An Empirical Approach for Determining Longitudinal Dispersion Coefficients in Rivers

Gene expression models for prediction of longitudinal dispersion coefficient in streams

Among several indicators for river engineering sustainability, the longitudinal dispersion coefficient () is the main parameter that defines the transport of pollutants in natural streams. Accurate estimation of has been challenging for hydrologists due to the high stochasticity and non-linearity of this hydraulic-environmental parameter. This study presents a new hybrid machine learning (ML) model integrating a Gaussian Process Regression (GPR) and an evolutionary feature selection (FS) approach (i.e. Covariance Matrix Adaptation Evolution Strategy (CMAES)) to estimate in natural streams. The dataset consists of geometric and hydraulic river system parameters from 29 streams in the United States. The modeling results showed that the proposed model outperformed other models in the literature, producing more stable and accurate estimations. The FS approach evidenced the significance of the cross-sectional average flow velocity (U), channel width (B), and channel sinuosity σ to estimate the dispersion coefficient. In quantitative terms, the integrated GPR model with feature selection approach attained the minimum root mean square error () and maximum coefficient of determination (). The proposed hybrid evolutionary ML model arises as robust, flexible and reliable alternative computer aid technology for predicting the longitudinal dispersion coefficient in natural streams.

Prediction of longitudinal dispersion coefficient in natural streams by prediction map

A method is given for predicting longitudinal dispersion coefficients in natural streams. The analysis is based on recognition that transverse velocity gradients (rather than the previously considered vertical velocity gradients) are the primary dispersive mechanism. The method consists of measuring the average shear and cross-sectional velocity distribution of the particular stream in the field. An integral formula can then be applied to predict the dispersion coefficient. Laboratory experiments are described which verify the theoretical results in flows whose velocity distributions are similar to those found in natural streams. Experiments in real streams will be described in future papers.

In nature, aquatic vegetation is one of the important factors—along with hydraulic characteristics and geometric configuration—determining the dispersion of scalars. Previous research has studied the effects of vegetation on longitudinal dispersion, varying plant population with uniform arrangement. However, since vegetation grows more often in clumped and heterogeneous rather than uniform patterns, the present work investigates the effects of the vegetation arrangement on the flows characteristics such as longitudinal dispersion, mean velocities, and drag coefficients. Several types of vegetation arrangement are described with the isometric and allometric concept and quantified with the standardized Morisita index. Laboratory experiments were performed to investigate variations of hydraulic parameters and longitudinal dispersion coefficients according to vegetation arrangements, which were quantified with the standardized Morisita index. The hydraulic parameters of mean velocity, turbulent kinetic energy, and drag depend on the vegetation arrangements, and in particular, the longitudinal dispersion coefficient varies with these arrangements by a factor of 4 or more.

Empirical models for longitudinal dispersion coefficient in natural streams

This study concerns the longitudinal dispersion of fluid particles which are initially distributed uninformly over one cross section of a uniform, steady, turbulent open channel flow. The primary focus is on developing a method to predict the rate of dispersion in a natural stream. Taylor's method of determining a dispersion coefficient, previously applied to flow in pipes and two-dimensional open channels, is extended to a class of three-dimensional flows which have large width-to-depth ratios, and in which the velocity varies continuously with lateral cross-sectional position. Most natural streams are included. The dispersion coefficient for a natural stream may be predicted from measurements of the channel cross-sectional geometry, the cross-sectional distribution of velocity, and the overall channel shear velocity. Tracer experiments are not required. Large values of the dimensionless dispersion coefficient D/rU* are explained by lateral variations in downstream velocity. In effect, the characteristic length of the cross section is shown to be proportional to the width, rather than the hydraulic radius. The dimensionless dispersion coefficient depends approximately on the square of the width to depth ratio. A numerical program is given which is capable of generating the entire dispersion pattern downstream from an instantaneous point or plane source of pollutant. The program is verified by the theory for two-dimensional flow, and gives results in good agreement with laboratory and field experiments. Both laboratory and field experiments are described. Twenty-one laboratory experiments were conducted: thirteen in two-dimensional flows, over both smooth and roughened bottoms; and eight in three-dimensional flows, formed by adding extreme side roughness to produce lateral velocity variations. Four field experiments were conducted in the Green-Duwamish River, Washington. Both laboratory and flume experiments prove that in three-dimensional flow the dominant mechanism for dispersion is lateral velocity variation. For instance, in one laboratory experiment the dimensionless dispersion coefficient D/rU* (where r is the hydraulic radius and U* the shear velocity) was increased by a factory of ten by roughening the channel banks. In three-dimensional laboratory flow, D/rU* varied from 190 to 640, a typical range for natural streams. For each experiment, the measured dispersion coefficient agreed with that predicted by the extension of Taylor's analysis within a maximum error of 15%. For the Green-Duwamish River, the average experimentally measured dispersion coefficient was within 5% of the prediction.

