Sediment transport is the movement of solid particles (sediment), typically due to a combination of gravity acting on the sediment, and the movement of the fluid in which the sediment is entrained. Sediment transport occurs in natural systems where the particles are clastic rocks (sand, gravel, boulders, etc.), mud, or clay; the fluid is air, water, or ice; and the force of gravity acts to move the particles along the sloping surface on which they are resting. Sediment transport due to fluid motion occurs in rivers, oceans, lakes, seas, and other bodies of water due to currents and tides. Transport is also caused by glaciers as they flow, and on terrestrial surfaces under the influence of wind. Sediment transport due only to gravity can occur on sloping surfaces in general, including hillslopes, scarps, cliffs, and the continental shelf—continental slope boundary. Sediment transport is important in the fields of sedimentary geology, geomorphology, civil engineering, hydraulic engineering and environmental engineering (see applications, below). Knowledge of sediment transport is most often used to determine whether erosion or deposition will occur, the magnitude of this erosion or deposition, and the time and distance over which it will occur.
Environments
Aeolian
Aeolian or eolian (depending on the parsing of æ) is the term for sediment transport by wind. This process results in the formation of ripples and sand dunes. Typically, the size of the transported sediment is fine sand (<1 mm) and smaller, because air is a fluid with low density and viscosity, and can therefore not exert very much shear on its bed. Bedforms are generated by aeolian sediment transport in the terrestrial near-surface environment. Ripples and dunes form as a natural self-organizing response to sediment transport. Aeolian sediment transport is common on beaches and in the arid regions of the world, because it is in these environments that vegetation does not prevent the presence and motion of fields of sand. Wind-blown very fine-grained dust is capable of entering the upper atmosphere and moving across the globe. Dust from the Sahara deposits on the Canary Islands and islands in the Caribbean, and dust from the Gobi Desert has deposited on the western United States. This sediment is important to the soil budget and ecology of several islands. Deposits of fine-grained wind-blown glacial sediment are called loess.
Fluvial
Coastal
Coastal sediment transport takes place in near-shore environments due to the motions of waves and currents. At the mouths of rivers, coastal sediment and fluvial sediment transport processes mesh to create river deltas. Coastal sediment transport results in the formation of characteristic coastal landforms such as beaches, barrier islands, and capes.
Glacial As glaciers move over their beds, they entrain and move material of all sizes. Glaciers can carry the largest sediment, and areas of glacial deposition often contain a large number of glacial erratics, many of which are several metres in diameter. Glaciers also pulverize rock into "glacial flour", which is so fine that it is often carried away by winds to create loess deposits thousands of kilometres afield. Sediment entrained in glaciers often moves approximately along the glacial flowlines, causing it to appear at the surface in the ablation zone.
Hillslope In hillslope sediment transport, a variety of processes move regolith downslope. These include:
Soil creep Tree throw Movement of soil by burrowing animals Slumping and landsliding of the hillslope These processes generally combine to give the hillslope a profile that looks like a solution to the diffusion equation, where the diffusivity is a parameter that relates to the ease of sediment transport on the particular hillslope. For this reason, the tops of hills generally have a parabolic concave-up profile, which grades into a convex-up profile around valleys. As hillslopes steepen, however, they become more prone to episodic landslides and other mass wasting events. Therefore, hillslope processes are better described by a nonlinear diffusion equation in which classic diffusion dominates for shallow slopes and erosion rates go to infinity as the hillslope reaches a critical angle of repose.
Debris flow Large masses of material are moved in debris flows, hyperconcentrated mixtures of mud, clasts that range up to boulder-size, and water. Debris flows move as granular flows down steep mountain valleys and washes. Because they transport sediment as a granular mixture, their transport mechanisms and capacities scale differently from those of fluvial systems.
