Sediment transport

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Mesquite Dunes, Death Valley, California. Ripples and dunes form as a natural self-organizing response to sediment transport.

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

Toklat River, East Fork, Polychrome overlook, Denali National Park, Alaska. This river, like other braided streams, rapidly changes the positions of its channels through processes of erosion, sediment transport, and deposition.

Sand ripples, Laysan Beach, Hawaii. Coastal sediment transport results in these evenly-spaced ripples along the shore. Monk seal for scale.

A glacier joining the Gorner Glacier, Zermatt, Switzerland. These glaciers transport sediment and leave behind lateral moraines.

Sediment transport is the movement of solid particles (sediment) due to the movement of the fluid in which they are entrained. This is typically studied in natural systems, where the particles are clastic rocks (sand, gravel, boulders, etc.), mud, or clay, and the fluid is air, water, or ice. This occurs in rivers, glaciers, wind, ocean currents, and tides. (Transportation does not apply to the passive movement of material under the influence of gravity, as might happen to boulders on a scree.) As the particles are transported, their edges can be smoothed, and is described as rounding. Deposition occurs when the speed of movement of the transporting medium becomes insufficient to hold the particles. Examples of such deposits are: bunter, moraine, overbank, dunes, and tombolo.

Sediment transport is important in the fields of sedimentary geology, geomorphology, civil engineering and environmental engineering (see applications, at bottom). It is most often used to know whether erosion or deposition will occur, the magnitude of this erosion or deposition, and the time and distance over which it will occur.

Mechanisms

Eolean

Main article: Aeolian processes

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 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 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.

Applications

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

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.

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.

Mechanics

The majority of the following equations were designed for fluvial sediment transport of particles carried along in a liqued flow, such as that in a river, canal, or other open channel.

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 do not fit the geometric simplifications in these equations, and also interact thorough electrostatic forces.

The following equations describe the initiation of sediment motion, give the vertical location within the flow (in terms of bed load, suspended load, and wash load), and show a simple introduction to sediment discharge and sediment transport rate.

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 $τ b$ exerted by the fluid must exceed the critical shear stress $τ c$ for the initiation motion of grains at the bed. This basic criterion can be for the initiation of motion can be written as:

$τ b = τ c$ .

This is typically represented by a comparison between a dimensionless shear stress ( $τ b *$ )and a dimensionless critical shear stress ( $τ c *$ ). 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, $τ *$ , is given by:

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

And the new equation to solve becomes:

$τ b * = τ c *$ .

Critical shear stress

The Shields empirically shows how the dimensionless critical shear stress required for the initation of motion is a function of a particular form of the particle Reynolds number, $R e p$ or Reynolds number related to the particle. This allows us to re-write the criterion for the initiation of motion in terms of only needing to solve for a specific version of the particle Reynolds number, which we call $R e p *$ .

$\tau_b*=f\left(Re_p*\right)$

This equation can then be solved by using the empirically-derived Shields curve to find $τ c *$ as a function of a specific form of the particle Reynolds number called the boundary Reynolds number.

Boundary Reynolds Number

In general, a particle Reynolds Number has the form:

$Re_p=\frac{U_p D}{\nu}$

Where $U p$ is a characteristic particle velocity, $D$ is the grain diameter (a characteristic particle size), and $ν$ is the kinematic viscosity, which is given by the dynamic viscosity, $μ$ , divided by the fluid density, $ρ$ .

$\nu=\frac{\mu}{\rho}$

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, $u *$ , which is a way of re-writing shear stress in terms of velocity.

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

where $τ b$ is the bed shear stress (described below), and $κ$ is the von Kármán constant, where

$κ = 0.407$ .

The boundary Reynolds number is therefore given by:

$Re_p*=\frac{u* D}{\nu}$

Bed shear stress

The boundary Reynolds number can be used with the Shields diagram to empirically solve the equation

$\tau_c*=f\left(Re_p*\right)$ ,

which solves the right-hand side of the equation

$τ b * = τ c *$ .

