CIVE 530 - OPEN-CHANNEL HYDRAULICS
LECTURE 17: SEDIMENTATION ENGINEERING

17.0 INTRODUCTION

The study of sediments can be divided into: (1) production, (2) transport, and (3) deposition.
Sediment production is primarily the result of raindrop impact at the watershed level.
Surface runoff acts to transport sediment downslope.
Deposition occurs where the kinetic energy of the water is insufficient to continue to entrain sediment in the flowing water.

17.1 SEDIMENT PROPERTIES

SEDIMENT FORMATION

Sediments are the products of disintegration and decomposition of rocks.
The disintegration is caused either by large temperature changes or by alternate cycles of freezing and thawing.
Decomposition refers to the breaking down of mineral components of rocks by chemical reaction.
Decomposition includes the processes of: (1) carbonation, (2) hydration, (3) oxidation, and (4) solution.
Carbon dioxide in the atmosphere readily combines with water to form carbonic acid.
Carbonic acid reacts with feldspars to produce clay minerals, silica, calcite, and other relatively soluble carbonates containing potassium, soidum, iron, and magnesium.
The addition of water to many of the minerals present in igneous rocks results in the formation of clay minerals such as allumimum silicates.
Oxidation is accelerated by the presence of moisture in the air.
Oxygen combines with other elements to form sulfates, carbonates, and nitrates, most of which are relatively soluble.
The amount of dissolved solids carried by streams in the contiguous United States has been estimated at more than 50% of the amount of suspended sediment.

PARTICLE CHARACTERISTICS

The characteristics are: (1) size, (2) shape, (3) specific weight and specific gravity, and (4) fall velocity.
Size
A classification of sediments according to size is shown in Table 15-1.
Five groups are included: (1) boulders and cobbles, (2) gravel, (3) sand, (4) silt, and (5) clay.
A No. 200 sieve is used to separate sand particles from the finer particles such as silt and clay.
Shape
Particle shape is numerically defined in terms of its sphericity and roundness.
True sphericity is the ratio of the surface area of a sphere having the same volume as the particle, to the surface area of the particle.
Alternate definition: sphericity is the ratio of the diameter of the sphere having the same volume as the particle, to the diameter of a sphere circumscribing the particle.
A sphere has a sphericity of 1, whereas all other shapes have a sphericity of less than 1.
Roundness is defined as the ration of the average radius oc urvature of the particles edges to the radius of the largest inscribed circle.
It refers to the sharpness of the edges of sediment particles and is commonly used as an indication of particle wear.
The shape factor is often used as an indicator of particle shape:
SF = c / (ab)^1/2

a, b, and c are three orthogonal particle length dimensions.
According to Corey, a is the longest, b is the intermediate, and c is the shortest length dimension.
According to McNown and Malaika, c is measured in the direction of motion, and a and b are perpendicular to c.
Specific weight and specific gravity
Specific weight of a sediment particle is the weight per unit volume.
Specific gravity of a sediment particle is the ratio of its weight to the weight of an equal volume of water.
Most sediment particls consists of quartz or feldspar, which are about 2.65 times heavier than water.
The specific gravity of sediments is generally considered to be about 2.65.
Exceptions are heavy minerals (for instance, magnetite, with specific gravity of 5.18), but these occur infrequently.
Fall velocity
The fall velocity of a sediment particle is its terminal rate of settling in still water.
Fall velocity is a function of size, shape, and specific gravity of the particle, and the specifie weight and viscosity of the surrounding water.
For spherical particles, the fall velocity (derived from a balance of submerged weight and drag) can be expressed as follows:
w = [(4/3) (gd_s/C_D) (γ_s - γ)/γ ] ^1/2

in which
w = fall velocity,
g = gravitational acceleration,
d_s = particle diameter,
C_D = drag coefficient (dimensionless),
γ_s = specific gravity of sediment particles, and
γ = specific weight of water.
The drag coefficient C_D is a function of the particle Reynolds number R, defined as:
R = = wd_s/ν

in which ν = kinematic viscosity of the fluid.
For particle Reynolds numbers less than 0.1, the drag coefficient is equal to C_D = 24/R (laminar flow).
Substituting the value of C_D into the equation of fall velocity leads to Stokes' law:
w = [(gd_s²/(18ν)](γ_s - γ)/γ

