A New General Method for Predicting the Frictional Characteristics of Alluvial Streams

(1)

(2)

I

A NEW GENERAL METHOD FOR PREDICTING

THE

FRICTIONAL

CHARACTERISTICS

OF ALLUVIAL

STREAMS

W R White BSe PhD CEng MICE

E

Paris

Ingegnere Civile

R Bettess BSe PhD

Report

No Ir

187 July 1979

Crown Copyright

Hydraulics

Research Station

Wallingford

Oxon

OX10 8BA

(3)

I

ABSTRACT

I

'

I

'I

I

A new method for predicting the frictional resistance of alluvial

channels is developed using experimental data. The method is

exhaustively tested on an extensive range of field and flume data

and compared with the three existing methods due to Einstein and

Barbarossa, Engelund and

Raudkivi,

(4)

I

NOTATION

A

d

(m)

Value of Fgr at initial motion (Ackers, White)

Mean depth of flow

I

d'

D

(m)

Mean depth associated with grain roughness (Engelund)

Sediment diameter

I

D

3s,Dso,D6s

(m)

Sediment diameters for which 35%, 50%,65% ofthe

sample is finer

Dgr

Dimensionless grain size (Ackers,

White)

I

F

Fcg

Froude number

Sediment mobility, coarse grains (Ackers, White)

I

Ffg

Fgr

Sediment mobility,

Sediment mobility (Ackers, White)

fine grains (Ackers,

White)

I

g

ks

(m/s

(m)

2)

Acceleration due to gravity

Roughness height

I

~'

n

(m)

Roughness height (Einstein)

Transition exponent (Ackers, White)

R

(m)

Hydraulic radius

I

R'

(m)

Hydraulic radius associated with the grain roughness

R"

(m)

Hydraulic radius associated with form roughnesses

I

s

Specific gravity of sediment

I

Sf

Friction slope

v.

(mis)

Shear velocity

,

(mis)

Shear velocity associated with grain roughness (Einstein)

v.

I

v.

"

(mis)

Shear velocity associated with form roughness (Einstein)

(v·)cr

(mis)

Critical shear velocity

I

V

(mis)

Mean velocity

y

Mobility number

I

Z

Ratio of depth to partiele diameter

(J

Bed shear (Engelund)

I

(J'

_{Shear due to}

skin

friction (Engelund)

v

(m

2

Is)

Kinematic viscosity of fluid

I

P

(kg/I)

Density of fluid

Ps

(kg/I)

Density of sediment

I

ÀcaJc

Predicted friction factor

Àobs

Observed friction factor

I

t/I'

Intensity of shear on representative partiele (Einstein)

T

Tractive shear

I

T

,

Tractive shear associated with grain roughness

"

Tractive shear associated with form roughness

T

I

(5)

I

Page

I

INTRODUCTION

1 I

PREVIOUS RESEARCH

1 I

DEVELOPMENT OF NEW METHOD

4 COMPARISON OF METHODS

6 I

CONCLUSIONS

8 I

ACKNOWLEDGEMENTS

9 I

I

REFERENCES

₉

APPENDICES:

1 Details of the new method

2 Summary of data and performance of new method

11

13 I

I

TABLES

I

1 Flume data used to develop new method (sand)

2 Additional flume data (sand)

3 Flume data (lightweight sediments)

4 Field data

I

FIGURES

I

1 Friction loss due to channel irregularities as a function of sediment transport

(Einstein)

2 Relationship between bed shear and total shear (Engelund)

I

3 Resistance in an alluvial channel as a function of the entrainment function

(Raudkivi)

I

4 FCgversus Fgr for selected data

5 Shear relationship based on D35 of the parent material (New method)

6 Shear relationship based on D

6S

of the surface material (New method)

7 Comparison of predicted and observed friction factors (Einstein)

8 Comparison of predicted and observed friction factors (Engelund)

9 Comparison of predicted and observed friction factors (Raudkivi)

10 Cornparison of predicted and observed friction factors (New method)

11 Discrepancy ratio versus Fgr (New method, selected data)

I

(6)

CONTENTS (Continueel)

FIGURES (Continued)

12

13

14 Discrepancy ratio versus Z

(New method, selected data)

Distribution of discrepancy ratios (Einstein, Engelund, Raudkivi)

Distn1Jution of discrepancy ratios (New method)

I

,

I

(7)

I

INTRODUCTION

To calculate flow in an open channel an engineer is almost inevitably

faced with the problem of determining the frictional losses on the

boundary of the channel. For example, a knowledge of the frictional

resistance is required for the design of irrigation channels, river

improve-ment works or for the determination of sediimprove-ment transport rates. In this

report we propose a new approach to the prediction of channel roughness

and we compare this new technique with a nurnber of better known

methods.

I

For artificial, regular channels which are fixed in shape and carry little

sediment there are data readily available which can be used as a basis

for the estimation of appropriate friction factors. When natural channels

are considered the problems of estimating the friction losses grow

considerably. In this case, not only must the frictional losses due to the

composition of the banks and bed of the channel be estimated but also

due allowance must be given for the effects of channel irregularities and

other factors. If one considers channels with movable beds the problems

are greatly compounded. The frictional losses are dependent upon the

shape of the bed but this is influenced by the transport of the sediment.

The sediment transport, however, depends upon the fluid motion and is

hence inseparable from the determination of the frictional losses.

Sediment movement has a number of effects upon the velocity

distribu-tion in a channel. Under different condidistribu-tions the bed may take up various

configurations such as ripples or dunes which influence the resistance of

the bed. The difficulties, however, are not just restricted to the problems

associated with the determination of the type and size of bed features.

Experiments have indicated that the motion of sediment can influence

the velocity distribution within the flow. It would seem that the presence

of a significant suspended load dampens turbulence close to the bed and

may lead to a reduction in the friction factor (Yalin, 1972). Thus studies

have shown that the resistance of a mobile plane bed is different from

that of a plane bed with no motion.

All the methods discussed are steady state methods in which the friction

is dependent upon the local values of the variables involved, that is, the

friction is determined by the values of the variables at that point. They

also assume that the friction does not depend upon the history of previous

flows but only on the conditions prevalent at that time. The significance

of this assumption should be appreciated. In the cases where the transport

rate is zero it is clear that the friction will depend upon the past flows as

the friction will be influenced by bed features present which were created

by earlier, larger flows. These bed features will remain and influence the

flow until the sediment transport rate is large enough to remove them.

It may be that the type and size of bed feature depends not only on the

flow conditions at the present but also on the history of previous flows.

I

PREVIOUS RESEARCH

We will consider the three existing methods due to Einstein and Barbarossa

(1

9 52), Engelund (1966, 1967) and Raudkivi (1967). In two of the methods,

those due to Einstein and Barbarossa and due to Engelund, the total

resistance is divided into the sum of the resistance of the bed surface and

the resistance of the bed formations. In the remaining method due to

Ra

udkivi no such division is made.

I

It will be assumed that any property concemed with sediment transport in

tw

o

-dimensional, free

-

surface flow can be completely specified by the

f

o

ll

o

wing

quantities:-I

P, v, Ps'

D , d , v. and g ,

I

where

P

is the density of fluid (kg/I)

v

is the kinematic viscosity (m

2

/s)

(8)

I

Ps is the density of solids (kg/I),

D

is the equivalent partiele diameter (m),

d

is the water depth (m),

v.

is the shear velocity

(mIs),

and

g

is the acceleration due to gravity

(m/s

2).

I

Four non-dimensional numbers may be associated with these quantities

and for the purpose of this work we will use the following

:3

b~

-b

vz.

I

(~_1\ 1/3

Dgr=

h71

D,

v

....(1)

v.

2

y

=

(s-l)gD

Z

_ d

_-5

,

and

s

=

Ps

P

I

....(2)

....(3)

I

....(4)

We now consider the significanee of these non-dimensional variables in

the problem of determining the roughness. One would expect that the

roughness of the surface depends upon the size of the bed material which

is

indicated by the value of Dgr. The friction should also be affected by

the quantity of sediment in motion and hence should depend upon the

mobility number Y and the density ratio, s. The value of Z will be

significant if the friction is dependent upon how the sediment is

distri-buted in the bulk of the flow.

