Chapter 4 Sediment Transport - SFU.cahickin/RIVERS/Rivers4(Sediment transport).pdf · sediment-transport modes: suspended-sediment load and bed-material load. Suspended-sediment load

Chapter 4

Sediment Transport 1. Introduction: Some Definitions 2. Modes of Sediment Transport Dissolved Load Suspended-Sediment Load Wash load Intermittently-suspended or saltation load Suspended-sediment rating curves Typical suspended-sediment loads in BC rivers Relation of sediment concentration to sediment load/discharge Bed Load (Traction Load) Bed load equations 3. The Theory of Sediment Entrainment Impelling forces Inertial forces The Shields entrainment function 4. Applications of Critical Threshold Conditions for Sediment Motion

Bedload transport equations The threshold (equilibrium) channel

5. Some further reading 6. What’s next?

1. Introduction: Some Definitions Not all channels are formed in sediment and not all rivers transport sediment. Some have been

carved into bedrock, usually in headwater reaches of streams located high in the mountains.

These streams have channel forms that often are dominated by the nature of the rock (of varying

hardness and resistance to mechanical breakdown and of varying joint definition, spacing and

pattern) in which the channel has been cut. Such channels often include pools that trap sediment

so that long reaches of channel may carry essentially no sediment at all. Such channels, known

as non-alluvial channels, will not be examined in any detail here. Instead, we will focus on

alluvial channels, the class of river channel forming the vast majority of rivers on the earth’s

surface. Alluvial channels are self-formed channels in sediments that the river typically has at

one time or another transported downstream in the flow.

Sediment transport is critical to understanding how rivers work because it is the set of processes

that mediates between the flowing water and the channel boundary. Erosion involves the

Hickin: River Geomorphology: Chapter 4

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removal and transport of sediment (mainly from the boundary) and deposition involves the

transport and placement of sediment on the boundary. Erosion and deposition are what form the

channel of any alluvial river as well as the floodplain through which it moves.

The amount and size of sediment moving through a river channel are determined by three

fundamental controls: competence, capacity and sediment supply.

Competence refers to the largest size (diameter) of sediment particle or grain that the flow is

capable of moving; it is a hydraulic limitation. If a river is sluggish and moving very slowly it

simply may not have the power to mobilize and transport sediment of a given size even though

such sediment is available to transport. So a river may be competent or incompetent with respect

to a given grain size. If it is incompetent it will not transport sediment of the given size. If it is

competent it may transport sediment of that size if such sediment is available (that is, the river is

not supply-limited).

Capacity refers to the maximum amount of sediment of a given size that a stream can transport

in traction as bedload. Given a supply of sediment, capacity depends on channel gradient,

discharge and the calibre of the load (the presence of fines may increase fluid density and

increase capacity; the presence of large particles may obstruct the flow and reduce capacity).

Capacity transport is the competence-limited sediment transport (mass per unit time) predicted

by all sediment-transport equations, examples of which we will examine below. Capacity

transport only occurs when sediment supply is abundant (non-limiting).

Sediment supply refers to the amount and size of sediment available for sediment transport.

Capacity transport for a given grain size is only achieved if the supply of that calibre of sediment

is not limiting (that is, the maximum amount of sediment a stream is capable of transporting is

actually available). Because of these two different potential constraints (hydraulics and sediment

supply) distinction is often made between supply-limited and capacity-limited transport. Most

rivers probably function in a sediment-supply limited condition although we often assume that

this is not the case.


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Much of the material supplied to a stream is so fine (silt and clay) that, provided it can be carried

in suspension, almost any flow will transport it. Although there must be an upper limit to the

capacity of the stream to transport such fines, it is probably never reached in natural channels

and the amount moved is limited by supply. In contrast, transport of coarser material (say,

coarser than fine sand) is largely capacity limited.

2. Modes of Sediment Transport

The sediment load of a river is transported in various ways although these distinctions are to

some extent arbitrary and not always very practical in the sense that not all of the components

can be separated in practice:

1. Dissolved load

2. Suspended load

3. Intermittent suspension (saltation) load

4. Wash load

5. Bed load

Dissolved Load

Dissolved load is material that has gone into solution and is part of the fluid moving through the

channel. Since it is dissolved, it does not depend on forces in the flow to keep it in the water

column.

The amount of material in solution depends on supply of a solute and the saturation point for the

fluid. For example, in limestone areas, calcium carbonate may be at saturation level in river

water and the dissolved load may be close to the total sediment load of the river. In contrast,

rivers draining insoluble rocks, such as in granitic terrains, may be well below saturation levels

for most elements and dissolved load may be relatively small. Obviously, the dissolved load is

also very sensitive to water temperature and other things being equal, tropical rivers carry larger

dissolved loads than those in temperate environments. Dissolved loads for some of the world’s

major rivers are listed in Figure 4.1.

Dissolved load is not important to the geomorphologist concerned with channel processes


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River Drainage area,

103 km2

Mean Discharge 103m3s-1

Mean suspended

Sediment load 106 t y-1

Mean dissolved

load 106 t y-1

Percentage of total load carried in solution

AFRICA Congo 3500 41.1 48 37 44 Niger 1200 4.9 25 14 36 Nile 3000 1.2 2 12 86

Orange 1000 0.4 0.7 1.6 70 Zambezi 540 2.4 20 25 56

ASIA Ganges/Brahmaputra 1480 30.8 1670 151 8

Huang Ho 752 1.1 900 22 2 Indus 1170 7.5 100 79 44

Irrawaddy 430 13.6 265 91 26 Lena 2440 16.0 12 56 82

Mekong 795 21.1 160 59 27 Ob 2990 13.7 13 46 78

Yenisei 2500 17.6 15 60 80

AUSTRALIA Murray-Darling 1060 0.7 30 8.2 21

EUROPE

Danube 817 6.5 83 53 39 Dnieper 527 1.7 2.1 11 84 Rhone 99 1.9 40 56 58 Volga 1459 7.7 27 54 67

NORTH AMERICA

Columbia 670 5.8 14 21 60 Mackenzie 1810 7.9 100 44 31 Mississippi 3267 18.4 210 142 40

St Lawrence 1150 13.1 5.1 70 93 Yukon 840 6.7 60 34 36

SOUTH AMERICA

Amazon 6150 200 900 290 24 Magdalena 260 6.8 220 20 8

Orinoco 990 36 150 31 17 Parana 2800 15 80 38 32

Figure 4.1: Sediment loads of major world rivers (after Knighton, 1998: Figure 3.2). Sources: Degens et al (1991), Meybeck (1976) and Milliman and Meade (1983).

because it is simply part of the fluid. On the other hand, dissolved load is very important to

geomorphogists concerned with sediment budgets at a basin scale and with regional denudation

rates. In many regions most of the sediment is removed from a basin in solution as dissolved

load and must be accounted for in estimating erosion rates.


