Chapter 8 Stream Power Theory

148

Chapter 8

STREAM POWER THEORY Stream power concepts can be used profitably to explain a variety of sediment transport phenomena, including those occurring in reservoir sedimentation. When using stream power theory in considering techniques to calculate profiles of deposited sediment, it. is necessary to distinguish between stable and

unstable non-equilibrium conditions.

Stable conditions

occur when

applied stream power is minimized (Chang, 1979; Yang, 1976a).

Under such

conditions uniform flow develops, sediment concentration remains constant throughout the reservoir and the bed profile does not change with time (Annandale, 1 9 8 4 ; Chang, 1982). Unstable conditions on

the other hand are characterized by continuously

changing flow conditions and bed profiles.

These changes are caused by

accumulation of sediment in the reservoir basin.

In recognizing these two possibilities, the material presented in this chapter is divided into two main subsections, viz. one dealing with stable and the other with unstable non-equilibrium conditions. 8.1

STABLE NON-EQUILIBRIUM CONDITIONS

Stable non-equilibrium conditions of reservoir sedimentation are dealt with by first presenting the theory and verification thereof, whereafter calculation procedures using basic theory and a semi-empirical technique are presented 8.1.1

.

Theory

Basic principles of non-equilibrium thermodynamics are used to show that applied stream power approaches a constant minimum value throughout a nonequilibrium system when stable conditions are approached. This conclusion is then used to derive a criterion with which such conditions can be identified. (i)

Non-equilibrium thermodynamics.

By viewing a reservoir basin subject

to sediment deposition as an open system, it is possible to describe the process in terms of non-equilibrium thermodynamics.

A system is in a state

of non-equilibrium when certain limitations prevent it from being in a state of equilibrium.

149

i t i s d i s t r i b u t e d i n such a way

When s e d i m e n t i s d e p o s i t e d i n a r e s e r v o i r , t h a t a c e r t a i n k i n d of o r d e r i s e s t a b l i s h e d . to

the

c o n c e p t of

Various meanings a r e a t t a c h e d

The Boltzman p r i n c i p l e f o r example s t a t e s

order.

i n a n e q u i l i b r i u m s y s t e m i s e q u i v a l e n t t o a s t a t e of

"order" However,

in

system i s

the

a n o n - e q u i l i b r i u m s y s t e m "order"

i n a stable condition

approached when steady.

case of all

(Nicolis

the processes within

low e n t r o p y .

o c c u r s when t h e

1977).

and P r i g o g i n e ,

t h e non-equilibrium

S t a b i l i t y and t h u s " o r d e r " i n n o n - e q u i l i b r i u m

that

This

is

s y s t e m are

systems c a n only b e

m a i n t a i n e d by c o n t i n u o u s exchange of energy w i t h t h e s u r r o u n d i n g environment, r e s u l t i n g i n such systems b e i n g c a l l e d d i s s i p a t i v e systems

to

distinguish

them from e q u i l i b r i u m systems. N o r e q u i l i b r i u m systems a r e o f t e n s t u d i e d p r o f i t a b l y

by

investigating

the

b e h a v i o u r of macroscopic phenomena, phenomena which a r e made up of a l a r g e

I n t h e c a s e of

number of m i c r o s c o p i c f l u c t u a t i o n s .

sediment t r a n s p o r t

the

m i c r o s c o p i c f l u c t u a t i o n s a r e r e p r e s e n t e d by t h e t u r b u l e n t a c t i o n of water and by d e p o s i t i o n and e n t r a i n m e n t of i n d i v i d u a l sediment p a r t i c l e s , which c o l l e c tively

contribute

to

represent

a

macroscopic

current

of

total

sediment

discharge. The

fluctuating

collectively

to

behaviour

of

individual

create a structure

f u n c t i o n , which

i n t h e c a s e of

t r a n s p o r t water

effectively.

to

sediment

aid

the

particles

system

in

co-operates

fulfilling

its

a c h a n n e l such a s a r e s e r v o i r b a s i n i s t o Changes

i n fluctuations

due

to

changes

in

d i s c h a r g e of sediment and w a t e r w i l l t h e r e f o r e l e a d t o a change i n s t r u c t u r e i n order

to

fulfil

t h e f u n c t i o n of

the

system,

so

that

the

interactive

r e l a t i o n s h i p between s t r u c t u r e , f l u c t u a t i o n and f u n c t i o n i s always m a i n t a i n e d ( F i g u r e 8. I ) ,

Structure

+

Fluctuation

Function Fig. 8.1

R e l a t i o n s h i p between s t r u c t u r e , f l u c t u a t i o n and f u n c t i o n .

150

Order in an open non-equilibrium system can hardly be studied profitably by researching the behaviour of individual microscopic fluctuations.

It is

therefore advisable to use the customary approach of studying the macroscopic current in order to improve understanding of reservoir sedimentation.

the processes

involved in

This is done by first investigating the temporal

change in total entropy of a small volume of fluid during its movement through the reservoir, whereafter the findings are interpreted macroscopically throughout the reservoir basin. Temporal

change in total entropy can be written as

(e.g.

Nicolis and

Prigogine, 1977) dSe

dS = _ dt

dt

+-

dSi dt

where dS/dt

=

temporal change in total entropy per unit volume; dS./dt

of internal entropy production per unit volume; dS /dt

entropy supply per unit volume; and t

=

=

=

rate

rate of external

time.

This equation can conveniently be rewritten as,

where W

=

dS/dt; U

=

dSe/dt; and P

=

dS./dt.

For a given temperature the rate of external entropy supply to a reservoir system is a function of hydrology, e.g. the discharge of water and sediment through the system, whereas rate of internal entropy production represents rate of irreversible friction losses. When the limitations imposed on a system prevent it from being in a state of equilibrium, it will endeavour to create "order" by assuming a stable nonequilibrium condition.

This will be reached when W does not change with

time, a condition that will exist if dP _dU= _ = dt

dt

For analysis of a stable non-equilibrium condition it must therefore be assumed that steady state flow conditions exist, i.e. dU/dt

=

0.

As

the

151

hydrology of r i v e r systems i s i r r e g u l a r , such a c o n d i t i o n c a n o n l y e x i s t i f t h e f l o w c o n d i t i o n s are such t h a t they a r e homogenous i n t h e l o n g term, i . e . i f a c o n s t a n t moving a v e r a g e i s approached. Assuming t h e r a t e of e x t e r n a l e n t r o p y s u p p l y through t h e system t o b e t i m e i n d e p e n d e n t , a l l t h a t remains i s t o i n v e s t i g a t e t h e b e h a v i o u r of t h e r a t e of i n t e r n a l entropy production.

The l a t t e r , b e i n g t i m e and d i s t a n c e d e p e n d e n t ,

c a n b e expanded a s f o l l o w s :

where x , y , d z / d t = flow

z = directions velocities

i n Cartesian co-ordinate

in

x,

y,

z

directions;

system; and

dxldt,

dyldt,

aP/ax,

aP/ay,

aP/az = change i n r a t e of i n t e r n a l e n t r o p y p r o d u c t i o n w i t h d i s t a n c e i n x , y and z d i r e c t i o n s . A s t h e v e l o c i t i e s w i l l b e unequal t o z e r o under n o n - e q u i l i b r i u m c o n d i t i o n s ,

i.e. dx

-d#t

0

*d#t O

a

s+&ie

(8.5)

n o n - ~ q l ~ 1i i h r i i i m

c o n d i t i o n w i l l only b e r e a c h e d when

and

(8.7)

The q u a l i t a t i v e meaning of e q u a t i o n (8.6) can b e i n v e s t i g a t e d m a c r o s c o p i c a l l y by v i e w i n g t h e s p a t i a l d i s t r i b u t i o n of t h e r a t e of i n t e r n a l e n t r o p y product i o n i n a s i m p l e model o f a r e s e r v o i r i n which f l o w i s one-dimensional and i n which sediment can b e d e p o s i t e d .

I n t h e c a s e where f l u i d flows through t h e

152

r e s e r v o i r b e f o r e d e p o s i t i o n o f s e d i m e n t , t h e d i s t r i b u t i o n of r a t e o f i n t e r n a l e n t r o p y p r o d u c t i o n t h r o u g h o u t t h e r e s e r v o i r is non-uniform.

