sedimentation and erosion

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6.1 INTRODUCTION

Since the beginning of mankind, sedimentation processes have affected water supplies,irrigation, agricultural practices, flood control, river migration, hydroelectric projects,navigation, fisheries, and aquatic habitat. In the last few years, sediment also has beenfound to play an important role in the transport and fate of pollutants; thus, sedimentation

control has become an important issue in water quality management. Toxic chemicals canbecome attached to, or adsorbed by, sediment particles and then be transported to anddeposited in other areas. By studying the quantity, quality, and characteristics of sedimentin rivers and streams, scientists and engineers can determine the sources of the sedimentand evaluate the impact of pollutants on the aquatic environment. In the United States,sedimentation control is a multibillion-dollar issue. For example, approximately $500 mil-lion are spent every year to dredge waterways and harbors for navigation purposes. Mostof the dredged sediment is the result of substantial soil erosion in watersheds. Estimatesby the U.S. Department of Agriculture indicate that annual offside costs of sedimentderived from copland erosion are on the order of $2 billion to $6 billion, with an addi-tional $1 billion arising from loss in compared productivity.

The sediment cycle starts with the process of erosion, where by particles or fragmentsare weathered from rock material. Action by water, wind, glaciers, and plant and animalactivities all contribute to the erosion of the earths surface. Fluvial sediment is the termused to describe the case where water is the key agent for erosion. Natural, or geologic,erosion takes place slowly, over centuries or millennia. Erosion that occurs as a result ofhuman activity may take place much faster. It is important to understand the role of eachcause when studying sediment transport.

Any material that can be dislodged is ready to be transported. The transportationprocess is initiated on the land surface when raindrops result in sheet erosion. Rills, gul-lies, streams, and rivers then act as conduits for the movement of sediment. The greater

the discharge, or rate of flow, the higher the capacity for sediment transport.The final process in the cycle is deposition. When there is not enough energy to transportthe sediment, it comes to rest. Sinks, or depositional areas, can be visible as newly deposit-ed material on a floodplain, on bars and islands in a channel, and on deltas. Considerabledeposition occurs that may not be apparent, as on lake and river beds. A knowledge of sed-iment dynamics is an integral part of understanding the aquatic ecosystem.

This chapter presents fundamental aspects of the erosion, transport, and deposition ofsediment in the environment. The emphasis is on the hydraulics of bedload and suspend-

CHAPTER 6

SEDIMENTATION ANDEROSION HYDRAULICS

Marcelo H. GarcaDepartment of Civil and Environmental Engineering

University of Illinois at Urbana-Champaign

Urbana, IL

6.1

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Source: HYDRAULIC DESIGN HANDBOOK


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ed load transport in rivers, with the goal of establishing the background needed for sedi-mentation engineering. Because of their relevance, the hydraulics of both reservoir sedi-mentation and turbidity currents also is considered. Emphasis is placed on noncohesive

sediment transport, where the material involved can be silt, sand, or gravel. When possi-ble, the behavior of both uniform-sized material and sediment mixtures is analyzed.Although such topics as cohesive sediment transport, debris and mud flows, alluvial fans,river meandering, and sediment transport by wave action are not discussed here, it ishoped that the material covered in this chapter will provide a firm foundation to tackleproblems in those.

For more information on sediment transport and sedimentation engineering, readersare referred to Allen (1985), Ashworth et al. (1996), Bogardi (1974), Bouvard (1992),Carling and Dawson (1996), Chang (1988), Coussot (1997), Fredse and Deigaard(1992), Garde and Ranga Raju (1985), Graf (1971), Jansen et al. (1979), Julien (1992),

Mehta (1986), Mehta et al. (1989a, 1989b), Morris and Fan (1998), Nakato and Ettema(1996), National Research Council (1996), Nielsen (1992), National Research council(1996), Parker and Ikeda (1989), Raudkivi (1990, 1993), Renard et al. (1997), Sieben(1997), Simons and Senturk (1992), Sloff (1997), van Rijn (1997), Yalin (1972, 1992),Yang (1996), and Wan and Wang (1994).

6.2 HYDRAULICS FOR SEDIMENT TRANSPORT

6.2.1 Flow Velocity Distribution

Consider a steady, turbulent, uniform, open-channel flow having a mean depthHand amean flow velocity U(Fig. 6.1). The channel is extremely wide and its bottom has a meanslope S and a surface roughness that can be characterized by an effective height ks(Brownlie, 1981b). When the bottom of the channel is covered with sediment having amean size or diameter D, the roughness height ks will be proportional to that diameter.Because of the weight of the water, the flow exerts on the bottom a tangential force perunit bed area known as the bed shear stress b, which can be expressed as:

b

gHS (6.1)

where is the water density and g is the gravitational acceleration. With the help of theboundary shear stress, it is possible to define the shear velocity u* as

6.2 Chapter Six

FIGURE 6.1 Definition diagram for open-channel flow over an erodible bed.



SEDIMENTATION AND EROSION HYDRAULICS


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u* b/ (6.2)

The shear velocity, and thus the boundary shear stress, provides a direct measure of the

intensity of flow and its ability to entrain and transport sediment particles. The size of thesediment particles on the bottom determines the surface roughness, which in turn affectsthe flow velocity distribution and its sediment transport capacity. Since flow resistanceand sediment transport rates are interrelated, the ability to determine the role played bythe bottom roughness is important.

Research has shown (Schlichting, 1979) that the flow velocity distribution is well rep-resented by:

uu

*

1

lnz const. (6.3)

where u is the time-averaged flow velocity at distancez above the bed and is known asVon Karmans constant and is equal to 0.4. For obvious reasons, the above law is knownas the logarithmic law of the wall. It strictly applies only in a thin layer near the bed. It isempirically found to apply as a reasonable approximation throughout most of the flow inmany rivers.

If the bottom boundary is sufficiently smooth (a condition rarely satisfied in rivers),turbulence will be drastically suppressed in an extremely thin layer near the bed. In thisregion, a linear velocity profile will hold:

uu

*

uv*z (6.4)

where is the kinematic viscosity of water. This law merges with the logarithmic law nearz v, where

v 11.6 u

* (6.5)

denotes the height of the viscous sublayer. In the logarithmic region, the constant of inte-gration introduced above has been evaluated from data to yield

uu

*

1ln

u

*z

5.5 (6.6)

Most boundaries in river flow are rough. Let ks denote an effective roughness height.Ifks/v 1, then no viscous sublayer will exist. The corresponding logarithmic velocityprofile is given by

uu

*

1 ln

kz

s

8.5 1 ln

30

kz

s

(6.7)

As noted above, this relation often holds as a first approximation throughout the flow in ariver. It is by no means exact.

The conditions ks/ 1 for rough turbulent flow and ks/ 1 for smooth turbulentflow can be rewritten to indicate that u*ks/ should be much larger than 11.6 for turbulent

rough flow and much smaller than 11.6 for turbulent smooth flow. A composite form thatrepresents both ranges, as well as the transitional range between them, can be written as

uu

*

1 ln

kz

s

Bs (6.8)

withBs as a function ofRe* u*ks/, which can be estimated with

Sedimentation and Erosion Hydraulics 6.3





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Bs 8.5 [2.5 ln(Re*) 3]e0.127[ln(Re*)]2 (6.9)

as proposed by Yalin (1992).

6.2.2 Relations for Channel Resistance

Most river flows are indeed hydraulically rough. Equation (6.7) can be used to obtain anapproximate expression for depth-averaged velocity U that is reasonably accurate formany flows. Using the following integral:

U H1

H

0

udz (6.10)

but changing the lower limit slightly to avoid the fact that the logarithmic law is singularatz 0, the following result is obtained:

uU

*

H1

H

ks

1 ln

kz

s

8.5

dz (6.11)

or, performing the integration

uU

*

1 ln

Hks

6 1 ln

11

Hks

(6.12)

This relation is known as Keulegan's resistance relation for rough flow.An approximation to Keulegan's relation is theManning-Strickler power form

uU

*

8

Hks

1/6(6.13)

Between Eqs. (6.2) and (6.12), a resistance relation can be found for bed shear stress:

b CfU2 (6.14)

where the friction coefficient Cf is given by

Cf1 ln

11

Hks

2(6.15)

If Eq. (6.13) is used instead of Eq. (6.12), the friction coefficient takes the form

Cf8

Hks

1/6

2

(6.16)

It is useful to show the relationship between the friction coefficient Cf and the rough-ness parameters in open-channel flow relations commonly used in practice. Between Eqs.(6.1) and (6.14), the following form ofChezy's law can be derived:

U CcH1/2S1/2 (6.17)

where the Chezy coefficient Cc is given by the relation

6.4 Chapter Six





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Cc

C

g

f

1/2(6.18)

A specific evaluation of Chezy's coefficient can be obtained by substituting Eq. (6.15) intoEq. (6.18). It is seen that the coefficient is not constant but varies as the logarithm ofH/ks.A logarithmic dependence is typically a weak one, partially justifying the commonassumption that Chezy's coefficient in Eq. (6.17) is a constant. Substituting Eq. (6.16) intoEqs. (6.17) and (6.18),Manning's law is obtained:

U= 1nH2/3S1/2 (6.19)

where Manning's n is given by

n =

8

k

g

s1

1

/6

/2(6.20)

The above relation is often called theManning-Stricklerform of Manning's n.

