Bed Load Transport Equations Research Articles

Evaluation of bedload transport predictions using flow resistance equations to account for macro‐roughness in steep mountain streams

Steep mountain streams typically feature macro‐roughness elements like boulders, step‐pool sequences, and a varying channel width. Flow resistance because of such roughness elements appears to be an important control on bedload transport rates. Many commonly used bedload transport equations overestimate the transport in steep streams by orders of magnitude. Few approaches take into account the typical macro‐roughness elements, and systematic tests of these models with field observations are lacking. In the present study several approaches were considered that allow calculating the contribution of macro‐roughness elements to flow resistance. These approaches were combined with bedload transport equations and the predictions were compared to field measurements of discharge, transported bedload volumes, and channel characteristics in 13 Swiss mountain streams. The streams have channel slopes ranging from 2% to 19%, and catchment areas of 0.5 to 170 km2. For six streams there were time series of sediment yields, mostly measured annually, and for the other seven streams sediment volume estimates were available for large flood events in 2000 and 2005. All tested equation combinations achieved an improvement in bedload prediction compared to a reference equation that was uncorrected for macro‐roughness. The prediction accuracy mainly depended on the size and density of the macro‐roughness and on flow conditions. The best performance overall was achieved by an empirical approach accounting for macro‐roughness, on the basis of an independent data set of flow resistance measurements.

Reformulation of the bed load equation of Meyer‐Peter and Müller in light of the linearity theory for alluvial channel flow

Due to a lack of sufficient theoretical relations, equations for bed load transport in alluvial channel flow have been developed using a variety of empirical assumptions. A linear relationship exhibited in the complex interactions among the three essential components of a river (channel geometry, bed load transport and flow resistance) has led this study to develop a more accurate form of the Meyer‐Peter and Müller (1948, hereafter MPM) relationship, one of the world's most widely used bed load transport equations. This study demonstrates that the application of linearity theory can significantly improve its performance. On the basis of the dataset originally used by MPM, it is shown that the applicable form of the Manning‐Strickler flow resistance relationship can be very satisfactorily determined. The MPM equation is consequently found to be expressible more accurately as Φ = 6[ητb* − 0.047]5/3, where Φ, η and τ*b are the normalized, dimensionless bed load transport rate, bed‐form correction, and flow‐strength parameters, respectively.

Sediment loads due to fluvial transport and debris flows during the 2005 flood events in Switzerland

AbstractSevere floods in August 2005 in Switzerland were associated with enormous morphologic changes along the stream channels and with large sediment transport, including many debris flows in the headwater catchments and fluvial bedload transport along the mountain rivers. We compiled data on transported sediment volumes and estimated surface runoff volumes in the streams determined from radar‐based rainfall data and stream gauge measurements lower in the catchments. A quantitative analysis of the data shows that the transported bedload volumes are related to the flood runoff volume and the relevant channel gradient. For channel gradients flatter than about 5%, our data agree with a simplified version of the Meyer‐Peter and Müller equation but two other equations tested tend to under predict bedload volumes. For channel gradients steeper than about 5%, the observed sediment volumes transported by fluvial processes are over predicted by all bedload transport equations tested, though two of them were specifically developed for this slope range, which likely results from non‐accounted energy losses and supply limitation. For events with debris flows as the dominant sediment transfer mechanism, sediment volumes show a large variability for given flood runoff volume and channel gradient. The sediment volumes of the torrent catchments show a similar functional dependence on catchment parameters as for two empirical power law equations for the estimation of a ‘design’ event magnitude of debris flows based on data from Northern Italy but our events are associated with a much smaller coefficient. Finally, the majority of our debris‐flow data is in line with the trend defined by our fluvial transport events, indicating a continuous transition to debris‐flood and debris‐flow conditions. Copyright © 2010 John Wiley & Sons, Ltd.

Sediment Transport on Arbitrary Slopes: Simplified Model

The paper presents a study on the influence of gravity on the incipient motion and the bed-load transport of sediment. The computation of critical bed-shear stress is revisited considering the balance of forces (hydrodynamic forces and submerged self-weight) acting on a solitary sediment particle lying on an arbitrary sloping bed. Modified effective bed-shear stress and the corresponding critical bed-shear stress, which are defined to assess the incipient motion of sediment in the direction of resultant force, are applied for the estimation of bed-load transport rate in the direction of resultant force. The sediment transport induced by the gravitational force, which is oblique to the direction of the drag force induced by flow, is incorporated into the bed-load transport equation. This modified model provides a reasonable prediction of the critical bed-shear stress and the bed-load transport rate. The model is validated by experimental data. It can be applied to steep slopes and can also avoid the problem of singularity that arises in numerically calculation of sediment transport rate. Additionally, the vectorial transport rate obtained in the model calculation can be implemented in a numerical simulation of channel bed evolution.