F i g u r e 8 . 2 shows

t h a t t h e h i g h e s t r a t e of i n t e r n a l e n t r o p y p r o d u c t i o n o c c u r s a t t h e i n f l o w t o the reservoir,

where t h e r i v e r f l o w i s suddenly r e t a r d e d by t h e v i r t u a l l y

s t a t i o n a r y f l u i d i n t h e r e s e r v o i r , and a h i g h d e g r e e of t u r b u l e n c e d e v e l o p s . As

t h e r e t a r d e d f l u i d f l o w s t h r o u g h t h e r e s t of t h e r e s e r v o i r , t h e r a t e of

i n t e r n a l e n t r o p y p r o d u c t i o n i s much lower and a l s o e x h i b i t s a l e s s pronounced s p a t i a l variation.

Fig. 8.2 sediment.

Stream power

distribution

in

a

reservoir

with

no

deposited

As s e d i m e n t i s d e p o s i t e d i n t h e r e s e r v o i r t h e f l u i d v e l o c i t y w i t h i n t h e b a s i n

w i l l increase,

l e a d i n g t o a more uniform d i s t r i b u t i o n of

r a t e of i n t e r n a l

e n t r o p y p r o d u c t i o n u n t i l a c o n s t a n t v a l u e i s r e a c h e d t h r o u g h o u t when a s t a b l e nowequilibrium condition develops (Figure 8 . 3 ) . Under t h e s e c o n d i t i o n s t h e mean sediment d i s c h a r g e t h r o u g h o u t t h e r e s e r v o i r b a s i n w i l l approach a c o n s t a n t v a l u e .

This w i l l r e s u l t i n t h e longitudinal

p r o f i l e and r a t e of i n t e r n a l e n t r o p y p r o d u c t i o n b e i n g time-independent,

This

equation

implies

that

t h e r a t e of

i n t e r n a l entropy production i s a

minimum under s t a b l e n o n - e q u i l i b r i u m c o n d i t i o n s when t h e r a t e of e n t r o p y s u p p l y through t h e s y s t e m i s t i m e - i n v a r i a n t , r a t e of

external

Prigogine,

1977).

i.e.

e n t r o p y s u p p l y i n t o o r o u t of The a c t u a l minimum v a l u e of

i.e.

external

t h e r e i s no n e t

t h e system (Nicolis r a t e of

internal

and

entropy

p r o d u c t i o n w i l l n o t b e a n u n i v e r s a l c o n s t a n t b u t w i l l vary from c a s e t o c a s e

153

Sediment Fig.

8.3

Stream power d i s t r i b u t i o n

under s t a b l e non-equilibrium as

it

is

'

dependent

i n a r e s e r v o i r w i t h d e p o s i t e d sediment

conditions.

on t h e e x t e r n a l

entropy

supply and o t h e r l i m i t a t i o n s

imposed on t h e system. A t a given temperature t h e r a t e of i n t e r n a l entropy p r o d u c t i o n can be quan-

t i f i e d by c a l c u l a t i n g t h e a p p l i e d power which i s r e q u i r e d t o overcome f r i c t i o n w i t h i n a f l u i d and between f l u i d and channel boundary.

The d e r i v a t i o n s

which follow w i l l b e s i m p l i f i e d by assuming one-dimensional flow. (ii)

I n p u t and a p p l i e d s t r e a m power.

I t i s u s e f u l t o d i s t i n g u i s h between

i n p u t and a p p l i e d stream power i n a n e f f o r t t o q u a n t i f y t h e r a t e of i n t e r n a l entropy production.

I n p u t s t r e a m power, which r e p r e s e n t s t h e r a t e of e x t e r -

n a l entropy

i s t h e r a t e a t which p o t e n t i a l energy i s r e l e a s e d

supply,

to

m a i n t a i n flow, whereas a p p l i e d s t r e a m power can be viewed a s t h e r a t e of work required

t o overcome f l u i d f r i c t i o n .

The l a t t e r

represents

the r a t e

of

i n t e r n a l entropy production. The r e l a t i o n s h i p between i n p u t and a p p l i e d stream power f o r r e a l f l u i d s can be d e r i v e d from Newton's second law of motion and t h e second law of thermodynamics by w r i t i n g f o r a f l u i d p a r t i c l e moving along a s t r e a m l i n e ( S t r e e t e r ,

1971): dp + pvdv + pgdz = d ( 1 o s s e s )

(8.9)

154 where dp = change i n p r e s s u r e ;

ty;

p = mass d e n s i t y of

g = a c c e l e r a t i o n due t o g r a v i t y ;

fluid;

v = flow veloci-

dz = change i n e l e v a t i o n ;

and d(1os-

ses) = irreversible f r i c t i o n losses. By d i v i d i n g equation (8.9) by a small time i n t e r v a l d t , t h i s equation can be changed t o a power r e l a t i o n s h i p a s follows:

dp + pvdv + p g e dt dt dt

=

h(1osses) dt

(8.10)

which, f o r steady flow, can be w r i t t e n as dz dx pg- = -d( l o s s e s ) dx d t dt or pgvs = -d( l o s s e s )

(8.11)

dt

where s = dz/dx = s l o p e ; Equation (8.11)

and v = dx/dt = v e l o c i t y .

can, f o r open channel flow with c r o s s - s e c t i o n a l a r e a A , be

w r i t t e n as

!,p,vs

=

I,

(8.12)

%(losses)

where jApgvs = t o t a l i n p u t stream power per u n i t

l e n g t h ; and

iAd / d t

(los-

ses) = t o t a l applied stream power per u n i t l e n g t h . Although t h i s equation s t a t e s t h a t t h e t o t a l i n p u t stream power equals t h e t o t a l applied stream power over an u n i t l e n g t h , t h e r e i s a d i f f e r e n c e i n t h e v e r t i c a l d i s t r i b u t i o n of

these variables.

Whereas t h e i n p u t stream power

w i l l have a logarithmic v e r t i c a l d i s t r i b u t i o n i n an open channel, t h e v e r t i c a l d i s t r i b u t i o n of the a p p l i e d stream power can be i n f e r r e d from the equat i o n of u n i t applied stream power (Rooseboom, 1974; Yang and Molinas, 1982) 1.e.

Unit a p p l i e d stream power =

T

dv dY

(8.13)

155 The d i f f e r e n c e i n v e r t i c a l d i s t r i b u t i o n between t h e s e two v a r i a b l e s f o r open channel f l o w i s shown i n F i g u r e 8 . 4 .

T h i s f i g u r e shows t h a t i n p u t s t r e a m

power d i s t r i b u t i o n i s p r o p o r t i o n a l t o t h e v e l o c i t y d i s t r i b u t i o n , whereas t h e major

portion

is

applied a t

t h e boundary t o overcome f r i c t i o n .

In

the

s e c t i o n which f o l l o w s i t i s shown t h a t i t i s indeed t h e a p p l i e d stream power t h a t must b e minimized t o e n s u r e t h a t t h e s y s t e m performs i t s f u n c t i o n w e l l , i . e . t h e e f f e c t i v e t r a n s p o r t a t i o n of water, when i n a s t a b l e non-equilibrium condition.

Fig. 8 . 4 flow.

(iii)

D i s t r i b u t i o n of a v a i l a b l e and a p p l i e d stream power i n open c h a n n e l

E f f i c i e n c y of flow.

The f u n c t i o n of t h e s y s t e m under d i s c u s s i o n i s ,

a s s t a t e d p r e v i o u s l y , t h e e f f i c i e n t t r a n s p o r t a t i o n of water. that

t h i s o b j e c t i v e i s reached

when

applied

stream

power

It c a n b e shown

is

minimized.

Conservation of energy between p o i n t s 1 and 2 on a s t r e a m l i n e can b e w r i t t e n as E l = E2 + hf

where El = t o t a l energy a t p o i n t

(8.14) 1;

E2 = t o t a l

energy a t p o i n t

hf = i r r e v e r s i b l e energy losses between p o i n t s 1 and 2.