6.2.3 Fixed-Bed and Movable-Bed Roughness

It is clear that to use the above relations for channel flow resistance, a criterion for evalu-ating ks is necessary. Nikuradse (1933) proposed the following criterion: Suppose a roughsurface is subjected to a flow. The equivalent roughness ks of that surface is equal to thediameter of sand grains that, when glued uniformly to a completely smooth wall and then

subjected to the same external conditions, yields the same velocity profile. Nikuradse usedsand glued to the inside of pipes to conduct this evaluation. Extending Nikuradse's con-cept of equivalent grain roughness to the case of rivers and streams, ks can be assumed tobe proportional to a representative sediment sizeDx,

ks = sDx (6.21)

Suggested values ofs, which have appeared in the literature, are listed in Table 6.1 (Yen,1992). Different sizes of sediment have been suggested forDxin Eq. (6.21). Statistically,D50(the grain size for which 50% of the bed material is finer) is most readily available andmeaningful. Physically, a representative size larger thanD50 is more meaningful to estimate


TABLE 6.1 Ratio of Nikuradse Equivalent Roughness Size and Sediment Size for Rivers.

Investigator Measure of Sediment Size, Dx s = ks/Dx

Ackers and White (1973) D35 1.23

Strickler (1923) D50 3.3

Keulegan (1938) D50 1

Meyer-Peter and Muller (1948) D50 1

Thompson and Campbell (1979) D50 2.0

Hammond et al. (1984) D50 6.6

Einstein and Barbarossa (1952) D65 1

Irmay (1949) D65 1.5

Engelund and Hansen (1967) D65 2.0

Lane and Carlson (1953) D75 3.2





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flow resistance because of the dominant effect by large sediment particles.In flow over a geometrically smooth, fixed boundary, the apparent roughness of the

bed ks can be computed using Nikuradse's approach. However, once the transport of bedmaterial has been instigated, the characteristic grain diameter and the thickness of the vis-

cous sublayer no longer provide the relevant length scales. The characteristic length scalein this situation is the thickness of the layer where the sediment particles are being trans-ported by the flow, usually referred to as the bedload layer.

Once the bed shear stress b exceeds the critical shear stress for particle motion c, theapparent bed roughness ka can be estimated as follows (Smith and McLean, 1997):

ka 0((

s

b

)c)

g ks (6.22)

where 0 26.3, ks is Nikuradse's fixed-bed roughness, and s is the bed sediment densi-ty. This approach is particularly suitable for sand bed rivers.

Under intense sediment transport conditions, bedforms, such as dunes, can develop. Inthis situation, the apparent roughness also will be influenced by the form drag caused bythe presence of bedforms. Nikuradse's approach is valid only for grain-induced roughness.Methods for flow resistance in the presence of both bedforms and grain roughness are pre-sented later.

6.3 SEDIMENT PROPERTIES

6.3.1 Rock Types

The solid phase of the problem embodied in sediment transport can be any granular sub-stance. In engineering applications, however, the granular substance in question typicallyconsists of fragments ultimately derived from rockshence the name sedimenttransport.The properties of these rock-derived fragments, taken singly or in groups of many parti-cles, all play a role in determining the transportability of the grains under fluid action. The

6.6 Chapter Six

TABLE 6.1. (Continued)

Investigator Measure of Sediment Size, Dx s = ks/Dx

Gladki (1979) D80 2.5

Leopold et al. (1964) D84 3.9

Limerinos (1970) D84 2.8

Mahmood (1971) D84 5.1

Hey (1979), Bray (1979) D84 3.5

Ikeda (1983) D84 1.5

Colosimo et al. (1986) D84 3 6Whiting and Dietrich (1990) D84 2.95

Simons and Richardson (1966) D85 1

Kamphuis (1974) D90 2.0

van Rijn (1982) D90 3.0

SOURCE: Adapted from Yen (1992)





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important properties of groups of particles include porosity and size distribution. The mostcommon rock type one is likely to encounter in the river or coastal environment is quartz.Quartz is a highly resistant rock and can travel long distances or remain in place for long

periods without losing its integrity. Another highly resistant rock type that is often foundtogether with quartz is feldspar. Other common rock types include limestone, basalt, gran-ite, and more esoteric types, such as magnetite. Limestone is not a resistant rock; it tendsto abrade to silt rather easily. Silt-sized limestone particles are susceptible to solutionunless the water is buffered sufficiently. As a result, limestone typically is not a majorcomponent of sediments at locations distant from its source. On the other hand, it oftencan be the dominant rock type in mountain environments.

Basaltic rocks tend to be heavier than most rocks composing the earths crust and typ-ically are brought to the surface by volcanic activity. Basaltic gravels are relatively com-mon in rivers that derive their sediment supply from areas subjected to vulcanism in recent

geologic history. Basaltic sands are much less common. Regions of weathered graniteoften provide copious supplies of sediment. Although the particles produced by weather-ing are often in the granule size, they often break down quickly to sand size.

Sediments in the fluvial or coastal environment in the size range of silt, or coarser, aregenerally produced by mechanical means, including fracture or abrasion. The clay miner-als, on the other hand, are produced by chemical action. As a result, they are fundamen-tally different from other sediments in many ways. Their ability to absorb water meansthat the porosity of clay deposits can vary greatly over time. Clays also display cohesivi-ty, which renders them more resistant to erosion.

6.3.2 Specific Gravity

The specific gravity of sediment is defined as the ratio between the sediment density sand the density of water . Some typical specific gravities for various natural and artifi-cial sediments are listed in Table 6.2.

6.3.3 Size

Herein, the notation D is used to denote sediment size, the typical units of which are

millimeters (mm) for sand and coarser material or microns () for clay and silt.Another standard way of classifying grain sizes is the sedimentological scale,according to which

TABLE 6.2 Specific Gravity of Rock Typesand Artificial Material

Rock type or Specific gravitymaterial s/

quartz 2.60 2.70limestone 2.60 2.80basalt 2.70 2.90magnetite 3.20 3.50plastic 1.00 1.50coal 1.30 1.50walnut shells 1.30 1.40





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6.8 Chapter Six

D 2 (6.23)

Taking the logarithm of both sides, it is seen that

log2(D) 11nn((D2)

) (6.24)

Note that the size 0 corresponds toD 1 mm. The usefulness of the scale willbecome apparent upon a consideration of grain size distributions. The minus sign has beeninserted in Eq. (6.24) simply as a matter of convenience to sedimentologists, who are moreaccustomed to working with material finer than 1 mm than they are with coarser materi-al. The reader should always recall that larger implies finer material. The scale pro-vides a simple way of classifying grain sizes into the following size ranges in descendingorder: boulders, cobbles, gravel, sand, silt, and clay. (Table 6.3).

Note that the definition of clay according to size (D 2) does not always correspondto the definition of clay according to mineral. That is, some clay-mineral particles can becoarser than this limit, and some silt-sized particles produced by grinding can be finer thanthat. In general, however, the effect of viscosity makes it difficult to grind up particles inwater to sizes finer than 2.

