Critical Shear Stress of Bimodal Sediment in Sand-Gravel Rivers

A new model for the critical shear stress and the transport of graded sediment is presented. The model is based on the size distribution of the bed surface and can be used to compute sediment transport rates in numerical simulations with an active layer model. This model makes a distinction between unimodal and bimodal sediments. It is assumed that all size fractions of unimodal sediments have the same critical shear stress while there is selective transport for the gravel fractions of bimodal sediments. A recently published laboratory transport data set is used to calibrate our model.     

Modeling Bed-Load Rates from Fine Grain-Size Patches during Small Floods in a Gravel-Bed River

This study investigates the applicability of five bed-load-transport formulas (the Meyer-Peter and Müller, Schoklitsch, Bagnold, Smart and Jaeggi, and Rickenmann equations) to predict bed-load transport rates of frequent, low-magnitude flood events (maximal bankfull discharge) for a mountainous, poorly sorted gravel-bed river characterized by a bimodal sediment-size distribution and spatially distributed patches. For model parametrization, special emphasis was placed on the spatial composition of the grain-size distribution (GSD) to evaluate the impact of preferential removal of sediments from patches with finer sediments on bed-load transport. Three parametrization approaches to the choice of an appropriate sediment size that considered the apparent bimodality of the GSD to varying degrees were tested. The modeling study demonstrated that the incorporation of spatial structure of GSD and its bimodal character has an important impact on model performance—a unimodal parametrization failed to reproduce measured bed-load rates for all tested bed-load formulas; a threshold parametrization approach that considered only finer sediments from the small patches as bed-load source material in combination with the Schoklitsch, Smart and Jaeggi, and Rickenmann equations yielded the best results, whereas the Meyer-Peter and Müller and the Bagnold equations failed to predict bed-load rates for all parametrization approaches. The modeling study thus showed that bed-load formulas are sensitive to the spatial structure of the GSD, which should not be treated as a continuum of sediment size fractions but rather as composition of finer sediment patches to enable an adequate reproduction of measured bed-load data from low-magnitude floods in gravel-bed rivers.     

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.     

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.     

Effect of Sand Supply on Transport Rates in a Gravel-Bed Channel

In a series of flume experiments using constant discharge, flow depth, and gravel feed rate, sand feed rates were varied from 0.16 to 6.1 times that of gravel. The bed slope decreased with increasing sand supply, indicating that the gravel could be transported at the same rate, along with increasing amounts of sand, at smaller shear stresses. Prediction of river response to an increase in sediment supply requires prediction of mutual changes in bed composition and transport, and therefore a transport model defined in terms of the grain size of the bed surface. A recent model provides satisfactory prediction of the experimental observations and indicates the general response of gravel beds to increased sand supply. An increase in sand supply may increase the sand content of the river bed and the mobility of gravel fractions, which can lead to bed degradation and preferential evacuation of these sediments from the river.     

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.     

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.     

Turbulence and Secondary Flow over Sediment Stripes in Weakly Bimodal Bed Material

Longitudinal stripes are a common bed form in heterogeneous alluvial sediments and consist of periodic, spanwise variations in bed texture and elevation that are aligned parallel to the mean flow direction. This paper quantifies mean and turbulent flow structures over self-formed sediment stripes in a weakly bimodal sand and gravel mixture. Turbulence anisotropy generates two secondary circulation cells across the channel half-width, which produce a cross-stream perturbation in boundary shear stress. The interaction between this flow structure and the selective transport of bed material generates spanwise sediment sorting that is symmetrical about the centerline. Finer sediments are entrained from regions of high shear stress, transported laterally by the secondary flow, and deposited in regions of lower shear stress. Lateral changes in bed texture further enhance the near-bed secondary flow, which provides a positive feedback mechanism for stripe growth. In bimodal sediments, at shear stresses just above the entrainment threshold, stripes may replace lower-stage plane beds. At higher shear stresses the coarser sediment becomes more mobile and the stripes are replaced by flow transverse bed forms.     

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.     