I

Einstein and Barbarossa (1952)

The earliest of the three methods we consider is due to

Einstein and

Barbarossa (1952).

The fundamental assumption on which their work is

based is that the friction loss due to the roughness of the bed and that

due to the irregularities in the channel are independent and additive,

that is, if

To

is the total shear stress on the bed then

ps

f_

!

...--i:

~

-0

?

'i

xi

er:

_r

d

R

5 c ::c TO

=

To'

+

To" ,

....(5)

where

TO'

is the shear stress due to the roughness of the bed surface

and

To"

is

the shear stress due to all other causes. Since

r-

Sf-!g =

gsR ,

....(6)

P

where R is the hydraulic radius,

we can define R' and Ril such that

I

'

I

, 11

To

=

gsR'

and

!2....

=

gsR".

P

\

4,

P

~$

Equation (5) then implies

f

....(7)

I

R

=

R'

+

Ril.

I

....(8)

It may then be assumed that the average velocity is given by any of the

standard formulae such as Manning's equation using R' as the hydraulic

radius and using

D65

as representative grain roughness. Einstein and

Barbarossa recommended the use of the Manning-Strickler equation

v _

R'

1/6

-,

- 7.66 (~)

,

v.

Ks

....(9)

I

where v..

=

y'gSR'

_....(10)

....(11)

I

It then remains to consider the friction loss due to all other causes.

Einstein and Barbarossa assume that this friction loss is a function of the

2

(9)

I

/l-<'

J

g_ ~

f

G'

-{ ,/

f2.Y

-/y.r

l

' ~

I

Engelund (1966, 1967)

sediment transport and as such must be a unique

f

unction of Einstein's

shear parameter

1/1',

where

1/1'

=

Ps-P 0

35

P

R'S

f

....(12)

Thus we have

v" =

f

(

1/I

'

)

,

v.

..

.

.(13)

where

v."

=

ygSfR"

....(14)

and f is some unknown

f

unction.

The relation between

vt-;:

and ",' was determined empirically from data

for a number of rivers most of whi

c

h

a

re in the Missouri River Basin. A

representative sample of the data used in our work

,

together with Einstein

and Barbarossa's curve

,

is shown in Fig 1

.

In practice the application of

the method involves using an iterat

i

ve

m

ethod to find val

u

es of R

'

and R

"

such that

R

=

R'

+

R"

.

...(15)

Equation (I3)

,

is given by Einstein and Barbarossa only for a restricted range of ",',

approximately 0

.

5 :s;;; ",'

:s;;;

40. In performing the calcula

t

ions it was found

necessary to extrapolate the curve beyond th

i

s range for a small proportion

of the data

.

For flow over a dune covered bed Engelund (I966

,

1967) claimed that

the friction loss could be considered as the sum of the loss due to skin

friction and the loss due to a series of expansions of the flow on the

downstream sides of the dunes. Using a sealing argument based on a belief

that all flows over a dune bed are similar

,

Engelund showed that the

dimensionless bed shear due to the skin friction 8' depends only on the

total dimensionless bed shear 8, that is

,

8'

=

8'(8)

.

....(16)

Engelund then determined this relationship empirically using flume data.

A representative sample of the data used in our work together with

Engelund's curve is shown in Fig 2

.

It

proved necessary to extrapolate Engelund's curve for values of 8 less

than 0.1. Since, on physical grounds, 8' is always less than 8 and the

curve given by Engelund approaches the line 8

=

8'

,

it was decided to

extrapolate the curve in such a way

.

that it was asymptotic to the line

8 =

8', with the two curves being indistinguishable for 8 less than 0.03,

which corresponds, approximately, to the threshold of movement.

If the total bed shear is known equation (I6) may be used to determine

the bed shear due to the skin friction and hence the corresponding shear

velocity v.'. Engelund then assumed that the mean velocity

V

can be

given by the equation

': =

2.50 log, d'

+

6.0 ,

v.

ks

....(17)

where k,

=

20

65

and

d' is determined from v.'

.

....(I8)

3

(10)

Raudkivi

(1967)

DEVELOPMENT

OF NEW

METHOD

I

It must be emphasised that it was assumed that the effects of viscosity

may be ignored and, as Engelund stresses, this may imply that the

method will not be applicable when the bed features are ripples. The

uncertainty lies in the fact that though previous work indicates that

viscosity is significant in the formation and amplitude of ripples

(Yalin, 1972) it is unclear what effect it has upon the corresponding

roughness.

I

Raudkivi (1967) was the first to suggest that significant quantities

associated with the problem rnight involve the amount by which certain

variables exceed those critical values at which sediment motion fust takes

place. For example, the important variabie may not be the shear velocity,

v., but by how much it exceeds the shear velocity corresponding to

initiation of motion. Raudkivi plotted values of V/";v.2 -{_V.)2

Cl

against

mobility for a large quantity of data. As Raudkivi points out

,

thls

indicates a trend though there is an appreciable amount of scatter at

high transport rates. For lower transport rates

,

ie smaller values of

mobility, however, a satisfactory relationship can be given. The curve

used in the calculations is shown in Fig 3 together with a representative

sample of data that was used in our work. This gives a simple method

of detemtining the friction factor provided the critical shear velocity is

known. This was deterrnined using Shields' curve for the initiation of

sediment transport.

I

Ackers and White (1973) have put forward a theory for the prediction

of sediment transport rates in alluvial channels. They assumed that fine

sediments travelled mainly in suspension and that the quantity in motion

would correlate with the tot al boundary shear stresses. Coarse sediments

were assumed to travel mainly as bed load and quantities were found

to correlate with the grain shear stresses whether or not the bed of the

channel was plain. Fine and coarse sediments were defined in terms of

the dimensionless grain size Dgr and analysis of the data suggested that

sediments became truly 'fme' at Dgr

=

1 (~ 0.04 mm for sand) and

'coarse' when Dgr exceeded 60 (~ 2.5 mm for sand). Intermediate sizes

were termed 'transitional' and the relative influences of total and grain

shear forces were established in terms of Dgr.

Sediment movement was predicted in terms of a sediment mobility number

based on the ratio of the shear forces to the immersed weight of the

particles. The innovation at this stage was to set up the equations defining

this mobility number in such a way that only the relevant shear forces

were used, ie total shear for fine sediments, grain shear for coarse

sedi-ments and an intermediate value, depending on Dgr, for the transitional

sediments.

I

"

I

,

I

The general form of their mobility number is

I

V n

[

V

]l-n

Fgr

=

v'gD(:-I)

. ..;n

IOgl0(10d/D)

..

..(19)

'

I

where n is an exponent which varies from 1

.

0 for fine sediments

(Dgr

=

1.0) to 0.0 for coarse sediments (Dgr

=

60). Thus for fine

sediments

F

-

v.

fg -

v'gD(s-l)

.

..(20)

I

and for coarse sediments

F

=

V

cg

v'gD(s-l)..;n

loglo(10d/D)

I

....(21)

Engelund suggested a unique correlation between grain shear and total

shear but his analysis ignored the effects of viscosity and hence is probably

only applicable to coarse sediments forrning dunes as bed features. For

4

(11)

I

transitional sizes

variabIe, Dgr' and to introduce the concept of the effective shear for this

it

is probably necessary to introduce at least one additional

ra

nge of partiele sizes.

I

T

o

test this hypothesis a selection of flume data covering a wide range of

parti

ele sizes was plotted in terms of total shear, as indicated by F

fg,

and

e

ff

e

ctive

s

hear as indicated by Fgr. The results are shown in Fig 4

.

The

se

diments for each data set were nominally uniform in size but where

va

ria

ti

o

ns occurred values of Dgr

,

Ffg and Fgr have been based on the

D

35 s

ize of the parent mate rial. There is clearly a progression away from the

F

fg

=

Fgr line with increasing values of Dgr and the line through each data

se

t c

o

nverges towards the Ffg

=

Fgr line at

,

or ar

o

und, the value of th

e

mobility number corresponding to the threshold of movement. Thus one

o

f

Ra

udkivi's assumptions, namely that the frictional characteristies of

all

uvial channels rnight depend on 'excess' shears, appears to be valid in

terms of this new form of presentation. Data for coarse sediment

s

only

c

o

ver a lirnited range of mobilities, say 0.2

<

Fgr

<

0.3 ,

and hence it is

di

fficult to confirm the general F

fg

,

Fgr pattern for Dgr values in excess of

6

0 . All that can be said at the present stage is that the limited amount

o

f data available does not contradiet the general pattern.

