A reservoir storage yield analysis for arid and semiarid climates

Transcription

1 Optimal Allocation of Water Resources (Proceedings of the Exeter Symposium, July 1982). IAHSPubl.no A reservoir storage yield analysis for arid and semiarid climates INTRODUCTION Y, P, PARKS & A, GUSTARD Institute of Hydrology, Walling ford, Oxon OX10 8BB, UK ABSTRACT Storage yield analyses were required for four reservoir sites in Botswana. The semiarid climate provided particular constraints to the analysis as the average annual open water evaporation is more than three times the average annual rainfall. The choice of technique is briefly discussed with particular reference to the short river flow series available for investigation. A behaviour analysis of flow data at Gaborone reservoir illustrates the technique in determining a storage requirement for a particular yield and probability of failure. A new technique for estimating the probability of failure from the distribution of non-failures is described incorporating monthly evaporation estimated over the changing surface area of the reservoir. The importance of accounting for evaporation in reservoir yield determination is demonstrated by comparing the storage yield relationships of three other sites. This paper describes the development of a new reservoir yield analysis and its application in Botswana, a semiarid country in southern Africa. The analysis was required to investigate the relative yields of a raised existing reservoir with those of new reservoirs at one of three possible locations. Fig.l shows the site of each of the reservoirs including Gaborone reservoir which has a 2 drainage basin of 4300 km", an annual average potential evaporation of 1730 mm and average annual rainfall of only 540 mm. Under these conditions accurate allowance for evaporation losses in the reservoir yield analysis is essential as these often constitute a larger proportion of gross yield than the net yield itself. CHOICE OF RESERVOIR YIELD DESIGN PROCEDURE The hydrological design of reservoirs is concerned with determining the storage capacity required to maintain a yield with a given probability of failure. McMahon & Mein (1978) have classified the large number of different design procedures into three broad groups. The first group is termed "critical period techniques" which rely on analysing those events when the yield exceeds demand. Examples are provided by the methods proposed by Rippl (1883), Alexander (1962) and Stall (1962). The second group is based on probability matrix 49

2 50 Y.P.Parks s A.Gustard DAM IN BOPHUTHATSWANA FIG.l Location map of reservoir sites. methods when the probabilities of the reservoir reaching a given storage condition from a previous condition are analysed (Gould, 1964). The third group embraces methods which, although using conventional techniques for assessing capacity, make use of sequences of stochastically generated flow data and thus enable an estimate to be made of the error of assessing the required capacity. The limited flow data available for Gaborone reservoir illustrate the problem of reservoir design in areas of scarce flow data and influenced the choice of reservoir yield analysis technique. A record representing the inflows to the reservoir was estimated from the spills and water level changes in the existing reservoir from 1965 to This 15 year monthly flow record was extended by applying a conceptual model, developed by Pitman (1973) for nearby South African basins, to a longer record ( ) of monthly rainfall over the river basin. The model fitting is based on

3 Reservoir storage yield analysis 51 optimizing the model parameters in order to preserve the mean, standard deviation and seasonal distribution of monthly flow data for the calibration period. The 59 year record used in this paper thus consisted of synthetic flows from 1922 to 1964 followed by recorded inflows from 1965 to As the record of inflows had already been extended it was considered inappropriate to calibrate a time series model using synthetic data and derive additional flow sequences. Initial trials made use of Gould's probability matrix method; however, this method assumes that there is zero serial correlation of annual streamflow data. In southern Africa there is evidence of serial correlation in the rainfall statistics data (Tyson et al., 1975) and this may be reflected in the river flow data. For this study the 16 years of flow data for Gaborone were considered insufficient to provide a reliable estimate of the serial correlation of annual flow data and thus a behaviour analysis was preferred (Institute of Hydrology, 1980). It is necessary in the semiarid climate of Botswana to allow for the variation of evaporation losses with the reservoir surface area. In addition to monthly inflow, inputs to the model include monthly rainfall on the surface of the reservoir while outputs include the net yield for supply and mean monthly surface water evaporation calculated by the Penman method (Pike, 1971). The following sections describe the estimation of the probability of annual maximum storage requirements when the reservoir does not fail and also the importance of evaporation losses and their effect on the storage yield relationship of different reservoir sites. SIMULATION ANALYSIS In this study it is convenient to adopt as the basic time-varying quantity, S-^, the volume of water that would be required to fill the reservoir (representing the storage needed to maintain a given yield). So when Sj_ = 0 the reservoir is full, when Sj_ is negative the reservoir is spilling (and is set to zero in the analysis). S^ is calculated by carrying out the following monthly water balance: s i+l = s i ~ I i + Y i + A i E i (For S^+1 < O the reservoir spills and S i + -^ is set to O; and for s i+l > v the reservoir fails and Sj_ + 2 is set to V.) ^i+1 storage requirement at the end of the i-th month, Sj_ storage requirement at the beginning of the i-th month, Ij_ inflow in i-th month, Y. yield in i-th month, A i E i V evaporation E^ from surface area A± in the i-th month, capacity of the reservoir. A simulation program was written which imitates the behaviour of the reservoir under given imposed yield conditions and reservoir capacities. The output is a set of start-of-month Sj_ values. The simulation was carried out on the 59 years of monthly flow data with yields ranging from 4.8 to 18.O x lo m year x (18-67% of mean annual runoff, MAR) and reservoir capacities from 34 to 200 x 10 6 m 3 ( % MAR). The initial storage was determined by using the median end of

