Sediment transport
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Sediment transport is the movement of solid particles ("sediment") due to the movement of the fluid in which they are immersed. This is typically studied in natural systems, where the particles are clastic rocks (sand, gravel, boulders, etc.) or clay, and the fluid is air, water, or ice.

Sediment transport is important in the fields of sedimentary geology, geomorphology, civil engineering and environmental engineering (see [applications], below). Our understanding of it is most often used to know whether erosion or deposition will occur, and in what magnitude it will occur.

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1 Mechanisms for Sediment Transport
2 Modes of Entrainment
- 2.1 Settling velocity
3 Initiation of motion
4 Transport Rate
5 Applications of Sediment Transport
- 5.1 References

Mechanisms for Sediment Transport

Eolean

Mesquite Dunes, Death Valley, CA, USA

A plume of dust blows off of the Sahara and sails over Atlantic Ocean towards the Canary Islands

Eolean 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 is a fluid with low density and viscosity, and can therefore not exert very much shear on its bed.

Aeolean sediment transport is common on beaches and in the arid regions of the world, because it is in these enviornments 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 Carribbean^{citation needed}, and dust from the Gobi desert has deposited on the western United States^{citation needed}. This sand is important to the soil budget and ecology of several islands; the soil formed by this wind-blown sediment is called loess.

Fluvial

In geology, physical geography, and sediment transport, fluvial processes relate to flowing water in natural systems. This encompasses rivers, streams, periglacial processes, flash floods and Glacial lake outburst floods. Sediment moved by water can be larger than sediment moved by air because water has both a higher density and viscosity. In typical rivers the maximum size of this sediment is of sand and gravel (<32 mm), but larger floods can carry boulders.

Glacial

Glaciers can carry the largest sediment, and areas of glacial deposition often contain a large number of glacial erratics, many of which are several meters in diameter.

Modes of Entrainment

Sediment entrained in a flow can be Sediment can be transported along the bed as bed load, in suspension as suspended load, or along the top (air-water) surface of the flow as wash-load.

The location in the flow in which a particle is entrained is determined by the Rouse number, which is determined by the density $ρ s$ 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.

$\textbf{Rouse}=\frac{w_s}{\kappa u_*}$

The term in the numerator is the (downwards) sediment the sendiment settling velocity $w s$ , which is discussed below. The upwards velocity on the grain is given as a product of von Kárman's constant,

$κ = 0.407$

and the shear velocity

$u_*=\sqrt{(\frac{tau_b}{\rho_w})}=\kappa z \frac{\partial u}{\partial z}$

The following table gives the required Rouse numbers for transport as bed load, suspended load, and wash load.

Mode of Tranpsort	Rouse Number
Bed load	>2.5
Suspended load: 50% Suspended	>1.2, <2.5
Suspended load: 100% Suspended	>0.8, <1.2
Wash load	<0.8

Settling velocity

Streamlines around a sphere falling through a fluid. This illustration is accurate for the laminar situation, in which the particle Reynolds number is small. This is typical for small particles falling through a viscous fluid; larger particles would result in the creation of a turbulent wake.

The settling velocity (also called the "fall velocity") can be calculated with Stokes' Law for small particles (with small particle Reynolds numbers and with the Drag Law for large particles. Ferguson and Church (2006)¹ analytically combined these two expressions into a single equation that works for all sizes of sediment.

$w_s=\frac{RgD^2}{C_1 \nu + (0.75 C_2 R g D^3)^(0.5)}$

In this equation w_s is the sediment settling velocity, g is acceleration due to gravity, and D is mean sediment diameter. R is the submerged specific gravity of the sediment, which is given by:

$R=\frac{\rho_s-\rho_w}{\rho_w}$

where ρ is density and the subscripts s and w indicate sediment and water, respectively. For quartz grains in water (a typical situation),

$ρ s = 2650 k g / m 3$

$ρ w = 1000 k g / m 3$

$R = 1.65$

\nu is the kinematic viscosity of water, which is approximately 1.0 x 10^-6 m²/s for water at 20ºC.

C_1 and C_2 are constants related to the shape and smoothness of the grains.

Constant	Smooth Spheres	Natural Grains: Sieve Diameters	Natural Grains: Nominal Diameters	Limit for Ultra-Angular Grains
$C 1$	18	18	20	24
$C 2$	0.4	1.0	1.1	1.2

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, $g = 9.8$ , $ν = 1.0$ , and $R = 1.65$ . From these parameters, the fall velocity is given by the expression:

$w_s=\frac{16.17D^2}{18 + (12.1275D^3)^(0.5)}$

Below are explanations of some standard equations that relate to sediment transport. These equations describe the initiation of sediment motion and the vertical location within the flow in a channel, such as a river, that sediment will occupy. These equations are designed for the transport of sediment in water or air. They only work for clastic, granular sediment: floccular sediment (including clays and muds) do not fit the geometric simplifications in these equations, and also interact thorough electrostatic forces.