2;

and

156

T h i s e q u a t i o n c a n b e w r i t t e n a s a power r e l a t i o n s h i p by d i f f e r e n t i a t i n g w i t h respect t o t i m e , i.e. dE1

dE2

dt

dt

- =-

dhf

(8. 15)

+dt

where d E l / d t = dE2/dt = t o t a l s t r e a m power a v a i l a b l e a t p o i n t s 1 and 2 ; and dh / d t = a p p l i e d s t r e a m power r e q u i r e d t o overcome f r i c t i o n between p o i n t s I f and 2. E f f i c i e n c y of f l o w ( r l )

between p o i n t s 1 and 2 c a n t h e n b e d e f i n e d f o r condi-

t i o n s where no a d d i t i o n a l power i s i n t r o d u c e d between t h e p o i n t s , i . e .

The h i g h e s t f l o w e f f i c i e n c y power i s a minimum.

is

t h e r e f o r e reached when t h e a p p l i e d s t r e a m

A s t h i s c o n d i t i o n i s r e a c h e d when t h e s y s t e m i s i n a

s t a b l e n o n - e q u i l i b r i u m c o n d i t i o n i t i s proposed t h a t sediment i s d e p o s i t e d i n r e s e r v o i r s t o e n a b l e w a t e r t o f l o w through a r e s e r v o i r i n t h e most e f f i c i e n t manner.

The

following

hypothesis

is

t h e r e f o r e proposed

to describe

the

b e h a v i o u r of a f l o w i n g f l u i d : When a l t e r n a t i v e modes of

flow e x i s t , a f l u i d w i l l always f o l l o w

t h a t mode t h a t r e q u i r e s t h e l e a s t amount of a p p l i e d stream power. A l t e r n a t i v e modes r e p r e s e n t a l t e r n a t i v e ways by which a system c a n yield.

A s t a b l e non-equilibrium condition i s

o n l y r e a c h e d when

f l o w i s s u c h t h a t a p p l i e d stream power i s a mininxim, s u b j e c t t o t h e l i m i t a t i o n s imposed on t h e system. Flow i n l o o s e boundary c h a n n e l s c o n s i s t s of y i e l d i n g of t h e f l u i d and of t h e s u r f a c e of

the

channel

boundary.

Minimization

of

applied

stream

power

t h e r e f o r e t a k e s p l a c e i n b o t h t h e f l u i d and a t t h e boundary, w i t h t h e l a t t e r p l a y i n g t h e dominant r o l e .

Rubey (1933) i n d e e d e s t i m a t e d t h a t 96 p e r c e n t of

t h e s t r e a m power i s a p p l i e d a t t h e bed.

I f one,

t h e r e f o r e wants t o d e r i v e a c r i t e r i o n t o

identify

f o r p r a c t i c a l purposes, s t a b l e non-equilibrium

c o n d i t i o n s of r e s e r v o i r s e d i m e n t a t i o n , i t i s r e a s o n a b l e t o i g n o r e t h e e f f e c t

157

of

minimization

of

s t r e a m power

minimization a t t h e boundary of

i n the fluid itself flow.

The e r r o r

and c o n c e n t r a t e on

introduced

in

t h i s way

should b e n e g l i g i b l e . Minimization of a p p l i e d stream power a t t h e bed c o u l d b e e x p r e s s e d mathemati c a l l y a s (Rooseboom and Mclke, 1982), minimize

[T$

]

=

b where

K =

minimize pgDs-JgDs Kk

von Karman c o e f f i c i e n t ;

(8.17)

k = absolute roughness;

and D = d e p t h of

flow. Minimization of e q u a t i o n ( 8 . 1 7 )

c o u l d t h e r e f o r e b e o b t a i n e d by changing t h e

channel geometry ( i . e . by v a r y i n g D , s and k ) a n d / o r by changing t h e properties

the f l u i d

of

(i.e.

by v a r y i n g K ) by

e.g.

entraining or depositing

sediment. When

a

s t a b l e non-equilibrium

condition has

r e g a r d s sediment t r a n s p o r t i s a l s o r e a c h e d ,

been

reached,

s t a b i l i t y as

i n d i c a t i n g t h a t t h e Von Karman

c o e f f i c i e n t h a s approached a c o n s t a n t v a l u e , from which f o l l o w s

(8. 1 8 )

where G = c o e f f i c i e n t Applied stream power a t

t h e bed of

t h e channel w i l l

therefore

assume a

minimum v a l u e when

(8. 19)

The c h a n n e l p r o f i l e w i l l t h e n b e s t a b l e and no change i n t h e n e t volume of deposited

sediment w i l l o c c u r

i n the

long

term.

The p r o p e r t i e s

of

the

sediment and t h e r e f o r e t h e v a l u e of t h e a b s o l u t e roughness k w i l l under t h e s e c o n d i t i o n s assume a c o n s t a n t v a l u e o v e r

t h e whole l e n g t h of

the

channel

p r o f i l e and i t c a n t h e r e f o r e b e concluded from e q u a t i o n (8.19) t h a t JgDs = c o n s t a n t

(8.20)

158

for a loose boundary channel as a whole under conditions of stability and minimum applied stream power.

As this parameter is derived from the minimi-

zation of applied stream power, representing the rate of internal entropy production, it will, as in the latter case, assume a constant value throughout the reservoir and not only at a particular point.

The parameter will

also not assume an universal constant, but its value will, as in the case of minimization of

internal entropy production, depend

on

the

limitations

imposed on the system.

Fig. As

the exchange of

sediment particles between fluid and channel boundary

would also be stable, it is possible to make use of conditions that prevail under such circumstances to derive the same criterion.

159 By using stream power theory it can be shown theoretically (Annandale, 1984; Rooseboom, 1974) and also experimentally (Yang, 1976) that

* J

= constant

(8.21)

ss

where v

ss

=

settling velocity of

sediment under conditions that prevail

during incipient motion of sediment (Figure 8.5). Following the same argument that led to equation (8.20) and assuming a direct relationship between absolute roughness and settling velocity, it can also be concluded from this point of view that equation (8.20) is valid and that shear velocity assumes a constant value under stable non-equilibrium conditions when applied stream power is a minimum. Verification of equation (8.20) as a criterion to identify non-equilibrium conditions of sediment transport in loose boundary channels is found by investigating case studies. 8.1.2

Verification

The principle of minimization of applied stream power which was derived from basic principles of non-equilibrium thermodynamics can be applied to explain the behaviour of rivers in general.

Using these principles, the cross-

sectional shape of rivers flowing around bends can be explained theoretically (Annandale, 1984) and it can be shown that applied stream power approaches a constant minimum value throughout a stable river reach. Such observations in regime theory have already been made by Langbein and Leopold (1957) who concluded that a stable channel represents a state of balance with a minimum rate of energy expenditure or an equal rate of energy expenditure along the channel.

Chang came to the same conclusion by applying

the principle of minimum stream power in mathematical modelling of rivers (Chang, 1982a and 1984).

In discussing a laboratory study of delta formation

in a reservoir, he also concluded that stream power approaches a constant minimum value under stable conditions (Chang, 1982b).

Griffiths

(1983),

using the theory of Chang (1979 and 1980b), also derived constant parameters with which stability of various types of rivers can be checked. Verification of

the theory presented herein, with specific reference to

reservoir sedimentation, is found by discussion of case studies of

three

South African reservoirs. The three reservoirs concerned are Lake Mentz, Van

160 Rhyneveldspass r e s e r v o i r and Welbedacht r e s e r v o i r .

Of t h e s e t h r e e r e s e r v o i r s

t h e l a s t two a r e a p p r o a c h i n g s t a b l e c o n d i t i o n s , whereas t h e r e s u r v e y h i s t o r y of Lake Mentz i s used t o i l l u s t r a t e t h e p r i n c i p l e . Lake Mentz, which was b u i l t i n 1924 and h a s l o s t more t h a n 40 p e r c e n t of i t s o r i g i n a l volume due t o sediment d e p o s i t i o n , l i e s i n t h e s a m e r i v e r as t h e Van Rhyneveldspass r e s e r v o i r .

