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Grain shape effects in bed load sediment transport

Nature volume 613, pages 298–302 (2023)Cite this article

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An Addendum to this article was published on 15 November 2023

Abstract

Bed load sediment transport, in which wind or water flowing over a bed of sediment causes grains to roll or hop along the bed, is a critically important mechanism in contexts ranging from river restoration1 to planetary exploration2. Despite its widespread occurrence, predictions of bed load sediment flux are notoriously imprecise3,4. Many studies have focused on grain size variability5 as a source of uncertainty, but few have investigated the role of grain shape, even though shape has long been suspected to influence transport rates6. Here we show that grain shape can modify bed load transport rates by an amount comparable to the scatter in many sediment transport datasets4,7,8. We develop a theory that accounts for grain shape effects on fluid drag and granular friction and predicts that the onset and efficiency of transport depend on the coefficients of drag and bulk friction of the transported grains. Laboratory experiments confirm these predictions and reveal that the effect of grain shape on sediment transport can be difficult to intuit from the appearance of grains. We propose a shape-corrected sediment transport law that collapses our experimental measurements. Our results enable greater accuracy in predictions of sediment transport and help reconcile theories developed for spherical particles with the behaviour of natural sediment grains.

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Fig. 1: Competing effects of grain shape on bed load sediment transport.
Fig. 2: Granular materials used in the experiments.
Fig. 3: Comparison of grain shape theory with laboratory flume data.

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Data availability

The experimental flume data and measurements of grain properties and code used to support the conclusions and generate the figures in the main text and extended data items are available in an online repository. Flume data: https://doi.org/10.7910/DVN/GBSC2U; grain properties: https://doi.org/10.7910/DVN/31KT36; main text figures: https://doi.org/10.7910/DVN/5PYJFP; and extended data items: https://doi.org/10.7910/DVN/NQ33OD. Source data are provided with this paper.

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Acknowledgements

We thank C. Johnson and M. Jellinek for logistical support, M. Rushlow, M. Cantine, A. Perron and M. Perron for assistance with grain shape measurements, and M. Church for discussions of grain shape and flume experiments. Research was sponsored by the Army Research Laboratory and was accomplished under grant number W911NF-16-1-0440. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. S.J.B. was partly supported by a grant from the NASA FINESST programme. The experimental facility was constructed using a Canadian Foundation for Innovation Leaders Opportunity Fund grant to J.G.V.

Author information

Author notes

  1. Eric Deal

    Present address: Department of Earth Sciences, ETH Zurich, Zurich, Switzerland

  2. Santiago J. Benavides

    Present address: Mathematics Institute, University of Warwick, Coventry, UK

Authors and Affiliations

  1. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

    Eric Deal, Santiago J. Benavides & J. Taylor Perron

  2. School of Environmental Science, Simon Fraser University, Burnaby, British Columbia, Canada

    Jeremy G. Venditti & Ryan Bradley

  3. Northwest Hydraulic Consultants, North Vancouver, British Columbia, Canada

    Ryan Bradley

  4. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Qiong Zhang & Ken Kamrin

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Contributions

E.D., J.T.P., K.K. and J.G.V. conceived the project. E.D. developed the grain shape theory with input from J.T.P. E.D., J.T.P., J.G.V., S.J.B. and R.B. performed laboratory flume experiments. E.D. and J.T.P. measured grain density, shape, drag coefficients and grain friction coefficients. E.D., J.T.P., S.J.B. and Q.Z. analysed the experimental data. E.D. and J.T.P. wrote the paper with input from all other authors.

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Correspondence to Eric Deal.

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Extended data figures and tables

Extended Data Fig. 1 Free body diagram of a single grain on a bed with inclination θ.

Force vectors shown correspond to forces in equations (4) and (5).  The various forces are a gravitational force, Fg, a buoyant force, Fb, a fluid drag force, FD, a lift force, FL, a bed contact force, Fc, and a frictional force, Ff.

Extended Data Fig. 2 Effects of different aspects of grain shape on fluid drag coefficient.

The measured drag coefficient of a grain settling in still water, CDsettle, relative to the calculated drag coefficient of the volume-equivalent sphere, Co, as a function of the Corey shape factor, a measure of gross grain shape. Coloured points show the materials used in our flume experiments and the additional materials in Extended Data Fig. 3a–e. Grey lines are fits to a large compilation32 of single-grain settling experiments that have been sorted by grain angularity, a measure of small-scale grain shape. Sketches in key show idealized grains of different angularity. Red dashed line is the trend 1/Sf for comparison. Error bars show one standard error of the mean.

Extended Data Fig. 3 Additional granular materials and their shape properties.

a, Tempered glass chips. b, Shell fragments. c–f, Natural gravels 2–5, respectively, with differences in gross grain shape, angularity and surface properties. All materials have similar average sizes and densities to each other and to the five granular materials used in the flume experiments (Extended Data Table 1). g, Normalized coefficient of friction, μ*, as a function of the Corey shape factor for the five granular materials used in the flume experiments and the six additional granular materials shown in a–f. h, Normalized coefficient of drag, C*, as a function of Corey shape factor for the same 11 granular materials as in g. i, The ratio of the two normalized coefficients from g and h, C*/μ*, as a function of Corey shape factor for the same 11 granular materials as in g and h. The parameters C* and μ* tend to vary similarly for many of the measured materials, such that their ratio is close to one. However, several materials have ratios of C*/μ* distinctly different from one. This highlights the difficulty of guessing the net effect of grain shape on sediment transport based on qualitative inspection of grains. Error bars show one standard error of the mean.

