage zone area and exchange coefficient are 0.01 m 3 s −1 , 0.2 m 2 s −1 , 1 m 2 and 0.00002 s −1 , respectively.The DaI number can be calculated from the Eq. ( 19) equal to 0.8.This example is implemented for two different types of upstream boundary conditions: (a) continuous and (b) Heaviside.

(a) Continuous boundary condition
In this case, a solute concentration of 5 mg m −3 is injected continuously for 10 h at the inlet.The time and space steps are considered equal to 30 s and 1 m, respectively.Figure 1 shows the TOASTS model results compared to the analytical solution at 50, 75 and 100 m from the inlet.Note that both axes have been nondimensionalized with respect to the maximum values.Also, square of correlation coefficient (R 2 ), root mean square error (RMSE), mean absolute error (MAE) and mean relative error (MRE) are given in Table 3.According to Fig. 1 and the error indices given in Table 3, it is clear that the trends of numerical and analytical solutions are similar, and the TOASTS model shows a good accuracy in this example.
In order to show the model capability and assess the model accuracy in a case without transient storage, the model is executed for α = 0 for this example, and the result at the distance of 100 m from the inlet is compared to the analytical solution www.hydrol-earth-syst-sci.net/21/99/2017/ Hydrol.Earth Syst.Sci., 21, 99-116, 2017 of the classical advection-dispersion equation.The results are shown in Fig. 2 and Table 3. Figure 2 also illustrates that in the case of transient storage, the concentration-time curve has a lower peak than the one without storage (α = 0), which matches the previously mentioned transient storage concept.

(b) Heaviside boundary condition
In this case a solute concentration of 5 mg m −3 is injected at the inlet for a limited time of 100 min.The time and space steps are considered equal to 30 s and 1 m, respectively.Comparison of the model results and the analytical solution at the distance of 50, 75 and 100 m from the inlet is presented in Fig. 3 and Table 4. Also, corresponding results at the distance of 100 m for the case without storage (α = 0) are given in Fig. 4 and Table 4.It is obvious that the TOASTS model results in both cases (with and without storage) have a reasonable agreement with the analytical solution.

Verification by 2-D model
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 2 s −1 , 22 m 2 and 1.8 × 10 −4 s −1 , respectively.For this example the DaI number is calculated equal to 0.

Test case 1: pure advection
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 −1 was assumed.Total simulation time was 5 h and space and time steps were 100 m and 10 s respectively.Note that advection is the only transport mechanism.The results of this test case are depicted in Fig. 8.It is clear that, for the pure advection simulation, the CTQS scheme has less oscillation than the other two schemes.In particular, this figure indicates that the result of the CTCS scheme, which is used in the OTIS model, shows high oscillations.Therefore, it can be concluded that for advection-dominant sim-ulation the TOASTS model has a better performance.It is interesting to note that in mountain rivers where the transient storage mechanism is more observed, due to relatively high slope, higher flow velocities occur which lead to advectiondominant solute transport.

Test case 2: transport with first-order decay
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 −3 .Also, in order to assess the TOASTS model capability in the case of high-flow velocity and advection-dominant transport, this example implemented for three cases with different Peclet numbers.The simulation parameters for different cases are given in    8 and 9.It is obvious from Fig. 9a-c that in the first case (Peclet number less than 2), all methods simulated concentration-time curves accurately.Also, Fig. 9d-f show that the MIKE 11 model cannot simulate a concentration longitudinal profile accurately, because it does not consider the transient storage effect on solute transport.
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).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.

Test case 3: conservative solute transport with transient storage
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 they are implemented 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.

Test case 4: non-conservative solute transport with transient storage
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 lowflow 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     This test case shows an application of the TOASTS model to simulate solute transport in a stream with irregular crosssections, 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 −1 (cubic feet per second) during the experiments.Since this stream does not have obvious surface storage zones, the cross-sectional area of storage zones and exchange rate of this area actually represent the rate of hyporheic exchange and interaction between surface and subsurface water (Runkel et al., 1998).Details of the experiments can be found in Runkel et al. (1998).Table 15 shows the simulation parameters.Figure 20a-c and Table 16 represent simulation results of lithium concentration at the 213 and 457 m stations, by the three subjected models.The results of the TOASTS model have a slightly better fitness to the observed data than the two other models.This figure also indicates that the general shape of the concentration-time curve for this example is a little different from the other examples.Figure 20c represents the results of the MIKE 11 model.As seen in this figure, results have large differences with the observed data in peak concentrations and general shape of the curve.Figure 21 shows the corresponding storage zone concentrations at 213 and 457 m stations.It can be seen that solute concentration-time curves in the storage zone have lower peaks and much longer tails, which imply longer residence time of solute in these areas compared to the main channel.

