Estimation of the Longitudinal Dispersion Coefficient Using the Velocity Profile in Natural Streams

In this study, a theoretical method for predicting the longitudinal dispersion coefficient is developed based on the transverse velocity distribution in natural streams. Equations of the transverse velocity profile for irregular cross sections of the natural streams are analyzed. Among the velocity profile equations tested in this study, the beta distribution equation, which is a probability density function, is considered to be the most appropriate model for explaining the complex behavior of the transverse velocity structure of irregular natural streams. The new equation for the longitudinal dispersion coefficient that is based on the beta function for the transverse velocity profile is developed. A comparison of the proposed equation with existing equations and the observed longitudinal dispersion coefficient reveals that the proposed equation shows better agreement with the observed data compared to other existing equations.     

Dispersion Coefficient of Streams from Tracer Experiment Data

The paper deals with a method for optimal identification of the dispersion coefficient for streams from observed concentration profiles at downstream sections (at least two sections are required) following injection of an environmentally safe tracer at an upstream section. The method makes use of the exact solution of the one-dimensional advection-dispersion equation. A reliable and accurate procedure is proposed for routing an arbitrary concentration variation further downstream using the convolution equation and pulse kernels. The new method does not require the frozen cloud approximation and avoids the error due to numerical integration of the convolution integral used for routing the concentration. It employs the objective criterion of minimum integral squared errors between observed and computed concentrations. Application of the method to field data sets shows that reliably accurate estimates of the dispersion coefficient are obtained.     

Predicting Longitudinal Dispersion Coefficient in Natural Streams by Artificial Neural Network

An artificial neural network (ANN) model was developed to predict the longitudinal dispersion coefficient in natural streams and rivers. The hydraulic variables [flow discharge (Q), flow depth (H), flow velocity (U), shear velocity (u*), and relative shear velocity (U/u*)] and geometric characteristics [channel width (B), channel sinuosity (σ), and channel shape parameter (β)] constituted inputs to the ANN model, whereas the dispersion coefficient (Kx) was the target model output. The model was trained and tested using 71 data sets of hydraulic and geometric parameters and dispersion coefficients measured on 29 streams and rivers in the United States. The training of the ANN model was accomplished with an explained variance of 90% of the dispersion coefficient. The dispersion coefficient values predicted by the ANN model satisfactorily compared with the measured values corresponding to different hydraulic and geometric characteristics. The predicted values were also compared with those predicted using several equations that have been suggested in the literature and it was found that the ANN model was superior in predicting the dispersion coefficient. The results of sensitivity analysis indicated that the Q data alone would be sufficient for predicting more frequently occurring low values of the dispersion coefficient (Kx<100?m2/s). For narrower channels (B/H<50) using only U/u* data would be sufficient to predict the coefficient. If β and σ were used along with the flow variables, the prediction capability of the ANN model would be significantly improved.     

Master Diagnostic Curve for Dispersion Coefficient of Soils

A master diagnostic curve (MDC) method is proposed for identifying the dispersion coefficient from observed breakthrough curve in soil–column experiments. The method uses matching of diagnostically plotted points to the MDC. Accurate identification of the dispersion coefficient is possible with a parallel shift of only one axis. Another advantage of the method is that the MDC is an invariant curve and once prepared, can be used for different data sets characterized by high Péclet number. The new method is simple, does not require large computations as in conventional least-squares approach, and yields a quick and accurate estimate of the dispersion coefficient. The new method does not require an initial guess for the dispersion coefficient. It can also yield a value of the dispersion coefficient from a single observed concentration. Although subjectivity is involved in matching, there is a transparent visual realization of the reliability of the estimated dispersion coefficient, and the points with substantial errors. The new method has several advantages over the least-squares approach and is suited for an advanced study on the subject. An estimate of the dispersion coefficient obtained using the new method is as good as obtained using a good optimization method.     

