A morphological time-scale for rivers

a morphological time- scale f.or rivers

paper presented at the XVI th IAHR Congress

Sao Paulo, July 1975

M.de Vries

publication no .. 147

INTERNATIONAL ASSOCIATION FOR HYDRAULIC RESEAH~H

A morphological time-scale for rivers (Subject Ba)

by

M. de Vries

Professor of Fl~id Mechanics, Delft University of Technology and advisor, Delft Hydraulics Laboratory, Delft, the Netherlands.

Summary

Processes on degradation and aggradation of rivers have a speed depending on the characteristics of the rivers. Starting from the basic equations a morphologica,l time-scale, characterising these processes, is defined. This morphological time-scale is estimated for various rivers .

.Hesume

Les phenomenes d'erosion et sedimentation des rivieres se produisent

a

une vitesse qui est fonction des caracte-ristiques de ces rivieres.

Partant d'equations fondamentales, on a etabli une echelle de temps morphologique qui caracterise ces processus. Cette echelle a ete evaluee pour divers types de rivieres.

1. Introduction

Morphological processes in rivers are complicated due to the interaction between water-movement and sediment-movement of a strongly three-dimensional nature. For engineering purposes it is of importance to have a thourough knowledge of' river-morphology in order to be able:to predict changes in the geometry of the river due to human interference.

Based on the knowledge available the river-engineer c.an construct physical and/or mathematical models in order to quantify the morphological changes of the river. The quantitative knowledge of morphological processes in rivers is restricted in spite of the fact that many investigators have studied the topic during many years .. Only rough schematisations of the time-depending processes are possible. For in-stance for aggradation and degradation a set of partial differential equations can be deduced; these can be solved numerically. In an even more simplified ver-sion the equations can be solved analytically.

In this paper these analytical solutions are used to define a morphologi~al time-scale for the degradation and aggradation-processes in rivers. This makes it pos-sible to characterise for a river by means of a single figure. the speed at which these processes take place.

2. Analysis

The mathematical description of morphological processes in rivers is for the time being almost exclusively restricted to a one-dimensional approach. This implies that the average bed-level over the width of the river is a function of space (x) and time ( t ) .

It has been shown

B,

2] that the problem is described by a set of equations (reference is made to the symbols at the end of this paper). See Fig. 1. Water: ( 1) 9v + av ah az V _lvl v - + g - + g -

= -

g at ax ax ax

_c

2_R ah ah _{h av}

₌

0 at+ v - · + _{ax ax} (2) Sediment: s = f (v,parameters) az as at+ ax = O (3)

(4)

Fig. 1 : Definition Sketch

The transportfunction (Eq.3) contains parameters like roughness (C), grainsize (D) and relative graindensity

(8), they are supposed to be constant. For many prac-tical cases the celerities of di..sturbance at the water-level (c 1 2 ) are in an absolute sense large compared to the celerity (c

3) of a disturbance at the bed.

de Vries, Transients :i.n bed-load transport. Delft Hydr. Lab. Report R 3 (presented at IJJIR Seminar, Montreal, 1959).

de Vries, Considerations about non-steady bed-load transport. IAHR Lenin-grad, 1965.

-From:

c3 <<

I

c 1 ,2

I

(

5)

it can be assumed that for morphological computations for these cases:

le

I

-+ a, (6)

1,2

This implies that the flow can be considered quasi-steady. The basic equations then reduce to:

av + ah+ g

h

= _ g vlvl v

ai

g ax ax C2h ·

v

ah+ h av = O ax ax h+~=O at ax s = s(v)

(7)

(8)

(9)

( 10)

Numerical techniques are used [3,

~

to solve these equations numerically for a specific case for which sufficient boundary conditions are available. A review has been presented recently[~ •

Numer,ical techniques of the equations require data and time. They are not very handy to make quick guesses on some aspects of morphological problems.

Therefore a study has been carried out [6] to investigate on the potentials of schematizations of Eqs.(7 ... 10) that allow analytical solutions.

It can be shown that the equations can be combined into one differential equation for the bedlevel z.

