Bed Load Transport Equations Research Articles

Stochastic bedload transport in mountain streams

Describing bedload transport as a stochastic process is an idea that emerged in the 1930s with the pioneering work of Einstein. For a long time, the stochastic approach attracted marginal attention, but the situation has radically changed over the last decade with the recent advances in the theory of bedload transport. In parallel, the implementation of bedload monitoring techniques at high temporal resolution has produced a wealth of interesting results showing, among other things, that classic empirical bedload transport equations do not capture neither the mean behavior of sediment transport rates qs nor its order of magnitude, especially at low sediment transport rates (a case that is most frequent in mountain streams). We have developed a stochastic model, which takes inspiration from population dynamics and provides a stochastic partial differential equation for the number of moving particles. Taking the ensemble average leads to a fairly simple advection diffusion equation for particle activity (i.e., the number of moving particles per unit streambed area). The model has a number of unique features. For instance, it yields the probability distribution of the bedload transport rate and predicts bedform formation for a wide range of Froude numbers.

Bed-load transport: From the beginning into the future

This research addresses bed-load transport in the fluvial environment. Based on the current knowledge, it is first noted that this problem is by far not solved. The eminent questions ask for a large experimental campaign by which the basic issues are solved first, and then expanded to more challenging topics relating to questions not included in the first stage involving uniform water and sediment flows. A number of currently available experimental techniques is mentioned by which a high-quality laboratory experimentation should be feasible. Next, the effects of various fundamental transport parameters should be cleared using both systematic experimentation and data analyses. An improved bed-load transport equation should be established first, by which the current tools are replaced to produce better agreement with prototype data. The role of computational hydraulics is also highlighted, to apply results to natural rivers. Froude similitude should be carefully applied so that the laboratory data can safely be transferred to prototype conditions, if the limitations are respected.

Parsimonious sediment transport equations based on Bagnold's stream power approach

AbstractIt is increasingly recognized that effective river management requires a catchment scale approach. Sediment transport processes are relevant to a number of river functions but quantifying sediment fluxes at network scales is hampered by the difficulty of measuring the variables required for most sediment transport equations (e.g. shear stress, velocity, and flow depth). We develop new bedload and total load sediment transport equations based on specific stream power. These equations use data that are relatively easy to collect or estimate throughout stream networks using remote sensing and other available data: slope, discharge, channel width, and grain size. The new equations are parsimonious yet have similar accuracy to other, more established, alternatives. We further confirm previous findings that the dimensionless critical specific stream power for incipient particle motion is generally consistent across datasets, and that the uncertainty in this parameter has only a minor impact on calculated sediment transport rates. Finally, we test the new bedload transport equation by applying it in a simple channel incision model. Our model results are in close agreement to flume observations and can predict incision rates more accurately than a more complicated morphodynamic model. These new sediment transport equations are well suited for use at stream network scales, allowing quantification of this important process for river management applications. Copyright © 2017 John Wiley & Sons, Ltd.

Bedload transport and flow resistance in steep channels – introduction to the issues in the context of mountain basins of the Central European region

Although sediment transport in the uppermost parts of mountain basins may be supply- limited, bedload transport in small torrents represents a potential hazard during extraordinary floods. The authors attempt to review recent worldwide research of bedload and hydraulics the results of which could potentially be applied in local steep channels. Approaches to the calculation of flow resistance (Manning relationship, logarithmic and power-law relationships) and critical conditions of the incipient motion of a bed particle (critical shear stress, stream power, flow velocity and discharge) are closely discussed in order to describe hydraulic and bedload transport conditions in steep headwater streams. Difficulties are pointed out in connection with the estimation of critical conditions of bedload transport in headwater streams with poorly sorted bed material and more or less developed bedforms. In the second part of the paper the authors present two pilot studies from the Western Carpathians and test potential applicability of specific formulas in this region. Results of the formulas point to relatively nonselective character of bedload transport during high magnitude floods in Beskydian headwater streams. An important role is played by the intensity of sediment delivery to active channels in limited sediment-supply conditions. The authors also point out that a majority of bedload transport and flow resistance equations were originally created on the basis of flume experiments or they were tested in Alpine torrents and therefore it is supposed that they might work well in local torrents after being calibrated to local conditions.