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Total dissolved-material transport, Qs(d)(kg/s), depends on the dissolved load concentration Co

(kg/m3), and the stream discharge, Q (m3/s): Qs(d) = CoQ

In sediment-transport theory an important distinction is made between dissolved material and

clastic material. Clastic material is all the particulate matter (undissolved material) carried by a

river regardless of the grain size. Clastic material includes boulders and clay. The clastic load of

a river is moved by several mechanisms that are the basis for recognizing the two principal

sediment-transport modes: suspended-sediment load and bed-material load.

Suspended-sediment load

Suspended-sediment load is the clastic (particulate) material that moves through the channel in

the water column. These materials, mainly silt and sand, are kept in suspension by the upward

flux of turbulence generated at the bed of the channel. The upward currents must equal or

exceed the particle fall-velocity (Figure 4.2) for suspended-sediment load to be sustained.

Figure 4.2: Fall velocity in relation to diameter of a spherical grain of quartz

0.1

0.01 Grain diameter, cm

Fall

velo

city

, cm

s-1

0 G r av

e l Sil t

1

100

10

S and Silt Sand Gravel

0.001 0.01 0.1 1.0


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Figure 4.3: A cable-mounted DH48 suspended-sediment sampler attached to the underside of a heavy lead “fish” for collecting samples on a large river from a boat.

Suspended-sediment concentration in

rivers is measured with an instrument

like the DH48 suspended-sediment

sampler shown in Figure 4.3. The

sampler consists of a cast housing with

a nozzle at the front that allows water

to enter and fill a sample bottle. Air

evacuated from the sample bottle is

bled off through a small valve on the

side of the housing. The sampler can

be lowered through the water column

on a cable (as shown in Figure 4.3) or

it can be attached to a hand-held rod if

the stream is small enough to wade. In

either case the sampler is lowered from

the water-surface to the bed and up to

the surface again at a constant rate so

that a depth-integrated suspended-

sediment sample is collected. The

instrument must be lowered at a constant rate such that the sample bottle will almost but not

quite fill by the time it returns to the surface. The sample bottle is then removed and capped and

returned to the laboratory where the fluid volume and sediment mass is determined for the

calculation of suspended-sediment concentration.

The DH48 suspended-sediment sampler is the standard instrument in use throughout North

America and much of the world. It has the advantages of being robust, inexpensive and simple

to use. Disadvantages include the fact that the maximum grain-size that can be sampled is

limited by the nozzle size and that the instrument cannot be used in very deep flow (in excess of

about 5 metres) because of the necessity to raise and lower the sampler at a reasonable rate while

not overfilling the bottle. Another problem is that the instrument must just touch the bed before

being raised and that is sometimes difficult to achieve in deep, fast-flowing muddy water in


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which the instrument is not visible. It is easy, for example, to lower the sampler to the bed and

be unaware that it has been lowered in such a way that it ploughs a dune front and so drastically

oversamples the sediment. Nevertheless, these limitations are not encountered in the vast

majority of streams where such measurements are made.

Suspended-sediment concentration is conventionally measured as milligrams per litre (mg/L)

which is the same number as gm/m3. So 1000 mg/L = 1000 gm/m3 = 1 kg/m3.

Suspended-sediment concentration and the grain size of suspended sediments typically appear in

the vertical water column distributed as depicted in Figure 4.4.

Figure 4.4: Typical vertical profiles of suspended-sediment concentration (A) & grain size in open-channel flows.

The size and concentration of suspended-sediment typically varies logarithmically with height

above the bed. That is, concentration and grain size form linear plots with the logarithm of

height above the bed. Coarse sand is highly concentrated near the bed and declines with height

at a faster rate than does fine sand. Fine silt is so easily suspended that it is far more uniformly

Hei

ght a

bove

the

bed,

m

Concentration, mg/L

Fine silt

Fine sand

Coarse sand

A

Grain size, D50, cm

Hei

ght a

bove

the

bed,

m Fine

silt Fine sand

Coarse sand

B


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distributed in a vertical section than is the coarser material. Similarly, the grain-size distribution

within a sample of sand displays far more vertical variation than does the vertical distribution of

grain size within the silt range. The former is too large for the flow to move much of it into the

upper water column and the latter is so small and easily suspended that it is well represented at

all levels thus giving rise to a more uniform grain-size profile.

Wash load

Although wash load is part of the suspended-sediment load it is useful here to make a

distinction. Unlike most suspended-sediment load, wash load does not rely on the force of

mechanical turbulence generated by flowing water to keep it in suspension. It is so fine (in the

clay range) that it is kept in suspension by thermal molecular agitation (sometimes known as

Brownian motion, named for the early 19th-century botanist who described the random motion of

microscopic pollen spores and dust). Because these clays are always in suspension, wash load is

that component of the particulate or clastic load that is “washed” through the river system.

Unlike coarser suspended-sediment, wash load tends to be uniformly distributed throughout the

water column. That is, unlike the coarser load, it does not vary with height above the bed.

“Glacial flour” is a common source of wash load present in glacial rivers and lakes. In lakes

deposition of the wash load may eventually occur but only after months of settling in an

undisturbed pond or lake. The repeated annual deposit of wash load from glacial meltwater

streams to lakes produces the “varved clays” that form the bottoms of glacial lakes.

Because all clastic sediment in the water column is measured simultaneously with a suspended-

sediment sampler, in practice it is not possible to distinguish between wash load and the rest of

the suspended load.

Intermittently-suspended or saltation load

Some river scientists make the not especially useful distinction between fully-suspended load

and bed load by classifying the intermediate and transient transport state as saltation load

transport. These are particles that bounce along the channel, partly supported by the turbulence

in the flow and partly by the bed. They follow a distinctively asymmetric trajectory (see Figure


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water surface

Flow direction

river bed

Figure 4.5: The trajectory of saltating (intermittently-suspended) sediment grains moving in the flow.

4.5). Saltation load may be measured as

suspended load (when in the water column)

or as bedload (when on the bed). Although

the distinction between saltating load and

other types of sediment load may be

important to those studying the physics of

grain movement, most geomorphologists are

content to ignore it as a special case.

Suspended-sediment rating curves

You will recall from the discussion in Chapter 2 that measuring the discharge on a river is a

time-consuming and expensive operation. For this reason we make considerable use of

discharge rating curves in the routine measurement of the flow in our rivers. As you might

imagine, measuring the average suspended-sediment concentration in streamflow is no less time-

consuming and expensive and for these same reasons we make considerable use of suspended-

sediment rating curves. Typically a suspended-sediment rating curve is based on an empirical

correlation between sediment concentration and discharge. They commonly take the form: log

load (Qs) or log concentration (Co) versus log discharge, or Co =kQP where P>1.0 and often ~

2.0. Examples are shown in Figure 4.6.

One of the first things that you might notice about the curves in Figure 4.6 is that they are not at

all as well-defined as the discharge rating curve in Figure 2.18, for example. This is typical and

should caution us to use the suspended-sediment variety with care. On some rivers it is possible

to define a strong relationship between discharge and sediment concentration so that suspended-

sediment rating curves are useful for calculating such properties as the annual sediment load. On

other rivers, however, it is very difficult to find a stable sediment-concentration/discharge

relation. Why is this so?

