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.     

Bed-Load Transport Flume Experiments on Steep Slopes

Experiments were conducted over uniform gravel bed materials to obtain 143 friction factor values under bed-load equilibrium flow conditions in an attempt to add to the scarce data available on slopes between 1 and 9% for Shields numbers between 0.08 and 0.29. Analyses showed that when only flows over flat beds are considered, a distinction must be made between flows with and without bed load. More particularly, fitting flow resistance equations indicated that the roughness parameter increases by a factor of 2.5 from clear water flow to intense bed-load transport. Between these two states, the flow resistance can be approximated by a constant for a given slope.     

Effect of Seepage-Induced Nonhydrostatic Pressure Distribution on Bed-Load Transport and Bed Morphodynamics

Bed-load transport is commonly evaluated in the condition of a hydrostatic pressure distribution of the flow field; while this condition is reasonable for quasi-steady, quasi-uniform rectilinear flows, it cannot be satisfied in a large variety of flow conditions, i.e., near an obstacle as in the case of a bridge pier. The dimensionless Shields number, which contains the assumption of a hydrostatic pressure distribution in its denominator, therefore cannot be strictly applied to evaluate bed-load transport in all the configurations where nonhydrostatic pressure distributions are observed. In the present work, a generalization of the Shields number is proposed for the case of nonhydrostatic pressure distribution produced by groundwater flow. Experiments showing the effects of vertical groundwater flow on the bed morphodynamics are presented. The comparison between the experimental observations and numerical results, obtained by means of a morphodynamic model which employs the new formulation of the Shields number, suggests that the proposed generalization of the Shields number is able to account the effect of the nonhydrostatic pressure distribution on the bed-load transport.     

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.     

Transition between Two Bed-Load Transport Regimes: Saltation and Sheet Flow

In the saltation regime where bed-shear stress is low, bed load moves by sliding, rolling, and saltating along the bed, while in the sheet-flow regime where bed-shear stress is high, it travels by a combination of saltation and sheet flow. In this paper a theoretical model is developed for predicting the onset of the sheet-flow regime as shear stress increases. This model is based on a new variable Pb representing the proportion of grains on the bed that are entrained as bed load. The model yields the equation Pb = 2.56θG3 in which G = 1?θc/θ, θ = dimensionless bed-shear stress; and θc = critical value of θ at which grains begin to move. The equation shows that θt, which is the value of θ at the onset of the sheet-flow regime and is assumed to occur when Pb = 1, is around 0.5 with the exact value controlled by θc. For example, when θc = 0.045, θt = 0.52. The theoretical model is verified by performing a nonlinear regression analysis on data from 285 flume experiments. Additional flume experiments with a high-speed video (HSV) system result in consistent values of θ for the onset of the sheet-flow regime, which support the theoretical model. The HSV images further reveal that: (1) the sheet-flow regime is characterized by granular sheets or laminations; and (2) a zone of mixed saltation and rolling grains exists not only in the saltation regime but also in the sheet-flow regime.     

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.     

Measurements of Sediment Erosion and Transport with the Adjustable Shear Stress Erosion and Transport Flume

Soil and sediments play an important role in water management and water quality. Issues such as water turbidity, associated contaminants, reservoir sedimentation, undesirable erosion and scour, and aquatic habitat are all linked to sediment properties and behaviors. In situ analysis is necessary to develop an understanding of the erosion and transport of sediments. Sandia National Laboratories has recently patented the Adjustable Shear Stress Erosion and Transport (ASSET) Flume that quantifies in situ erosion of a sediment core with depth while affording simultaneous examination of transport modes (bedload versus suspended load) of the eroded material. Core erosion rates and ratios of bedload to suspended load transport of quartz sediments were studied with the ASSET Flume. The erosion and transport of a fine-grained natural cohesive sediment were also observed. Experiments using quartz sands revealed that the ratio of suspended load to bedload sediment transport is a function of grain diameter and shear stress at the sediment surface. Data collected from the ASSET Flume were used to formulate a novel empirical relation for predicting the ratio of bedload to suspended load as a function of shear stress and grain diameter for noncohesive sediments.     

Modes of Bed-Load Movement on a Smooth Rigid Surface

Sediment grains transported as supply-limited bed load on a rigid surface move either discretely or collectively as bed forms, with significantly different effective grain speeds and active storage volumes. The adopted mode has implications for sediment sorting and heavy mineral placer formation, dispersal of grain-associated pollutants, and accumulation and flushing of sediment deposits in unlined canals and sewers. The threshold condition between the two modes has been established for a smooth surface from flume experiments with different sediment types, flow conditions, and sediment supply rates. This is expressed in terms of a relationship between the sediment movability number, a dimensionless bed load parameter, and a grain shear Reynolds number.     