Extended Data Fig. 4 Competing effects of grain shape on bed load sediment transport.

Same as Fig. 1, but including the six granular materials in Extended Data Fig. 3. a, Comparison of bulk coefficient of static friction with a measure of grain circularity, Sc = 4πA/P2, where A is the projected grain area and P is the projected perimeter (values closer to 1 indicate more-circular grains), for a compilation of observations47,48,49 and the materials measured here. b, Comparison of the still-water-settling drag coefficient, CDsettle, normalized by the drag coefficient for a sphere of the same volume (Methods) with another measure of grain shape, the Corey shape factor, Sf = c/(ab)1/2, where a, b, and c are the long, intermediate, and short axes of a grain (values closer to 1 indicate more-spherical grains), for a compilation of observations32 and the materials measured here. The coefficients of both friction and drag decrease with increasingly spherical grains. Error bars show one standard error of the mean.

Extended Data Fig. 5 Schematic diagram of laboratory flume.

Measurements of bed and water surface slope were made in the middle 2.5 m of the flume, where there were no visible entry or exit effects on grain motion. The flume is inclined 3°, but the sediment bed can develop a slope that is either steeper or less steep than the flume.

Extended Data Fig. 6 Shape distributions of granular materials with variable grain shapes.

a–c, Histograms of the three axes (a, b, and c) used to characterize grain shape. d–f, Corresponding histograms of the Corey shape factor. n is the sample size for each grain type.

Extended Data Fig. 7 Distributions of settling velocities for the grain types used in flume experiments.

n is the sample size for each grain type.

Extended Data Fig. 8 Measurement of the angle of repose of experimental materials.

a, Spheres. b, Faceted ellipsoids. c, Rounded chips. d, Painted natural gravel. Painted gravel was used in the experiments to aid automated grain identification. e, Rectangular prisms. Blue and red lines are the right and left edges of the pile silhouette extracted with image analysis. Yellow lines are least-squares fits to these edges used to estimate the angle of repose. Vertical red line at the centre of each image is a plumb line used to determine the direction of gravity.

Extended Data Fig. 9 Comparison of boundary shear stress estimates from different methods.

For a subset of the flume experiments with spheres, flow velocity was measured using laser particle image velocimetry (PIV). a, Profiles of fluid velocity in the downstream direction as a function of distance above the grain bed (blue dotted lines), offset on the x-axis for visual clarity, are fit with the law of the wall (black lines), u = (u*/κ)ln(30z/do), where κ = 0.4 is the von Karman constant, do is the grain diameter, and u* = √(τ/ρ) is the shear velocity, which yields an estimate of the shear stress. The law of the wall is fit to the part of each profile between 20% and 80% of the maximum velocity (solid blue lines). b, Plot of the nondimensional bed shear stress estimated from τ = ρgRS against the nondimensional shear stress estimated from the Law of the Wall (blue points) and the shear stress calculated by applying a wall correction factor53 to the original estimates of τ = ρgRS (green points) for flume experiments with glass spheres. c, Same as b, but for the flume experiments with natural gravel, and without PIV-derived shear stress. Dashed lines are least-squares fits. The wall-corrected shear stress estimates for spheres and natural gravel are within error of each other and of the PIV-derived estimates. The average wall correction factor for the two grain types is (2.7 + 2.1)/2 = 2.41. Error bars show best estimate of uncertainty in shear stress estimates. For PIV-derived estimates this is the uncertainty of the log-linear fits in a; for the other estimates, it is the propagated standard error of the mean.

Extended Data Table 1 Grain properties
Full size table

Supplementary information

Supplementary Video 1

A video of spheres undergoing low-intensity bed load transport. Dimensionless shear stress is τ* = 0.056, and dimensionless transport rate is q* = 0.018.

Supplementary Video 2

A video of natural gravel 1 undergoing low-intensity bed load transport. Dimensionless shear stress is τ* = 0.081, and dimensionless transport rate is q* = 0.025.

Supplementary Video 3

A video of spheres undergoing moderate-intensity bed load transport. Dimensionless shear stress is τ* = 0.106, and dimensionless transport rate is q* = 0.185.

Supplementary Video 4

A video of natural gravel 1 undergoing moderate-intensity bed load transport. Dimensionless shear stress is τ* = 0.148, and dimensionless transport rate is q* = 0.230.

Supplementary Video 5

A video of faceted ellipsoids undergoing moderate-intensity bed load transport. Dimensionless shear stress is τ* = 0.127, and dimensionless transport rate is q* = 0.262.

Supplementary Video 6

A video of rectangular prisms undergoing moderate-intensity bed load transport. Dimensionless shear stress is τ* = 0.123, and dimensionless transport rate is q* = 0.224.

Supplementary Video 7

A video of rounded glass chips undergoing moderate-intensity bed load transport. Dimensionless shear stress is τ* = 0.124, and dimensionless transport rate is q* = 0.262.

Supplementary Video 8

A video of spheres undergoing high-intensity bed load transport. Dimensionless shear stress is τ* = 0.160, and dimensionless transport rate is q* = 0.598.

Supplementary Video 9

A video of natural gravel 1 undergoing high-intensity bed load transport. Dimensionless shear stress is τ* = 0.206, and dimensionless transport rate is q* = 0.543.

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Deal, E., Venditti, J.G., Benavides, S.J. et al. Grain shape effects in bed load sediment transport. Nature 613, 298–302 (2023). https://doi.org/10.1038/s41586-022-05564-6

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