I

Th

bou

ere is the usual amount of scatter associated with measurements of loose

ndary fluvial processes in Fig 4 but the data seem to suggest a

f

u

nc

t

i

o

nal relationship of the form

I

F

gr-

A

= ti>

{

D

j

F

fg-

A

gr

wi

t

hin the range F

<

0.8 and 1

<

Dgr

<

60. F

ig

5 shows this functi

o

nal relationship for nominally uniform flume data

in which

D

gr is based on the D35 size of the parent material. A curve was

fitted

as

follows:-..

..(22)

I

...

.

(23)

I

D35 o

f the parent mate rial was chosen as the repre

s

entative partiele size

s

in

c

e this is the size of sediment on which the Ackers and White transport

met

hod is b

a

sed

.

It

may be argued, however, that the properties of the

m

a

teri

a

l on the surface of the bed are more relevant to the deterrnination

o

f the roughness than those of the parent materia

l.

The composition

o

f

t

h

e s

urf

ace

of a bed is different from, but depen

d

ent upon, that of the

pa

rent material. It would seem, therefore, that the r

e

p

r

esentative partiele

s

iz

e sh

o

uld be

det

e

rmined from th

e

surface

s

ize

-

d

i

stri

b

uti

o

n rat

h

er than

f

r

om

th

at o

f the bulk material

.

Experiments

b

y

D

ay

(1

9

79 )

on

t

he

mo

v

e

-ment o

f

g

r

ad

e

d

s

e

diments have indicated that the appropriate

p

artie

l

e s

ize

f

r

om w

hi

c

h

t

o

de

termine transport phen

o

mena may

b

e

D6S o

f

the su

r

f

a

ce

ma

t

erial a

n

d

hence the calculations were r

e

peated using thi

s

as t

h

e re

presen-tati

v

e

s

i

ze

.

I

_{Fig 6 sho}

_ws

_t

_{he same data plotted in terms of these}

_D

_{6S siz}

_e

_{s. In th}

_i

_s

case the

c

urve w

as:

I

F

-

A

[

1 - ~

=

0.70

1 Ffg-A

.

..(2

4)

I

Each point plot

t

e

d in Figs 5 and 6 represents the mean of a se

rie

s of

measurements

carried o

u

t in a giv

e

n flume wi

t

h

a

gi

v

en s

ed

im

e

nt

s

ize

as

defined b

y

D

gr.

Having establish

e

d

equat

i

ons (23) and (2

4 ) from s

elected flume

d

ata (sands)

,

see Table 1

,

we

ha

v

e

t

e

s

t

e

d thes

e

r

e

lation

s

hip

s ag

ain

st an extended range of

flume

data (

s

and

s

), see

T

able 2, against da

ta

f

o

r

lightwe

i

ght sediments

,

see

I

5

(12)

,I

Table 3, and against field data

,

see Table 4. The data in Tables 2 to 4

represent a gross extrapolation from the formulative data and hence

provide a good check on the form and general application of the new

method; see later for details.

In the proposed method there is no explicit mention of the typ

e

of bed

features that are formed. Brooks (1958) believes that the average flow

velocity is not a unique function of depth, slope and sediment size but

also depends upon the form of the bed features, which would imply that

there is no unique value of Fgr for given values of F

fg

and Dgr' contrary

to equation (22). Einstein and Chien (1958), however, believe that the

multiplicity of velocity occurs over too narrow a range of depth to be

significant for field conditions. Consideration was therefore given to the

type of bed feature associated with each point in Fig 4 to investigate

whether such multi-value effects could be determined and associated with

types of bed feature. The data, however, presented no discernible trend

which indicated multiple values dependent upon the type of bed feature

present. Nor does it seem possible, using this approach, to predict which

type of bed feature might be expected.

I

It

will

have been noted that all the methods rest on some empirical

relationship derived from appropriate observations. Thus the reliability of

the methods depends upon the reliability of the data that were used and

the range of applicability of the method may be restricted to that of the

data. The curves obtained by the various authors may be extrapolated

but this may lead to significant errors

.

An important consideration then

is the range of conditions over which the methods may be applied.

I

COMP ARISON O

F

METHODS

The methods have been evaluated on a large quantity of field and flume

data from a number of different sources. These data cover a large range

of Dgr, Y and Z. Data which have a range of values of s are difficult to

obtain but a number of measurements for lightweight materials have been

included. For each flume or natural channel the value of Dgr and s are

virtually constant since these variables depend upon the sediment and

water properties. The data cover the range 1.0 ~ Dgr ~ 1450 which for

sand in water corresponds to a range of partiele sizes 0.04 ~ D (mm)

~ 68. For each data set from a particular flume or channel there may be

a large range of values of Y and Z depending upon the range of flow

conditions that were measured.

I

'

I

To avoid the complexities of critical and supercritical flow data were

selected with a Froude number less than or equal to 0.8. Thus this work

is only applicable to subcritical flow.

I

There are uncertainties in the data associated with hysteresis effects.

Experiments have shown that bed features, and hence frictional forces,

take time to develop. In a recirculating flume, running with constant

discharge, equilibrium may take up to 72 hours to establish.

In

a river

equilibrium may never be established because of continuous changes in

flows. Hence some of the data used for developing and testing the

methods

will

inevitably fall into the 'non-equilibrium' category and this

will

be revealed in terms of random differences between predicted and

observed values. We have no way of ascertaining which data fall into

this category.

The data used were an augmented set of that used by White

,

Milli and

Crabbe (1973) in their study of sediment transport theories and an

indication of the range of the data used is given there

.

I

Each method was applied to every friction measurement and the

discrep-ancy ratio, given by

I

6

(13)

I

,

I

Àcak:

Àobs

was

determined, where

....(24)

In a few cases it was necessary to extrapolate the appropriate method.

For each data set the mean discrepancy ratio was determined as well as

the maximum and minimum values. These values were then plotted

against Dgr, Figs 7 to 10. The spread of errors may be random or may

be regarded as being due to the failure of the method to take proper

account of variations in Y, Z and s or, possibly, errors in the observed

values. Def1cienciesin the method in taking account of Y and Z may

be indicated by plotting values of the discrepancy ratio against either Y

or Z for particular values of Dgr.

There is a difference between the comparisons of the three previous

methods and that of the new method. The empirical relationships within

the three established methods were derived on a limited spectrum of data

covering only a small range of partiele sizes. The new method has been

developed on a wider range of partiele sizes but the data used were

restricted to flume experiments with sand. Thus, testing the methods

against all the readily available data including lightweight sediments and

field data from large rivers represents a larger extrapolation for the

established methods than for the new method. This does not imply,

however, that one should dismiss the level of agreement between theory

and observation as inevitabie because of the approach used. To obtain

good agreement it is necessary that the form of the equations should be

appropriate. If a method has been developed on a large range of data and

good agreement has been obtained on an even larger set of data there is

a clear indication that the method is of general application.

Einste

in

and Barbarossa

(1

952 )

The Einstein and Barbarossa method, see Figs 7 and 13, tends to

over-predict the friction factor though the scatter of results is large. For only

21 %

of the data was the friction factor calculated by the Einstein and

Barbarossa method within a factor of 2 of the observed value. These

findings are in line with those of Garde and Ranga Raju (1966) and

Yalin (1972) who have criticised the method on theoretical grounds.

As mentioned previously, to perform the calculations for some of the data

it wa

s

necessary to extrapolate the curve given by Einstein and

B

arbarossa

beyond the range, 0.5 ~

vl ~

40. We were concerned lest the errors were

introduced by the extrapolation and were not inherent in the method.

Using the data available, therefore, we calculated the values of both

V/v." and

1/1'.