In order to solve the left-hand side, expanded as

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

we must find the bed shear stress, $τ b$ . There are several ways to solve for the bed shear stress. First, we develop the simplest approach, in which the flow is assumed to be steady and uniform and reach-averaged depth and slope are used. Due to the difficulty of measuring shear stress in situ, this method is also one of the most-commonly used. This method is known as the depth-slope product.

Depth-slope product

Main article: 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:

$τ b = ρ g h S$

Shear velocity, velocity, and friction factor

For the steady case, by extrapolating the depth-slope product and the equation for shear velocity:

$τ b = ρ g h S$

$u_*=\sqrt{\left(\frac{\tau_b}{\rho_w}\right)}$ ,

We can see that the depth-slope product can be re-written as:

$\tau_b=\rho u_*^2$ .

$u *$ is related to the flow velocity, $u$ , through a friction factor, $c f$ . Inserting this friction factor,

$\tau_b=\rho \left(c_f u^2 \right)$ .

Unsteady flow

For all flows that cannot be simplified as a single-slope infinate channel (as in the depth-slope product, above), the bed shear stress can be locally found by applying the Saint-Vennant equations for continuity, which consider accelerations within the flow.

Solution

Set-up

The criterion for the initiation of motion, established earlier, states that

$τ b * = τ c *$ .

In this equation,

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

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

$τ c *$ is a function of boundary Reynolds number, a specific type of particle Reynolds number.

$\tau_c*=f \left(Re_p* \right)$ .

For a particular particle Reynold's number, $τ c *$ will be an emprical 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 that we seek to solve is:

$\frac{\tau_b}{(\rho_s-\rho)(g)(D)}=f \left(Re_p* \right)$ </math>.

Solution

We make several assumptions to provide an example that will allow us to bring the above form of the equation into a solved form.

First, we assume that the a good approximation of reach-averaged shear stress is given by the depth-slope product. We can then re-write the equation as

$ρ g h S = 0.06(ρ s - ρ)(g)(D)$ .

Moving and re-combining the terms, we obtain:

${h S}={\frac{(\rho_s-\rho)}{\rho}(D)}\left(f \left(Re_p* \right) \right)$

We then make our second assumption, which is that the particle Reynolds number is high. This is typically applicable to particles of gravel-size or larger in a stream, and means that the critical shear stress is a constant. The Shields curve shows that for a bed with a uniform grain size,

$τ c * = 0.06$ .

Later researchers (e.g., Parker, REF) have shown that this value is closer to

$τ c * = 0.03$

for more uniformally-sorted beds. Therefore, we will simply insert

$\tau_c*=f \left(Re_p* \right)$

and insert both values at the end.

The equation now reads:

${h S}=\frac{(\rho_s-\rho)}{\rho}(D)\tau_c*$ .

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 $\left(\rho_s=2650 \frac{kg}{m^3} \right)$ in water $\left(\rho=1000 \frac{kg}{m^3} \right)$ , the submerged specific gravity is equal to 1.65.

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

Plugging this into the equation above,

$h S = 1.65(D)τ c *$ .

For the Shield's criterion of $τ c * = 0.06$ . 0.06 * 1.65 = 0.099, which is well within standard margins of error of 0.1. Therefore, for a uniform bed,

$h S = 0.1(D)$ .

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 $τ c * = 0.03$ , which is supported by more recent research as being more broadly applicable because most natural streams have mixed grain sizes. 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:

$h S = 0.05(D 50)$

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

Sediment entrained in a flow 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.

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 $ρ s$ 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 the von Kármán constant, $κ = 0.407$ , and the shear velocity, $u *$ .

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 laminar flow, 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" 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. 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),

$ρ s = 2650 k g / m 3$

$ρ w = 1000 k g / m 3$

$R = 1.65$

$ν$ is the kinematic viscosity of water, which is approximately 1.0 x 10^-6 m²/s for water at 20°C.

$C 1$ 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:

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

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, $b$ ("breadth"):

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

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:

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

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.

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

External links

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This page was last modified on 5 January 2009, at 23:26.

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