Morphological Response of Rivers to Withdrawal of Water

Morphological

Response of

Rivers to Withdrawal

of Water

Ir. A Crosato

-

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KORFHOLOGICAL RESPONSE OF RlVERS TO VITHDRAWAL OF WATER

December 1989

A1essandra Crosato

Delft University of Techno1ogy Faculty of Civil Engineering Hydraulic Engineering Group

CONTENTS

1. Introduction 1

1. 1 General 1

1.2 Objective of this study 1

2. Theory 2

2.1 General 2

2.2 The Equilibrium Model 2

2.3 One-Dimensional Time-Dependent Model. 7

3. Computational example of the morpho1ogical response to water

witbdrawa1 10

3.1 General 10

3.2 Input data 10

3.3 Dimensionless parameters 13

3.4 Representation of the results 14

4. Discussions and Conclusions 23

4.1 Discussions of results 23

4.2 Applicability of the performed analysis 25

5. Conclusions 27

HOTATIOHS

MORPHOLOGICAL RESFONSK OF RlVERS Ta A VITHDRAWAL OF WATER

1. Introduction

1.1 General

It is very convenient to make use of rivers as water supply for various purposes, i.e. irrigation, industrial and domestic uses. Even if it is a very common practice, the utilization of river waters must be carefully planned and its consequences analysed. For example, before starting on a new withdrawal of water the need of water supply more downstream must be considered. Futhermore, whenever those needs are satisfied, the morphological changes of the river must be taken into account. Charac-teristic consequences of withdrawal of water are: aggradation of the river bed; eventual rise of the water level and decrease of the water depth down-stream of the extraction point. All these morphological changes can affect the existing structures along the river and the navigability of the chan-nel. For this reason the morphological response of the river must be pre-dicted. In this way it is possible to take measures in advance, when neces-sary.

1.2 Objective of th is study

Here the river response to water withdrawal is analyzed by means of mathematical modeis. After a short description of the underlying theory, the morphological changes are investigated computationally by using the computer program "ODIRMO" (Vermeer [1985]) available at the Delft University of Technology. In order to give a more general insight into the involved processes the computations are carried out for several conditions. Furthermore, the input data and the output layout are comparable with those adopted in the study made by Hendrickx [1988]. In this way the morphological changes due to water withdrawal can be compared with the

2. Theory

2.1 General

The need of predicting the morphological changes of rivers due to water withdrawal is a classical problem. Nowadays mathematical models are much more convenient than scale models for their relative1y low costs, their general applicability, and the absence of scale effects. Due to need of verification of the developed mathematical models the sedimentation process caused by water withdrawal has been recently investigated experimenta1ly by Kerssens

&

Van Urk [1986].

In this chapter two simple ways of computing the final configuration of a river following water withdrawal are described. The first is the "equili-brium model". This is a very useful tooI for the extimation of the final equilibrium configuration (no information is given about the transitional period). The final formulas are very simple and handy and they can be solved with the use of a simple pocket computer.

In the same chapter a more complete model, but still very simple, is des-cribed in its main lines. This one-dimensional time-dependent model gives information about the transitional period also. However, due to the amount of computing work necessary to predict morphological changes with time, the adoption of this model requires a micro computer.

2.2 The Equilibrium Model

A very simple way to compute the new equilibrium conditions following (in time) a withdrawal of water is described here (the problem is schematized in Figure 1). The analysis applies for those rivers having a regular regime. The new equilibrium conditions are simply determined as a function of those immediately before the withdrawal is started. This simple analysis is not time-dependent, which means that no information can be obtained about the time preceding the establishment of the new equilibrium conditions (transitional period). For a more general understanding refer to De Vries [1986].

The main assumptions are:

the channel is in equilibrium at the starting time, t=O (the initial conditions are identified with subscript 0);

after the transitional period, at t = ~, the channel has reached a new equilibrium (the final situation is identified with subscript ~);

AQ

8

QO

r-t

L

1

-Figure 1 Schematization of the prob1em

The fol1owing equations are considered:

*

the Chézy equation for the flow: i (1)

in which:

Q discharge;

C Chézy coefficient;

B channel width; a water depth;

i water level slope.

