Bed Load Equations Research Articles

An Attempt to Take Into Account Natural Variability in 1D Bedload Prediction

AbstractBedload transport can fluctuate considerably over relatively short periods of time and for a given quasi‐constant flow rate. What are the implications of replacing the fluctuating signal with a smoothed signal when calculating bedload transport using averaged values, as is common practice? This question was investigated with the BedloadR code, which allows 1D bedload calculation as well as Monte Carlo simulations using a new data set collected in the Severaisse River (French Ecrins massif). Four bedload equations (Camenen & Larson, 2005, https://doi.org/10.1016/j.ecss.2004.10.019; Meyer‐Peter & Mueller, 1948; Parker, 1990, https://doi.org/10.1080/00221689009499058; Recking, 2013a, https://doi.org/10.1061/(asce)hy.1943‐7900.0000653) were selected for their performance relative to the measured bedload (except for and Meyer‐Peter and Mueller) and because each equation has a different mathematical form and degree of nonlinearity. They were used in a Monte Carlo approach, with input probability distributions fitted to the measured river width, slope, bed grain‐size distribution, and to the associated (computed) Shields stress. The results show that accounting for natural variability in the calculation reproduces bedload fluctuations well. But overall, when calculating the bedload volume transported by a flow event, accounting for variability systematically leads to higher estimated volumes (of the order of 20%) than those obtained with a deterministic approach using average input parameters. This is a direct consequence of the nonlinearity of the equations.

Coupling of sediment transport phenomena with turbulent surface flows: mathematical modeling, finite volume approximation and test simulations

Several sediment transport models coupled with shallow water equations have been developed in the literature over the past few decades. The water flow in the presence of abrupt bottom slopes distorts and cause turbulence. This phenomenon is well-known in the sediment transport theory. Sediment transport models based on shallow water equations do not include turbulence effects or the phase-lag between water velocity and bottom velocity. In this study, we consider a one-dimensional free surface turbulent water flow with sediment and morphodynamic. We propose a new one-dimensional acceleration-dissipation mathematical sediment transport model that considers the distortion of horizontal profile velocity in transport. The model is an improved version of existing averaged sediment transport models. In the new model, there is a distinction between fluid velocity and bottom velocity through a new bedload equation. The mathematical model is a system of nonlinear hyperbolic partial differential equations that are approximated by a stable and robust finite volume scheme presented in the path-conservative framework. Numerical tests are performed to demonstrate the efficiency of the developed theory.

The effectiveness parameters analysis for piers scour calculation

AbstractA computational fluid dynamics numerical model addressed the problem of local scouring and deposition calculation for non-cohesive sediment and clear water conditions near single and double cylindrical piers. The numerical results of single cylindrical piers correlate very well with the physical model's results while are higher than the case of the double pier, especially when the large-eddy turbulence model, the van Rijn bed-load transport equation, and fine mesh size are considered. Additionally, the final numerical predictions are compared to experimental data after parameters effectiveness explores the range of results based on projected user inputs like the bed-load equation, mesh cell size, and turbulence model.

The contribution of grain sorting to the dynamics of the bedload active layer

AbstractDuring the last 20 years, flume and field experiments have shown that grain sorting contributes to bed‐level fluctuations and bedload pulses. In this work, we propose a new analysis of these experimental data. From the flume data, we derive a model for gravel‐bed rivers where both local (bedform‐scale) slope and bedload are known to fluctuate through space and time, in the so‐called ‘bedload active layer’. The model uses standard concepts and empirical tools with reach‐averaged data for the hydraulics and sediment transport. It considers a maximum slope for local armouring equal to the mean bed slope (reach scale) affected by a coefficient which expresses the difference in mobility of the coarse fraction considered alone or in a mixture. The minimum local slope for bed erosion is the mean bed slope corrected by a coefficient that depends on the armour ratio Ar (ratio of the surface to the subsurface grain diameter) and the reach‐averaged transport rate. The model is compared with a compilation of scour–fill depths measured in the field. Results suggests that the slope fluctuations in 1D flume experiments are consistent with in‐channel bed‐level fluctuations associated with scour–fill processes in the active layer. The model also suggests that although the length scale of the maximum scour depth δ is on the order of the bed surface D90, it is well explained by the product between the mean bed slope S and the active channel width W, with δ ≈ 1.4SW. For the pulse intensity, we provide a justification for the simplified squared slope equation for solid concentration C = Qs/Q ∝ S2 (with Qs the solid discharge, Q the water discharge and S the slope), which has often been used in place of standard bedload equations for modelling highly concentrated bedload transport events in mountain streams.

