A STUDY ON MAXIMUM FLOOD DISCHARGE FORMULAS

Tae Sang Won, PhD.CE., Dr. En.${

ABSTRACT

This paper describes a new formula for the calculation of approaching velocity of rain water, and a number of new formulas for the estimation of maximum flood discharge which have been deve- loped by the author.

Many empirical formulas, which have limited application, exist. However, in devising his formulas, the author derived theo- retically the form of the basic maximum discharge formula for the case of rivers with no tributaries, and determined stochastically the value of the coefficients in his basic formulas using the re- cords of observed measurements. Then the author derived theoretic2 lly many differe,it formulas for the case of rivers with tributa- ries to fit in the actual localities of the site under considera- tion, besides the basic formulas. So the author's formulas would be widely applicable for rivers or sewer nets, and also for any regions, countries, with different locality. The author could con firm these facts through the numerical examples. The author's fo: mulas may be used not only for estimating the design flood, but also in flood routing. The author believes that his formulas would be very helpful in the planning of water resources development pro jects -specially for those with inadequate data.

1

RESUME

L'auteur présente une nouvelle formule pour la vitesse de con calcul du d g centration d'un bassin et en suggere d'autres pour le

bit de la crue maximale.

On trouve de nombreuses formules empiriques dans de nombreux manuels, mais ces formules sont d'une application limitée. L'auteur parvient cependant à asseoir la forme de sa formule sur des bases théoriques, lorsqu'il s'agit de cours d'eau sans affluents; il pro cede à l'évaluation des paramètres qu'elle contient par ajustement statistique aux données d'observation disponibles. I1 generalise ensuite à différents cas de cours d'eau avec affluents. Les formu- les proposées de'vraient pouvoir être appliquées n'importe où, aussi bien pour les cours d'eau naturels que pour les réseaux d'assainissement; c'es ce que l'auteur peut confirmer par des applications numériques. Les formules peuv:nt servir non seulement au calcul des crues de projet, mais aussi a celui de la propagation des crues. L'auteur pense que ses formules devraient rendre de grands services dans la planification de l'aménagement des eaux, spécialement lorsque les données disponibles sont insuffisantes.

fg Professor of Civil Engineering, Seoul National University, Seoul, Korea.

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I e XNTR001';TION a

Charles F. Ruff defin& B m x i m m probable floodn as follows. "The maximum probable flood does not mean the largest flood possible but a flood so large that the chance of its being exceeded is no greater than the hazards normal to all of man's activities." The author will use here the term of 'maximum flood discharge" with the same meaning of "maximum probable flood" as defined by Ruff.

It is very important to calculate maximum flood discharge cor- rectly, and also it i5 a very difficult problem theoretically and practically. It m y be impossible to establish a plan for flood con- trol an8 water resources development or sewer nets projects without reckoning correctly the maximum flood discharge or the design flood.

charge, and we have to adopt the most suitable method in accordance with the completeness of the data. However,the method of calculation by the maximum flood discharge formulas,especially for the case of those with inadequate data, is easy and simple for practicing engi- nßers. There are many empirical formulas devised by many authors such 88 Kuichling,Mead,KresnikeDickens,Metcalf and Eddy,Brix,Lauter- burg,Possenti,Buerkli-Ziegler,Dr.Hisanaga,Kajiyama,and many others. These old fosmulas have been devised empirioally and have limited application. It will be clear that one may be unable to apply them generallj.. Aliso it will not be strange to obtain results which may be 10 or liio time8 of the correct values, according to selection OP the coefficieqts in these formulais when these formulas are actually applied to practical problems.

Generally speaking,the flood discharge depends upon the shape of catchment,drainage area,amount of rainfall and the position at- taoked by the heavy rainfall,pemneability,slope of the catchment, shape of the water cours6,status of the surface,geological status, etc. Strictly epeaking,such statua of catchment differs from others from se68on to season,for every floo8,even in the same catchment as well as in different draimge basins. In other words,flood disoharge üepends also upon the inteneity of Fainfall which causes the flood, duration of the rainfall and the position of the oater of the lows, or statua of the ground in case of heavy rainfall,vie.,dry ground or saturated qround,etc.

stated above, it may be very difficult to express it in a formula. However,if we can consider theoretioally correct value of approach- ing velocity of rain water and intensity of rainfall, we may deduct the maximum probable flood by getting the rainfall for a certain dis- trictc The principle of derivation of the author's formulas belongs to this process, and it may be said that this is an approach differ- e.it from many scholars who had derived the old formulas.