A simple estimation of the growth rate and equilibrium size of bedforms created by a turbulent wall jet

A simple procedure for predicting the time evolution and equilibrium size of the two dimensional bedform created by a turbulent wall jet is presented. The model applies to a planar jet discharging flush with and parallel to an initially horizontal bed of cohesionless sediment. Additionally, the water is assumed to be deep relative to the initial jet thickness. The model is developed using existing free- and wall-jet boundary layer relations and existing sediment incipient motion and bedload transport equations. The model is validated against live-bed experiments from previous studies and from a new set of laboratory experiments. The agreement between the predictions for and observations of equilibrium size is quite good. The agreement regarding the time evolution is also good. These results suggest that several key characteristics of jet-induced bedforms are captured with a limited physical approach.

Predicting fractional bed load transport rates: Application of the Wilcock‐Crowe equations to a regulated gravel bed river

Bed load samples from four locations in the Trinity River of northern California are analyzed to evaluate the performance of the Wilcock‐Crowe bed load transport equations for predicting fractional bed load transport rates. Bed surface particles become smaller and the fraction of sand on the bed increases with distance downstream from Lewiston Dam. The dimensionless reference shear stress for the mean bed particle size (τ*rm) is largest near the dam, but varies relatively little between the more downstream locations. The relation between τ*rm and the reference shear stresses for other size fractions is constant across all locations. Total bed load transport rates predicted with the Wilcock‐Crowe equations are within a factor of 2 of sampled transport rates for 68% of all samples. The Wilcock‐Crowe equations nonetheless consistently under‐predict the transport of particles larger than 128 mm, frequently by more than an order of magnitude. Accurate prediction of the transport rates of the largest particles is important for models in which the evolution of the surface grain size distribution determines subsequent bed load transport rates. Values of τ*rm estimated from bed load samples are up to 50% larger than those predicted with the Wilcock‐Crowe equations, and sampled bed load transport approximates equal mobility across a wider range of grain sizes than is implied by the equations. Modifications to the Wilcock‐Crowe equation for determining τ*rm and the hiding function used to scale τ*rm to other grain size fractions are proposed to achieve the best fit to observed bed load transport in the Trinity River.

EVALUATION OF BED LOAD EQUATION USING TAPI RIVER DATA, INDIA

ABSTRACT The bed load transport in the rivers has been studied by various investigators in the past. They developed bed load transport equation for particular size of sediment in a non uniform bed material, where as the influence of other sizes of sediments has been neglected. In the present paper the rate of bed load transport in weight per unit width for non uniform bed material considering the various variables like discharge, hydraulic mean depth, flow velocity, bed slope etc. by collecting the field data of Tapi River for 15 years, Monsoon season, Sarangkheda gauging station has been computed. Tapi River is alluvial, westward flowing river of India. On the down stream side of gauging station an Ukai dam is constructed which is a multipurpose dam used for irrigation and power development purposes. The sedimentation of the reservoir is one of the problems faced by the dam and which may reduce the life of dam. The majority of the bed load formulae represent a functional relation between bed load discharge and shear stress. The formulae are characterized by three aspects namely the basic function qb = A f (t0-tc)., the characteristic grain-size to be used and the correction applied to bed load equations. The main objectives of this paper are to estimate bed load of Monsoon season, using various measured parameters. The rate of bed load transport in weight per unit width is computed using Shields equation. The mathematical model has been developed using effective shear stress and bed load. The statistical analysis, multiple regression and curve fitting (by nonlinear square fitter) is carried out by using allometric function of Micro cal Origin 6.1. The new bed load equation for Tapi River, Monsoon season for Sarangkheda gauging station is proposed. The proposed bed load equation is tested using five years data of Tapi River. Also, model is tested using data of seven world rivers. The statistical analysis is carried out by computing rmse, inequality ratio and discrepancy ratio. The analysis suggests good agreement between measured and predicted bed load for 8 out of 10 rivers data.