Evaluation and calibration of bedload equation for the mountain ephemeral stream of Gujarat, India

Evaluation and calibration of bedload equation for the mountain ephemeral stream of Gujarat, India

A Novel Sediment Transport Model (STM) Accounting Phase Lag Effect. A Resonance Condition

The classical Exner model coupled with a bed-load sediment flux formula is widely used to describe the morphodynamics of coastal environments. However, the main drawbacks of this model are (i) Lack of robustness, (ii) Lack of differentiation between sediment and fluid velocities, and (iii) Generation of instabilities when the interactions between sediment and fluid flow become more important. Moreover, Exner's model does not allow us to know with which characteristic velocity the bottom is moving. This set of drawbacks weakens the effectiveness of most sediment transport models proposed in the literature, particularly the Exner model. In this work, we reformulate the bed-load equation and we propose a new averaged sediment transport model for application in coastal or estuarine environments. The proposed model incorporates phase shift effects into the bed-load equation. The bedform's characteristic velocity, sediment, and fluid velocity are differentiated. We developed a new first-order, well-balanced, positivity-preserving, path-preserving, and central wind (WBPP-PCCU) scheme to solve the proposed hyperbolic sediment transport model (HSTM). We used the Averaging Essentially Non-Oscillatory (AENO) reconstruction coupled with the third-order Runge-Kutta Semi-Implicit (SI-RK3) method to achieve second-order accuracy. The balance and positivity of the water depth properties were proven. In this work, a resonance condition is proposed. The model facilitates the application of several other schemes such as Roe, HLLC, HLLEM, PVM (polynomial viscosity matrix), RVM (rational viscosity matrix), which require the diagonalization of the Jacobian matrix. The accuracy, robustness, positivity preservation, and equilibrium properties of the resulting model are evaluated using a series of carefully selected test cases. The proposed model provides an excellent ability to simulate sediment transport in a wide range of coastal environments.

Source‐to‐sink response to high‐amplitude lake level rise driven by orbital‐scale climate change: An example from the Pleistocene Lake Malawi (Nyasa) Rift, East Africa

AbstractSedimentary systems respond to environmental forcings in dissimilar ways over different timescales. The Lake Malawi (Nyasa) Rift is an ideal natural laboratory for evaluating the overall functioning of lacustrine source‐to‐sink systems on orbital or younger scales. These closed sedimentary systems exhibited high spatiotemporal climate variability, and their responses to two late Pleistocene lake‐level stillstands are evaluated. The coarse‐grained deposits documented a dramatic transgression from 350 to 200 m below present lake‐level (BPLL) developed during an important climate transition in tropical Africa. Based on an integrated analysis of a digital elevation model and high‐resolution single/multi‐channel seismic profiles, catchment geomorphology has been linked with sediment delivery in the sink area. The coarse‐grained deposition of each source‐to‐sink system is quantified through a sediment mass calculation. A modified empirically‐derived ‘BQART’ predictor with a bedload equation to assess the sediment discharge is employed based on a Monte Carlo simulation, considering temperature lapse rate and topographic effects. The river discharges are estimated by specific empirical relationships that associate catchment area to various climate systems, developed using a global modern river database. The results show that the total sediment discharge increases from 7.53 to 9.50 Mt year−1; likewise, the preserved coarse‐grained deposits also record a significant increase in deposition rate from 350 to 200 m BPLL stage, indicating that the short length‐scale source‐to‐sink systems are sensitive to the high‐amplitude lake transgression developed from the climate shift. The volume of upstream buffered deposits may decrease within the progressively wetter climate, while the buffering degree was substantially influenced by the pre‐existing landforms. Moreover, the substantial deep‐water mud dispersal is not well‐developed, despite the relatively higher lake‐level and slightly wetter climate. This quantitative source‐to‐sink analysis with the modified sediment predictor yields preliminary constraints for system functioning in response to high‐amplitude climate change in a closed sedimentary system.