Conclusions
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.

Data availability
In order to access the data, we kindly ask researchers to contact the corresponding author.
Edited by: Y. Chen Reviewed by: three anonymous referees

Figure 1 .
Figure 1.Results of the TOASTS model verification by the analytical solution for continuous boundary condition (α = 0).

Figure 2 .
Figure 2. Results of the TOASTS model verification by the analytical solution for continuous boundary condition (α = 0).

Figure 3 .Figure 4 .
Figure 3. Results of the TOASTS model verification by the analytical solution for Heaviside boundary condition (α = 0).

Figure 5 .
Figure 5. Bed elevation contours of the 2-D hypothetical example.

Figure 6 .
Figure 6.Bed elevation three-dimensional view of the 2-D hypothetical example.

Figure 7 .
Figure 7. Results of the TOASTS model verification using the 2-D model.

Figure 9 .
Figure 9.Comparison of the TOASTS, OTIS and MIKE 11 models in test case 2 for Pe = 0.24.

Figure 10 .
Figure 10.Comparison of the TOASTS, OTIS and MIKE 11 models in test case 2 for Pe = 2.4.

Figure 11 .
Figure 11.Comparison of the TOASTS, OTIS and MIKE 11 models in test case 2 for Pe = 10.
4. The upstream boundary condition for transport submodel is a 3 h lasting step loading pulse with 20 mg m −3 pick concentration.The results of the TOASTS model for simulating with and without transient storage were compared to the 2-D model at the distance of 800 m from the inlet.Figure 7 and

Figure 13 .
Figure 13.Observed and simulated chloride concentrations in the main channel (test case 3).

Figure 15 .
Figure 15.Observed and simulated storage zone concentrations computed by the TOASTS model (test case 3).

Figures 9 -
Figures 9-11 show simulation results of the three numerical models in comparison with analytical solution.Error indices are given in Tables8 and 9.It is obvious from Fig.9a-cthat in the first case (Peclet number less than 2), all methods simulated concentration-time curves accurately.Also, Fig.9d-fshow that the MIKE 11 model cannot simulate a concentration longitudinal profile accurately, because it does not consider the transient storage effect on solute transport.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 Table9show 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 Figures 9-11 show simulation results of the three numerical models in comparison with analytical solution.Error indices are given in Tables8 and 9.It is obvious from Fig.9a-cthat in the first case (Peclet number less than 2), all methods simulated concentration-time curves accurately.Also, Fig.9d-fshow that the MIKE 11 model cannot simulate a concentration longitudinal profile accurately, because it does not consider the transient storage effect on solute transport.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 Table9show 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

Figure 16 .Figure 17 .
Figure 16.Observed and simulated strontium concentrations in the main channel (test case 4).

Figure 18 .
Figure 18.Simulation results for the Athabasca River experiment (test case 5).

Table 3 .
Error indices of verification by the analytical solution for continuous boundary condition.

Table 4 .
Error indices of verification by the analytical solution for Heaviside boundary condition.

Table 5 .
Error indices of verification by the 2-D model.

C/C in t/t max
where A and A S are the main channel and storage zone crosssectional area respectively, C, C L and C S are the main chan-

Table 6 .
Properties of the test cases used for the TOASTS model application.

Table 7 .
Simulation parameters related to test case 2.

Table 8 .
Error indices of concentration time series in test case 2.

Table 9 .
Error indices of concentration longitudinal profiles in test case 2.

Table 10 .
Simulation parameters for the Uvas Creek experiment (test case 3).

Table 11 .
Error indices of simulation of the Uvas Creek experiment (test case 3).
Table5illustrates these results.This figure shows that with appropriate choice of A S and α, concentrationtime curves given by the TOASTS model are close to those given by the 2-D model.These results also imply the necessity of considering transient storage term in the advectiondispersion equation for more accurate simulation of solute transport, especially in natural rivers and streams.
ing a variety of hypothetical examples and several sets of observed data are presented.Some properties of these test cases are given in Table6.As shown in this table, the test cases include a wide variety of solute transport simulation applications at different conditions.

Table 12 .
Simulation parameters related to test case 4.

Table 13 .
Error indices of simulation of the Uvas Creek experiment (test case 4).

Table 14 .
Error indices of the Athabasca River experiment (test case 5).

Table 15 .
Simulation parameters related to test case 6.