The reason why suspended-sediment rating curves work on some rivers is perhaps easier to

understand than why they fail. The assumption implicit in suspended-sediment rating curves is

that Co = f(Q). But this is not a direct physical relationship; Q is simply a surrogate measure of


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the forces in the flow throwing the sediment into motion. The real forces involved are the

moment to moment shear stress (τo = ρgds) or stream power (Ω = ρgQs) conditioned by the

turbulence in the flow. So Q is simply a convenient and very general shortcut to a very complex

pattern of forces. Nevertheless, the relationship between Qs and Q is sufficiently strong in many

cases for a rating curve to work as a summary measure. Implicit in the Qs/Q correlation is the

notion that suspended-sediment concentration/load is capacity limited.

The reason why suspended-sediment rating curves do not work on some rivers can be

summarized in the corresponding inequalities:

(a) Qs ≠ f(Ω) ≠ f(τo) ≠ f(turbulent flux);

(b) Qs ≠ f (hydraulics); that is, Qs is supply limited (ie, not capacity limited)

In other words, in the case (a) for some reason the assumed dependence of sediment discharge on

the force of flowing water expressed as the discharge or mean shear-stress driving the turbulent

A

B

C

Figure 4.6: Examples of suspended-sediment rating curves. A: Rio Grande, Albuquerque, New Mexico (after Nordin & Beverage, 1965, in Knighton, 1998). B: River Bolin, UK (from Knighton, 1998). C: Squamish River, BC (from Hickin, 1989)


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flux supporting the suspension simply is more complex than we have allowed. Reasons for this

might include discontinuities in the fluid mechanics such as those associated with changing

bedforms in a sand-bed channel or with sudden changes in turbulence structure in rapids or

channel constrictions, in a gravel-bed river, for example.

But a well-documented and probably more common cause of sediment-rating curve failure is the

sediment-supply limitations of case (b). Many (perhaps most) rivers transport suspended

sediment at the rate it is supplied to the channel rather than at the quantity limited by flow

capacity. Unfortunately, our ability to predict quantitatively the amount and timing of sediment

delivery from tributaries, hillslopes and channel banks to a mainstem river is primitive at best.

One of the most compelling indications that external sediment-supply variation is controlling the

amount of suspended-sediment being transported in a river is sediment-load hysteresis in the

suspended–sediment rating curve. Hysteresis refers to the fact that the event rating-curve (the

rating curve based on a single discharge event such as a storm or a single seasonal cycle), instead

of being a single-valued function, is better described as a loop (see Figure 4.7). That is, the

suspended-sediment concentration on the rising discharge is different from the loads at the

corresponding discharges as the discharge declines.

Figure 4.7: A hypothetical suspended-sediment rating curve exhibiting hysteresis in which concentrations for given discharges are higher on the rising limb of the hydrograph than they are on the falling limb.

Discharge

Concentration-discharge values on the falling limb of the hydrograph

Concentration-discharge values on the rising limb of the hydrograph

Con

cent

ratio

n


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Hysteresis in the suspended-sediment rating curve is typical of those on Fraser River and reflects

the varying availability of transportable sediment. During the winter months when the discharge

is relatively low, sediment accumulates in the channel at the base of the banks but especially in

small fans formed at the tributary junctions. When the river rises in the Spring as snow in the

basin melts, the accumulated-sediment supply is abundant and sediment concentrations are

relatively high. But as discharge reaches a seasonal peak in early to mid-July much of this

readily-available sediment has been flushed through the system and the river becomes sediment-

starved and concentrations therefore are supply limited. Consequently, as discharge declines

through late July into August the concentrations for the same discharges experienced earlier in

the seasonal event are systematically lower.

Examples of the role of sediment supply in limiting sediment concentration is shown in Figure

4.8. In general, it is the relationship between the supply-limited suspended-sediment

concentration curve and the discharge curve that determines the nature of the sediment

concentration hysteresis (Figure 4.8D).

Hysteresis in the seasonal suspended-sediment rating curve of the type described for Fraser River

is common but there are causes other than seasonally-varied sediment-storage/supply that can

produce hysteresis. For example, on Squamish River, hysteresis is caused by contrasting

conditions of water and sediment delivery to the channel. In the spring and summer water

supplied to the channel is from melting snow and ice and the meltwater runs off the slopes

without disturbing greatly the surface sediments. Consequently, over the course of the year,

small to moderate discharges produced by meltwater carry only modest amounts of suspended

sediment. In contrast, the fall is a season of intense rainstorms that generate very high discharges

and greatly disturb the sediment on the valley sides. Thus moderate to high discharges in fall are

associated with higher suspended-sediment concentrations than are the same-magnitude flows in

the spring-summer months (Figure 4.9b). When hydrologic populations are mixed in this way,

suspended-sediment rating curves based on discharge are poorly defined (Figure 4.9a) and often

are greatly improved by separating the measurements by season so that we have, in the case of

Squamish River, an April-July rating curve and another for August-March (Figure 4.9c & d).


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Figure 4.8: A scanned image of Figure 4.8 from Knighton (1998) illustrating the role of sediment supply in controlling suspended sediment in various rivers.


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Figure 4.9: Suspended-sediment rating curves for Squamish River at the Brackendale gauging station (from Hickin, E.J. 1989. Contemporary Squamish River sediment flux to Howe Sound, British Columbia. Canadian Journal of Earth Sciences, 26: 1953-1963)

Typical suspended-sediment loads in BC rivers

In British Columbia typical suspended-sediment concentrations vary between a lower limit of

zero and an upper limit of about 5000 mg/L:

Load Range (mg/L) Example

0-100 mg/L: Low Capilano River, Seymour River at low flow

100-500 mg/L: Medium Fraser River at winter low flow

500-1000 mg/L High Fraser River in freshet

1000-5000 mg/L Exceptionally high Nelson and Muskwa Rivers, NE BC

During the hydrologically quiet months (December-April) river water is relatively clear and

suspended-sediment concentration typically falls well below 100 mg/L. During the freshet (the

snowmelt peak) or during rainstorms suspended-sediment concentrations often exceed 500 mg/L

and may rise to 2-3 times this value. In rivers like Fort Nelson (Figure 4.10), Prophet and


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Figure 4.10: Fort Nelson River in northeastern British Columbia is incised into a plateau formed of an ancient lake bed and fine sediment is abundantly supplied to the muddy river.

Figure 4.11: Collapsing banks of Fort Nelson River contribute massive inputs of fine lake sediments to the river.

Muskwa in northeast British Columbia the channels are incising into old lake-bed sediments and

sediment is so abundantly available that concentrations commonly rise to 3000-5000 mg/L. The

concentrations measured in these rivers are among the highest in British Columbia and peak


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concentrations often coincide with local bank collapses (Figures 4.10 & 4.11) or with tributary

inflow events such as arrival in the mainstem channel of a debris flow.

Relation of sediment concentration to sediment load/discharge

The basic equation of suspended-sediment discharge is

Qs = CoQ (with appropriate units) (4.1)

For example, on Lower Fraser River at Mission the average annual discharge is 3500 m3s-1 and

the average annual suspended-sediment concentration is 186 mgL-1 (=186 gm/m3 = 0.186kg/m3).

