Some theoretical considerations in reservoir design

Journal of Hydrology, 29 (1976) 115--120 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

115

SOME THEORETICAL CONSIDERATIONS IN RESERVOIR DESIGN

SUBHASH C. MEHROTRA Bechtel Inc., San Francisco, Calif. (U.S.A.) (Received February 3, 1975; accepted for publication July 2, 1975)

ABSTRACT Mehrotra, S.C., 1976. Some theoretical considerations in reservoir design. J. Hydrol., 29: 115--120. The concept of a critical duration of a drought of given probability of occurrence is elucidated. It is also shown that the unit time distribution of a drought of a given duration and probability of occurrence obtained by the method first proposed by Stall (1962) represents a sequence of events which has a lower probability of occurrence than that of the postulated event. A rigorous mathematical formulation for finding the unit time distribution is presented.

INTRODUCTION

A s t u d y o f t h e r e s p o n s e o f a reservoir t o d r o u g h t s o f various intensities a n d d u r a t i o n s f o r m s a vital p a r t o f t h e overall s t u d y leading t o its successful design. This is especially t r u e w h e n t h e reservoir is i n t e n d e d t o s u p p l y cooling w a t e r for a n u c l e a r - p o w e r p l a n t u n d e r b o t h n o r m a l a n d a c c i d e n t a l c o n d i t i o n s . T h e s t u d y is b a s e d on t h e f u n d a m e n t a l storage e q u a t i o n : dS --

= I(t)

--

dt

O(t)

(1)

in w h i c h S is t h e storage, I(t) and O(t) are, r e s p e c t i v e l y , t h e i n f l o w and o u t f l o w rates, a n d t r e p r e s e n t s t i m e . T h e c o n s t i t u e n t s o f t h e i n f l o w t e r m are s t r e a m f l o w , rainfall, a n d p u m p i n g t o t h e reservoir, if a n y , a n d t h o s e o f t h e o u t f l o w t e r m are d o w n s t r e a m releases, f o r c e d a n d n a t u r a l e v a p o r a t i o n , seepage a n d m a k e - u p water. F o r a k n o w n g e o m e t r y o f t h e reservoir, t h e t e r m dS/dt can be r e l a t e d t o dh/dt, in w h i c h h is t h e w a t e r surface e l e v a t i o n in t h e reservoir. In reservoir design e m p l o y i n g n u m e r i c a l m e t h o d s t h e discretized f o r m o f t h e s t o r a g e e q u a t i o n is used: AS At

- I --

O

in which AS is t h e i n c r e m e n t in storage in u n i t t i m e At, a n d o v e r b a r represents average values.

(2)

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NOTATION T h e f o l l o w i n g s y m b o l s a r e u s e d in t h i s p a p e r : h = reservoir water surface elevation I = inflow rate J = joint probability O = outflow rate p, r = probability of occurrence Q, q = f l o w r a t e s S = storage T = duration of drought Tc = critical duration of drought t = time A = increment

Two important considerations enter in the design of a reservoir under drought conditions. These are, first, the critical duration of a (synthetic) drought of a given recurrence interval which would yield the lowest level to which the reservoir can be drawn down, and second, the unit time distribution of the drought. (To clarify the term " u n i t time distribution," consider the following example. If a drought of 1-year duration is being considered and the unit time employed is a month, then the " u n i t time distribution" of this drought refers to its m o n t h l y distribution.) The concept of a critical duration of a drought has not received hitherto the attention it deserves. At present no theoretical basis exists for determining the critical duration of a drought. In the present paper this shortcoming has been addressed. The unit time distribution of a drought of a postulated frequency has been customarily done following a method first proposed by Stall (1962). This paper shows that the distribution so obtained represents a sequence of events that has a lower frequency of occurrence than the postulated frequency of the drought. Since elementary concepts of probability theory will be invoked in the following discussion, the term "probability of occurrence" shall henceforth replace "recurrence interval" or "frequency of occurrence" for convenience, although the latter terms are generally used by practicing engineers. CRITICAL

DURATION

OF A DROUGHT

The critical duration of a drought of a given probability o f occurrence is defined as one which results in the maximum possible reservoir drawdown (assuming, of course, that the reservoir does not become dry during the drought). Before further elucidation of the above definition, a few remarks on the intensity--duration relationship of a drought of a given probability of occurrence are in order. Choosing low flow as a typical drought parameter it is a truism to state that for a given probability of occurrence of a drought, the longer the duration of it, the greater is the average low flow. In other words, a shorter duration drought is more intense than a longer one of the same probability of occurrence. This seemingly trivial observation is of utmost

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significance in the understanding of the mathematical definition of the critical duration of a drought. In comparing reservoir responses to droughts of different intensities, i.e., of different durations for the same probability of occurrence, the reservoir level at the beginning of each drought will be assumed to be the same, say, at the normal pool elevation. If a long-duration drought is considered the reservoir may reach its lowest level during the drought, so that at the end of it the reservoir level is above the lowest level attained. The duration of the drought is longer than the "critical". (For the m o m e n t the exact definition of the term "critical" will be held in abeyance.) On the other hand, if a shortduration drought of the same probability of occurrence is considered, the reservoir may not reach its lowest level at the end of it. The duration of the drought in this case is less than the "critical". If T is the duration of the drought considered, and Tc is the "critical" duration, the response of the reservoir for the cases T > Tc and T < Te will be as shown schematically in Fig. 1 a and b. h

h

\

W~ter Surfoce ELevation

I I I

I l

0

T

G

~t

0

I I '

~t

T

b

l~'ig. 1. S c h e m a t i c o f t h e r e s e r v o i r r e s p o n s e , a. T > Te. b. T < T c.