For particle Reynolds numbers greater than 0.1, the drag coefficient is still a function of Reynolds number, but the relationship cannot be expressed in analytical form.
The relationship of C_D versus R for a wide range of particle Reynolds numbers is shown in Fig. 15-1.
Since fall velocities vary with fluid temperature and viscosity, two particles of the same size, shape, and specific gravity, falling on two different fluids of different viscosity, or in the same fluid at different temperatures, will have different fall veolocities.
To provide a measure of comparison, the concept of standard fall velocity was developed.
The standard fall velocity of a particle is the average rate of fall that it would attain if falling alone in quiescent water of infinite extent at the temperature of 24^oC.
The standard fall diameter of a particle is the diameter of an equivalent sphere having the same standard fall velocity and specific gravity.

Example 5-1.
Calculate the fall velocity of a spherical quartz particle of diameter d_s = 0.1 mm and drag coefficient C_D = 4.
Solution: Use the equation for fall velocity:
w = [(4/3) (gd_s/C_D) (γ_s - γ)/γ ] ^1/2

with γ_s = 2.65 g/cm³, γ = 1 g/cm³, g = 9.81 m/s², d_s = 0.0001 m, to find w = 0.023 m/s.
Example 5-1A.
Calculate the fall velocity of a spherical sediment particle, given d_s = 0.15 mm, and temperature T = 24^oC.
Solution: Proceed by iteration. Assume R, calculate C_D from Fig. 15-1, calculate w, verify assumed R or iterate. Use online fall velocity. Answer: 0.01619 m/s.

SIZE DISTRIBUTION OF SEDIMENT DEPOSITS

Particle size distribution is the key to predicting the behavoir of a sediment deposit and estimating its specific weight.
A sediment sample containing a wide range of particle sizes is well graded, or poorly sorted.
A sediment sample containing a narrow range of particle sizes is poolr graded, or well sorted.
The size distribution can be measured in many ways.
The coarsest fraction can be separated by direct measurement for boulders and cobbles, and by sieving for sands and gravels.
The Visual accumulation tube (VA tube) is a fast, economical, and accurate method of determining the size distribution of sediment samples.
In the VA tube method, the particles start falling from a common source and become stratified according to their relative settling velocities.
At a given instant, the particles coming to rest at the bottom of the tube are of a certain sedimentation size, finer than particles than have already settled, and coarser that those still remaining in suspension.

VA tube assembly.

SPECIFIC WEIGHT OF SEDIMENT DEPOSITS

The specific weight of a sediment deposit is the dry weight of sedimentary material per unit volume.
The specific weight of a sediment deposit is always less than the specific weight of the individual particles.
A knowledge of the specific weight of a sediment deposit allows the conversion of weight to volume.
Factors influencing the specific weight of a sediment deposit are:

its mechanical composition,
the environment in which the deposits are formed,
time.

The specific weight of coarse materials such as boulders, gravel and coarse sand change very little with time.
The specific weight of silt and clay deposists may reduce considerably in time.
The Lane and Koelzer relation for specific weight:
W = W₁ + B log T

in which:

W = specific weight of a deposit after T years,
W₁ = initial specific weight, measured after 1 year of consolidation, and
B = a constant.

Table 15-2 shows values of W₁ and B.

VA tube assembly.

Drying or aeration of a sediment deposit helps to accelerate consolidation through removal of the water from the pore spaces.
Table 15-3 shows the effect of aeration on the specific weight of sediment deposits for several types of soil mixtures.

VA tube assembly.