To calculate the sediment discharge, we have:

Qs = (0.186)(3500) = 650 kg/s

To report the sediment discharge in more practical units of kg/year or tonnes/year we have to

complete the conversion and calculation as follows:

Qs = 650 kg/s

= (650)(60) kg/minute

= (650)(60)(60) kg/hr

= (650)(60)(60)(24) kg/day

= (650)(60)(60)(24)(365) = 2.0 x 1010 kg/year = 2.0 x 107 metric tonnes/year

That is, the sediment discharge is about 20 million metric tonnes/year.

Squamish River carries about the same concentration of sediment but only about 1/10th of the

discharge so the total annual sediment discharge from that river is about 2 million metric

tonnes/year. The Fort Nelson and Muskwa Rivers average about 3000 mg/L concentration.

Suspended-sediment concentrations in rivers elsewhere in the world can be higher still. An

extreme case is the Huang Ho River in China. Where the Huang Ho River traverses the Loess

Plateau (wind-blown or aeolian sediments; http://en.wikipedia.org/wiki/Loess_Plateau) sediment

concentration becomes so high that on occasion there is actually as much or more suspended-

sediment flowing in the river as water! During floods typical concentrations of suspended-

sediment in the Huang Ho tributaries reach hyperconcentrated loads in excess of 40% by weight

(>400 kg/m3 or 400 000 gm/m3 or 400 000 mg/L). These muddy slurries possess some very


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unusual properties (such as laminar-like flow because turbulence is dampened by the heavy

suspended sediment) but we need not be concerned with these special types of fluids here.

Figure 4.1 lists the annual suspended-sediment load for Huang Ho river as 900 x 106 tonnes/year

and average annual discharge is Q = 1.1 x 103 m3/s. Suspended-sediment concentration can be

calculated, thus:

€

loaddischarge

=900x106 ty−1

1.1x103m3s−1

Co =900x106 x103 x103

1.1x103 x60x60x24x365=9.0x1014

3.5x1010= 2.6x104 gm /m3

Co = 26 000 mg/L

Bed Load (Traction Load)

Bed load is the clastic (particulate) material that moves through the channel fully supported by

the channel bed itself. These materials, mainly sand and gravel, are kept in motion (rolling and

sliding) by the shear stress acting at the boundary. Unlike the suspended load, the bed-load

component is almost always capacity limited (that is, a function of hydraulics rather than

supply). A distinction is often made between the bed-material load and the bed load.

Bed-material load is that part of the sediment load found in appreciable quantities in the bed

(generally > 0.062 mm in diameter) and is collected in a bed-load sampler. That is, the bed

material is the source of this load component and it includes particles that slide and roll along the

bed (in bed-load transport) but also those near the bed transported in saltation or suspension.

Bed load, strictly defined, is just that component of the moving sediment that is supported by the

bed (and not by the flow). That is, the term “bed load” refers to a mode of transport and not to a

source.

Bed load is extremely difficult to measure directly because the measuring instrument (bed-load

sampler) invariably interferes with the flow. A commonly used type of bed-load sampler is

shown in Figure 4.12. In small streams where the sampler can often be placed on the bed so that

it is appropriately oriented to the flow, the sample collected may be meaningful although there is


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A

B

Figure 4.12: (A) A commonly used type of bed-load sampler. The sampler base usually has a heavy flat weight attached and the fins keep the instrument oriented into the flow. (B) When the bed-load sampler is appropriately oriented in the flow, bed-load material enters the sampler through the inlet and the divergent flow within the sampler reduces the flow velocity, allowing the sediment to accumulate. A fine mesh at the rear of the sampler allows water to pass through but not the bed-load sediment. After an appropriate measured time-interval the sampler is recovered and the trapped sediment is removed for weighing.

always some bed scour at the inlet that distorts the actual bed-load transport in the vicinity of the

instrument. In large rivers where the sampler must be lowered from a boat by cable to an unseen

bed, however, measurements can be highly inaccurate and must be repeated many times before

one can have confidence in the results. The problems relate largely to the fact that the operator is

unable to see the position of the sampler on the bed. If the sampler settles on a boulder or dune


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face, for example, it may push the sampler inlet into the bed and drastically oversample the rate

of bed-load transport. At other times the sampler position and bed morphology may be such that

scouring of the bed at the sampler inlet is severe, also leading to oversampling. A different

problem (undersampling) occurs if the bed-load sampler settles on the back of a dune or perhaps

the front of the sampler settles on an object that keeps the inlet from contacting the bed. For

these reasons river scientists often prefer to rely on other methods to estimate bed-load transport

rates in rivers.

Methods other than direct measurement by bed-load sampler include:

i. Bed-load pits or traps (used to calibrate bed-load equations)

ii. Morphological methods

a. Bedform surveys

b. Channel surveys

c. Sedimentation-zone surveys (delta progradation)

Bed-load pits or traps operate on the principle illustrated in Figure 4.13. These are major

Figure 4.13: Schematic of a bed-load measurement station involving removal of bed-load sediment from a channel and its return to the sediment transport system after weighing.


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installations that divert sediment from a channel and convey it to a measurement facility where it

is weighed and then returned once again to the channel so that the sediment-transport system is

not unduly disrupted. Obviously such a facility is expensive to build and operate and there are

few of them. The main purpose of such a facility is to calibrate bed-load transport equations for

use on other river channels.

Morphological methods have seen increasing use in recent years because they completely avoid

the problems related to direct bed-load sampling. Where bed-material is moving as bedforms

such as dunes, bedform surveys can be used to track the downstream movement of sediment

(Figure 4.14). This technique relies on high-resolution sonar imaging of the river bed to

Figure 4.14: Bedform surveys track the downstream translation of features such as prograding dunes on the bed of a river as a basis for determining volumetric bed-load transport rates.

construct profiles that can be differenced to determine the volumetric bed-load sediment

transport rate. Individual dunes can be tracked in this way or even entire dune fields. Bed-load

transport rates have been measured on lower Fraser River using this technique. An important

assumption of the method is that all bed-load sediment is transported within the prograding dune.

That is, there is no throughput of bed-material (no grains that roll up the back of the dune and

simply bounce along onto the next dune downstream, and so on, without actually residing in the

dune itself). Laboratory experiments do indicate that the vast majority of grains in bed-load

transport over dunes do indeed become deposited on the dune face rather than being maintained

in continuous motion.


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Channel surveys can be used to produce sequential morphologic maps of a reach of river that can

be differenced (using GIS) to yield amounts of erosion and deposition over time. The principle

here is the same as that for bedform surveys but in this case involves the entire three-dimensional

channel morphology (Figure 4.15). Like the bedform-based calculation, differencing channel

Figure 4.15: Channel surveys showing channel alignment at two points in time (2003 & 2004). Morphologic differencing reveals zones of erosion and deposition that can be used to construct a sediment budget for the channel reach.

morphology as a basis for calculating bed-load sediment transport relies on the assumption that

there is no sediment throughput. That is, all transported bed-load is involved in local deposition

and erosion and not simply transported through the reach without contributing to the changing

channel morphology. Some geomorphologists have argued that this assumption often may not

be met and that this morphologic method can only yield a minimum bed-load transport rate.