In all cases the value of

1/1'

lay within the range,

0 .5 ~

1/1' ~

40, but

in

certain cases the values of

vt-;:

and

1/1'

were such

that the points were some considerable distance from the curve given by

Einstein and Barbarossa. The method, however, assumes that all the points

lie on the given curve and it was in. trying to force this condition that it

was necessary to use extremely large or small values of

1/1'

in the

calcula-tion of the friccalcula-tion factor. It would therefore appear to us that the souree of

the errors lies not in the extrapolation of the curve but is inherent in the

method which forces a relationship between

vt-:'

and

1/1'

which is not

always substantiated by the data.

Engelund

(

1966

,

1967)

The results, see Figs 8 and 13, are very good for large Dgr (Dgr

>

2

0 ) but

there is a significant amount of scatter for small Dgr. There is a slight

tendency to over-predict, which is more noticeable for lightweight materials

though the method under-predicts on field data for small values of Dgr. 83%

o

f the calculated results lie within a factor of 2 of the observed value

.

38% of the results tie with 0.8 and 1.25 of the observed value.

7

(14)

Randkivi (1967)

New approach

(1979)

CONCLUSIONS

1 I

The results, see Figs 9 and 13, are good for large Dgr (Dgr

>

20) but

there is a marked scatter for small Dgr. As smaU Dgr are more likely

to be associated with large mobilities the problems with smaU values of

Dgr may be associated with the large scatter encountered in the Raudkivi

curve (Fig 3) for large mobilities. There is a tendency to over-predier

which could be associated with the asymptotic value of the curve chosen

for mobility greater than 1. There is no noticeable difference between the

results for the lightweight materials and the rest of the data. 73% of the

calculated results lie within a factor of 2 of the observed value. 25% of

the results lie within 0.8 and 1.25 of the observed value.

I

The results, see Figs 10 and 14, for large Dgr (Dgr

>

20) are very good

but there is scatter for smaU Dgr. For Dgr greater than 40 aU the results,

with the exception of one data set, lie within a factor of two of the

observed value. The method appears to

over-predict

slightly for lightweight

material, the method over-prediering on average by approximately

50%.

Therefore, the errors for each lightweight data set were studied to

deter-mine if they depended in a systematic way upon the

specific

gravity but

no trend was discernible. 89% of the total calculated results lie within a

factor of two of the observed value and 44% within 0.80 and 1.25 of

the observed value.

I

A study was made to see

if

the discrepancy ratio depended upon Fgr and

Zinsome

defmite

manner,

which would imply that the theory did not

take proper account of variations in Fgr and Z, or whether the variation

was random which would imply that there was no systematic bias in the

method. The appropriate values are plotted for a selected range of data

in Figs 11 and 12 which indicate that any systematic trend is smaU.

I

2 A new general method is proposed for predicting the frictional characteristics

of aUuvial streams using the non-dimensional variables proposed by Ackers

and White (1973) in their approach to sediment transport. The method is

developed using flume experiments with sand, see Table 1.

The performances of the new method and the established methods of

Einstein and Barbarossa, Engelund and Raudkivi are evaluated using an

enlarged set of data which included lightweight sediments and information

from large rivers, see Tables I, 2, 3 and 4. These data cover the range

,

I

1 ~ Dgr

<

1450,

(0.04 mm to 68 mm sand sizes)

F

<

0.8 ,

1.07 ~

s

<

2.7 I

'

I

3 The present review suggests that the method proposed by Einstein and

Barbarossa (1952) is unreliable. Only 21% of the calculated friction factors

are within a factor of 2 of the observed values, see Figs 7 and 13. The

method has asystematic and significant tendency to predict friction factors

which are too high.

The performance of the method proposed by Engelund (1966, 1967) is

found to be generally reliable, There is excellent agreement between

calculated and observed friction factors for larger sediments (Dgr

>

20).

Some scatter occurs at the lower end of the Dgr range but

this

may be

associated with the quality of the data and, in particular, the possibility

that equilibrium conditions had not been established, see Figs 8 and 13.

83% of calculated friction factors are within a factor of 2 of the observed

values; 38% have discrepancy ratios between 0.8 and 1.25.

The method proposed by Raudkivi (1967) is less satisfactory. Reasonable

agreement is obtained for coarse sediments but excessive scatter develops

I

4 I

I

5

8 I

(15)

I

towards the lower end of the Dgr range, see Figs 9 and 13. The method

appears to be particularly unreliable at high rates of sediment transport.

6 The new approach (I979) has performance characteristics similar to the

Engelund approach but with some improvement in the overall agreement

with the data, see Figs 10 and 14. When the method is based on the D35

size of the parent bed material 89% of calculated friction factors are

within a factor of 2 of the observed values; 48% have discrepancy ratios

between 0.8 and 1.25. The figures based on the D65 size of the surface

material are marginally inferior. A study to see whether the discrepancy

ratios varied systematically with either the sediment mobility or the

depth/diameter ratio proved unfruitful, see Figs 11 and 12.

7 The new method gives little information on the type of bed feature which

will occur for a specific type of sediment subjected to specific hydraulic

conditions.

I

'

8 Appendix I to this report gives step by step instructions on the use of

the new method.

9 Appendix 2 provides a summary of the data and the performance of the

new method.

I

ACKNOWLEDGEMENTS The investigation of which this is the official HRS account was carried out

in Mr A J M Harrison's Fluvial Hydraulics Division by Dr W R White's

Section. Enio Paris gratefully acknowledges support under U.022 del

ProgelloFinalizzalo del CNR 'Conservazione del Suolo', subprogello

'Dimanica Fluviale', Firense.

I

REFERENCES

Ackers P and White W R, 1973, Sediment transport: new approach and

analysis. Proc ASCE, JHD, 99, HY11, pp 2041-2060.

I

Brooks N H, 1958, Mechanics of streams with movable beds of fine sands.

Trans ASCE, 123, pp 526-549.

Day T J, 1979, A study of the transport of graded sediments. HRS

Report, to be published.

I

Einstein H A and Barbarossa N L, 1952, River channel roughness. Trans

ASCE, 117, pp 1121-1132.

Einstein H A and Chien N, 1958, Discussion of Brooks (1958), Trans

ASCE, 123, pp 553-562.

Engelund F, 1966, Hydraulic resistance of alluvial streams. Proc ASCE,

JHD, 92, HY2, pp 315-326.

I

Engelund F, 1967, Hydraulic resistance of alluvial streams (Closure to

Discussion). Proc ASCE, JHD, 93, HY4, pp 287-296.

I

Garde R J and Ranga Raju KC, 1966, Resistance relationships for alluvial

channel flow. Proc ASCE, JHD, 92, HY4, pp 77-100.

I

Raudkivi A J, 1967, Analysis of resistance in fluvial channels. Proc ASCE

JHD, 93, HY5.

I

White W R, Milli H and Crabbe A D, 1973, Sediment transport: an appraisal

of available methods. Volume 1: Summary of existing theories, Volume 2:

Performance of theoretical methods when applied to flume and field data.

HRS Report INT 119.

I

Yalin M S, 1972, Mechanics of sediment transport. Pergamon Press.

9 I

(16)

I

APPENDICES

I

(17)

I

_Details

_{of the new method}

APPENDIX 1 _{Friction factors are detennined using the following}

procedure:-I

I

1 Calculate

v

*

=

ygas;

2 Calculate

Dgr

=

D [gc:-;n}/3

,

where D is D3S (bed material)

or D6S (surface material) ,

I

and then calculate the parameters n and A

I

n

=

0.0 A

=

0.17 J

Dgr ~ 60

J

I~Dgr<60

I

'

I

n

=

1.0 - 0.56 IOglODgr

A

=

.JiÇ

+

0.14

3 Calculate

F

-

v*

fg -

v'gO(s-l)

4 Calculate

Fgr using appropriate equation

I

a) if D

=

D3S (bed material)

I

Fgr

-

A

F

_fg

-

A

=

1.0 - 0.76[1.0

-

e(logloDgr).

1 17]

b) if D

=

D6S (surface mate rial)

I

Fgr - A

Ffg - A

1.0 - 0.70 [1.0 -

e

1.4

(loglO

1 D

gr) .