*

A transport formu1a for the sediment (simp1ified to a power law):

n

s

=

m u (2)

in which:

s sediment transport per unit of width inc1uding pores; m coefficient;

n exponent.

When apdopting the Engelund

&

Hansen [1967] sediment transport formu1a:

n

=

5

with:

ä relative density of sediment; DSO mean grain-size;

g acceleration due to gravity.

The total amount of sediment transport in a cross-section is:

s

(4)

in which: S total amount of sediment transport including pores.

from Eq. [4] :

(l.

_-

1)

(l.)

(l.)

B n n S n _Q

a

=

m

Combining Eqs. [ 1] and [5] :

(2n)

_Cl

_{- -)}n n n i3 Q3

S m C 3 B 3

(5 )

(6)

This equation is valid for any value of the discharge. When the discharge in the stretch upstream of the intake is constant and equal to the initial value Q, and the amount of withdrawn water is constant and equal to öQ, the discharge downstream of the intake is:

Q

=

Q - äQ 1 0

Considering the two successive states of equilibrium characterized by the same S, m, C and B, but (due to the upstream withdrawal of water) with different Q, and assuming that the upstream boundary conditions, Q and S ,

o 0

remain unchanged two simple relations are obtained for the slope and water depth at the new equilibrium situation (downstream of the extract ion point):

Q - äQ o

Q - AQ

o (8)

From Eqs. [7] and [8] one notices immediately that a withdrawal of water leads to an increase of the slope (deposition) and a decrease of the water-depth.

When the value of the discharge is relatively strong variabie, the river regime with all the occuring values of the discharge, and not a unique value of the discharge, should be considered. In this case every value of the discharge must be multiplied by its probability density and the obtai-ned relation for the new equilibrium slope is the following:

[

.::....f_Q_n...,..,/3:-pQ_}_dQ_]3In ....::;O_{

f Qn/3 PI {Q}dQ

(9)

in which: Po and PI are the probability densities at initial time (and at every time upstream of the extraction point) and after the withdrawal, res-pectively.

When the monthly hystogram is used, the relation for the new equilibrium slope becomes: [ 12 i1 ...

=

i~l . 12 1. o .E1 1.= (10 )

Water level and bed level changes, Ah and AZb respectively, are variabie along the channel axis. In order to better visualize those changes it is convenient to consider the coordinate X, positive in the upstream direct ion and equal to zero at the river mouth, which represents the distance of a general section from the river mouth. In this case Ah(X,t) and Azb(x,t) at the intake section and at the new equilibrium conditions can be written as Ah(L,"')and Azb(L,"')respectively.

To compute Ah(X,"')and Azb(x,"')for X ~ L the following relations can be adopted:

(12)

for X

>

L:

(13)

(14)

The presented simple analysis can be rather useful for the case of rivers with a relatively regular regime.

~ h( L

,

C%:»

-

L

L

o

2.3 One-Dimensional Time-Dependent Hodel

The equilibrium model described in Chapter [2.2] can be used only for the

prediction of the new equilibrium state following the water withdrawal. No

information is given about the transitional period preceding the establis

h-ment of the new equilibrium conditions.

In order to study the transitional period it is necessary to use a

time-dependent model. For large scale predictions a one-dimensional model is the

most suitable. If x is the longitudinal coordinate, positive in the down

-stream direction, t is time, and assuming a straight channel the set of

equations is:

*

One-dimensional momentum equation:

au au aa aZb

at + u ax + g ax + g ax (15)

*

water continuity equation:

aa êa au

at + u ax + a ax

o

(16)

*

sediment balance equation:

o

(17)

*

sediment transport equation:

s = f (u, other parameters) (18)

the variables are function of space and time, namely: a

=

a(x,t), u

=

u(x,t) etc.