Thus, it should be used with caution. Another limitation of the method is that field data on the

vertical distribution of sediment in the eroded/deposited material must be known and some

criterion to distinguish between bed-load and other materials must also be employed.

This morphologic method of determining bed-load transport has the distinct advantage over

direct measurement by sampler of averaging the transport rate over a long time period. Indeed,


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the longer the time interval between surveys, the more accurate the results. Thus, it potentially

yields a more representative bed-load transport rate than a sampler-based measurement at one

point in time. Morphologic surveys have been used on Chilliwack and Fraser rivers by Professor

Mike Church and his research collaborators at UBC and he concludes that they yield realistic

results (for example, see Ham and Church, 2000).

Sedimentation-zone surveys are perhaps one of the most reliable methods of determining

representative long-term bed-load transport rates in rivers. This morphologic method relies on

measuring the accumulating sediment in a feature such as a delta that a river is gradually

building into a lake or embayment. Sequential aerial photography allows the mapping of the

delta position over time and field surveys of the delta depth (based on bathymetry of the

receiving basin or subsurface imaging of the delta using geophysical methods such as ground-

penetrating radar, GPR) complete the picture of three-dimensional volume changes. This

technique has been used to estimate total sediment transport rate on Squamish River at Squamish

and the bed-load transport rate on Fitzsimmons Creek at Whistler (Figure 4.16). Ideal sites for

this type of delta survey are those for which the morphologic changes in the delta are occurring

rapidly. These sites include deltas building into shallow lakes (Green Lake into which

Figure 4.16: Fitzsimmons Creek fan delta is building into Green Lake at Whistler. The isochrone map indicating the delta boundary over time (1947-1999) is the basis for determining the volumetric sediment-transport rate for Fitzsimmons Creek. After Pelpola & Hickin (2004).


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Fitzsimmons Creek delta is prograding) and confined receiving basins such as Howe Sound (into

which Squamish River delta is prograding).

Bedload equations In many circumstances direct or indirect measurements of bedload are not possible and it is

necessary to estimate bed-load transport rates using general capacity-limiting sediment-transport

equations. In all cases these equations have been calibrated against actual measurements.

Almost all bedload formulae are one of three types in which the sediment transport rate per unit

channel width (qsb) is related to either

(i) excess shear stress

€

τ 0 − τ cr( )

€

qsb = ′ X τ 0 τ 0 − τ cr( ) [Du Boys type]

(ii) excess discharge/unit width

€

q − qcr( )

€

qsb = ′ ′ X sk q − qcr( ) [Schoklitsch type]

(iii) excess stream power/unit width

€

ω −ωcr( )

€

qsb ≅ ω −ωcr( )3 / 2d−2 / 3D−1/ 2 [Bagnold, 1980]

where

€

′ X and

€

′ ′ X are sediment coefficients, d is flow depth, and D is grain size.

An equation which has found wide application in the gravel-bed rivers of coastal British

Columbia is the Meyer-Peter & Muller (M-P&M) equation, a simplified version of which is

€

gb = 0.253(τ 0 − τ cr)3 / 2 (4.2)

where gb is the mass rate of transport per unit width of channel (kg/m).

In general, bed-load transport rates have been estimated/measured to be between 1% and 10% of

the suspended load. On Fraser River at Mission, for example, about 1 million tonnes of the total

22 million tonnes of total sediment transport moves as bedload (about 5%).

Equation (4.2) and others of its type utilize a critical condition (denoted by the subscript cr) for

initiating sediment movement. Indeed, this threshold is critical in more ways than one! Once

sediment begins to move on the bed of a channel the sediment-transport rate increases in a rather

orderly fashion as discharge (and shear stress and velocity and stream power) increases and most

sediment-transport equations do a fair and similar job of predicting the rate of increase in

sediment transport. When sediment-transport equations fail (as they often do) it is almost always


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because they fail to predict the beginning of sediment transport: the critical threshold conditions

for initiating sediment motion. It is on this sediment-entrainment threshold condition that we

must now focus our attention.

3. The Theory of Sediment Entrainment

At a very simple deterministic level of analysis, a particle of sediment will begin to move when

the force of the flowing water applied to it equals its submerged weight (Figure 4.17). This

simple analysis leads to:

€

τ cr = Kg(ρs − ρ)D (4.3)

where

€

K =η

tanφ and

€

η = nD2 (a packing coefficient) and φ is the internal angle of friction of the

sediment.

Although this simple analytical approach (called the White analysis, after its originator) is useful

because it highlights the general structural relationships (balance of forces) involved in this

problem, it is not of much practical use because it greatly oversimplies the actual complex forces

involved. That is, there is more to this problem of specifying the entrainment conditions than

merely balancing mean boundary shear stress and the submerged weight of the particle. Mean

boundary shear stress is just one of several impelling forces and the particle submerged weight is

just one of several inertial forces. Unfortunately, the other forces are very difficult to

characterize in a precise quantitative way but it is important to our purpose here to at least

understand their nature at a qualitative level.

The impelling forces tending to move a sediment grain on the bed of a river channel include the

following (the parenthetic italicized notations refer to their inclusion/non-inclusion in the White

analysis in Figure 4.17):

1. Shear stress (included)

2. Impact force (momentum transfer) (not included)

3. Lift forces

a. Bouyancy (included)

b. Vertical velocity-gradient pressure force (not included)

c. Upward turbulence (eddying) forces (not included)


-94-

Figure 4.17: Derivation of the critical shear stress for sediment entrainment (scanned from Knighton, 1998, Figure 4.4)

Shear stress is included in the analysis presented in Figure 4.17 and is regarded by many river


-95-

scientists as the most important of the impelling forces. It is argued by those favouring a simple

mechanical analysis that, although there are other forces at work here, shear stress is correlated

with them all and therefore provides a surrogate measure of them. Those that are unconvinced

argue that the other forces are not strongly enough correlated to shear stress for it alone to

capture the net force acting on the grain.

The impact force refers to the momentum transfer that occurs when the horizontal column of

water upstream strikes the grain. This force is a function of the projected cross-sectional area

presented by the grain to the flow from upstream, the mean flow-velocity within the fluid

column, and the density of the fluid. Predicting the near-boundary flow velocity with the

precision necessary to specify the impact force presently is beyond the science of fluid

mechanics. In any case, as we will see below, even it we could predict the near-boundary mean

velocity at the bed, there remains another problem: the degree of velocity fluctuation about the

mean may be even more important to initiating sediment motion!

The lift forces include buoyancy, accounted for in the White analysis by the submerged weight of

the particle, and two other sets of forces that are not included.

The vertical velocity-gradient pressure force is not an intuitively obvious force and relates to an

energy-conservation principle described by the Bournoulli Theorem. We need not worry about

the derivation of this theorem but it is a fundamental relationship of fluid mechanics and can be

stated thus:

€

pρ

+v 2

2+ gz = H where p = fluid pressure, ρ = fluid density, v = flow velocity,

g = gravity, z = fluid height above an arbitrary datum and H = a constant hydraulic head.