The estimation of bedload in poorly-gauged mountain rivers

Bedload transport is one major driver of gravel-bed river morphodynamics, and its quantification is capital for many environmental issues and river engineering applications, as well as for landscape evolution studies. To this point, bedload transport rates and volumes have been classically computed by means of sediment transport formulae. The most used bedload transport equations compute the bulk bedload mass based on section-averaged hydraulic parameters. However, due to the non-linear behavior of sediment transport, bedload formulae are sensitive to the input parameters. Then, some doubts arise when applying bedload equations on poorly gauged river reaches, i.e. rivers where there are no hydrological records and rating curves. In this paper, we assess the application of bedload equations in the case of poorly gauged river reaches, and we test a workflow to follow in such situations. This workflow consists of three steps: (i) Reconstructing the flow duration curve, based on gauging records from neighboring river basins; (ii) Solving the hydraulic geometry relations of the study-case river-reach, based on a flow friction equation; and finally, (iii) Computing bedload with a sediment transport equation. We tested this approach against the bedload information available in the literature for Idaho streams and we found that it could potentially approximate annual bedload volumes in ungauged reaches under certain conditions. To illustrate the potential of this workflow, we also computed bedload volumes for two ungauged river reaches from the Cantabrian mountains (NW Spain).

Performance of bedload transport equations in a mixed bedrock–alluvial channel environment

Mixed bedrock–alluvial channel beds are common in mountain rivers. It is well-known that in such system the balance between sediment supply and transport capacity defines either the level of the exposure of bedrock (in the riverbed) or its coverage by alluvial material. We investigate whether applying the representation of the energy losses (caused by macro-roughness) developed for alluvial channels improve the capacity of bedload transport equations to predict the bedload transport in a mixed bedrock–alluvial stream. Therefore, the purpose of this study was to evaluate the performances of five bedload equations against field measurements, while considering the effects of macro-roughness on the equations and the limited availability of mobile sediment. We analyzed the data collected in a stream located in southern Brazil, which has a 2.3% gradient and mixed bed conditions. Optimal results in estimating the bedload transport rate were obtained when using the reduced shear stress τ' and reduction factor of the available bed material (Frm) together in the five tested equations. Analyzing the two approaches separately, the implementation of τ' proved to be more critical for improving the performance of the equations than using Frm alone. Among all the equations, Recking (2013) presented the best result for the case in which 85.2% of the estimates fell within one order of magnitude of the measured transport rates (0.1 < r > 10.0); this was followed by the Recking (2010) (70.4% of estimates within one order of magnitude) and Meyer-Peter and Müller (1948) (37% of estimates within one order of magnitude) equations. In conclusion, we observed that even by implementing flow resistance and with the limited availability of mobile sediment, the equations overestimated the bedload transport rate in the studied river reach, indicating an underestimation of energy loss in the flow resistance equations of mixed bedrock–alluvial channels.

Rational Evaluation Methods of Topographical Change and Building Destruction in the Inundation Area by a Huge Tsunami

In the case of huge tsunamis, such as the 2004 Great Indian Ocean Tsunami and 2011 Great East Japan Tsunami, the damage caused by ground scour is serious. Therefore, it is important to improve prediction models for the topographical change of huge tsunamis. For general models that predict topographical change, the flow velocity distribution of a flood region is calculated by a numerical model based on a nonlinear long wave theory, and the distribution of bed-load rates is calculated using this velocity distribution and an equation for evaluating bed-load rates. This bed-load rate equation usually has a coefficient that can be decided using verification simulations. For the purpose, Ribberink’s formula has high reproducibility within an oscillating flow and was chosen by the authors. Ribberink’s formula needs a bed-load transport coefficient that requires sufficient verification simulations, as it consumes plenty of time and money to decide its value. Therefore, the authors generated diagrams that can define the suitable bed-load coefficient simply using the data acquired from hydraulic experiments on a movable bed. Subsequently, for the verification purpose of the model, the authors performed reproduced simulations of topography changes caused by the 2011 Great East Japan Tsunami at some coasts in Northern Japan using suitable coefficients acquired from the generated diagrams. The results of the simulations were in an acceptable range. The authors presented the preliminary generated diagrams of the same methodology but with insubstantial experimental data at the time at the International Society of Offshore and Polar Engineers (ISOPE), (2018 and 2019). However, in this paper, an adequate amount of data was added to the developed diagrams based on many hydraulic experiments to further raise their reliability and their application extent. Furthermore, by reproducing the tsunami simulation on the Sendai Natori coast of Japan, the authors determined that the impact of total bed-load transport was much bigger than that of suspension loads. Besides, the simulation outputs revealed that the mitigation effect of the cemented sand and gravel (CSG) banks and artificial refuge hills reduced tsunami damage on Japan’s Hamamatsu coast. Since a lot of buildings and structures in the inundation area can be destroyed by tsunamis, building destruction design was presented in this paper through an economy and simplified state. Using the proposed tsunami simulation model, we acquired the inundation depth at any specific time and location within the inundated area. Because the inundation breadth due to a huge tsunami can extend kilometers toward the inland area, the evaluation of building destruction is an important measure to consider. Therefore, the authors in this paper presented useful threshold diagrams to evaluate building destruction with an easy and cost-efficient state. The threshold diagrams of “width of a pillar” for buildings or “width of concrete block walls” not breaking to each inundation height were developed using the data of damages due to the 2011 Great East Japan Tsunami.