2

6SJ

5 Calculate

V using the equation

I

6 Calculate

À

=

B(V)*

2

V

I

11 I

(18)

I

_{Summary of data and}

APPENDIX 2 I

performance of the new

method

I

Note

I

In

this

Appendix the following computer notation is

used:-D

D(m)

Sediment diameter

IX;R

_Dgr

Dimensionless gain size

FFO

À

obs

Observed friction factor

FFC

Àcak;

Calculated friction factor

RFO(M)

ks

(m)

Observed roughness height

RFC(M)

ks

(m)

Calculated roughness height

V(MjS)

V (mis)

Meao velocity

V.(M/S)

v« (mis)

Shear velocity

Y(M)

_{d (m)}

_{Mean depth of flow}

DDB Dd 650449 7/80

13

(19)

I

FLUME DATA USED TO DEVELOP NEW METHOD (SAND)

(see Table 1)

(20)

- - - -

_

'

_{.. .. ..}

-DATA REFERENCE 34 _{DATA REFERENCE} ₅₆

MEAN SEDIMENT DIAMETER

=

0.04 MM SPECIFIC GRAVITY

=

2.65 DGR

=

1.10 MEAN SEDIMENT DIAMETER

=

2.65 DGR

=

2.57

SEDIMENT GRADING CURVE SEDIMENT GRADING CURVE

D D D D D D D D D D D D D D D D D D D D

5 15 25 35 45 55 65 75 85 95 5 15 25 35 45 55 65 75 85 95

0.010 0.016 0.021 0.030 0.039 0.048 0.057 0.070 0.084 0.125 MM 0.062 0.075 0.088 0.090 0.095 0.100 0.110 0.120 0.130 0.140 MM

Y(M} V(M/S} V*(M/S} FFO FFC FFC/FFO RFO(M} RFC(M} RFC/RFO Y(M} V(M/S} V*(M/S} FFO FFC FFC/FFO RFO(M} RFC(M} RFC/RFO

0.173 0.572 0.0418 0.0427 0.0139 0.326 0.0097 0.0001 0.010 0.274 0.589 0.0509 0.0597 0.0371 0.622 0.0364 0.0102 0.281 0.143 0.531 0.0402 0.0459 0.0144 0.314 0.0097 0.0001 0.010 0.335 0.482 0.0413 0.0588 0.0332 0.565 0.0429 0.0089 0.207 0.205 0.583 0.0421 0.0416 0.0135 0.325 0.0107 0.0001 0.009 0.335 0.416 0.0381 0.0672 0.0322 0.478 0.0584 0.0080 0.137 0.146 0.650 0.0390 0.0288 0.0143 0.497 0.0024 0.0001 0.042 0.305 0.686 0.0387 0.0254 0.0330 1.295 0.0032 0.0079 2.435 0.173 0.798 0.0439 0.0242 0.0140 0.578 0.0015 0.0001 0.067 0.290 1.360 0.0653 0.0184 0.0394 2.137 0.0008 0.0129 15.369 0.138 0.376 0.0332 0.0625 0.0143 0.229 0.0204 0.0001 0.005 0.244 1.234 0.0520 0.0142 0.0382 2.694 0.0002 0.0099 58.826 0.116 0.260 0.0296 0.1037 0.0147 0.141 0.0481 0.0001 0.002 0.247 1.125 0.0511 0.0165 0.0379 2.300 0.0004 0.0098 24.254 0.202 0.740 0.0445 0.0290 0.0136 0.469 0.0034 0.0001 0.030 0.244 1.048 0.0518 0.0196 0.0382 1.952 0.0009 0.0099 11.004 0.244 0.953 0.0440 0.0170 0.0361 2.122 0.0005 0.0084 18.197

MEAN ERROR 0.360 MEAN ERROR 0.022 0.238 0.879 0.0354 0.0130 0.0333 2.565 0.0001 0.0063 63.113

MIN. ERROR 0.141 MIN. ERROR 0.002

MAX. ERROR 0.578 MAX. ERROR 0.067 0.238 0.782 0.0354 0.0164 0.0333 2.027 0.0004 0.0063 17.860

STD. DEV. 0.136 STD. DEV. 0.021 0.287 0.486 0.0453 0.0694 0.0354 0.510 0.0535 0.0093 0.173

0.287 0.571 0.0498 0.0609 0.0366 0.600 0.0399 0.0102 0.256

0.247 0.752 0.0437 0.0270 0.0360 1.335 0.0032 0.0084 2.602

DATA REFERENCE 33 0.290 0.401 0.0450 0.1007 0.0353 0.350 0.1137 0.0092 0.081

0.262 0.358 0.0277 0.0477 0.0289 0.605 0.0198 0.0043 0.217

=

2.65 DGR

=

2.55 0.259 0.448 0.0425 0.0719 0.0353 0.491 0.0523 0.0083 0.158

0.232 1.203 0.0542 0.0162 0.0391 2.412 0.0003 0.0101 29.036

SEDIMENT GRADING CURVE _0.162 _0.863 ₀_.₀₄₁₃ ₀_.₀₁₈₄

0.0384 2.088 0.0004 0.0066 16.026

D ₅ D₁₅ D₂₅ D₃₅ D₄₅ D₅₅ D₆₅ D₇₅ D₈₅ D₉₅ 0.158 0.950 0.0417 0.0154 0.0386 2.505 0.0001 0.0066 45.444

0.067 0.088 0.099 0.102 0.110 0.120 0.130 0.140 0.148 0.160 MM 0.189 0.919 0.0443 0.0185 0.0381 2.056 0.0005 0.0076 14.521

0.183 1.016 0.0444 0.0153 0.0384 2.516 0.0002 0.0075 43.740

0.186 1.059 0.0466 0.0155 0.0389 2.515 0.0002 0.0080 40.580

Y(M} V(M/S} V*(M/S} FFO FFC FFC/FFO RFO(M} RFC(M} RFC/RFO _0.232 _0.401 _0.0362 ₀_.₀₆₅₁ _0.0338

0.519 0.0375 0.0064 0.172 0.232 0.501 0.0443 0.0624 0.0366 0.587 0.0340 0.0083 0.243 0.171 0.515 0.0451 0.0614 0.0390 0.636 0.0242 0.0074 0.306 0.229 0.415 0.0347 0.0562 0.0329 0.586 0.0263 0.0059 0.223 0.229 0.561 0.0441 0.0494 0.0366 0.742 0.0190 0.0082 0.432 0.283 0.405 0.0439 0.0938 0.0349 0.372 0.0974 0.0088 0.090 0.207 0.784 0.0354 0.0163 0.0342 2.103 0.0003 0.0060 21.431 0.283 0.704 0.0527 0.0449 0.0373 0.830 0.0182 0.0107 0.588 0.213 0.871 0.0396 0.0166 0.0357 2.154 0.0003 0.0070 21.189 0.157 0.585 0.0482 0.0542 0.0407 0.751 0.0164 0.0076 0.465

MEAN ERROR 1.530 MEAN ERROR 15.999

0.158 0.518 0.0472 0.0665 0.0404 0.607 0.0268 0.0075 0.280 _{MIN. ERROR} 0.350 MIN. ERROR 0.081 0.162 0.402 0.0408 0.0824 0.0379 0.460 0.0435 0.0064 0.148 _MAX_. _ERROR 2.694 MAX. ERROR 63.113 0.230 0.527 0.0451 0.0585 0.0367 0.628 0.0292 0.0083 0.285 _STD_. _DEV. 0.842 STD. DEV. 18.641 0.303 0.485 0.0421 0.0603 0.0339 0.562 0.0411 0.0085 0.208 0.116 0.472 0.0463 0.0769 0.0428 0.556 0.0270 0.0065 0.241 0.095 0.326 0.0387 0.1127 0.0416 0.369 0.0454 0.0049 0.108 0.116 0.390 0.0408 0.0877 0.0407 0.464 0.0352 0.0056 0.160 0.221 0.515 0.0421 0.0533 0.0361 0.676 0.0223 0.0075 0.338 0.076 0.351 0.0396 0.1022 0.0441 0.431 0.0307 0.0046 0.150 0.216 0.671 0.0472 0.0397 0.0379 0.953 0.0098 0.0085 0.869

MEAN ERROR 0.592 MEAN ERROR 0.297

MIN. ERROR 0.369 MIN. ERROR 0.090

MAX. ERROR 0.953 MAX. ERROR 0.869

(21)

