Applicability of Sediment Transport Capacity Formulas to Dam-Break Flows over Movable Beds

In this paper, we investigate the extent to which well-known sediment transport capacity formulas can be used in one-dimensional (1D) numerical modeling of dam-break waves over movable beds. The 1D model considered here is a one-layer model based on the shallow-water equations, a bed update (Exner) equation, a space-lag equation for the nonequilibrium sediment transport and an empirical formula calculating the sediment transport capacity of the flow. The model incorporates a variety of sediment transport capacity formulas proposed by Meyer-Peter and Müller, Bagnold, Engelund and Hansen, Ackers and White, Smart and Jaeggi, van Rijn, Rickenmann, Cheng, Abrahams and Camenen, and Larson. We examine the performance of each formula by simulating four idealized laboratory cases on dam-break waves over sandy beds. Comparisons between numerical results and measurements show that for each case better predictions are obtained using a particular formula, but overall, formulas proposed by Meyer-Peter and Müller (with the factor 8 being replaced by 12), Smart and J?ggi, Cheng, Abrahams and Camenen, and Larson rank as the best predictors for the entire range of conditions studied here. Moreover, results show that in the cases where a bed step exists, implementing a mass failure mechanism in the numerical modeling plays an important role in reproducing the bed and water profiles.     

Reanalysis and Correction of Bed-Load Relation of Meyer-Peter and Müller Using Their Own Database

The pioneering predictor of fluvial bed-load transport rate proposed by Meyer-Peter and Müller in 1948 is still extensively used in basic research and engineering applications. A review of the basis for its formulation reveals, however, that an unnecessary bed roughness correction was applied to cases of plane-bed morphodynamic equilibrium. Its inclusion followed a flow resistance parameterization in terms of the Nikuradse roughness height, which has been shown (well after the publication of their work) to be inappropriate for the characterization of mobile bed rough conditions in rivers. Removing the unnecessary correction and incorporating an improved correction of the boundary shear stress due to sidewall effects allow elucidation of the most parsimonious form of the bed-load relation of Meyer-Peter and Müller that is dictated by their own data set. The new predictor is presented in terms of two alternative power law forms. These amended forms show that, in the case of lower-regime plane-bed equilibrium transport of uniform bed sediment, the new estimates of volume bed-load transport rates are less than or equal to half the values that would be obtained with the original relation of Meyer-Peter and Müller in the absence of the unnecessary bed roughness correction. The meticulous database and clear analysis of the original work of Meyer-Peter and Müller greatly aided the present writers in their reanalysis, which liberally uses the hindsight offered by 58 years of subsequent research.     

Bedload Transport in Alluvial Channels

Hydraulic, sediment, land-use, and rock-erosivity data of 22 alluvial streams were used to evaluate conditions of bedload transport and the performance of selected bedload-transport equations. Transport categories of transport-limited (TL), partially transport-limited (PTL), and supply-limited (SL) were identified by a semiquantitative approach that considers hydraulic constraints on sediment movement and the processes that control sediment availability at the basin scale. Equations by Parker et al. in 1982, Schoklitsch in 1962, and Meyer-Peter and Muller in 1948 adequately predicted sediment transport in channels with TL condition, whereas the equations of Bagnold in 1980, and Schoklitsch, in 1962, performed well for PTL and SL conditions. Overall, the equation of Schoklitsch predicted well the measured bedload data for eight of 22 streams, and the Bagnold equation predicted the measured data in seven streams.     