Numerical simulation of scour and flow field over movable bed induced by a submerged wall jet

Abstract Dam construction continues its rapid expansion around the world primarily for the purpose of hydropower generation. One important consequence of such projects is local scour at the downstream of the dam caused by outflow of excess reservoir water through spillways or bottom outlets that is associated with high velocities. The scour development endangers the dam foundation and river banks and undermines the stability of the hydraulic structures. In this study, a detailed three-dimensional (3D) flow simulation is conducted to investigate the complex fluid–sediment interactions leading to the formation of the scour hole and ridge systems downstream of a near-bottom jet. Three different bed-load equations, including Meyer-Peter–Müller, Nielsen, and Van Rijn formulas, are applied for calculating the bed-load transport rate. Comparison with a series of available experimental data shows that the Meyer-Peter–Müller equation results in better predictions than the two other relations. The performance of different turbulence models to reproduce vertical profiles of velocity and scour characteristic against the experimental data were evaluated. The vertical and horizontal profiles of the scour hole-ridge system are also compared with the corresponding experimental ones. The numerical model satisfactorily reproduces the geometric parameters representing the scour hole. However, the model overestimates the length of the scour hole.

Numerical modeling of local scour due to submerged wall jets using a strict vertex-based, terrain conformal, moving-mesh technique in OpenFOAM

This study presents the implementation and validation of a new sediment-scour model with a strict vertex-based, terrain conformal, moving-mesh technique within the framework of OpenFOAM. OpenFOAM lacks the ability to simulate large-amplitude motion needed for analysis of sediment-scour problems, and, thus, its application normally is restricted to small-amplitude cases to prevent computational divergence due to mesh deterioration. The proposed simple, moving-mesh technique in OpenFOAM is implemented to overcome the shortcomings of the conventional automatic mesh-motion techniques in handling large-amplitude moving geometries. The model is used to simulate a simple case of prescribed boundary motion, a previous experiment in the literature, and a new laboratory experiment for local scour due to submerged wall jets. The results are compared with both the experimental and other numerical results. The comparisons demonstrate that the proposed model has the novel advantage of allowing for more severe topographic variations, and can provide more reliable predictions for the key characteristics and evolution of the bed profiles in wall jet scour problems. Furthermore, to improve the practice of modeling wall jet scour, various turbulence modeling approaches and bedload equations also are evaluated and compared.

Uniform and graded bed-load sediment transport in a degrading channel with non-equilibrium conditions

Bed-load transport plays a critical role in river morphological change and has an important impact on river ecology. Although there is good understanding of the role of the variation of river bed grain size on transport dynamics in equilibrium conditions, much less is understood for non-equilibrium conditions when the channel is either aggrading or degrading. In particular, the relative role of different grain sizes in the promotion and hindering of the transport of coarse and fine fractions in a degrading channel has yet to be investigated. The current study attempts to provide new understanding through a series of flume experiments done using uniform and graded sediment particles. The experiments revealed coarser grain-size fractions for a poorly-sorted sediment, relative to uniform-sized sediment, reduced the transport of finer grains and finer fractions enhanced the transport of coarse grains. This hindering-promotion effect, caused by relative hiding and exposure of finer and coarse fractions, increased with bed slope and decreased with relative submergence. In particular, as relative submergence increased, the graded fractions tended towards behaving more like their uniform-sized counterparts. Also, the bed-load parameter of the graded fractions increased more with a rise in bed slope than observed for the uniform-sized counterparts. These results revealed, for degrading channel conditions, such as downstream of a dam, bed-load equations developed for uniform bed sediment are inappropriate for use in natural river systems, particularly in mountain streams. Furthermore, changes in river bed composition due to activities that enhance the input of hill-slope sediment, such as fire, logging, and agricultural development, are likely to cause significant changes in river morphology.