When the celerity of a bed disturbance is much lower than the celerity of a flow disturbance, which is true for a small to moderate Froude number (F

<

0.6), a quasi-steady approach can be appropriate. In this case it is assumed that the flow is adapting itself instantaneously to the bed configuration (which is changing in time). The quasi-steady approach con-sists in neglecting the time-dependency of velocity and of water depth, that is:

au aa at = 0; at 0

while azb/at is retained.

The approach is applicable for most rivers, except for those characterized by rather steep flood waves and in tidal areas. For a better description of this approach and its applicability refer to Barneveld [1988].

The above described one-dimensional model for river morphology is the ma-thematical base of ODIRMO, a computer program available at TU Delft, (see Vermeer [1985]). This program is adopted in this study to estimate the morphological variations due to water withdrawal as function of time. The

computations in the program are performed as follows:

INITIAL

WATER MOVEMENT

t

SEDIMENT TRANSPORT

+

SEDIMENT BALANCE

~t

BED TOPOGRAPHY

The water movement and the sediment transport and balance are computed se-parately (decoupling of flow and bed). This approach is valid when the changes of the bed level are relatively slow so that they do not affect the flow characteristics.

The numerical solution of the equations requires four boundary conditions:

*

initial conditions (t = 0), corresponding to the hydraulic, sediment transport's and morphological conditions along the stretch of interest at the initial time;

*

upstream boundary conditions (x = 0), corresponding to the hydraulic, sediment transport's and morphological conditions at the upstream

section, as a function of time;

*

downstream boundary conditions (x X), corresponding to the

hydraulic, sediment transport's and morphological conditions at the

downstream section, as a function of time; and

*

internal boundary conditions (x = X-L), corresponding to the

disconti-nuities of one or more parameters at a distance L from the river

mouth, as a function of time.

Furthermore, the analysis of stability of the numerical scheme adopted in

the computer program ODIRMO, shows that the computations are stabie when

the Courant number, defined as:

(19)

in which:

cb propagation celerity of a small bed perturbation;

At time step;

Ax space step.

is smaller than 1.7 (the numerical scheme adopted in the program is not

completely explicit).

This condition implies that the time-step and the space-step cannot be

chosen arbitrarily, but that they are related to each other.

For a more detailed description of the use of the computer program ODIRMO,

3. Computational example of the morphological response to water withdrawal

3.1 General

The morphological response of a river to water withdrawa1 is analysed com -putationally in this chapter. In order to give a more general insight into the process computations are performed for various conditions. Furthermore, to make this example comparable with the analysis performed by Hendrickx [1988] on the morphological response of rivers to sediment extraction, the same initial conditions, and the same output layout are adopted here.

The computational example is schematized in Figure 3.

t.Q

I

g

\

~

}

-1-- -_-_-_-_-_-_-_-_--_Q_;;:O~

~

I

---1~

_

-I

~

I

,L. -_

---

--rl

-=-

~

O~---4---+- x 10 L (X-L) X I

x

_--__j'--_J L 0

Figure 3 Schematization of the computational example

3.2 Input data

Computations are performed for eight àifferent situations. The parameters which distinguish them are the amount of water withdrawn, t.Q, and the posi -tion of the intake, defined by the distance from the river mouth, L.

The following parameters are fixed for all the computations:

channel width initial discharge Chézy coefficient

initial energy slope: grain size relative density water depth B

=

100 (m) Q = 1250 (m3Is) o C

=

50 (m1/J./s) _4 = 5

*

10 (-) _4 = 5

*

10 (m) 1.64 (-) a = 5.00 (m) o

The following values of the distanee of the intake from the river mouth are considered:

L = 2000; 5000; 10.000; 20.000 (m)

For each position of the intake point computations are performed for two different amounts of water withdrawal:

~Q = 125 and 250 (m3/s) (10% and 20% of the total discharge)

The values of the discharge, Q and ~Q, are kept constant in time as weIl o

as the input of sediment at the upstream boundary, S . The sediment o

transport is assumed to be simply determined by the local conditions. The sediment transport formula which is adopted for the computations is the one designed by Engelund

&

Hansen [1967].

In this way the influence of the amount of water withdrawal and of the distance of the intake from the river mouth is investigated.