Sand Transport in Nile River, Egypt

Measurements of bed-load and suspended-load transport rates were carried out successfully at four cross sections of the Nile River, in Egypt, along the entire length from Aswan to Cairo using a mechanical sampler called the Delft Nile Sampler. The measured transport rates were compared to similar data sets from two other large scale rivers: the Rhine-Waal River in the Netherlands and the Mississippi River in the USA. The bed-load transport rates in the Nile River and in the Rhine-Waal River are in very good agreement. Comparison of suspended transport rates in the Nile River and in the Mississippi River shows that both data sets are complementary, revealing a very consistent trend of suspended transport against current velocity; suspended transport is roughly proportional to (Vav)3?to?4. Three formulas for the prediction of bed-load transport were tested using the Nile data: Meyer-Peter–Muller, Bagnold, and Van Rijn. The prediction formula of Van Rijn produced significantly better results than the other two formulas; the average relative error was about 60%. The formula of Van Rijn was modified to extend it to conditions with slightly nonuniform sediment mixtures by introducing a correction factor for the bed shear parameter. Based on a limited number of flume experiments, the correction factor was found to be dependent on the characteristics of the sediment mixture (d10, d50, d90, and σg). Comparison of bed-load transport measured in the Nile River with computed transport rates of the modified formula showed improved results; the average relative error decreased to about 30%. The formulas of Bagnold and Van Rijn were also used to compute the suspended transport rates in the Nile River. The computed transport rates were found to be within a factor of 2 of measured values; the formula of Bagnold performed slightly better. The total load transport formula of Engelund–Hansen was also successfully used (computed values within a factor of about 2 of measured values).     

Simple Model for Downstream Variation of Median Sediment Size in Chilean Rivers

A simple theory is developed to account for the observed downstream variation of the median sediment size in Chilean rivers based on a reach-wise equilibrium sediment transport concept. The theory makes use of a bedload transport equation linked with a resistance equation to estimate the median sediment diameter as a function of channel slope, with the flow Reynolds number and bedload concentration as parameters. Both, Meyer-Peter and Müller’s and Ackers and White’s formulas are used alternatively as bedload equations. A Manning–Strickler type of formulation is used as a resistance relationship. The resulting model is validated against field data corresponding to 150 rivers in Central Chile, covering slopes in the range of 0.04–8.61%, with median sediment size in the range of 0.3–250 mm. Despite the simplicity of the present theory and the somewhat bold assumptions made in its derivation, the estimated variation of the median sediment size with channel slope follows the same trend as the field data. Most of the scatter of these data falls within the theoretical limits given by the estimated range of values of the parameters of the model.     

Approach to Separate Sand from Gravel for Bed-Load Transport Calculations in Streams with Bimodal Sediment

The bed material found in gravel-bed streams is nonuniform in terms of grain size and can typically be classified as unimodal or bimodal. The latter type of sediment distribution is usually represented by two modes, one of sand size and another of gravel. For this case, the movement of one mode becomes nonlinearly influenced by the other. As a result, the presence of the two modes in a bimodal material complicates the calculation of bed-load transport rates. The present study proposes an approach to separate the calculation of bed-load transport rates for bimodal materials into two independent fractions of sand and gravel, thereby rendering the bed sediment into two unimodal components. This approach is accomplished by decoupling the two fractions through scaling the reference Shields stresses of the sand and gravel modes to match the value of the mode of unimodal materials. Consequently, the contribution of each fraction to bed load can be estimated using a suitable relation derived for unimodal materials. Laboratory and field bed-load data available in the literature are used to examine the validity of the overall approach.     

Machine Learning Approach to Modeling Sediment Transport

Inaccuracies of sediment transport models largely originate from our limitation to describe the process in precise mathematical terms. Machine learning (ML) is an alternative approach to reduce the inaccuracies of sedimentation models. It utilizes available domain knowledge for selecting the input and output variables for the ML models and uses modern regression techniques to fit the measured data. Two ML methods, artificial neural networks and model trees, are adopted to model bed-load and total-load transport using the measured data. The bed-load transport models are compared with the models due to Bagnold, Einstein, Parker et al., and van Rijn. The total-load transport models are compared with the models due to Ackers and White, Bagnold, Engelund and Hansen, and van Rijn. With the chosen data sets on bed-load and total-load transport the ML models provided better accuracy than the existing ones.     