Applications
Sediment transport is applied to solve many environmental, geotechnical, and geological problems. Measuring or quantifying sediment transport or erosion is therefore important for coastal engineering. Several sediment erosion devices have been designed in order to quantify sediment erosion (e.g., Particle Erosion Simulator (PES)). One such device, also referred to as the BEAST (Benthic Environmental Assessment Sediment Tool) has been calibrated in order to quantify rates of sediment erosion. Movement of sediment is important in providing habitat for fish and other organisms in rivers. Therefore, managers of highly regulated rivers, which are often sediment-starved due to dams, are often advised to stage short floods to refresh the bed material and rebuild bars. This is also important, for example, in the Grand Canyon of the Colorado River, to rebuild shoreline habitats also used as campsites. Sediment discharge into a reservoir formed by a dam forms a reservoir delta. This delta will fill the basin, and eventually, either the reservoir will need to be dredged or the dam will need to be removed. Knowledge of sediment transport can be used to properly plan to extend the life of a dam. Geologists can use inverse solutions of transport relationships to understand flow depth, velocity, and direction, from sedimentary rocks and young deposits of alluvial materials. Flow in culverts, over dams, and around bridge piers can cause erosion of the bed. This erosion can damage the environment and expose or unsettle the foundations of the structure. Therefore, good knowledge of the mechanics of sediment transport in a built environment are important for civil and hydraulic engineers. When suspended sediment transport is increased due to human activities, causing environmental problems including the filling of channels, it is called siltation after the grain-size fraction dominating the process.
Initiation of motion
Stress balance For a fluid to begin transporting sediment that is currently at rest on a surface, the boundary (or bed) shear stress τ b {\displaystyle \tau _{b}} exerted by the fluid must exceed the critical shear stress τ c {\displaystyle \tau _{c}} for the initiation of motion of grains at the bed. This basic criterion for the initiation of motion can be written as:
τ b = τ c {\displaystyle \tau _{b}=\tau _{c}} . This is typically represented by a comparison between a dimensionless shear stress τ b ∗ {\displaystyle \tau _{b}*} and a dimensionless critical shear stress τ c ∗ {\displaystyle \tau _{c}*} . The nondimensionalization is in order to compare the driving forces of particle motion (shear stress) to the resisting forces that would make it stationary (particle density and size). This dimensionless shear stress, τ ∗ {\displaystyle \tau *} , is called the Shields parameter and is defined as:
τ ∗ = τ ( ρ s − ρ f ) ( g ) ( D ) {\displaystyle \tau *={\frac {\tau }{(\rho _{s}-\rho _{f})(g)(D)}}} . And the new equation to solve becomes:
τ b ∗ = τ c ∗ {\displaystyle \tau _{b}*=\tau _{c}*} . The equations included here describe sediment transport for clastic, or granular sediment. They do not work for clays and muds because these types of floccular sediments do not fit the geometric simplifications in these equations, and also interact thorough electrostatic forces. The equations were also designed for fluvial sediment transport of particles carried along in a liquid flow, such as that in a river, canal, or other open channel. Only one size of particle is considered in this equation. However, river beds are often formed by a mixture of sediment of various sizes. In case of partial motion where only a part of the sediment mixture moves, the river bed becomes enriched in large gravel as the smaller sediments are washed away. The smaller sediments present under this layer of large gravel have a lower possibility of movement and total sediment transport decreases. This is called armouring effect. Other forms of armouring of sediment or decreasing rates of sediment erosion can be caused by carpets of microbial mats, under conditions of high organic loading.
Critical shear stress
The Shields diagram empirically shows how the dimensionless critical shear stress (i.e. the dimensionless shear stress required for the initiation of motion) is a function of a particular form of the particle Reynolds number, R e p {\displaystyle \mathrm {Re} _{p}} or Reynolds number related to the particle. This allows the criterion for the initiation of motion to be rewritten in terms of a solution for a specific version of the particle Reynolds number, called R e p ∗ {\displaystyle \mathrm {Re} _{p}*} .
τ b ∗ = f ( R e p ∗ ) {\displaystyle \tau _{b}*=f\left(\mathrm {Re} _{p}*\right)}
This can then be solved by using the empirically derived Shields curve to find τ c ∗ {\displaystyle \tau _{c}*} as a function of a specific form of the particle Reynolds number called the boundary Reynolds number. The mathematical solution of the equation was given by Dey.
Particle Reynolds number In general, a particle Reynolds number has the form:
R e p = U p D ν {\displaystyle \mathrm {Re} _{p}={\frac {U_{p}D}{\nu }}}
Where U p {\displaystyle U_{p}} is a characteristic particle velocity, D {\displaystyle D} is the grain diameter (a characteristic particle size), and ν {\displaystyle \nu } is the kinematic viscosity, which is given by the dynamic viscosity, μ {\displaystyle \mu } , divided by the fluid density, ρ f {\displaystyle {\rho _{f}}} .
