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Chapter 4
Sediment Transport
1. Introduction: Some Definitions
2. Modes of Sediment TransportDissolved LoadSuspended-Sediment Load
Wash loadIntermittently-suspended or saltation loadSuspended-sediment rating curvesTypical suspended-sediment loads in BC riversRelation of sediment concentration to sediment load/discharge
Bed Load (Traction Load)Bed load equations
3. The Theory of Sediment EntrainmentImpelling forcesInertial forces
The Shields entrainment function4. Applications of Critical Threshold Conditions for Sediment MotionBedload transport equationsThe threshold (equilibrium) channel
5. Some further reading6. Whats next?
1. Introduction: Some Definitions
Not all channels are formed in sediment and not all rivers transport sediment. Some have beencarved 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 earths
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
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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 iscompetent 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 sedimentsupply) 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 load3. 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 worlds
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 Drainagearea,
10 3 km 2
MeanDischarge
10 3m 3s-1
Meansuspended
Sediment load10 6 t y -1
Meandissolved
load10 6 t y -1
Percentage oftotal loadcarried insolution
AFRICACongo 3500 41.1 48 37 44Niger 1200 4.9 25 14 36Nile 3000 1.2 2 12 86
Orange 1000 0.4 0.7 1.6 70Zambezi 540 2.4 20 25 56
ASIAGanges/Brahmaputra 1480 30.8 1670 151 8
Huang Ho 752 1.1 900 22 2Indus 1170 7.5 100 79 44
Irrawaddy 430 13.6 265 91 26Lena 2440 16.0 12 56 82
Mekong 795 21.1 160 59 27Ob 2990 13.7 13 46 78
Yenisei 2500 17.6 15 60 80
AUSTRALIAMurray-Darling 1060 0.7 30 8.2 21
EUROPEDanube 817 6.5 83 53 39Dnieper 527 1.7 2.1 11 84Rhone 99 1.9 40 56 58Volga 1459 7.7 27 54 67
NORTH AMERICAColumbia 670 5.8 14 21 60Mackenzie 1810 7.9 100 44 31Mississippi 3267 18.4 210 142 40
St Lawrence 1150 13.1 5.1 70 93Yukon 840 6.7 60 34 36
SOUTH AMERICAAmazon 6150 200 900 290 24
Magdalena 260 6.8 220 20 8Orinoco 990 36 150 31 17Parana 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, Q s(d)(kg/s), depends on the dissolved load concentration C o
(kg/m 3), and the stream discharge, Q (m 3/s): Q s(d) = C oQ
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 upwardflux 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
0.01Grain diameter, cm
F a l
l v e
l o c i t y , c m s -
1
0 G r av
l Sil t
1
100
10
S and Silt Sand Gravel0.001 0.01 0.1 1.0
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Figure 4.3: A cable-mounted DH48 suspended-sedimentsampler attached to the underside of a heavy lead fish forcollecting 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) orit 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/m 3. So 1000 mg/L = 1000 gm/m 3 = 1 kg/m 3.
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
H e i g h
t a b o v e
t h e
b e d , m
Concentration, mg/L
Finesilt
Finesand
Coarsesand
A
Grain size, D 50, cm
H e i g h
t a b o v e
t h e
b e d , m Fine
siltFinesand
Coarsesand
B
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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 ( !o = " gds) or stream power ( # = " gQs) 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 Q s and Q is sufficiently strong in many
cases for a rating curve to work as a summary measure. Implicit in the Q s/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(# ) $ f( ! o) $ f(turbulent flux);
(b) Qs $ f (hydraulics); that is, Q s 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, NewMexico (after Nordin & Beverage,1965, in Knighton, 1998).
B: River Bolin, UK (from Knighton,1998).
C: Squamish River, BC (fromHickin, 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 theamount of suspended-sediment being transported in a river is sediment-load hysteresis in the
suspendedsediment 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 inwhich concentrations for given discharges are higher on the rising limb of thehydrograph than they are on the falling limb.
Discharge
Concentration-discharge valueson the falling limb of the hydrograph
Concentration-discharge valueson the risin l imb of the h dro ra h
C o n c e n
t r a
t i o 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 Figure4.8. 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 ofsediment 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 ancientlake 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 = C oQ (with appropriate units) (4.1)
For example, on Lower Fraser River at Mission the average annual discharge is 3500 m 3s-1 and
the average annual suspended-sediment concentration is 186 mgL -1 (=186 gm/m 3 = 0.186kg/m 3).