The h i g h sediment y i e l d of

this river,

i.e.

the

Sondags R i v e r , has a l s o c l a i m e d 39 p e r c e n t of t h e o r i g i n a l volume o f t h e Van Welbedacht reser-

Rhyneveldspass r e s e r v o i r s i n c e i t s c o n s t r u c t i o n i n 1925.

v o i r , l y i n g on a n o t h e r h i g h sediment y i e l d r i v e r v i z . t h e Caledon R i v e r , has a n even more d r a m a t i c h i s t o r y i n t h e s e n s e t h a t more t h a n 50 p e r c e n t of i t s volume h a s b e e n l o s t due t o sediment d e p o s i t i o n s i n c e i t s c o n s t r u c t i o n i n 1973.

Other r e l e v a n t d e t a i l of t h e s e r e s e r v o i r s a p p e a r i n T a b l e 8.1.

Plan

views a p p e a r i n F i g u r e s 8 . 6 , 8 . 7 and 8 . 8 . TABLE 8.1

Original capacity

Sediment volume Mean Original (% of o r i g i n a l annual capacity/MAR capacity) runoff ratio (x 1 0 ~ ~ ~ ) (x 1 0 6 ~ ~ )

Reservoir

Van Rhyneveldspass Lake Mentz Welbedacht

76,3

39,O

35,6

327,6 114.1

41.5 51 .O

159,5 2 422.9

Shear v e l o c i t i e s f o r one-in-five-year

Catchment size

(km2 3 680 16 300 15 245

291

0.05

flow conditions (being regarded as the

dominant flow) were c a l c u l a t e d a t v a r i o u s l o c a t i o n s t h r o u g h o u t t h e s e reserv o i r s i n o r d e r t o v e r i f y equation (8.20).

I n s p i t e of t h e f a c t t h a t Lake

Mentz h a s n o t a t t a i n e d a s t a b l e c o n d i t i o n , v e r i f i c a t i o n of

e q u a t i o n (8.20)

can be found by a n a l y z i n g t h e r a t e of sediment d e p o s i t i o n i n t h i s r e s e r v o i r over various periods

and by

comparing

observed f o r t h e c u r r e n t condition.

the

latter

with

shear

velocities

The h i s t o r y of sediment d e p o s i t i o n a t

two l o c a t i o n s f o r t h i s r e s e r v o i r i s found i n T a b l e 8 . 2 and a p l o t of velocities

i n Figure 8.9.

The a v e r a g e s h e a r v e l o c i t y

shear

a t l o c a t i o n 1 1 is

c l o s e r t o t h e c r i t i c a l v a l u e t h a n t h e a v e r a g e s h e a r v e l o c i t y a t l o c a t i o n 5. By comparing t h i s w i t h t h e r a t e s of sediment d e p o s i t i o n a t t h e two l o c a t i o n s , i t i s s e e n t h a t t h e r a t e of d e p o s i t i o n a t l o c a t i o n 5 i s much h i g h e r t h a n t h a t at location 11.

T h i s i n d i c a t e s t h a t s e d i m e n t d e p o s i t i o n i s such t h a t s h e a r

v e l o c i t y w i l l u l t i m a t e l y approach a c o n s t a n t v a l u e under s t a b l e c o n d i t i o n s .

161

0.5

f . . . .

0

.

0.5 SCALE

1

1.5

2

- krn

DAM

Fig. 8.6

Plan view of Van Rhyneveldspass reservoir.

0

1

2

SCALE

- km

- - I

Fig. 8.7

Plan view of Lake Mentz.

The plot of shear velocities for the van Rhyneveldspass reservoir in Figure

8.10 indicates however that this reservoir is closer to stability than Lake Mentz.

These values are approaching a constant average shear velocity of

approximately 6 x 1 0-3m/s throughout the reservoir.

162

0

1

2-

3

1

4

Scale-km

Fig. 8.8

Plan view of Welbedacht r e s e r v o i r .

20

-

16

c?

0 l-

Y

12

\ u)

g 8 4 0 0

2

1

3

4

5

DISTANCE FROM DAM

Fig. 8 . 9 Welbedacht

6

- km

7

Relationship between shear v e l o c i t y and d i s t a n c e f o r Lake Mentz. reservoir,

allowed c a l c u l a t i o n of

being

much

longer

than

the

enough shear v e l o c i t i e s

other

two r e s e r v o i r s ,

t o construct

a histogram.

This histogram, which i s presented i n Figure 8.11 with a curve of t h e r e l e vant normal d i s t r i b u t i o n superimposed on i t , i n d i c a t e s t h a t shear v e l o c i t i e s approach a c o n s t a n t v a l u e i n t h i s r e s e r v o i r .

Equation (8.20)

i s therefore

approached, implying t h a t t h i s r e s e r v o i r i s approaching s t a b l e conditions.

163

20

-

16

Y

12

m

z

G8 \ u)

E

4

0

1

0

2

3

DISTANCE FROM DAM

- km

Fig. 8.10 Relationship between shear velocity and distance for Van Rhyneveldspass reservoir.

Relative Frequency ('3.) 29.6

25.9

f1

x- 0,104 s = 0,0288 coefficient of variation-O,28 curtosis = 2,8 113

22.2

1

18.5

14.8

11.1

7.4

3.7

0.0 0

9

0

Fig. 8.11

x

N

t

8

x

In

Q

8

2

0

.

-

N

0

2 0

:

0

:

0

&F (m/s)

Histogram of shear velocities for Welbedacht reservoir.

164 TABLE 8 . 2

Rate of sediment d e p o s i t i o n a t t w o l o c a t i o n s i n Lake Mentz (m3/m/year)

~~

Year

Position 5

1924- 1926

Position 1 1

434

15

19 26- 1929

834

21

1929- 1935

1 356

149

1935- 1946

1 175

29

860

29

1946-1978

8.1.3

C a l c u l a t i o n procedure:

A n a l y t i c a l approach

Both a n a l y t i c a l and semi-empirical

procedures f o r c a l c u l a t i n g p r o f i l e s of

deposited sediment f o r s t a b l e non-equilibrium conditions can be developed by using t h e p r i n c i p l e of minimization of applied stream power. procedure w i l l be d e a l t w i t h

in this

subsection,

and

The a n a l y t i c a l

t h e semi-empirical

approach w i l l be explained i n t h e next. The o b j e c t of t h e a n a l y t i c a l procedure i s t o determine t h e p r o f i l e of r i v e r bed t h a t w i l l r e s u l t i n a constant value of dominant flow conditions

(equation ( 8 . 2 0 ) ) .

the

the shear v e l o c i t y f o r

This can be achieved by per-

forming a modified backwater c a l c u l a t i o n with two moving boundaries, v i z . t h e bed p r o f i l e and t h e f r e e water s u r f a c e . assuming a

cross-sectional

p r o f i l e of

The procedure c o n s i s t s the r i v e r

of

first

channel a t a p a r t i c u l a r

chainage, whereafter t h e energy equation i s balanced by t h e standard s t e p method f o r backwater c a l c u l a t i o n (Henderson,

1966) t o e s t a b l i s h t h e water

stage. Once t h e l a t t e r has been e s t a b l i s h e d , t h e shear v e l o c i t y can be c a l c u l a t e d t h e v a l u e of

and compared with t h e assumed constant value.

If

velocity

value,

does

not

match

the

assumed

constant

t h e shear

the calculation

is

repeated with a new assumed c r o s s - s e c t i o n a l p r o f i l e a t t h e chainage under consideration.

This procedure i s repeated u n t i l t h e shear v e l o c i t i e s match,

whereafter t h e c a l c u l a t i o n i s c a r r i e d out a t t h e next cross-section. Two problems concerning t h e c a l c u l a t i o n procedure s t i l l have t o b e resolved however v i z . ,

t h e manner by which t h e bed p r o f i l e i s a d j u s t e d and the con-

s t a n t v a l u e t o be assumed f o r t h e shear v e l o c i t y .

Adjustment of

p r o f i l e must be made t o approximate t h e a c t u a l d e p o s i t i o n of closely

as

possible.

Inspection

of

resurveyed

sections

the bed

sediment as of

sediment

165

deposition in reservoirs reveals that the major proportion of sediment is deposited from the bottom up, i.e. the deepest part of the cross-section at each position in the reservoir basin is filled before sediment is deposited on the sides.