Because fluid density and gravity are sensibly constant and height z varies little over short

distances, the Bournoulli theorem dictates that pressure in flowing water is inversely related to

the velocity squared. Because flow velocity increases with height above the river bed, relatively

low velocity at the base of a grain resting on the bed and higher velocity at the top of the grain

imposes an upward decreasing vertical pressure gradient that tends to lift the grain and allows it

to move. The tendency increases as the velocity increases (and the velocity gradient near the bed

steepens).


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The upward turbulence (eddying) forces, the third kind of lift force, may be among the most

important of all controls on initiating sediment movement. Turbulence refers to the unsteady

motion in the flow related to the intense shearing and eddy generation at the boundary that

diffuses upward in the water column and is transported downstream by the mean flow.

Turbulence can be resolved into three rectilinear components: a downstream vector, a cross-

channel vector, and a vertical vector. Each is characterized by a range of velocity fluctuations

about the mean flow for each vector. The intensity of turbulence (the size of the velocity

excursions from the mean flow) for the vertical vector likely plays a critical role in dislodging

grains from the bed so that they can move. Overcoming the inertial force holding a grain on the

bed likely always requires a larger flow force that that needed to keep it in motion once it starts

to move. This notion is the same as the familiar procedure in which full thrust is needed to get

an aircraft to lift off the ground on takeoff (overcoming the friction of the undercarriage wheels

rolling on the runway), then allowing the pilot to throttle back somewhat because less thrust is

needed to keep the aircraft airborne. The peak “thrust” needed to dislodge a sediment grain may

be provided by the high instantaneous velocity of turbulence even if the mean velocity is not

sufficient to initiate motion. Our ability to measure these turbulent forces in a field setting is

primitive at best and practical models of turbulence in relation to incipient motion simply do not

exist.

The inertial forces that must be overcome in order for a grain to move are also much more

complex than the White analysis allows. They include (again, the parenthetic italicized notations

refer to their inclusion/non-inclusion in the White analysis in Figure 4.17):

1. Submerged weight of the grain (included)

2. Grain sheltering (dependent in grain-size distribution and on grain shape) (not included)

3. Packing (included for same-size grains)

4. Grain fabric effects (imbrication) (not included)

5. Adhesion forces between small grains (electrochemical forces) (not included)

6. Organic mats (not included)


-97-

The moment of the submerged weight of the grain is the only inertial force included in the White

analysis and is assumed to be overwhelmingly important relative to other factors. Clearly, this is

often not the case.

Grain sheltering cannot occur in a sediment sample of same-sized spheres such as that assumed

in the White analysis but real rivers never possess this degree of sedimentological uniformity.

Grains vary in size and larger grains shelter smaller ones. Depending on the degree of sheltering,

some larger force will be needed to move a grain of given size than would be the case if it simply

rested on the surface of the bed exposed to the full force of the flow. Of course, some of the

grains in the lee of a large clast (sediment grain) might experience the opposite of sheltering as

turbulent eddies shed from edges of the clast impinge on the sediment downstream and markedly

increase shear stress above the average for the bed and cause local scouring there.

Packing of grains also influences the ease with which they can be dislodged by the flow. As we

note above, the case of same-size spherical grains assumed in the White analysis never occurs in

nature. Instead a mixture of grain size means that the grains settle on the bed of a real river so

that the interstitial spaces between the larger self-supporting grains are filled with smaller

diameter grains and these all contribute additional friction holding each grain in place among its

neighbours. The force required to move a single spherical grain in the White analysis is certainly

going to be less that that required to move the same-sized grain in a packed mixed-sized

sediment.

Grain-fabric effects refer to the way in which grains organize themselves into interlocking

arrangements in which the whole structure is stronger than the component grains. Natural grain

shape may tend to be spherical but there are as many non-spherical plates, blades, stems and

wedges that get incorporated into the mix, depending on lithology of the particles. Elongated

particles often overlap each other so that they display imbrication (Figure 4.18).

Little imagination is needed to see that other non-spherical grains, if appropriately wedged next

to a grain, will quite firmly lock it in place so that the fabric has to be broken up before an

individual particle can move.


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Figure 4.18: Elongated particles on a river bed displaying imbricated fabric.

We can imagine a case of average packing where all the grains are packed together in a mutually

self-supporting matrix relative to which packing might be underloose or overloose. An

underloose packing is one in which the state of packing and fabric is so well developed that it

requires forces well above average to move the component particles. Such packing is thought to

develop when declining flow (on the falling limb of the flood hydrograph is prolonged at about

the threshold for movement so that grains have time to mutually jiggle into a structured

arrangement. An overloose fabric is one in which structure and fabric are poorly developed.

Such packing is thought to develop in sediments when transport is suddenly halted by a

precipitous decline in flow so that there is no time for a fabric to develop. These qualitative bed

states find some application in assessing incipient-motion thresholds in certain engineering

problems to be considered later in our discussion.

Adhesion forces refer to surface-tension effects and to the mutual electrochemical attraction that

some very fine particles such as clay exhibit. These forces have no practical relevance for

channels formed in sand or gravel but they are very important in muddy (silt and clay) channels.

Organic mats represent another factor independent of grain size that can affect the force required

to initiate movement of a particle on the bed of a river. Mud and sand-bed rivers commonly

develop a surface mat of algae during prolonged low flows. If a flood flow is unable to break up

the surficial algal mat sediment will not move regardless of the size of the flood.

These lists of impelling and inertial forces are not exhaustive and you may be able to think of

other factors at work here. But these represent the principal reasons why the White analytical

approach in its simplest form (Figure 4.17) does not work very well in practice. But the

Flow direction

Imbricated channel bed

Water surface


-99-

conceptualization of the problem of defining the threshold of sediment motion (impelling force =

inertial force) is sound. We simply need to temper our theory with a little empirical reality!

Shields entrainment function

The most widely used semi-empirical approach to defining the threshold of sediment motion was

proposed in the early 1900s by the German physicist Albert F. Shields. Shields (1936) plotted

the dimensionless shear stress (

€

θ =τ cr

g(ρs − ρ)D) against the dimensionless particle Reynolds

number (

€

Re p =Dδ0

) where

€

δ0 is the thickness of the laminar sublayer (Figure 4.19A).

For our purposes, the particle Reynolds number can be taken as an approximate measure of grain

size (read mm for

€

Re p =Dδ0

in Figure 4.19A). The dimensionless shear stress in the Shields

diagram is commonly termed the Shields stress or the Shields parameter.

The structure of the Shields parameter should be vaguely familiar. The White analysis (Figure

4.17) concluded that

€

τ cr = Kg(ρs − ρ)D, from which it follows that

€

τ crg(ρs − ρ)D

= K (the Shields

parameter is

€

τ crg(ρs − ρ)D

= θ ). You will recall that the constant K in the White analysis includes

just a packing term and the internal angle of friction of the sediment (

€

K =ηtanφ

); we now

realize that K (or Shields stress,

€

θ ) includes all those factors that affect the magnitude of the

impelling and inertial forces and that K and

€

θ will only be constant for certain constrained

circumstances (such as a limited range of grain size).