ν = μ ρ f {\displaystyle \nu ={\frac {\mu }{\rho _{f}}}}
The specific particle Reynolds number of interest is called the boundary Reynolds number, and it is formed by replacing the velocity term in the particle Reynolds number by the shear velocity, u ∗ {\displaystyle u_{*}} , which is a way of rewriting shear stress in terms of velocity.
u ∗ = τ b ρ f = κ z ∂ u ∂ z {\displaystyle u_{*}={\sqrt {\frac {\tau _{b}}{\rho _{f}}}}=\kappa z{\frac {\partial u}{\partial z}}}
where τ b {\displaystyle \tau _{b}} is the bed shear stress (described below), and κ {\displaystyle \kappa } is the von Kármán constant, where
κ = 0.407 {\displaystyle \kappa ={0.407}} . The particle Reynolds number is therefore given by:
R e p ∗ = u ∗ D ν {\displaystyle \mathrm {Re} _{p}*={\frac {u_{*}D}{\nu }}}
Bed shear stress The boundary Reynolds number can be used with the Shields diagram to empirically solve the equation
τ c ∗ = f ( R e p ∗ ) {\displaystyle \tau _{c}*=f\left(\mathrm {Re} _{p}*\right)} , which solves the right-hand side of the equation
τ b ∗ = τ c ∗ {\displaystyle \tau _{b}*=\tau _{c}*} . In order to solve the left-hand side, expanded as
τ b ∗ = τ b ( ρ s − ρ f ) ( g ) ( D ) {\displaystyle \tau _{b}*={\frac {\tau _{b}}{(\rho _{s}-\rho _{f})(g)(D)}}} , the bed shear stress needs to be found, τ b {\displaystyle {\tau _{b}}} . There are several ways to solve for the bed shear stress. The simplest approach is to assume the flow is steady and uniform, using the reach-averaged depth and slope. because it is difficult to measure shear stress in situ, this method is also one of the most-commonly used. The method is known as the depth-slope product.
Depth-slope product
For a river undergoing approximately steady, uniform equilibrium flow, of approximately constant depth h and slope angle θ over the reach of interest, and whose width is much greater than its depth, the bed shear stress is given by some momentum considerations stating that the gravity force component in the flow direction equals exactly the friction force. For a wide channel, it yields:
τ b = ρ g h sin ( θ ) {\displaystyle \tau _{b}=\rho gh\sin(\theta )}
For shallow slope angles, which are found in almost all natural lowland streams, the small-angle formula shows that sin ( θ ) {\displaystyle \sin(\theta )} is approximately equal to tan ( θ ) {\displaystyle \tan(\theta )} , which is given by S {\displaystyle S} , the slope. Rewritten with this:
τ b = ρ g h S {\displaystyle \tau _{b}=\rho ghS}
Shear velocity, velocity, and friction factor For the steady case, by extrapolating the depth-slope product and the equation for shear velocity:
τ b = ρ g h S {\displaystyle \tau _{b}=\rho ghS}
u ∗ = ( τ b ρ ) {\displaystyle u_{*}={\sqrt {\left({\frac {\tau _{b}}{\rho }}\right)}}} , The depth-slope product can be rewritten as:
τ b = ρ u ∗ 2 {\displaystyle \tau _{b}=\rho u_{*}^{2}} .
u ∗ {\displaystyle u*} is related to the mean flow velocity, u ¯ {\displaystyle {\bar {u}}} , through the generalized Darcy–Weisbach friction factor, C f {\displaystyle C_{f}} , which is equal to the Darcy-Weisbach friction factor divided by 8 (for mathematical convenience). Inserting this friction factor,
τ b = ρ C f ( u ¯ ) 2 {\displaystyle \tau _{b}=\rho C_{f}\left({\bar {u}}\right)^{2}} .
Unsteady flow For all flows that cannot be simplified as a single-slope infinite channel (as in the depth-slope product, above), the bed shear stress can be locally found by applying the Saint-Venant equations for continuity, which consider accelerations within the flow.
Example
Set-up The criterion for the initiation of motion, established earlier, states that
τ b ∗ = τ c ∗ {\displaystyle \tau _{b}*=\tau _{c}*} . In this equation,
τ ∗ = τ ( ρ s − ρ ) ( g ) ( D ) {\displaystyle \tau *={\frac {\tau }{(\rho _{s}-\rho )(g)(D)}}} , and therefore
τ b ( ρ s − ρ ) ( g ) ( D ) = τ c ( ρ s − ρ ) ( g ) ( D ) {\displaystyle {\frac {\tau _{b}}{(\rho _{s}-\rho )(g)(D)}}={\frac {\tau _{c}}{(\rho _{s}-\rho )(g)(D)}}} .