Input‐variable sensitivity assessment for sediment transport relations

AbstractA methodology to assess input‐variable sensitivity for sediment transport relations is presented. The Mean Value First Order Second Moment Method (MVFOSM) is applied to two bed load transport equations showing that it may be used to rank all input variables in terms of how their specific variance affects the overall variance of the sediment transport estimation. In sites where data are scarce or nonexistent, the results obtained may be used to (i) determine what variables would have the largest impact when estimating sediment loads in the absence of field observations and (ii) design field campaigns to specifically measure those variables for which a given transport equation is most sensitive; in sites where data are readily available, the results would allow quantifying the effect that the variance associated with each input variable has on the variance of the sediment transport estimates. An application of the method to two transport relations using data from a tropical mountain river in Costa Rica is implemented to exemplify the potential of the method in places where input data are limited. Results are compared against Monte Carlo simulations to assess the reliability of the method and validate its results. For both of the sediment transport relations used in the sensitivity analysis, accurate knowledge of sediment size was found to have more impact on sediment transport predictions than precise knowledge of other input variables such as channel slope and flow discharge.

IMPLICATIONS OF PLUME DISCHARGE FOR TIDAL CHANNELS MORPHODYNAMICS: A COUPLED ONSHORE AND OFFSHORE SYSTEM

This contribution investigates the morphodynamic equilibrium of a funnel-shaped, well-mixed tidal channel taking into account the existing dynamical coupling between the tidal channel itself and the related offshore sediment-laden plume. We use a quasi two-dimensional numerical model that resolves the fully nonlinear unsteady shallow water, sediment bed load transport and suspended sediment advection-diffusion equations along with the Exner equation for the bathymetric changes. We close this model by including a dynamic boundary condition at the channel mouth that transfers the offshore plume sediment concentration to the channel dynamics. This model reveals that the offshore plume reduces the timescales to reach equilibrium of the channel and plays a crucial role on shaping it. At equilibrium, the non-plume influence case attains a quasi-linear profile of constant slope in the seaward part. However, the bottom profile in the case that includes the offshore plume tends to increase the concavity of the bottom profile, reducing the final channel mouth depth. Finally, numerical results suggest that the plume characteristics are altered as a consequence of tidal channel evolution.

Behavioral of sediment transport at lowland and mountainous rivers: a special reference to selected Malaysian rivers

Mountain rivers and lowland rivers differ in many ways. Some of the many elements that distinguish both river systems apart are the rivers’ geomorphological appearances, hydraulic geometry, pollution transport, sediment transport, and its own roughness and coarseness. In this particular study, the element of sediment transport is given distinctive attention too. This study employs the deterministic approach for bed load prediction entailing the use of the parameterization concept where particle size and flow variation is taken into full consideration. Apart from that, the classical approach of Shields number is also used to determine the transport rates at the fluvial system. However, due to some limitation and range of applicability, the Shields number approach was re-modified to suit the range of applicability. Changes were made to the alleviating critical stress term and formulating exponential approach. For this research, it was important to deduce the underlying principles which are universal and common to all river systems, due to obvious and distinct differences between the mountain and lowland rivers. A special attribute was given to the Malaysian natural rivers because of the limited recorded data available. All datasets were compiled and tested with the bed load predictors to observe the commonality pattern between the lowland and mountain river systems. Most of the bed load transport equations limit the range of applicability by isolating the flow regime or bed roughness individually. Thus, it was vital for researchers to find the commonality pattern between these two river systems which needed to be statistically sound in its form. It was postulated that the particle densimetric Froude number is graphically and statistically fit for both river systems. Hence, this predictor is used as it has a high potential to be included in the sediment transport parameterization for Malaysian natural rivers.

Effect of bed load supply on sediment transport in mountain streams

Mountain streams with their tributary torrents build the upper part of the fluvial network. They are important regarding the transfer of sediment from headwaters to lower basins. Channels are typically steep with wide grain size distributions, ranging from fine sand up to large boulders, and a stabilized bed surface. Mountain streams often are supply-limited with respect to mobile bed load, which needs to be addressed when bed load transport equations are applied to such streams. To better understand supply limitation, laboratory experiments highlighting the effect of bed load supply on incipient motion and bed load transport rate are discussed. Experimental tests were done in which fine bed load was supplied to a previously armored channel bed, with flow conditions ranging from one-third to twice the critical discharge for the bed surface. At flows not exceeding the critical discharge, the time series of the bed load transport rate at the downstream model boundary featured consistent patterns which are attributed to distinct phases: (i) a temporal lag, (ii) an equilibrium state, and (iii) a post-supply phase. Bed load transport occurred even at flows distinctly below that for incipient motion of the bed surface. But, with the mass of total bed load outflow approaching the supply amount, the mass did not exclusively consist of supplied grains. The coarser the supplied bed load, the more sediment was mobilized from the bed surface. At higher flows, processes differed. Total bed load outflow exceeded the supply amount and the break-up of the armor layer caused a refining of the bed surface.