In practical terms, there are several ways to determine grain size. The most popularway for grains ranging from 4 to 4 (0.0625 to 16 mm) is with the use ofsieves. Each sieve has a square mesh, the gap size of which corresponds to the diameterof the largest sphere that would fit through it. Thus, the grain sizeD so measured corre-sponds exactly to the diameter only in the case of a sphere. In general, the sieve sizeD

corresponds to the smallest sieve gap size through which a given grain can be fitted.For coarser grain sizes, it is customary to approximate the grain as an ellipsoid. Threelengths can be defined. The length along the major (longest) axis is denoted as a, thelength along the intermediate axis is denoted as b, and the length along the minor (small-est) axis is denoted as c. These lengths are typically measured with a caliper. The value bis then equated to grain sizeD.

For grains in the silt and clay sizes, many methods (hydrometer, sedigraph, and soforth) are based on the concept of equivalent fall diameter. That is, the terminal fall veloc-ity vs of a grain in water at a standard temperature is measured. The equivalent fall diam-eterD is the diameter of the sphere having exactly the same fall velocity under the sameconditions. Sediment fall velocity is discussed in more detail below.

A variety of other more recent methods for sizing fine particles rely on blockage oflight beams. The blocked area can be used to determine the diameter of the equivalent cir-cle: i.e., the projection of the equivalent sphere. It can be seen that all the above methodscan be expected to operate consistently as long as grains shape does not deviate too great-ly from a sphere. In general, this turns out to be the case. There are some important excep-tions, however. At the fine end of the spectrum, mica particles tend to be platelike; thesame is true of shale grains at the coarser end. Comparison with a sphere is not necessar-ily an especially useful way to characterize grain size for such materials.

6.3.4 Size Distribution

Any sample of sediment normally contains a range of sizes. An appropriate way to char-acterize these samples is by grain size distribution. Consider a large bulk sample of sedi-ment of given weight. Let pf(D)orpf()denote the fraction by weight of material inthe sample of material finer than sizeD(). The customary engineering representation of





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TABLE6.3

Sedim

entGradeScale

Approximate

Sieve

Mesh

Class

Name

Size

Range

Openingsper

Inch

Millimeters

Micro

ns

Inches

Tyler

U.S.st

andard

Verylargeboulders

4,096

2,048

160

80

Largeboulders

2,048

1,024

80

40

Mediumboulders

1,024

512

40

20

Smallboulders

512

256

9

8

20

10

Largecobbles

256

128

8

7

10

5

Smallcobbles

128

64

7

6

5

2.5

Verycoarsegravel

64

32

6

5

2.5

1.3

Coarsegravel

32

16

5

4

1.3

0.6

Mediumgravel

16

8

4

3

0.6

0.3

2

1/2

Finegravel

8

4

3

2

0.3

0.16

5

5

Veryfinegravel

4

2

2

1

0.16

0.08

9

10

Verycoarsesand

2.000

1.000

1

0

2,000

1,000

16

18

Coarsesand

1.000

0.500

0

1

1,000

500

32

35

Mediumsand

0.500

0.250

1

2

500250

60

60

Finesand

0.250

0.125

2

3

250125

115

120

Veryfinesand

0.125

0.062

3

4

12562

250

230

Coarsesilt

0.062

0.031

4

5

6231

Mediumsilt

0.031

0.016

5

6

3116

Finesilt

0.016

0.008

6

7

168

Veryfinesilt

0.008

0.004

7

8

8

4

Coarseclay

0.004

0.0020

8

9

4

2

Mediumclay

0.0020

0.0

010

2

1

Fineclay

0.0010

0.0

005

1

0.5

Veryfineclay

0.0005

0.0

0024

0.50.24

SOURCE:AdaptedfromVanoni,1975.





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6.10 Chapter Six

the grain size distribution consists of a plot ofpf 100 (percentage finer) versus log10(D):that is, a semilogarithmic plot is used. The same size distribution plotted in sedimento-logical form would involve plottingpf 100 versus on a linear plot.

The size distributionpf() and size densityp() by weight can be used to extract use-ful statistics concerning the sediment in question. Letxdenote some percentage, say 50%;the grain size x denotes the size such thatxpercent of the weight of the sample is com-posed of finer grains. That is, x is defined such that

pf(x) 10x

0 (6.25)

It follows that the corresponding grain size of equivalent diameter is given byDx, where

Dx 2 x (6.26)

The most commonly used grain sizes of this type are the median size D50 and the sizeD90: i.e., 90% of the sample by weight consists of finer grains. The latter size is especial-ly useful for characterizing bed roughness.

The densityp() can be used to extract statistical moments. Of these, the most usefulare the mean size m and the standard deviation . These are given by the relations.

m p()d; 2 ( m)2p()D (6.27a, b)The corresponding geometric mean diameter Dg and geometric standard deviation g

are given as

Dg 2m; g 2 (6,28a,b)

Note that for a perfectly uniform material, 0 and g 1. As a practical matter, a sed-iment mixture with a value ofg less than 1.3 is often termed well sorted and can be treat-ed as a uniform material. When the geometric standard deviation exceeds 1.6, the mater-ial can be said to be poorly sorted (Diplas and Sutherland, 1988).

In fact, one never has the continuous function p() with which to compute themoments of Eqs. (6.27a, and b). Instead, one must rely on a discretization. To this end, thesize range covered by a given sample of sediment is discretized using n intervals bound-ed by n 1 grain sizes 1, 2,, n 1 in ascending order of. The following defini-

tions are made from i 1 to n:

i 12(i i1) (6.29a)

pi pf(i) pf(i1) (6.29b)

Eqs. (6.27a and b) now discretize to

m ni1

ipi 2 ni1

(i m)2pi (6.30)

In some cases, especially when the material in question is sand, the size distributioncan be approximated as gaussian on the scale (i.e., log-normal inD). For a perfectlyGaussian distribution, the mean and median sizes coincide:

m 50 12(84 16) (6.31)





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Furthermore, it can be demonstrated from a standard table of the Gauss distribution thatthe size displaced one standard deviation larger that m is accurately given by 84; bysymmetry, the corresponding size that is one standard deviation smaller than m is 16.The following relations thus hold:

12(84 16) (6.32a)

m 12(84 16) (6.32b)

Rearranging the above relations with the aid of Eqs. (6.28a and b) and Eqs. (6.31 and6.32a),

g

D

D

8

1

4

6

1/2

(6.33a)

Dg (D84D16)1/2 (6.33b)

It must be emphasized that the above relations are exact only for a gaussian distributionin . This is not often the case in nature. As a result, it is strongly recommended thatDgand g be computed from the full size distribution via Eqs. (6.30a and b) and (6.28a andb) rather than the approximate form embodied in the above relations.

6.3.5 Porosity

Theporosity p quantifies the fraction of a given volume of sediment that is composed ofvoid space. That is,

p

If a given mass of sediment of known density is deposited, the volume of the depositmust be computed, assuming that at least part of it will consist of voids. In the case ofwell-sorted sand, the porosity often can take values between 0.3 and 0.4. Gravels tend to

be more poorly sorted. In this case, finer particles can occupy the spaces between coarserparticles, thus reducing the void ratio to as low as 0.2. Because so-called open-work grav-els are essentially devoid of sand and finer material in their interstices, they may haveporosities similar to sand. Freshly deposited clays are notorious for having high porosi-ties. As time passes, the clay deposit tends to consolidate under its own weight so thatporosity slowly decreases.

The issue of porosity becomes of practical importance with regard to salmon spawn-ing grounds in gravel-bed rivers, for example (Diplas and Parker, 1985). The percentageof sand and silt contained in the sediment is often referred to as the percentage of fines inthe gravel deposit. When this fraction rises above 20 or 26 percent by weight, the deposit

is often rendered unsuitable for spawning. Salmon bury their eggs within the gravel, anda high fines content implies a low porosity and thus reduced permeability. The flow ofgroundwater necessary to carry oxygen to the eggs and remove metabolic waste productsis impeded. In addition, newly hatched fry may encounter difficulty in finding enoughpore space through which to emerge to the surface. All the above factors dictate loweredsurvival rates. Chief causes of elevated fines in gravel rivers include road building andclear-cutting of timber in the basin.

volume of voids

volume of total space





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6.3.6 Shape

Grain shape can be classified in a number of ways. One of these, theZingg classification

scheme, is illustrated here (Vanoni, 1975). According to the definitions introduced earlier,a simple way to characterize the shape of an irregular clast(stone) is by lengths a, b, andc of the major, intermediate, and minor axes, respectively. If the three lengths are equal,the grain can be said to be close to a sphere in shape. Ifa and b are equal but c is muchlarger, the grain should be rodlike. Finally, ifc is much smaller than b, which in turn, ismuch larger than a, the resulting shape should be bladelike.