SPECIFIC GRAVITY

=

2.65 MEAN SEDIMENT DIAMETER

=

0.10 MM

SEDIMENT GRADING CURVE

D D D 5 15 25 0.062 0.075 0.088 YCM) 0.320 0.283 0.290 0.290 0.305 0.299 0.271 0.274 0.259 0.177 0.198 0.195 0.119 0.143 0.168 0.210 0.210 0.287 0.308 0.271 0.143 0.137 VCM/S) 0.581 0.819 1.043 1.123 1.024 0.622 1.113 1.185 0.538 0.788 0.469 0.533 0.688 0.486 0.416 0.994 1.104 0.892 0.981 1.284 0.480 0.759 V*CM/S) 0.0453 0.0426 0.0501 0.0572 0.0596 0.0474 0.0511 0.0551 0.0504 0.0379 0.0396 0.0429 0.0394 0.0441 0.0402 0.0408 0.0450 0.0441 0.0365 0.0726 0.0430 0.0413 D 35 0.090 FFO 0.0487 0.0216 0.0185 0.0207 0.0271 0.0464 0.0169 0.0173 0.0703 0.0185 0.0571 0.0517 0.0262 0.0657 0.0749 0.0135 0.0133 0.0195 0.0111 0.0256 D 45 0.095 FFC 0.0355 0.0355 0.0375 0.0390 0.0391 0.0365 0.0382 0.0390 0.0383 0.0372 0.0370 0.0383 0.0410 0.0412 0.0385 0.0370 0.0384 0.0359 0.0328 0.0420 0.0643 0.0409 0.0238 0.0406 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV. D 55 0.100 FFC/FFO 0.729 1.644 2.025 1.880 1.440 0.787 2.264 2.254 0.546 2.016 0.648 0.741 1.565 0.628 0.515 2.749 2.893 1.838 2.961 1.641 0.635 1.709 1.550 0.515 2.961 0.781 DATA REFERENCE 24 D 65 0.110 RFO(M) 0.0256 0.0016 0.0008 0.0014 0.0041 0.0211 0.0005 0.0006 0.0497 0.0005 0.0236. 0.0182 0.0014 0.0237 0.0368 0.0001 0.0001 0.0011 0.0001 0.0030 D 75 0.120 D 85 0.130 DGR

=

2.65

o

95 0.140 MM RFC/RFO 0.407 5.805 13.189 8.989 3.237 0.505 21.752 20.248 0.214 14.171 0.310 0.439 4.337 0.306 0.189 77.682 86.879 9.188 78.361 4.909 0.312 6.103 16.251 0.189 86.879 26.510 0.0226 0.0071 0.0011 0.0066 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV. RFCCM) 0.0104 0.0092 0.0111 0.0125 0.0133 0.0106 0.0110 0.0118 0.0107 0.0066 0.0073 0.0080 0.0059 0.0072 0.0070 0.0078 0.0087 0.0097 0.0078 0.0145

=

0.09 MM SPECIFIC GRAVITY

=

2.65 SEDIMENT GRADING CURVE

D D D 5 15 25 0.050 0.075 0.080 YCM) VCM/S) V*CM/S) 0.072 0.072 0.058 0.069 0.057 0.085 0.070 0.087 0.085 0.086

-0.640 0.640 0.597 0.408 0.375 0.250 0.302 0.402 0.649 0.329

-0.0402 0.0393 0.0375 0.0436 0.0430 0.0329 0.0405 0.0451 0.0399 0.0430

-D 35 0.084 FFO 0.0316 0.0302 0.0315 0.0911 0.1051 0.1388 0.1444 0.1006 0.0303 0.1364 D 45 0.087 FFC 0.0467 0.0463 0.0477 0.0486 0.0505 0.0413 0.0471 0.0468 0.0449 0.0461 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

-D 55 0.090 FFC/FFO 1.478 1.534 1.515 0.534 0.480 0.298 0.326 0.465 1.483 0.338 0.845 0.298 1.534 0.541

-D 65 0.092 RFOCM) 0.0016 0.0013 0.0012 0.0224 0.0241 0.0571 0.0503 0.0339 0.0016 0.0560

-D 75 0.097 D 85 0.104 RFC(M) 0.0051 0.0050 0.0043 0.0054 0.0049 0.0043 0.0051 0.0062 0.0054 0.0059 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

-DGR

=

D 95 0.123 MM RFC/RFO 3.255 3.724 3.523 0.242 0.205 0.075 0.101 0.183 3.367 0.105 1.478 0.075 3.724 1.629

_

,

2.70

-MEAN SEDIMENT DIAMETER SEDIMENT GRADING CURVE

D D D 5 15 25 0.062 0.075 0.088 YCM) 0.256 0.302 0.305 0.195 0.280 0.277 0.305 0.317 0.308 0.305 0.320 0.366 0.375 0.378 0.305 0.317 0.314 0.314 0.262 0.271 0.274 0.274 0.229 0.241 0.232 0.229 0.235 0.146 VCM/S) 1.089 0.924 0.914 0.838 1.242 1.340 1.143 1.026 0.905 0.838 0.726 0.571 0.498 0.420 0.381 0.440 0.506 0.518 0.794 0.856 0.931 1.016 0.813 0.868 1.003 0.764 1.089 0.438 V*CM/S) 0.0610 0.0523 0.0455 0.0442 0.0717 0.0547 0.0497 0.0470 0.0373 0.0401 0.0388 0.0431 0.0445 0.0358 0.0303 0.0379 0.0442 0.0435 0.0439 0.042·3 0.0461 0.0466 0.0469 0.0442 0.0439 0.0359 0.0488 0.0417 D 35 0.090 FFO 0.0251 0.0256 0.0198 0.0222 0.0267 0.0133 0.0151 0.0168 0.0136 0.0183 0.0229 0.0455 0.0638 0.0580 0.0508 0.0594 0.0610 0.0566 0.0244 0.0196 DATA REFERENCE 58 0.10 MM SPECIFIC GRAVITY

=

2.65 D 45 0.095 FFC 0.0420 0.0389 0.0370 0.0400 0.0430 0.0401 0.0382 0.0372 0.0341 0.0352 0.0345 0.0350 0.0353 0.0322 0.0309 0.0342 0.0364 0.0362 0.0376 0.0368 0.0196 0.0379 0.0169 0.0381 0.0266 0.0396 0.0207 0.0383 0.0153 0.0385 0.0177 0.0356 0.0161 0.0399 0.0723 0.0414 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

MEAN SEDIMENT DIAMETER SEDIMENT GRADING CURVE

D D D 5 15 25 0.066 0.099 0.114 YCM) 0.168 0.134 0.114 0.105 0.104 0.077 0.072 0.071 VCM/S) 0.277 0.347 0.411 0.442 0.448 0.600 0.649 0.655

-V*CM/S) 0.0305 0.0445 0.0485 0.0454 0.0402 0.0436 0.0375 0.0381

-D 35 0.124 FFO 0.0966 0.1312 0.1110 0.0845 0.0645 0.14 MM D 55 0.100 FFC/FFO 1.674 1.518 1.868 1.799 1.611 3.007 2.522 2.210 2.509 1.927 1.504 0.770 0.553 0.555 0.610 0.575 0.597 0.640 1.538 1.883 1.937 2.261 1.487 1.848 2.517 2.009 2.479 0.573 1.606 0.553 3.007 0.723 DATA REFERENCE 20 D 65 0.110 RFOCM) 0.0026 0.0033 0.0012 0.0012 0.0036 0.0001 0.0003 0.0006 0.0002 0.0008 0.0023 0.0244 0.0580 0.0469 0.0271 0.0416 0.0439 0.0366 0.0024 0.0010 0.0010 0.0005 0.0028 0.0011 0.0002 0.0005 0.0003 0.0299 D 75 0.120 D 85 0.130 RFCCM) 0.0137 0.0129 0.0113 0.0090 0.0160 0.0130 0.0124 0.0119 0.0089 0.0097 0.0095 0.0115 0.0120 0.0091 0.0064 0.0092 0.0111 0.0109 0.0102 0.0099 0.0109 0.0111 0.0103 0.0099 0.0096 0.0075 0.0109 0.0075 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV. SPECIFIC GRAVITY

=

2.65

o

45 0.130 FFC 0.0405 0.0528 0.0570 0.0563 0.0531 0.0422 0.0591 0.0267 0.0556 0.0270 0.0562 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

-D 55 0.139 FFC/FFO 0.419 0.403 0.514 0.666 0.824 1.402 2.083 2.078 1.048 0.403 2.083 0.666

-D 65 0.150 RFOCM) 0.0610 0.0828 0.0530 0.0295 0.0164 0.0041 0.0008 0.0009

..

