HYDROLOGICAL DESIGN OF A TRANSDANUBIAN WATERCOURSE NAMED "CSASZARVtZ", BASED ON 26 YEARS OF HYDROLOGICAL DATA *

HYDROLOGICAL DESIGN OF A TRANSDANUBIAN WATERCOURSE NAMED "CSASZARVtZ", BASED ON 26 YEARS OF HYDROLOGICAL DATA *

by

G. ^PAPP

Institute of Water Management and Hydraulic Engineering, Technical University, Budapest (Received February 8, 1972)

Presented by Prof. 1. V. NAGY

Hydrological data and theoretical (Pearson Ill), negative binomial as well as empirical distributions of the examined watercourse are seen in Table 1

and Fig. 1, respectively.

Year

1934 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

!

Table 1

Discharges of "Csaszarvlz" (1934 to 1959)

Discharge during the summer half-year

m:l/sec

0,047 0,079 0,196 0,174 0,150 0,164 0,392 0,111 0,184 0,106 0,174 0,096 0,112 0,006 0,135 0040 , 0,024 0,240 0,136 0,169 0,179 0,224 0,184 0,151 0,169 0,073

i I

Q(IO'm')

0,741 1,246 3,091 2,744 2,365 2,586 6,181 1,750 2,901 1,671 2,744 1,514 1,766 0,095 2,129 0.631 0,378 3,784 2,144 2,665 2,822 4,532 2,901 2,381 2,665 1,151

I

Discharge during the winter half-year mS/sec

0,850 0,747 0,963 1,445 1,598 0,609 1,420 1,955 1,814 0,494 0,789 1,508 0,710 1,862 0,482 0,418 0,976 1,102 0,677 1,330 0,443 0,874 1,585 1,065 0,723 0,662

•

i

!

,

!

Q (lO'm')

13,403 11,779 15,185 22,785 25,197 9,603 22,390 30,826 28,603 7,789 12,583 23,778 11,195 29,360 7,600

-

6,;,91 15,390 17,376 10,675 20,971 6,985 13,781 24,992 16,793 11,400 10,438

!

-

Annual discharge

mSJsec Q(IO'm')

0,897

I

^14,144

0,826 13,025 1,159 18,276 1,619 25,529 1,748 27,563 0,773 12,189 1,812 28,517 2,066 32,576 1,998 31,504

0,600 9,460

0,972 15,327 1,604 25,292 0,822 12,961 1,868 29,455

0,617 9,729

0458 , 7222 ,

1,000 15,768 1,342 21,160 0.813 12,819 1;499 23,636

0.622 9,807

1;098 17,313 1,769 27,893 1,216 19,174 0,892 14.065

0,735 1(589

*

Lecture delivend at the Conference of Reservoirs and Storage, Gyor, Sept. 13 to 17 1971.

228

P

(%)

'100 90 80 70 50

30 2"

10

10

G. P.-JPP

, - -

'-\N-~clive binomiOl ,disrriburion function (n=7; p;;6.2~f; q=O,71)

Empirical distribution :uncti0r:.

Pearson ill distribution fundicr.

20 30

I ig. 1

Mathematical construction of the model

50

Let the random variable set {~k}

;:=0

denote the rainfall in the wet seasons of consecutive hydrological years, i.e. by ~k the rainfall in the wet season of the k-th hydrological year. The set of random variables is assumed

1. to be independent and uniformly distributed:

2. there being no water withdrawal from but only water inflow into the reservoir during the wet season. If the water level is beyond a maximum level, constant during the storage process, the entering water is discharged through the spillway:

3. to be no inflow during the dry period but only withdrawal of a water volume lVI. It should he noted that the water volume lVI can be withdrawn at any rate of flow, hence, at any instant of the dry season.

Denote hy Ih the water content in the reservoir at the heginning of the k-th hydrological year (k = 0, 1, ... , n) i.e. the water volume in the resen:oir before filled up volume ~k hecomes fed in (Fig. 2).

Let K denote the reservoir capacity.

In conformity with the above, the following relationships hold:

I

^{Ih~ ;k - lVI, for}

lh+l = 0 , for

K M , for

M

<

^Ih.;- ~k

<

K

Ih.;- ~k

<

lVI

Ih.;-

;k >

^K

Values assumed for the random variahles {rh }L~~\f indicate the degree of fullness of the reservoir. According to assumption 1, the set of random variables

{17,j constitutes a homogeneous lVIarkov chain, that is, the probability of

DES/GS OF A TlUSSD_L\TBI.-1S lrATERCOUlSE

Fullness condition of reservoir

K (m3)

K ---V~~r~~---~

K-In hydrological year

Wet period Dry period Fig. 2 transition

i)

IS independent of ^11.

229

D(~nott' the probability matrix of the Markoy chain transition by :-r

-BI.

^{1\ -.If}

-JI

~.I\-.\f

~\-_\l. ^{I . . .} PI, --_\f." -.\f

Here an elt'mcnt t'.g., Pij denotes the probability of the reserYoir condition changing from i to j in one step (Fig. 2).