A reasonable assumption would therefore be to adjust the bed

of reservoir basin cross sections with horizontal lines from the "bottom up". The fact that the constant value for shear velocity approached during stable non-equilibrium conditions of

reservoir sedimentation is

(compare e.g. Figures 8.9, 8.10 and 8.11), correct value.

As

not universal

presents a problem in choosing the

the value sought indirectly represents minimization of

applied stream power, it is a function of fluid and sediment properties as well as of discharge and channel geometry.

However, a practical way which is

proposed f o r such estimation is to calculate shear velocities in the original river for

dominant flow conditions (say one-in-two or one-in-five-year

discharges) at various cross-sections and use the average value as criterion. Strictly speaking this should hold only for rivers that are approaching stability themselves, but ought to yield representative values for other rivers. Example : The example presented here is designed to illustrate the principles of the calculation procedure and therefore represents a very simple model of a

100,000rn

7

Fig. 8.12.

Model of reservoir basin to illustrate calculation procedure.

reservoir in which sediment can be deposited.

The river channel and reser-

voir basin both have widths of 1 m and slopes of 0,002 and 0,OZ respectively. Flow depth in the original river channel is I m and varies in the reservoir

TABLE 8.3

A n a l y t i c a l C a l c u l a t i o n Procedure

Stage

Qlainage

IaJ bed level 'depth

Area

2/29

0,0225

Total

R43

head

Friction Average s Length of slope over reach reach

hf

head

JgDs

0,1326 0,1400

0 0

100,980 100.980

99,900 99,980

1,080 1

1,080

1,ooO 0,0262

101,0325 101,0062

3,160 3,000

0,3418 0,3333

0,2390 0,2311

0,0017 0,0020

4 4

99,990 99,990 99,988

1,010 0,998

1,010 0,0257 0,998 0,0263 1,ooO 0,0262

101,0257 101,0143 101,0142

3,020 2,996

L

101,000 100,988 100,988

0,3344 0,3331 0,3333

0,2321 0,2309 0,2311

0,0020 0,0020 0,0320

0,0020 0,0020 0,0020

4 4 4

0,0078 0,0081 0,0080

101,0140 101,0143 101,0142

0,1403 0,1400

8 8 8

101,100 100,997 100,996

99,990 99,990 99,996

1,110 1,007

1,110 1,oM)

101,1213 101,0229 101,0222

3,220 3,014

1

0,0213 0,0259 0,0262

0,3447 0,3341 0,3333

0,2417 0,2318 0,2311

0,0316 0,0020 0,0320

0,0018 0,0320 0,0020

4 4 4

0,0071 0,0079 0,0080

101,0213 101,0221 101,0222

0,1394 0,1400

10

101,00

1,ooO

0,0262

101,0262

3,coO 0,3333 0,2311 0,0320

0,0020

2

0,0040 101,0262 0,1403

100,oo

1 ,ooO

1,007

3,000

3,O

Notes:

# 0,1400

new ass&

(I)

Shear velocity

(2)

Shear velocity = 0,1400 m / s , proceed t o next reach.

(3)

Total head in c o l m (7) # total head i n colm (15), adjust water stage

(4)

Total heads balance but shear velocity # 0,140c m/s, adjust bed level and repeat calculation.

m/s, repeat calculation with

(5) Total heads balance and shear velocity

= 0,1403 m/s,

bed level.

proceed to next reach.

repeat calculation.

Notes

167

basin.

The

stage close to the dam wall is assumed to be

100,980

m

(Figure 8.12). River and reservoir properties could therefore be summarized as follows: Width

=

B

= I

m

Flow depth in river

=

D

Bed slope of river = s Bed slope of reservoir Manning's n

= 0,030

= 1

m

= 0,002 = s'

0 =

0,02

(assumed constant)

Hydraulic radius of river reach Flow velocity in river Discharge

Q

=

=

=

v

=

=

R

B x D +

= -= ZD

R43 s g2

= n

0,333 m

0,717 m/s

vA = 0,717m3/s

Shear velocity in river reach

=

Go = 0,140 m/s.

An assumption is made that the river is in a stable non-equilibrium condition and the constant value assumed for the shear velocity for similar conditions in the reservoir basin is therefore set at 0,140 m/s. The calculation procedure to establish the stable non-equilibrium profile of deposited sediment in the reservoir basin is presented in Table 8.3.

The

final levels for the stable non-equilibrium condition are therefore: Reach (m)

Bed level (m) 99,980 100,988 100,996 100,000

0

4

8 10

8.1.4

A

Calculation procedure:

semi-empirical

Semi-empirical approach

calculation procedure developed from the principle of

minimization of applied stream power is presented in this subsection.

In an

effort to obtain a simple functional relationship which can be used to compile a semi-empirical graph representing sediment distribution during stable non-equilibrium conditions, use can be made of equation (8.12).

This

equation states that total input stream power equals total applied stream power at all times, therefore implying that input stream power will also have a minimum value when applied stream power has been minimized during stable

168 non-equilibrium conditions.

The most relevant parameters can therefore be

obtained by differentiating input stream power (pi, represented by

P = PgQs

(8.22)

where p

=

and s

energy slope with respect to longitudinal distance (x) and setting

=

mass density;

g

=

acceleration due to gravity;

Q

=

discharge;

the result equal to zero. The following relationship is then obtained:

(8.23)

where A

=

cross-sectional area of flowing water; and P

=

wetted perimeter.

Chang (1982b) and Annandale (1984) concluded that flow tends towards uniform conditions when stream power approaches a minimum, i.e.

3 + 0

(8.24)

dx

which implies a relationship between longitudinal sediment distribution in a reservoir and the rate at which the wetted perimeter changes with distance in the direction of flow under conditions of minimum stream power

(equa-

tion (8.23)). This conclusion can be used t o compile a semi-empirical graph relating dimensionless cumulative volume of

deposited

sediment to

dimensionless

longitudinal distance in a reservoir basin, measured from the dam wall. Compilation of such a graph requires a number of reservoirs which are in the stable non-equilibrium condition. As such data are relatively scarce, reservoirs with large volumes of accumulated sediment and different values of dP/dx were used instead.

Relation-

ships thus obtained should approach that of stable non-equilibrium conditions as the profiles of deposited sediment will approach stable conditions asymptotically.

The reservoirs used to compile the dimensionless relationship in

Figure 8.13 for various values of dP/dx are presented in Table 8.4.

169 TABLE 8.4 Reservoirs used in compilation of Figure 8.13

Symbol

Kopp ies Hartebeespoort Wentzel Van Rhyneveldspass G amkapo o r t Leeugamka Lake Mentz Grass ridge Welbedacht Glen Alpine Flor is kraal

A

0

0 0 A 0

0 A

v

0

02

dP -

Sediment volume %

Reservoir

0,4

dx

0,02-0,05 0,75-1,33 0,09-0,lO 1,22 0 , I1-0,67 0,80 0,80

22,74 15,57 20,34 39,03 13,84 35,52 41,47 43,61 51,54 7,85 22,96

0,6

0,10

0,16 0,lO-0,20 0,50

0,8

1,o

Relative Distance from Dam Wall (L/LFSL) Fig. 8.13 Dimensionless cumulative mass curves explaining sediment distribution as a function of dP/dx for stable conditions.

170

The general behaviour of this relationship can be verified by observing the limits. The condition when

dP + o

(8.25)

dx

represents a situation where only a small disturbance exists in the channel. Sediment will under such circumstances be deposited in the proximity of the disturbance with very little build-up in the upstream direction.

The dimen-

sionless cumulative curve will then have a shape as shown by curve A in Figure 8.14.

Fig. 8 . 1 4

Sediment deposition for extreme values of dP/dx.

When, however, dP dx

- + m

(8.26)

a condition similar to a river flowing into an ocean exists, and the major volume of sediment will be deposited in the vicinity of the river mouth (curve B in Figure 8 . 1 4 ) . It should be observed that Figure 8.13 accounts only for sediment deposited below full supply level.

In an effort to compile an empirical graph to

171

estimate sediment deposition above full supply level, five reservoirs from Table 8 . 4 , viz.