Bedload Transport in Gravel-Bed Streams with Unimodal Sediment

Bedload transport in many gravel-bed streams becomes highly complicated because of the nonuniformity of the grain size and the vertical stratification of channel bed material. A new relation for computing bedload transport rates in gravel-bed streams is proposed here. In an effort to account for the variation of the makeup of the surface material within a wide range of Shields stresses, the relation employs a two-parameter approach, one related to the material in the pavement and the other to that in the subpavement layers. The mode is used to represent the grain sizes of each layer. Available bedload transport data from gravel-bed streams with unimodal sediment are used to test the accuracy of the relation. A comparison with other bedload transport relations is also considered.     

Effect of Coarse Surface Layer on Bed-Load Transport

Existing bed-load transport formulas may overestimate the transport rate in mountain rivers by two orders of magnitude or more. Recently published field data sets provide an opportunity to take a fresh look at the bed-load transport relationship and it is hypothesized that the overestimate is due to a failure to account for the effect of a coarse surface layer of bed material inhibiting the release of fine subsurface material. Bed-load transport is determined as gs = aρ(q?qc) where q=water discharge per unit width; qc=critical value for initiation of bed material movement; ρ=water density; and a=coefficient. The gs/q relationship is typically piecewise linear, characterized by two transport phases with, respectively, low and high rates of change. Twenty-one flume and 25 field data sets were used to quantify the relationship for Phase 2. The flume data confirm the dependence of a on S1.5, where S=channel slope, in agreement with earlier studies. The field data additionally show that a varies inversely with the degree of bed armoring, given by the ratio of surface to subsurface bed material size. The finding is consistent with the hypothesis and suggests the need to account for the bed material supply limitation in the bed-load transport formula. However, the available data are not entirely sufficient to rule out an alternative dependency, or codependency, on flow resistance. The critical conditions for initiation of Phase 2 transport are also quantified as a function of bed material size and channel slope. The resulting set of equations allows a more accurate estimation of Phase 2 bed-load transport rates. However, the equations are empirical and should be restricted for use within the range of conditions used in their development, to determine mean rather than instantaneous transport rates and to determine bulk transport rates, not transport by size fraction.     

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.     

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.     

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.     

Surface-based Transport Model for Mixed-Size Sediment

We present a transport model for mixed sand/gravel sediments. Fractional transport rates are referenced to the size distribution of the bed surface, rather than subsurface, making the model completely explicit and capable of predicting transient conditions. The model is developed using a new data set of 48 coupled observations of flow, transport, and bed surface grain size using five different sediments. The model incorporates a hiding function that resolves discrepancies observed among earlier hiding functions. The model uses the full size distribution of the bed surface, including sand, and incorporates a nonlinear effect of sand content on gravel transport rate not included in previous models. The model shares some common elements with two previous surface-based transport models, but differs in using the full surface size distribution and in that it is directly developed from a relatively comprehensive data set with unambiguous measurement of surface grain size over a range of flow, transport rate, and sediments.     

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.     

Pollution of Gravel Spawning Grounds by Deposition of Suspended Sediment

Observations have shown that accumulation of fine sediment in the pores of spawning open-work gravel have a detrimental effect on stream biota. The rate of deposition is intimately linked to the concentration of suspended fines near the gravel bed. If interstitial voids in coarse sediment deposits are filled or covered with sand or inorganic fine materials, their habitat value is greatly reduced. In this paper, a simple method is proposed to predict analytically the concentration profile and transport of fine suspended sediment when a steady, uniform suspension flows from a sediment-covered bed to an open-work gravel bed. Comparisons of the analytical model predictions with previous laboratory observations show reasonable agreement. The proposed solution can be used to estimate “clarification distances” for streams carrying fine sediments over open-work gravel beds.     

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.     

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.     

Effects of Lifting Force on Bed Topography and Bed-Surface Sediment Size in Channel Bend

In bed-load sediment transport, the lifting force plays an important role in reducing the friction between sediment particles and the bed surface, and it makes particle transportation by the shear force easier. Because the lifting force is related to vorticity, a three-dimensional (3D) numerical model incorporating large eddy simulations was applied to simulate the vorticity field in a channel bend. The results show that the distribution of vorticity is highly nonuniform, and it can lead to significant variations in lifting force and bed-load sediment transport per unit width in a channel bend. Relevant theories are modified on the basis of physical reasoning and then incorporated into numerical models to investigate the lifting-force effects on the bed topography and bed-surface sediment size gradation in a channel bend. With the lifting-force effects considered, it is shown that the errors in simulated bed topography can be reduced by approximately 40% and in bed-surface sediment size by 50%.