6.3.7 Fall Velocity

A fundamental property of sediment particles is theirfall velocity. The relation for termi-nal fall velocity in quiescent fluid vs can be presented as

Rf 43CD1

(Rp)1/2 (6.34)

where

Rf R

vs

gD (6.35a)

Rp v

sv

D (6.35b)

and the functional relation CD CD(Rp) denotes the drag curve for spheres. This relationis not particularly useful because it is not explicit in vs; one must compute fall velocity bytrial and error. One can use the equation for CD given below

CD 2R

4

p

(1 0.152Rp1/2 0.0151Rp) (6.36)

and the definition

Rep

Rg

D

D

(6.37)

to obtain an explicit relation for fall velocity in the form ofRf versusRep. In Fig. 6.2, theranges for silt, sand, and gravel are plotted for 0.01 cm2/s (clear water at 20C) andR 1.65 (quartz). A good summary of relations for terminal fall velocity for the case ofnonspherical (natural) particles can be found in Dietrich (1982), who also proposed thefollowing useful fit:

Rf exp{b1 b2ln(Rep) b3[ln(Rep)]2 b4[ln(Rep)]3 b5[ln(Rep)]4} (6.38)

where b1

2.891394, b2

0.95296, b3

0.056835, b4

0.002892, and b5

0.000245

6.3.8 Relation Between Size Distribution and Stream Morphology

The study of sediment properties and, in particular, size distribution is most relevant to thecontext of stream morphology. The following discussion points out some of the moreinteresting issues.

6.12 Chapter Six





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FIGURE6.2

Sedimentfallvelocitydiagram





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6.14 Chapter Six

FIGURE 6.3 Particle size distribution of bed materials in Kankakee River, Illinois.(Bhowmik et al., 1980)

In Fig. 6.3, several size distributions from the sand-bed Kankakee River in Illinois, areshown (Bhowmik et al., 1980). The characteristic S shape suggests that these distributionsmight be approximated by a gaussian curve. The median sizeD50 falls near 0.3 to 0.4 mm.

The distributions are tight, with a near absence of either gravel or silt. For practical pur-poses, the material can be approximated as uniform.

In Fig. 6.4, several size distributions pertaining to the gravel-bed Oak Creek in Oregon,are shown (Milhous, 1973). In gravel-bed streams, the surface layer (armor or pave-ment) tends to be coarser than the substrate (identified as subpavement in the figure).Whether the surface or substrate is considered, it is apparent that the distribution rangesover a much wider range of grain sizes than is the case in Fig. 6.3. More specifically, in





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FIGURE

6.4

Sizedistributionofbedmate

rialsamplesinOakCreek.

Oregon

.Source:(Milhous,1973)





16/113

6.16 Chapter Six

the distributions of the sand-bed Kankakee River, varies from about 0 to about 3, where-as in Oak Creek, varies from about 8 to about 3. In addition, the distribution of Fig.6.4 is upward-concave almost everywhere and thus deviates strongly from the gaussian

distribution.These two examples provide a window toward generalization. A river can be loosely

classified as sand-bed or gravel-bed according to whether the median size D50 of the sur-face material or substrate is less than or greater than 2 mm. The size distributions of sand-bed streams tend to be relatively narrow and also tend to be S shaped. The size distribu-tions of gravel-bed streams tend to be much broader and to display an upward-concaveshape. Of course, there are many exceptions to this behavior, but it is sufficiently generalto warrant emphasis.

More evidence for this behavior is provided in Fig. 6.5. Here, the grain size distribu-tions for a variety of stream reaches have been normalized using the median size D50.

Four sand-bed reaches are included with three gravel-bed reaches. All the sand-beddistributions are S shaped, and all have a lower spread than the gravel-bed distributions.The standard deviation is seen to increase systematically with increasingD50(White et al.,1973).

The three gravel-bed size distributions differ systematically from the sand-bed distrib-utions in a fashion that accurately reflects Oak Creek (Fig. 6.4). The standard deviation inall cases is markedly larger than any of the sand-bed distributions, and the distributions

FIGURE 6.5 Dimensionless grain-size distribution for different rivers (White et al., 1973)

are upward-concave except perhaps near the coarsest sizes.

6.4 THRESHOLD CONDITION FOR SEDIMENT MOVEMENT





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When a granular bed is subjected to a turbulent flow, virtually no motion of the grains isobserved at some flows, but the bed is mobilized noticeably at other flows. Factors thataffect the mobility of grains subjected to a flow are summarized below:

randomness

grain placement

turbulence

forces on grain

fluid

liftmean & turbulent

drag

gravity

In the presence of turbulent flow, random fluctuations typically prevent the clear defini-tion of a critical, or threshold condition for motion: The probability for the movement ofa grain is never precisely zero (Lavelle and Mofjeld, 1987). Nevertheless, it is possible todefine a condition below which movement can be neglected for many practical purposes.

6.4.1 Granular Sediment on a Stream Bed

Figure 6.6 is a diagram showing the forces acting on a grain in a bed of other grains. Whencritical conditions exist and the grain is on the verge of moving, the moment caused by

the critical shear stress c about the point of support is just equal to that of the weight ofthe grain. Equating these moments gives (Vanoni, 1975):

FIGURE 6.6 Forces acting on a sediment particle on an inclined bed





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6.18 Chapter Six

c c

c1

2

a

a1

2

(s )Dcos (tan tan) (6.39)

in which s specific weight of sediment grains, specific weight of water,D diam-eter of grains, is the slope angle of the stream, the angle of repose of the sediment, c1and c2 are dimensionless constants, and a1 and a2 are lengths shown in Fig. 6.6. Any consis-tent set of units can be used in Eq. (6.39). For a horizontal bed, Eq. (6.39) reduces to

c c

c1

2

a

a1

2

(s )D tan (6.40)

For an adverse slope (i.e., 0),

c c

c1

2

a

a1

2 (s )D cos (tan tan ) (6.41)

Equations (6.39), (6.40), and (6.41) cannot be used to give c because the factors c1, c2,a1, and a2 are not known. Therefore, the relation between the pertinent quantities isexpressed by dimensional analysis, and the actual relation is determined from experimen-tal data. Figure 6.7 is such a relation, first presented by Shields (1936) and carries hisname. The curve is expressed by dimensionless combinations of critical shear stress c,sediment and water specific weights s and , sediment sizeD, critical shear velocity u*c c/and kinematic viscosity of water.

These quantities can be expressed in any consistent set of units. Dimensional analysisyields,

c* (s

c)Df

u*

cD

(6.42)

The Shields values ofc* are commonly used to denote conditions under which bedsediments are stable but on the verge of being entrained. Not all workers agree with theresults given by the Shields curve. For example, some workers give c* 0.047 for thedimensionless critical shear stress for values ofR* u*D/ in excess of 500 instead of0.06, as shown in Fig. 6.7. Taylor and Vanoni (1972) reported that small but finite amountsof sediment were transported in flows with values ofc* given by the Shields curve.

The value ofc to be used in design depends on the particular case at hand. If the sit-uation is such that grains that are moved can be replaced by others moving from upstream,

some motion can be tolerated, and the Shields values can be used. On the other hand, ifgrains removed cannot be replaced, as on a stream bank, the Shields value ofc are toolarge and should be reduced.

The Shields diagram is not especially useful in the form of Fig. 6.7 because to find c,one must know u* c/. The relation can be cast in explicit form by plotting c* ver-susRep, noting the internal relation

u*

D

Ru*

gD (*)1/2Rep (6.43)

whereR s is the submerged specific gravity of the sediment. A useful fit is given

by Brownlie (1981a):

*c 0.22Rep0.6 0.06 exp(17.77Rep0.06) (6.44)

RgDD





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FIGUR

E6.7

Shieldsdiagramforinitiationofmotion.

SourceVanoni(1975)





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6.20 Chapter Six

FIGURE 6.8 Angle of repose of granular material. (Lane,1955)

With this relation, the value ofc* can be computed readily when the properties of thewater and the sediment are given.

The value of bed-shear stress b for a wide rectangular channel is given by b HS,as shown earlier. The average bed-shear stress for any channel is given by b RhS, inwhichRh the hydraulic radius of the channel cross section.