( 11 )

In this sim le wave e uation c represents the celerity of a disturbance at the bed. The f:mction f

1 v) will ndt .be specified here as Eq. (11) is too simple for the goal aimed here.

Another simplification is reached if less restrictions a.re made [ 6

J .

It leads to a hyperbolic model: 2 2 K ~ -

.!L

~ = 0 ax2 c 3 axat az at ( 12) Here c

3 is again the celerity of a disturbance at the bed with ., ds/dv

c3 2

1 -Fr

( 13)

[3JC.B.

Vreugdenhil and M. de Vries, Computation of non-steady bed-load

trans-[4

JM.

de Vries,

port by a pseudo-viscosity method. !AHR Fort Collins, 1967. Solving riverproblems by hydraulic and mathematical models. Lecture Conference Jablonna, Poland, 1969 (also Delft Hydr. Lab. Publ. No. 76-II).

[5]

M. de Vries, River-bed variations - aggradation and degradation. Lecture IAHR Int. Sem. on Hydr. of Alluvial Streams. New Delhi, 1973 (also Delft Hydr. Lab. Publ. No. 107),

Vreugdenhil and M. de Vries, Analytical approaches to non-steady bed-load transport. Delft Hydr. Lab., Report S 78-IV, 1973.

-The parameter K will be discussed later. After linearisation c and K become constants. It can be remarked tpa"ti Eq.(12) was deduced by Poly~ to describe the ~tochastic model that Einstein L7J used to describe the dispersion of tracers

1.n bedload. ,

The hyperbolic model will not be discussed here as it has two important dis-advantages. Firstly the equation can not be solved analytically for a varying discharge (q), therefore the regime of the river can not be introduced. In the second place also for a constant discharge only for specific boundary conditions analytical solutions can be found.

An approach situated "between" the simple wave approach and the hyperbolic model serves better. This is the parabolic model derived intuitively in a qualitative sense by Culling [8] and in a quantitative sense by Ashida and Michue r97 . As the latter seem to linearise too early:,during the derivation, the compfete

analy-sis will be given here. ·

It is assumed that the watermovement can be considered uniform during the non-steady process of the change of the riverbed. Hence Eq. (7) reduces to

:tltl

az v3 - C2h

=

ax

= -

c

2q

Differentiation of Eq. (14) with respect to x gives a2z _ 2

3 y__ av

--

-

-ax C q ax 2

From Eqs. ( 9)' ( 10) and ( 15) it yields az _{K ~} 2 ₌ 0 at ax2 with K

=

c

2 gds/dv

=

3v2 1 (ds/dv )v

3

_l. ( 14) ( 15) ( 16) (17)

In this derivation only differentiation with respect to x takes place.

Therefore all ~arameters may still be functions oft. This implies that the

general form of Eq. (16) 1.s

( 18) at

3. Morphological time-scale

The "diffusion" equation (Eq. 17) containing the time depending diffusion coeffi-cient K (t) can be used to define a morphological time scale for a river with respect to degradation and/or aggradation.

The river considered is supposed to discharge into a lake (Fig. 2). If further the lake-level is supposed to drop at t ~ 0 over a distance 6z, then a degradation

[ 7] H.A. Einstein, 11_{Das Geschiebetrieb als Wahrscheinlichk.eitsproblem}11 • E.T.H. Zurich, 1936. ,

[ B]

W.E.H. Culling, Analytical theory of erosion. Journal of Geology, 68, 3, 1960. [9] K. Ashida and M. Michue, An investigation of riverbed degradation downstream

of a dam. I.A.H.R. Paris 1971, Paper C30.

4

-process is starting which lasts untill the riverbed is finally lowered by an amount L'lz over its entire length.

This simple boundary condition for Eq. (8) leads to an analytical solution. Ac-cording to the definitions of Fig. 2:

with

----X z (x,T) = - L'lz erfc 2) 03/K(t)dt erfc y = -2- / 00 exp (-~2)~d

Vn

y

----

~ : - - -

~

~ ---... V

r-c:o----

... - ... RIVE"R Fig. 2 Degradation . ( 19) (20)

Considering now a standard length L, the question can

m. .,

now be raised: how long does it take(T) before at the station xm= L the riverbed is lowered by 50% of the final value, thus by ~L'lz?'' This is the case for

erfc L m _:::

~

_{or L}

= ✓

_To/K(t)dt m 0/ · ( 21 )

It is convenient to express T as a number of years (N), Defining with Eq. (17) m

year 1 year 0 /K(t)dt a:

i

B:

0 /S(t)dt y = ₍₂₂₎ • . ( ) b .