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.

Inferring bedload transport from stratigraphic successions: examples from Cenozoic and Pleistocene rivers, south central Pyrenees, Spain

Abstract Geologists and geomorphologists have long been concerned with rates of sediment transfer as bedload in gravel-bed rivers, especially as rates of sediment transfer are important factors controlling river aggradation and incision. Bedload transport equations, originally derived for Holocene streams, have been used widely in modern gravel-bed river systems. However, palaeohydraulic reconstructions have received less attention and are generally dismissed as inaccurate since most are estimated to be at least an order of magnitude out. This study focuses on deriving stream power, bedload transport rates and efficiency estimates for Oligo–Miocene and Plio–Pleistocene gravel-bed river deposits from the south central Pyrenees, Spain. The basic data used in the palaeohydraulic calculations are estimates of palaeoslope, palaeovelocity, palaeodepth and the volume of sediment accreted in yearly flood events on gravel bars. Analyses of data from these ancient river systems yield more accurate estimates of relative stream power, bedload transport rates and efficiency parameters. This study illustrates the need for understanding the palaeohydraulics of river systems in order to characterize ancient rivers. Gravel-bed rivers with low sediment supply and high bedload transport rates incise. Conversely, when sediment supply is abundant, bedload transport rates and efficiency are low and the river system aggrades.

Bed-Load Transport Equation on Arbitrarily Sloping Beds

This work presents a simple tool to evaluate bed-load transport intensity and direction on arbitrarily sloping beds with local longitudinal and transversal inclinations up to 25° and in the case of uniform sediments and low values of the applied Shields stress. The tool is composed of a set of equations which fit the results obtained via an iterative procedure by the semiempirical model recently proposed in 2003 by Parker, Seminara, and Solari. The tool provides a fully nonlinear description of bed-load transport which overcomes the limitations of linear formulations developed in the case of negligible local bed inclinations. The proposed tool can be easily implemented in any morphodynamic model to describe the evolution of the bottom topography and to capture the dynamics of relevant sloping beds.

Calculating bed load transport in steep boulder bed channels

Steep, rough channels occupy a large fraction of the total channel length in mountainous regions. Most sediment mobilized on hillslopes must pass through these streams before reaching lower‐gradient channels. Steep channels have wide grain size distributions that are composed of finer, more mobile sediment and large, rarely mobile grains. The large grains can bear a significant portion of the total shear stress and thereby reduce the stress available to move the finer sediment. Conventional bed load transport equations often overpredict the sediment flux in steep channels by several orders of magnitude. We hypothesize that sediment transport equations overpredict the sediment flux because they do not (1) account for the stress borne by rarely mobile grains, (2) differentiate between highly and rarely mobile sediment, and (3) account for the limited availability of mobile sediment. Here we modify a conventional bed load transport equation to include these three effects. We use measurements of the flow, bed properties, and sediment flux in a small, steep flume to test this equation. We supply gravel at a constant rate through fields of regularly spaced immobile spheres and measure the bed coverage by gravel and sphere protrusion (the percent of the sphere that protrudes above the gravel deposit). For a given sphere spacing, the proportion of the bed covered by gravel increases and the sphere protrusion decreases with greater sediment supply. Thus bed coverage and immobile grain protrusion may serve as proxies for sediment availability in steep, rough streams. Unlike most transport equations that we tested, our modified bed load equation predicts sediment fluxes to within an order of magnitude of the measured values. Our results demonstrate that accurately predicting bed load transport in steep, rough streams may require accounting for the effects of local sediment availability (coverage by mobile sediment) and drag due to rarely mobile particles.

Examining the dynamics of grain size distributions of gravel/sand deposits in the Sandy River, Oregon with a numerical model