6.4.2 Granular Sediment on a Bank

A sediment grain on a bank is less stable than one on the bed because the gravity forcetends to move it downward (Ikeda, 1982). The ratio of the critical shear stress wc for a par-ticle on a bank to that for the same particle on the bed c is (Lane, 1955)

w

c

c = cos 1

1

tta

an

n

1

2

(6.45)

where 1 is the slope of the bank and is the angle of repose for the sediment. Values of are





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given in Fig. 6.8 after Lane (1955) and also can be found in Simons and Senturk (1976).

6.4.3 Granular Sediment on a Sloping Bed

Equation (6.39) shows that c diminishes as the slope angle increases. For extremelysmall s, c is given by Eq. (6.40). Taking the ratio between Eqs. (6.39) and (6.40) yields

c

c

cos

1

t

t

a

a

n

n

(6.46)

c is the critical shear stress for sediment on a bed with a slope angle , and co is the crit-ical shear stress for a bed with an extremely small slope. The value of co can be found

from the Shields diagram or with Eq. (6.44). Equation (6.46) is for positive , which ispositive for downward sloping beds. For beds with adverse slope, is negative and theterm tan /tan in Eq. (6.46) is positive.

6.4.4 Sediment Mixtures

Several authors have offered empirical or quasi-theoretical extensions of the above rela-tions to the case of mixtures (e.g., Wilcock, 1988). LetDi denote the characteristic grainsize of the ith size range in a mixture. Furthermore, letDsg denote the geometric mean size

of the surface (exchange, active) layer. Most of the generalizations can be written in thefollowing form (Parker, 1990):

*ci *cgD

D

s

i

g

(6.47)

Here

*ci Rbgc

Di

i

(6.48a)

and

*

cg

Rbg

c

D

sg

sg

(6.48b)

where bci and bcsg denote the values of the dimensioned critical shear stress required tomove sediment of sizesDi andDsg in the mixture, respectively, and is an exponent tak-ing a value given below;

0.9 (6.49)

Figure 6.9 shows the similarity between four different published expressions havingthe general form given by Eq. (6.47), which is of interest because it includes the effect ofhiding. For uniform material, the critical Shields stress is defined by Eq. (6.44). Consider

two flumes, one with uniform sizeDa and the other with uniform sizeDb. For sufficient-ly coarse material (u*D/ 1 orRep 1), the critical Shields stress must be the same forboth sizes (Fig. 6.7). It follows from Eq. (6.42) that where bca and bcb denote the dimen-sioned boundary shear stresses for cases a and b respectively,

bcb bca

D

Db

a

(6.50)





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6.22 Chapter Six

For the case of mixtures, on the other hand, it is seen from Eqs. (6.47) and (6.48) that

bci bcsgD

D

s

i

g

1 bcsg

D

D

s

i

g

0.1(6.51)

Comparing Eqs. (6.50) and (6.51), it is seen that a finer particle (Db Da, or alternative-ly,Di Dsg) is more mobile than a coarser particle. For example, suppose that one grainsize is four times coarser than another. If two uniform sediments are being compared, it

follows from Eq. (6.50) that the critical shear stress for the coarser material is four timesthat of the finer material. In the case of a mixture, however, the critical shear stress for thecoarser material is only about 40.1, or 1.15 times that for the finer material.

A finer particle in a mixture is thus seen to be only a little more mobile than its coars-er-sized brethren, where uniform beds of fine material are much more mobile than are uni-form beds of coarser material. The reason is that finer particles in a mixture are relativelyless exposed to the flow; they tend to hide in the lee of coarser particles. By the same token,a particle is relatively more exposed to the flow when most of its neighbors are finer.

A method to calculate the critical shear stress for motion of uniform and heterogeneoussediments was proposed by Wiberg and Smith (1987) on the basis of the fluid mechanicsof initiation of motion, which takes into account both roughness and hiding effects.

6.5 SEDIMENT TRANSPORT

FIGURE 6.9 Critical shear stress for sediment mixture (Source: Misri et

al., 1983)





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6.5.1 Sediment Transport Modes

The most common modes of sediment transport in rivers are bedload and suspended load.

In the case of bedload, the particles roll, slide, or saltate over each other, never deviatingtoo far above the bed. In the case of suspended load, the fluid turbulence comes into playcarrying the particles well up into the water column. In both cases, the driving force forsediment transport is the action of gravity on the fluid phase; this force is transmitted tothe particles via drag.

The same phenomena of bedload and suspended load transport occur in a variety ofother geophysical contexts. Sediment transport is accomplished in the near-shore lake andoceanic environment by wave action. Turbidity currents carry sediment into lakes, reser-voirs, and the deep sea.

The phenomenon of sediment transport can sometimes be disguised in rather esoteric

phenomena. When water is supercooled, large quantities of particulate frazil ice can form.As this water moves under a frozen ice cover, the phenomenon of sediment transport inrivers is stood on its head. The frazil ice particles float rather than sink and thus tend toaccumulate on the bottom side of the ice cover rather than on the river bed. Turbulencetends to suspend the particles downward rather than upward.

In the case of a powder snow avalanche, the fluid phase is air and the solid phase con-sists of snow particles. The dominant mode of transport is suspension. These flows areclose analogies of turbidity currents, insofar as the driving force for the flow is the actionof gravity on the solid phase rather than the fluid phase. That is, if all the particles dropout of suspension, the flow ceases. In the case of sediment transport in rivers, it is accu-rate to say that the fluid phase drags the solid phase along. In the case of turbidity currents

and powder snow avalanches, the solid phase drags the fluid phase along.Desert sand dunes provide an example for which the fluid phase is air, but the domi-

nant mode of transport is saltation rather than suspension. Because air is so much lighterthan water, quartz sand particles saltate in long, high trajectories, relatively unaffected bythe direct action of turbulent fluctuations. The dunes themselves are created by the effectof the fluid phase acting on the solid phase. They, in turn, affect the fluid phase by chang-ing the resistance.

Among the most interesting sedimenttransport phenomena are debris flows, slurries,and hyperconcentrated flows. In all these cases, the solid and fluid phases are present insimilar quantities. A debris flow typically carries a heterogeneous mixture of grain sizes

ranging from boulders to clay. Slurries and hyperconcentrated flows are generally restrict-ed to finer grain sizes. In most cases, it is useful to think of such flows as consisting of asingle phase, the mechanics of which are highly non-Newtonian.

The study of the movement of grains under the influence of fluid drag and gravitybecomes even more interesting when one considers the link between sediment transportand morphology. In the laboratory, the phenomenon can be studied in the context of a vari-ety of containers, such as channel and wave tanks, specified by the experimentalist. In thefield, however, the fluid-sediment mixture constructs its own container. This new degreeof freedom opens up a variety of intriguing possibilities.

Consider the river. Depending on the existence or lack of a viscous sublayer and therelative importance of bedload versus suspended load, a variety of rhythmic structures canform on the river bed. These include ripples, dunes, antidunes, and alternate bars. The firstthree of these can have a profound effect on the resistance to flow offered by the river bed.Thus, they act to control river depth. River banks themselves also can be considered to bea self-formed morphological feature, thus specifying the entire container.

The container itself can deform in plan. Alternate bars cause rivers to erode their banksin a rhythmic pattern, thus allowing for the onset of meandering. Fully developed rivermeandering implies an intricate balance between sediment erosion and deposition. If a





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stream is sufficiently wide, it will braid rather than meander, dividing into several inter-twining channels.

Rivers create morphological structures at much larger scales as well. These include

canyons, alluvial fans, and deltas. Turbidity currents create similar structures in the ocean-ic environment. In the coastal environment, the beach profile itself is created by the inter-action of water and sediment. On a larger scale, offshore bars, spits, and capes constituterhythmic features created by wave-current-sediment interaction. The boulder levees oftencreated by debris flows provide another example of a morphologic structure created by asediment-bearing flow.

The floodplains of most sand-bed rivers often contain copious amounts of silt and clayfiner than approximately 50 . This material is often called wash loadbecause it movesthrough the river system without being present in the bed in significant quantities.Increased wash load does not cause deposition on the bed, and reduced wash load doesnot cause erosion because it is transported well below capacity. This is not meant to implythat the wash load does not interact with the river system. Wash load in the water columnexchanges with the banks and the floodplain rather than the bed. Greatly increased washload, for example, can lead to thickened floodplain deposits, with a consequent increasein bankfull channel depth.