Mixing is the process that homogenizes initially segregated miscible constituents, increases the volume occupied by a solute, and decreases concentration peaks. It is important for the assessment of contamination levels and biogeochemical reactions in groundwater and soils. Mixing processes are governed by the interplay of fluid advection, molecular diffusion and local-scale dispersion at Darcy scale. Here we study the mechanisms of mixing in three-dimensional Darcy scale porous media with different heterogeneity structure. We analyze the role of medium and flow topology on the mixing and dispersion behavior. To this end, we perform Darcy-scale numerical simulations of incompressible flow and transport in heterogeneous three-dimensional porous media. Hydraulic conductivity is represented as a multi-Gaussian random field with lognormal marginal distribution. We consider isotropic and anisotropic correlation structures and scalar and tensorial conductivity. Flow is solved using a finite volume two-point method and transport using a Lagrangian approach. The flow topology is quantified by the helicity of the velocity field. We consider a planar injection of particles. Dispersion is quantified by the longitudinal and transverse dispersion coefficient, which are determined by the evolution along time of the position’s variance in the respective direction divide by two. It is also quantified by the breakthrough curves, which measure the distribution of arrival times at a given position from the initial one. Mixing is quantified by the ability of the flow to stretch and elongate a fluid strip which enhances diffusion through the creation and sustaining of concentration gradients. Results show that for a helical flow, a finite transverse dispersion coefficient is observed at long times and that the elongation of elemental strips follow an exponential stretching  (for large logK variances). On the contrary, on non-helical flows, transverse dispersion tends asymptotically to zero and the stretching rate is algebraic. The longitudinal dispersion coefficient seems unaffected by the helicity of the flow. These results shed light on the relation between medium structure and flow topology on mixing, making an important step towards the control, upscaling and large scale representation of mixing in porous media .  

Longitudinal dispersion coefficients in natural channels

Abstract. The effective longitudinal dispersion is a primary tool for determining property distributions in estuaries. Most previous studies have examined the longitudinal dispersion coefficient for the average tidal condition. However, information on spatial and temporal variations of this coefficient at low and high tides is scarce. Three years of hydrographic data taken at low and high tide along the main axis of the Sumjin River Estuary (SRE), Korea are used to estimate the spatial and temporal variation of the effective longitudinal dispersion coefficient. The range of the dispersion coefficient is rather broad at high water slack (HWS) and narrower at low water slack (LWS) because of the different tidal amplitudes. The spatially varying dispersion coefficient has maximal values (>300 m2 s−1) near the mouth at high water and decreases gradually upstream, with fluctuations. The temporally varying dispersion coefficient appears to be positively correlated with river discharges at both low and high tide. The dispersion varies with the square root of river discharges at HWS and LWS. The dispersive salt fluxes increases with increasing river discharges and decreases with decreasing river discharges at HWS and LWS. Estimation of the numerical values of the effective longitudinal dispersion coefficient in the SRE can be useful for better understanding of the distributions of other tracers in the SRE as well as for developing and testing hypotheses about various mixing mechanisms.

The conceptualized hybrid-cells-in-series model, consists of a plug flow zone and two thoroughly mixed unequal reservoirs, all connected in series, has three time parameters, namely: (1) residence time of solute in the plug flow zone; and (2) residence times of solute in the two thoroughly mixed reservoirs. The model simulates closely advection-dispersion solute transport in natural streams. The resident time parameters are related to the velocity of flow, width of water surface, and depth of flow in the stream. Through the Peclet number, defined as Pe =(Δxu)∕ DL (in which Δx =process unit size; u =mean flow velocity; and DL =longitudinal dispersion coefficient), the relations of the model parameters with the longitudinal dispersion coefficient and with the bulk stream flow characteristics have been established. For a given reach of a stream, the parameters are inversely proportional to the flow velocity. By decoupling of pure advection by the plug flow component and dispersion of tracer by the two thorou...