Initiation of motion

For a fluid to begin transporting sediment in rest on a surface, the boundary, or bed, shear stress $τ b$ exerted by the fluid must exceed the critical shear stress $τ c$ for the motion of grains at the bed. This is typically represented by a comparison between a dimensionless shear stress and a dimensionless critical shear stress for the initiation of particle motion. The nondimensionalization is in order to compare the driving forces of particle motion (shear stress) to the forces that would make it stationary (particle density and size). This dimensionless shear stress, $τ *$ , is given by:

$\tau*=\frac{\tau_b}{(\rho_s-\rho)(g)(D)}$

Shields (REF) showed that for a bed of uniformly-sized grains, the dimensionless critical shear stress is a function of the particle Reynolds number.

For high particle Reynolds numbers, he and other researchers concluded that:

$τ c * = 0.06$

This is typically applicable to particles of gravel-size or larger in a stream.

Later researchers (e.g., Parker, REF) have shown that this value is closer to 0.03 for more uniformally-sorted beds. Parker (1982), (Wilcock, 1990's?) also found [equal mobility]

Considering the critical dimensionless shear stress as a result of the Shields criterion, one may calculate the shear stress and flow properties required to move a grain of a particular size.

Bed shear stress

Depth-slope product

For a river undergoing approximately steady, uniform flow, of approximately constant depth h and slope &theta over the reach of interest, and whose width is much greater than its depth, the bed shear stress is given by the depth-slope project:

$τ b = ρ g h sin(θ)$

For shallow slopes, which are found in almost all natural lowland streams, the small-angle formula shows that $sin(θ)$ is approximately equal to $tan(θ)$ , which is given by S, the slope. Rewritten with this:

$τ b = ρ g h S$

Other methods of calculating bed shear stress

For all flows that cannot be simplified as a single-slope infinate channel (above), the bed shear stress can be found by

Critical shear stress

The critical shear stress that a particle must overcome to enter motion can be given by a variety of formulas. Shield's Diagram

Solution

By setting the two sides of the equation equal to one another, we have for initiation of motion:

$τ * = τ c *$

$\frac{\tau_b}{(\rho_s-\rho)(g)(D)}=\frac{\tau_{b,c}}{(\rho_s-\rho)(g)(D)}$

For a particular particle Reynold's number, $τ c *$ will be an emprical constant. We pick 0.06 for this example, because it is the Shield's criterion for particles of gravel-size or larger. (More recent studies indicate that, for a bed of mixed-size sediment, 0.03 is a better value.)

$\frac{\tau_b}{(\rho_s-\rho)(g)(D)}=0.06$

Solving for $τ b$ ,

$τ b = 0.06(ρ s - ρ)(g)(D)$

Inserting the depth-slope product,

$ρ g h S = 0.06(ρ s - ρ)(g)(D)$

And moving and combining the constants

${h S}=0.06{\frac{(\rho_s-\rho)}{\rho}(D)}$

This final expression shows that the product of the channel depth and a 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 in water, the submerged specific gravity is equal to 1.65.

${\frac{(\rho_s-\rho)}{\rho}}=1.65$

Plugging this into the equation above, 0.06 * 1.65 = 0.099, which is well within standard margins of error of 0.1,

$h S = 0.1(D)$

And therefore, for these situations, the product of the depth and slope of the flow should be 10% of the diameter of the median grain diameter.

However the alternate critical shields stress of 0.03 is more appropriate for mixed-grain-size beds and supported by more recent research. Using this value, and changing D to D_50 ("50" for the 50th percentile, or the median grain size, as we are now looking at a mixed-grain-size bed), the equation becomes:

$h S = 0.05(D 50)$

Transport Rate

Bed Load

Bed load moves by rolling, sliding, and hopping (or saltating) over the bed. Bed load transport rates are usually expressed as being related to excess shear stress (dimensional or nondimensional), which is a measure of bed shear stress about the threshold for motion,

$(τ b - τ c)$ or $(\tau^*_b-\tau^*_c)$ ,

or by a ratio of bed shear stress to critical shear stress, which is equivalent in both the dimensional and nondimensional cases.

$\frac{\tau_b}{\tau_c}$ .

The relations for bedload transport are given for bedload transport per unit stream width, $b$ :

$q_s=\frac{Q_s}{b}$ .

In these relations, $q s$ is given as a constant times either the excess shear stress or the ratio of bed shear stress to critical shear stress, raised to a high power:

$q_s={a}{\left(\frac{Q_s}{b}\right)}^b$ or $q_s={a}{\left(\tau_b-\tau_c\right)}^b$ .

Due to the difficulty of estimating bed load transport rates, these equations are typically only suitable for the situations for which they were designed.

Suspended Load

Suspended load is carried in the lower to middle parts of the flow, and moves at a large fraction of the mean flow velocity in the stream.

Wash Load

Wash load is carried high in the water column as part of the flow, and therefore moves with the mean velocity of the upper layers of the flow in the stream.

Applications of Sediment Transport

Sediment transport is applied to solve many environmental, geotechnical, and geological problems.

It has been shown that 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 campsites for the multimillion-dollar river trip industry.
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.
The flow under the culvert is turbulent which leeds to the erosion of soil and exposed the foundation and thus unsettle the structure. Thus the maximum depth of erosion and its longitudanal distance from the structure has great importance in the field of civil engineering.

References

^ Ferguson, R. I., and M. Church (2006), A Simple Universal Equation for Grain Settling Velocity, Journal of Sedimentary Research, 74(6) 933-937, doi: 10.1306/051204740933

http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-090Fall-2006/CourseHome/index.htm