Bed load transport in a very steep mountain stream (Riedbach, Switzerland): Measurement and prediction

AbstractCompared to lower‐gradient channels, steep mountain streams typically have rougher beds and shallower flow depths, making macro‐scale flow resistance (due to, e.g., immobile boulders and irregular bedforms) more important as controls on sediment transport. The marked differences in hydraulics, flow resistance, and grain mobility between steep and lower‐gradient streams raise the question of whether the same equations can predict bed load transport rates across wide ranges of channel gradients. We studied a steep, glacier‐fed mountain stream (Riedbach, Ct. Valais, Switzerland) that provides a natural experiment for exploring how stream gradients affect bed load transport rates. The streambed gradient increases over a 1 km stream reach by roughly one order of magnitude (from 3% to 38%), while flow discharge and width remain approximately constant. Sediment transport rates were determined in the 3% reach using Bunte bed load traps and in the 38% reach using the Swiss plate geophone system. Despite a ten‐fold increase in bed gradient, bed load transport rates did not increase substantially. Observed transport rates for these two very different bed gradients could be predicted reasonably well by using a flow resistance partitioning approach to account for increasing bed roughness (D84 changes from 0.17 m to 0.91 m) within a fractional bed load transport equation. This suggests that sediment transport behavior across this large range of steep slopes agrees with patterns established in previous studies for both lower‐gradient and steep reaches, and confirms the applicability of the flow resistance and bed load transport equations at very steep slopes.

Numerical simulation of sediment transport in coastal waves and wave-induced currents

Prediction of coastal sediment transport is of particularly importance for analyzing coast erosion accurately and solving the corresponding coast protection engineering problems. The present study provided a numerical scheme for sediment transport in coastal waves and wave-induced currents. In the scheme, the sand transport model was implemented with wave refraction-diffraction model and near-shore current model. Coastal water wave was simulated by using the parabolic mild-slope equation in which wave refraction, diffraction and breaking effects are considered. Wave-induced current was simulated by using the nonlinear shallow water equations in which wave provides radiation stresses for driving current. Then, sediment transport in waves and wave-induced currents was simulated by using the two-dimensional suspended sediment transport equations for suspended sediment and the bed-load transport equation for bed load. The numerical scheme was validated by experiment results from the Large-scale Sediment Transport Facility at the US Army Corps of Engineer Research and Development Center in Vicksburg. The numerical results showed that the present scheme is an effective tool for modeling coastal sediment transport in waves and near-shore currents.

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.

Numerical Simulation of Bed Level Variation in Open Channels Under Steady Flow Conditions

Bed load transport is an important process in maintaining balance and stabilising channel geometry for restoring the form and function of river ecosystems. The amount and spatial distribution of bed load sediment particles contribute significantly to riverbed level changes. The prediction of bed load sediment transport evolution is an important aspect of catchment planning. This work can be effectively supported through numerical simulation by detailed analyses of flow components and sediment transport inside watersheds. The purpose of this research is to develop a twodimensional depth-averaged numerical model for flow and sediment transport using eight bed load transport equations to predict the time variation of bed deformation in steep slope, torrents and mountain river areas. The two-dimensional depth-averaged shallow water equations, along with the sediment continuity equation, are solved by using the Marker and Cell explicit scheme. Applying the eight bed load transport formulas to both ADM and MLSHM experimental flumes. After we will choice to the most appropriate formula to simulate the bed load transport rate and bed elevation change in the Yang yang mountains river in South Korea. The differences found between the measured experimental data and the numerical simulation for both flow and the time variations of bed deformation showed that the numerical model used in this research is useful for the analysis and prediction of riverbed level variations.

Taking the river inside: Fundamental advances from laboratory experiments in measuring and understanding bedload transport processes

Bedload transport in gravel-bed rivers impacts channel stability, the lifespan of hydraulic structures, physical components of aquatic habitat, and long-term channel evolution. Field measurements of bedload transport are notoriously difficult, which precludes understanding many of the processes and mechanics associated with grain motion. Such uncertainties are exacerbated when using bedload transport equations, most of which were derived using data from a single river or set of laboratory flume experiments. Recently, laboratory experiments have focused on better quantifying the processes that impact bedload fluxes, which can then be used to improve sediment transport predictions. We highlight recent advances in laboratory instrumentation that can be used in bedload transport studies. In particular, more accurate ways to measure bedload fluxes, near-bed turbulence, bed grain sizes, and topography hold great promise. Laboratory experiments have also fundamentally improved our understanding of the influence of sediment supply and armoring processes on bedload fluxes and channel conditions. The importance of flow hydrographs in controlling total bedload transport rates and bedload hysteresis has also been demonstrated using flume experiments. Finally, many details about the mechanics of grain motion including flow turbulence, bed arrangement, and particle transport statistics are only possible through laboratory investigations, and we feature key knowledge gaps that can be improved with further study.