Several aspects of the Shields diagram are particularly noteworthy:

1. The lowest Shields stress occurs in the sand range (0.06-2.00 mm). Sand is small enough

to have small mass but too large for adhesion forces to come into play. Shields diagram

confirms what every gardener knows: sand is the most easily worked and eroded

sediment.


-100-

Figure 4.19: (A) Critical Shields stress as a function of particle Reynolds number; (B) Hjulstrom’s critical entrainment velocity versus grain size; (C) Sediment motion states in relation to the bed shear stress/grain size domain; (D) Critical Shields stress versus grain-size distribution characteristics.

2. Silt/clay, in spite of the smaller size, requires a higher shear stress for motion than sand.

Here adhesion forces become overwhelmingly large and bind the sediment together into a

mass that is very resistant to erosion.


-101-

3. The Shields parameter for gravel is constant at 0.06, implying that Shields stress here

becomes a simple function of grain size. This is a quite remarkable finding and allows

us, as we will see below, to derive a simple relationship between the size of gravel and

the shear stress required to move it.

4. From engineering studies, particularly here in British Columbia, we now know that the

Shields parameter applies well to natural gravel-bed rivers, the particular Shields stress

value depending on the state of the bed packing/fabric: normal (0.06), underloose (>0.06)

or overlooseloose (<0.06). Some recent field data suggest that a more meaningful

average is lower (0.05) than Shields original formulation although data from real rivers to

refine this modification are very limited.

Figure 4.19B illustrates another approach to defining the critical flow conditions for initiating

sediment movement, this time based on mean flow-velocity (more artistic renderings of this

graph are available at http://en.wikipedia.org/wiki/Hjulstrom_curve). This graph is known as the

Hjulstrom curve after the Swedish geoscientist Filip Hjulstrom who proposed it in 1935, a year

before the Shields diagram was published. It has much in common with the Shields diagram:

lowest velocity for eroding sediment is in the sand range and higher velocity is required to erode

silt/clay sediments and materials in the gravel range and the size of gravel eroded is a linear

function of velocity. Furthermore, the velocity required to erode sediment is higher than that

required to keep the same-size material in motion. All rather familiar notions now. Right?

The Shields diagram rather than Hjulstrom diagram has become the preferred basis for predicting

eroding-flow conditions. Reasons for this relate in part to the appeal of the dimensionless

elegance of the Shields criterion and to the fact that shear stress is more easily and reliably

estimated than is velocity. No doubt scientific fashion and bandwagon effects also play a part

here.

It is important to remember when viewing the Shields and Hjulstrom diagrams, however, that

their trend lines have the effect of pushing to the less accessible parts of our conciousness the

reality that these are semi-empirical constructs based on data that are untidy at best. We should

closely examine Figure 4.19 C; it will keep us honest! The change for mixed-size sediment from


-102-

stability to motion is not as abrupt as a single curve implies; it is a transitional state that reflects

the natural variability in all those impelling and inertial forces.

Given the above caveat we can say something about the mean state of incipient motion for that

part of the Shields diagram where the curve becomes horizontal (in the gravel range). Here we

can say that:

€

τ crg(ρs − ρ)D

= 0.06 (4.4)

If we make some reasonable substitutions for the constants in equation (4.3), gravitational

acceleration g = 9.81 ms-2 and the densities are respectively 2650 kgm-3 (for quartz, the most

commonly occurring mineral in nature) and 1000 kgm-3 (for water), the Shields stress for gravel

becomes

€

τ cr9.81(2650 −1000)D

= 0.06

and reduces to

€

τ cr ≅ 970D (4.5)

Equation (4.4) can be adjusted as the bed state of the channel dictates. Equation (4.4)

incorporates a Shields parameter equal to 0.06 and we might consider that, in light of subsequent

observations, to be normal. But we have noted earlier that some beds are overloose with very

little surface structure while others are highly structured in a way that resists erosion and so are

underloose. The table below gives the range of Shields equations that we might consider for

different bed states.

Bed state Shields parameter, θ Shields equation

Overloose (no structure) 0.05

€

τ cr ≅ 810D (4.5A)

Normal (weak structure) 0.06

€

τ cr ≅ 970D (4.5B)

Underloose (strong structure) 0.07

€

τ cr ≅1130D (4.5C)

Underloose (very strong structure) 0.08

€

τ cr ≅1290D (4.5D)

Table 4.1: Shields parameter and Shields equation for various bed states.


-103-

5. Applications of Critical Threshold Conditions for Sediment Motion There are numerous applications of the Shields stress and these are perhaps best illustrated by

working through a couple of examples.

Incipient Motion

Problem 4.1: What shear stress will move a 2.0 cm particle resting on the bed of a river?

Solution 4.1: From equation (4.5B) we can say that particle motion will occur if shear stress exceeds

€

τ cr = 970D = (970)(0.02) =19.4Nm−2 Problem 4.2: A channel with an imbricated bed of 1.0 cm gravel has a slope of 0.005. What depth of flow is required to initiate particle motion on the bed?

Solution 4.2: From Table 4.1 we might guess that imbrication represents the upper limit of bed structure and choose equation (4.5D) to yield an appropriate Shields stress:

€

τ cr =1290D = (1290)(0.01) =12.9Nm−2 From equation (3.3) for shear stress we can say that, in this case: τo = ρgds = (1000)(9.81)(d)(0.005) = 49.05d Setting the actual shear stress (τo) equal to the critical shear stress (τcr) yields: 40.05d = 12.9 and so d = 0.30 m That is, for these channel conditions sediment on the bed will begin to move once the flow depth exceeds 0.30 m. Problem 4.3: A 20.0 m-long laboratory flume is to discharge water over a coarse sand bed at a depth of 0.50 m and the flume operator wants to know to what degree the flume has to be tilted in order to ensure that the sediment on the bed is mobile. What is the appropriate slope and height by which the end of the flume must be lowered?

Solution 4.3: This is a variant of Problem 4.2. Again we might guess that the state of the bed is as unstructured as it gets (markedly overloose) and choose equation (4.5A) to estimate the Shields stress for coarse sand (D = 2.00 mm):

€

τ cr = 810D = (810)(0.002) =1.62Nm−2 Again, from equation (3.3) for shear stress we can say that, in this case: τo = ρgds = (1000)(9.81)(0.5)(s)= 1.62 Nm-2 Solving for the slope (s) yields s = 0.00033 Thus the end of the 20.0 m-long flume would have to be lowered by no less than (0.00033)(20.0) = 0.0066 m or about 7 mm.


-104-

Bedload transport formulae

We now have sufficient theory to return to the business of predicting bed-load transport in a river

channel. You will recall that equation (4.2) is the Meyer-Peter and Muller bedload transport

equation but its use requires the critical shear stress for motion to be specified. Here is how it

works:

Problem 4.4: A 50.0 m-wide channel carries a flow 2.50 m deep at a slope of 0.0015 over a gravel bed in which D50 = 3.2 cm. If these condition are sustained for 12 days, how much bed load will the river have transported at the end of the period?