The propagation celerity of a small bed perturbation, cbo' is to be deter-mined when checking the stability of the numerical scheme adopted in the computer program ODIRMO. The numerical seheme is stabie when the Courant number, a, is Iess than 1.7, Eq. [19]. The celerity of a small bed perturbation ean be determined by the following expression:

as ~

au

a (20)

which, when the sediment transport rate is determined by a power law, Eg. [2], becomes:

n-1 u

cb

=

Cm nu) a = n s

a (21)

The obtained celerity is then corrected to take into account the flow con-ditions:

s ( 1 )

cb = n a 1 - F2 (22)

in which F Froude number (F

<

0.6).

cb is variable both with space and time. The condition of stability of the numerical scheme, a

<

1.7, must be true for every point and every time. Being the values of cb(x,t) known only at the initial and final stages and

in order to avoid a long prediction correct ion procedure for the

computations, the Courant number is derived using the most critical value

between cbo (initial conditions) and cb1~ (final equilibrium conditions

downstream of the intake, defined in Chapter 2.2), being: s o cbo = n A o 1 (1 F2) (23) o and (24) with UZ o

ga;

o s o n m u . o ' and sl~ so

The boundary conditions adopted in the computationa1 example are:

*

initial conditions (t

=

0):

at the initial state it is assumed that the channel is in equilibrium, corresponding to: Q(x,o) i(x,o) s(x,o) Qo (m3/s) i (-) o s (mZ/s) o

*

upstream boundary conditions (x = 0):

discharge and sediment transport are assumed constant with time, fur -thermore at the upstream boundary the flow conditions are assumed uni -form: Q(O,t) s(O,t) Qo (m3/s) s (m2Is) o

This implies that the upstream boundary is chosen far enough from the intake section (no backwater effects). According to Hendrickx [1988] when the distance between the upstream boundary, x = 0, and the intake, x = X - L, is greater than 10 times L, the computations show that the upstream section is not influenced by backwater effects. For this reason lOL will be the distance between the upstream section and the intake section (where the water withdrawal takes part) for all the computations, see Figure 3.

*

downstream boundary conditions (x = X):

The water level is assumed constant with time: h(X,t)

=

h(X,O)

*

internal boundary conditions (x = X-L):

At the intake point part of the discharge is extracted from the river. The amount of water withdrawn AQ, is kept constant with time:

AQ(t) = AQ(O)

3.3 Dimensionless parameters

The results of the computations are made dimensionless so as to give a more

general information on the process. x, t, Az, etc. are made dimensionless

according to Ribberink

&

Der Sande [1985] and Hendrickx [1988]:

i

o

x x (dimensionless longitudinal coordinate positive in the

a o downstream direction) i o

x

_a o

x

(dimensionless distance from the river mouth)

i

o L

a o

(dimensionaless distance of the intake from the river mouth)

t i 3 0 t cbo a o (dimensionless time) 1

-in this case x

3

t corresponds to x

=

cbot, propagation law of a shock

front at the channel bed; the dimensionless time

T

=

3L

corresponds to the

time necessary for the shock front to propagate along

L.

Water level and bed level changes, Ah and Azb, are both variable along the

channel axis and with time. It is convenient to express them as function of

It is logical to make the values 6h(X,E), 6zb(x,E) dimensionless with their values at the expected new equilibrium conditions: 6h(X,~) and 6zb(X,~), respectively, which can be computed according to the equilibrium model, described in Chapter [2.2].

The discharge 6Q is made dimensionless with the initial value Q . o

3.4 Representation of the results

The computational results show the importance of 6Q/Q and of t for the

o

morphological changes and their time scale. The results are summarized in graphs using the dimensionless parameters described in Chapter [3.3].

The representation of the results in the graphs is the following:

(i) water level and bed level variation at the intake as a function of L and E, for 6Q/Q = 0.10 as weIl as 6Q/Q = 0.20;

o 0

(ii) waterlevel variation as function of X (distance from the river mouth)

and E, for L=0.2; L=0,5; L=1.0 and L 2.0 and 6Q/Q = 0.10. o

In the graphs representing the bed level variation, an additional dashed line is drawn through the four points

T

=

3L. This is made so as to compare the time necessary to reach the new equilibrium and the time necessary for a bed shock front to propagate along

L.