-

Leeugamka

-

van Rhyneveldspass

-

Welbedacht

Grassridge Lake Mentz

in which sediment deposits above full supply level were observed were used to compile Figure 8.15. A

o

Leeugarnka Grassridge Van Rhyneveldspas Mentz Welbedacht

1

L Relative Distance from Dam wall (LFSL

Fig. 8.15

Sedlment distribution above full supply level.

This graph does not indicate any significant trends in deposition of sediment above full supply level as regards wetted perimeter changes, an observation which could be expected as these changes in river reaches are relatively insignificant compared with those in reservoirs. Example: This example is designed to merely illustrate the calculation procedure involved in using Figures 8.13 and 8.15 to estimate sediment distribution in reservoirs, and therefore one of the reservoirs used to compile these figures, viz. Lake Mentz, i s used as prototype. appears in Figure 8.6.

The plan view of this reservoir

172 After estimating the volume of sediment expected to deposit in the reservoir by methods discussed in Chapter 5, in this case assumed to be 129 x 106m3, the first step in the calculation procedure for estimating sediment distribution by the semi-empirical method is to establish the value of dP/dx.

This

is done by compiling a graph representing the relationship between wetted perimeter and distance.

I n the case of wide reservoirs the wetted perimeter

can be replaced by the width of the reservoir at particular sections.

By

making this assumption for Lake Mentz the relationship between width of water surface at full supply level and distance was compiled as indicated in Figure 8.16.

6000

-

. '\

5000-

-

p

d_p =

dx

4 m -

2ooo

2500

= 0,80

5

g

3ooc-

2Mx)

-

a00 -

I

0

lob0

2dOO

3000

4000

5000

6000

7000

8000

goo0

KIWO

11 OOo

I

12000

Distance (In)

Fig. 8.16 Determination of dP/dx for Lake Mentz. From this graph it is then estimated that -dP =

dx

0,80

This information is used to select a dimensionless cumulative sediment volume curve from Figure 5.13 and set up a table relating cumulative sediment volume and distance.

This is done in Table 8.5, a table that also presents infor-

mation regarding sediment distribution above full supply level, obtained from Figure 8.15.

173 8.2

UNSTABLE NON-EQUILIBRIUM CONDITIONS

An approach for calculating the shapes of deposited sediment during unstable non-equilibrium conditions is also required.

The principle of

minimum

applied stream power cannot be used here as the magnitude of the stream power is continuously changing with the changing shape of the sediment profile. This progression of course comes to an end when a stable non-equilibrium TABLE 8.5 Sediment distribution for Lake Mentz obtained with the aid of Figures 8.13 and 8.15.

Relative distance

Actual distance

L/LFSL

(m)

Dimensionless cumulative sediment volume 1 (V/VFSL)

(x106m3)

0

0 9,92 24,81 44,65 62,02 74,42 86,83

0 1 200 2 400 3 600 4 800 6 000 7 200 8 400 9 600 10 800 1 2 000 13 200 14 400 15 600 16 800

0 0,1 092

093

114

Estimated cumulative sediment volume

0,08 0,20

0,36 0,50

0,60 0,70 0,82 0,90

0,95 1

,oo

1,02 1,03

1.04 1104

01,71 I I ,63

17,84 24,04 26,52 27,76 29,OO 129,oo

Estimated sediment volume between sections (XI06,~)

0 9,92 14,89 19,84 17,37 12,40 12,41 14,88 9,92 6,21 6,20 2,48 1,24 1,24 0,oo

condition is approached asymptotically and stream power is ultimately minimized.

However, the sediment carrying capacity of a stream acts as

an

important limiting factor of sediment transport through a reservoir during unstable conditions and has a major influence on determining the shape of deposited sediment profiles.

By using a parameter representing sediment

carrying capacity, such as stream power, it is possible to relate the slope of deposited sediment to the carrying capacity.

This is done by first

presenting the theory, verifying it and then explaining calculation procedures to be followed when calculating shapes of sediment profiles during unstable conditions.

8.2.1

Theory

Sediment in reservoirs can be transported by three different modes viz.

174 (i)

colloidal suspension,

(ii)

turbulent suspension and

(iii) density currents. Turbulent suspension is considered to be the dominant mode of transportation of sediment in reservoirs. The small percentage of particles transported as colloids combined with the special conditions required for deposition of such particles contributes to its negligible effect on the shape of deposited sediment profiles.

Rooseboom (1975) further showed that density currents

containing sediment only occur under special conditions in steep and deep reservoirs.

This mode of transportation of sediment can therefore be re-

garded as a special case and can be ignored in general analysis of sediment deposition in reservoirs.

The approach followed is therefore only to con-

sider turbulent suspension by deriving an equation relating the profile of deposited sediment to sediment carrying capacity of a stream. Stream power can be used profitably to express the relationship between the sediment carrying capacity of turbulent flow and sediment concentration (Rooseboom, 1974; Yang and Molinas, 1982).

Yang (197613) also showed that

such a relationship is superior to most of the sediment transport theories currently used.

It was therefore decided to use such an equation as basis to

express the relationship between the profile of deposited sediment in a reservoir and sediment carrying capacity of a current within a reservoir. The sediment transport equation which is used therefore is the most basic form of that proposed by Yang (1972), viz. log c

=

n + B log

where c = sediment concentration;

-

power; v

=

(8.27)

(GS)

a,

B

average flow velocity; and s

= =

coefficients;

vs

=

unit stream

energy slope.

An advantage of using this equation is that the coefficients n and 6 do not vary significantly with varying sediment size, especially for diameters less than 1,7 mm (Yang, 1972). A mathematical relationship between the profile of deposited sediment and stream power can be derived by observing flow through a wide rectangular channel.

Figure 8.17 represents such a channel in which small changes in

total flow area (A), suspended sediment area (A ) , sediment discharge (Q ) , sediment concentration (c) and elevation ( z ) sediment over a small distance (dx).

occur due to deposition of

175

Continuity of sediment discharge without lateral inflow over a small distance dx can be written as, dA dQs+ s = 0

(8.28)

dt

dx

B

SEDIMENT

Fig. 8.17

/

Sediment deposition in a wide rectangular channel.

By using the relationship between total discharge area and cross-sectional area of suspended sediment, viz. A

=

(8.29)

c.A

it follows from equation (8.28) that

(8.30)

Equation (8.30) can then be expanded as follows:

dAs dt

=

c d A +A dt

d c dt

(8.31)

As the variables in equation (8.31) are functions of both space and time, this equation can be rewritten in partial differential form as,

176

(8.32)

which after a small time-interval can be written as

-aAs - - c&?.+A- ac ax ax ax

(8.33)

A general relationship between the profile of deposited sediment and other

variables of the cross-section can therefore be written as

_ ax az - I - A where A

=

[ c %ax+

A&

ax

1

porosity of deposited sediment; and B

(8.34)

=

width of flow.

For the special case when

(8.35)

it is possible to write equation ( 8 . 3 4 ) in much simpler form as,

(8.36)

where y

=

depth of flow.

The partial differential of sediment concentration c can be obtained by partially differentiating equation ( 8 . 2 7 ) with respect to x, i.e.

(8.37)

Substituting equation ( 8 . 3 7 ) into equation ( 8 . 3 6 ) yields

(8.38)

for small values of A.

177 8.2.2

Verification

Equation (8.38) is verified by applying it for mean annual flow conditions to Glen Alpine and Wentzel reservoirs, two South African reservoirs with 7,9 and 20,3

per cent accumulated sediment by volume.

TABLE 8 . 6

aA Comparison o f the products c -and ax

ac A-for ax

A comparison between the

Glen Alpine reservoir.