Evaluation of bed load equations using field measured bed load and bed material load

ABSTRACT Bed load transport can be determined by direct measurement of bed load being transported by a stream. In India, measurement of bed load is rarely done. In the present study, bed load has been measured by unequal width incremental method using United States Geological Survey (USGS) bed load sampler (Modified-Helley–Smith bed load sampler) and bed material has been measured using scoop type sampler for the Ver stream in Mandvi region of Surat, Gujarat, India. From the field measurements and analysis, it is seen that the Ver stream has dominance of bed load transport with a Rouse number higher than 2.5 for all the selected cross sections. The measured average bed load transport rate is found to be 7.77 t/day. The measured bed load transport rate has been tested using two prominent bed load models. In 1950 Brown proposed the reach averaged approach which was augmented and supported by Recking in 2013. Both Brown's and Recking's reach averaged approaches have been used for computing bed load of Ver stream using particle size of bed load and bed material. Bed load computed using Recking's approach matches well with the observed bed load while Brown's model over-predicted bed load transport rate using bed load particle size. Overall, Recking's reach-averaged approach performs well using bed load particle size and bed material particle size.

An Analytical Approach to Estimate Optimum River Channel Dimensions

Extremal hypotheses without bank stability constraint typically over-predict and under-predict channel width in large rivers and natural streams, respectively. In general, results obtained from unconstrained extremal hypotheses are indicative of inappropriate agreement between predicted and observed dimensions of the rivers. One of the important factors in disparity of the data may be lack of appropriate relationships to assess bank vegetation of the rivers. For this reason, a modified analyticalmodel has been developed to reduce the effect of bias by considering bank stability and vegetation. The model takes into account channel shape factor, a wide range of bed load equations in the form excess shear stress and vegetation quantification is able to predict optimal channel geometry dimensions. Finally, developed model was calibrated using the field data of the United Kingdom and Iran. In addition to indicating the effect of bank stability and vegetation on estimation of the geometric characteristics of the channel, obtained results also confirmed the efficiency of the constrained model in comparison to the unconstrained model. This study also provides support for the use of the concepts of maximum sediment transporting capacity and minimum stream power for understanding the operation of alluvial rivers.

A total load approach to predict bed load transport of Indian alluvial river

Bed load data are scarcely measured continuously for Indian rivers. In the absence of measured bed load data, the present study tests bed load computed using selected bed load sediment transport functions. The computed bed load is compared with widely accepted thumb rule estimation of the bed load, i.e. the Maddock’s approach. To check the suitability of these bed load equations for Indian river (Purna River), the computed bed load (BLc) has been added to observed suspended load and total load of the Purna River is derived. On other hand, the total load for the Purna River is worked out using Yang’s (1973) unit stream power approach for total load prediction. The total load computed (TLc) for the Purna river is then compared with the total load predicted (TLp) using Yang’s approach. It is observed that best prediction in total load transport is obtained when bed load is computed as 10% of observed suspended load. A similar performance is obtained when bed load is computed using Recking’s bed load function.

Numerical study on influences of breakwater layout on coastal waves, wave-induced currents, sediment transport and beach morphological evolution

This study provides a numerical model to investigate the influences of breakwater layout on coastal waves, wave-induced currents, sediment transport and beach morphological evolution in the vicinity of breakwater. The numerical model is developed based on the sub-models for nearshore wave, wave-induced current, sediment transport and beach morphological evolution. Nearshore wave is simulated based on the parabolic mild-slope equation considering wave refraction, diffraction and breaking effects. Wave-induced nearshore current is modeled using the nonlinear shallow water equations in which wave radiation stresses are provided by wave model for driving current. Then, the two-dimensional suspended sediment transport equation, bed-load equation and coast beach morphological evolution are coupled with the wave and current models for simulating sediment transport and morphological evolution in coastal waves and wave-induced currents. The numerical model is firstly tested by the experiment results for coastal waves, near-shore currents, sediment transport and beach morphological evolution around breakwater from the Large-scale Sediment Transport Facility at the US Army Corps of Engineer Research and Development Center (Gravens and Wang, 2007). Then the model is used to study the influences of breakwater layout on coastal waves, wave-induced currents, sediment transport and beach morphological evolution by set several breakwater layouts in the LSTF basin.