Bed-Load Transport Equation for Sheet Flow

When open-channel flows become sufficiently powerful, the mode of bed-load transport changes from saltation to sheet flow. Where there is no suspended sediment, sheet flow consists of a layer of colliding grains whose basal concentration approaches that of the stationary bed. These collisions give rise to a dispersive stress that acts normal to the bed and supports the bed load. An equation for predicting the rate of bed-load transport in sheet flow is developed from an analysis of 55 flume and closed conduit experiments. The equation is ib = ω where ib = immersed bed-load transport rate; and ω = flow power. That ib = ω implies that eb = tan?α = ub/u, where eb = Bagnold’s bed-load transport efficiency; ub = mean grain velocity in the sheet-flow layer; and tan?α = dynamic internal friction coefficient. Given that tan?α ≈ 0.6 for natural sand, ub ≈ 0.6u, and eb ≈ 0.6. This finding is confirmed by an independent analysis of the experimental data. The value of 0.60 for eb is much larger than the value of 0.12 calculated by Bagnold, indicating that sheet flow is a much more efficient mode of bed-load transport than previously thought.     

Concentration Profiles and Solids Transport above Stationary Deposit in Enclosed Conduit

Concentration profiles and overall flow parameters were measured in a 0.37-mm-sand-mixture flow above a plane stationary deposit in a pressurized circular pipe of the inner diameter of 150 mm. The new data and additional data from the literature are used to propose a tentative transport formula of the Meyer-Peter and Müller type, and a slip-ratio formula, for flows at high shear stress in enclosed conduits.     

Performance of Bed-Load Transport Equations Relative to Geomorphic Significance: Predicting Effective Discharge and Its Transport Rate

Previous studies assessing the accuracy of bed-load transport equations have considered equation performance statistically based on paired observations of measured and predicted bed-load transport rates. However, transport measurements were typically taken during low flows, biasing the assessment of equation performance toward low discharges, and because equation performance can vary with discharge, it is unclear whether previous assessments of performance apply to higher, geomorphically significant flows (e.g., the bankfull or effective discharges). Nor is it clear whether these equations can predict the effective discharge, which depends on the accuracy of the bed-load transport equation across a range of flows. Prediction of the effective discharge is particularly important in stream restoration projects, as it is frequently used as an index value for scaling channel dimensions and for designing dynamically stable channels. In this study, we consider the geomorphic performance of five bed-load transport equations at 22 gravel-bed rivers in mountain basins of the western United States. Performance is assessed in terms of the accuracy with which the equations are able to predict the effective discharge and its bed-load transport rate. We find that the median error in predicting effective discharge is near zero for all equations, indicating that effective discharge predictions may not be particularly sensitive to one’s choice of bed-load transport equation. However, the standard deviation of the prediction error differs between equations (ranging from 10% to 60%), as does their ability to predict the transport rate at the effective discharge (median errors of less than 1 to almost 2.5 orders of magnitude). A framework is presented for standardizing the transport equations to explain observed differences in performance and to explore sensitivity of effective discharge predictions.     

Assessment of Methods Used in 1D Models for Computing Bed-Load Transport in a Large River: The Danube River in Slovakia

Comprehensive measurements of bed-load sediment transport through a section of the Danube River, located approximately 70?km downstream from Bratislava, Slovakia, are used to assess the accuracy of bed-load formulas implemented in 1D modeling. Depending on water discharge and water level, significant variations in the distribution of bed load across the section were observed. It appeared that, whatever the water discharge, the bed shear stress τ is always close to the estimated critical bed shear stress for the initiation of sediment transport τcr. The discussion focuses on the methods used in 1D models for estimating bed-load transport. Though usually done, the evaluation of bed-load transport using the mean cross-sectional bed shear stress yields unsatisfactory results. It is necessary to use an additional model to distribute the bed shear stress across the section and calculate bed load locally. Bed-load predictors also need to be accurate for τ close to τcr. From that point of view, bed-load formulas based on an exponential decrease of bed-load transport close to τcr appear to be more appropriate than models based on excess bed shear stress. A discussion on the bed-load formula capability to reproduce grain sorting is also provided.     