τ c ∗ {\displaystyle \tau _{c}*} is a function of boundary Reynolds number, a specific type of particle Reynolds number.
τ c ∗ = f ( R e p ∗ ) {\displaystyle \tau _{c}*=f\left(\mathrm {Re} _{p}*\right)} . For a particular particle Reynolds number, τ c ∗ {\displaystyle \tau _{c}*} will be an empirical constant given by the Shields Curve or by another set of empirical data (depending on whether or not the grain size is uniform). Therefore, the final equation to solve is:
τ b ( ρ s − ρ ) ( g ) ( D ) = f ( R e p ∗ ) {\displaystyle {\frac {\tau _{b}}{(\rho _{s}-\rho )(g)(D)}}=f\left(\mathrm {Re} _{p}*\right)} .
Solution Some assumptions allow the solution of the above equation. The first assumption is that a good approximation of reach-averaged shear stress is given by the depth-slope product. The equation then can be rewritten as:
ρ g h S = f ( R e p ∗ ) ( ρ s − ρ ) ( g ) ( D ) {\displaystyle {\rho ghS}=f\left(\mathrm {Re} _{p}*\right){(\rho _{s}-\rho )(g)(D)}} . Moving and re-combining the terms produces:
h S = ( ρ s − ρ ) ρ ( D ) ( f ( R e p ∗ ) ) = R D ( f ( R e p ∗ ) ) {\displaystyle {hS}={{\frac {(\rho _{s}-\rho )}{\rho }}(D)}\left(f\left(\mathrm {Re} _{p}*\right)\right)=RD\left(f\left(\mathrm {Re} _{p}*\right)\right)}
where R is the submerged specific gravity of the sediment. The second assumption is that the particle Reynolds number is high. This typically applies to particles of gravel-size or larger in a stream, and means the critical shear stress is constant. The Shields curve shows that for a bed with a uniform grain size,
τ c ∗ = 0.06 {\displaystyle \tau _{c}*=0.06} . Later researchers have shown this value is closer to
τ c ∗ = 0.03 {\displaystyle \tau _{c}*=0.03}
for more uniformly sorted beds. Therefore the replacement
τ c ∗ = f ( R e p ∗ ) {\displaystyle \tau _{c}*=f\left(\mathrm {Re} _{p}*\right)}
is used to insert both values at the end. The equation now reads:
h S = R D τ c ∗ {\displaystyle {hS}=RD\tau _{c}*}
This final expression shows the product of the channel depth and slope is equal to the Shield's criterion times the submerged specific gravity of the particles times the particle diameter. For a typical situation, such as quartz-rich sediment ( ρ s = 2650 k g m 3 ) {\displaystyle \left(\rho _{s}=2650{\frac {kg}{m^{3}}}\right)} in water ( ρ = 1000 k g m 3 ) {\displaystyle \left(\rho =1000{\frac {kg}{m^{3}}}\right)} , the submerged specific gravity is equal to 1.65.
R = ( ρ s − ρ ) ρ = 1.65 {\displaystyle R={\frac {(\rho _{s}-\rho )}{\rho }}=1.65}
Plugging this into the equation above,
h S = 1.65 ( D ) τ c ∗ {\displaystyle {hS}=1.65(D)\tau _{c}*} . For the Shield's criterion of τ c ∗ = 0.06 {\displaystyle \tau _{c}*=0.06} . 0.06 * 1.65 = 0.099, which is well within standard margins of error of 0.1. Therefore, for a uniform bed,
h S = 0.1 ( D ) {\displaystyle {hS}={0.1(D)}} . For these situations, the product of the depth and slope of the flow should be 10% of the diameter of the median grain diameter. The mixed-grain-size bed value is τ c ∗ = 0.03 {\displaystyle \tau _{c}*=0.03} , which is supported by more recent research as being more broadly applicable because most natural streams have mixed grain sizes. If this value is used, and D is changed to D_50 ("50" for the 50th percentile, or the median grain size, as an appropriate value for a mixed-grain-size bed), the equation becomes:
h S = 0.05 ( D 50 ) {\displaystyle {hS}={0.05(D_{50})}}
Which means that the depth times the slope should be about 5% of the median grain diameter in the case of a mixed-grain-size bed.