A generalized threshold model for computing bed load grain size distribution

AbstractFor morphodynamic studies, it is important to compute not only the transported volumes of bed load, but also the size of the transported material. A few bed load equations compute fractional transport (i.e., both the volume and grain size distribution), but many equations compute only the bulk transport (a volume) with no consideration of the transported grain sizes. To fill this gap, a method is proposed to compute the bed load grain size distribution separately to the bed load flux. The method is called the Generalized Threshold Model (GTM), because it extends the flow competence method for threshold of motion of the largest transported grain size to the full bed surface grain size distribution. This was achieved by replacing dimensional diameters with their size indices in the standard hiding function, which offers a useful framework for computation, carried out for each indices considered in the range [1, 100]. New functions are also proposed to account for partial transport. The method is very simple to implement and is sufficiently flexible to be tested in many environments. In addition to being a good complement to standard bulk bed load equations, it could also serve as a framework to assist in analyzing the physics of bed load transport in future research.

A probabilistic formulation of bed load transport to include spatial variability of flow and surface grain size distributions

AbstractBed load fluxes are typically calculated as a function of the reach averaged boundary shear stress and a representative bed grain size distribution. In steep, rough channels, heterogeneous bed surface texture and macro‐roughness elements cause significant local deviations from the mean shear stress but this variability is often omitted in bed load calculations. Here we present a probabilistic bed load transport formulation that explicitly includes local variations in the flow field and grain size distribution. The model is then tested in a 10% gradient stream, to evaluate its predictive capability and to explore relations between surface grain size sorting and boundary shear stress. The boundary shear stress field, calculated using a quasi‐3D hydraulic model, displayed substantial variability between patch classes, but the patch mean dimensionless shear stress varied inversely with patch median grain size. We developed an empirical relation between the applied shear stress on each patch class and the reach averaged shear stress and median grain size. Predicted sediment volumes using this relation in our bed load equation were as accurate as those using complete shear stress distributions and more accurate than current bed load transport equations. Our results suggest that when spatially variable grain size distributions (e.g., patches of sediment) are present they must be explicitly included in bed load transport calculations. Spatial variability in shear stress was relatively more important than grain size variations for sediment transport predictions.

Quantifying the Morphological Print of Bedload Transport

AbstractBedload and river morphology interact in a strong feedback manner. Bedload conditions the development of river morphology along different space and time scales; however, by concentrating the flow in preferential paths, a given morphology controls bedload for a given discharge. As bedload is a non‐linear response of shear stress, local morphology is likely to have a strong impact on bedload prediction when the shear stress is averaged over the section, as is usually done.This was investigated by comparing bedload measured in different bed morphologies (step‐pool, plane bed, riffle‐pool, braiding, and sand beds), with bedload measured in narrow flumes in the absence of any bed form, used here as a reference. The initial methodology consisted of fitting a bedload equation to the flume data. Secondly, the morphological signature of each river was studied as the distance to this referent equation.It was concluded that each morphology affects bedload in a different way. For a given average grain shear stress, the larger the river, the larger the deviation from the flume transport. Narrow streams are those morphologies that behave more like flumes; this is particularly true with flat beds, whereas results deviate from flumes to a greater extent in step‐pools. The riffle‐pool's morphology impacts bedload at different levels depending on the degree of bar development, considered here through the ratio D84/D50 which is used as a proxy for the local bed patchiness and morphology. In braiding rivers morphological effects are important but difficult to assess because width is dependent on transport rate. Bed morphology was found to have negligible effects in sand bed rivers where the Shields stress is usually sufficiently high to minimize the non‐linearity effects when hydraulics is averaged over the section. Copyright © 2015 John Wiley &amp; Sons, Ltd.