Estimating Dispersion Coefficient and Porosity from Soil-Column Tests

A simple method for simultaneous and explicit estimation of dispersion coefficient and porosity from laboratory data on soil-column tests is presented. It makes use of the peak derivative of the observed breakthrough curve (BTC). An objective procedure for approximate location of the peak is suggested based upon a parabolic fit through three consecutive points around the peak. The computations involved are simple and can be done on a calculator. The method does not require either the rate of flow or the cross-sectional area of the soil-column for estimating the dispersion coefficient. This completely avoids the possible errors involved in the measurement of these quantities. The new method does not require the full BTC to be observed, however, its application does require data slightly beyond the time at which the exit concentration reaches half the input concentration. Application of the method on a number of laboratory data sets shows that it can yield a reliable estimate of the dispersion coefficient and porosity using only a few points near the peak. The method is applicable for high values (>100) of the piclet number.     

Empirical Relations for Longitudinal Dispersion in Streams

Although several methods are available for dispersion in natural streams, no method is accurate enough to satisfactorily predict the time variation of stream pollution concentration. Further, limited studies exist for dispersion of nonconservative pollutants. In this paper a six-parameter concentration equation for dispersion of conservative and nonconservative pollutants has been proposed. The parameters of the equation have been related to hydraulic variables and stream geometry. Using these predictors, the equation is fairly accurate for concentration predictions. It is hoped that the equation is useful in water quality management studies.     

Longitudinal Dispersion Coefficient in Straight Rivers

An analytical method is developed to determine the longitudinal dispersion coefficient in Fischer's triple integral expression for natural rivers. The method is based on the hydraulic geometry relationship for stable rivers and on the assumption that the uniform-flow formula is valid for local depth-averaged variables. For straight alluvial rivers, a new transverse profile equation for channel shape and local flow depth is derived and then the lateral distribution of the deviation of the local velocity from the cross-sectionally averaged value is determined. The suggested expression for the transverse mixing coefficient equation and the direct integration of Fischer's triple integral are employed to determine a new theoretical equation for the longitudinal dispersion coefficient. By comparing with 73 sets of field data and the equations proposed by other investigators, it is shown that the derived equation containing the improved transverse mixing coefficient predicts the longitudinal dispersion coefficient of natural rivers more accurately.     

Stage–Discharge Relations for Low-Gradient Tidal Streams Using Data-Driven Models

Development of stage–discharge relationships for coastal low-gradient streams is a challenging task. Such relationships are highly nonlinear, nonunique, and often exhibit multiple loops. Conventional parametric regression methods usually fail to model these relationships. Therefore, this study examines the utility of two data-driven computationally intensive modeling techniques namely, artificial neural networks and local nonparametric regression, to model such complex relationships. The results show an overall good performance of both modeling techniques. Both neural network and local regression models are able to predict and reproduce the stage–discharge multiple loops that are observed at the outlet of a 28.5?km2 low-gradient subcatchment in southwestern Louisiana. However, the neural network model is characterized with higher prediction ability for most of the tested runoff events. In agreement with the physical characteristics of low-gradient streams, the results indicate the importance of including information about downstream and upstream water levels, in addition to water level at the prediction site.     

Eulerian-Lagrangian Numerical Scheme for Simulating Advection, Dispersion, and Transient Storage in Streams and a Comparison of Numerical Methods

Past applications of one-dimensional advection, dispersion, and transient storage zone models have almost exclusively relied on a central differencing, Eulerian numerical approximation to the nonconservative form of the fundamental equation. However, there are scenarios where this approach generates unacceptable error. A new numerical scheme for this type of modeling is presented here that is based on tracking Lagrangian control volumes across a fixed (Eulerian) grid. Numerical tests are used to provide a direct comparison of the new scheme versus nonconservative Eulerian numerical methods, in terms of both accuracy and mass conservation. Key characteristics of systems for which the Lagrangian scheme performs better than the Eulerian scheme include: nonuniform flow fields, steep gradient plume fronts, and pulse and steady point source loadings in advection-dominated systems. A new analytical derivation is presented that provides insight into the loss of mass conservation in the nonconservative Eulerian scheme. This derivation shows that loss of mass conservation in the vicinity of spatial flow changes is directly proportional to the lateral inflow rate and the change in stream concentration due to the inflow. While the nonconservative Eulerian scheme has clearly worked well for past published applications, it is important for users to be aware of the scheme’s limitations.     

