Sediment transport

The phrase Sediment transport is used to describe the movement of solid particles (sediment) and the processes that govern their motion. Sediment transport is typically due to a combination of the force of gravity acting on the sediment, and/or the movement of the fluid in which the sediment is entrained. This is typically studied 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 is due to the sloping surface on which the particles are resting. This occurs in rivers, glaciers, wind, ocean currents, tides, and on hillslopes.

Sediment transport is important in the fields of sedimentary geology, geomorphology, civil engineering and environmental engineering (see applications, below). Knowledge of sediment transport 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.

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.

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.

Soil formed from wind-blown glacial 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 flows, 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 largest carried sediment is of sand and gravel size, but larger floods can carry cobbles and even boulders.

Fluvial sediment transport can result in the formation of ripples and dunes, in fractal-shaped patterns of erosion, in complex patterns of natural river systems, and in the development of floodplains.



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 transport processes also include fluvial processes.

Coastal sediment transport results in the formation of characteristic coastal landforms such as beaches and barrier islands. In coastal-fluvial systems, river deltas form.



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.

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-down profile, which grades into a concave-up profile around valleys.

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 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.

Stress balance
For a fluid to begin transporting sediment that is currently at rest on a surface, the boundary (or bed) shear stress $$\tau_b$$ exerted by the fluid must exceed the critical shear stress $$\tau_c$$ for the initiation motion of grains at the bed. This basic criterion can be for the initiation of motion can be written as:


 * $$\tau_b=\tau_c\,$$.

This is typically represented by a comparison between a dimensionless shear stress ($$\tau_b*$$)and a dimensionless critical shear stress ($$\tau_c*$$). 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, $$\tau*$$, is given by:


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

And the new equation to solve becomes:


 * $$\tau_b*=\tau_c*\,$$

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. They were also designed fluvial sediment transport of particles carried along in a liquid flow, such as that in a river, canal, or other open channel.

Critical shear stress
The Shields diagram empirically shows how the dimensionless critical shear stress required for the initiation of motion is a function of a particular form of the particle Reynolds number, $$Re_p$$ or Reynolds number related to the particle. This allows us to rewrite 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 $$Re_p*$$.


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

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

Particle 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 $$\nu$$ is the kinematic viscosity, which is given by the dynamic viscosity, $$\mu$$, divided by the fluid density, $$\rho$$.


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

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 rewriting shear stress in terms of velocity.


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

where $$\tau_b$$ is the bed shear stress (described below), and $$ \kappa $$ is the von Kármán constant, where


 * $$ \kappa = {0.407}\,$$

The particle 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


 * $$\tau_b*=\tau_c*\,$$.

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, $${\tau_b}$$. 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
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:


 * $$\tau_b=\rho g h \sin(\theta)\,$$

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


 * $$\tau_b=\rho g h S\,$$

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


 * $$\tau_b=\rho g h S\,$$


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

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


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

$$u*$$ is related to the mean flow velocity, $$\bar{u}$$, through the generalized Darcy-Weisbach friction factor, $$C_f$$, which is equal to the Darcy-Weisbach friction factor divided by 8 (for mathematical convenience). Inserting this friction factor,


 * $$\tau_b=\rho C_f \left(\bar{u} \right)^2$$.

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-Vennant equations for continuity, which consider accelerations within the flow.

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


 * $$\tau_b*=\tau_c*\,$$.

In this equation,


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


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


 * $$\tau_c*$$ 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 Reynolds number, $$\tau_c*$$ 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:


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

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 rewrite the equation as


 * $${\rho g h S}=0.06{(\rho_s-\rho)(g)(D)}\,$$

Moving and re-combining the terms, we obtain:


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

where R is the submerged specific gravity of the sediment.

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,


 * $$\tau_c*=0.06\,$$.

Later researchers have shown that this value is closer to


 * $$\tau_c*=0.03\,$$

for more uniformly-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:


 * $${h S}=R D \tau_c*\,$$

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


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

Plugging this into the equation above,


 * $${h S}=1.65(D)\tau_c*\,$$.

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


 * $${h S}={0.1(D)}\,$$.

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 $$\tau_c*=0.03$$, 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 $$\rho_s$$ and diameter$$d$$ of the sediment particle, and the density $$\rho$$ and kinematic viscosity $$\nu$$ 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, $$ \kappa = {0.407}$$, and the shear velocity, $$u_*$$.

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

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. Ferguson and Church (2006) 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 ws is the sediment settling velocity, g is acceleration due to gravity, and D is mean sediment diameter. $$\nu$$ 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.

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, $$g=9.8$$, and values given above for $$\nu$$ and $$R$$. From these parameters, the fall velocity is given by the expression:


 * $$w_s=\frac{16.17D^2}{1.8\cdot10^{-5} + (12.1275D^3)^{(0.5)}}$$

Bed Load
Bed load moves by rolling, sliding, and hopping (or saltating) over the bed, and moves at a small fraction of the fluid flow velocity. Bed load is generally thought to constitute 5-10% of the total sediment load in a stream, making it less important in terms of mass balance. However, the bed material load (the bed load plus the portion of the suspended load which comprises material derived from the bed) is often dominated by bed load, especially in gravel-bed rivers. This bed material load is the only part of the sediment load that actively interacts with the bed. As the bed load is an important component of that, it plays a major role in controlling the morphology of the channel.

Bed load transport rates are usually expressed as being related to excess dimensionless shear stress which is a measure of bed shear stress about the threshold for motion,


 * $$(\tau_b-\tau_c)\,$$ or $$(\tau^*_b-\tau^*_c)$$,

and/or by a ratio of bed shear stress to critical shear stress, which is equivalent in both the dimensional and nondimensional cases. This ratio is called the "transport stage" $$T_s$$ and is an important in that it shows bed shear stress with respect to a criterion for the initiation of motion.


 * $$T_s=\frac{\tau_b}{\tau_c}$$

The majority of the published relations for bedload transport are given in dry sediment weight per unit channel width, $$b$$ ("breadth"):


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

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

Meyer-Peter Müller and derivatives
The transport formula of Meyer-Peter and Müller, originally developed in 1948, was designed for well-sorted fine gravel at a transport stage of about 8. The formula uses the above nondimensionalization for shear stress,


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

and Hans Einstein's nondimensionalization for sediment volume


 * $$q_s* = \frac{q_s}{D \sqrt{\frac{\rho_s-\rho}{\rho} g D}} = \frac{q_s}{Re_p \nu}$$.

Their formula reads:


 * $$q_s* = 8\left(\tau*-\tau*_c \right)^{3/2}$$.

Their experimentally-determined value for \tau*_c is 0.047, and is the third commonly used value for this (in addition to Parker's 0.03 and Shields' 0.06).

Because of its broad use, some revisions to the formula have taken place over the years that show that the coefficient on the left ("8" above) is a function of the transport stage:


 * $$T_s \approx 2 \rightarrow q_s* = 5.7\left(\tau*-0.047 \right)^{3/2}$$
 * $$T_s \approx 100 \rightarrow q_s* = 12\left(\tau*-0.047 \right)^{3/2}$$

The variations in the coefficient were later generalized as a function of dimensionless shear stress:


 * $$\begin{cases} q_s* = \alpha_s \left(\tau*-\tau_c* \right)^n \\ n = \frac{3}{2} \\ \alpha_s = 1.6 \ln\left(\tau*\right) + 9.8 \approx 9.64 \tau*^{0.166} \end{cases}$$

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.