* Ruff,Charles F.;nMaximuni probable floods in Pennsylvania Streamst' Transactions ,American Society of Civil Engineers .Vol .i06,1g4l ,p . 11 53

There are many methods for calculation of maximum flood dis-

As the maximum flood discharge depends upon many factors, as

~

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In the first step, the author thought out a method to ascertain correctly the approaching velocity of rain water. At the same time, the author found that the Rizha's (Germany) formula,the only complete one for this purpose, could not be applicable to solve practical problem8 as it gives too small values. In the second step, the author studied the rainfall intensity curve comprehensively, and found out theoretically when the maximum flood discharge may occur. In the third step, the author has theoretically derived the maximum dia, charge formulas for rivers with many tributaries by appljing the general rules which he has determined by the first and second step. In the fourth step, the author determined the discharge coefficient in his formula from the actual records. The author was then able to calculate the value of the discharge coefficient, with a great degree of acouracy, of the rivers in Korea and Manchuria.

II. APPROACHING VELOCITY OF RAIN WATER

The approaohing velocity of rain water (U) is defined as the mean velocity of rain, water approaching from the farthest point F in a river basin to the point O where the maximum flood discharge is to be ascertained,in other words, the mean velocity of flow between F and O (Fig-1). There is only one formula to find such approaching velocity so far expressed in equation, given by Rizha,Germany, and a table given by Kraven,Germany.

1) RIZYA'S FORMULA

0" 72 So'' (1) where

a= Approaching velocity of rain water (km/hr)

H = Difference of elevation of height between O and F L = Distance of OF (Length of water course) s = H/L

2) KRAVEN'S TABLE

C above 0.01 0.01 - 0.005 below 0.005

Kraven had expressed only about approximate limite of Cd , the author tried to formulate his table, to pass through the medium points as follows.

w (kln/hr) 12.6 10.8 7.56

(2)

3) THE AUTHOR'S FORMULA

approaching velocity of rain water e3 ,theoretically which may be applicable for rivers where the hydrographic surveying was completed. Neglecting the principle and the process of derivation here, the re- sult of the author's formula is illustrated as follows.

The author succeasfully devised a new method to determine the

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The value of for rivers in Manchuria calculated using eq(3) is shown in Table-i. As we see Table-1, the value of lies between 133 and í77, and we may be able to recognize that the Rizhass formula wil not be of practical use because of the reason that the salue of iCt in his formula is too smll,apparently,compared with the normal salues. ide can also see from Table-1 that the value of K represents the bottom slope OP the hydraulic siope of the river, and this fact coin- cides with practice. Also it can be seen that the value of &, de- ureases gradually according aa approaching the downstream of a river, and this fact skok's that the bottom slope or the hydraulic slope of a rivep generally decreases gradually as we approach the downstream.

III. RAPFALL INTENSITY CURVE

h'e can express the rainfall intensity by the following equation. Table-I. The values of & and others fop rivers in Manchuria Name of Rivers

Tumen 8. Whancheng R. Rohe H. Seasamorin R. Sealeog R. Tongleog 3. The upatrean of m i n Lacg R. The middle of

the

the Leog GhsnF: R. Icwaslg R. Van R. Pa B. Csnkai R.

I =

t = I =

wher0

/c 277 207 177 171 149 207 150

158 1.62 159 3.90 137 4-36 149 7.07 150 3.68 134 1.43

33,400 487.6 68.50. .LW

29,927 412.0 72.64 .i76 4,000 160.7 24.92 .l52 31,455 444.0 70.84 -159

510165 767.0 66.71 .O86

178,699 1040.5 171.74 .O65 10,318 33345 313.13 -093

157.49 30 . 42 22.65 10.46 13.26 10.10

Range of S (400)

4.82 - 8,06 6.72 - 9.64 3.54 - 4.22 2.81 - 2.97 8.37 - 3.48 1.70 - 3.16 1.83 - 2.30 .

