Sediment transport

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 ^[1] and dunes ^[2] 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, ^[3] and dust from the Gobi desert has deposited on the western United States. ^[4] 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

This section is an excerpt from Fluvial sediment processes.[ edit ]

In geography and geology, fluvial sediment processes or fluvial sediment transport are associated with rivers and streams and the deposits and landforms created by sediments. It 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 and the occurrence of flash 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.

When the stream or rivers are associated with glaciers, ice sheets, or ice caps, the term glaciofluvial or fluvioglacial is used, as in periglacial flows and glacial lake outburst floods. ^{[5] [6]} Fluvial sediment processes include the motion of sediment and erosion or deposition on the river bed. ^{[7] [8]}

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. ^[9]

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. ^[10]

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.

Stress balance

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

${\displaystyle \tau _{b}=\tau _{c}}$.

This is typically represented by a comparison between a dimensionless shear stress ${\displaystyle \tau _{b}*}$ and a dimensionless critical shear stress ${\displaystyle \tau _{c}*}$. 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, ${\displaystyle \tau *}$, is called the Shields parameter and is defined as: ^[12]

${\displaystyle \tau *={\frac {\tau }{(\rho _{s}-\rho _{f})(g)(D)}}}$.

And the new equation to solve becomes:

${\displaystyle \tau _{b}*=\tau _{c}*}$.

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. ^[13] 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. ^[14]

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, ${\displaystyle \mathrm {Re} _{p}}$ 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 ${\displaystyle \mathrm {Re} _{p}*}$.

${\displaystyle \tau _{b}*=f\left(\mathrm {Re} _{p}*\right)}$

This can then be solved by using the empirically derived Shields curve to find ${\displaystyle \tau _{c}*}$ 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. ^[15]

Particle Reynolds number

In general, a particle Reynolds number has the form:

${\displaystyle \mathrm {Re} _{p}={\frac {U_{p}D}{\nu }}}$

Where ${\displaystyle U_{p}}$ is a characteristic particle velocity, ${\displaystyle D}$ is the grain diameter (a characteristic particle size), and ${\displaystyle \nu }$ is the kinematic viscosity, which is given by the dynamic viscosity, ${\displaystyle \mu }$, divided by the fluid density, ${\displaystyle {\rho _{f}}}$.

${\displaystyle \nu ={\frac {\mu }{\rho _{f}}}}$

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, ${\displaystyle u_{*}}$, which is a way of rewriting shear stress in terms of velocity.

${\displaystyle u_{*}={\sqrt {\frac {\tau _{b}}{\rho _{f}}}}=\kappa z{\frac {\partial u}{\partial z}}}$

where ${\displaystyle \tau _{b}}$ is the bed shear stress (described below), and ${\displaystyle \kappa }$ is the von Kármán constant, where

${\displaystyle \kappa ={0.407}}$.

The particle Reynolds number is therefore given by:

${\displaystyle \mathrm {Re} _{p}*={\frac {u_{*}D}{\nu }}}$

Bed shear stress

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

${\displaystyle \tau _{c}*=f\left(\mathrm {Re} _{p}*\right)}$,

which solves the right-hand side of the equation

${\displaystyle \tau _{b}*=\tau _{c}*}$.

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

${\displaystyle \tau _{b}*={\frac {\tau _{b}}{(\rho _{s}-\rho _{f})(g)(D)}}}$,

the bed shear stress needs to be found, ${\displaystyle {\tau _{b}}}$. 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.

Example

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. ^[19]

${\displaystyle P={\frac {w_{s}}{\kappa u_{\ast }}}}$

Here, the Rouse number is given by P. The term in the numerator is the (downwards) sediment the sediment settling velocity w_s, 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. ^{[19] [20]}

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. ^[21] 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. ^[22] Their equation is

${\displaystyle w_{s}={\frac {RgD^{2}}{C_{1}\nu +(0.75C_{2}RgD^{3})^{(0.5)}}}}$.

In this equation w_s is the sediment settling velocity, g is acceleration due to gravity, and D is mean sediment diameter. ${\displaystyle \nu }$ is the kinematic viscosity of water, which is approximately 1.0 x 10⁻⁶ m²/s for water at 20 °C.