Solution 4.4: We know that the two-dimensional transport rate is given by equation (4.2) as

€

gb = 0.253(τ 0 − τ cr)3 / 2

We need to determine τo thus: τo = (1000)(9.81)(2.50)(0.0015) = 36.79 Nm-2 τcr is estimated by the Shields stress for a normal bed:

€

τ cr = 970D = (970)(0.032) = 31.04Nm−2 So, substituting in equation (4.2):

€

gb = 0.253(36.79 − 31.04)3 / 2 = 0.253(5.75)3/2 = 3.49 kgs-2 m-1

The three-dimensional sediment transport rate Qb = gbw = (3.49)(50.0) = 174.5 kgs-1. Over the 12-day period the total amount of bed load transported is (174.5)(60)(60)(24)(12) = 1.81 x 108 kg ≅ 181 000 tonnes. Problem 4.5: What is the rate of two-dimensional bed-load transport if a 1.5 m-deep flow has a water-surface slope = 0.0001 and flows over a gravel bed in which D50 = 5 cm?

Solution 4.5: We know that the two-dimensional transport rate is given by equation (4.2) as

€

gb = 0.253(τ 0 − τ cr)3 / 2

We need to determine τo thus: τo = (1000)(9.81)(1.50)(0.0001) = 1.47 Nm-2 τcr is estimated by the Shields stress for a normal bed:

€

τ cr = 970D = (970)(0.05) = 48.5Nm−2 So, substituting in equation (4.2):

€

gb = 0.253(1.47 − 48.5)3 / 2 = 0.253(-47.03)3/2 = ??? kgs-2 m-1

The mathematics here fails and is telling us that shear stress is not high enough to initiate motion (τo - τcr is negative) so gb = 0; no bed-load transport is possible.

The threshold or equilibrium channel form

We are now in a position to revisit the business of predicting the basic dimensions (form ratio,

w/d) of an equilibrium channel. In an equilibrium channel w/d will be a function of boundary-

material strength in terms of

(a) bank angle and height and

(b) the maximum allowable (threshold) flow depth.


-105-

If we ignore the question of bank strength by simply assuming that the banks are strong enough

to stand vertically, we now have enough theory to solve this problem for an equilibrium

rectangular channel cross-section. Remember that an equilibrium channel is one in which the

cross-section is adjusted so that the boundary particles are poised at a threshold between erosion

and deposition. Once again, an example provides the most direct route to understanding how

theory works in practice:

Problem 4.6: Determine the form ratio of a rectangular channel that will carry a bankfull discharge of 1000 m3s-1 if the bed consists of gravel (D50 = 2.5 cm) and the downstream slope of the bed (and water surface) is 0.001.

Solution 4.6: The solution to this problem comes in four steps:

1. Determine the critical shear stress required to move the boundary material on the bed. Shields criterion for a normal bed yields: τ0 =970D50 = (970)(0.025) = 24.25 Nm-2

2. Determine the maximum depth of flow allowable at threshold conditions (τ0 = τcr):

τ0 = τcr

ρgds = (1000)(9.81)(d)(0.001): 9.81d = 24.25 dmax = 2.47 m 3. Determine the mean flow-velocity using the Manning-Strickler relation (equation 3.10):

€

v =d2 / 3s1/ 2

n=

d2 / 3s1/ 2

0.0478D501/ 6 =

(2.47)2 / 3(0.001)1/ 2

(0.0478)(0.025)1/ 6= 2.25m

4. From the continuity relation determine the channel width and the form ratio of the threshold channel: Q = 1000 = vdmaxw = (2.25)(2.47)w

€

w =1000

(2.25)(2.47)=180m The form ratio of the channel (w/d) is 180/2.47 = 73

5. Some Further Reading

I suggest that you limit your further reading (beyond these notes) of more advanced material to

the following sections of Knighton (1998): Knighton, D. 1998: Fluvial Forms and Processes: A New Perspective, Hodder Arnold Publication: pages 80-95; 107-136.

Other cited references: Bagnold, R.A., 1980: An empirical correlation of bedload transport rates in flumes and natural rivers. Proceedings of the Royal Society 372A, 453-73.


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Colby, B.R. 1963: Fluvial sediments – a summary of source, transportation, deposition, and measurement of sediment discherge. United States Geological Survey Bulletin 1181A. Degens, E.T., Kempe, S. and Richey, J.E. 1991: Summary: Biogeochemistry of major world rivers. In Degens, E.T., Kempe, S. and Tichey, J.E. (eds), Biogeochemistry of major world rivers, SCOPE 42. Chichester: Wiley, 323-47. Ham, D.G. and Church, M., 2000: Bed material transport estimated from channel morphodynamics: Chilliwack River, British Columbia. Earth Surface Processes and Landforms, 24 (10) 1123-1142. Hickin, E.J. 1989: Contemporary Squamish River sediment flux to Howe Sound, British Columbia. Canadian Journal of Earth Sciences, 26: 1953-1963 Hjulstrom, F. 1935: Studies of the morphological activity of rivers as illustrated by the River Fyris. Bulletin of the Geological Institute University of Uppsala, 25, 221-527. Meybeck, M. 1976: Total mineral dissolved transport by world major rivers. Hydrological Sciences Bulletin, International Association of Scientific Hydrology, 21, 265-84. Milliman, J.D. and Meade, R.H. 1983: World-wide delivery of river sediment to the oceans. Journal of Geology, 91, 1-21. Nordin, C.F. and Beverage, J.P. 1965: Sediment transport in the Rio Grande, New Mexico. U.S. Geological Survey Professional Paper 462F, 35. Pelpola, C. and Hickin, E.J. 2004: Long-term bed-load transport rate based on air photo and ground-penetrating radar surveys of fan-delta growth, Coast Mountains, British Columbia. Geomorphology, 57,169-181. Shields, A. 1936: Anwendung der Ahnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Mitteilung der preussischen Versuchsanstalt fur Wasserbau und Schiffbau, 26, Berlin. Walling, D.E. and Webb, B.W. 1987: Suspended load in gravel-bed rivers: UK experience: In Thorne, C.R., Bathurst, J.C. and Hey, R.D. (eds), Sediment transport in gravel-bed rivers. Chichester: Wiley, 691-723. Williams, G.P. 1989: Sediment concentration versus water discharge during single hydrologic events in rivers. Journal of Hydrology, 111, 89-106

6. What’s Next?

Now that we have mastered some basic concepts of open-channel fluid mechanics and sediment

transport theory, we are now ready to consider various aspects of channel morphology and river

planforms and the processes that seem to govern them. This is the morphology in

geomorphology and our investigations into these natural complex three-dimensional fluvial

forms will become more qualitative and descriptive as we encounter increasing complexity for

which a quantitative theoretical basis has yet to be developed. One of the important descriptive

tools used by geomorphologists to characterize channel morphology is known as hydraulic

geometry and this topic leads us into Chapter 5.


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Documents

Chapter 4 Sediment Transport - SFU.cahickin/RIVERS/Rivers4(Sediment transport).pdf · sediment-transport modes: suspended-sediment load and bed-material load. Suspended-sediment load