1

.0r---~---.----.----.---.----,---r---~--~----~

flheL. t) Clhel, co)

1

-1.0~1~~---~---1~0~O--~---~--~~1----L_----~--~L2--~

10-

2

5

2

5

10

2

5

10

2 Ilo t

Figure 4 Water level variation at the intake section as a function of [ and t for

àQ/Q

= 0.10

1.0r----,---r----r----.---.----.----.---.---~--~ 0.2 6h(L.

t)

6h(L.oo)

r

0.0

-0.2

Figure 5 Water level variation at the intake section as a function of L

and

E

for AQ/Q = 0.20 o

L\Zb(l,t) L\ Z bel, Cl) )

I

-I

~-r-

L=O

.

2

O'~O~-~1~_'2----~5~--1~O~O--~2---~5----1~O~1--~2----~5~--1~O~2~~2

Figure 6 Bed level variation at the intake section as a function of Land

E

for AQ/Q = 0.10 o

1.or----.---.----.----.---~----~----._----._--~----~

Á

Zb(L,

t)

0.6

t----+----t--~~-+__::JIIIC--_+_-__+-~:}_--+_-__+_~~ ÁZb(L,co)

r

0.0~-~1~-.---~--1-0~O~--~----~--~~1----~----~--~~2--~

10

2

5

2

5

10 2

5

10 2

"t

Figure 7 Bed level variation at the intake section as a function of Land

E

for

àQ/Q

=

0.20

t

I

11h(X

. t)

11h(X.

CXl)

0.75

Figure 8 Water level variation as function of X and

E

for

L

0.2 and

àQ/Q

..

0.10 o 0.50 100~ ~ ~~ __ ~~~~~ 0.25 ...~:iiiII""I 0.10

5~---~---~~

2.2 1.2 X ...~ I---0.2 0.0

5 ... t

r

2 _0.75 101 0.50 5 _0.25 0.10 2 10° Äh(X, t) Äh(X,CD)

5~---+---~~

5.5

3

.

0

X

....

~I-- --0.5 0.0

Figure 9 Water level variation as function of

i

and

E

for

L

0.5 and

A.Q/Q = 0.10

2 6 h(X.

t)

Ah(X.

co )

102 0.75 5 '" t _0.50

I

10'2 0.25 0.10 5 11 6 1

o

X ....4t--

--Figure 10 Water level variation as function of

i

and

E

for

L

1.0 and

b.Q/Q = 0.10 o

2 A h(X. t) A h(X.

CD)

102 0.50 5 0.25 t

r

2 0.10 10'

5~---~---+---+-

~

2

2

12 X ... ~t----2

o

Figure 11 Water level variation as function of

i

and

E

for [ b.Q/Q

=

0.10

o

4. Discussions and Conclusions

4.1 Discussions of results

From the graphs representing the water level variation at the intake point

(

X

L) as funciton of Land

E

(Figures 4 and 5) it is easy to detect the initial drop of the water level immediately after the start of water wit h-drawing. This is caused by the decrease of the discharge without any morphological adaptation. The initial drop of water level can be easly computed (i.e. with the Chézy equation). lts intensity and duration must be taken into account when planning a new water withdrawal.

lt can be observed that the graphs corresponding to ~Q/Q_o = 0.1 and ~Q/Q₀ = 0.2 are nearly identical. This means that the morphological process is

(nearly) independent of the values of ~Q/Q ,which is true as long as this o

value is not too large (~Q/Q

<

0.3). This can be proved by observing that o

the equations describing the morphological conditions at the new equilibrium stage, Eqs. [11] and [12], become linear when (~Q/Q)2

«

1.

o

From the graphs representing the water level variation as function of

X

and

E,

for L = 0.2, 0.5, 1.0 and 2.0, (Figures 8, 9 10 and 11) it is easy to observe that the farther from the intake and the slower the water reaches the level of the new equilibrium (apart from very close to the intake).