SedFlow – a tool for simulating fractional bedload transport and longitudinal profile evolution in mountain streams

Abstract. Especially in mountainous environments, the prediction of sediment dynamics is important for managing natural hazards, assessing in-stream habitats and understanding geomorphic evolution. We present the new modelling tool {sedFlow} for simulating fractional bedload transport dynamics in mountain streams. sedFlow is a one-dimensional model that aims to realistically reproduce the total transport volumes and overall morphodynamic changes resulting from sediment transport events such as major floods. The model is intended for temporal scales from the individual event (several hours to few days) up to longer-term evolution of stream channels (several years). The envisaged spatial scale covers complete catchments at a spatial discretisation of several tens of metres to a few hundreds of metres. sedFlow can deal with the effects of streambeds that slope uphill in a downstream direction and uses recently proposed and tested approaches for quantifying macro-roughness effects in steep channels. sedFlow offers different options for bedload transport equations, flow-resistance relationships and other elements which can be selected to fit the current application in a particular catchment. Local grain-size distributions are dynamically adjusted according to the transport dynamics of each grain-size fraction. sedFlow features fast calculations and straightforward pre- and postprocessing of simulation data. The high simulation speed allows for simulations of several years, which can be used, e.g., to assess the long-term impact of river engineering works or climate change effects. In combination with the straightforward pre- and postprocessing, the fast calculations facilitate efficient workflows for the simulation of individual flood events, because the modeller gets the immediate results as direct feedback to the selected parameter inputs. The model is provided together with its complete source code free of charge under the terms of the GNU General Public License (GPL) (www.wsl.ch/sedFlow). Examples of the application of sedFlow are given in a companion article by Heimann et al. (2015).

Using repeat lidar to estimate sediment transport in a steep stream

Sediment fluxes in steep mountain streams remain difficult to quantify, despite their importance in geomorphology, ecology, and hazard analysis. In this work, aerial lidar surveys, acquired in 2002, 2008, and 2012, are used to quantify such fluxes in Tahoma Creek, a proglacial stream on Mount Rainier, Washington. As these surveys encompass all coarse sediment sources in the basin, we are able to translate geomorphic change into total bed material transport volumes for the time steps between surveys. By assuming that the relationship between daily sediment transport and daily mean discharge is of the form Qs=a(Q−Qc)b, our two observed total loads and estimates of daily mean discharge allow us to numerically solve for values of a and b to create a bed material sediment rating curve. Comparisons of our transport estimates with sediment deposition in a downstream reservoir indicate that our transport estimates and derived rating curve are reasonable. The method we present thus represents a plausible means of estimating transport rates in energetic settings or during extreme events, applicable whenever at least two cumulative sediment loads and the driving hydrology are known. We use these results to assess the performance of several bed load transport equations. The equations generally overpredict transport at low to moderate flows but significantly underpredict transport rates during an extreme event. Using a critical shear stress value appropriate for steep streams improves agreement at lower flows, whereas a shear-partitioning technique accounting for form drag losses significantly underpredicts transport at all flows.

A bedload transport equation for the Cerastoderma edule cockle

Hydrodynamics play an important role in the structure of many marine ecosystems of bivalves. After severe storm periods, large amounts of the Cerastoderma edule stocks were transported from the Lombos do Ulla shellfish bed (Spain). This paper presents the results of laboratory experiments carried out to analyze the bedload transport of this bivalve emulating the stormy shellfish bed conditions. Flow velocities were measured using particle image velocimetry and the double averaged methodology was applied to determine the main flow characteristics over different cockle patches. The flow structure exhibits properties of skimming and isolated flows depending on the density of bivalves. Bed shear stress was determined from the log-law and the cockles were geometrically characterized in order to derive specific bedload transport equations in a conventional deterministic sediment transport framework. The obtained formulas can be implemented in common numerical codes to further analyze mollusk stability, bedload transport and dispersal in their aquatic systems.