AbstractThis paper describes the application of The Unified Gravel‐Sand (TUGS) model for the simulation of the Sandy River, Oregon. TUGS model employs Wilcock and Crowe's (2003) bedload transport equation for simulation of gravel and sand transport, and is capable of simulating the dynamics of bed material grain size distributions, including the fractions of sand in the deposits. The model has been examined with three large‐scale flume experiments and a flushing flow flume experiment with good agreements in bed profiles, gravel characteristic grain sizes and fractions of sand in the deposits. These examinations, along with descriptions of model development, are presented in a manuscript submitted concomitantly elsewhere (Cui, 2007), and this paper expands model examinations to a natural river. Eight runs are conducted with the Sandy River, Oregon as the prototype river, focusing on the responses of bed material grain size distributions under different hydrologic and sediment supply conditions. Simulation with the recorded discharge and the best understanding of sediment supply in the Sandy River produced good agreement in river longitudinal profile. This simulation also produced decreased median grain size and increased sand fraction in the downstream direction that match well with the general observations in the field. Examinations with varied sand supply indicate that bed material sand fractions are positively correlated with sand supply. Examinations with varied water discharge indicate that increased discharge under the same sediment supply conditions results in decreased bed material sand fraction. Examination of a hypothetical backwater effect from the Columbia River indicates that backwater effect results in increases in bed material sand fractions. Simulation of the sedimentation process upstream of a 14‐m tall dam during dam operation produced similar current bed profile and a stratified sediment deposit very similar to those observed in the field. Copyright © 2007 John Wiley & Sons, Ltd.

Application of surface-based bed load transport equations to a desert gravel-bed stream

Bed load measurements during low flow in a desert gravel-bed stream, LasVegasWash, were used to evaluate fractional bed load transport equations. Since the stream is an urban perennial channel that conveys primarily treated waste effluent, bed load transport is limited by the supply of transportable bed material. The Parker, Wilcock and Crowe, and Wu et al. surface-based fractional bed load transport equations were evaluated against the field data. The ratio between the calculated and measured bed load transport rate (C/O) as well as the data correlation coefficient with respect to each formula were calculated to judge the applicability of these equations to this field data set. Comparisons showed that both the Parker and Wu et al. equations satisfactorily predicted the transport rate, while theWilcock and Crowe equation has a larger scatter. The apparent success of Parker and Wu et al. equations can be attributed to two factors: a hiding function based on physical mechanism of the particle hiding and exposure, and the empirical coefficients calibrated using field data sources. The study results indicate that these two fractional bed load transport equations derived for perennial streams in wet regions are applicable to the field data collected at this arid region channel

A bed‐load transport model for rough turbulent open‐channel flows on plane beds

AbstractData from flume studies are used to develop a model for predicting bed‐load transport rates in rough turbulent two‐dimensional open‐channel flows moving well sorted non‐cohesive sediments over plane mobile beds. The object is not to predict transport rates in natural channel flows but rather to provide a standard against which measured bed‐load transport rates influenced by factors such as bed forms, bed armouring, or limited sediment availability may be compared in order to assess the impact of these factors on bed‐load transport rates. The model is based on a revised version of Bagnold's basic energy equation ibsb = ebω, where ib is the immersed bed‐load transport rate, ω is flow power per unit area, eb is the efficiency coefficient, and sb is the stress coefficient defined as the ratio of the tangential bed shear stress caused by grain collisions and fluid drag to the immersed weight of the bed load. Expressions are developed for sb and eb in terms of G, a normalized measure of sediment transport stage, and these expressions are substituted into the revised energy equation to obtain the bed‐load transport equation ib = ω G 3·4. This equation applies regardless of the mode of bed‐load transport (i.e. saltation or sheet flow) and reduces to ib = ω where G approaches 1 in the sheet‐flow regime. That ib = ω does not mean that all the available power is dissipated in transporting the bed load. Rather, it reflects the fact that ib is a transport rate that must be multiplied by sb to become a work rate before it can be compared with ω. It follows that the proportion of ω that is dissipated in the transport of bed load is ibsb/ω, which is approximately 0·6 when ib = ω. It is suggested that this remarkably high transport efficiency is achieved in sheet flow (1) because the ratio of grain‐to‐grain to grain‐to‐bed collisions increases with bed shear stress, and (2) because on average much more momentum is lost in a grain‐to‐bed collision than in a grain‐to‐grain one. Copyright © 2006 John Wiley & Sons, Ltd.