17.2 SEDIMENT PRODUCTION

The presence of sediment in rivers has its origin in soil erosion.
Erosion encompasses a series of complex and interrelated natural processes that have the effect of looseing and moving away soil and rock materials under the action of water, wind, and other geologic factors.
In the long term, the effect of erosion is the denudation of the land surface.
The rate of landscape denudation can be quantified from a geologic perspective.
The number of cm per 100 years can be used as a measure of the erosive activity of a region.
Geologic measures of landscape denudation appear insignificant whe compared to the typical timespan of human activity.
However, the quantities of sediment removed may concentrate downstream and negatively impact the operation of hydraulic structures.
SEDIMENT PRODUCTION AND SEDIMENT YIELD

A distinction should be made between the quantity of sediment removed at the source(s) and the quantity of sediment delivered to a downstream point.
Usually the sediment delivery is a fraction of the sediment produced.
Gross sediment production refers to the amount of sediment eroded and removed from the source.
Sediment yield refers to the actual delivery of eroded soil particles to a given downstream point.
The ratio of sediment yield to gross sediment production is the sediment delivery ratio (SDR).
Gross sediment production is measured in metric tons per hectare per year.
Sediment yield is measured in metric tons per day at the catchment outlet or other point of interest.
NORMAL AND ACCELERATED EROSION

Erosion can be classified as:
Normal erosion has been occurring at variable rates since the first solid materials formed on the surface of the Earth.
Normal erosion is extremely slow in most places.
It is a function of climate, parent rocks, precipitation, topography, and vegetative cover.
Accelerated erosion occurs at a much faster rate than normal, usually through reduction of vegetative cover.
Deforestation, overgrazing, overcultivation, forest fires, and urban sprawl, result in accelerated erosion.
SEDIMENT SOURCES

Erosion can be classified as:
sheet erosion,
rill erosion,
gully erosion, and
channel erosion.

Sheet erosion is the wearing away of a thin laer on the land surface, primarily by overland flow.
Rill erosion is the removal of soil by small concentrations of flowing water (rills).
Gully erosion is the removal of soil from incipient channels that are large enough so they cannot be removed by normal cultivation.
Channel erosion refers to erosion occurring in stream channels in the form of streambank erosion or channel degradation.
Upland erosion is made of sheet and rill erosion.
Channel erosion excludes sheet and rill erosion.
UPLAND EROSION AND THE UNIVERSAL SOIL LOSS EQUATION

The prediction of upland erosion is made by the Universal Soil Loss Equation (USLE):

A = R K L S C P

in which
A = annual soil loss due to sheet and rill erosion in tons per acre per year,
R = rainfall factor,
K = soil eodibility factor,
L = slope-length factor,
S = slope gradient factor,
C = crop management factor, and
P = erosion control practice factor.

Rainfall factor

When factors other than rainfall are held constant, soil losses from cultivated fields are shown to be directly proportional to the product of the storm's total kinetic energy E and its maximum 30-minute intensity I.
The product EI reflects the combined potential of raindrop impact and runoff turbulence to transport dislodged soil particles.
The sum of EI products for a given year is an index of the erosivity of all rainfall for that year.
The rainfall factor R is the average value of the series of annual sdums of EI products.
Values of R applicale to the contiguous United States are shown in Fig. 15-2.

VA tube assembly.

Soil erodibility factor

The soil erodibility factor is a measure of the resistance of a soil surface to erosion.
Its is defined as the amount of soil loss, in tons per acre per year, per unit of rainfall factor R for a unit plot.
A unit plot is 72.6 ft long, with a uniform lengthwise gradient of 9%, in continuous fallow, tilled up and down the slope.
Values of K for 23 major soils are shown in Table 15-4.
K factors for other soils are estimated by comparison with those values in Table 15-4.

VA tube assembly.

Slope-length and slope-gradient factors

The rate of soil erosion by flowing water is a function of slope length (L) and gradient (S).
For practical purposes, there two topographic characteristics are combined into a single topographic factor (LS).
The factor LS is defined as the ratio of soil loss from a slope of given length and gradient to the soil loss from a unit plot of 72.6 ft length and 9% gradient.
Values of LS are shown in Figue 15-3.

VA tube assembly.