D 75 0.169 D 85 0.195 RFCCM) 0.0080 0.0132 0.0135 0.0121 0.0103 0.0100 0.0079 0.0081 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

.. ..

DGR

=

2.76 D 95 0.140 MM RFC/RFO 5.286 3.923 9.391 7.462 4.503 92.815 37.293 20.026 55.414 11.628 4.173 0.469 0.207 0.194 0.235 0.220 0.252 0.297 4.247 9.888 10.678 21.675 3.614 8.694 39.016 14.593 31.897 0.251 14.227 0.194 92.815 20.452 DGR

=

3.70 D 95 0.250 MM RFC/RFO 0.132 0.159 0.254 0.408 0.628 2.415 9.459 9.220 2.834 0.132 9.459 3.820

..

(22)

-MEAN SEDIMENT DIAMETER SF.DIMENT GRADING CURVE

D D D 5 15 25 0.074 0.096 0.110 Y(M) 0.073 0.074 0.073 0.073 0.072 0.076 0.092 0.071 0.165 0.161 0.167 0.163 0.169 0.166 V(M/S) 0.235 0.274 0.326 0.390 0.427 0.439 0.424 0.649 0.244 0.317 0.372 0.454 0.524 0.771 V*(M/S) 0.0314 0.0381 0.0448 0.0448 0.0445 0.0430 0.0424 0.0381 0.0253 0.0332 0.0424 0.0439 0.0405 0.0424

-D 35 0.120 FFO 0.1431 0.1543 0.1510 0.1055 0.0870 0.0767 0.0800 0.0276 0.0861 0.0879 0.1038 0.0747 0.0478 0.0241

D D D 5 15 25 0.060 0.095 0.110 Y(M) 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 V(M/S) 0.244 0.259 0.277 0.299 0.317 0.366 0.421 0.472 0.515 0.558 0.628 V*(M/S) 0.0381 0.0387 0.0415 0.0433 0.0448 0.0442 0.0415 0.0405 0.0387 0.0381 0.0405 D 35 0.120 FFO 0.1953 0.1786 0.1787 0.1680 0.1598 0.1168 0.0777 0.0589 0.0452 0.0373 0.0333

-0.14 MM D 45 0.130 FFC 0.0502 0.0561 0.0611 0.0611 0.0611 0.0594 0.0564 0.0566 0.0358 0.0436 0.0496 0.0507 0.0484 0.0497 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV. 0.15 MM

o

45 0.142 FFC 0.0593 0.0598 0.0621 0.0635 0.0646 0.0641 0.0621 0.0613 0.0598 0.0593 0.0613 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

-DATA REFERENCE 21 SPECIFIC GRAVITY

o

55 0.144 FFC/FFO 0.350 0.363 0.405 0.579 0.702 0.774 0.706 2.055 0.416 0.497 0.478 0.679 1.012 2.058 0.791 0.350 2.058 0.546 DATA REFERENCE 22 SPECIFIC GRAV ITY

o

55 0.163 FFC/FFO 0.304 0.335 0.347 0.378 0.404 0.549 0.799 1.041 1.324 1.588 1.839 0.810 0.304 1.839 0.531

-o

65 0.155 RFOCM) 0.0511 0.0584 0.0559 0.0312 0.0215 0.0175 0.0232 0.0009 0.0481 0.0489 0.0694 0.0357 0.0128 0.0014

o

65 0.189 RFOOO 0.0802 0.0712 0.0713 0.0654 0.0610 0.0374 0.0174 0.0094 0.0048 0.0027 0.0019

-2.65

o

75 0.168

o

85 0.189 RFCCM) 0.0062 0.0084 0.0102 0.0102 0.0101 0.0099 0.0106 0.0082 0.0054 0.0095 0.0140 0.0145 0.0132 0.0139 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV. 2.65

o

75 0.225

o

85 0.279 RFC(M) 0.0095 0.0097 0.0106 0.0112 0.0116 0.0115 0.0106 0.0103 0.0097 0.0095 0.0103 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

-OGR

=

o

95 0.260 MM RFC/RFO 0.121 0.144 0.183 0.327 0.468 0.565 0.459 8.745 0.113 0.195 0.202 0.405 1.033 9.856 1.630 0.113 9.856 3.147 OGR

=

D 95 0.402 MM RFC/RFO 0.119 0.137 0.149 0.171 0.191 0.307 0.610 1.101 2.049 3.491 5.397 1.247 0.119 5.397 1.657 3.74 4.10

-MEAN SEDIMENT DIAMETER

SEDIMENT GRAOING CURVE

o

D 0 5 15 25 0.096 0.130 0.136 Y(M) 0.074 0.072 0.059 0.074 0.075 0.076 0.091 0.060 VCM/S) 0.622 0.634 0.610 0.622 0.411 0.283 0.427 0.381 V*CM/S) 0.0427 0.0411 0.0375 0.0390 0.0436 0.0387 0.0445 0.0454 D 35 0.140 FFO 0.0377 0.0337 0.0303 0.0315 0.0898 0.1492 0.0870 0.1137

SEDIMENT GRAOING CURVE

D D D 5 15 25 0.130 0.144 0.159 Y(M) 0.305 0.177 0.311 0.168 0.314 0.290 0.283 0.165 0.323 0.171 0.168 0.332 0.271 0.262 0.241 0.158 0.311 0.186 0.207 0.158 0.219 0.155 0.149 VCM/S) 0.338 0.226 0.396 0.283 0.469 0.546 0.607 0.317 0.597 0.344 0.360 0.716 0.942 0.981 1.055 0.512 0.820 0.509 0.543 0.552 1.170 0.881 0.640 V*CM/S) 0.0290 0.0244 0.0363 0.0305 0.0424 0.0433 0.0442 0.0357 0.0512 0.0375 0.0390 0.0567 0.0515 0.0521 0.0515 0.0445 0.0631 0.0488 0.0533 0.0479 0.0579 0.0509 0.0533

-0.16 MM

o

45 0.144 FFC 0.0642 0.0633 0.0627 0.0610 0.0648 0.0603 0.0626 0.0698 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

-DATA REFERENCE 23 SPECIFIC GRAVITY D 55 0.147 FFC/FFO 1.704 1.879 2.071 1.936 0.722 0.404 0.719 0.614 1.256 0.404 2.071 0.654 DATA REFERENCE

-o

65 0.149 RFOCM) 0.0028 0.0019 0.0011 0.0016 0.0236 0.0571 0.0272 0.0291 0.19 MM SPECIFIC GRAVITY

o

35 0.173 FFO 0.0586 0.0935 0.0670 0.0925 0.0652 0.0503 0.0425 0.1013 0.0588 0.0948 0.0941 0.0501 0.0239 0.0226 0.0191 0.0604 0.0474 0.0734 0.0773 0.0602 0.0196 0.0267 0.0556

o

45 0.186 FFC 0.0363 0.0347 0.0432 0.0430 0.0478 0.0493 0.0501 0.0488 0.0530 0.0502 0.0518 0.0555 0.0551 0.0559 0.0565 0.0570 0.0591 0.0580 0.0594 0.0594 0.0611 0.0617 0.0638 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

o

55 0.199 FFC/FFO 0.620 0.371 0.645 0.465 0.734 0.979 1.180 0.482 0.902 0.529 0.550 1.108 2.305 2.477 2.962 0.943 1.247 0.790 0.768 0.987 3.120 2.310 1.148 1.201 0.371 3.120 0.807

o

65 0.215 RFOCM) 0.0386 0.0604 0.0538 0.0562 0.0510 0.0252 0.0156 0.0652 0.0413 0.0599 0.0581 0.0286 0.0023 0.0018 0.0008 0.0216 0.0231 0.0392 0.0487 0.0214 0.0008 0.0019 0.0166