The emphasis here is the understanding of bedload and suspended load transport inrivers, with the goal of providing the knowledge needed to do sound sedimentation engi-neering, particularly with problems involving stream restoration and naturalization.

6.5.2 Shields Regime Diagram

In the context of rivers, it is useful to have a way to determine what kind of sediment-transport phenomena can be expected for different flow conditions and different charac-teristics of sediment particles. In Fig. 6.10, the ordinates correspond to bed shear stresseswritten in the dimensionless form proposed by Shields

* gR

b

D

RH

DS (6.52)

and the particleRep, defined by Eq. (6.37) is used for the abscissa values. There are threecurves in the diagram which make it possible to know, for different values of (*,Rep), if the given bed sediment will go into motion, and if this is the case whether or not the pre-vailing mode of transport will be in suspension or as bedload. The diagram also can be usedto predict what kind of bedforms can be expected. For example, ripples will develop in thepresence of a viscous sublayer and fine-grained sediment. If the viscous sublayer is dis-rupted by coarse sediment particles, then dunes will be the most common type of bedform.

The Shields regime diagram also shows a clear distinction between the conditionsobserved in sand-bed rivers and gravel-bed rivers at bankfull stage. If one wanted to mod-el in the laboratory sediment transport in rivers, the experimental conditions would be dif-ferent, depending on the river system in question. As could be expected, the diagram alsoshows that in gravel-bed rivers, sediment is transported as bedload. In sand-bed rivers, on

the other hand, suspended load and bedload transport coexist most of the time.The regime diagram is valid for steady, uniform, turbulent flow conditions, where thebed shear stress b can be estimated with Eq. (6.1). The ranges for silt, sand, and gravelalso are included. In the diagram, the critical Shields stress for motion was plotted withthe help of Eq. (6.44). The critical condition for suspension is given by the following ratio:

u

v*

s

1 (6.53)

6.24 Chapter Six





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FIGURE6.10

Shieldsregimediagram.

(Source:GaryParker)





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6.26 Chapter Six

where u* is the shear velocity and vs is the sediment fall velocity. Equation (6.53) can betransformed into

s R12f (6.54)where

s gRu2*D (6.55)

andRf is given by Eq. (6.35a) and can be computed for different values ofRep with thehelp of Eq. (6.38).

Finally, the critical condition for viscous effects (ripples) was obtained with the helpof Eq. (6.5) as follows:

11.6 u*

D 1 (6.56)

which in dimensionless form can be written as

*v

1R1

e

.

p

6

2(6.57)

Relations (6.44), (6.54), and (6.57) are the ones plotted in Fig. 6.10. The Shields regimediagram should be useful for studies concerning stream restoration and naturalizationbecause it provides the range of dimensionless shear stresses corresponding to bankfullflow conditions for both gravel- and sand-bed streams.

6.6 BEDLOAD TRANSPORT

6.6.1 The Bed Load Transport Function

Bedload particles roll, slide, or saltate along the bed. The transport thus occurs tangentialto the bed. In a case where all the transport is directed in the streamwise, or s direction,the volume bedload-transport rate per unit width (n direction) is given by q; the units are

length3/length/per time, or length2/time. In general, q is a function of boundary shear stressb and other parameters; that is,

q q(b, other parameters) (6.58)

In general, bedload transport is vectorial, with components qs and qn in the s and n direc-tions, respectively.

6.6.2 Erosion Into and Deposition from Suspension

The volume rate of erosion of bed material into suspension per unit time per unit bed areais denoted asE. The units ofEare length3/length2/time, or velocity. A dimensionless sed-iment entrainment rateEs can thus be defined with the sediment fall velocity vs:

E vsEs (6.59)





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In general,Es can be expected to be a function of boundary shear stress b and other para-meters. Erosion into suspension can be taken to be directed upward normal: i.e., in thepositivez direction.

Let c denote the volume concentration of suspended sediment (m3 of sediment/m3 ofsediment-water mixture), averaged over turbulence. The streamwise volume transport rateof suspended sediment per unit width is given by

qs H

0

cudz (6.60)

In a two-dimensional case, two components, qSs and qSn, result, where

qSs H

0

cudz (6.61a)

qSn H

0

cvdz (6.61b)

Deposition onto the bed is by means of settling. The rate at which material is fluxedvertically downward onto the bed (volume/area/time) is given by vscb, where cb is a near-bed value ofc. The deposition rate D realized at the bed is obtained by computing thecomponent of this flux that is actually directed normal to the bed:

D vscb (6.62)

6.6.3 The Exner Equation of Sediment Mass Conservation forUniform Material

Consider a portion of river bottom, where the bed material is taken to have a (constant)porosity p. Mass balance of sediment requires the following equation to be satisfied:

t [mass of bed material] net mass bedload inflow rate

net mass rate of deposition from suspension.A datum of constant elevation is located well below the bed level, and the elevation of thebed with respect to such datum is given by . Then, bed level changes as a result of bed-load transport, sediment entrainment into suspension, and sediment deposition onto thebed can be predicted with the help of

(1 p) t

q

ss

q

nn

vs (cb Es) (6.63)

To solve the Exner equation, it is necessary to have relations to compute bedload transport(i.e., q

s

and qn

), near-bed suspended sediment concentration cb

, and sediment entrainmentinto suspensionEs. The basic form of Eq. (6.63) was first proposed by Exner (1925).

6.6.4 Bedload Transport Relations

A large number of bedload relations can be expressed in the general form






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6.28 Chapter Six

q* q*(*, Rep, R) (6.64)

Here, q* is a dimensionless bedload transport rate known as the Einstein number, first

introduced by H. A. Einstein in 1950 and given by

q* Rg

q

DD (6.65)

The following relations are of interest. In 1972, Ashida and Michiue introduced

q* 17(* *c) [(*)1/2 (c)1/2] (6.66)

and recommend a value ofc* of 0.05. It has been verified with uniform material rangingin size from 0.3 mm to 7 mm. Meyer-Peter and Muller (1948) introduced the following:

q* 8(* *c)3/2 (6.67)

where *c 0.047. This formula is empirical in nature and has been verified with data foruniform gravel.

Engelund and Fredse (1976) proposed,

q* 18.74(* *c) [(t*)1/2 0.7(c)1/2] (6.68)

where *c 0.05. This formula resembles that of Ashida and Michiue because the deriva-tion is almost identical.

Fernandez Luque and van Beek (1976) developed the following,

q* 5.7(* *c)3/2 (6.69)

where c varies from 0.05 for 0.9 mm material to 0.058 for 3.3. mm material. The relationis empirical in nature.

Wilson (1966):

q* 12(* c)3/2 (6.70)

where c was determined from the Shields diagram. This relation is empirical in nature;most of the data used to fit it pertain to very high rates of bedload transport.

Einstein (1950):q* q*(*) (6.71)

where the functionality is implicitly defined by the relation

1 (0.143/*)2(0.413/*)2

et2dt1

434.53q.5

*

q* (6.72)

Note that this relation contains no critical stress. It has been used for uniform sand andgravel.

Yalin (1963):

q* 0.635s(*)1/21

1n(1

a

2s

a2s)

(6.73)

where

a2 2.45(R 1)0.4 (c)1/2; s *

*c

*c (6.74)

1





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and c is evaluated from a standard Shields curve. Two constants in this formula have beenevaluated with the aid of data quoted by Einstein (1950), pertaining to 0.8 mm and 28.6mm material.

Parker (1978):

q* 11.2(*

0*3

.03)4.5 (6.75)

developed with data sets pertaining to rough mobile-bed flow over gravel.Several of these relations are plotted in Fig. 6.11. They tend to be rather similar in

nature. Scores of similar relations could be quoted.To date, only few research groups have attempted complete derivations of the bedload

function in water. They are Wiberg and Smith (1989), Sekine and Kikkawa (1992), Garcaand Nio (1992), Nio and Garca, (1994, 1998), and Nio et al., (1994).