Furthermore the farther the intake is from the river mouth (bigger L) and the slower the process is. This implies that the distance of the intake section from the river mouth influences the time scale of the phenomenon. From the graphs representing the bed level variation at the intake

(X=

L)

as funciton of Land

E

(Figures 6 and 7) an oscillation of the bed can be observed especially for the smaller values of L. This oscillation is present for both cases ~Q/Q

=

0.10 and ~Q/Q 0.20, a little bit stronger

o 0

in the latter case, and is function of L: it is very strong for L 0.2,

weaker but still present for L

=

0.5, just visible for L

=

1.0 while it is completely absent for L = 2.0.

This oscillation is not due to any instability related to the value of the Courant number, which for those computations was relatively small (between 0.2 and 0.3). However, such a low Courant number yields to non-optimal

accuracy conditions, which have been proved to be optimal for a Courant number around 1, see Vermeer [1985].

There is no physical explanation for the observed bed oscillation: for the

case considered the bed is expected to rise asymptotically to the level

corresponding to the new equilibrium conditions.

A part of the deformation of the curves is due to the inaccuracy of the

computations. Every graph is the result of various computations having

different time-steps (this cannot be avoided when using the computer

program ODIRMO, due to the impossibility of restarting the computations

while contemporarly changing the time-step). Therefore the impossibility of

obtaining exactly the same values for a certain time when using different

time-steps. For the case considered those computations with the smaller

values of t are more affected by accuracy problems.

However, the main oscillation can have also another cause, i.e. the schematization of the computational model (see the computational scheme in Chapter [2.2]), the water flow and the bed (sediment balance ~ bed level changes) computations beeing decoupled. This schematization is valid untill the bed level changes are relatively slow so that they do not affect the flow field. This might not be the case of the computations performed here, when at the initial stage the bed rises with a relatively large speed. Table 1 and 2 summarize the values of the celerity of the bed level rise at the intake at the starting of the process. In the Tables the values of the bed level change, ~zb' are made dimensionless with the initial waterdepth, ao' while the time is made dimensionless with a characteristic time T. The dimensionless celerity of bed level rise is given by the following expression:

celerity of bed level rise

~Zb(L,i)/ao ~t./T

~ in which:

~zb(L,i) is the bed level rise during the time-step, a t· T __ 0 sec ~onj 3io Cbo' ~t. at the intake ~

For the example performed here the most critical values are at the begin of the process, just after the water withdrawal is started, consequently the celerity of bed level rise is computed for the first time step.

r.

0.2 0.5 1.0 2.0

cbi 1.32 0.88 0.26 0.12

Table 1 Dimensionless celerity of bed level rise as function of t for AQ/Q - 0.10 and i = 1.

0

r.

0.2 0.5 1.0 2.0

2.56 1.13 0.57 0.26

Table 2 Dimensionless celerity of bed level rise as function of

r.

for AQ/Q z 0.20 and i = 1.

o

The value of cbi can give an indication on the possibility of having a bed oscillation due to the decoupling of the flow and bed computations. From the example performed here a value around 0.5 can be suggested as thres-hold. However, one example is not enough to derive any conclusions on es-tablishing a critical value of

ë

bi.

Another cause of the formation of the observed oscillation can be the non-optimized flow computation routine adopted in the computed program ODIRMO in the version available at the Delft University (an improved version of this routine has already been designed by Vermeer and will be soon installed).

4.2 Applicability of the performed analysis

It would be very handy to have the possibility of making use of the obtained dimensionless graphs (Figure 4...11) for predictions of the morphological response of a whole class of rivers. This could be useful, for example, during the feasibility stage of a project whenever there is no time or mean available to perform new computations. The general applicability of those graphs is analyzed in this chapter.

A major limitation of applicability lies in the fact that the graphs have been derived on the basis of computations for prototype conditions which have been strongly simplified, as due for the example the assumptions of a constant width in space and time (it is probable, for the case of an

uncontrolled river, that the river will also react in changing its width, when not even its planform characteristics!).