- 1.77 187,250 1199.0 - 5.51 4,958 163.0 - 5.26 2,129 94.0 -18,66 1,072 102.5 4.06 2,361 178.0 - 1.96 515 51.0

0132 .i86 231 .lo2 e 074 .198

F:catchmtmt area ß/(t + 1 ( 4) L:length of main water oourse

Bwatim Average intensity of rainfall during duration t

OC,,^ = Any-constant Eq(4) represents a kind of hyperbola, and the constants a ar3.B can be found by eq(5) by the principles of the method of the least squame

n(12t) - (ï)(ït) d =

LI)' - n(I') (Il(P2t) - (It)(12) ( I ) ~ - n(I')

E = nwnber of observations B =

Next let R be total. amount of rainfall aurin$ the buration t, R = It = p t/(t + QL 1 (4)

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T~ble-2 illustrates the values of the constants d and fi in eq(4) for various regions. In this table, those for the regions of Korea and Manchuria show the absolute maximum rainfall intensity curves during those periods, Those for the regions marked with the asterisk (*I were calculated by the author himself by the records of the re- cording gauges.

IV. THE AUTHOR'S MAXIMUM FLOOD DISCHARGE FORMULAS

1) FUNDAMENTAL FORMULA FOR THE CASE OF A RIVER WITH NON-TRIBUTARY

(a) Retardation of Run-off Table-2. The values of cc and P in eq(4)

Period taken Range of t Region d a

Seoul ,K 59 0.938 7,860 131 1905 - 1920 5-60 min Inchon , K 37.5 0.625 8,640 144 ditto 5-240 PymgYanR , K 41 0.683 6,000 100 1914 - 1920 ditto Pusan , K 106.1 1.77 14,015 233.6 1914 - 1953 10min-24hr Wonsan , K 75 1.250 7,740 129. 1914 - 1920 5-240 min Taegu,K * 40.2 0.67 8,711 145.2 1929 - 1953 10min-24hr

Kwangju,K * 90.4 1.51 10,866 181.2 1938 - 1954 ditto Mokpo,K * 101.8 1.70 11,398 190 1916 - 1953 ditto ChmgCheng,): * 40.5 0.675 5,929 98.8 1937 - 1943 10min-48hr Sping,r? * 45.1 0.752 8,487 141.4 i934 - 1944 ditto Tokyo, J 50 0.833 5,500 91.6 1891 - 1911 5-60min

where a d/60 b P 1'3 /60 KæKorea M=Manchuria JsJapan

Prior to deecribing the flood discharge formula, the definition of "retardationR must be understood. Now let O and F be the point under coneiäeration and the farthest point of a catchment respective- ly, 1 be the length of water course between O and F , CL) the approaoh- ing velocity of rain water flowing from F to O, tc the time of con- centration,i.s.,the time necessary for reaching O from F, tr the du- ration OP rainfall,i.e.,the perlob between the beginning and enâing of a reinfall (see Fig-i), then,

P b the records (minutes) (min) (hour)

Chonj:i,K * 81.1 1.35 15,160 252.7 1918 - 1954 ditto

t, = l/Ld (79 T tr + t c * tr + l/W í 8)

where T P The time of the period between the beginning of a rain-

fall and the ending of the run-off due to the rainfall at the point O

It may be better to use the author's formula for determining tc When it becomes tr<t,, the run-off is subjecteä to retardation, and in this case the rain water falling the whole catchment would not reach the proposed point simultaneously. Because as the rain water

640

falling at F reached O, the rainfall would have ceased. In other words, the rainfall causing the maximum flood discharge is the rain- fall which falls in a part of the catchment.

(b) Fundamental Formula for the case of non-Retardation

The fundamental principles of the author's maximum flood dis- charge formulas have already been described. The author adopted de- ductive and inductive theories for the derivation.

discharge hydrograph, the author expressed the peak of the flood discharge hydrograph for the case of non-tributary and non-retar- dation by the following equation.