Numerical Model for Sediment Transport and Bed Degradation in the Yangtze River Channel Downstream of Three Gorges Reservoir

This paper presents a two-dimensional morphological model for unsteady flow and both suspended-load and bed-load transport of multiple grain size to simulate transport of graded sediments downstream from the Three Gorges Reservoir. The model system includes a hydrodynamic module and a sediment module. The hydrodynamic module is based on the depth-averaged shallow water equations in orthogonal curvilinear coordinates. The sediment module describing nonuniform sediment transport is developed to include nonequilibrium transport processes, bed deformation, and bed material sorting. The model was calibrated using field observations through application to a 63-km-long alluvial river channel on the middle Yangtze River in China. A total of 16 size groups and a loose layer method of three sublayers were considered for the transport of the nonuniform bed materials in a long-term simulation. Predictions are compared with preliminary results of field observations and factors affecting the reliability of the simulated results are discussed. The results may be helpful to the development of more accurate simulation models in the future.     

Validation of Existing Bed Load Transport Formulas Using In-Sewer Sediment

Granular sediment in pipe inverts has been reported in a number of sewer systems in Europe. Given the range of flow conditions and particle characteristics of inorganic sewer sediments the mode of transport may normally be considered as bed load. Current commercial software for modeling the erosion and transport of sediments in sewer pipes still utilizes well-known, or modified versions of transport equations that were derived for transport of noncohesive sediment in alluvial streams. In this paper the performances of the equations of Ackers and White (originally developed for the transport of river sediments) and of May (derived from laboratory pipe experiments) are examined against two separate data sets. One set is from laboratory erosion experiments on sewer sediment obtained in Paris. A second data set has bed load transport rate measurements recorded in a sewer inlet pipe. The formulas were selected because of their widespread use in the prediction of in-sewer sediment transport both in commercial software and in the latest United Kingdom design guidance for new sewers. The results indicated that both the relationships performed poorly, even in such well-controlled conditions. These formulas have significant difficulties in predicting the erosion thresholds and fractional transport rates for non-uniformly sized in-sewer sediments. An empirical formula to adjust the threshold of motion for individual grain size fractions was developed which significantly improved predictions. Although such techniques have been used in gravel bed rivers, the threshold adjustment function for in-sewer deposits was significantly different from these previously published for fluvial gravels, indicating that a direct transfer of fluvial relationships to sewers may be inappropriate without further research.     

Simulating Bed-Load Transport in a Complex Gravel-Bed River

An existing two-dimensional mobile-bed hydrodynamic model has been modified to simulate bed-load transport in a complex gravel-bed river. We investigated the sensitivity of predicted bed load to control parameters, and compared model predictions of flow depth, shear stress, and gravel transport with field measurements made from the river. The predictions are based on concurrent field data of flow discharge, water level, and sediment for model input. The model takes into account multiple-fraction transport rates, and continuously updates the river bed and surface grain-size distribution. The model predictions are in reasonable agreement with field measurements.     

Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and Bed-Load Transport

Attention is given to the properties of sediment beds over the full range of conditions (silts to gravel), in particular the effect of fine silt on the bed composition and on initiation of motion (critical conditions) is discussed. High-quality bed-load transport data sets are identified and analyzed, showing that the bed-load transport in the sand range is related to velocity to power 2.5. The bed-load transport is not much affected by particle size. The prediction of bed roughness is addressed and the prediction of bed-load transport in steady river flow is extended to coastal flow applying an intrawave approach. Simplified bed-load transport formulas are presented, which can be used to obtain a quick estimate of bed-load transport in river and coastal flows. It is shown that the sediment transport of fine silts to coarse sand can be described in a unified model framework using fairly simple expressions. The proposed model is fully predictive in the sense that only the basic hydrodynamic parameters (depth, current velocity, wave height, wave period, etc.) and the basic sediment characteristics (d10, d50, d90, water temperature, and salinity) need to be known. The prediction of the effective bed roughness is an integral part of the model.     