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 10 10 kg/year = 2.0 x 10 7 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 1/10 th 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/m 3 or 400 000 gm/m 3 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 10 6 tonnes/year
and average annual discharge is Q = 1.1 x 10 3 m3/s. Suspended-sediment concentration can be
calculated, thus:
load disch arg e
=
900 x10 6 ty " 1
1.1 x10 3 m 3s" 1
Co =900 x10 6 x10 3 x10 3
1.1 x10 3 x60 x60 x24 x365=
9.0 x1014
3.5 x1010= 2.6 x10 4 gm / m 3
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 flatweight attached and the fins keep the instrument oriented into the flow. (B) When the bed-load sampleris appropriately oriented in the flow, bed-load material enters the sampler through the inlet and thedivergent 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. Afteran 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 ofbed-load sediment from a channel and its return to the sediment transportsystem 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 featuressuch as prograding dunes on the bed of a river as a basis for determiningvolumetric 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 andsimply 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 thevertical 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 mapindicating the delta boundary over time (1947-1999) is the basis for determining the volumetricsediment-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 equationsIn 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 (q sb) is related to either
(i) excess shear stress " 0 # " cr ( ) qsb = " X # 0 # 0 $ # cr( ) [Du Boys type]
(ii) excess discharge/unit width q " q cr( ) qsb = " " X sk q # qcr( ) [Schoklitsch type]
(iii) excess stream power/unit width " # " cr ( ) qsb " # $ # cr( )3 / 2
d $ 2 / 3 D$ 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( " 0 # " cr )3 / 2 (4.2)
where g b 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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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
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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 meanvelocity 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: p
" +
v2
2+ gz = H where p = fluid pressure, " = 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, relativelylow 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 wheelsrolling 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 )
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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-sizedsediment.
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 todevelop 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
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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 (
" =# cr
g($ s % $ ) D) against the dimensionless particle Reynolds
number (
Re p
=
D"
0
) where " 0 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 = D"
0
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 " cr = Kg(# s $ # ) D , from which it follows that" cr
g(# s $ # ) D= K (the Shields
parameter is" cr
g(# s $ # ) D=
% ). 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 ="
tan # ); we now
realize that K (or Shields stress, " ) includes all those factors that affect the magnitude of the
impelling and inertial forces and that K and " 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.
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Figure 4.19: (A) Critical Shields stress as a function of particle Reynoldsnumber; (B) Hjulstroms critical entrainment velocity versus grain size;(C) Sediment motion states in relation to the bed shear stress/grain sizedomain; (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.
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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 (
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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:
" cr
g(# s $ # ) 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 " cr9.81(2650 # 1000) D
= 0.06
and reduces to " cr # 970 D (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 areunderloose. The table below gives the range of Shields equations that we might consider for
different bed states.
Bed state Shields parameter, Shields equation
Overloose (no structure) 0.05 " cr # 810 D (4.5A)
Normal (weak structure) 0.06 " cr # 970 D (4.5B)
Underloose (strong structure) 0.07 " cr # 1130 D (4.5C)
Underloose (very strong structure) 0.08 " cr # 1290 D (4.5D)
Table 4.1: Shields parameter and Shields equation for various bed states.
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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 bankfulldischarge of 1000 m 3s-1 if the bed consists of gravel (D 50 = 2.5 cm) and the downstream slope ofthe 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. Shieldscriterion for a normal bed yields: !0 =970D 50 = (970)(0.025) = 24.25 Nm -2
2. Determine the maximum depth of flow allowable at threshold conditions ( ! 0 = ! cr ):
! 0 = ! cr" gds = (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 =d 2 / 3 s1/ 2
n=
d 2 / 3 s1/ 2
0.0478 D 501/ 6
=
(2.47) 2 / 3 (0.001) 1/ 2
(0.0478)(0.025) 1/ 6= 2.25 m
4. From the continuity relation determine the channel width and the form ratio of the thresholdchannel: Q = 1000 = vd max w = (2.25)(2.47)w
w =1000
(2.25)(2.47)= 180 m 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. Proceedingsof the Royal Society 372A, 453-73.
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Colby, B.R. 1963: Fluvial sediments a summary of source, transportation, deposition, and measurement ofsediment 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: ChilliwackRiver, 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 theGeological Institute University of Uppsala , 25, 221-527.
Meybeck, M. 1976: Total mineral dissolved transport by world major rivers. Hydrological Sciences Bulletin, International Association of Scientif ic 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 SurveyProfessional Paper 462F, 35.
Pelpola, C. and Hickin, E.J. 2004: Long-term bed-load transport rate based on air photo and ground-penetratingradar 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. Whats 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
geo morphology and our investigations into these natural complex three-dimensional fluvial
forms will become more qualitative and descriptive as we encounter increasing complexity forwhich 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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