Since it is unable to derive the rational equation of the flood

q m - 9. a C f A R / T (9) where

qm = Peak discharge in flood time q o = Discharge of run-off in normal time T = Duration of flood - tr+ tc tr = Duration of rainfall tc = Approaching time or time of concentration R = Total amount of rainfall during duration of tr A = Catchment area 9 = Average run-off factor C P A coefficient depending upon the shape of the flood

discharge hydrograph Now let Fig-,? show a discharge hydrograph during a flood period.Then the, peak disoharge

AR T. As this also represents the area of DMED of Fig-2 geometrically, we may affirm that the authorls fundamental formula is reasonable anal tically or graphically. Replacing the value of R of eq(6) into

qm-qO may be represented by eq(9)and the product of eq(9) shows the total run-off during the period of flood

qm - q o = C 9 AB / T = C 9 A b t r /(tr + t, )(tr+ a (10) ea(9 9 , We h o w through eq(1O) that the peak discharge is a function of tr, anä it will take a limiting value to make the peak discharge maximum. So differentiating eq(l0) with respect to tr

and. tr = (unit in hours) (11)

T=t,+ tG= 6 +t, (12)

Hence we know that the maximum flood discharge will occur when the duration of rainfall t r satisfies eq(l1). Up-to-datesue have taken tr generally without definite reason as follows: 5 or 10 minutes for de- sign of sewers, 3 or 4 hours for small rivers flowing the vicinity of a city, 24 home or more for big rivers. However aceording to the author's theory, the value of tr must satisfy eq(l1) to cause the maximum flood discharge. Substituting eq(l1) into eq(lO),

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If we express in metric units,i.e. ,A(km2),R(~) ,t (hr) ,q (cms), (13) becomes,

1 where (14)

qm-q,= 0.2778C 9 b a A / (t,+fic)( a +

a,b = any constants depending upon rainfall (see Table-2) The value of tc can be found from eq(3) .(see Table-1)

(c) The value of the coefficient C As stated above, the coefficient C in eq(9) depends upon the

8hape of the discharge hydrograph of the region . The relation be- tween the kinds of the cuPve consisting the discharge hydrograph and the value of C is illustrated deductively as fOllOW6.

Table-3. The value of C found by deduction

Kind of curve C kind of curve C

parabol a 1.5 cosine curve 2.0 triangle(strai ht 2.0 probability 2.394

1 ins? curve

Also the value of C can be calculated inductively from a dis- charge hydrograph by using eq(91,whioh gives,

nhere

The value of C for the rivers in Manchuria found by the author using eq(l5) are given in Table-4.

Table-4. The value of C for Manchrian rivers found by induction

C 0 (qhi-Qo)T /$ARa(Qm=Q,)T/ V

V = The volume of run-off represented by the area DMED of Fig-2. (15)

Name of river

Tongleog Ho

Whan Ho Main stream of Leog Ho ditto

Taitse Ho

n

n

n n n

I

Site of measurement

Tidathergtse Sankankeu P eidakeng C hengs enkong

ditto Whelongbo

n n n H

Duration of flood taken from the records ( hr , day-hr ,day, month, r 15,3rd-21 ,4th,Aug,l9ii 12,10 .e 5,21 ,Sep,1939

19,2nd- 6,6th,Sep,1939

12,31Jul-3,3~,Aug,1940

16,6th-l9,8th, I

15,24. -1 9 , 27, AUg,l94O

797th- 7,103 Se~r1939

9,4th-i7,6th,A~g,l 940

17,2nd- 8 s 5thSSQp ,1939 9,jth-l6,9th, " " 5,6th-l3,9th, Jul ,

Value of' T

30 257 76 83

72 63 56 51 63

103 80

Value of .c 1.664 1 .697 1.543 2.090

1.966 1.754 1 0975 1.087 2.137 1.806 2 . 204*

* show6 the value oalculated by estimation because of non-measure- ment at the vicinity of the peak discharge.