Dam Break in Channels with 90° Bend

In practice, dam-break modeling is generally performed using a one-dimensional (1D) approach for its limited requirements in data and computation. However, for valleys with multiple sharp bends, such a 1D model may fail for predicting as well the maximum water level as the wave arrival time. This paper presents an experimental study of a dam-break flow in an initially dry channel with a 90° bend, with refined measurements of water level and velocity field. The measured data are compared to some numerical results computed with finite-volume schemes associated with Roe-type flux calculation. The 1D approach reveals the expected limits, while a full two-dimensional (2D) approach provides fine level prediction and rather satisfactory information about the arrival time. A hybrid approach is now proposed, mixing the 1D model for the straight reaches and local 2D models for the bends. The compatibility of the Roe fluxes at the interfaces requires a careful formulation, but the resulting scheme seems able to capture reflection and diffraction processes in such a way that the results are really good in what concerns the water level.     

Discharge Coefficient for a Water Flow through a Bottom Orifice of a Conical Hopper

Discharge coefficients for water flow through a vertical, circular orifice at the bottom of a conical hopper were experimentally studied in the present paper. The conical hopper consists of a cylindrical hopper of inside diameter of 48 cm and a bottom cone of side slope of 45°. Experiments were carried out under different orifice diameters and water heads. The dependence of the discharge coefficient on the orifice diameter and water head was analyzed, and then an empirical relation was developed by using a dimensional analysis and a regression analysis. The results show that the larger orifice diameter or higher water head have a smaller discharge coefficient and the orifice diameter plays more significant influence on the discharge coefficient than the water head does. The discharge coefficient of water flow through a bottom orifice is larger than that through a sidewall orifice under the similar conditions of the water head, orifice diameter, and hopper size.     

Transverse Dispersion Caused by Secondary Flow in Curved Channels

A new theoretical equation is proposed to describe the streamwise variations of the transverse velocity along a curved channel with a constant curvature. Furthermore, based on this theoretical equation for the transverse velocity, a new equation for the transverse dispersion coefficient is developed to incorporate the effect of the secondary flow on the transverse dispersion in curved channels. The new equations for the transverse velocity and dispersion coefficient are verified with experimental data sets that were obtained from laboratory experiments conducted in two different channels. The results show that the proposed velocity equation properly describes the streamwise variations of the secondary flow developed in the curved channels. The reach-averaged values of the transverse dispersion coefficient calculated by the new equation are in relatively good agreement with the observed values from the laboratory channels. Sensitivity analysis reveals that both the secondary flow and the transverse dispersion coefficient are proportional to the roughness factor, and in inverse proportion to the aspect ratio of the channel.     

Flow Depletion of Semipervious Streams Due to Unsteady Pumping Discharge

Analytical expressions for rate and volume of flow depletion of semipervious streams due to sinusoidal variation in pumping rate are obtained. An analytical but approximate method is developed for obtaining the rate and volume of stream flow depletion due to arbitrary unsteady pumping discharge. The method uses the ramp kernel and convolution. The use of ramp kernels permits linear interpolation between two consecutive discretized discharge values. The analytical equations for the ramp kernels for the rate and volume of stream flow depletion are derived. The proposed method is applicable for homogeneous and isotropic aquifers that are hydraulically connected to streams.     

Dimensional Analysis of Reaeration Rate in Streams

Atmospheric reaeration at the free surface of lakes and streams is a relevant process for water quality, thus the amount of oxygen transferred to the water body should be carefully estimated. Recent studies have demonstrated that available equations for estimation of the reaeration rate offer a poor fit with field data different from those for which each equation was originally developed. Thus, none of the available equations is applicable to all stream hydrodynamic conditions; on the contrary, they remain stream-specific, probably since some parameters involved in the process have been neglected in their formulation and their expressions are too simplistic. This paper proposes a comprehensive approach to the mass-transfer process at the air-water interface that is based on dimensional analysis. Careful inspection of equations in the literature shows that the mass-transfer process at the air-water interface has been affected by 14 different parameters. The application of dimensional analysis produces, for a wide rectangular section if wind speed is negligible, a dimensionless equation for the mass-transfer rate, where this rate is a function of the Froude number, channel slope, Reynolds number, Sherwood number, Weber number, and relative roughness. This expression is further developed to address the reaeration process in streams and rivers. As a result, at a fixed temperature, the dimensionless reaeration rate KaND (where ND denotes nondimensional) is finally a function of only the Froude number, channel slope, Reynolds number, and relative roughness. Moreover, the application of the Darcy-Wiesbach equation allows this dimensionless rate KaND to be considered as a function of only three of the aforementioned parameters. This result provides a comprehensive approach to the reaeration process that can also explain the unreliability of the literature equations available up to now.     