Effect of Sand Movement on a Cohesive Substrate

Flume experiments investigated the effect of mobile sand on the erosion of cohesive beds. The fluid-induced stress alone was not enough to cause erosion, and sand motion as bed load was needed. Erosion rates and suspended sediment concentration were found to increase with increasing sand transport and to decrease with increasing median grain size. The erosion rate was found to be at a maximum during saltation, intermediate during creep, and lowest during suspension.     

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.     

Simulation of Bed-Load Dispersion Process

Two general dispersion models suitable for nonequilibrium bed-load transport were constructed. The first one, called the P model, is based on the probability of migration for specific groups of sediment particles. The second one, called the D model, is derived from the advection equation discretized in finite-difference form, which is equivalent to the general dispersion equation. By comparing these models, it is found that the D model can be treated like the P model in some respects. The Courant number, Cr, in the D model has the same physical meaning as the probability of migration, P, in the P model. Although the D model and P model were based on different concepts, the simulated bed-load transport rates, which result from their application, are the same. Therefore, the dispersion equation was replaced by the numerical algorithm of the advection equation (D model) to examine several dispersion phenomena of bed-load transport. To explore further the nonequilibrium dispersion process, a series of flume experiments was conducted by using color-painted fine gravels. Having compared model simulation results and experimental data, it is shown that the models derived in this study have a reasonably good agreement with the experimental results. In summary, this study has indirectly proven that the D model, which is equivalent to the dispersion equation, is capable of simulating the nonequilibrium bed-load transport.     

One-Dimensional Model for Transient Flows Involving Bed-Load Sediment Transport and Changes in Flow Regimes

This study presents a novel, simple, but rather accurate approximation of the eigenvalues of the system formed by the Saint-Venant–Exner equations, based on the comparison between eigenvalues for the complete system and eigenvalues for the water phase only. Moreover, a strategy is proposed to compute efficiently the intercell fluxes by properly adapting a Harten, Lax, and van Leer scheme for each equation. Two examples of transient transcritical flows are developed: the erosive migration of a knickpoint induced by an increase in the bed slope, and the evolution of a hydraulic jump over a mobile bed.     

Influence of Turbulence on Bed Load Sediment Transport

This paper summarizes the results of an experimental study on the influence of an external turbulence field on the bed load sediment transport in an open channel. The external turbulence was generated by (1) a horizontal pipe placed halfway through the depth h; (2) a series of grids with a clearance of about one-third of the depth from the bed, and extending over a finite length of the flume; and (3) a series of grids with a clearance in the range (0.1–1.0)h from the bed, but extending over the entire length of the flume. Two kinds of experiments were conducted: plane-bed experiments and ripple-covered-bed experiments. In the former case, the flow in the presence of the turbulence generator was adjusted so that the mean bed shear stress was the same as in the case without the turbulence generator in order to single out the effect of the external turbulence on the sediment transport. In the ripple-covered-bed case, the mean and turbulence quantities of the streamwise component of the velocity were measured, and the Shields parameter, due to skin friction, was determined. The Shields parameter, together with the RMS value of the streamwise velocity fluctuations, was correlated with the sediment transport rate. The sediment transport increases markedly with increasing turbulence level.     

Bedload Layer Thickness and Disturbance Depth in Gravel Bed Streams

A field investigation in ten gravel bed stream reaches determined that substrate disturbance depth associated with a moving bedload layer was a small multiple of the bed surface D90. Disturbance depth during plane bed transport of coarse, heterogeneous mixtures appeared similar in magnitude to particle exchange depth and moving layer thickness. Maximum disturbance depth was distributed approximately uniformly over the most active areas of the streambed when local scour and fill were negligible. The distribution upper bound was the smaller of approximately 1.5 times the competent grain size or twice the surface D90, and was invariant with flow strength once the largest grains present were mobilized. Disturbance depth did not scale with grain sizes smaller than D50 when larger grains were mobilized. Thicker traction carpets were not predicted to occur because much larger shear stresses then observed naturally were needed to mobilize two or more layers of the bed simultaneously. Bedload transport rate in coarse streambeds is suggested to increase primarily with mobile fraction of bed surface area and grain velocity, than with layer thickness.     