Distance from dam (m) (1)

3 255

6,8 x

1 , 7 x 10-l

4 395

],I

1 , 3 x 10-I

5 265

2,4

1,2 x 10-1

7 665

2,7

6,O x

9 122

3,6

10 250 1 1 345

12 620

2,4

1 3 a20

3,9

10-5

5 , 5 x 10-2

8,7

10-4

2 , l x 10-2

2,1

10-4

1 , 3 x 10-2 8,3

10-3

lob4

4,4

10-3

TABLE 8.7

aA ac Comparison of the products c- and A- for Wentzel Reservoir. ax ax

Distance from dam (m>

aA cax

ac Aax

1 760

1,2 x 100

2 117

1 , 2 x 100

2 461

7 , 1 x 10-1

2 830

6 , 3 x 10-l

3 255

4 , 8 x lo-'

3 718

4 , 3 x 10-1

4 180

1 , 9 x 10-l

4 705

9 , 5 x 10-2

5 218

9,5 x

5 768

6.3 x

10-2

178

E

4

a

.r(

4 C

c7

r-

M $=.

.r(

179 aA ac p r o d u c t s c - and A f o r t h e s e two r e s e r v o i r s i s p r e s e n t e d i n T a b l e s 8.6 ax ax and 8 . 7 , v e r i f y i n g t h e assumption made i n e q u a t i o n ( 8 . 3 5 ) . The v a l u e s of c

ac

i n t h e s e t a b l e s were c a l c u l a t e d by u s i n g e q u a t i o n s (8.27) and (8.37). ax P l a n views of t h e s e r e s e r v o i r s a l s o a p p e a r i n F i g u r e s 8.18 and 8.19. A

and

comparison between observed Figures

8.20

and 8 . 2 1 .

and

calculated

The broken

lines

sediment

profiles

appears

in

i n these figures represent the

c a l c u l a t e d p r o f i l e s , whereas t h e f u l l l i n e s r e p r e s e n t t h e observed sediment profiles.

m500

0, 1000 , 2000 2500m -

Scale

F i g . 8.19

P l a n view of Wentzel r e s e r v o i r .

L -

GLEN ALPINE

CALCULATED

-ACTUAL

DISTANCE

F i g . 8.20

- km

A c t u a l and c a l c u l a t e d sediment p r o f i l e s f o r Glen A l p i n e r e s e r v o i r .

180

96

E I

II 0

iii

I

-

FSL WENTZEL

94-

92-

-

90

-

DISTANCE

Fig. 8.21

-

km

Actual and calculated sediment profiles for Wentzel reservoir,

8.2.3 Calculation procedure:

Analytical approach

The calculation procedure for the analytical approach is based on applying equation (8.38) viz.

for mean annual flow conditions.

This one-dimensional equation, which does

not allow for transverse distribution of sediment, represents the sediment profile that will develop over a particular period provided sufficient sediment is available for deposition.

The

gradually

changing sediment

profile, which is characteristic of the unstable non-equilibrium condition, can be traced by repeating the calculation procedure as many

times as

required. For purposes of application it is necessary to distinguish the three compo-

(%)@I

nents of the equation, viz. the sediment concentration:stream power ratio (c/(;s)@), depth (y).

the rate of change of stream power

[&

The latter two of these parameters are

and the average flow variable and a function

of the flow conditions and reservoir under consideration, whereas the sediment concentrati0n:stream power ratio can be regarded as virtually constant for most cases.

181

By using the relationship between c and s;

which was determined for Hendrik

Verwoerd dam (Rooseboom, 1 9 7 5 ) viz. log c

= 0,9024

+ 0 , 2 4 3 8 l o g (Gs)

(8.39)

it can be shown that,

(8.40)

Yang ( 1 9 7 2 ) found that the relationship between sediment concentration and stream power (equation ( 8 . 2 7 ) ) was not very sensitive to sediment diameter, especially diameters less than

1,7

nun,

and the numerical value of the

constant in equation ( 8 . 4 0 ) can therefore be used with relative confidence for smaller sediment sizes.

a

Numerical values for the parameters - ( i s ) ’ and y can only be determined by ax considering the reservoir in which sediment is to be deposited. Calculation of such values will be demonstrated by taking Loskop dam as an example.

Assumptions that are made in the calculation procedure are:

-

the reservoir is full;

-

the water surface is horizontal;

-

the discharge is equal to the mean annual runoff and is constant through-

out the reservoir basin. The calculation, which i s

llustrated in detail in Table 8.8, is briefly

explained.

aa x (vs)~, which represents the rate at which Determination of the value stream power changes in the reservoir, is considered first.

The value of

this parameter is essentially the slope of a curve representing the relationship between stream power and distance in a reservoir.

By plotting

average stream power versus distance on log-log graph paper (Figure 8 . 2 2 ) , the relationship, (8.41)

can be determined.

182

TABLE 8 . 8 Calculation of stream power

Segment

2 3 4 5 6 7 8 9 10

11

12 13 14 15 16 17

18 19

Length of segment

Volume of segment

(L)

(V)

900

1 110

1 060 900 1 200 1 300 I 100 1 250

800

1 200

890 1 010 1 160

810

690 960 1 260 995

16 735 260 17 522 765 16 0 7 0 737 1 1 196 4 4 5 18 6 4 0 5 5 0 1 7 965 321 9 129 952 8 877 5 6 3 3 758 527 5 973 863 5 219 161 3 770 6 0 5 3 400 5 5 0 1 224 964 867 490 6 5 2 588 454 049 6 3 034

Mean annual runoff (MAR)

=

Mean crosssectional area (A = V/L)

594,73 786,27 161,07 440,49 533,79 819,48 299,96 102,05 698,16 978,22 864,23 733,27 931,51 1 512,30 1 257,23 679,78 360,36 63,35

18 15 15 12 15 13 8 7 4 4 5 3 2

1 4 , 7 7 m3/s

Mean velocity (V = MAR/A)

7,945~101~ 9 , 3 5 4 ~ 0-4 1 9,745~10-~ 1,187~10-~ 9,508~10-~ 1,069~10-~ 1,780~10-~ 2, O ~ O X I O - ~ 3,144xlO-3 2 , 9 6 7 ~ 10-3 2,51 ~ x I O - ~ 3,956~105,038~10 9 , 7 6 7 1~0: 1,175XlO-2 2,173xlO-2 4 , 0 9 9 ~1 0-2 2,331x10 1

Surface area of segment (A' )

97,51 104,67 101,34 89,74 115,55 170,13 83,90 74,17 40, I3 67,51 74,22 53,70 43,08 24, I6

18,OO 14,65 11,Il 8,95

Average depth (y

=

V/A')

17,16 16,74 15,86 12,48 16,13 10,56 10,88 11,97 9,37 8,85 7,03 7,02 7,89 5,07 4,82 4,45 4,09 0,70

Average stream power (V3/PY)

2,893 4,997 5,954 1,369 5,443

x x

3,014 2,321 9,009 1,656 1,377 3,436 2,354 1,719 1,849

x x x x x x 10 x IOI7 x x 10

x x lo-"

x 1,181 x 10-l' 5 , 2 9 0 x 10-l' 7 , 6 7 4 x 10-l' 3,387 x

Distance from origin

450

1 455

2 540 3 520 4 570 5. 820 7 020 8 195 9 220 10 220 1 1 265 12 215 13 300 14 285 15 035 15 860 16 970 18 097

184

It follows from equation ( 8 . 4 1 ) that

(8.42)

.

a - 0 which is representative of ax (vs)

In spite of the fact that B # 0 , it has been found that this method of

a

determining - ( G s ) ax

'

works relatively well.

Calculation of stream power at various locations in the reservoir, which is required to compile equation ( 8 . 4 1 ) , is explained in Table 8 . 8 .

By using the

Chezy equation, stream power can be written as

vs

=

3;

-

(8.43)

C2Y where C

=

Chezy coefficient; and y

=

average flow depth.

If the assumption is made that the Chezy coefficient is constant, C 2 can be replaced by another constant, e.g. g (acceleration due to gravity) without changing the slope of the stream power/distance curve or the dimensions of The parameter calculated in column 8 of Table 8.8 i.e.

equation ( 8 . 4 3 ) . v3 SY

(8.44)

therefore represents average stream power at particular segments. The values plotted in Figure 8.22 indicate a distinct change in slope between the sixth and seventh segments. approximately this position.