Crop management factor

The crop management factor C is defined as the rate of soil loss from a certain combination of vegetative cover and management practice to the soil loss resulting from tilled, continuous fallow.
Values of C range from as little as 0.0001 for undisturbed forest land to a maximum of 1 for disturbed areas with no vegetation.
Values of C for cropland are estimated on a local basis.
Values of LS are shown in Tables 15-5 and 15-6.

VA tube assembly.

VA tube assembly.

Erosion control practice factor

The erosion control practice factor is defined as the ration of soil loss under a certain erosion-control practice to the soil loss resulting from straight row farming.
Values of P have been establsished for contouring and contour strip cropping.
In contour strip cropping, strips of sod or meadow are alternated with strips of row crops or small grains.
Values of P used for contour strip cropping are also used for contour-irrigated furrows.
Values of P are shown in Table 15-7.

VA tube assembly.

Use of the Universal Soil Loss Equation

The USLE cannot be used to compute sediment yield.
For instance, for a 1000-km² basin, only 5% of the soil loss computed by the USLE may appear as sediment yield at the basin outlet.
The remaining 95% is redistributed on uplands or flood plains, and it does not constitute a net loss from the drainage basin.

Example 15-3.
Assume a 600-ac watershed in Fountain County, Indiana. Compute the average annual soil loss by the USLE for the following conditions:

cropland, 280 ac, contour strip-cropped, soil is Fayette silt loam, slopes are 8% and 200 ft long;
pasture, 170 ac, 50% canopy cover, 80% groundcover with grass, soil is Fayette silt loam, slopes are 8% and 200 ft long;
forest, 150 ac, soil is Marshall silt loam, 30% canopy cover, slopes are 12% and 100 ft long.
Solution:

From Fig. 15-2, R = 185
From Table 15-4, K = 0.38
From Fig. 15-3, LS = 1.4
Value of C for cropland is obtained from local sources. Assume C = 0.12
From Table 15-7, P = 0.25
A = R K SL C P = 2.95 tons/ac/yr

From Fig. 15-2, R = 185
From Table 15-4, K = 0.38
From Fig. 15-3, LS = 1.4
No value of P has been established for pasture. Assume P = 1.
A = R K SL C P = 1.18 tons/ac/yr
From Fig. 15-2, R = 185
From Table 15-4, K = 0.33
From Fig. 15-3, LS = 1.8
No value of P has been established for pasture. Assume P = 1.
A = R K SL C P = 0.66 tons/ac/yr
The total sheet and rill erosion from the 600-ac watershed is:
(280 × 2.95) + (170 × 1.18) + (150 × 0.66) = 1126 tons/yr
Verify Example 15-3 with online usle 2.
[Watershed] soil loss = 1126.406 tons/yr
Channel erosion

Channel erosion includes gully erosion, streambank erosion, streambed degradation, floodplain scour, and other sources of sediment, excluding upland erosion.
Gullies are incipient channels in process of development.
Gully growth is usually accelerated by severa climatic events, improper land use, or changes in stream base levels.
Significant gully activity is found in regions of moderate to steep topography with thick soil mantles.
Camp Creek, Oregon.
The total sediment outflow from gullies is usually less than sheet and rill erosion.
Streambank erosion and streambed degradation can be significant in certain cases.
Downstream of a newly constructed dam, "hungry" water will cause streambed degradation.

Degradation to bedrock downstream of a sediment retention dam.

Changes in channel alignment and/or removal of natural vegetation from streambanks may cause streambank erosion.
Methods for determining soil loss due to various types of channel erosion include the following:

comparing aerial photos taken at different times,
performing river cross sectional surveys to determine changes in cross section,
assembling historical data,
performing field studies to evaluate annual growth.

Field surveys may provide sufficient data to estimate streanbak erosion as follows:

S = H L R
in which:
S = annual volume of streambank erosion,
H = average height of bank,
L = lenght of eroded bank, and
R = annual rate of bank recession (net rate).