Here the sedimenttransport functions= s v has been approached bys= av in which a and b·contain all parameters except v. As an approximation in Eq. (22)

b, Band i are supposed to be independent .of time. It is, however, quite possi-ble to leave these parameters under the integral as the latter has to be deter-mined numerically anyway.

Thus ~he number o~ ye~rs (N

50) necessary to lower at x = the final amount is given by

5

(23)

Remarks

(i)

The jntegral of Eq. (22) contains the yearly sedimenttransport of the river. The accuracy of N

50 is therefore directly proportional to the accuracy by which this yearly transport can be determined~

The standard length (L) should not be too small. It has been shown [6] that the parabolic modfu used here is only a good representation for the complete equations if

(ii)

X > 3h/i (24)

4.

Examples

For a number of rivers the morphological time-scales have·been computed (see Table I). A standard length L = 200 km has been used. With the derivation of Chapter 3 this Table seems self-explaining. The following explanations on the rivers can be given.

Rhine-River: The results are based on a large amount of transport measurements carried-out by Rijkswaterstaat, in the Netherlands. The Meyer-Peter formula can be applied for this river.

I

Magdalena-River: According to a large series of field data the Engelund-Hansen ['i5J-I·ormuia-is applicable here

Qi]

Danube River: The basic data have been recieved from the Hungarian Research

Institute-for Water Resources Development (VITUKI). They are based on long series of field observations.

Tana River: For this river only very little data are available. However, according to-experience with the Magdalena and with some rivers in Central Java

(Indonesia) it seems justified to apply the Engelund-Hansen formula to estimate Y with the available data on bedmaterial discharges and slope.

Apure_River: Also here the Engelund-Hansen formula is applied.

RIVER STATION D _i

_-4

3h/i

N~O

(approx. distance mm *10 km y ars

from sea)

Rhine River Zaltbommel 2 1. 2 100 1000

(Netherlands)

Magdalena R. Puerto Berrio 0.33 5 30 100

(Colombia) (730 km)

Dunaremete 2 3.5 40 500

( 1826 km)

Danube River Nagymaros 0.35 0.8 180 130

(Hungary) (1695 km)

Kuna uj varos 0.35 0.8 180 75

(1581 km)

Baja 0.26 0.7 210 30

( 1480 km)

Tana River Bura

o.

32 3.5 50 120

(Ken.va)

Apure River San Fernando 0.35 0.7 200 110

(Venezuela)

Table I. Morphological time-scale for some rivers.

[Jo]

F. Engelund and E. Hansen·, ·11

A monograph on sedimenttransport 1n alluvial streams", Teknisk Forlag, Copenhagen, 1967.

G1] MITCH, Rio Magdalena and Canal del Dique survey project. Mission Tecnica Colombo-Holandesa, NEDECO report, 1973

-Remarks

(i) The rivers mentioned in Table I show a considerable difference in time-scale.

(ii) The Hungarian Section of the Danube shows an interesting change of the morphological time-scale. Slope and grainsize vary significant along this part of the Danube.

(iii) The values of N'50 have accuracies similar to the ones of the yearly seaimenttranspoH (c.f. Eqs. 22 and 23).

Acknowlegdment. The writer likes to express his thanks to :br. Stelczer, Director of VITUKI for the data on the Danube River made available.

Symbols b exponent of transportrelation B width of river c celerity C Chezy coefficient D grainsize Fr Froude number= v/f;; g acceleration of gravity h waterdepth i slope K "diffusion" coefficient L length

q discharge per unit width R hydraulic radius

s sedimenttransport (bulkvolume including pores) S sedimenttransport over entire width

t T V X z A time time flow velocity ordinate in flowdirection bed-level = (p _s - p)/p _, relative density density of sediment density of water 7 -Dimension