Reply to comment by Claude Michel on “A general power equation for predicting bed load transport rates in gravel bed rivers”

[1] We thank Michel [2005] for the opportunity to improve our bed load transport equation [Barry et al., 2004, equation (6)] and to resolve the dimensional complexity that he identified. However, we do not believe that the alternative bed load transport equation proposed by Michel [2005] provides either the mechanistic insight or predictive power of our transport equation. [2] Although some bed load transport data exhibit nonlinear trends in log-log plots of transport rate versus discharge, a simple linear function is sufficient to describe our data [Barry et al., 2004, paragraph 43]. The Figure 7 data of Barry et al. [2004] could be fit by a nonlinear function as suggested by Michel [2005], but we believe this to be an unnecessary complication, particularly given how well our simple equation predicts observed transport rates compared to other more complex equations, such as Parker's [1991] three-part bed load transport function [Barry et al., 2004, Figure 11]. Furthermore, an important aspect of our equation, that is not preserved in Michel's alternative, is the between-site variation in the exponent of the transport function that results from supply related channel armoring (i.e., transport capacity in excess of bed load sediment supply) which provides a mechanistic understanding of the bed load transport process [Barry et al., 2004]. Michel [2005] proposes a bed load transport equation that mimics our equation in terms of the range of exponents that we observe (i.e., 1.5–4 [Barry et al., 2004, Figure 8a] but lacks the mechanistic insight and consequent predictive power. Moreover, Michel's equation requires a sufficient number of bed load transport observations across a broad range of discharges to empirically calibrate his a and β values.

A general power equation for predicting bed load transport rates in gravel bed rivers

A variety of formulae has been developed to predict bed load transport in gravel bed rivers, ranging from simple regressions to complex multiparameter formulations. The ability to test these formulae across numerous field sites has, until recently, been hampered by a paucity of bed load transport data for gravel bed rivers. We use 2104 bed load transport observations in 24 gravel bed rivers in Idaho to assess the performance of eight different formulations of four bed load transport equations. Results show substantial differences in performance but no consistent relationship between formula performance and degree of calibration or complexity. However, formulae containing a transport threshold typically exhibit poor performance. Furthermore, we find that the transport data are best described by a simple power function of discharge. From this we propose a new bed load transport equation and identify channel and watershed characteristics that control the exponent and coefficient of the proposed power function. We find that the exponent is principally a factor of supply‐related channel armoring (transport capacity in excess of sediment supply), whereas the coefficient is related to drainage area (a surrogate for absolute sediment supply). We evaluate the accuracy of the proposed power function at 17 independent test sites.

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.

Extension of ENO and WENO schemes to one-dimensional sediment transport equations

Essentially nonoscillatory and weighted essentially nonoscillatory schemes are high order resolution schemes constructed for the hyperbolic conservation laws. In this paper we extend these schemes to the one-dimensional bed-load sediment transport equations. The difficulties that arise in the numerical modelling come from the fact that a nonconservative product is present in the system. Our specific numerical approximations for the nonconservative product are based on two ideas. First is to include the influence of that term in the system upwinding and the second is to define the numerical approximation in such a way that the obtained scheme solves the system for the quiescent flow case exactly. As a consequence, the resulting schemes give excellent results, as it can be seen from the numerical tests we present. On the opposite, the numerical results obtained by applying the pointwise evaluation of nonconservative product on the same tests present unacceptably large numerical errors.

Numerical simulation of graded sediment transport

River engineers commonly need to predict sediment transport rates in the absence of adequate field data, and so make frequent use of numerical sediment transport models. This paper compares the performance of two such computer models for the prediction of graded sediment transport. The intention is to inform model users of the level of accuracy that can be expected for typical applications. One model uses the Parker bedload transport equation and the other uses the Meyer-Peter and Müller sediment transport formula modified for use with graded sediments. The computer model predictions are compared with detailed experimental data collected for the purpose of computer model validation. The Meyer-Peter and Müller-based model produces better predictions of sediment transport rates, whereas the Parker-based model performs better in terms of the grain sizes of both the bedload and the bed material. These results illustrate the limitations of present models and indicate areas requiring further investigation.

Numerical simulation of graded sediment transport

River engineers commonly need to predict sediment transport rates in the absence of adequate field data, and so make frequent use of numerical sediment transport models. This paper compares the performance of two such computer models for the prediction of graded sediment transport. The intention is to inform model users of the level of accuracy that can be expected for typical applications. One model uses the Parker bedload transport equation and the other uses the Meyer-Peter and Müller sediment transport formula modified for use with graded sediments. The computer model predictions are compared with detailed experimental data collected for the purpose of computer model validation. The Meyer-Peter and Müller-based model produces better predictions of sediment transport rates, whereas the Parker-based model performs better in terms of the grain sizes of both the bedload and the bed material. These results illustrate the limitations of present models and indicate areas requiring further investigation.