Streambed degradation can be estamated as follows:

S = W L D
in which:
S = annual volume of streambed degradation,
W = average bottom width of degrading channel reach,
L = length of degrading channel reach, and
R = annual rate of streambed degradation.
Accelerated erosion due to strip mining and construction activities

Strip mining and construction activities greatly accelerate erosion rates.
Human induced land distrubances have a substantial impact on sediment production.
EPA's BMP's (best management practices) are used to control erosion from anthropogenically disturbed sites.
The USLE may be used to compute erosion from disturbed sites.
Sediment yield

In engineering applications, the quantity of sediment eroded at the sources is not as important as the quantity of sediment delivered to a downstream point, i.e., the sediment yield.
Sediment yield is calculated by multiplying the gross sediment production by a sediment delivery ratio that varies in the range 0-1.
Sediment delivery ratio

The sediment delivery ratio (SDR) is largely a function of:
sediment source,
proximity of sediment source to the fluvial transport system,
density and condition of the fluvial transport system,
sediment size and texture, and
catchment characteristics.

The sediment source has an influence on the SDR.
Not all sediments originating in sheet and rill erosion are likeyl to enter the transport system.
Sediments originating in channel bank erosion are more likely to be delivered to downstream points.
The amount of sediments delivered to downstream points will depend on the ability of the fluvial transport system to entrain and hold on to the sediment particles.
Silt and clay particles can be transported much more readily than sand particles.
High catchment relief often indicates both high erosion and high SDR.
High channel density is an indication of an efficient transport system and, therefore, of a high SDR.
Estimation of sediment delivery ratios

The SDR is the ratio of sediment yield to gross sediment production.
Estimates of sediment yield can be obtained by reservoir sedimentation surveys.
Alternatively, sediment yield can be evaluated by a direct measurement of sediment load at the point of interest.
Estimates of gross sediment production can be obtained with the USLE.
When warranted, this estimate can be augmented by field measurements of gully and channel erosion
Regional SDR equations can be derived with data.
The simplest SDR prediction equation is based soleyl on drainage area.

Degradation to bedrock downstream of a sediment retention dam.

This figure shows that SDR varies in inverse proportion of the 1/8 power of the drainage area.
The greater the drainage area, the smaller tha catchment relief, and the greater the chances for sediment deposition within the catchment.
Rough estimates can be obtained from Fig. 15-8, but caution is recommended for more detailed studies.
An example of a regional SDR equation:

SDR = 31,623 (10 A)^-0.23(L/R)^-0.51 B^-2.79
in which:
SDR = sediment delivery ratio,
A= drainage area, in sq. mi.,
L/R = ratio of catchment length to relief,
B = weighted mean bifurcation ratio, defined as the ratio of number of streams of a given order to the number of streams in the next higher order.
Empirical formulas for sediment yield

Statistical analysis has been used to develop regional equations for the prediction of sediment yield.
The Dendy and Bolton formula is a good example of a sediment yield equation.
Sediment yield vs drainage area

Dendy and Bolton studied sedimentation data for about 1500 reservoirs, ponds, and detention basins.
They used reservoirs with drainage areas greater than 1 sq mi.
For drainage areas between 1 and 30,000 sq mi, Dendy and Bolton found that the annual sediment yield per unit area was inversely related to the 0.16 power of the drainage area:

S/S_R = (A/A_R)^-0.16
in which:
S = sediment yield in tons per acre per sq mi,
S_R = reference sediment yield corresponding to 1 sq mi area, equal to 1645 tons/yr,
A = drainage area in sq mi, and
A_R = reference drainage area, equal to 1 sq mi.
Sediment yield vs mean annual runoff

Dendy and Bolton studied sedimentation data from 505 reservoirs having mean annual runoff data.
Annual sediment yield per unit area was shown to increase sharply as mean annual runoff Q increased from 0 to 2 in.
Thereafter, for mean annual runoff from 2 to 50 in, annual sediment yield per unit area decreased exponentially.
This lead to the following equations for sediment yield:
For Q ≤ 2 in:

S/S_R = 1.07 (Q/Q_R)^0.46

For Q > 2 in:

S/S_R = 1.19 e ^-0.11(Q/Q_R)
in which:
Q_R = reference mean annual runoff, equal to 2 in.
Dendy and Bolton further combined the equations for sediment yield in terms of drainage area and mean annual runoff into the following:
For Q ≤ 2 in:

S/S_R = 1.07 (Q/Q_R)^0.46 [1.43 - 0.26 log(A/A_R)]

For Q > 2 in:

S/S_R = 1.19 e ^{-0.11(Q/Q_R) [1.43 - 0.26 log(A/A_R)]}

For S_R = 1645 tons/yr, Q_R = 2 in, and A_R = 1 sq mi, the Dendy and Bolton equations reduce to:
For Q ≤ 2 in:

S = 1280 Q^0.46(1.43 - 0.26 log A)

For Q > 2 in:

S = 1965 e ^{-0.055Q (1.43 - 0.26 log A)}

with Q in in., A in sq. mi., and S in tons/sq. mi./yr.
These equations should be used with caution.
In certain cases, local factors such as soils, geology, topography, land use, and vegetation may have a greater influence on sediment yield than either mean annual runoff or drainage area.

Example 15-4.
Calculate the sediment yield by the Dendy and Bolton formula for a 150-sq mi watershed with 3.5 in of mean annual runoff.
Solution:
The application of the second Dendy and Bolton formula leads to:
S = 210,000 tons/yr.
Verify Example 15-3 with online dendy and bolton.
Sediment yield = 210,123.47 tons/yr

17.3 SEDIMENT TRANSPORT

Sediment transport refers to the entrainment and movement of sediments by flowing water.
An understanding of the principles of sediment transport is essential for the interpretation and solution of many hydraulic, hydrologic, and water resources engineering problems.
The study of sediment transport can be divided into:

sediment transport mechanics,
sediment transport prediction, and
sediment routing.

Sediment transport mechanics refers to the fundamental processes by which sediment is entrained and transported by flowing water.
Sediment transport prediction refers to the methods and techniques to precit the equilibrium or steady rate of sediment transport in streams and rivers.
The prediction of sediment transport is accomplished by a sediment transport formula.
Sediment routing refers to the nonequilibrium or unsteady sediment transport processes, the net result of which is either the aggradation or degradation of streams or river beds.
The energy and turbulence of the flow gives streams and rivers the capacity to entrain and transport sediment.
Entrainment and transport is a function of the vertical velocity gradient near the streambed.
The sediment being transported can originate in either upland or channel sources.
A significant feature of sediment transport is the entrainment and transport by the flow of the material constituting the streambed.
Thus, sediment transport is the mechanism by which streams and rivers determine their own cross sectional shapes and boundary roughness.
SEDIMENT TRANSPORT MECHANICS

Sediment load or sediment discharge is the total amount of sediment transported by a stream or river past a given point, expressed in terms of weight per unit volume.
Based on the predominant mode of transport, sediment load can be classified into:
bed load, and
suepended load.

Bed load is the fraction of sediment load that moves by saltation and rolling along the streambed, primarily by action of bottom shear stresses caused by vertical velocity gradients.
Suspended load is the fraction of sediment load that moves in suspension by the action of turbulence.
Particles transported as bed load are coarser than particles transported as suspended load.
Some particles may move as bed load at one point, and as suspended load at another.
Based on whether the particle sizes are represented in the streambed, sediment load can be classified into:
bed material load, and
fine material load.

Bed material load is the fraction of sediment load whose particle sizes are significantly represented in the streambed.
Fine material load, commonly referred to as "wash load," is the fraction of sediment load whose particle sizes are not significantly represented in the streambed.
Bed material load is the coarser fraction of sediment load, which may have originated in the channel bed, and that may be subject to deposition under certain conditions.
Wash load is the finer fraction of sediment load, that has not originated in the channel bed, and that is not likely to deposit.
Wash load is largely unaffected by the hydraulics of the flow.
The relationship between the two classifications of sediment load is shown in Fig. 15-5.
This figure shows that the concepts of bed load and wash load are mutually exclusive.
The middle overlap is the suspended bed material load, i.e., the fraction of sediment load that moves in suspended mode and is composed of particle sizes that are represented in the channel bed.

Degradation to bedrock downstream of a sediment retention dam.