Modes of entrainment The sediments entrained in a flow can be transported along the bed as bed load in the form of sliding and rolling grains, or in suspension as suspended load advected by the main flow. Some sediment materials may also come from the upstream reaches and be carried downstream in the form of wash load.
Rouse number The location in the flow in which a particle is entrained is determined by the Rouse number, which is determined by the density ρs and diameter d of the sediment particle, and the density ρ and kinematic viscosity ν of the fluid, determine in which part of the flow the sediment particle will be carried.
P = w s κ u ∗ {\displaystyle P={\frac {w_{s}}{\kappa u_{\ast }}}}
Here, the Rouse number is given by P. The term in the numerator is the (downwards) sediment the sediment settling velocity ws, which is discussed below. The upwards velocity on the grain is given as a product of the von Kármán constant, κ = 0.4, and the shear velocity, u∗. The following table gives the approximate required Rouse numbers for transport as bed load, suspended load, and wash load.
Settling velocity
The settling velocity (also called the "fall velocity" or "terminal velocity") is a function of the particle Reynolds number. Generally, for small particles (laminar approximation), it can be calculated with Stokes' Law. For larger particles (turbulent particle Reynolds numbers), fall velocity is calculated with the turbulent drag law. Dietrich (1982) compiled a large amount of published data to which he empirically fit settling velocity curves. Ferguson and Church (2006) analytically combined the expressions for Stokes flow and a turbulent drag law into a single equation that works for all sizes of sediment, and successfully tested it against the data of Dietrich. Their equation is
w s = R g D 2 C 1 ν + ( 0.75 C 2 R g D 3 ) ( 0.5 ) {\displaystyle w_{s}={\frac {RgD^{2}}{C_{1}\nu +(0.75C_{2}RgD^{3})^{(0.5)}}}} . In this equation ws is the sediment settling velocity, g is acceleration due to gravity, and D is mean sediment diameter. ν {\displaystyle \nu } is the kinematic viscosity of water, which is approximately 1.0 × 10−6 m2/s for water at 20 °C.
C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are constants related to the shape and smoothness of the grains.
The expression for fall velocity can be simplified so that it can be solved only in terms of D. We use the sieve diameters for natural grains, g = 9.8 {\displaystyle g=9.8} , and values given above for ν {\displaystyle \nu } and R {\displaystyle R} . From these parameters, the fall velocity is given by the expression:
w s = 16.17 D 2 1.8 ⋅ 10 − 5 + ( 12.1275 D 3 ) ( 0.5 ) {\displaystyle w_{s}={\frac {16.17D^{2}}{1.8\cdot 10^{-5}+(12.1275D^{3})^{(0.5)}}}}
Alternatively, settling velocity for a particle of sediment can be derived using Stokes Law assuming quiescent (or still) fluid in steady state. The resulting formulation for settling velocity is,
w s = g ( ρ s − ρ ρ ) d s e d 2 18 ν , {\displaystyle {\displaystyle w_{s}={\frac {g~({\frac {\rho _{s}-\rho }{\rho }})~d_{sed}^{2}}{18\nu }}},}
where g {\displaystyle g} is the gravitational constant; ρ s {\displaystyle \rho _{s}} is the density of the sediment; ρ {\displaystyle \rho } is the density of water; d s e d {\displaystyle d_{sed}} is the sediment particle diameter (commonly assumed to be the median particle diameter, often referred to as d 50 {\displaystyle d_{50}} in field studies); and ν {\displaystyle \nu } is the molecular viscosity of water. The Stokes settling velocity can be thought of as the terminal velocity resulting from balancing a particle's buoyant force (proportional to the cross-sectional area) with the gravitational force (proportional to the mass). Small particles will have a slower settling velocity than heavier particles, as seen in the figure. This has implications for many aspects of sediment transport, for example, how far downstream a particle might be advected in a river.
Hjulström–Sundborg diagram
In 1935, Filip Hjulström created the Hjulström curve, a graph which shows the relationship between the size of sediment and the velocity required to erode (lift it), transport it, or deposit it. The graph is logarithmic. Åke Sundborg later modified the Hjulström curve to show separate curves for the movement threshold corresponding to several water depths, as is necessary if the flow velocity rather than the boundary shear stress (as in the Shields diagram) is used for the flow strength. This curve has no more th