Validation and implications of an energy‐based bedload transport equation

AbstractA recently developed bedload equation (Abrahams & Gao, 2006) has the form ib = ωG3˙4, where ib is the immersed bedload transport rate, ω is the stream power per unit area, G = 1−θc/θ, θ is the dimensionless shear stress and θc is the associated threshold value for the incipient motion of bed grains. This equation has a parsimonious form and provides good predictions of transport rate in both the saltation and sheetflow regimes (i.e. flows with low and high θ values, respectively). In this study, the equation was validated using data independent of those used for developing it. The data represent bedload of identical sizes transported in various steady, uniform, fully rough and turbulent flows over plane, mobile beds. The equation predicted ib quite well over five orders of magnitude. This equation was further compared with six classic bedload equations and showed the best performance. Its theoretical significance was subsequently examined in two ways. First, based on collision theory, the parameter G was related to the ratio of grain‐to‐grain collisions to the total collisions including both grain‐to‐grain and grain‐to‐bed collisions, Pg by Pg = G2, suggesting that G characterizes the dynamic processes of bedload transport from the perspective of granular flow, which partly accounts for the good performance of the equation. Moreover, examining the ability of two common equations to predict bedload in gravel‐bed rivers revealed that G can also be used to simplify equations for predicting transport capacities in such rivers. Second, a simple dimensionless form of the equation was created by introducing B = ib/ω. The theoretical nature of the term B was subsequently revealed by comparing this equation with both the Bagnold model and two commonly used parameters representing dimensionless bedload transport rates.

RECTANGULAR STORM SEWER DESIGN UNDER EQUAL SEDIMENT MOBILITY

Rectangular storm sewers are conduits delivering water to outfall locations and have sizes that are typically larger than standard pipes, by which they become subject to more sediment deposits during operation and thus require higher flow strengths to maintain the bed clean. This study suggests a design procedure of self-cleansing rectangular sewers based on maintaining a lower limit of Shields stress and an upper limit of dimensionless bedload transport capacity. The lower limit of Shields stress is proposed under two considerations: to sustain equal sediment mobility at the channel bottom instead of selective transport and to avoid progressive deposition of finer grains due to low and reducing flows. The upper dimensionless bedload capacity is determined rationally and confirmed by using experimental data obtained from the literature. An existing bedload transport equation developed under equal sediment mobility is modified to provide a basis for the design method. It is shown that the proposed design procedure can practically be applied for a channel to estimate sediment concentrations by setting the required flow strength of Shields tress and particle size. Charts are given as an example for determining explicitly the channel design parameters. The study demonstrates that despite the high flows imposed, the design specifications determined according to this criterion can reasonably be achieved in practice for a given project.

Testing bedload transport equations with consideration of time scales

ABSTRACTBedload transport is known to be a highly fluctuating temporal phenomenon, even under constant (mean) flow conditions, as a consequence of stochasticity, bedform migration, grain sorting, hysteresis, or sediment supply limitation. Because bedload transport formulas usually refer to a single mean transport value for a given flow condition, one can expect that prediction accuracy (when compared to measurements) will depend on the amplitude and duration of fluctuations, which in turn depend on the time scale used for observations. This paper aims to identify how the time scale considered can affect bedload prediction. This was done by testing 16 common bedload transport formulas with four data sets corresponding to different measurement period durations: (i) highly fluctuating (quasi‐)instantaneous field measurements; (ii) volumes accumulated at the event scale on two small alpine gravel‐bed rivers, potentially affected by seasonal fluctuations; (iii) volumes accumulated at the interannual scale in a meandering gravel bed river, thought to be weakly subject to fluctuations; (iv) time‐integrated flume measurements with nearly uniform sediments. The tests confirmed that the longer the measurement period, the better the precision of the formula's prediction interval. They also demonstrate several consequential limitations. Most threshold formulas are no longer valid when the flow condition is below two times the threshold condition for the largest elements' motion on the bed surface (considering D84). In such conditions, equations either predict zero transport, or largely overestimate the real transport, especially when D84 is high. There is a need for new sediment data collected with highly reliable techniques such as recording slot bedload samplers to further investigate this topic. Copyright © 2012 John Wiley & Sons, Ltd.

Assessment of bed-load predictors based on sampling in a gravel bed river

Bedload transport in alluvial channels has been extensively studied and different equations based on field and/or experimental data have been proposed. Prediction of bed-load transport rate using different equations results in wide ranges which are not always reliable. In this study, some of the universal bedload predictors were evaluated with measured load by a Helley-Smith sampler in the Node River, a gravel bed river in the northeast part of Iran. From 19 sets of data, 14 series of data were used to evaluate the bed-load transport equations. The results show that the equations presented by Van Rijn, Meyer-Peter and Mueller, and Ackers and White may adequately predict bedload transport in the range of field data.