Rachichuela wash, in La Leche basin, northern Peru, showing very large bed material sizes on the streambed..

Initiation of Motion

Water flowing over a streambed has a marked vertical velocity gradient near the streambed.
This velocity gradient exerts a shear stress on the particles lying on the streambed, i.e., a bottom shear stress.
For wide channels, the bottom shear stress can be approximated by the following formula:

τ_o = γ d S
in which:
τ_o = bottom shear stress;
γ = specific weight of water;
d = flow depth; and
S = equilibrium or energy slope.

There is a threshold value of bottom shear stress above which the particles actually begin to move.
This threshold value is referred to as the critical bottom shear stress, or critical stractive stress.
The Shields curve, shown in Fig. 5-6, represent the earliest attempt to combine theoretical and empirical approaches to estimate critical stractive stress.
The Shields curve depicts the threshold of motion, i.e., the condition separating motion (above the curve) from no motion (below the curve).

Shields diagram for initiation of motion.

The abscissa in the Shields diagram is the boundary Reynolds number, defined as:

R_* = U_*d_s/ν

in which:
R_* = boundary Reynolds number;
U_* = shear velocity;
d_s = mean particle diameter; and
ν = kinematic viscosity of the water.
The shear velocity is defined as:

U_* = (&tau_o/ρ)^1/2

in which ρ = density of water.
The ordinate in the Shields diagram is the dimensionless tractive stress, defined as:

&tau_* = &tau_o/[(γ_s - γ) d_s]

in which τ_o = dimensionless tractive stress.
The Shields diagram indicates that, within a midrage of boundary Reynolds numbers (approximately 2-200), the dimensionless critical tractive stress can be taken as a constant for practical purposes.
Within this range, the critical tractive stress is proportional to the sediment particle size.
For instance, assuming τ_* = 0.04 from the Shields diagram:

&tau_c = 0.04 (γ_s - γ) d_s]

in which &tau_c = = critical tractive stress.
For quartz particles, for which γ_s = 2.65 * 62.4 lb/ft³:

&tau_c = 0.34 d_s

in which τ_c is in lbs/ft² and d_s in inches.
Extensive experimental studies by Lane have shown that the coefficient in the above equation is around 0.5.
However, Lane used the d₇₅ (diameter for which 75% by weight is finer) instead of the d₇₅ used by Shields.

Example 15-5.
Based on the Shields criterion, determine whether a 3-mm quartz particle is at rest or moving under a 7-ft flow depth with channel slope S_o = 0.0001. Asssume T = 70^oF.
Solution:
The bottom shear stress is:
&tau_o = γ d S = 62.4 × 7.0 × 0.0001 = 0.0437 lb/ft²
The dimensionless tractive stress is:
τ_* = &tau_o/[(γ_s - γ) d_s] = 0.0437/[(2.65 - 1.00) × 62.4 × 3.0/(25.4 × 12)] = 0.0431.
The shear velocity is:
U_* = (&tau_o/ρ)^1/2 = (0.0437/1.94)^1/2 = 0.150 ft/s.
The kinematic viscosity is:
ν = 1.058 × 10^-5 ft²/s.
The boundary Reynolds number is:
R_* = U_*d_s/ν = 0.150 × [3.0/(25.4 × 12)] /(1.058 × 10^-5) = 140.
For R_* = 140, the dimensionless critical tractive stress is: τ_*c = 0.048.
Since τ_* = 0.0431 < τ_*c = 0.048
Thus, the particle is at rest.

Example 15-5A.
Solve this example using online shields.
Boundary Reynolds number R_* = 139.6
Dimensionless tractive stress τ_* = 0.0431
Dimensionless critical tractive stress τ_*c = 0.0482
τ_* = 0.0431 < τ_*c = 0.0482
Thus, the particle is at rest.

17.4 SEDIMENT DEPOSITION IN RESERVOIRS

The k
The water

∂Q/∂x + ∂A/∂t = 0

17.5 SEDIMENT MEASUREMENT TECHNIQUES

The k
The water

∂Q/∂x + ∂A/∂t = 0

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