Determination and processing of limit probabilities offered by the model

The analysis is intended to determine limit probabilities {P,J L'~;;-o·\1 meeting the equation sYstem

} \ - M

If = .::E

^{Plc ~;j;}

/;=0

K M and the equation

2 If

⁼ 1.

j=O

j

=

0, 1, .. . ,K ilI

230 G. PAPP

The value of limit probabilities {Plc} £(~"o",r indicates t!J.e probability for the reservoir to assume conditions 0, I, ... , K - lVI, after a long series of condition changes.

Rather than by solving the established equation system, the limit pro- babilities have been determined by raising the matrix 7C to its power, a much simpler and faster procedure.

Elements of matrix 7C are taken from the empirical distribution function determined from the hydrological data series for the wet season (Fig. 3).

p

(%)

1,0 0,9 0,8 0,7 0,6 0,5

...---

,

,...----J

,

,....J r-'

-

r-'

---

,

r---1 ,

r-'

, r-'

,...,

r'

,

,...

, ,

,..-J

,...

,

,

0,0

° ⁵

^{1'0 15 20 25 30}

1,0,0,0,0,0,0,22,023,1201 2 D2 00011 112 DOD 2 01, ' 0'1 " 5 " 10" 15'" 20 ' 25 ' 30'

Fig. 3

35

Q (10⁶m3) Frequenc~

) ';

--

35

Column elements of the 16th power of the matrix agreed at an accuracy of 3 decimals. The calculations involved the following K and NI values:

K

=

10 . 10⁶ m³

{ ^M= M=

^{8 . 109 . 106 6 mm3 3}

1

¹¹¹

⁼

^{9 . 10}

6 m:l K

=

15 .10⁶ m³ lvI

=

10 . 106 m;l

I ^{M =}

^{II .106 m3}

I ^{M =}

^{II .106 m3}

K

=

20 . 106 m;l lyI

=

12 . 10⁶ m³ lvI

=

13 . 10⁶ m³

I ^{M =}

^{12 .106 m3}

K

=

25 . 10⁶ m³ lkI

=

13 . 10⁶ m³ lvI

=

14 . 10⁶ m3•

DESIG,Y OF A TRA1YSDANUBI.flY WATERCOURSE 231

Diagrams of limit probabilities for a reservoir of K = 25 10⁶ cU.m capacity and a discharge 1VI

=

14 . 10⁶ cu.m/half year are shown in Fig. 4.

Discharge NI for probabilities p = 0.01 P

=

0.05 P

=

0.10

has been determined by linear interpolation from limit probabilities. Reservoir Ilapacity curves ale indicated by circle3 in Fig. 5.

p

(%)

0,8 0,7 0,6 0,5 0,4

o 2b

Fig. 4

40

Plotting the capacity diagram permits the hydrological design of the reservoir. According to Fig. 5, if e.g., a water volume lVI

=

10 . 10⁶ cn.m/half a year is to he provided at a probability p = 0.90, a reservoir of capacity K

=

13.8 . 10⁶ cU.m is to be huilt.

Checking the model For checking the model, the function

has been processed in a digital computer for correlated K and .i~I values.

For rh +1 <:; 0 the output was zero, for 7)" + ~k :> K the NI value "was deduced from the fixed K value.

232

q. (m³/s) 0,9

0,8 0,7 0,6 0,5

O,L,.

0,3 0,2

011 0,0631,.

G. 1'..11'1'

15 14-

13 12 11

p

=

^0,95

.~--

--

.

--

^/

P=090 / _{1 /} ... _ - _ - ... ₀ .,..". .",.. :It 0

. ; - - - I _~_-_-

! I."".". 0

I " ' " _ - _ - 0

+ ,_-.;... .--t 1'"' G+---~ __ ---_~~~ ... ~~O::,-~

~--- .--

_:::;e-~ ... -.

9 8 7

D-

5

f,.

3 2 1

"...",. ..

^~ ,"""""'- ~

- ~ Fitting straight lines plotted from the storage

I

theory and the statistical estimation

+ statistically e'stimated plots I

o Plots from the storage theory

!

!i 7 8 9 10 11 12 13 if,. 15 16 17 18 19 20 21 22 23 21. 25

The number of zeros indicated the frequency of emptying the reservoir.

i\!I values for relative frequencies

p

=

0.01 , P

=

0.05, and p

=

0.10,

have heen obtained by linear interpolation. Correlated NI and J( values are shown in Fig. 5 by (+) evidencing that the real reservoir conditions during the period of 26 years fairly approximate the conditions determined according to the reservoir theory.

Acknowledgements are due to Prof. Dr. A. Prekopa and Senior Ass. E. B6keffi for their assistance in design.

Summary

The mathematical model for 1Ioran's storage theory has been presented and applied to a design based on the concrete hydrological data set in Table 1. The theory has been checked on a simple, realistic model and the results plotted in a graph intended for the use of design engineers. The graph simplifies hydrological design.

DESIG_'- OF A TR.-ISSDASCBIAS WATERCOL-RSE 233

References

l\IORAN, P. A. P.: The Theory of Storage. - Methueu Monographs on Appl. Prob. and Stat.

Methuen and Co. Ltd. London 1959.

PREKOPA, A.: Theory of Probability with Technical Applications. (In Hungarian). l\Hiszaki Konyvkiad6, Budapest 1962.

Senior Ass. Gahor PAPP, fIll Budapest, Muegyetem rkp. 3, Hungary