The best currently available description of the longitudinal mixing properties of a porous medium is an equation of the formEquation 1 which relates the effective longitudinal dispersion coefficient Dl to the molecular diffusion coefficient D0, the electrical resistivity factor F, the porosity f and a Peclet number. If the parameters dps and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method, featuring on-line gas analysis by thermal conductivity and on-line data reduction by analog computation, was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two dolomite samples. The exponent m of the above equation was found to vary between 1.0 and 1.5. The characteristic length dp s in the above equation was found to vary between 0.25 and 1.9 cm, with an average value of 0.4 cm for sandstones. Measurements were made on two cores in which paraffin wax had been deposited by evaporation from a pentane solution. They indicated that the presence of an immobile phase such as connate water could increase the dispersion coefficients significantly. INTRODUCTION While the petroleum and chemical industries have studied the mixing of miscible liquids flowing in consolidated porous media and of miscible gases flowing in unconsolidated porous media, relatively little data have been presented to describe the mixing of gases flowing through consolidated porous media. Such data are of particular interest to the gas storage industry. For instance, the U.S. Bureau of Mines is storing large quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15 percent of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure and rate of movement. This investigation was concerned only with the determination of longitudinal dispersion coefficients. It is understood that a transverse dispersion coefficient, which characterizes mixing perpendicular to the direction of flow, may be an order of magnitude less than the coefficient characterizing mixing in the direction of bulk flow.5,19 It should also be recognized that the use of any dispersion coefficient is in itself a simplification. It is necessary to assume that mixing in a porous medium may be characterized by the equationEquation 2 for flow in a single direction. A number of authors1 have pursued the mixing problem, not in terms of the so-called "dispersion model" described by Eq. 1, but in terms of a "mixing cell model". This model supposes that a porous medium is constructed of a large number of small mixing chambers and that the concentration of the diffusing component within each mixing chamber is uniform. Fick's law (Eq. 1) assumes that there is no gross by-passing of one fluid by another, and that there are not stagnant pockets of gas in the system under consideration as discussed by Coats and Smith.8 These assumptions are not always valid for flow through porous media and it is important to recognize the limitations upon Eq. 1.

ABSTRACTThe focus of this paper is on the ice-cover effects on the longitudinal dispersion coefficient in rivers. The dispersion coefficients due to either the vertical or transverse velocity shear are calculated using Elder’s or Fisher’s triple integral equation, respectively, based on the assumptions of a parabolic cross section and a vertical logarithmic velocity distribution in both the riverbed and the ice-cover zones. The results show that in ice-covered natural rivers, where the aspect ratio is usually larger than 10, the dispersion coefficient due to the transverse velocity shear is 50 times greater than that due to the vertical velocity shear. An analytical formula for longitudinal dispersion coefficient is further developed by the simplification of Fischer’s triple integral equation, and it is validated against field tracer tests with discrepancy between observations and predictions less than 150%. Comparing the proposed formula to the most widely-used Fischer’s formula demonstrates that for wide and shallow rivers the presence of ice-cover will lead to about a three-fold increase in the magnitude of the longitudinal dispersion.

Rolling friction representing the energy dissipation mechanism with the elastic deformation at the contact point could act directly on particle percolation. The present investigation intends to elucidate the influence of rolling friction coefficient on inter-particle percolation in a packed bed by discrete element method (DEM). The results show that the vertical velocity of percolating particles decreases with increasing the rolling friction coefficient. With the increase of rolling friction coefficient, the transverse dispersion coefficient decreases, but the longitudinal dispersion coefficient increases. Packing height has a limited effect on the transverse and longitudinal dispersion coefficient. In addition, with the increase of size ratio of bed particles to percolation ones, the percolation velocity increases. The transverse dispersion coefficient increases with the size ratio before D/d<14. And it would keep constant when the size ratio is greater than 14. The longitudinal dispersion coefficient decreases when the size ratio increases up to D/d=14, then increases with the increase of the size ratio. External forces affect the percolation behaviours. Increasing the magnitude of the upward force (e.g. from a gas stream) reduces the percolation velocity, and decreases the dispersion coefficient.

Investigation of the water quality of rivers is a key point in Water Resources Engineering. The longitudinal dispersion coefficient is one of the foremost significant parameters in river water quality monitoring. Several parameters such as hydraulic, morphology, total dissolved solids, and total suspended solids are effective parameters in the determination of the longitudinal dispersion coefficient as revealed by this study. The assessment of the river shows mean hydraulic and geometric properties such as flow, depth, velocity, longitudinal slope, and width to be 354.17 m3/s, 9.61 m, 0.69 m/s, 0.0079,101.63 m and the range of the longitudinal dispersion coefficient as (72–104.4) m2/s. Results obtained by employing the established equations revealed standard error indices and RMSE of the developed equation, and Kashefipour and Falconer equation gives correlation coefficient of about 0.819 and 4.182 and 0.421 and 12.186, respectively, as coefficient of determination and RMSE, and they are more accurate among the empirical equations. However, the newly derived equation for the longitudinal dispersion coefficient performed better when compared with others, indicating the fitness of the developed equation to estimate longitudinal dispersion coefficient.