The value of (;s)

curve 1:

1,321 x 3 , 7 7 9 x 10-5

Equation ( 8 . 3 8 ) can therefore be enumerated as:

Segments 2-6

: dz - I ,055

dx

/L therefore changes at

The numerical values of ( i s )B /L for these two

curves are:

curve 2:

B

10-1hY

I

I I

I I

I

(

185

0

Q

+a

a

186

Segments 7-19:

=

dx

3,019 x IOb4y

The s l o p e a t segments 2-6 a l l y horizontal

i s s o s m a l l t h a t i t c a n b e c o n s i d e r e d t o be v i r t u -

and r e q u i r e s no f u r t h e r

calculation.

Table

8.9

p r e s e n t s t h e c a l c u l a t e d v a l u e s of t h e bed l e v e l s f o r segments 7-19.

however These

v a l u e s a r e p l o t t e d i n F i g u r e 8.23. TABLE 8 . 9 C a l c u l a t i o n of bed l e v e l s

(1)

(2)

(3)

Segment

Average depth (Y)

Segment length

(m)

(m)

10,56

1 300 1 100 1 250

7 8 9 10

10,88

11,97 9,37 8,85 7,03 7,02 7,89 5,07 4,82 4,45 4,09 0,70

11 12

13 14 15 16 17 18

19

(4) Elevation difference (Az)

(AL)

(m)

3,613 4,517 2,263 3,206 1,889 2,141 2,763 1,240 I ,004 1,290 1,556 0,210

800

1 200

890

1 010 1 160

810 690 960 I 260 995

(5) Bed l e v e l s (m)

66,308 69,921 74,438 76,701 79,907 81,796 83,937 86,700 87,940 88,944 90,234 91,790 92,000

Notes: 1.

Assume bed l e v e l a t 19 t o b e 92,000 m.

2.

E l e v a t i o n d i f f e r e n c e Az = 3 , 0 1 9 ~ 1 0 - ~ y ( A L )

A p p l i c a t i o n of t h e method a s s e t o u t i n t h i s example does n o t a l l o w c a l c u l a t i o n of t h e sediment p r o f i l e above f u l l s u p p l y l e v e l .

8.2.4

C a l c u l a t i o n procedure:

The r e l a t i v e l y

Semi-empirical approach

good c o r r e l a t i o n s

o b t a i n e d between c a l c u l a t e d and observed

s e d i m e n t p r o f i l e s when a p p l y i n g e q u a t i o n (8.38)

i n d i c a t e d t h a t i t might b e

p o s s i b l e t o r e l a t e a v e r a g e stream power i n a r e s e r v o i r t o a v e r a g e s l o p e of d e p o s i t e d sediment.

Such a r e l a t i o n s h i p

17 r e s e r v o i r s p r e s e n t e d i n T a b l e 8.10.

( F i g u r e 8.24) w a s o b t a i n e d f o r t h e Stream power i s r e p r e s e n t e d on t h e

0.1 I

I

I

I

I

I

I

I

I

I

I

I

I

I

Fig. 8 . 2 4

I

I

I

I

I

I

I

I

Semi-empirical relationship between stream power factor and slope of deposited sediment.

1

I

i

TABLE 8.10 Reservoir d a t a used t o compile r e l a t i o n s h i p between average stream f a c t o r and s l o p e of deposited sediment.

w r t Rietvlei Fast de Winter Wentzel Nwitgedacht Tiepmrt Bmnkhorstspmit Klasserie Beervlei hskap Welbedacht (1976) klbedacht (1978) Lake MEntz Van IUqwveldspass Hartebeespoort Glm Alpire Kamnassie Wies Pietersfmtein

9,35 931 54.37 40,35 63,32 20,15 51.69 46,24 69,78 449,33 2 422,99 2 422.99 159,54

35,s 162.11 97,92 33 I03,68 0,33

I I 033

9030 7 m 69M) 10 320 10 786 9 703 1 1 103 13 703 18 703 36 540 35 820 I 1 503

5 503 7 420

15 5M) 10 550

16 850 1 640

13,s 52 4,4 20,3 OJ 0,I 11,7 7.5 7.3 5,9 32,O 4499 41,5 39,o l0,O

73 77 8 22.7 5,3

46,423 12.197 28;483

5,105 79,489 34,343 58,577 5,789 88,749 179,794 77,587 62,811 191,758 46,538 194,627 21,928 36,276 40,715 2,491

O,M214 o,aI222 O,033!X

O,DYJ73 0,03307 O,co422 0,03233 0,00127 0,084 O,aI188 O,c0333 0,03027 O,m4 0,03224 0,03338 O,m3 o,aI222 0,03124 O,C0403

624,92 187,70 493,27 265,78 75739 910,64 848,27 118,79 2 299,61 1 582.78 I744,20 I €87,68 3 371,02 I 039,41 2 034,36 490,67 350,78 1 366,75 35,Ol

7,43 6,543 5,77 1,92 10,49 3,77 6,91 4,87 3,86 I1,36 4,45 3,72 5,69 4.61 9,57 4,47 lO,34

2,98 7.12

4 220,o 1 355,O 3 797.7 739,9 7 696,O 3 184,o 6 038,9 521,5 6 478.0 9 614;7 2 123,3 I 753,5 16 674,6 8 461,5 26 229,9 I 414,7 3 438,5 2 416,3 1 519,O

7,026 x 1015 2,295 x 4,540 lo-'+ 1,729 10-3 2,m

10-15

A

1013 lo-' lo-'

l,65 x 2.75 x I0 1,73 10113 2 , ~10-13 2,95 10-13 4,66 x 1,05 x 2,92 x lo-" I , @ x 10-6 2,31 x 10

C D E F G H J K I M

10-4

10113

10-4

2,031 x 10 2,714 x

4

1014

2,812 x 10 3

3.416 x i;a2 3,618 x 4,382 x 3,034 I ,334 1,960 2,195 x 3,536 1,321 6,835x

4,73

lo-'+

10-3 10

I,%

5,oi 5 3

10-l~ 8,03 1 0 - l ~ 2.41 x 10 l o

B

N

o P 0

189

a b s c i s s a of

this

g r a p h by

a p a r a m e t e r s i m i l a r t o t h a t proposed by equa-

tion (8,44) viz. (8.45)

where v ' = a v e r a g e f l o w v e l o c i t y through r e s e r v o i r = Q/A; flow

through

reservoir;

A = average

D = a v e r a g e d e p t h of r e s e r v o i r = V / A ' ;

level;

cross-sectional

Q = mean a n n u a l

a r e a of

V = r e s e r v o i r volume a t

A' = s u r f a c e a r e a of water a t f u l l supply l e v e l ;

flow = V/L; f u l l supply

and L = length of

r e s e r v o i r a t f u l l supply l e v e l . The c o r r e l a t i o n c o e f f i c i e n t of

t h e r e l a t i o n s h i p between sediment s l o p e and

stream power a p p e a r i n g i n F i g u r e 8 . 2 4

i s 0,80 and t h e d o t t e d l i n e s i n d i c a t e

t h e 95 p e r c e n t c o n f i d e n c e l i m i t s . No example on t h e a p p l i c a t i o n of t h i s approach i s g i v e n a s t h e d a t a p r e s e n t e d

i n T a b l e 8 . 1 0 , which w a s used t o compile F i g u r e 8 . 2 4 , i s s e l f - e x p l a n a t o r y . The s l o p e o b t a i n e d from t h e g r a p h i s t h a t of d e p o s i t e d sediment below f u l l s u p p l y l e v e l , and i s t h e r e f o r e r e p r e s e n t e d by a s t r a i g h t l i n e o r i g i n a t i n g a t t h e p o i n t where t h e h o r i z o n t a l water l e v e l

a t full

supply

l e v e l and

o r i g i n a l r i v e r bed meet ( F i g u r e 8 . 2 5 ) .

Fig. 8.25

P l a c i n g of sediment s l o p e i n a c c o r d a n c e w i t h F i g u r e 8 . 2 4 .

the

190

a.3

CONCLUSION

Procedures to calculate sediment profiles in reservoirs can be divided into two categories, viz. those for stable and those for unstable non-equilibrium conditions.

It is proposed that estimates of sediment profiles be obtained

by first applying techniques for determining the profile for stable nonequilibrium conditions.

In so doing the outer boundary of the sediment

profile is determined and the time-dependent profiles of the unstable nonequilibrium conditions can be established within this limit.