Another limitation is due to the fact that the computations were performed for a single value of the discharge. In reality both the discharge of the river and the magnitude of the withdrawal will vary through the year (whereby of ten withdrawal is limited during the flood season).

The applicability of the graphs is also limited by the fact that they hold only for one value of the parameter ~Q/Q . This limitation lies in the

non-o

linearity of the morphological response, which can be observed from Eqs. [11] and [12] for values of ~Q/Q

>

0.3 (see Chapter [4.1]).

o

The possibility of superimposing more withdrawals located at different distances from the mouth (which is a very common situation) is also restricted by the non-linearity of the governing equations. As a principle, being the equations quasi-linear for small values of ~Q/Q , the

super-o imposition is possible when the total withdrawal, ~QTOT (= L ~Qi)' satisfies: ~QTOT

<

0.3 Qo·

5. ConclusioDs

The results of the morphological changes due to water withdrawal show the importance of investigating the transitional period and not only the final situation. During the transitional period the water level initially drops and then rises. The decrease of the waterlevel can temporarely affect, for example, the intake of irrigation canals further downstream. Consequently

its intensity and duration must be taken into account when planning a new water withdrawal.

The performed computations show how effecting the schematization of the computational model can be on the results. The decoupling of flow and bed computations might yield to undesired oscillations of the bed when the celerity of the bed rise is two high. Here a criterion for a restriction of validity of that schematization via a dimensionless parameter,

ë

bi, is sug-gested. The lack of computational examples, together with the uncertainty added by (i) the problems related to the accuracy of the computations, (ii) the non-optimization of the flow computation routine, lead to the impossi-bility of extimating a critical value of

ë

bi by means of the computational results of this study only.

The applicability of the obtained dimensionless graphs for predictions of the morphological response of a class of rivers (i.e. those having a

extreme simplifications made for the performed example and

by the by the

>

0.3. regular regime no suspended sediment etc.) is strongly limited

NOTATIONS a water depth channel width B F g h

propagation celerity of a small bed perturbation Chézy coefficient

mean grain size Froude number

i

acceleration due to gravity water level

water surface slope

distance of the intake from the river mouth coefficient

exponent of transport law probability density

discharge

sediment tranport per unit of width including pores total sediment transport including pores

time L m n p Q s S t

T (dimensionless) time necessary for the shock front to popagate along L velocity

longitudinal coordinate, positive in the downstream direction bed level

u

x

relative density of sediment finite increase

(1 courant number

longitudinal coordinate, positive in the upstream direction (distance from the river mouth)

x

subscript subscript subscript superscript

0: initial equilibrium conditions ~: final equilibrium condition

1: stretch between the intake and the river mouth dimensionless

REFERKNCES

Barneveld, H.J. (1988),

"Numerieke methoden voor morfologische berekeningen tijdens kortdurende hoogwatergolven",

Delft University of Delft, Faculty of Civil Engineering (in Dutch).

Engelund, F.

&

Hansen, E. (1967),

"A monograph on sediment transport in alluvial streams", Copenhagen, Danish Technical Press.

Hendrickx P.H.A. (1988),

"Morphological reaction of rivers due to sediment mining", Delft University of Technology, Faculty of Civil Engineering.

Kerssens, P.J.M.

&

Urk, A. van (1986)

"Experimental studies on sedimentation due to water with drawal", Journalof Hydraulic Engineering, 112 (1986)7.

Ribberink, J.S.

&

Sande, J.T.M. v.d. (1985),

"Aggradation due to overloading-analytical approaches", Journalof Hydraulic Research, Vol. 23, No. 3, pp. 273;283.

Vermeer, K. (1985),

"A one dimensional model for river morphology",

Delft University of Technology, Faculty of Civil Engineering.

Vries, M. de (1986),

"Morphological predictions for rivers with special reference to drought

"

prone areas ,

Proc. of Int. Conference on Water Resources Needs and Planning in Drought Prone Areas, Khartoum Dec. 1986.