4 52 Y.P.Parks S A.Gustard year storage calculated by running the simulation program through the set of data. The annual maximum values of S^ were extracted from the series. In a given year this annual maximum represents the annual minimum volume of storage required to maintain the yield. If V is reached the reservoir will fail. ESTIMATION OF PROBABILITY OF RESERVOIR FAILURE The annual maximum values of Sj_ are ranked in increasing order and the non-exceedance probability FJ of the j-th smallest storage requirement is estimated using the Blom plotting position, - J ~ F j " n where j is the rank of the annual maximum Sj_ series (j = 1 for the smallest), and n is the number of years in the record. Values of the normal reduced variate yj calculated from F-; are shown in Table 1 and plotted on Fig.2 for a typical reservoir simulation. In general, as the drought severity increases, Sj_ increases until the point is reached when Sj_ exceeds the reservoir capacity V. For droughts of greater severity the reservoir empties and the annual maximum S- series all take on the value V. This is shown in Fig.2 by the dashed line. We may read from Fig.2 a return period of failure, Tp. For the reservoir of 60 x ÎO m capacity and a yield of 9.6 x lo m year -, T p is about 10 years. It must be realized TABLE 1 Extract from example of series of annual maximum storages required for Gaborone reservoir (yield of 9.6 x 10 6 m 3 l year and capacity of 60 x 10 m ) Year Annual maximum storage required Sj_ (xl0 e m 3 ) Rank j Reduced variate y j

5 Reservoir storage yield analysis 53 RETURN PERIOD IVEABS] 10 T, 1.0 Reduced 1.2 variate y, FIG.2 Annual maximum storage requirement (yield of 9.6 x 10 " m - year" and reservoir capacity V = 60 x 10 & m 3 ) that reservoir design cannot be based on lower return periods than Tp. This is because the surface area calculated relates to the reservoir size V (and not the smaller reservoir) leading to an overestimate of evaporation losses and resulting in an overestimate of the storage requirements. Thus the Tp point is a correctly evaluated design condition i.e. where V is the true storage required to sustain the given yield. Nevertheless, this point is most accurately estimated from the sample of non failures shown in Fig.2. (This departs from the conventional technique where probability of failure is often estimated by counting failures.) Thus we have estimated a return period of failure associated with a true reservoir of capacity V required to sustain the yield Y. SIMULATION WITH AREA RELATED EVAPORATION This process is repeated for several different reservoir capacities until there are sufficient points to define the true storage-return period curve for a given yield. Fig.3 illustrates the results derived for the reservoir at Gaborone. This curve provides an estimate of the return period of failure expected for each storage considered, supplying a yield of 9.6 x 10 m 3 year _1. This figure also illustrates that the (V,Tp) point is the only true point on each curve. Tp for a capacity of 60 x lo m is actually 10 years, but if 6 3 we consider the curve drawn for V = 80 x 10 m it would appear that the storage required for this return period is much higher at about 75 x 10 m 3. Fig.3 also shows how the repetition of the analysis for several reservoir sizes assists the Tp estimation and improves the estimation of the shape of the storage-return period relationship. The range of average annual evaporation for the reservoir trials varies from 12.5 x 10'm for 60 x lo m capacity to 20.7 x ÎO m' for

6 54 Y.P.Parks S A.Gustard Design storage-return period relationship o o o.. o o o Capacities V m 3 RETURN PERIOD jyearsl 10T, Reduced variate i FIG.3 Construction of storage yield curve for Gaborone reservoir (yield of 9.6 x 10 m 3 year ). 125 x 10 m 3 capacity, that is, a range of annual runoff. to 77% of the mean Comparison with simplified evaporation estimation The results from the storage yield analysis described above, with a varying reservoir area, have been compared with results obtained using evaporation losses estimated from the average monthly evaporation rate over the mean reservoir surface area. The curves shown in Fig.4 are strikingly different and demonstrate that a method of analysis which does not include evaporation based on the changing surface area in the design can overestimate the required capacity by 200%. The average evaporation method tends to overestimate evaporation from the reservoir during droughts (when the surface area is in practice relatively small) and this increases the evaporation and hence the tendency to fail. This emphasizes the need for accurate consideration of evaporation losses, even in preliminary reservoir yield investigations, in arid and semiarid regions. A COMPARISON OF STORAGE YIELD RELATIONSHIPS BETWEEN FOUR RESERVOIR SITES The importance of reservoir evaporation in this investigation is further illustrated by the results for the three other reservoirs considered at Morwa, Thamaga and Kumukwane (all shown in Fig.l) by comparing their storage-area curves. Fig.5 illustrates the different geometrical characteristics of the three sites. The Kumukwane site is situated in a steep gorge with a relatively small