-2.65

o

75 0.154 D 85 0.160 RFCCM) 0.0116 0.0109 0.0088 0.0103 0.0119 0.0103 0.0135 0.0113 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV. 2.65

o

75 0.236 D 85 0.263 RFCOO 0.0106 0.0053 0.0180 0.0095 0.0240 0.0239 0.0244 0.0132 0.0322 0.0147 0.0156 0.0370 0.0297 0.0297 0.0280 0.0188 0.0403 0.0230 0.0272 0.0208 0.0307 0.0223 0.0231 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

-DGR

=

4.29

o

95 0.172 MM RFC/RFO 4.078 5.622 8.011 6.449 0.506 0.180 0.495 0.388 3.216 0.180 8.011 2.999 OGR

=

4.63

o

95 0.310 MM RFC/RFO 0.274 0.087 0.334 0.170 0.470 0.947 1.563 0.202 0.778 0.245 0.269 1.293 13.023 16.940 35.540 0.869 1.741 0.587 0.558 0.970 38.039 11.485 1.387 5.555 0.087 38.039 10.637

(23)

D D D 5 15 25 0.068 0.139 0.159 Y(M) 0.165 0.201 0.091 0.140 0.110 0.155 0.122 0.198 0.122 0.171 0.146 0.192 0.223 0.238 0.210 0.137 0.421 0.256 0.314 0.183 0.125 0.210 0.238 0.210 0.186 0.198 0.232 0.229 0.171

-V(M/S) 0.216 0.226 0.229 0.250 0.283 0.299 0.363 0.363 0.399 0.408 0.424 0.424 0.433 0.436 0.442 0.457 0.488 0.500 0.533 0.735 0.780 0.786 0.792 0.817 0.835 0.866 0.911 0.914 0.981

-V*(M/S) 0.0283 0.0280 0.0378 0.0351 0.0415 0.0369 0.0436 0.0393 0.0408 0.0387 0.0415 0.0411 0.0418 0.0457 0.0430 0.0448 0.0536 0.0475 0.0430 0.0463 0.0384 0.0518 0.0530 0.0573 0.0558 0.0591 0.0628 0.0616 0.0515

-D 35 0.170 FFO 0.1373 0.1237 0.2187 0.1573 0.1711 0.1220 0.1155 0.0940 0.0837 0.0719 0.0766 0.0755 0.0745 0.0880 0.0756 0.0768 0.0968 0.0724 0.0519 0.0318 0.0194 0.0347 0.0358 0.0394 0.0357 0.0373 0.0380 0.0363 0.0220

-0.18 MM D 45 0.179 FFC 0.0410 0.0389 0.0586 0.0504 0.0598 0.0511 0.0602 0.0507 0.0578 0.0518 0.0560 0.0526 0.0514 0.0536 0.0530 0.0596 0.0520 0.0539 0.0487 0.0571 0.0552 0.0589 0.0581 0.0619 0.0627 0.0636 0.0632 0.0629 0.0614 MEAN ERROR MIN. ERROR MAX. ERROR STD. OEV.

-SPECIFIC GRAVITY D 55 0.185 FFC/FFO 0.299 0.315 0.268 0.321 0.349 0.419 0.521 0.539 0.690 0.721 0.732 0.696 0.690 0.609 0.700 0.776 0.537 0.745 0.938 1.793 2.848 1.695 1.621 1.572 1.758 1.704 1.665 1.733 2.786 1.036 0.268 2.848 0.710

-D 65 0.194 RFO(M) 0.1088 0.1125 0.1153 0.1138 0.1003 0.0850 0.0609 0.0685 0.0336 0.0344 0.0337 0.0429 0.0483 0.0725 0.0472 0.0318 0.1537 0.0524 0.0296 0.0042 0.0004 0.0064 0.0080 0.0093 0.0061 0.0075 0.0093 0.0080 0.0010

-2.65

o

75 0.209

o

85 0.228 RFC(M) 0.0082 0.0086 0.0116 0.0122 0.0145 0.0141 0.0165 0.0175 0.0149 0.0160 0.0166 0.0186 0.0204 0.0242 0.0208 0.0181 0.0399 0.0266 0.0252 0.0218 0.0137 0.0270 0.0295 0.0304 0.0277 0.0305 0.0351 0.0342 0.0242 MEAN ERROR MIN. ERROR MAX. ERROR STO. OEV.

-OGR

=

4.70 D 95 0.360 MM RFC/RFO 0.075 0.076 0.100 0.107 0.145 0.165 0.270 0.256 0.444 0.465 0.494 0.435 0.423 0.334 0.441 0.569 0.259 0.507 0.849 5.193 34.758 4.221 3.707 3.252 4.505 4.054 3.794 4.298 23.652 3.374 0.075 34.758 7.379

-MEAN SEDIMENT DIAMETER SEDIMENT GRADING CURVE

D D D 5 15 25 0.140 0.170 0.180 Y(M) 0.082 0.108 0.124 0.136 0.130 0.090 0.114 0.146 0.180 0.208 0.116 0.136 0.165 0.184 V(M/S) 0.229 0.302 0.366 0.521 0.689 0.283 0.317 0.344 0.387 0.433 0.277 0.341 0.372 0.482 V*(M/S) 0.0401 0.0474 0.0493 0.0515 0.0504 0.0364 0.0408 0.0463 0.0515 0.0553 0.0338 0.0364 0.0401 0.0425

D D D 5 15 25 0.144 0.177 0.200 Y(M) 0.271 0.239 0.222 0.242 0.207 0.200 0.200

-V(M/S) 0.379 0.407 0.390 0.429 0.498 0.512 0.524 V*(M/S) 0.0365 0.0415 0.0399 0.0428 0.0480 0.0476 0.0603

-D 35 0.200 FFO 0.2456 0.1974 0.1454 0.0780 0.0428 0.1321 0.1326 0.1448 0.1418 0.1307 0.1188 0.0909 0.0932 0.0622 D 35 0.222 FFO 0.0741 0.0831 0.0837 0.0796 0.0743 0.21 MM D 45 0.220 FFC 0.0641 0.0670 0.0665 0.0668 0.0667 0.0583 0.0601 0.0617 0.0628 0.0633 0.0517 0.0530 0.0546 0.0554 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV. 0.25 MM D 45 0.242 FFC 0.0490 0.0557 0.0549 0.0569 0.0637 DATA REFERENCE 27 SPECIFIC GRAVJTY D 55 0.240 FFC/FFO 0.261 0.340 0.457 0.856 1.559 0.441 0.453 0.426 0.443 0.484 0.435 0.583 0.585 0.891 0.587 0.261 1.559 0.318 DATA REFERENCE 55 SPECIFIC GRAVJTY D 55 0.264 FFC/FFO 0.661 0.670 0.656 0.714 0.858 0.924 0.695 0.740 0.656 0.924 0.099 0.0691 0.0638 0.1059 0.0737 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

-D 65 0.250 RFO(M) 0.1183 0.1198 0.0898 0.0325 0.0073 0.0561 0.0712 0.1049 0.1255 0.1276 0.0609 0.0440 0.0560 0.0268 D 65 0.288 RFO(M) 0.0583 0.0651 0.0614 0.0604 0.0447 0.0370 0.0860

..

2.65 D 75 0.280 D 85 0.310 RFC(M) 0.0127 0.0187 0.0211 0.0233 0.0221 0.0112 0.0152 0.0209 0.0269 0.0316 0.0108 0.0134 0.0175 0.0203 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV. 2.65 D 75 0.316 D 85 0.328 RFC(M) 0.0220 0.0268 0.0241 0.0286 0.0319 0.0310 0.0425 MEAN ERROR MIN. ERROR MAX. ERROR STD. DEV.

-DGR = 5.10 D 95 0.500 MM RFC/RFO 0.107 0.156 0.235 0.717 3.047 0.200 0.214 0.199 0.214 0.248 0.177 0.305 0.313 0.759 0.492 0.107 3.047 0.733 DGR

=

6.30 D 95 0.415 MM RFC/RFO 0.377 0.412 0.393 0.473 0.714 0.837 0.494 0.528 0.377 0.837 0.164