FIGURE 6.11 Bedload transport relations. (Parker, 1990)





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6.30 Chapter Six

6.6.5 Bedload Transport Relation for Mixtures.

Relatively few bedload relations have been developed specifically in the context of mix-

tures (e.g., Bridge and Bennett, 1992). One of these is presented below as an example.The relationship of Parker (1990) applies to gravel-bed streams. The data used to fit

the relation are solely from two natural gravel-bed streams: Oak Creek in Oregon and theElbow River in Alberta, Canada. The relation is surface-based; load is specified per unitof fractional content in the surface layer. The surface layer is divided intoNsize ranges,each with a fractional content Fi by volume, and a mean phi size i;Di 2i. The arith-metic mean of the surface size on the phi scale and the corresponding arithmetic stan-dard deviation are given by

Fii; 2 Fi(i )2 (6.76a, b)

The corresponding geometric mean size Dsg and the geometric standard deviation sg ofthe surface layer are given by

Dsg 2 sg 2

(6.77a, b)

In the Parker relation, the volume bedload transport per unit width of gravel in the ithsize range is given by the product qiFi (no summation), where qi denotes the transport perunit fraction in the surface layer. The total volume bedload transport rate of gravel per unitwidth is qT, where

qT qiFi (6.78)

The relation does not apply to sand. Thus, before using the relation for a given surfacedistribution, the sand content of the grain-size distribution must be removed and Fi mustbe renormalized so that it sums to unity over all sizes in excess of 2 mm.

Ifpi denotes the fraction volume content of material in the ith size range in the bed-load, it follows that

pi q

qiF

iFi

i

(6.79)

The parameter qi is made dimensionless as follows:

W*si (b

R

/g

)

q3/

i

2Fi (6.80)

A dimensionless Shields stress based on the surface geometric mean size is defined asfollows:

*sg Rg

b

Dsg (6.81)

Let sgo denote a normalized value of this Shields stress, given by

sgo

*

*

r

s

s

g

g

o

o

(6.82)

where

*rsgo 0.0386 (6.83)

corresponds to a near-critical value of Shields stress. The Parker relation can then be





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expressed in the form

W*si 0.0218 G [sgogo(i)] (6.84a)

In the above relationship, go denotes a hiding function given by

go(i) 0.0951i ; i DD

s

i

g

(6.84b)

The parameter is given by the relationship

1

o

(o 1) (6.84c)

where

o and o are specified as functions ofsgo in Fig. 6.12. The function G is speci-fied as

5474(1 0.853/)4.5 1.65

G[]

exp[14.2( 1) 9.28( 1)2] 1 1.65

Mo 1 (6.85)

and is shown in Fig. 6.13. Here, Mo 14.2 and is a dummy variable for the argumentin Eq. (6.84) and is not to be confused with the grain-size scale.

An application of Eq. (6.84) to uniform material with sizeD

results in the relation

q* 0.0218(*)3/2G

0.0

3

*

86

(6.86)

where

q* gR

q

DD ; * g

R

b

D (6.87)

and q denotes the volumetric sediment transport per unit width. In Fig. 6.11, Eq. (6.86) iscompared to several other relations and selected laboratory data for uniform material. Thefigure is adapted from Figs. 6b and 7 in Wiberg and Smith (1989), where reference to thedata and equations can be found. The data pertain to 0.5 mm sand and 28.6 mm gravel.Equation (6.86) shows a reasonable correspondence with the data and with several otherrelations for uniform material.

The Parker relationship (Eq. 6.84) can be used to predict mobile or static armor ingravel streams. Note that there is no formal critical stress in the formulation; instead for 1, the transport rates become extremely small. For the computation of bedload trans-port in poorly sorted gravel-bed rivers, the above formulation has been used to implementa series of programs named ACRONYM (Parker, 1990). The program ACRONYM1provides an implementation of the surface-based bedload transport equation presented in

Parker (1990). It computes the magnitude and size distribution of bedload transport overa bed surface of given size distribution, on which a given boundary shear stress isimposed. The program ACRONYM2 inverts the same bedload transport equation,allowing for calculation of the size distribution at a given boundary shear stress. The pro-gram was used to compute mobile and static armor size distributions in Parker (1990) andParker and Sutherland (1990).





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6.32 Chapter Six

FIGURE6.12

Plotsof0and

0versus

sg0,

theasymptotesaren

otedontheplot.(Parker,1990)





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FIG

URE6.13

PlotofGandG

Tvers

us

50.

(Parker,1990)





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The program ACRONYM3 allows for the computation of aggradation or degrada-tion to a specified active or static equilibrium final state. To this end, Parkers method(1990) is combined with a resistance relation of the Keulegan type. In the program, bothconstant width and water discharge are assumed.

The program ACRONYM4 is directed toward the wavelike aggravation of self-sim-ilar form discussed in Parker (1991a, 1991b). It uses Parkers method and a resistancerelation of the Manning-Strickler type to compute downstream fining and slope concavi-ty caused by selective sorting and abrasion.

6.7 BEDFORMS

The formation and behavior of sediment waves produced by moving water are, in equalmeasure, intellectually intriguing and of great engineering importance. Because of thecentral role they play in river hydraulics, fluvial ripples, dunes, and bars have receivedextensive attention from engineers for at least the past two centuries, and even more inten-sive descriptive study from geologists. Such studies can be divided into three categoriesaccording to the approach followed: analytical, empirical, or statistical.

Analytical models for bedforms have been proposed since 1925 (Anderson, 1953;Blondeaux et al., 1985; Colombini et al., 1987; Engelund, 1970; Exner, 1925; Fredsoe,1974, 1982; Gill, 1971; Haque and Mahmood, 1985; Hayashi, 1970; Kennedy, 1963,1969; Parker, 1975; Raudkivi and Witte, 1990; Richards, 1980; Smith, 1970; Tubino andSeminara, 1990).

Empirical methods include the following works (Coleman and Melville, 1994;Colombini et al., 1990; Garca and Nio, 1993; Garde and Albertson, 1959; Ikeda, 1984;Jaeggi, 1984; Kinoshita and Miwa, 1974; Menduni and Paris, 1986: Ranga Raju and Soni,1976; Raudkivi, 1963; van Rijn, 1984; Yalin, 1964; Yalin and Karahan, 1979).

Statistical models for bedforms have been advanced by the following authorsAnnambhotla et al.,1972; Hino, 1968; Jain and Kennedy, 1974; Nakagawa and Tsujimoto,1984; Nordin and Algert, (1966).

Despite all the research that has been done, there is presently no completely reliablepredictor for the conditions of occurrence and characteristics of the different bed config-urations (ripples, dunes, flat bed, antidunes).

6.7.1 Dunes, Antidunes, Ripples, and Bars

The ripples, dunes, and antidunes illustrated in Fig. 6.14 are the classic bedforms oferodible-bed open-channel flow. On the one hand, they are a product of the flow andsediment transport; on the other hand, they profoundly influence the flow and sedimenttransport. In fact, all the bedload formulas quoted previously are strictly invalid inthe presence of bedforms. The adjustments necessary to render them valid are discussedlater.

Ripples, dunes, and antidunes are undular (wavelike) features that have wavelengths and wave heights that scale no larger than on the order of the flow depthH, as definedbelow.

6.7.1.1 Dunes. Well-developed dunes tend to have wave heights D scaling up to aboutone-sixth of the depth: i.e.,

6.34 Chapter Six





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H

16 (6.88)

Dune wavelengths can vary considerably. A fairly typical range can be quantified asdimensionless wave number k, where

k2

H (6.89)

This range is

0.25 k 4.0 (6.90)

Dunes invariably migrate downstream. Typically, they are approximately triangular inshape and usually (but not always) possess a slip face, beyond which the flow is separat-ed for a certain length. A dune progresses forward as bedload accretes on the slip face.Generally, little bedload is able to pass beyond the face without depositing on it, whereasmost of the suspended load is not directly affected by it.

Let c denote the wave speed of the dune. The bedload transport rate can be estimated asthe volume of material transported forward per unit bed area per unit time by a migratingdune. If the dune is approximated as triangular in shape, the following approximation holds:

q 12 c(1 p) (6.91)

Dunes are characteristic of subcritical flow in the Froude sense. In a shallow-water (longwave) model, the Froude criterion (Fr) dividing subcritical and supercritical flow is

FIGURE 6.14 Schematic of different bedforms. (Vanoni, 1975)





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6.36 Chapter Six

Fr 1 (6.92)

where

Fr (6.93)

Dunes, however, do not qualify as long waves because their wavelength is of the order ofthe depth. A detailed potential flow analysis over a wavy bed yields the following (wave-number dependent) criterion for critical flow over a bedform (Kennedy, 1963).