7 Reservoir storage yield analysis 55 D Area related evaporation O Average evaporation RETURN PERIOD I YEARS I Reduced variate yk FIG.4 Comparison of results for methods incorporating average and surface area related evaporation. surface area compared with the site at Morwa which is much flatter with a relatively large surface area. The storage yield analysis has been carried out for the three sites and the dimensionless yield-storage curves have been superimposed in Fig.6. The steepest site, at Kumukwane, can sustain a greater yield than the flatter sites for any chosen reservoir capacity, the difference in available yield being due to the evaporation component of the behaviour analysis. The importance of the storage-area relationship in reservoir design is also reflected --THAMAGA GABORONE ^~ KUMUKWANE Reservoir capacity /MAR FIG.5 Comparison of dimensionless storage-area curves at alternative reservoir sites.

8 56 Y.P.Parks & A.Gustard KUMUKWANE Reservoir capacity/mar FIG.6 Comparison of dimensionless storage-yield relationships at alternative reservoir sites. in work carried out by Midgley & Pitman (1969), where the evaporation losses for reservoir yield calculation were related to the exponent of the reservoir area-volume relationship. CONCLUSIONS The paper describes a method for estimating the probability of reservoir failure from the annual maximum series of storage requirements when the reservoir does not fail. The technique demonstrates that for each reservoir capacity V there is a unique storage-return period relationship given by (V,Tp). By repeating the analysis for different yields and storages a family of storage probability curves can be drawn. This method makes full use of all the available flow data and is preferred to that of counting failures. From Fig.4 it can be seen that an analysis incorporating an average evaporation rate and average reservoir surface area (rather than a variable area) is conservative for storage yield design and leads to a gross overestimate of the storage required for a given probability of failure. The importance of the accurate assessment of evaporation is further demonstrated in Figs 5 and 6 where the difference in yields available from four reservoir sites, under similar conditions, is closely related to the geometry of the reservoir site. This study is part of a current investigation into reservoir yield analysis which will include further work on the Gould probability matrix method and the effect of serial correlation on

9 annual maximum storage requirements. Reservoir storage yield analysis 57 ACKNOWLEDGEMENTS The analysis described in this paper was carried out during an investigation of Gaborone reservoir undertaken by the Institute of Hydrology for Sir Alexander Gibb and Partners (Botswana). Although the views expressed in this paper are those of the authors, thanks are due to the consultants both for allowing us to publish this paper and for their assistance during the study. We would also like to thank our colleague, F.A.K.Farquharson, for assisting with the comparison of storage yield relationships between reservoir sites. REFERENCES Alexander, G.N. (1962) The use of the Gamma distribution in estimating regulated output from storages. Civ. Engr Trans., The Institute of Engineers, Report no. 8 (Australia). Gould, B.W. (1964) Statistical methods for reservoir yield estimation. Wat. Res. Foundation of Australia, Report no. 8. Institute of Hydrology (1980) Low Flow Studies. Institute of Hydrology, Wallingford, Oxon, UK. McMahon, T.A. & Mein, R.G. (1978) Reservoir Capacity and Yield. Developments in Water Science no. 9, Elsevier, Amsterdam. Midgley, D.C. & Pitman, W.V. (1969) Surface water resources of South Africa. Hydrological Research Unit Report no. 2/69, University of Witwatersrand, Johannesburg. Pike, J.G. (1971) Rainfall and Evaporation in Botswana. UNDP/FAO Tech. Document no. 1, Gaborone, Botswana. Pitman, W.V. (1973) A mathematical model for generating monthly river flows from meteorological data in South Africa. Hydrological Research Unit Report no. 2/73, University of Witwatersrand, Johannesburg. Rippl, W. (1883) Capacity of storage reservoirs for water supply. Minutes of Proc., ICE, 71. Stall, J.B. (1962) Reservoir mass analysis by a low-flow series. J. Sanit. Engng Div., ASCE. Tyson, P.D., Dyer, T.G.S. & Mametse (1975) Secular changes in South African rainfall: Quart. J. Roy. Met. Soc. 101.

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