${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ 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
${\displaystyle C_{1}}$	18	18	20	24
${\displaystyle C_{2}}$	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, ${\displaystyle g=9.8}$, and values given above for ${\displaystyle \nu }$ and ${\displaystyle R}$. From these parameters, the fall velocity is given by the expression:

${\displaystyle w_{s}={\frac {16.17D^{2}}{1.8\cdot 10^{-5}+(12.1275D^{3})^{(0.5)}}}}$

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,

${\displaystyle {\displaystyle w_{s}={\frac {g~({\frac {\rho _{s}-\rho }{\rho }})~d_{sed}^{2}}{18\nu }}},}$

where ${\displaystyle g}$ is the gravitational constant; ${\displaystyle \rho _{s}}$ is the density of the sediment; ${\displaystyle \rho }$ is the density of water; ${\displaystyle d_{sed}}$ is the sediment particle diameter (commonly assumed to be the median particle diameter, often referred to as ${\displaystyle d_{50}}$ in field studies); and ${\displaystyle \nu }$ 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.

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 raised to some power. Excess dimensionless shear stress is a nondimensional measure of bed shear stress about the threshold for motion.

${\displaystyle (\tau _{b}^{*}-\tau _{c}^{*})}$,

Bed load transport rates may also be given 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" ${\displaystyle (T_{s}{\text{ or }}\phi )}$ and is an important in that it shows bed shear stress as a multiple of the value of the criterion for the initiation of motion.

${\displaystyle T_{s}=\phi ={\frac {\tau _{b}}{\tau _{c}}}}$

When used for sediment transport formulae, this ratio is typically raised to a power.

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

${\displaystyle q_{s}={\frac {Q_{s}}{b}}}$.

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

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.

A common characterization of suspended sediment concentration in a flow is given by the Rouse Profile. This characterization works for the situation in which sediment concentration ${\displaystyle c_{0}}$ at one particular elevation above the bed ${\displaystyle z_{0}}$ can be quantified. It is given by the expression:

${\displaystyle {\frac {c_{s}}{c_{0}}}=\left[{\frac {z\left(h-z_{0}\right)}{z_{0}\left(h-z\right)}}\right]^{-P/\alpha }}$

Here, ${\displaystyle z}$ is the elevation above the bed, ${\displaystyle c_{s}}$ is the concentration of suspended sediment at that elevation, ${\displaystyle h}$ is the flow depth, ${\displaystyle P}$ is the Rouse number, and ${\displaystyle \alpha }$ relates the eddy viscosity for momentum ${\displaystyle K_{m}}$ to the eddy diffusivity for sediment, which is approximately equal to one. ^[38]

${\displaystyle \alpha ={\frac {K_{s}}{K_{m}}}\approx 1}$

Experimental work has shown that ${\displaystyle \alpha }$ ranges from 0.93 to 1.10 for sands and silts. ^[39]

The Rouse profile characterizes sediment concentrations because the Rouse number includes both turbulent mixing and settling under the weight of the particles. Turbulent mixing results in the net motion of particles from regions of high concentrations to low concentrations. Because particles settle downward, for all cases where the particles are not neutrally buoyant or sufficiently light that this settling velocity is negligible, there is a net negative concentration gradient as one goes upward in the flow. The Rouse Profile therefore gives the concentration profile that provides a balance between turbulent mixing (net upwards) of sediment and the downwards settling velocity of each particle.

Bed material load

Bed material load comprises the bed load and the portion of the suspended load that is sourced from the bed.

Three common bed material transport relations are the "Ackers-White", ^[40] "Engelund-Hansen", "Yang" formulae. The first is for sand to granule-size gravel, and the second and third are for sand ^[41] though Yang later expanded his formula to include fine gravel. That all of these formulae cover the sand-size range and two of them are exclusively for sand is that the sediment in sand-bed rivers is commonly moved simultaneously as bed and suspended load.

Wash load

Wash load is carried within the water column as part of the flow, and therefore moves with the mean velocity of main stream. Wash load concentrations are approximately uniform in the water column. This is described by the endmember case in which the Rouse number is equal to 0 (i.e. the settling velocity is far less than the turbulent mixing velocity), which leads to a prediction of a perfectly uniform vertical concentration profile of material.

Total load

Some authors have attempted formulations for the total sediment load carried in water. ^{[42] [43]} These formulas are designed largely for sand, as (depending on flow conditions) sand often can be carried as both bed load and suspended load in the same stream or shoreface.