Fr2 1k tanh(k) (6.94)

Note that as k 0( ) tanh(k) k, and condition (6.92) is recovered in the long-wave limit. For dunes to occur, then, the following condition must be satisfied:

F2 1k tanh (k) (6.95)

Both dunes and antidunes cause the water surface as well as the bed to undulate. In thecase of dunes, the undulation of the water surface is usually of much smaller amplitudethan that of the bed; the two are nearly 180o out of phase.

Dunes also can occur in the case of wind-blown sand. Barchan dunes are commonlyobserved in the desert. In addition, they can be found in the fluvial environment in the caseof sand (in supply insufficient to cover the bed completely) migrating over an immobilegravel bed.

6.7.1.2 Antidunes. Antidunes are distinguished from dunes by the fact that the undula-tions of the water surface are nearly in phase with those of the bed. They are associatedwith supercritical flow in the sense that

F2 1k tanh (k) (6.96)

Antidunes may migrate either upstream or downstream. Upstream-migrating antidunesare usually rather symmetrical in shape and lack a slip face. Downstream-migratingantidunes are rarer; they have a well-defined slip face and look rather like dunes. The dis-tinguishing feature is the water surface undulations, which are pronounced in the case of

antidunes.The potential-flow criterion dividing upstream-migrating antidunes from downstream-

migrating antidunes is

F2 ktan

1

h (k) (6.97)

Values lower than the value in Eq. 6.97 are associated with upstream-migrating antidunes.

6.7.1.3 Ripples. Ripples are dunelike features that occur only in the presence of a vis-cous sublayer. They look much like dunes because they migrate downstream and have apronounced slip face. They generally are much more three-dimensional in structure than

are dunes, however, and have little effect on the water surface.

A criterion for the existence of ripples is the existence of a viscous sublayer. Recallingthat the thickness of the viscous sublayer is given by v 11.6v/u*, it follows that ripplesform when

U

gH





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u*D 11.6 (6.98)

6.7.1.4 Bars. Bars are bedforms in rivers that scale the channel width. They include

alternate bars in straight streams, point bars in meandering streams, and pool bars in braid-ed streams. In straight streams, the minimum channel slope S necessary for alternate-barformation is given by

S (6.99)

(Jaeggi, 1984), whereB is the channel width,Dg is the geometric mean size of the bed sed-iment, as given by Eq. (6.82a), andMis a parameter that varies from 0.34 for uniform-

sized bed material to 0.7 for poorly sorted material.Scour depth (Sd) caused by alternate bar formation can be estimated with

Sd 0.76AB , (6.100)

where AB is the total height of the alternate bar.

6.7.1.5 Progression of bedforms. Various bedforms are associated with various flowregimes. In the case of a sand-bed stream with a characteristic size less than about 0.5 mm,

a clear progression is evident as flow velocity increases. This is illustrated in Fig. 6.14.

The bed is assumed to be initially flat. At low imposed velocity U, the bed remains flatbecause no sediment is moved. As the velocity exceeds the critical value, ripples areformed first. At higher values, dunes form and coexist with ripples. For even higher veloc-ities, well-developed dunes form in the absences of ripples.

At some point, the velocity reaches a value near the critical value in the Froude sense:

B

6

DB

g

0.15

exp1.07

DB

g

0.15M

12.9

DB

g

Ripples

Bars

Wavelength less thanapprox 1 ft; height lessthan approx 0.1 ft.

Lengths comparable tothe channel width.Height comparable tomean flow depth.

Roughly triangular inprofile, with gentle,slightly convexupstream slopes anddownstream slopesnearly equal to theangle of repose.Generally short-crestedand three-dimensional.

Profile similar toripples. Plan formvariable.

Move downstream withvelocity much less thanthat of the flow.Generally do not occurin sediments coarserthan about 0.6 mm.

Four types of bars aredistinguished: (1)point, (2) alternating,(3) transverse, and (4)tributary. Ripples mayoccur on upstreamslopes.

Table 6.4 Summary of Bedform Effects on Flow Configuration

Bed Form or Behavior andConfiguration Dimensions Shape Occurrence

(1) (2) (3) (4)





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6.38 Chapter Six

Dunes

Transition

Flat bed

Antidunes

Wavelength and heightgreater than ripples butless than bars.

Vary widely

Wave length 2V2/g(approx)* Height

depends on depth andvelocity of flow.

Similar to ripples.

Vary widely.

Nearly sinusoidal inprofile. Crest length

comparable towavelength.

Upstream slopes ofdunes may be coveredwith ripples. Dunesmigrate downstream inmanner similar toripples.

A configurationconsisting of a

heterogeneous array ofbed forms, primarilylow amplitude ripplesand dunes interspersedwith flat regions.

A bed surface devoidof bed forms. May notoccur for some rangesof depth and sand size.

In phase with andstrongly interact with

gravity watersurfacewaves. May moveupstream, downstream,or remain stationary,depending on propertiesof flow and sediment.

Table 6.4 (Continued)

Bed Form or Behavior and

Configuration Dimensions Shape Occurence(1) (2) (3) (4)

*Reported by Kennedy (1969).Source: Vanoni (1975).

i.e., Eq. (6.94). Near this point, the dunes often are suddenly and dramatically washed out.This results in a flat bed known as an upper-regime (supercritical) flat bed. Furtherincreases in velocity lead to the formation of antidunes and, finally, to the chute and poolpattern. The last of these is characterized by a series of hydraulic jumps.

In the case of a bed coarser than 0.5 mm, the ripple regime is replaced by a zone char-acterized by a lower-regime (subcritical) flat bed. Above this lies the ranges for dunes, theupper-regime flat bed, and antidunes.

The effect of bedforms on flow resistance is summarized in Table 6.4. As noted earli-er for equilibrium flows in wide straight channels, the relation for bed resistance can beexpressed in the form

b CfU2 (6.101)

where Cf denotes a bed-friction coefficient. If the bed were rigid and the flow wererough, Cf would vary only weakly with the flow, according to the logarithmic lawembodied in Eq. (6.12). As a result, the relation between b and Uis approximately par-abolic.





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The effect of bedforms is to increase the bed shear stress to values often well abovethat associated with the skin friction of a rough bed alone. In Fig. 6.15, a plot ofbversus U is given for the case of an erodible bed. At extremely low values of U, theparabolic law is followed. As ripples, then dunes are formed, the bed shear stress risesto a maximum value. At this maximum value, the value ofCf is seen to be as muchas five times the value without dunes. It is clear that dunes play an important roleregarding bed resistance. The increased resistance results from form drag in the lee of

the dune.As the flow velocity increases further, dune wavelength gradually increases and dune

height diminishes, leading to a gradual reduction in resistance. At some point, the dunesare washed out, and the parabolic law is again satisfied. At even higher velocities, theform drag associated with antidunes appears; it is usually not as pronounced as that ofdunes.

6.7.2 Dimensionless Characterization of Bedform Regime

Based on the above arguments, it is possible to identify at least three parameters govern-ing bedforms at equilibrium flow. These are Shields stress *, shear Reynolds numberRe u*D/, and Fr. A characteristic feature of sediment transport is the proliferation ofdimensionless parameters. This feature notwithstanding, Parker and Anderson (1977)showed that equilibrium relations of sediment transport for uniform material in a straightchannel can be expressed with just two dimensionless hydraulic parameters, along with aparticle Re (e.g.,Rep orRe) and a measure of the denstiy difference (e.g.,R).

FIGURE 6.15 Variations of bed shear stress b and Darcy-Weisbach friction fac-torfwith mean velocity Uin flow over a fine sand bed. (Raudkivi, 1990)





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6.40 Chapter Six

FIGURE 6.16 Bedform predictor proposed by Simons and

Richardson (1966).

In the case of bedforms, then, the following classification can be proposed:

bedform type function (1, 2;Rep,R) (6.102)

Here, any independent pair of hydraulic variables 1, 2 applicable to the problem can bespecified because any one pair can be transformed into any other independent pair. Forexample, the pair * and Fr might be used or, alternatively, SandH/D.

One popular discriminator of bedform type is not expressed in dimensionless form atall. It is the diagram proposed by Simons and Richardson (1966), (Fig. 6.16). In the dia-gram, regimes for ripples and dunes, transition to the upper-regime plane bed, and upper-regime plane bed and antidunes are shown. The two

sedimentation and erosion

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