Sorption Behavior and Long-Term Retention of Reactive Solutes in the Hyporheic Zone of Streams

This paper analyzes the transport of sorbing solutes by extending the advective storage path model developed for longitudinal transport of inert solutes in streams coupled with flow-induced uptake in the hyporheic zone. Independent observations of a conservative (3H) and a reactive (51Cr) tracer in both the stream water and the hyporheic zone were used to differentiate between hydraulic and sorption processes. The method of temporal moments was found to be inadequate for parameter determination, whereas fitting versus the entire tracer breakthrough curves with special emphasis on the tail indicates that the proposed model could be used to represent both conservative and reactive transport. Information on the tracer inventory of the conservative tracer in the hyporheic zone was found to be of vital importance to the evaluation of the hydraulic exchange. A model evaluation based on stream water data alone can yield predictions of a wash-out in the hyporheic zone that deviates markedly from the observed wash-out. This prohibits long-term predictions of the wash-out from the hyporheic zone as well as the evaluation of sorption properties. The sorption in the hyporheic zone was found to follow a two-step model, where the first step is instantaneous and the second kinetic. A model with a single-step sorption process could not reproduce the observed breakthrough curves. An evaluation of the relative importance of including sorption kinetics in solute stream transport models is elucidated by means of the analytical expressions for the temporal moments. The omission of the kinetics in the second sorption step in the hyporheic zone will result in relative errors in the moments of second order or higher. The error will increase with decreasing residence time in the hyporheic zone. Especially, long-term predictions of the wash-out from the hyporheic zone require consideration of the rate-limited sorption.     

Linking Pathogen Sources to Water Quality in Small Urban Streams

Alternative measures of terrestrial pollutant loading were investigated to identify those that are better predictors of water quality in urban streams. Results from 18 watersheds with the same climatic conditions show that the density of terrestrial fecal-coliform loading is a better indicator of median instream concentrations than total terrestrial fecal-coliform loading. Watersheds with fecal-coliform loading densities less than around 2×1011?cfu?km?2?day?1 generally had median instream concentrations less than the reference water-quality standard of 400 cfu/100 mL. Median instream concentrations were also less than the reference water-quality standard for population densities less than around 400?persons?km?2. For any given terrestrial loading intensity or population density, summer conditions of high rainfall and high temperature generally resulted in the greatest water-quality impacts. These results are particularly useful in determining terrestrial loading reductions in support of TMDLs, and in focusing best management practices.     

Flow Depletion of Semipervious Streams Due to Pumping

Expressions for the rate and volume of flow depletion of semipervious streams due to pumping are presented in computationally simple forms. Analytical expressions have been proposed to take into account both partial penetration and semipervious bed and banks of the stream. Graphs suitable for engineering applications are presented for siting wells, and the effect of an intermittent pumping cycle on the rate and volume of stream flow depletion has also been discussed. The exclusive volume of flow depletion during a cycle is shown to vary with the selection of the end of the cycle.     

Calculation of Contraction Coefficient under Sluice Gates and Application to Discharge Measurement

The contraction coefficient under sluice gates on flat beds is studied for both free flow and submerged conditions based on the principle of momentum conservation, relying on an analytical determination of the pressure force exerted on the upstream face of the gate together with the energy equation. The contraction coefficient varies with the relative gate opening and the relative submergence, especially at large gate openings. The contraction coefficient may be similar in submerged flow and free flow at small openings but not at large openings, as shown by some experimental results. An application to discharge measurement is also presented.