(d) The fundamental formula for the case of retardation of flow

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The basic formula for the case Of non-retarclation ,mentioned above, is applicable for the case of retardation of flow,too. But it is necessary to multiply the coefficientp due to retardation, viz.,

q,-qo=yCpbf& A / ( t c + G L )(a+ Gc 1 (16)

p = f(tc/tr) (17)

It is clear that the value of the Coefficient /3 equals to 1 for the case of non-retardation, but it beoomes less than 1 for the case of retardation. The value of p varies inversely with that of t c / t, . practical calculation. So the author tried to find out the general form of the function f(tL/ty) atoohastically using some data o b tained for rivers in Korea by some other methods. The author would like to assume the general form of the function o f p as follows.

It is necessary to find out a general form of f(tc/tr) for

J)= (1 + k 1 / ( tc/t,+ k 1 where

k = Any constant Finding the value of k in above equation by the method of the least squares, we get k = 4.802 . Accordingly,

p = 5.802 / (tc /tr+ 4.802) (18)

2) 'THE MAXIYUM FLOOD DISCHARGE FORMULAS FOR THE CASE OF RIVERS WITH TRIBCTARIES

(a) The maximum flood discharge st the confluence of a trlbutary

The author found that existence of tributaries affect greatly the peak discharge of flood flow at the proposed site of the main stream. Su the author derived many different formulas of maximum dis- charm for the case of rivers with tributaries, besides the basic formula for the case of thoee with non-tributary. Therefore it would be said that this is a great approach different from many scholars who never considered the Influence of tributaries in their tradition- al formulas.

KOW assume one of the simplest case as Fig-3. The discharge hydrograph for this case may be illustrated as Fig-&. The value of q, in ~ig-4 shows the peak discharge of the triùutaryíI), and the value of q 2 shows that of the main river (II) alone,excluding that of the tributary(1) , also Qm shows that of the composed maximum discharge to be occurred at the proposed site. The rational equation of the curve ,i.e.,the true shape of the discharge hydrograph is unknown. But the author would like to discuss about the shape of the curve in the following. Let us consider two cases, one of them the simplest case,i.e.,the case assumed that the discharge hydrograph consista of an isosceles triangle, and the other the case assumed that it con- sists of a parabolio ourve, to seek the effect of the nature of the discharge hydrograph which influences on the peak disoharge Qn

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( i ) 'The case of an isosceles triangle

In this case, it evident from Fig-5,

(19) TE T, Um" qp+q,(2 --1

(ii) The case of a parabola

Since it is evident as the nature of the parabola,at Fig-6,

q = 4qot/T - 4q,(t/TI2 í a) we can get the following equation for Fig-?,

and by dQ/dt = O Q,/TI + q2 /T2 (20)

to= 2 (q,/T,' + q,/T:) Accordingly, substituting eq( 201 into eq( b) , we get

NUMERlCAL EXAMPLE

An ilìwtration is given here to compare the degree of accuracy of the two cbses mentioned above. Given T2 = 26 hr, TI = 20 hr, = 5000 cms, q = 3000 cms . Then since Te/T,= 26/20 = 1.3 from eq81) , the case of assuming as parabolic curve, &TA= (5000 + jOOOx1.3 Next from eq(19), the straight line formula,

Qm = 5000 + 3000x(2 - 1.3) = 7100 cms Hence,we h o w that there is not any remarkable difference on the re- sults of calculation of the maximum discharge whether we assume the discharge hydrograph as straight lines or a parabolic curve through this numerical example. Aliso we can imagine that we shall obtain the similar results with this numerical example even in the cases we adopt Borne other ourves else than parabola for the discharge hydro- graph,e.g.,cosine or probability curve. But adopting the oase as- sumed as a parabolic curve is safer,easier to handle,& reasonable. So the author would like to suggest those of the parabolic curve as the general formula in this paper.