The above observations suggest that if h ( t ) is the function describing the reservoir level, then the "critical" duration of the drought is defined by: ~h (3)

t=Tc = 0

The cases T > Tc, and T < Tc are governed by: T > Tc:

°2/

> 0

(4)

~-tt=T< 0

(5)

t=T

T< Tc:

The question, nevertheless, remains as to how it can be argued that the

118

reservoir level at the end of a "critical" drought is the lowest possible. The answer can be furnished in the light of the preceding discussion about the intensity--duration relationship of a drought of a given probability of occurrence. When T > Tc, a more intense drought of a shorter duration would have given a greater reservoir drawdown, whereas when T < T c a longer duration, less intense drought would have the same effect. Eq. 3 represents the precise optimum for the duration of the drought from the view-point of the worst drawdown of the reservoir. Thus the term "critical" used in the preceding is indeed critical in the sense of its definition given in the beginning. Theoretical establishment of the criterion for the critical duration of the drought does not, however, obviate the necessity of having to arrive at it in a trial and error fashion. This is due to the fact that the function h(t) is itself a function of the chosen length of the drought, and is more correctly represented by h(t; T). Nevertheless, it is important to set a fundamental parameter such as the critical duration of a drought on a firm footing, and it cannot but aid in determining the critical duration of a drought in a rational way. UNIT TIME DISTRIBUTION

In the foregoing all functions pertaining to the drought such as the stream r u n o f f which constitute known functions in the solution of the storage eq. 1 to yield h(t) were implicitly assumed as known continuous functions of time. This was necessary to establish a theoretical criterion for the critical duration of the drought. In practice, however, such functions are available as discrete points representing values averaged over the unit time. Furthermore, their unit time distribution is not known a priori. Again choosing low flow as a typical drought parameter, the unit time distribution of low flow of a given duration and probability of occurrence has customarily been done following a m e t h o d first proposed by Stall (1962). To illustrate his method, let ql and Q2 be, respectively, 4 t- and 24 t-duration low flows having a probability of occurrence of p. Suppose one is interested in a 24 t-duration low flow having a probability of occurrence of p and wants its unit time distribution. According to Stall's m e t h o d the unit time distribution is taken as ql and q2 (= Q2 - ql) for the first and second unit times. Simple concepts from probability theory can be used to demonstrate that the sequence of events represented by the above distribution has a probability of occurrence which is less than p. The proof is very simple and is as follows. The 4 t-duration low flow equal to q, has the given probability of occurrence o f p . Let r (<1) be the probability of occurrence of a A t-duration flow equal to q2. Then the joint probability J (q~, q2) of the two flows ql and q2 occurring in a sequence satisfies:

J(qb q2) < P

(6)

Thus the sequence of events represented by a distribution following the m e t h o d of Stall has a probability of occurrence less than that of the postulated

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event. Inasmuch as the sequence represents a more severe event, it would result in a more conservative design. The question now may be posed as to how a unit time distribution should be accomplished so that the resulting sequence of events has the same probability of occurrence as that of the postulated event. With low flow as a typical drought parameter, the general mathematical problem is the following. If Qn is the nA t-duration low flow having a probability of occurrence of p, find At-duration qi, such that: n

(7)

qi = Qn i=l

J ( q b q2 . . . . .

qn) =P

(8)

To elucidate eqs. 7 and 8, let us consider an example. Suppose one is interested in the m o n t h l y distribution of a 6-month low flow of a probability of occurrence of 0.01. The synthetic low flow of any duration, and of any probability of occurrence can be found by using a suitable m e t h o d of analyzing drought frequencies. (For a summary of available methods, see Chow, 1964.) Suppose, for our example the 6-month low flow of a probability of occurrence of 0.01 is 10 cfs. Then Q6 = 6 . 1 0 = 60 cfs. m o n t h Now one has to find m o n t h l y flows qi (i = 1, 2 , . . . , 6) such that 6

E

qi = 60

i=l

and J ( q l , q2 . . . . .

q6) = 0.01

It should be emphasized that each of the m o n t h l y flows qi has an i n d e p e n d e n t probability of occurrence, P i ( q i ) , which can be obtained in the same manner as that used for the 6-month low flow. Eq. 8 can now be written as: n

[~ P i ( q i ) = P

(8a)

i=1

where 1I is the multiplication sign, i.e. n

[~ Pi = - p I p 2 . .

• Pn

i=1

It is clear that Pi > P. Furthermore, eqs. 7 and 8 will not, in general, admit of unique solution. From the point of view of a practicing hydrologist it is seldom necessary to seek an exact solution of eq. 8. The purpose of presenting

120

a rigorous formulation of the problem is to alert him against overconservatism in selecting a distribution. CONCLUSIONS

Some important theoretical considerations in the design of a reservoir under drought conditions have been presented. The concept of a critical duration of a drought of a given probability of occurrence has been defined mathematically. It is shown that the unit time distribution of a drought following the method of Stall (1962) yields a sequence of events of a greater severity than the postulated event. A rigorous mathematical formulation for finding the correct distribution has been presented.

REFERENCES Chow, V.T. (Editor), 1964. Handbook of Applied Hydrology. McGraw Hill, New York, N.Y., Section 18, pp. 1--26. Stall, J.B., 1962. Reservoir mass analysis by a low flow series. ASCE J. Sanit. Eng. Div., 28: 21--40.