Measurement of Coarse Gravel and Cobble Transport Using Portable Bedload Traps

Portable bedload traps (0.3 by 0.2 m opening) were developed for sampling coarse bedload transport in mountain gravel-bed rivers during wadable high flows. The 0.9 m long trailing net can capture about 20 kg of gravel and cobbles. Traps are positioned on ground plates anchored in the streambed to minimize disturbance of the streambed during sampling. This design permits sampling times of up to 1 h, overcoming short-term temporal variability issues. Bedload traps were tested in two streams and appear to collect representative samples of gravel bedload transport. Bedload rating and flow competence curves are well-defined and steeper than those obtained by a Helley–Smith sampler. Rating curves from both samplers differ most at low flow but approach each other near bankfull flow. Critical flow determined from bedload traps is similar using the largest grain and the small transport rate method, suggesting suitability of bedload trap data for incipient motion studies.     

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.     

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.     

Experimental and Numerical Investigation of Bed-Load Transport under Unsteady Flows

The dynamic behavior of bed-load sediment transport under unsteady flow conditions is experimentally and numerically investigated. A series of experiments are conducted in a rectangular flume (18?m in length, 0.80?m in width) with various triangular and trapezoidal shaped hydrographs. The flume bed of 8?cm in height consists of scraped uniform small gravel of D50 = 4.8??mm. Analysis of the experimental results showed that bed-load transport rates followed the temporal variation of the triangular and trapezoidal hydrographs with a time lag on the average of 11 and 30?s, respectively. The experimental data were also qualitatively investigated employing the unsteady-flow parameter and total flow work index. The analysis results revealed that total yield increased exponentially with the total flow work. An original expression which is based on the net acceleration concept was proposed for the unsteadiness parameter. Analysis of the results then revealed that the total yield increased exponentially with the increase in the value of the proposed unsteadiness parameter. Further analysis of the experimental results revealed that total flow work has an inverse exponential variation relation with the lag time. A one-dimensional numerical model that employs the governing equations for the conservation of mass for water and sediment and the momentum was also developed to simulate the experimental results. The momentum equation was approximated by the diffusion wave approach, and the kinematic wave theory approach was employed to relate the bed sediment flux to the sediment concentration. The model successfully simulated measured sedimentographs. It predicted sediment yield, on the average, with errors of 7% and 15% of peak loads for the triangular and trapezoidal hydrograph experiments, respectively.     

Bed-Load Sediment Transport on Large Slopes: Model Formulation and Implementation within a RANS Solver

Standard bed-load sediment-transport formulas are extended using basic mechanical principles to include gravitational influence on large slopes of arbitrary orientation. The resulting sediment fluxes are then incorporated into a morphodynamics model in a general-purpose, three-dimensional, finite-volume, Reynolds-averaged Navier–Stokes (RANS) code. Major features are: (1) the downslope component of weight is combined with the fluid stress to form an effective bed stress (similar to the work of Wu in 2004); (2) the critical effective stress is reduced in proportion to the component of gravity normal to the slope; (3) a simple flux-based model for avalanching is implemented as a numerical means of preventing the local slope from exceeding the angle of repose; (4) an entirely vectorial formulation of bed-load transport is developed to account for arbitrary surface orientation; and (5) methods for reducing numerical instability in the morphodynamics equation are described. Sample computations are shown for scour and accretion in a channel bend and for the movement of sand mounds on erodible and nonerodible bases.     

Erosion of Cohesive Sediments: Resuspension, Bed Load, and Erosion Patterns from Field Experiments

New field data on cohesive sediment erosion is presented and discussed, with particular focus on partitioning the total erosion into resuspension and bed load. The data were obtained using a recently developed in situ flume of the National Institute of Water and Atmospheric Research, New Zealand. The erosion rate is estimated from direct measurements of bed surface elevations by acoustic sensors, whereas resuspension rate is obtained using data on sediment concentrations measured by optical backscatter sensors. The bed- load contribution to the total erosion rate is evaluated from the conservation equation for sediments. To test repeatability, the data from the in situ flume are compared with those from a previous version of the flume. The results show that comparative studies of in situ flumes and standardized deployment procedures enable direct comparison of experimental data on cohesive sediment erosion. Overall, the data show that a commonly used assumption that the erosion rate is equal to the resuspension rate is not always valid as bed load plays a significant role in cohesive sediment erosion. The data also highlight the importance of clay content and other sediment physical characteristics in the sediment mixture.