/( 5000 + 3000x1.3x1.3 1 - 7865 cms

(b) The maximum flood discharge formula at the confluence for the oaee of a river where n-1 tributaries flow into the confluence

(F ig-8 1 If we assme the discharge hydrograph consists of a parabolic

curve, by the srne priciple with that in the previous paragraph, we

(0) The maximum flood discharge at the proposed site which is located the downstream of a tributary

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Now let O is the proposed site, O' the confluence of the tribu- tary in Fig-10, and t, is the necesssry tirne for reaching of rain water from O' to O. If we assume the discharge hydrograph consists of a parabolic curve.FiR-11 ,then

( Q = Q;,+ qiez4q, (t-t, 1 /TI - 4 qi( t - t , l2 / TF+4 qr t/T+ - 4 kt2 /Ti

(d) The maximum flood discharge at 8 proposed site where n-1 tribu- taries join to the main river at its upstream side.(Fig-12)

if we assume the discharge hydrograph consists of 8 parabolic curve, by the same principle with that in the previous paragraph, we

(e) The ma>;Smum flood discharge at a proposed site where m tribu- taies flow into this site and n-1 tributaries join to the main river at its upetream side. (Fig-14)

This is the most general case. If we assume the discharge hydro- graph consists of a parabolic curve, by the same principles, we get

ìGYi V. CONCLUSION

The maximum flood discharge generally increase toward âown- stream, a6 the result of increment of the drainage area. But as the approaching time also increases approaching down stream,in other words,ss the nearer approaching downstrem,the greater effect of re- tardatlon. Accordingly the rate of increment of the peak discharge decreases generally approaching downstream; and sometimes,i.e.,in such oa8e18 where the approaching time remarkably increases compared with the inorement of the drainage area, not only the rate but also the actual absolute value of the peak discharge decreases at the d o m stream than those of the upstream. These fsots are experienced some- times in practice, In such cases,it was impossible to expreee this fa& by the old formulas. However by the author's formulas, it is

645

easy am2 theoretically sound to express this fact. Because as we see the author's basic formulas-eq( 9)-(14) , which represent the drainage area A in the numerator and the factor of the approaching time t c in the denominator. So it may also be said that the author's formulas are very theoretical from the point of view of this fact.

As mentioned above,the author derived theoretically,i.e.,ration- ally or stochastically many formulas of maximum flood discharge- the basic formulas for the case of a river with non-tributary and many other different formulas for the case of rivers with tributaries. Be- cause the author found that the existence of tributaries affect great- ly not only the peak discharge but also the entire shape of the dis- charge hydrograph at the proposed site of the downstream. Consequent- ly it may be posaible,by applying the author's formulas,to find the real shape of the discharge hydrograph at the point under consider- ation to be occurreti in some flood time.

Some scholars advocate that the actual shape of the flood dia- charge hydrograph resembles to ~ig-16. On the other hand,some other scholars insist that it should be resembled to Fig-17. But the author should say that these theories both advocated by the traditional scholars are those have not been touched to the core of the true theo- ries. The real shape of the discharge hydrograph depends upon the lo- cality Qf the point under consideration,in other words, it depends on the relat4ve position of the proposed point and those of the conflu- ences of th? tributaries on the mainstream under consideration. Conse- quently it is resembled to ~ig-16 in some cases, and also it takes a shape resembleel to Fig-17 in some casessin accordance with the locali- ty of the point under consideration. As stated above,it would be able to show the real shape of the discharge hydrgraph just fitted in the locality of the proposed site by applying the author's formulas.

The author's formulas also would be applicable not only for the purpose of reckoning of the design flood, but also for that of esti- mation of the flood routing for some floods. In the case of flood routing,it would be possible to obtain more correct results by taking the real value fop tr instead of that calculated from eq(l1) in some cases,i.e.,the real value of tr is greatly different from that calcu- lated from eq(i1).

The author's formulas would be widely applicable for rivers or sewer nets,& also for any regions,countries with different locality, and it would be possible to obtain correct and accurate results by selecting or assuming the values of the coefficieuits in his fQrUIUlaS appropriately. Aceoräingly the author should like to suggest that the author's formulas shall be applied in practioe in many regions and also for many purpose8 as far as possible.

646 (p F ¡y -I

e Fij-3 Y Fi 9 -2

ot P

e

t O

-+ t

c, 3

O

Fig- 8 Fig- 7

O

Fìg -9

647