One-Dimensional Modeling of Dam-Break Flow over Movable Beds

A one-dimensional model has been established to simulate the fluvial processes under dam-break flow over movable beds. The hydrodynamic model adopts the generalized shallow water equations, which consider the effects of sediment transport and bed change on the flow. The sediment model computes the nonequilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. In particular, a correction factor is proposed to modify the Van Rijn formulas of equilibrium bed-load transport rate and near-bed suspended-load concentration for the simulation of sediment transport under high-shear flow conditions. The governing equations are solved by an explicit finite-volume method with the first-order upwind scheme for intercell fluxes. The model has been tested in two experimental cases, with fairly good agreement between simulations and measurements. The sensitivities of the model results to parameters such as the sediment nonequilibrium adaptation length, Manning’s roughness coefficient and the proposed correction factor have been verified. The proposed model has also been compared to an existing model and the results indicate the new model is more reliable.     

Probabilistic Modeling of Bed-Load Composition

This paper proposes that the changes which occur in composition of the bed load during the transport of mixed-grain-size sediments are largely controlled by the distributions of critical entrainment shear stress for the various size fractions. This hypothesis is examined for a unimodal sediment mixture by calculating these distributions with a discrete particle model and using them in a probabilistic calculation of bed-load composition. The estimates of bed-load composition compare favorably with observations of fractional transport rates made in a laboratory flume for the same sediment, suggesting that the hypothesis is reasonable. The analysis provides additional insight, in terms of grain mechanics, into the processes that determine bed-load composition. These insights strongly suggest that better prediction methods will result from taking account of the variation of threshold within size fractions, something that most previous studies have neglected.     

Field Assessment of Alternative Bed-Load Transport Estimators

Measurement of near-bed sediment velocities with acoustic Doppler current profilers (ADCPs) is an emerging approach for quantifying bed-load sediment fluxes in rivers. Previous investigations of the technique have relied on conventional physical bed-load sampling to provide reference transport information with which to validate the ADCP measurements. However, physical samples are subject to substantial errors, especially under field conditions in which surrogate methods are most needed. Comparisons between ADCP bed velocity measurements with bed-load transport rates estimated from bed-form migration rates in the lower Missouri River show a strong correlation between the two surrogate measures over a wide range of mild to moderately intense sediment transporting conditions. The correlation between the ADCP measurements and physical bed-load samples is comparatively poor, suggesting that physical bed-load sampling is ineffective for ground-truthing alternative techniques in large sand-bed rivers. Bed velocities measured in this study became more variable with increasing bed-form wavelength at higher shear stresses. Under these conditions, bed-form dimensions greatly exceed the region of the bed ensonified by the ADCP, and the magnitude of the acoustic measurements depends on instrument location with respect to bed-form crests and troughs. Alternative algorithms for estimating bed-load transport from paired longitudinal profiles of bed topography were evaluated. An algorithm based on the routing of local erosion and deposition volumes that eliminates the need to identify individual bed forms was found to give results similar to those of more conventional dune-tracking methods. This method is particularly useful in cases where complex bed-form morphology makes delineation of individual bed forms difficult.     

Investigation of Cadmium Desorption from Different-Sized Sediments

Using the adsorptive reaction kinetics equation and the mass conservation equation, formulas for calculating the particulate heavy metal concentration (or the residual adsorption content of unit weight of sediment) and the dissolved heavy metal concentration were deduced. Furthermore, equations for calculating the equilibrium particulate heavy metal concentration (or the equilibrium residual adsorption content of unit weight of sediment) and the equilibrium dissolved heavy metal concentration were formulated. On the basis of these theoretically deduced formulas and through laboratory reactor modeling of artificially contaminated sediments with cadmium in suspension, heavy metal desorption rates from nonuniform sediment particles were studied. Both theoretical analysis and experimental results indicate that adsorption and desorption of heavy metal pollutants to and from sediment particles are partly reversible. It is also noticed from the experiments that the adsorption and desorption rates of heavy metals to and from sediments are dependent on the content of active adsorption components in the sediments. The greater the content of active adsorption components, the higher the adsorption rate of the sediments and the lower the desorption rate. The total amount of heavy metal desorbed from sediments is directly proportional to the concentration of suspended sediment particles.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.